ALGEBRA is that branch of the mathematical sciences which has for its object the carrying on of operations either in an order different from that which exists in arithmetic, or of a nature not contemplated in fixing the boundaries of that science. The circumstance that algebra has its origin in arithmetic, however widely it may in the end differ from that science, led Sir Isaac Newton to designate it “Universal Arithmetic,” a designation which, vague as it is, indicates its character better than any other by which it has been attempted to express its functions—better certainly, to ordinary minds, than the designation which has been applied to it by Sir William Rowan Hamilton, one of the greatest mathematicians the world has seen since the days of Newton—“the Science of Pure Time;” or even than the title by which De Morgan would paraphrase Hamilton’s words—“the Calculus of Succession.”
To express in few words what it is which effects the transition from the science of arithmetic into a new field is not easy. It will serve, probably, to convey some notion of the position of the boundary line, when it is stated that the operations of arithmetic are all capable of direct interpretation per se, whilst those of algebra are in many cases interpretable only by comparison with the assumptions on which they are based. For example, multiplication of fractions—which the older writers on arithmetic, Lucas de Burgo in Italy, and Robert Recorde in England, clearly perceived to be a new application of the term multiplication, scarcely at first sight reconcilable with its original definition as the exponent of equal additions,—multiplication of fractions becomes interpretable by the introduction of the idea of multiplication into the definition of the fraction itself. On the other hand, the independent use of the sign minus, on which Diophantus, in the 4th century, laid the foundation of the science of algebra in the West, by placing in the forefront of his treatise, as one of his earliest definitions, the rule of the sign minus, “that minus multiplied by minus produces/»?««”—this independent use of the sign has no originating operation of the same character as itself, and might, if assumed in all its generality as existing side by side with the laws of arithmetic, more especially with the commutative law, have led to erroneous conclusions. As it is, the unlimited applicability of this definition, in connection with all the laws of arithmetic standing in their integrity, pushes the dominion of algebra into a field on which the oldest of the Greek arithmeticians, Euclid, in his unbending march, could never have advanced a step without doing violence to his convictions.
In asserting that the independent existence of the sign minus, side by side with the laws of arithmetic, might have led to anomalous results, had not the operations been subject to some limitation, we are introducing no imaginary hypothesis, but are referring to a fact actually existing. The most recent advance beyond the boundaries of algebra, as it existed fifty years ago, is that beautiful extension to which Sir W. R. Hamilton has given the designation of Quaternions, the very foundation of which requires the removal of one of the ancient axioms of arithmetic, “that operations may be performed in any order.”
HISTORY.
At what period and in what country algebra was invented? is a question that has been much discussed. Who were the earliest writers on the subject? What was the progress of its improvement? And lastly, by what means and at what period was the science diffused over Europe? It was a common opinion in the 17th century that the ancient Greek mathematicians must have possessed an analysis of the nature of modem algebra, by which they discovered the theorems and solutions of the problems which we so much admire in their writings; but that they carefully concealed their instruments of investigation, and gave only the results, with synthetic demonstrations.
This opinion is, however, now exploded. A more intimate acquaintance with the writings of the ancient geo-meters has shown that they had an analysis, but that it was purely geometrical, and essentially different from our algebra.
Although there is no reason to suppose that the great geometers of antiquity derived any aid in their discoveries from the algebraic analysis, yet we find that at a considerably later period it was known to a certain extent among the Greeks.
About the middle of the 4th century of the Christian era, a period when the mathematical sciences were on the decline, and their cultivators, instead of producing original works of genius, contented themselves with commentaries on the works of their more illustrious predecessors, there was a valuable addition made to the fabric of ancient learning.
This was the treatise of Diophantus on arithmetic, consisting originally of thirteen books, of which only the first six, and an incomplete book on polygonal numbers, supposed to be the thirteenth, have descended to our times.
This precious fragment does not exhibit anything like a complete treatise on algebra. It lays, however, an excellent foundation of the science, and the author, after applying his method to the solution of simple and quadratic equations, such as to “find two numbers of which the sum and the sum or difference of the squares are given,” proceeds to a peculiar class of arithmetical questions, which belong to what is now called the indeterminate analysis.
Diophantus may have been the inventor of the Greek algebra, but it is more likely that its principles were not unknown before his time; and that, taking the science in the state in which he found it as the basis of his labours, he enriched it with new applications. The elegant solutions of Diophantus show that he possessed great address in the particular branch of which he treated, and that he was able to resolve determinate equations of the second degree. Probably this was the greatest extent to which the science had been carried among the Greeks. Indeed, in no country did it pass this limit, until it had been transplanted into Italy on the revival of learning.
The celebrated Hypatia, the daughter of Theon, composed a commentary on the work of Diophantus. This, however, is now lost, as well as a similar treatise, on the Conics of Apollonius, by this illustrious and ill-fated lady, who, as is commonly known, fell a sacrifice to the fury of a fanatical mob about the beginning of the 5th century.
About the middle of the 16th century, the work of Diophantus above referred to, written in the Greek language, was discovered at Rome in the Vatican library, having probably been brought there from Greece when the Turks possessed themselves of Constantinople. A Latin translation, without the original text, was given to the world by Xylander in 1575; and a more complete translation, by Bachet de Mezeriac (one of the earliest members of the French Academy), accompanied by a commentary, appeared in 1621. Bachet was eminently skilful in the indeterminate analysis, and therefore well qualified for the work he had undertaken; but the text of Diophantus was so much in jured, that he was frequently obliged to guess the meaning of the author, or supply the deficiency. At a later period, the celebrated French mathematician Fermat supplemented the commentary of Bachet by notes of his own on the writings of the Greek algebraist. These are extremely valuable, on account of Fermat’s profound knowledge of this particular branch of analysis. This edition, the best which exists, appeared in 1670.
Although the revival of the writings of Diophantus was an important event in the history of mathematics, yet it was not from them that algebra became first known in Europe. This important invention, as well as the numeral characters and decimal arithmetic, was received from the Arabians. That ingenious people fully appreciated the value of the sciences; for at a period when all Europe was enveloped in the darkness of ignorance, they preserved from extinction the lamp of knowledge. They carefully collected the writings of the Greek mathematicians; they translated them into their language, and illustrated them with commentaries. It was through the medium of the Arabic tongue that the elements of Euclid were first introduced into Europe; and a part of the writings of Apollonius are only known at the present day by a translation from the Arabic, the Greek original being lost.
The Arabians ascribe the invention of their algebra to one of their mathematicians, Mahommed-ben-Musa, or Moses, called also Mahommed of Buziana, who flourished about the middle of the 9th century, in the reign of the Caliph Almamoun.
It is certain that this person composed a treatise on this subject, because an Italian translation was known at one time to have existed in Europe, although it is now lost. Fortunately, however, a copy of the Arabic original is preserved in the Bodleian Library at Oxford, bearing a date of transcription corresponding to the year 1342. The titlepage identifies its author with the ancient Arabian. A marginal note concurs in this testimony, and further declares the work to be the first treatise composed on algebra among the faithful; and the preface, besides indicating the author, intimates that he was encouraged by Almamoun, commander of the faithful, to compile a compendious treatise of calculation by algebra.
The circumstance of this treatise professing to be only a compilation, and, moreover, the first Arabian work of the kind, has led to an opinion that it was collected from books in some other language. As the author was intimately acquainted with the astronomy and computations of the Hindoos, he may have derived his knowledge of algebra from the same quarter. The Hindoos, as we shall presently see, had a science of Algebra, and knew how to solve indeterminate problems. Hence we may conclude, with some probability, that the Arabian algebra was originally derived from India.
The algebraic analysis, having been once introduced among the Arabians, was cultivated by their own writers. One of these, Mahommed Abulwafa, who flourished in the last forty years of the 10th century, composed commentaries on the writers who had preceded him. He also translated the writings of Diophantus.
It is remarkable, that although the mathematical sciences were received with avidity, and sedulously cultivated during a long period by the Arabians, yet in their hands they received hardly any improvement. It might have been expected that an acquaintance with the writings of Diophantus would have produced some change in their algebra. This, however, did not happen: their algebra continued nearly in the same state, from their earliest writer on the subject, to one of their latest, Behaudin, who lived between the years 953 and 1031.
Writers on the history of algebra were long under a mistake as to the time and manner of its introduction into Europe. It has now, however, been ascertained that the science was brought into Italy by Leonardo, a merchant of Pisa. This ingenious man resided in his youth in Barbary, and there learned the Indian method of counting by the nine numeral characters. Commercial affairs led him to travel into Egypt, Syria, Greece, and Sicily, where we may suppose he made himself acquainted with everything known respecting numbers. The Indian mode of computation appeared to him to be by far the best. He accordingly studied it carefully; and, with this knowledge, and some additions of his own, and also taking some things from Euclid’s Geometry, he composed a treatise on arithmetic. At that period algebra was regarded only as a part of arithmetic. It was indeed the sublime doctrine of that science; and under this view the two branches were handled in Leonardo’s treatise, which was originally written in 1202, and again brought forward under a revised form in 1228. When it is considered that this work was composed two centuries before the invention of printing, and that the subject was not such as generally to interest mankind, we need not wonder that it was but little known; hence it has always remained in manuscript, as well as some other works by the same author. Indeed it was not known to exist from an early period until the middle of the last century, when it was discovered in the Magliabecchian library at Florence.
The extent of Leonardo’s knowledge was pretty much the same as that of the preceding Arabian writers. He could resolve equations of the first and second degrees, and he was particularly skilful in the Diophantine analysis. He was well acquainted with geometry, and he employed its doctrines in demonstrating his algebraic rules. Like the Arabian writers, his reasoning was expressed in words at length; a mode highly unfavourable to the progress of the art. The use of symbols, and the method of combining them so as to convey to the mind at a single glance a long process of reasoning, was a much later invention.
Considerable attention was given to the cultivation of algebra between the time of Leonardo and the invention of printing. It was publicly taught by professors. Treatises were composed on the subject; and two works of the oriental algebraists were translated from the Arabian language into Italian. One was entitled the Pule of Algebra, and the other was the oldest of all the Arabian treatises, that of Mahommed-ben-Musa of Corasan.
The earliest printed book on algebra was composed by Lucas Paciolus, or Lucas de Burgo, a minorite friar. It was first printed in 1494, and again in 1523. The title is Summa de Arithmetica, Geometria, Proportioni, et Prο- portionalita.
This is a very complete treatise on arithmetic, algebra, and geometry, for the time in which it appeared. The author followed close on the steps of Leonardo; and, indeed, it is from this work that one of his lost treatises has been restored.
Lucas de Burgo’s work is interesting, inasmuch as it shows the state of algebra in Europe about the year 1500: probably the state of the science was nearly the same in Arabia and Africa, from which it had been received.
The power of algebra as an instrument of research is in a very great degree derived from its notation, by which all the quantities under consideration are kept constantly in view; but in respect of convenience and brevity of expression, the algebraic analysis in the days of Lucas de Burgo was very imperfect: the only symbols employed were a few abbreviations of the words or names which occurred in the processes of calculation, a kind of short-hand, which formed a very imperfect substitute for that compactness of expression which has been attained by the modem notation.
The application of algebra was also at this period very limited; it was confined almost entirely to the resolution of certain questions of no great interest about numbers. No idea was then entertained of that extensive application which it has received in modern times.
The knowledge which the early algebraists had of their science was also circumscribed: it extended only to the resolution of equations of the first and second degrees; and they divided the last into cases, each of which was resolved by its own particular rule. The important analytical fact, that the resolution of all the cases of a problem may be comprehended in a single formula, which may be obtained from the solution of one of its cases, merely by a change of the signs, was not then known: indeed, it was long before this principle was fully comprehended. Dr Halley expresses surprise, that a formula in optics which he had found, should by a mere change of the signs give the focus of both converging and diverging rays, whether reflected or refracted by convex or concave specula or lenses; and Molyneux speaks of the universality of Halley’s formula as something that resembled magic.
The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in some cases the two sciences may serve to illustrate each other, there is not. now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former. It was otherwise in former times. Lucas de Burgo found it to be convenient, after the example of Leonardo, to employ geometrical constructions to prove the truth of his rules for resolving quadratic equations, the nature of which he did not completely comprehend; and he was induced by the imperfect nature of his notation to express his rules in Latin verses, which will not now be read with the kind of satisfaction we receive from the perusal of the well-known poem, “the Loves of the Triangles.”
As Italy was the first European country where algebra became known, it was there that it received its earliest improvements. The science had been nearly stationary from the days of Leonardo to the time of Paciolus, a period of three centuries; but the invention of printing soon excited a spirit of improvement in all the mathematical sciences. Hitherto an imperfect theory of quadratic equations was the limit to which it had been carried. At last this boundary was passed, and about the year 1505 a particular case of equations of the third degree was resolved by Scipio Ferreus, a professor of mathematics in Bononia. This was an important step, because it showed that the difficulty of resolving equations of the higher orders, at least in the case of the third degree, was not insurmountable, and a new field was opened for discovery. It was then the practice among the cultivators of algebra, when they advanced a step, to conceal it carefully from their contemporaries, and to challenge them to resolve arithmetical questions, so framed as to require for their solution a knowledge of their own new-found rules. In this spirit did Ferreus make a secret of his discovery: he communicated it, however, to a favourite scholar, a Venetian named Florido. About the year 1535 this person; having taken up his residence at Venice, challenged Tartalea of Brescia, a man of great ingenuity, to a trial of skill in the resolution of problems by algebra. Florido framed his questions so as to require for their solution a knowledge of the rule which he had learned from his preceptor Ferreus; but Tartalea had, five years before this time, advanced further than Ferreus, and was more than a match for Florido. He therefore accepted the challenge, and a day was appointed when each was to propose to the other thirty questions. Before the time came, Tartalea had resumed the study of cubic equations, and had discovered the solution of two cases in addition to two which he knew before. Florido’s questions were such as could be resolved by the single rule of Ferreus; while, on the contrary, those of Tartalea could only be resolved by one or other of three rules, which he himself had found, but which could not be resolved by the remaining rule, which was also that known to Florido. The issue of the contest is easily anticipated; Tartalea resolved all his adversary’s questions in two hours, without receiving one answer from him in return.
The celebrated Cardan was a contemporary of Tartalea. This remarkable person was a professor of mathematics at Milan, and a physician. He had studied algebra with great assiduity, and had nearly finished the printing of a book on arithmetic, algebra, and geometry; but being desirous of enriching his work with the discoveries of Tartalea, which at that period must have been the object of considerable attention among literary men in Italy, he endeavoured to draw from him a disclosure of his rules. Tartalea resisted for a time Cardan’s entreaties. At last, overcome by his importunity, and his offer to swear on the holy Evangelists, and by the honour of a gentleman, never to publish them, and on his promising on the faith of a Christian to commit them to cypher, so that even after his death they would not be intelligible to any one, he ventured with much hesitation to reveal to him his practical rules, which were expressed by some very bad Italian verses, themselves in no small degree enigmatical. He reserved, however, the demonstrations. Cardan was not long in discovering the reason of the rules, and he even greatly improved’'them, so as to make them in a manner his own. From the imperfect essays of Tartalea he deduced an ingenious and systematic method of resolving all cubic equations whatsoever; but with a remarkable disregard for the principles of honour, and the oath he had taken, he published in 1545 Tartalea’s discoveries, combined with his own, as a supplement to a treatise on arithmetic and algebra, which he had published six years before. This work is remarkable for being the second printed book on algebra known to have existed.
In the following year Tartalea also published a work on algebra, which he dedicated to Henry VIII., king of England.
It is to be regretted that in many instances the authors of important discoveries have been overlooked, while the honours due to them have been transferred to others having only secondary pretensions. The formulae for the resolution of cubic equations are now called Cardan’s rules, notwithstanding the prior claim of Tartalea. It must be confessed, however, that he evinced considerable selfishness in concealing his discovery; and although Cardan cannot be absolved from the charge of bad faith, yet it must be recollected that by his improvements in what Tartalea communicated to him, he made the discovery in some measure his own; and he had moreover the high merit of being the first to publish this important improvement in algebra to the world.
The next step in the progress of algebra was the discovery of a method of resolving equations of the fourth order. An Italian algebraist had proposed a question which could not be resolved by the newly invented rules, because it produced a biquadratic equation. Some supposed that it could not be resolved at all; but Cardan was of a different opinion. He had a pupil named Lewis Ferrari, a young man of great genius, and an ardent student in the algebraic analysis: to him Cardan committed the solution of this difficult question, and he was not disappointed. Ferrari not only resolved the question, but he also found a general method of resolving equations of the fourth degree, by making them depend on the solution of equations of the third degree.
This was another great improvement; and although the precise nature of an equation was not then fully understood, nor was it indeed until half a century later, yet, in the general resolution of equations, a point of progress was then reached which the utmost efforts of modern analysis have never been able to pass.
There was another Italian mathematician of that period who did something for the improvement of algebra. This was Bombelli. He published a valuable work on the subject in 1572, in which he brought into one view what had been done by his predecessors. He explained the nature of the irreducible case of cubic equations, which had greatly perplexed Cardan, who could not resolve it by his rule; he showed that the rule would apply sometimes to particular examples, and that all equations of this case admitted of a real solution; and he made the important remark, that the algebraic problem to be resolved in this case corresponds to the ancient problem of the trisection of an angle.
There were two German mathematicians contemporary with Cardan and Tartalea, viz., Stifelius and Scheubelius. Their writings appeared about the middle of the 16th century, before they knew what had been done by the Italians. Their improvements were chiefly in the notation. Stifelius, in particular, introduced for the first time the characters which indicate addition and subtraction, and the symbol for the square root.
The first treatise on algebra in the English language was written by Robert Recorde, teacher of mathematics and practitioner in physic at Cambridge. At this period it was common for physicians to unite with the healing art the studies of mathematics, astrology, alchemy, and chemistry. This custom was derived from the Moors, who were equally celebrated for their skill in medicine and calculation. In Spain, where algebra was early known, the title of physician and algebraist were nearly synonymous. Accordingly, in the romance of Don Quixote, when the bachelor Samson Carasco was grievously wounded in his rencounter with the knight, an algebristα was called in to heal his bruises.
Recorde published a treatise on arithmetic, which was dedicated to Edward VI.; and another on algebra, with the title, The Whetstone of Wit, &c. Here, for the first time, the modern sign for equality was introduced.
By such gradual steps did algebra advance in improvement from its first introduction by Leonardo, each succeeding writer making some change for the better; but with the exception of Tartalea, Cardan, and Ferrari, hardly any one rose to the rank of an inventor. At length came Vieta, to whom this branch of mathematical learning, as well as others, is highly indebted. His improvements in algebra were very considerable; and some of his inventions, although not then fully developed, have yet been the germs of later discoveries. He was the first that employed genera 1 characters to represent known as well as unknown quant, ties. Simple as this step may appear, it has yet led to important consequences. He must also be regarded as the first that applied algebra to the improvement of geometry. The older algebraists had indeed resolved geometrical problems, but each solution was particular; whereas Vieta, by introducing general symbols, produced general formulae, which were applicable to all problems of the same kind, without the trouble of going over the same process of analysis for each.
This happy application of algebra to geometry produced great improvements: it led Vieta to the doctrine of angular sections, one of the most important of his discoveries, which is now expanded into the arithmetic of sines or analytical trigonometry. He also improved the theory uf algebraic equations, and he was the first that gave a general method of resolving them by approximation. Aa he lived between the years 1540 and 1603, his writings belong to the latter half of the 16th century. He printed them at his own expense, and liberally bestowed them on men of science.
The Flemish mathematician Albert Girard was one of the improvers of algebra. He extended the theory of equations somewhat further than Vieta, but he did not completely unfold their composition; he was the first that showed the use of the negative sign in the resolution of geometrical problems, and the first to speak of imaginary quantities. He also inferred by induction that every equation has precisely as many roots as there are units in the number that expresses its degree. His algebra appeared in 1629.
The next great improver of algebra was Thomas Harriot, an Englishman. As an inventor he has been the boast of this country. The French mathematicians have accused the British of giving discoveries to him which were really due to Vieta. It is probable that some of these may be justly claimed for both, because each may have made the discovery for himself, without knowing what had been done by the other. Harriot’s principal discovery, and indeed the most important ever made in algebra, was, that every equation may be regarded as formed by the product of as many simple equations as there are units in the number expressing its order. This important doctrine, now familiar to every student of algebra, developed itself slowly. It was quite within the reach of Vieta, who unfolded it in part, but left its complete discovery to Harriot.
We have seen the very inartificial form in which algebra first appeared in Europe. The improvements of almost 400 years had not given its notation that compactness and elegance of which it is susceptible. Harriot made several changes in the notation, and added some new signs: he thus gave to algebra greater symmetry of form. Indeed, as it came from his hands, it differed but little from its state at the present time.
Oughtreed, another early English algebraist, was a contemporary with Harriot, but lived long after him. He wrote a treatise on the subject, which was long taught in the universities.
In tracing the history of algebra, we have seen, that in the form under which it was received from the Arabs, it was hardly distinguishable as a peculiar mode of reasoning, because of the want of a suitable notation; and that, poor in its resources, its applicability was limited to the resolution of a small number of uninteresting numeral questions. We have followed it through different'stages of improvement, and we are now arrived at a period when it was to acquire additional power as an instrument ol analysis, and to admit of new and more extended applications. Vieta saw the great advantage that might be derived from the application of algebra to geometry. The essay he made in his theory of angular sections, and the rich mine of discovery thus opened, proved the importance of his labours. He did not fully explore it, but it has seldom happened that one man began and completed a discovery. He had, however, an able and illustrious successor in Descartes, who, employing in the study of algebra that high power of intellect with which he was endowed, not only improved it as an abstract science, but, more especially by its application to geometry, laid the foundation of the great discoveries which have since so much engaged mathematicians, and have made the last two centuries ever memorable in the history of the progress of the humar mind.
Descartes’ grand improvement was the application of algebra to the doctrine of curve lines. As in geography we refer every place on the earth’s surface to the equator, and to a determinate meridian, so he referred every point of a curve to some 1ine given by position. For example, in a circle, every point in the circumference might be referred to the diameter. The perpendicular from any point in the curve, and the distance of that perpendicular from the centre or from the extremity of a diameter, were lines which, although varying with every change of position in the point from which the perpendicular was drawn, yet had a determinate relation to each other, which was the same for all points in the curve depending on its nature, and which, therefore, served as a characteristic to distinguish it from all other curves.
The relations of lines drawn in this way could be readily expressed in algebraic symbols; and the expression of this relation in general terms constituted what is called the equation of the curve.
This might serve as its definition; and from the equation by the processes of algebra, all the properties of the curve could be investigated.
Descartes’ Geometry (or, as it might have been named, the application of algebra to geometry) appeared first in 1637. This was six years after the publication of Harriot’s discoveries, which was a posthumous work. Descartes availed himself of some of Harriot’s views, particularly the manner of generating an equation, without acknowledgment; and on this account Dr Wallis, in his algebra, has reflected with considerable severity on the French algebraist. This spirit has engendered a corresponding eagerness in the French mathematicians to defend him. Montucla, in his history of the mathematics, has evinced a strong national prejudice in his favour; and, as usually happens, in order to exalt him, he hardly does justice to Harriot, the idol of his adversaries.
The new views which the labours of Vieta, Harriot, and Descartes opened in geometry and algebra were seized with avidity by the powerful minds of men eager in the pursuit of real knowledge. Accordingly, we find in the 17th century a whole host of writers on algebra, or algebra combined with geometry.
Our limits will not allow us to enter minutely into the claims which each has on the gratitude of posterity. Indeed, in pure algebra the new inventions were not so conspicuous as the discoveries made by its applications to geometry, and the new theories which were suggested by their union. The curious speculations of Kepler concerning the solids formed by the revolutions of curvilinear figures, the Geometry of Indivisibles by Cavalerius, the Arithmetic of Infinites of Wallis, and, above all, the Method of Fluxions of Newton, and the Differential and Integral Calculus of Leibnitz, are fruits of the happy union. All these were agitated incessantly by their inventors and contemporaries; by such men as Barrow, James Gregory, Wren, Cotes, Taylor, Halley, De Moivre, Maclaurin, Stirling, and others, in this country; and abroad by Roberva1, Fermat, Huyghens, the two Bernoullis, Pascal, and many others.
The first half of the 18th century produced little in the •way of addition either to pure algebra or to its applications. Men were employed rather in elaborating and working out what Newton, Leibnitz, and Descartes had originated, than in exercising themselves in independent investigations. There are, indeed, to be found some names of eminence associated with the science of algebra, such as Maclaurin, but their eminence will be found to depend on their connection with the extensions of the science, rather than with the science itself. It was reserved for Lagrange, in the latter part of the century, to give a new impulse to extension in pure algebra, in a direction which has led to most important results. Not only did he, in his Traité de la Resolution des Equations Numériques, lay the foundation on which Budan, Fourier, Sturm, and others, have built a goodly fabric after the pattern of the Universal Arithmetic of Newton, but in his Théorie des fonctions analytiques, and Calcul des fonctions, he endeavoured, and with a large amount of success, to reduce the higher analysis (the Fluxions of Newton), to the domain of pure algebra. Nor must the labours of a fellow-workman, Euler, be forgotten. In his voluminous and somewhat ponderous writings will be found a perfect storehouse of investigations on every branch of algebraical and mechanical science. Especially pertinent to our present subject is his demonstration of the Binomial Theorem in the Novi Commentarii, vol. xix,, which is probably the original of the development that Lagrange makes the basis of his analysis ( Calcul des fonctions, leçon seconde), and which for simplicity and generality leaves nothing to be desired.
This brings the history down to the close of the last century. We have been as copious as our limits would permit on the early history, because it presents the interesting spectacle of the progress of a science from an almost imperceptible beginning, until it has attained a magnitude too great to be fully grasped by the human mind.
It will be seen from what precedes, that we have not limited “algebra” to the pure science, but have retained the name when it has encroached on the territories of geometry, trigonometry, and the higher analysis. To continue to trace its course through all these branches during the present century, when it has extended into new directions within its own borders, would far exceed the limits of an introductory sketch like the present. We must, therefore, necessarily limit ourselves to what has been done in the Theory of Equations (which may be termed algebra proper), and in Determinants.
Theory of Equations.— That every numerical equation has a root—that is, some quantity in a numerical form, real or imaginary, which, when substituted for the unknown quantity in the equation, shall render the equation a numerical identity—appears to have been taken for granted by all writers down to the time of Lagrange. It is by no means self-evident, nor is it easy to afford evidence for it which shall be at the same time convincing and free from limitations. The demonstrations of Lagrange, Gauss, and Ivory, have for simplicity and completeness given way to that of Cauchy, published first in the Journal de V Ecole Polytechnique, and subsequently in his Cours d ’ Analyse Algébrique.
The demonstration of Cauchy (which had previously been given by Argand, though in an imperfect form, in Gergonnds Annales des Mathématiques, vol. v.) consists in showing that the quantity which it is wished to prove capable of being reduced to zero, can be exhibited as the product of two factors, one of which is incapable of assuming a minimum value, or, in other words, that a less value than one assigned can always be found, and therefore that it is capable of acquiring the value zero. This argument, if not absolutely free from objection, is less objectionable than any of the others. The reader may consult papers by Airy and De Morgan, in the tenth volume of the Transactions of the Cambridge Philosophical Society.
Admitting, then, that every equation has a root, it becomes a question to what extent are we in possession of an analysis by which the root can be ascertained. If the question be put absolutely, we fear the answer must be, that in this matter we are in the same position that we have held for the last three centuries. Cubic and biquadratic equations can be solved, whatever they may be; but equations of higher orders, in which there exists no relation amongst the several coefficients, and no known or assumed connection between the different roots, have baffled all attempts at their solution. Much skill and ingenuity have been displayed by writers of more or 1ess eminence in the attempt to elaborate a method of solution applicable to equations of the fifth degree, but they have failed; whether it be that, like the ancient problems of the quadrature of the circle, and the duplication of the cube, an absolute solution is an impossibility, or whether it is reserved for future mathematicians to start in the research in some new path, and reach the goal by avoiding the old tracks which appear to have been thoroughly traversed in vain.
It is scarcely necessary to refer to such writers as Hoene de Wronski, who, in 1811, announced a general method of solving all equations, giving formulae without demonstration. In 1817, the Academy of Sciences of Lisbon proposed as the subject of a prize, the demonstration of Wronski’s formulae. The prize was in the following year awarded to M. Torriani for the refutation of them.
The reader will find in the fifth volume of the Reports of the British Association, an elaborate report by Sir W. R. Hamilton on a Method of Decomposition, proposed by Mr G. B. Jerrard in his Mathematical Researches, published at Bristol in a work of great beauty and originality, but which Hamilton is compelled to conclude fails to effect the desired object. In fact, the method which is valid when the proposed equation is itself of a sufficiently elevated degree, fails to reduce the solution of the equation of the fifth degree to that of the fourth.
But although the absolute solution of equations of higher orders than the fourth remains amongst the things uneffected, and rather to be hoped for than expected, a very great deal has been done towards preparing the way for approximate, if not for absolute solutions.
In the first place, equations of the higher orders, when they assume certain forms, have been shown to be capable of solution. An equation of this kind, to all appearance of a very general and comprehensive form, had been solved by De Moivre in the Philosophical Transactions for 1737. Binomial equations had advanced under the skilful hands of Gauss, who, in his Disquisitiones Arithmeticae, which appeared in 1801, added largely to what had been done by Vandermonde in the classification and solution of such equations; and subsequently, Abel, a mathematician of Norwegian birth, who died too early for science, completed and extended what Gauss had left imperfect. The coÛected writings of Abel published at Christiania in 1839, contain original and valuable contributions to this and many other branches of mathematics.
But it is not in the solution of equations of certain forms that the greatest advance has been made during the present century. The basis of all methods of solution must evidently be found in the previous separation of the roots, and the efforts of mathematicians have been directed to the discovery of methods of effecting this. The object is not so much to classify the roots into positive and negative, real and imaginary, as to determine the situation and number of the real roots of the equation. The first writer on the subject whose methods appeared in print is Budan, whose treatise, entitled Nouvelle methode pour la résolution des équations numériques, appeared in 1807. But there is evidence that Fourier had delivered lectures on the subject prior to the publication of Budan’s work, and consequently, without detriment to the claims of Budan, we may admit that the most valuable and original contribution to the science is to be found in Fourier’s posthumous work, published by Navier in 1831, entitled Analyse des équations déterminées. The theorem which Fourier gave for the discovery of the position, within narrow limits, of a root of an equation, is one of two theorems, each of which is known by mathematicians as “Fourier’s Theorem.” The other is a theorem of integration, and occurs in the author’s magnificent work Théorie de la Chalcur. During the interval between the publication of Budan’s work and that of Fourier, there appeared a paper in the Philosophical Transactions of the Royal Society for 1819, by W. G. Horner, upon a new method of solving arithmetical equations. From its being somewhat obscurely expressed, the great originality of the memoir did not at once appear. Gradually, however, Mr Horner’s method came to be appreciated, and it now ranks as one of the best processes, approaching, in some points, to Fourier’s. In the Mémoires des savons étrangers for 1835, appears a memoir, which, if it does not absolutely supersede all that had been previously done in assigning the positions of the real roots of equations, yet in simplicity, completeness, and universality of application, surpasses them all. The author, Μ. Sturm, of French extraction, but born at Geneva, has in this memoir linked his name to a theorem which is likely to retain its place amongst the permanent extensions of the domain of analysis as long as the study of algebra shall last. It was presented to the Academy in 1829.
Determinants.— The solution of simultaneous equations of the first degree may be presented under the form of a set of fractions, the numerators and denominators of which are symmetric products of the coefficients of the unknown quantities in the equations. These products were originally known as resultants, a name applied to them by Laplace, and retained as late as 1841 by Cauchy in his Exercices d ’ analyse et de physique mathématique, vol. ii. p. 161, but now replaced by the title determinants, a name first applied to certain forms of them by Gauss. In his Cours d i analyse algébrique, Cauchy terms them alternate functions. The germ of the theory of determinants is to be found in the writings of Leibnitz, who, indeed, was far-seeing enough to anticipate for it some of the power which, about a century after his time, it began to attain. More than half that period had indeed elapsed before any trace of its existence can be found in the writings of the mathematicians who succeeded Leibnitz. The revival of the method is due to Cramer, who, in a note to his Analyse des lignes courbes algébriques, published at Geneva in 1750, gave the rule which establishes the sign of a product as plus or minus, according as the number of displacements from the typical form has been even or odd. Cramer was followed in the last century by Bézout, Laplace, Lagrange, and Vandermonde. In 1801 appeared the Disquisitiones Arithmeticae of Gauss, of which a French translation by M. Poullet-Delisle was given in 1807. Notwithstanding the somewhat obscure form in which this work was presented, its originality gave a new impetus to investigations on this and kindred subjects. To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant. Binet, Cauchy, and others followed, and applied the results to geometrical problems. In 1826, Jacobi commenced a series of papers on the subject in Crellds Journal. In these papers, which extended over a space of nearly twenty years, the subject was recast and made available for ordinary readers; and at the same time it was enriched by new and important theorems, through which the name of Jacobi is indissolubly associated with this branch of science. Following the steps of Jacobi, a number of mathematicians of no mean power have entered the field. Pre-eminent above all others are two British names, those of Sylvester and Cayley. By their originality, by their fecundity, by their grasp of all the resources of analysis, these two powerful mathematicians have enriched the Transactions of the Royal Society, Crellc ’ s Journal, the Cambridge and Dublin Mathematical Journal, and the Quarterly Journal of Mathematics, with papers on this and on kindred branches of science of such value as to have placed their authors at the head of 1iving mathematicians. The reader will find the subject admirably treated in Baltzer’s Theorie und Anwendung der Deter-minenten; and more briefly in Salmon’s Higher Algebra. Elementary treatises have also been published by Spottis-woode in 1851, by Brioschi in 1854, by Todhunter in his Theory of Equations in 1861, and by Dodgson in 1867.
The attention of the learned has, during the present century, been called to a branch of the history of algebra, in no small degree interesting; we mean the cultivation of the science to a considerable extent, and at a remote period, in India.
We are indebted, we believe, to Mr Reuben Burrow for some of the earliest notices which reached Europe on this very curious subject. His eagerness to illustrate the histoιy of the mathematical sciences led him to collect oriental manuscripts, some of which, in the Persian language, with partial translations, were bequeathed to his friend Mr Dalby of the Royal Military College, who communicated them to such as took an interest in the subject, about the year 1800.
In the year 1813, Mr Edward Strachey published in this country a translation from the Persian of the Bijα Ganita (or Vija Ganita), a Hindoo treatise on algebra; and in 1816 Dr John Taylor published at Bombay a translation of Ielawati (or Dilαυαti), from the Sanscrit original. This last is a treatise on arithmetic and geometry, and both are the production of an oriental algebraist, Bhascara Acharya. Lastly, in 1817, there came out a work entitled Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahrne-gupta and Bhascara, translated by Henry Thomas Cole-brooke, Esq. This contains four different treatises, originally written in Sanscrit verse, viz., the Vija Ganita and JÀlavati of Bhascara Achaιya, and the GanitaThayα and Cuttacad, ’ hyaya of Brahmegupta. The first two form the preliminary portion of Bhascara’s Course of Astronomy, entitled SidT hanta Siromani, and the last two are the twelfth and eighteenth chapters of a similar course of astronomy, entitled Brahma-sidd hanta.
The time when Bhascara wrote is fixed with great precision, by his own testimony and other circumstances, to a date that answers to about the year 1150 of the Christian era. The works of Brahmegupta are extremely rare, and the age in which he lived is less certain. Mr Davis, an oriental scholar, who first gave the public a correct view of the astronomical computations of the Hindoos, is of opinion that he lived in the 7th century; and Dr William Hunter, another di1igent inquirer into Indian science, assigns the year 628 of the Christian era as about the time he flourished. From various arguments, Mr Colebrooke concludes that the age of Brahmegupta was antecedent to the earliest dawn of the culture of the sciences among the Arabians, so that the Hindoos must have possessed algebra before it was known to that nation.
Brahmegupta’s treatise is not, however, the earliest work known to have been written on this subject. Ganessa, a distinguished astronomer and mathematician, and the most eminent scholiast of Bhascara, quotes a passage from a much older writer, Arya-Bhatta, specifying algebra under the designation of Vija, and making separate mention of Cuttaca, a problem subservient to the resolution of indeterminate problems of the first degree. He is understood by another of Bhascara’s commentators to be at the head of the older writers. They appear to have been able to resolve quadratic equations by the process of completing the square; and hence Mr Colebrooke presumes that the treatise of Arya-Bhatta then extant extended to quadratic equations in the determinate analvsis. and to indeterminate equations of the first degree, and probably to those of the second.
The exact period when Arya-Bhatta lived cannot be determined with certainty; but Mr Colebrooke thinks it probable that this earliest of known Hindoo algebraists wrote as far back as the fifth century of the Christian era, and perhaps earlier. He 1ived therefore nearly as early as the Grecian algebraist Diophantus, who is reckoned to have flourished in the time of the emperor Julian, or about A.D. 360.
Mr Colebrooke has instituted a comparison between the Indian algebraist and Diophantus, and found reason to conclude that in the whole science the latter is very far behind the former. He says the points in which the Hindoo algebra appears particularly distinguished from the Greek are, besides a better and more convenient algorithm, 1sí, the management of equations of more than one unknown quantity; 2√, the resolution of equations of a higher order, in which, if they achieved little, they had at least the merit of the attempt, and anticipated a modem discovery in the resolution of biquadratics; 3<Z, general methods for the resolution of indeterminate problems of the first and second degrees, in which they went far indeed beyond Diophantus, and anticipated discoveries of modem algebraists; and 4th, the application of algebra to astronomical investigations and. geometrical demonstration, in which also they hit upon some matters which have been re-invented in modem times.
When we consider that algebra made little or no progress among the Arabians, a most ingenious people, and particularly devoted to the study of the sciences, and that centuries elapsed from its first introduction into Europe until it reached any considerable degree of perfection, we may reasonably conjecture that it may have existed in one shape or other in India long before the time of Arya-Bhatta; indeed, from its close connection with their doctrines of astronomy, it may be supposed to have descended from a very remote period along with that science. Professor Playfair, adopting the opinion of Bailly, the eloquent author of the Astronomie Indienne, with great ingenu⅛y attempted to prove, in a Memoir on the Astronomy of the Brahmins, that the observations on which the Indian astronomy is founded were of great antiquity, indeed more than 3000 years before the Christian era. The very remote origin of the Indian astronomy had been strongly questioned by many in this country, and also on the Continent; particularly by Laplace, and by Delambre in his Histoire de V Astronomie Ancienne, tome i. p. 400, &c., and again in Histoire de V Astronomie du Moyen Age, Discours Préliminaire, p. 18, &c., where he speaks slightingly of their algebra; and in this country, Professor Leslie, in his Philosophy of Arithmetic, pp. 225 and 226, calls the Lilaυati “a very poor performance, containing merely a few scanty precepts couched in obscure memorial verses.” We are disposed to agree with Professor Leslie as to the value, and with Professor Playfair as to the antiquity of this Hindoo algebra. That it should have remained in a state of infancy for so many centuries is accounted for by the latter author in the following passage :—“In India everything [as well as algebra] seems equally insurmountable, and truth and error are equally assured of permanence in the stations they have once occupied. The politics, the laws, the religion, the science, and the manners, seem all nearly the same as at the remotest period to which history extends. Is it because the power which brought about a certain degree of civilisation, and advanced science to a certain height, has either ceased to act, or has met with such a resistance as it is barely able to overcome? or is it because the discoveries which the Hindoos are in possession of are an inheritance from some more inventive and more ancient people, of whom no memorial remains but some of their attainments in science!”
Writers on Algebra.
Diophantus, Arithmetico rum Libri sex, about a.d. 360
(First edition of his writings, 1575; the best, 1670.)
Leonardo Bonacci (hisworks described in Cossali) 1202
Lucas Paciolus, or De Burgo, Summa de Arithmetica, &c 1494
Rudolff, Algebra 1522
Stifelius, Arithmetica Integra, &c 1544
Cardan, Ars Magna quam vulgo Cossam vocant 1545
Ferreus 1545
Ferrari (first resolved biquadratic equations) 1545
Tartalea, Quesiti ed Inventioni diverse 1546
Scheubelius, Algebra Compendiosa 1551
Recorde, Whetstone of Wit 1557 Peletarius, Be Occulta parte Numerorum 1558
Buteo, Be Logistica 1559
Ramus, Arithmeticae Libri duo et totidem Algebrae... 1560 Pedro Nugnez or Nonnius, Libro de Algebra, &c. ...1567 Jossalin, Be Occulta parte Mathematicorum 1576
Bombelli 1579
Clavius 1580
Bernard Solignac, Arith. Libri ii. et Algebiae totidem 1580
Stevinus, Arithmétique, Ac. aussi l’ Algèbre 1585
Vieta, Opera Mathematica, 1600 Folinus, Algebra, sive Liber de Rebus Occultis 1619
Van Ceulen 1619
Bachet, Diophantus cum Commentariis 1621
Albert Girard, Invention Nouvelle en Algèbre 1629
Gbetaldus, Be Resolutione et Compositione Mathematica 1630
Harriot, Artis Analyticae Praxis 1631
Oughtreed, Clavis Mathematica 1631
Herigonius, Cursus hathe-matieus 1634
Cavalerius, Geometria Indivisibilibus Continuorum, Ac 1635
Descartes, Geometria 1637
Franciscus à Schooten, Florimond de Beaune, Erasmus Bertholinus, Joh.
Hudde, F. Rabuel, James Bernoulli, John de Witt, &c.— Commentators on Descartes.
Roberval, Be Recognitione Aequationum, &c 1640
De Billy, Nova Geometricae Claris Algebra 1643
Renaldinus, Opus Algebrai- cum 1644
Pascal 1654
Wallis, Arithmetica Infini torum, 1655; Algebra, 1685 Slusius, Mesolabum 1659
Rhonius, Algebra (trans lated into English) 1659
Kinckhausen, used as a text book by Sir I. Newton ...1661
Sir Isaac Newton, The Binomial Theorem 1666
Frenicle, Various papers in Mem. of French Acad. ...1666
Pell (translated and improved Rhonius’ Algebra) 1668
James Gregory, Exercitationes Geometricae 1668
Mercator, Logarithmotech-nia 1668
Barrow, in Lectiones Geometricae 1669
Kersey, Elements of Algebral673
Prescot, Nouveaux Elémens de Mathématiques 1675
Leibnitz, in Leipsic Acts, &c 1677
Fermat, in Varia Opera Mathematica 1679
Bulliald, Opus Novum ad Arithmeticam Infinitorum 1682
Tschirnhausen, in the Leipsic Acts 1683
Baker, Geometrical Key, &C.1684
Dr Halley, in Philosophical Transactions ...1687 and 1694
Rolle, Une Méthode pour la Résolution des Equations Indéterminées 1690
Raphson, Analysis Aequationum Universalis 1690
Dechales, Cursus seu Mundus Mathematicus 1690
De Lagny, various pieces on Equations 1692
Alexander, Synopsis Alge- braica 1693
Ward, Compendium of Algebra, 1695; Young Mathematician’s Guide 1706
De Moivre, various Memoirs in Phil. Trans. ...1697-1730
Sault, New Treatise of Algebra 1698.
Christopher, Be Constructione Aequationum
Ozanam, Nouveaux Elémens P Algèbre 1702
Harris, Lexicon Technicum 1704
Guisnée, Application de l'Algèbre.à la Géométrie.. 17 05
Jones, Synopsis Palmariorum Matheseos 1706
Newton, Arithmetica Universalis 1707
L’Hôpital, Traité Analytique des Sections Coniques 1707
Rey neau, Analyse Démontrée 1708
Brooke Taylor, Methodus Incrementorum 1715
Stirling, Lineae Tertii Ordinis, 1717; Methodus Differentiatis 1730
Nicole, On Cubic Equations, in Mém. Acad.des Scieηcesl717
S’Gravesande, Algebra 1727
Wolfius, Algebra: Cursus Mathematicus 1732
Kirby, Arithmetic and Algebra 1735
James Gregory 1736
Simpson, Algebra, and various works 1740, 1742
Saunderson, Algebra, 2 vols. 4to 1740
La Caille, Algebra in Leçons de Mathématiques 1741
De Gua, On the Roots of Equations, in Mém. Acad, des Sciences 1741
Clairaut, Elémens d' Algèbre 1746
Maclaurin, Algebra 1747
Fontaine, L’Art de Résoudre les Equations 1747
Donna Maria Gaetana Agnesi, Instituzioni Anali-tichi 1748
Boscovich, in Elementa Universae Matheseos 1754
Segner, Berlin Mem 1756
Castillon, Arithmetica Universalis Newtoni cum Commentario 1761
Emerson, Algebra, Ac 1763
Landen, Residual Analysis, &c 1764
Lagrange, Traité de la Résolution des Equations Numériques, Recueil des Mém. de I’Acad, de Berlin 1767
Do. republished with
Notes, Paris 1797
Euler, Algebra 1770
Waring, Meditationes Alge- braicae, &c 1770, 1776
Soladini, Compendio d'Ana-lisi 1775
Paoli, Elementi d'Algebra ..1794
Ruffini, Teoria delie Equa- zione Algeb. ...............1799
Budan, Nouvelle Méthode pour la Résolution des Equations Numériques ...1807
Gompertz, Principles and Application of Imaginary Quantities 1817 and 1818
Biot, Qergoηηeis Annales, vol. vi. Horner, Philos. Trans 1819
Dandelin, Mém. de I’Acad. Roy. de Bruxelles 1826
Swinburne and Tylecote, on the Binomial Theorem 1827
Warren, on the Geometric Representation of the Square Roots of Negative Quantities .....1828
Abel, Mémoire sur les Equations Algébriques, Christiania 1829
and in Crelle, vol. i. 4.
Fourier, Analyse des Equations Déterminées (posthumous), with preface by Navier 1831
Malfatti, Mem. della Soc. Ital., vol. xi.
Davies Gilbert, Philos. Trans......................... 1881
Sturm, Mém. présentés par les Savans Etrangers 1835
Lockhart, Resolution of Equations 1837
To the preceding list of writers, which contains almost ,all of an early date, the following are to be added:—
Arbogast, Calcul des Dérivations. The Bernoullis, Begnalt, Bertrand, Bezout (Cours des Mathématiques), Bossuet, Burja, Brunacci, Babbage, Bridges, Bland, Budan, Bonnycastle, Bourdon, Barlow (on the Theory of Numbers), Baltzer (on Determinants).
Cousin, Cauchy, Coignet, Carnot, Cayley, Cockle.
Degraave, Ditton, Dodgson (on Determinants).
Frisius, Francoeur, Frend.
Gauss, Disquisitiones Arithmeticae, 1801.
Hemischius, Hales, Hirsch, Hutton, Holdred, Horner, Hargreaves.
Kuhnius, Kramp, Kaestner.
Laloubre, Lorgna, Le Blonde, Lee, Lacroix, Ludlam, Le gendre (on the Theory of Numbers), L’Huillier, Leroy.
Mescher, Malebranche, Manfredi, Maseres, Murphy.
Nicholson, Nieuwentijdt, (Analysis Infinitorum).
Polleti, Poignard (on Magic Squares), Playfair.
Rowning, Reimer.
Suremain-Missery (on Impossible Quantities), Schonerus, Salig-nut, Sturm, Serret, Salmon, Sylvester.
Trail, Tedenat, Thacker.
Vilent, Vandermonde.
Warren, Wells, Wilson, Wood, Woodhouse.
Young.
Elementary Treatises of Bryce, Colenso, De Morgan, Hind, Kelland, Peacock, Todhunter.
Writers on the History of Algebra.
Wallis, in his Algebra; Maseres, Scriptores Logarithmiσi, 1791, Montucla, in Histoire des Mathématiques; Bossuet, Histoire des Mathématiques; Cossali, Origine, Trasporto in Italia, e Primi Progressi in Essa dell’ Algebra, 2 vols, printed in 1797; Hutton, in his Dictionary, and more diffusely in his Tracts, vol. ii.; Libri Histoire des Sciences Mathématiques en Italie, Paris 1838.
Terquem, Bulletin de Bibliographie.
Peacock, Report of British Association, 1833.
For the titles of works on Algebra, consult Murhard, Bibliotheca Mathematica; and for Memoirs on Algebra, in Academical Collections, see Reuss, Repertorium Commentationum, tom. vii.; Smith (on the Theory of Numbers), Brit. Assoc. 1859-60, 1862-63.
notation and first principles.
1. In arithmetic there are ten characters, which being variously combined, according to certain rules, serve to denote all numerical magnitudes whatever. But this method of expressing quantities (a phrase used to designate something more than mere numbers), is found to be inadequate, taken by itself, to the more difficult cases of mathematical investigation; and it is therefore necessary, in many inquiries concerning the relations of magnitude, to have recourse to that more general mode of notation, and more extensive system of operations, which constitute the science of algebra.
In algebra quantities of every kind may be denoted by any characters whatever, but those commonly used are the letters of the alphabet; and as in the simplest mathematica! problems there are certain magnitudes given, in order to determine other magnitudes which are unknown, the first 1etters of the alphabet, a, b, c, &c., are used to denote known quantities, while those to be found are represented by v, x, y, <tc., the last 1etters of the alphabet, i 2. The sign + (plus) denotes, in arithmetic, that the quantity before which it is placed is to be added to some other quantity. Thus, a + b denotes the sum of a and b; 3 + 5 denotes the sum of 3 and 5, or 8.
The sign - (minus) signifies that the quantity before which it is placed is to be subtracted. Thus, a - b denotes the excess of a above δ; 6 — 2 is the excess of 6 above 2, or 4.
Quantities which have the sign + prefixed to them are called positive, and such as have the sign - are called negative.
When no sign is prefixed to a quantity, + is always understood, or the quantity is to be considered as positive.
Quantities which have the same sign, either + or —, are said to have like signs. Thus, + a and + b have like signs, but + a and - c have unlike signs.
3. A quantity which consists of one term is said to be simple; but if it consist of severa! terms, connected by the signs + or —, it is then said to be compound. Thus, + a and -c are simple quantities; and b- ∖ -c, and a + b-d, are compound quantities.
4. To denote the product arising from the multiplication of quantities, they are either joined together, as if intended to form a word, or else they are connected together, with the sign × or . interposed between every two of them. Thus, ab, or a × b, or a. b, denotes the product of a and .δ∕ also αδc, or a × δ × c, or a . δ. c, denotes the product of a, b, and c. If some of the quantities to be multiplied be compound, each of these has a line drawn over it called a vinculum, and the sign × is interposed, as before. Thus, a× c + d× e-f denotes that a is to be considered as one quantity, the sum of c and d as a second, and the difference between e and f as a third; and that these three quantities are to be multiplied into one another. Instead of placing a line over such compound quantities as enter a product, we may enclose each of them between two parentheses, so that the last product may be otherwise expressed thus, α(c + d)(e-f); or thus, a× (c + d) × (e-f).
A number prefixed to a letter is called a numerical coefficient, and denotes how often that quantity is to be taken. Thus, 3α signifies that a is to be taken three times. When no number is prefixed, the coefficient is understood to be unity.
5. The quotient arising from the division of one quantity by another is often expressed by placing the dividend 12
above a line, and the divisor below it. Thus, — denotes the quotient arising from the division of 12 by 3, or 4; - denotes the quotient arising from the division of δ by a.
6. The equality of two quantities is expressed by putting the sign — between them. Thus, a+b=c-d denotes that the sum of a and δ is equal to the excess of c above d.
7. Simple quantities, or the terms of compound quantities, are said to be like, which consist of the same letter or letters taken together in the same way. Thus, + ab and — 5ab are like quantities, but + ab and + abb are unlike.
There are some other characters, such as > for greater than, < for less than, .∙. for therefore, which will bo explained when we have occasion to use them; and in what follows we shall suppose that the operations and notation of common arithmetic are sufficiently understood.
8. As the science extends itself beyond its original boundaries, it begins gradually to appear that the limits imposed by these definitions have been transgressed, so that almost insensibly the symbols have acquired for themselves significations much more comprehensive than those originally attached to them. Thus, were + a to signify a gain of £a, — a would signify a loss of the same sum; were + a to signify motion forwards through a feet, - a would signify motion backwards through the same space. The extended definitions of + and - may now be such as the following: + and - are collective symbols of operations the reverse qí each other. From similar considerations to those by which the signification of + and - has been extended, we extend that of × and ÷ to something like the following: × and ÷ are emulative symbols of operations the inverse of each other. We may now exhibit the most general definition of the four symbols in the following form: + and — are symbols of operations prefixed to algebraical symbols of quantity, and are such that +α-α=+0 or — 0, where + 0 means simply or very nearly increased by 0; — 0, diminished by 0. × and ÷ are symbols of operations prefixed to alge braical symbols of quantity, and are such that ×a÷a = × 1 or ÷ 1, where × 1 means simply or very nearly multiplied by 1; ÷ 1, divided by 1.
9. The laws by which the symbols are combined are the same as in arithmetic. It is desirable, however, to exhibit them. They are three,—
Law I. Q uant iti es affected by the signs + and - are in no way influenced by the quantities to which they are united by these signs.
Law II. The Distributive Law. —Additions and subtractions may be performed in any order.
Law III. The Commutative Law.— Multiplications and divisions may be performed in any order.
We may remark that these laws are assumed for algebra, so that the science is limited by their applicability. Algebra has been extended into the science of quaternions by freeing it from part of the limitation imposed by the third of these laws. In this new science ab is not the same thing as ba.
We add a few examples of the substitution of numbers for letters. (Ex. 3 and 4 involve processes that will be explained later.)
Ex. 1. If a = 1, b-2,c = 3, find the value of (a + δ + c). (a + 2δ - c) . (δ + 2c - a).
Itis(1 + 2 + 3).(1 + 4-3).(2 + 6-1) = 84.
Ex. 2. If α = ⅜, δ = ⅜, c = ∣, x = Q find the value of α , δ ι c ι 9 boa
It is f∙ + ⅜ + ⅜ + 0 = 3⅜
Ex. 3. With the same data as in example 2, find the α 2 -δ 2 δ 2 -c 2 value of τr~ x x 2
The first term is infinite, and the second is infinitely greater than the first, because x 2 = x×x .'. the answer is — ∞.
Ex. 4. If ae = - = - = 0; find the value of y ≈
_ ax 2y 3 fx ∖ fx ∖ Write down the expression in x by putting - for y, &c.
X
It becomes 2 -⅛+^+J-l(* ι +i)(- τ +D
= 2 · because x = 0.
Sect. I.— Fundamental Operations.
The primary operations in algebra are the same as in common arithmetic—name!y, addition, subtraction, multiplication, and division; and from the various combinations of these four, al! the others are derived.
I. Addition.
10. In addition there maybe three cases: the quantities to be added may be like, and have 1ike signs; or they may be like, and have unlike signs; or, lastly, they may be unlike.
Case 1. To add quantities which are like, and have like signs.
Pule. Add together the coefficients of the quantities, prefix the common sign to the sum, and annex the letter or 1etters common to each term. [table]
Case 2. To add quantities which are like, but have unlike signs.
Pule. Add the positive coefficients into one sum, and the negative ones into another; then subtract the less of these sums from the greater, prefix the sign of the greater to the remainder and annex the common letter or letters as before [table]
Case 3. To add unlike quantities.
Pule. Put down the quantities, one after another, in any order, with their signs and coefficients prefixed. [table]
II. Subtraction.
11. General Pule.— Change the signs of the quantities to be subtracted, or suppose them changed, and then add them to the other quantities, agreeably to the rules of addition. [table]
The reason of the rule for subtraction may be explained thus. Let it be required to subtract 2p — 3<? from m + n. If we subtract 2p from m + n, there will remain m + n - 2p, but if we are to subtract 2p — 3çr, which is less than 2p, it is evident that the remainder will be greater by a quantity equal to that is, the remainder will be m + n - 2υ + 3q, hence the reason of the rule is evident.
III. Multiplication.
12. General Pule for the Signs.— If the quantities to be multiplied have like signs, the sign of the product is +; but if they have unlike signs, the sign of the product is — .
This rule, which is given by Diophantus as the definition of + and -, may be said to constitute the basis of algebra as distinct from arithmetic.
If we admit the definitions given above, the rule may be demonstrated in the following way :—
(1.) + a× + δ = + αδ is assumed.
(2.) + a× - b will have the same value, whatever — b may be connected with, as it has when — δ is connected with +δ (Law 1).
Now +α×( + δ-δ1=+α× + 0 = 0 (Def.)
But +α×( + δδ)=+α× +δ, and + a × - δ (Law 2).
.∙. + a × + δ and + a × -b make up 0 ·
i.e., + ab and + a × - δ make up 0.
Now +αδ-αδ = 0, .∙.+ α× - δ= -aδ.
(3.) Similarly - a × - δ = + ab.
The examples of multiplication may be referred to two cases; the first is when both the quantities are simple, and the second when one or both of them are compound.
Case 1. To multiply simple quantities.
Pule. Find the sign of the product by the general rule, and annex to it the product of the numeral coefficients; then set down all the letters, one after another, as in one word. [table]
Case 2. To multiply compound quantities.
Pule. Multiply every term of the multiplicand by all the terms of the multiplier, one after another, according to the preceding rule, and collect their products into one sum, which will be the product required. [table]
When several quantities are multiplied together so as to constitute a product, each of them is caI1ed a factor of that product: thus a, b, and c are factors of the product abc; also, a- s cx and b-x are factors of the product (α + #). (δ-ar).
The products arising from the continuai multiplication of the same quantity are called powers of that quantity, which is called the root. Thus aa, aaa, aaaa, &c., are powers of the root a. These powers are commonly expressed by placing above the root, towards the right hand, a figure, denoting how often the root is repeated. This figure serves to denominate the power, and is called its index or exponent. Thus, the quantity a being considered as the root, or as the first power of a, we have aa or α 2 for its second power, aaa or α 3 for its third power, aaaa or α 4 for its fourth power, and so on.
The second and third powers of a quantity are generally called its square and cube.
By considering the notation of powers, and the rules for multiplication, it appears that powers of the same root are multiplied by adding their exponents. Thus a × a 3 = a i , also ae 3 × αf = x 7 ; and in general α" × α n = a m+n .
When the quantities to be multiplied appear under a symmetrical form, the operation of multiplying them may sometimes be shortened by detached coefficients, by symmetry, and by general considerations suggested by the particular examples under consideration.
13. Detached Coefficients. Ex. 1. Multiply αf - 3x 3 + 2x 2 - 7x + 3 by x 2 - 5x + 4. Here the powers of x occur in regular order, so that we need only write down the coefficients of the several terms during the operation, having it in our power to supply the x , s whenever we require them; we write, therefore,— [table]
The last line (for which the result might have been written down in full at once) is equivalent to x β - 8x 5 + 21x 4 - 29x 3 + 46≈ 2 - 43z +12.
When any terms are wanting, they may be supplied by zeros; thus, Ex. 2. Multiply x i -7x 3 + x- 1 by x 3 -x + 2.
We write [table]
the product required.
14. Symmetry.
We may take advantage of symmetry by two considerations either separately or combined.
(1.) Symmetry of a Symbol. Ex. Find the sum of (a + b - 2c) 2 + (a + c - 2b) 2 + (Z> + c-2α) 2 . V ,
Here α 2 occurs with 1 as a multiplier in the first square, with 1 as a multiplier in the second square, and with 4 as a multiplier in the third square,
.∙. 6α 2 is part of the result; ab occurs with 2 as a multiplier in the first square, with - 4 in the second, and with - 4 in the third
.∙. — 6αδ is part of the result.
But α 2 , δ 2 , c 2 , are similarly circumstanced, as also ab, ac, be; hence the whole result must be 6(α 2 + δ 2 + c 2 - ab - ac - be).
(2.) Symmetry of an Expression. Ex. Find the sum of (α + δ + c)(aj + y + z) + (α + δ-c) (x + y - z) + (a - b + c)(x - y + z) + ( - a + b + c)( - x + y + z);
First, the product of (α + δ + c) by ae + y + z is to be found by multiplying out term by term. '
It is ax + ay + az + bx + by + bz + cx + cy + cz.
The product of (α + δ-c) (z + y-z) is now simply written down from the above, by changing the sign of every term which contains one only of the two quantities affected with a — sign, i.e., in this case c and z.
Lastly, the four products may be arranged below each other, the signs alone being written down; thus, [table]
and the sum required is therefore a mx + H>y + 4cz.
15. General Considerations. Ex. Find (a + b + c) 3 .
By multiplying out we get (a + b) 3 = α 3 + 3a 2 b +
Now a, b, c are similarly involved in (α + δ + c) 3 ; .'. b a and c 3 must appear along with a 3 , 3a 2 c, 3b 2 a, &c., along with 3a 2 b, and hence we can at once write down all the terms except that which contains abc. To obtain the coefficient of abc, we observe that if a, b, and c, are each equal to 1, (α + δ + c) 3 is reduced to 3 3 or 27. In other words, there are 27 terms, if we consider 3a 2 b and every similar expression as three terms; and as the terms preceding abc are in this way found to be 21 in number, we require Gabe to make up the full number 27; . ∙. (a + b + c) 3 = α 3 + b 3 + c 3 + 3a 2 b + 3a 2 c + 3δ 2 a + 3δ 2 c + 3c 2 a + 3c 2 b + Gabe.
It is desirable to introduce here some examples of the application of the process of the substitution of a letter for any number or fraction to the properties of numbers, inequalities, &c.
16. Properties of Humbers. Ex. 1. If unity is divided into any two parts, the difference of their squares is equal to the difference of the parts themselves.
Let x stand for one part; 1 - x for the other.
Now (1 - x) 2 - x 2 = 1 - 2x + x 2 - x 2 ≈ 1 - 2x — (1 - χ) - χ. i.e., the difference of the squares of the parts is equal to the difference of the parts.
Ex. 2. The product of three consecutive even numbers is divisible by 48.
Let 2n, 2n + 2, 2n + 4, be the three numbers .∙. their product is 8n(n +l)(n +2). Now, of three consecutive numbers, n, n + l, n + 2, one must be divisible by 2, and one by 3, .’. n(n + l)(n + 2) is divisible by 6, whence the proposition.
Ex. 3. The sum of the squares of three consecutive odd numbers, when increased by 1, is divisible by 12, but never by 24.
Let 2n — 1, 2n + 1, 2n + 3, be the three odd numbers.
The sum‘of their squares when increased by 1 is 12» 2 +12» +12 = 12(n 2 + n +1) = 12(τz. n+I +1).
Now, either n or n + 1 is even, .∙. 7i(n + 1) + 1 is odd; hence the sum under consideration is 12 times an odd number, whence the proposition.
Additional Examples in Symmetry, dec. Ex. 1. (α + b + c) 2 + (a + b - c) 2 + (α + c - δ) 2 + (δ + c - α) 2 = 4(α 2 + δ 2 + c 2 ).
This is written down at once, from observing that α 2 occurs in each of the four expressions, and that 2ab occurs with a + sign in two, and with a - sign in the other two. There is no other form.
Ex. 2. (a+b+ c) 3 +(a + b — c) 3 + (a +c — b) 3 + (b + e — d) 3 = 2(α 3 + b 3 + c 3 ) + 6(α 2 δ + a 2 c 4- b 2 a + b 2 c + c 2 α + c 2 δ) -1 2abc. lsi, α 3 occurs + in three, and — in one term.
2d, 3α 2 δ occurs + in three, and — in one term.
⅛d, When a, b, c are all units, the number resulting is 30; . ’. there are 30 terms, and as (1sí) and (2cZ) make up 42, there fall to be subtracted 12, i.e., the coefficient of abc is —12.
Ex. 3. (ax + by + cz) 2 + (ax + cy + bz) 2 + (bx + ay + cz) 2 + (bx + cy + a?) 2 + (ex + ay 4- δz) 2 + (cx + by + az) 2 = 2(a 2 + δ 2 4- c 2 ) (x 2 + y 2 + z 2 ) + 4(ab + ac + be) (xy + xz + yz). Ex. 4. The difference of the squares of two consecutive numbers is equal to the sum of the numbers.
Ex. 5. The sum of the cubes of three consecutive numbers is divisible by the sum of the numbers.
Ex. 6. If a; is an odd number, x 3 - x is divisible by 24, and (x 2 + 3) (x 2 + 7) by 32.
Ex. 7. If (pq - r) 2 + 4(∕> 2 - q)(pr - q) 2 = 0, then will 4(∕> 2 - q) 3 = (2y> 3 - 3pq + r) 2 , and 4(g 2 -pr) 3 = (2g 3 - 3pqr + r 2 ) 2 .
Ex. 8. Given x + y + z = 0, X + Y + Z = 0, to prove that (x 2 + X 2 )yz + (y 2 + Y 2 )zic + (z 2 + Z 2 )zy = (x 2 + X 2 ) YZ +
(y 2 + Y 2 )ZX + (z 2 + Z 2 )XY.
Let the left hand side equal the right + u ; then multiplying out, xyz(x + y 4- z) + X 2 yz + Υ 2 zx 4- Z 2 xy — XYZ(X + Y + Z) + x 2 YZ + y 2 ZX + z 2 XY + u, i.e., X 2 yz + Υ 2 zx + (X + Υ) 2 xy ≈ x 2 YZ + y 2 ZX + (x + y) 2 X Y + u, or, X 2 y(z + x) + Y¾(z + y) = z 2 Y(Z + X) + y 2 X(Z + Y) + u, OT, _ X2 y 2 _ γ2 a ∙2 = _ z 2γ 2 _ y 2 X 2 + u .∙. u = 0. Ex. 9. If 4α 2 δ 2 c 2 (x 2 + y 2 + z 2 ) (a 2 x 2 + b 2 y 2 + c¾ 2 ) = {(δ 2 + e 2 )a 2 x 2 4- (c 2 + a 2 )b 2 y 2 + (a 2 + b 2 )e 2 z)} 2 , when a is greater than δ, and δ greater than c; then is y = 0. As the argument concerns y, multiply out, and arrange in order of powers of y. After reduction this results in
(α 2 — c 2 )δ 4 y 4 +2 {(α 2 — c 2 χδ 2 — c 2 )α⅛ 2 4- (α 2 — c 2 )(α 2 — δ 2 )c⅜ 2 } δ 2 y 2 4- {(δ 2 - c 2 )a¾ 2 - (a 2 - δ 2 )c¾ 2 } 2 = 0.
Now each of these three terms is a positive quantity, if it be not zero, and as the sum of three positive quantities cannot be equal to zero, it follows that each term must be separately equal to zero, i.e., y — 0, and (δ 2 — c 2 )a 2 x 2 = (α 2 - δ 2 )c 2 z 2 .
17. Inequalities.
The demonstrations of inequalities are of so simple and instructive a character, that a somewhat lengthened exhibition of them forms a valuable introduction to the higher processes of the science. In all that follows under this head, the symbols x, y, z stand for positive numbers or fractions, usually unequal. Ex. 1. x i + f i >2xy.
Because (x — y) 2 is 4-, whether x be greater or less than y, it follows that x 2 - 2xy + y 2 is 4-, i.e., is some positive number or fraction, .∙. x 2 + y 2 >2xy.
It will be remarked that when x and y are equal, the inequality rises into an equality, and this is common to all inequalities of the character under discussion.
X ’ u
Cor. - + ->2; i.e., the sum of a fraction and its recipro y x cal is greater than 2.
Ex. 2. x 2 + y 2 + z 2 >xy+xz+yz∙
For x 2 + y 2 >⅛xy, x 2 + z 2 >2xz, y 2 + z 2 >2yz', which being added and divided by 2, gives the result required.
Ex. 3. χ m +" + y m + n >x m y η + x η y m .
For (z m - y η ) (x η - y n ) is +, whether x be greater or less than y.
As a particular case x 3 + y 3 >x 2 y + xy 2 . Ex. 4. z 2n +1 >z 2n-2r + z 2r .
For (.τ 2n ~ 2r - 1) (x 2r — 1) is positive.
Cor. 1. a5 - + ξ> a *-* + -l-.
ae x n ~ 2r
Cor. 2. Similarly, ir n + - n >^ 1 + - κ ^ , X X i.e., as n increases x κ + - n increases, .∙. as a particular case
„ 1 1
z" + - n >x+-∙ X X Ex. 5. If a, b, c are the sides of a triangle, a 2 + b 2 + c 2 >ab + ac + δc< 2(ab + ac + be). The former inequality is proved in example 2. For the latter we have (Euclid, I. 20), α< δ + c . ∙. α 2 < ab + ac. Similarly, δ 2 < ab + be, c 2 < ac + be.
.∙. α 2 + δ 2 + c 2 <2(αδ + αc + δc).
Ex. 6. The arithmetic mean of any number of quantities (all positive) is greater than the geometric.
(The arithmetic mean is the sum of the quantities divided by their number; the geometric is that root of their product which is represented by their number.) Let the quantities be denoted by x 1 , x 2 , x 3 , . . . x n , the numbers 1, 2, 3, placed under the x, indicating order only, so that x 1 may be read the first x, x 2 the second x, &c. Ex-
íC ∙4∙ íCo -
ample 1 gives √∙¾a⅞, if we suppose the x and y Ji of that example to be jx 1 , f jx 2 of the present.
τ , . ≈ 1 +iC 9 X°+Xi.
It also gives 2 4 — ,— —
2 2 > ∕ z ι+¾ ⅞+¾
2 V 2 2
> ∖ ∕ ∣ fx l x 2 >Jχfc i
In the same way we prove the proposition for 8, 16, or any number of quantities which is a power of 2.
For any other number, such, for instance, as 5, the following process is employed :—The number is made up to 8 by the insertion of three quantities, each equal to the 1 ,∙ c n a∏÷a⅛4-⅞ + aι ⅜+ a ⅛
arithmetic mean of the other five, viz.,
Call this quantity y; then x l +x 2 + . . . α⅛4-3y 8 .
g > √z 1 ¾ . . . x b yyy 8y β , l ' β ' , 8" > V χ ι x i · · · x di i or, y s >x.x i . . . x i y 3 .∙. f>>x x x i . . . x s
X 1 + ‰ + · · · a ⅛ 5/
.∙. y or —— 4 -r ~> ,Jx i x. 1 x 3 x i x j .
Cor. Às a particular case, x 2 + y 3 + z 3 >3xyz. Ex. 7. Given x 1 x i . . . x κ = y n , to prove that (1+x 1 ) (1 +a> 2 ) . . . (1 + x n )>(1 +y)"∙
The demonstration will be perfectly general in fact, though limited in form, if we suppose the number of quantities to be 5; in which case, ic 1 aj 2 . . . αr s = y 5 .
Make the number up to 8 by introducing three fs ; then
(1 +tf 1 ) (1 +x 2 ) >(1 + V∙≈ c ι⅝) 2 by example 1.
(1+x 3 ) (1+x 4 )>(1 + √¾) 2
(l+a⅛)(l+y) >(1+ √x 3 y) 2
(1+3∕)(1+y) =(1+ √yy) 2
. ’. Multiplying these products together, and combining the right hand factors two and two,
(1+xJ(1+x 2 ).. .... (1+a⅛)(1+y)3
>{(1+ √^ 2 )(1+j√⅛)(1 + ⅞(1+y)} 2
>{(1+ ^x 1 x,pcx^(l + ^‰ s y 3 )} 4
>(1+ ^ 1 x 2 a⅛r 4 ¾y 3 ) 8
>(1 + 3∕) s
∙∙∙ (i+* 1 )(l+¾).... (l+* β )>(l+y) 5 .
Ex. 8. If the sum of n fractions makes up 1, the sum of their reciprocals is greater than the square of their number.
Let x i + x 2 +. . . x n ≈l, then, έ ⅛ + * * · ⅛ >n ⅞ c 1 a , (example 6). *1 - ^'2 - t, ∙ ⅛⅛ ····*'»
But >Z¾ 1 ⅞ 2 . . . · · ,g ∙ t (example 6)<-
1 ->n, ∖ t X l X 2 . . .Xη
, 11 1
whence —∣—+ . . . 4— >n 2 . x 1 x 2 x h
„ o l+≈ 2 +≈*+ · · · ¾ 2 " „n+1/ η t 1∖ n+lf 1∖ ae+x¼ . . . x 2η ~ 1 2n ∖ x λ ) 2n ∖ + x)' Let the numerator and denominator of this fraction be designated by N and D. N may be divided into pairs of terms, at the same distance from either end, viz., 1 + x 2 ", x 2 + x 2η ~ 2 , <fcc., with or without a middle term, each of which (after 1 + x 2λ ) is, by example 4, less than that quantity; the middle term, if there be one, being less than
⅜ (i+^ 2n ),
∙. in either case N<^p(l + x 2 ") . . . (1.)
Again (example 6), D >n λ Jxx 2 . . . ic 2 "- j >nx" . (2.) . λ í i ∙ N n+lf 1∖
• · the fraction — < ( χ η -1— ·
D 2n ∖ x η ∕
To prove the second proposition, that the fraction is greater than Ç χ + -), it is only necessary to multiply up and reduce the result; thus, ’ ^∕(≈+l)⅛+x≡+...x=-)l n+l, τ N ,.
< N (by 1)
n η ' j '
<N.
Whence the proposition.
Additional Examples. Ex. 1. (x + y+ zi 2 <3(x 2 + y 2 + z 2 ), and generally, (x + y + z) n < 3 n-1 (x n + f + z n ). (See Induction.) g Ex. 2. (x+y} (y + z) (z + x)>8xyz<-(x 3 + y 3 + z 3 ).
Μ
Ex. 3. (x i + y i + z 4 ) >xyz(x + y + z). Ex. 4. (α 2 + b 2 + c 2 ) (x 2 + y 2 + z 2 ) > (ax + by + cz) 2 .
Ex. 5. The arithmetic mean of the y>th powers of n positive quantities is greater than the ∕>th power of their mean, and also greater than the mean of their combinations p together.
Ex. 6. (ax + by + czf + (ax + cy + bzf + (bx + ay + cz) 2 + ∖ bx + cy + αz) 2 + (cx + ay + bz) 2 + (ex + by + az) 2 > G(ab + ac + δc) (xy + xz + yz <6(α 2 + δ 2 + z 2 ) (ir 2 + y 2 + z2),
18. Induction.
It will be noted that the numerical multiplier of the second term of the powers of a + x already obtained is the same as the index. It is easy to see that this law is general. To demonstrate the fact formally we employ the method of induction.
The argument may be divided into four distinct steps— 1. Inference; 2. Hypothesis; 3. Comparison; 4. Conclusion.
The first step, inference, is the discovery of the probable existence of a law.
The second step, hypothesis, is the assumption that that law holds to a certain point, up to which the opponent to the argument may be presumed to admit it.
The third step consists in basing on this assumption the demonstration of the law to a stage beyond what the opponent was prepared to admit.
The fourth step argues that as the law starts fair, and advances beyond a point at which any opponent is prepared to admit its existence, it is necessarily true.
Ex. 1. To prove that (α + x) n = α n + nα n ^⅛ + , &c.
I. By multiplication we get (a + x) 4 = α 4 + 4α 3 x +, &c.
II. Let it be granted that (a + xf = a m + ma ju ~ 1 x +, &c., where m is the extreme limit to which the opponent will admit of its truth.
III. By multiplying the equals by a + x, w get (a + x) m + 1 = α m + 1 + ma m x +, &c., + a^x + , <fcc., — α" +1 + (m + 1)<x β ⅛ +, &c., i.e., if the 1aw be admitted true for m it is proved true for m +1; in other words, at whatever point the opponent compels us to limit our assumption, we can advance one step higher by argument.
IV. Now, thelawis^ true for 4, .∙. it is proved true for 5; and being true for 5, it is proved true for 6, and so on, ad infinitum. Ex. 2. The sum of the cubes of the natural numbers is the square of the sum of the numbers,
1≡ + 2 3 =9 = (1+2) 2 = ^y .
I. Let us assume that
1 3 +2 3 +, &c., + x 3 =
II. If this be so, then by adding (z+ 1) 3 we get ι>+2>+... + (a , + i)∙>(⅛⅛Ωy + (a . +1 )≡ ∕ (x÷l) (x+2)∖ 2 ∖ 2 ) ·
III. Hence, if the law be true for any one number x, it is also true for x + 1.
IV. But it is true for 2, for 3, .∙. for 4, <fcc.
Ex. 3. To prove the inequality, (x + y + z)*< 3"~ 1 (aj" + y n + a") .
From the second example of inequalities we get at once (ic + y + z) 2 <3(x 2 + y 2 + z 2 ).
Let us assume that (x + y + z)* , < 3 m ^ 1 (x m + y m + z m ), then by multiplication we get {x + y + z)"* +1 < 3" -1 (ir" +1 + y" +1 + z m + 1 + x m y + y m x + x m z + z m x + y m z + z m y).
Now, inequality, example 3, gives x n y + y m x<x m+1 + y m+1 , &c.
.∙. x m y + y m x + x n z + z n x + y m z + z m y< 2(x"* +1 + y m + 1 + z"+ 1 ), and (x + y + z)"+ 1 < 3 m (x m + j + y m + 1 + z" +1 ), i.e., the law is true for m+ 1, if true for m> but it is true for 2, .∙. it is always true.
IV. Division.
19. General Pule for the Signs.— If the signs of the divisor and dividend be like, the sign of the quotient is +; but if they be unlike, the sign of the quotient is -.
This rule is derived from the general rule for the signs in multiplication, by considering that the quotient must be such a quantity as, when multiplied by the divisor, shall produce the dividend, with its proper sign.
This definition of division is the same as that of a fraction; hence the quotient arising from the division of one quantity by another may be expressed by placing the dividend above a line, and the divisor below it; but it may also be often redu'ced to a more simple form by the following rules.
Case 1. When the divisor is simple, and a factor of every term of the dividend.
Pule. Divide the coefficient of each term of the dividend by the coefficient of the divisor, and expunge out of each term the letter or letters in the divisor: the result is the quotient.
Ex. Divide 16a 3 xy — 28a 2 xz 2 + 4a 2 x 3 by 4a 2 x.
The process requires no explanation. It is founded on Laws II. and III., together with the rule of signs.
The quotient is 4ay - 7z 2 + x 2 .
If the divisor and dividend be powers of the same quantity, the division will evidently be performed by subtracting the exponent of the divisor from that of the dividend. Thus α 5 , divided by α 3 , has for a quotient α 5 ~ 3 = a 2 .
Case 2. When the divisor is simple, but not a factor of the dividend.
Pule. The quotient is expressed by a fraction, of which the numerator is the dividend, and the denominator the divisor.
Thus the quotient of 3αδ 2 , divided by 2mbc, is the frac- x . 3αδ 2 tl0n Ä·
It will sometimes happen that the quotient found thus may be reduced to a more simple form, as shall be explained when we come to treat of fractions.
Case 3. When the divisor is compound.
Pule. The terms of the dividend are to be arranged in the order of the powers of some one of its letters, and those of the divisor according to the powers of the same letter. The operation is then carried on precisely as for division of numbers.
To illustrate this rule, let it be required to divide 8α 2 + 2αδ - 15δ 2 by 2a + 3δ, the operation will stand thus:
2α + 3δ)8α 2 + 2ab - 15δ 2 (4a - 5δ
8a 2 + 12aδ
— 10aδ-15δ 2
-10aδ-15δ 2
Here the terms of the divisor and dividend are arranged according to the powers of the quantity a. We now divide 8α 2 , the first term of the dividend, by 2a, the first term of the divisor; and thus get 4α for the first term of the quotient. We next multiply the divisor by 4α, an» subtract the product 8α 2 + 12αδ from the dividend; we get — 10αδ - 15δ 2 for a new dividend.
By proceeding in all respects as before, we find — 5δ for the second term of the quotient, and no remainder: the operation is therefore finished, and the whole quotient is 4α - 5δ.
The following examples will also serve to illustrate the manner of applying the rule.
Ex. 1.
3α - δ)3α 3 - 12α 2 - a 2 b + 10αδ - 2δ 2 (α 2 - 4a + 2b
3α 3 — α 2 δ
- 12α 2 +Ι0αδ
- 12α 2 + 4αδ
———fHQfb-2b 2 ÷ 6aδ-2δ 2 Ex. 2. l-α>)1 (1+x + x 2 + &c.
1-x + x + x-x - + x 2 + x 2 — 0? + x 3 .
Sometimes, as in this last example, the quotient will never terminate; in such a case it may either be considered as an infinite series, the law according to which the terms are formed being in general sufficiently obvious; or the quotient may be completed as in arithmetical division, by annexing to it a fraction (with its proper sign), the numerator of which is the remainder, and denominator the divisor Thus the completed quotient, in last example, is—
1 9 χ8
1 +x + x 2 + - 1—x
If x be small compared with unity, the remainders, as we advance, continually become smaller and smaller. If, on the other hand, x be large compared with unity, the remainders continually become larger and larger. In this case the quotient is worthless. To obtain a quotient which shall be of any practical value, we must reverse the order of arrangement, putting - x +1 in place of 1 - x. The division then becomes
-+ 1 ) 1 (÷⅛- i0 ∙
1-1
x
1
+ -
X
+ ~--⅜
Γ 4 9
x 2
As it is generally the 1argest of the quantities that we desire to divide out, we observe that, in order to effect this, we have had to begin with that quantity. Hence the Rule—
The terms of the divisor and dividend are to be arranged according to the powers of that letter which it is wished (if possible) to divide out.
Ex. 3. Divide α 4 + b i + δ(α + δ) 3 by α 2 + δ 2 — ab, where a is large compared with δ.
We must arrange according to powers of a.
α 2 — ab + δ 2 )a 4 + a 3 δ + 3a 2 δ 2 + 3aδ 3 + 2δ 4 (a 2 + 2aδ + 4δ 2 a 4 — a z b + a 2 δ 2
+ 2a 3 δ + 2a 2 δ 2 + 3aδ 3
+ 2a 3 δ — 2a 2 δ 2 + 2aδ 3
+ 4a 2 δ 2 + aδ 3 + 2δ 4 + 4a 2 δ 2 -4aδ 3 + 4δ 4
+ 5aδ 3 -2δ 4
We have spoken as if magnitude alone was the circumstance which should determine the precedence of the letters in a division. In the more advanced processes of algebra there are other circumstances which give precedence to certain letters, such, for example, as the fact that x may and often does stand for the phrase “ quantity,” whilst a stands for some determinate numerical quantity. This leads us to exhibit a proposition in division of the greatest value and most extensive application. It is as follows :—
20. Proposition. —If any function of x, consisting of powers of that letter with numerical multipliers, is divided by x - a, the remainder, when all the a/s are divided out, is the same function of a that the dividend is of χ∙, in other words, the remainder is the dividend altered by writing a in place of x.
To prove this proposition we shall employ the following Axiom :—If two expressions in x are identical in form and value, but one multiplied out farther than the other, we may write any numerical quantity we please in place of x in both, and the results will be equal.
For example, (#-l) 2 + (# —1)-3 is identical with x 2 -2(x + C) + x-i; and it is evident that if we write any number (say 1) for x, the results are the same in both.
We now proceed to prove the proposition.
Let the dividend be x n + px n ~ 1 + qx n ~ 2 , &c., where n is a whole number, and p, q, &c., positive or negative numerical quantities.
Let the quotient, when this is divided by x - a, be Q, the remainder, which does not contain x, R; then x n +pαP~ l + qrf~ i +, &c. = Q(íc - a) + R
by the definition of Division.
Now this equality is in reality an identity in terms of the axiom. If then we write a in place of x, the results will be equal; this gives
α n +7>α n ^ 1 + + &c. =Q.0 + R
= R, which is the proposition to be proved.
Examples. Ex. 1. If n be any whole number, x* — a n is divisible by x — a without remainder.
For the remainder, by the proposition, is a n — α* = 0.
Ex. 2. If n be an even number, x* - α" is divisible by x+a without remainder.
For the remainder is ( - α)* - α" = 0, since n is even.
Observe that the divisor here has to be changed to x - ( - a), so that - a stands in place of the a qì the proposition.
Ex. 3. If n be an odd number, x* + a* is divisible by x + a without remainder.
For the remainder is ( - α) n + α" = 0, because n is odd.
Ex. 4. To prove that 4δ 2 c 2 - (δ 2 + c 2 - α 2 ) 2 is divisible by -a + b + c> and hence to resolve it into simple factors. Here the x - a of the proposition is replaced by a - (δ + c) (the negative sign of the whole divisor being of no consequence).
To determine the remainder, therefore, we write δ + c in place of a in the dividend, or thing to be divided; the result is,
4δ 2 c 2 -(δ 2 + c 2 -δ+^e) 2 = 0, hence 4δ 2 c i ∙ - (δ 2 + c 2 - α 2 ) 2 is divisible by - a + δ + c.
Now, since the dividend contains only squares of a, and δ, and c, any change in the sign of a, or δ, or c, produces no change in the dividend. What we have just proved then becomes (putting - a for α) the following :—
4δ 2 c 2 - (δ 2 + c 2 - α 2 ) 2 is divisible by a + δ + c.
This last becomes (putting - δ for δ, and then - c for c) :— 4δ 2 c 2 — (δ 2 + c 2 — α 2 ) is divisible by a — δ + c, and by a + b — c.
Hence finally, 4δ 2 c 2 - (δ 2 + c 2 - α 2 ) 2 = (α + δ + c) ( - a + δ + c) (α - δ + c) (α + δ - c).
The above example is a good exercise for the student. The result may be more simply arrived at by employing a proposition of very great value and frequent use—that the difference of the squares of two quantities is the product of the sum and difference of the quantities. Ex. 5. To prove that (1 - α 2 ) (1 - δ 2 ) (1 - c 2 ) - (c + ab) (δ + ac) (a + be) is divisible by 1 + abc.
It is simpler here to write a single letter x for abc, whereby the given quantity becomes
(1 - α 2 ) (1 - δ 2 )(1 - c 2 ) - l(i c + α 2 )(z + b 2 )(x + c 2 ), which is obviously under the form p— p, when - 1 is written for x, and . ∙. is divisible by 1 + x. Ex. 6. Prove that (<r 2 -α! + 1)(x 4 -ic 2 + 1)(<r 8 -ic 4 +l) (x 16 - zr 8 + 1)... (x 2n - x η +1) is the quotient of x 1λ + x 2 * +1 by x 2 + x +1 · n being any power of 2.
The divisor (x 2 + x + 1) being multiplied by x 2 - τ + 1 gives x i + x 2 +1; which, being again multiplied by x i — x 2 +1, gives a: 8 + rc 4 +1; and so on to the end.
Additional Examples in Division. Ex. 1. Divide 1 -10^ 3 +15# 4 - 6# 5 by (1 - x) 3 .
We must first multiply out (1 - x) 3 , and then divide the given expression by the product, 1 - 3x + 3x 2 - x 3 . The quotient is 1 + 3x + 6x 2 .
Ex. 2. Divide 65#¾/ 2 - (x i + 64y 4 ) by x 2 - 7xy - 8y 2 .
We must arrange dividend and divisor in terms of powers of one of the letters, say x ; the division will then assume the form x 2 — 7xy — 8y 2 ) — x 4 + 65x 2 y 2 — 64y 4 ( giving — x 2 — 7 xy + 8y 2 .
Ex. 3. Divide x 3 + y 3 + z 3 - 3xyz by x + y + z.
We must give exclusive attention to some one letter, say x, in dividing out; thus aj + y-f-z^a^ + j^ + z 3 - 3xyz (x 2 — x{y + z) + (y 2 + z 2 — yz) + x ∖ y + z) -ar 2 (y + zj
-¾ 2 (y + z)- ι r(y + 2) 2 χ{y 2 ~ y z + z ^)+y i + 3 ≡ the quotient being x 2 + y 2 + z 2 — xy - xz - yz. Ex. 4. Divide the product of x 2 + 3x + 2, x 2 - 5x + 4, íc 4 + 5x 2 - 14, by the product of α 2 -1, x 2 -2. Here we observe that x 2 - 1 is the product of x + 1, x - 1.
Now (Art. 20), x 2 + 3x + 2 is divisible by íc+1, and x 2 - 5x + 4 by x - 1. Hence, if the product is divisible by x 2 - 1, x 2 - 2, without remainder, the third factor, α 4 + 5aj 2 - 14 must be divisible by x 2 - 2, which is found to be the case. The quotient required is therefore the product of (x + 2) (x - 4) (x 2 + 7) = x! i - 2x 3 - x 2 - 14# - õ6.
Ex. 5. Divide 12x 4 - 10x 3 y- 3,r 2 y 2 + 30ary 3 - 25y 4 by 3λ 2 - 4xy + 5y 2 .
We will employ this example to indicate Homer’s method of synthetic division.
Let the dividend be represented by kx i + Bz 3 + Gx 2 + T)x + E, the divisor by ax 2 + bx + c, and the quotient by αx 2 + ßx + y + &c.
Then, multiplying the quotient by the divisor, we produce the dividend, which, exhibited by the method of detached coefficients, stands thus— [table]
The last line being the sum column by column of the three preceding lines. Now, as the upper of these three lines contains term by term the quantities required, we convert this addition into subtraction: thus, [table]
The first vertical column gives α; the second ß, and so on.
In the example before us we write,— [table]
whence 3α = 12 gives a = 4; 3β = -10 + 4a gives β = 2∙, 3γ = - 3 + 4/? - 5α gives y = - 5.
Therefore the quotient required is 4z 2 + 2x - 5.
Sect. II.— Involution and Evolution.
21. In treating of multiplication, we have observed, that when a quantity is multiplied by itself any number of times, the product is called a power of that quantity, while the quantity itself, from which the powers are formed, is called the root. Thus, α, α 2 , and α 3 are the first, second, and third powers of the root a; and in like manner 1, A, and A denote the same powers of the root - .
But before considering more particularly what relates to powers and roots, it will be proper to observe, that the quantities A, &c., admit of being expressed under a different form; for, just as the quantities α, α 2 , α 3 , <fcc., are expressed as positive powers of the root a, so the quantities 1, ∙^, <fcc,, may be respectively expressed thus α -1 , a~ 2 , a -3 , &c., and considered as negative powers of the root a.
This method of expressing the fractions ~, ∙^, as powers of the root a, but with negative indices, is a consequence of the rule which has been given for the division of powers; for we consider - as the quotient arising from the division of any power of α by the next higher power; for example, from the division of the 2d by the 3d, and so 1 (fl we have - = —; but since powers of the same quantity are divided by subtracting the exponent of the divisor from that of the dividend (Art. 19), it follows that = α 2 - 3 = a~' 1 ; therefore the fraction - may also be expressed
1 ^2
thus, a~ 1 . By considering as equal to , it will appear
1
in the same manner that — = — = a~ 2 ; and proceeding in CL~ , Cv
,, · 1 α 2 , 1 α 2 . _
this way, we get - τ = — = a~ 3 , ~∑ = -^ = α~ i , &c., and so on, j ° α 3 a 5 ’ a 4 a δ ’ ’ ’
as far as we please. It also appears that unity or 1 may be represented by α 0 , where the exponent is a cypher, for l=⅛ = α 2-2 = σ 0.
α 2
The rules which have been given for the multiplication: and division of powers with positive integral exponents' will apply in every case, whether the exponents be positive or negative, integral or fractional, provided we assume as the definition of the index in such cases, the law of combination α m × a n = α* ’ +*.
Involution.
22. Involution is the method of finding any power of any assigned quantity, whether it be simple or compound: hence its rules are easily derived from the operation of multiplication.
Case 1. When the quantity is simple.
Pule. Multiply the exponents of the letters by the index of the power required, and raise the coefficient to the same power.
Note. If the sign of the quantity be +, all its powers will be positive; but if it be -, then a∏ its powers whose exponents are even numbers are positive, and all its powers whose exponents are odd numbers are negative.
Ex. 1. Required the cube, or third power, of 2a 2 x. (2α⅜) 3 = 2 × 2 × 2α 2 ×⅛ 1 × 3 = 8α⅛ 3 , the answer.
Ex. 2. Required the fifth power of - 3α⅛ 3 . ( — 3α⅜ 3 ) 5 = - 243cι 1 ⅛ 15 , the answer.
Ex. 3. Required the fourth power of - ·
∕-2αx 2 M 16α 4 x 8 x , ) = ^iΓ i > the answer.
∖ 3δ 2 τ∕ ∕ g 81&V
Case 2. When the quantity is compound.
Pule. The powers must be found by a continual multiplication of the quantity by itself.
Ex. 4. Required the first four powers of the binomial quantity a + x. a + X the root, or first power.
a + x
α 2 + ax + ax + x 2 a 2 + 2ax + x 2 the square, or second power.
a +x a 3 + 2a 2 x + ax i + a 2 x + 2ax 2 + x 3 a 3 + 3α⅛ + 3ατ 2 + x 3 the cube, or third power.
α 4 + 3a 3 x + 3a 2 x 2 + ατ 3 + a 3 x + 3a 2 x 2 + 3ax 3 + x i a i + 4α⅛ + 6α 2 z 2 + 4αx 3 + <r 4 the fourth power.
If it be required to find the same powers of a — x, it will be found, writing — x for x i that a — X being the root, or first power; then α 2 - 2ax + x 2 is the square, or second power; α 3 - 3α⅛ + 3<xr 2 - ar 5 , the cube, or third power; α 4 - 4a 3 x + 6a 2 x 2 - 4ax 3 + x i , the fourth power.
Hence it appears that the powers of a + x differ from the powers of a — x only in this respect, that in the former the signs of the terms are all positive, but in the 1atter they are positive and negative alternately.
Besides the method of finding the powers of a compound quantity by multiplication, which we have just now explained, there is another more general, as well as more expeditious, by which a quantity may be raised to any power whatever without the trouble of finding any of the inferior powers, namely, by means of what is commonly called the binomial theorem, to be proved hereafter. This theorem may be expressed as foÛows :—Let a+x be a binomial quantity, which is to be raised to any power denoted by the number n, then (α + xf =
∣ « α -ι τ , < to ~ 1 ) α - 2j , 2 . n(n-U(^-2) 3 , a + -a x + 1 ' 2 a X + 1 2 - 3 a xr n(n - l)(n - 2)(n - 3)
. + l.2 . 3 . 4 α X n(η -1)(⅞ - 2)(n - 3)(n - 4)
+ 1 . 2 . 3 . 4 . 5 a ^ + &C ·
This series will always terminate when n is any whole positive number, by reason of some one of the factors 7i — 1, n — 2, &c., becoming = 0; but if n be either a negative or fractional number, the series will consist of an infinite number of terms. As, however, we mean to treat in this section only of the powers of quantities when their exponents are whole positive numbers, we shall make no further remarks upon any other. The nth power of a — x will not differ from the same power of a + x, except in the signs of the terms which compose it, for it will stand thus: (α — xf — a . _» ∙-⅛ + ⅛iW- n(n-l)(n-2)^ 1 1 . Ü 1 . Z · o η(n-lYn-2Yn-3) . p , ,, + ——;—-—-————-a i x i — &c., where the signs are + 1 . 2 . 3 . 4
and — alternately.
Let it be required, for instance, to raise a + x to the fifth power.
Here n, the exponent of the power, being 5, the first term a η of the general theorem will be equal to α 5 , the second na nr ~ 1 x = 5a*x, the third 1 ~- 1 -∙ a n ~ 2 x 2 = ∣^⅛z 3 a 2 =
’ 1.2 l×2
10a⅜ 3 , the fourth ( n ∙ ^α n ~ r sc 3 = ð- × × a 2 x 3 ≈
1.2.3 1 X 2 X 3
10aV,t⅛e≡ ^ ,1 T 1 ><∖ 2χ,, - 3 W =
’ 1.2.3.4 l×2×3×4
= 5aas 4 , and the sixth and last n(n-l) (n-2) (n-3) (n-4) = 5×4x3×2xl =
1.2.3. 4. 5 l×2×3×4×5 ’ the remaining terms of the general theorem all vanish, by reason of the factor n — 5 = 0 by which each of them is multiplied, so that we get (α + ic) 5 = α 5 + 5α⅛+10α⅛ 2 + 10a 2 s¾ + 5ax 4 + x δ . If the quantity to be involved consists of more than two terms, as, if p + q - r were to be raised to the second power, put p = a and q-r = b, then (p + q -r) 2 = (a + δ) 2 = α 2 + 2ab + b 2 =p 2 + 2p(q - r) + (q - r) 2 , but ^p{q-r) = 2pq - 2pr, and by the general theorem (q - r) 2 = q 2 - 2qr + r 2 , therefore we get (p + q-r) 2 =p 2 + 2pq-2pr + q 2 - 2qr + r 2 ∙, and by a similar method of proceeding a quantity consisting of four or more terms may be raised to any power.
Ex. 1. From the value of (a+xf found in example 4, to find that of (α + b + c) 4 . From example 4 we write at once, by symmetry, [table]
where R is the series of remaining terms denoting the three following forms, a 2 bc, b 2 ac, c 2 ab. Now when a, b, c arc each unity, there are 81 terms (viz. 3 4 ). But the number of terms already written down (4a 3 b being considered as 4 tenns, &c.) is 45. The quantity R must consequently make up the other 36 terms, .'. it can be nothing else than 12a 2 bc + 12b 2 ac + 12c 2 ab. Ex. 2. (p ÷ q + r) 2 =p 2 + q 2 + r 2 + 2(pq + qr + rp).
Cor. If y> + <7 + z∙ = 0; then/» 2 + cf + r 2 + 2(pq + qr + rp) = 0. Case 1. a-b + b-c + c-a = (j, gives
(α - δ) 2 + (δ - c) 2 + (c — α) 2 + 2{(α-δ)(δ -c) + {b- c)(c — α) + (c - α)(α - δ)} = 0 .
Case 2. α(δ — c) + δ(c — a) + c(a — δ) = 0, gives a 2 (δ - c) 2 + δ 2 (c - a) 2 + c 2 (a - δ) 2 + 2 {aδ(δ - c)(c - a) + δc(c - a)(a - δ) + ca(a - δ)(δ - c)} = 0 .
Ex. 3. Prove that (x 2 - yz) 3 + (y 2 - arz) 3 + (z 2 - xf) 3 - 3(x 2 — yz)(jy 2 — xz)(z 2 — xf) is a complete square.
The expression will assume symmetry if (x 2 - yz)(y 2 — xz) (z 2 -xf), instead of being multiplied by 3, be repeated three times, each being connected with one of the cubes in turn; this gives— [table]
Ex. 4. Prove that (α 2 + δ 2 + c 2 ) 3 + 2(αδ + be + cα) 3
- 3(α 2 + δ 2 + c 2 )(αδ + be + cα) 2 = (α 3 + δ 3 + c 3 - 3αδc) 2 .
Combine each of the cubes with each of the products in succession, and reduce, as in the last example.
Ex. 5. To find the condition that px 2 + 2qxy + ry 2 may be incapable of changing its sign through any change of sign or value of x and y. It is evident that p and r must have the same sign. Suppose it positive. By multiplying by p, the quantity may be thrown into the form (px + qf) 2 + {pτ - ⅛ 2 )y 2 , which is the sum of two positive quantities provided pr>ef. The condition required is, therefore, pr>cf∙, or as a particular case prr = q 2 . Ex. 6. To find the condition that ax 2 + by 2 + cz 2 + 2Pyz + 2Qzx + 2Rxy may be incapable of changing its sign through any change of sign or value of x, y, z.
We will suppose a, b, o to be all positive, in which case the whole result is also positive.
If we multiply the whole by a, we may write it under the form of a square and a supplement, viz., (ax + Q2 + Ry) 2 + (ac - Q 2 )s 2 + (αδ-R 2 )y 2 + 2(αP-QR>.
Now as the first term of this expression is a square, it is essentially positive. Hence the required condition can be satisfied only by rendering the remainder positive.
It follows that αc>Q 2 , αδ>R 2 , and
(Example 5) (ac - Q 2 )(αδ - R 2 ) >(αP - QR) 2 , i.e., abc + 2PQR >αP 2 + δQ 2 + cR 2 .
If we had begun by throwing the expression into the form of (δy + Pz + 1Lr) 2 ÷ <kc., a resulting condition would have been bc>P 2 . The four conditions are consequently
αδ>R 2 , αc>Q 2 , δc>P 2 , abc + 2PQR >αP 2 + δQ 2 + cR 2 .
Results of this kind are of the utmost value in the higher analysis.
Evolution.
23. Evolution is the reverse of involution, or it is the method of finding the root of any quantity, whether simple or compound, which is considered as a power of that root: hence it follows that its operations, generally speaking, must be the reverse of those of involution.
To denote that the root of any quantity is to be taken, the sign s ∕ (called the radical sign) is placed before it, and a small number placed over the sign to express the denomination of the root. Thus ^∕α denotes the square root of a, fa its cube root, fa its fourth root, and in general, fa its nth root. The number placed over the radical sign is called the index or exponent of the root, and is usually omitted in expressing the square root: thus, either fa or fa denotes the square root of a. Case 1. When roots of simple quantities are to be found.
Pule. Divide the exponents of the 1etters by the index of the root required, and prefix the root of the numéral coefficient; the resuit will be the root required.
Hole 1. The root of any positive quantity may be either positive or negative, if the index of the root be an even number; but if it be an odd number, the root can be positive only.
2. The root of a negative quantity is also negative when the index of the root is an odd number.
3. But if the quantity be negative, and the index of the root even, then no root can be assigned.
Ex. Required the cube root of 125α 6 x 9 .
Here the index of the root is 3, and the root of the coefficient 5, therefore i ^∕125α⅛ 9 = 5α¾c 3 , the root required; and in like manner the cube root of — 125α⅛ 9 is found to be - 5α⅛ 3 .
The root of a fraction is found by extracting the root of both numerator and denominator. Thus the square l . 4α 2 x 4 . 2αx 2 rootof w 13 W
Case 2. When the quantity of which the root is to be extracted is compound.
I. To extract the square root.
Range the terms of the quantity according to the powers of one of the 1etters, as in division.
Find the square root of the first term for the first part of the root sought, subtract its square from the given quantity, and divide the remainder by double the part already found, and the quotient is the second term of the root.
Add the second part to double the first, and multiply their sum by the second part; subtract tho product from the remainder, and if nothing remain, the square root is obtained. But if there is a remainder, it must be divided by the double of the parts already found, and the quotient will give the third term of the root, and so on.
3 ce 1
Ex. Required the square root of xf - 2z 3 + - - + — · [table]
To understand the reason of the rule for findiag the square root of a compound quantity, it is only necessary to involve any quantity, as a + b + c, to the second power, and observe the composition of its square; for we have (α + δ + c) 2 = α 2 + 2αδ + δ 2 + 2αc + 2δc + c 2 j but 2αδ + δ 2 = (2α + b)b and 2αc + 2bc + c 2 = (2α + 2b + c)c, therefore, (α + b + c) 2 = a 2 + (2a + δ)δ + (2α + 2δ + c)c; and from this expression the manner of deriving the rule is obvious.
As an illustration of the common rule for extracting the square root of any proposed number, we shall suppose that the root of 59049 is required.
Accordingly we have (a + δ + c) 2 ⅛= 59049, and from hence we are to find the values of a, b, and c. [table]
II. To extract the cube root.
Range the terms of the quantity according to the powers of some one of the letters.
Find the root of the first term, for the first part of the root sought; subtract its cube from the whole quantity, and divide the remainder by three times the square of the part already found, and the quotient is the second part of the root.
Add together three times the square of the part of the root already found, three times the product of that part and the second part of the root, and the square of the second part; multiply the sum by the second part, and subtract the product from the first remainder, and if nothing remain, the root is obtained; but if there is a remainder, it must be divided by three times the square of the sum of the parts already found, and the quotient is a third term of the root, and so on, till the whole root is obtained.
Ex. Required the cube root of α 3 + 3a¾: + 3αr 2 + ae 3 . [table]
The reason of the preceding rule is evident from the composition of a cube; for if any quantity, as a + δ + c, be raised to the third power, we have (α + δ + c) 3 = α 3 + (3α 2 + 3ab + δ 2 )δ + {3(α + δ) 2 + 3(α + δ)c + c 2 } c, and by considering in what manner the terms a, b, and c are deduced from this expression for the cube of their sum, we also see the reason for the common rule for extracting the cube Toot in numbers. Let it be required to find the cube root of 13312053, where the root will evidently consist of three figures; let us suppose it to be represented by a + δ + c, and the operation for finding the numerical values of these quantities may stand as follows :— [table]
III. To extract any other root.
Range the quantity of which the root is to be found, according to the powers of one of its letters, and extract the root of the first term; that will be the first member of the root required.
Involve the first member of the root to a power less by unity than the number that denominates the root required, and multiply the power that arises by the number itself; divide the second term of the given quantity by the product, and the quotient shall give the second member of the root required.
Find the remaining members of the root in the same manner by considering those already found as making one term.
24. In the preceding examples, the quantities whose roots were to be found have been all such as could have their roots expressed by a finite number of terms; but it will frequently happen that the root cannot be otherwise assigned than by a series consisting of an infinite number of terms. The preceding rules, however, will serve to determine any number of terms of the series. Thus, the square root of α 2 + χ 2 will be found to be a + -— 7—5 +
i 2a 8α 3
ÍCð ζ)JP
160 3 ~ι280^^^ & C- ’ an ^ cu ^ e γ ° 0 ^ α3 + iβ3 will stand ., ae 3 x β 5x 9 10x 12 . .
thus, o+ --- + - + * c , But a3 the extraction of roots in the form of series can be more easily performed by the aid of the binomial theorem, we shall ι , eferthe reader to the section where this subject is resumed.
Additional Examples.
3
Ex. 1. Write down the square root of x i - 2# 8 + - x z -
2
-x + W» which is given as a perfect square. 2 lb
Since the square contains 5 terms, the root must contain 3. Of these the first is x l on account of z 4 , the second - X on account of 2zr 3 , and the third ÷ 7 on account of A - 4 16
But as the last term but one of the square is —, and the last term but one of the root also -, the last term of the root must be +.
.∙. x" i - X + ∣ is the root required.
Ex. 2. Extract the square root of 25íc 4 + 16y 4 - ⅛xy (5a> 2 + 4y 2 ) + 49∙r 2 y 2 . We must first arrange the square in terms of some one quantity (say x^).
The first term of the square is 25ar t , which gives 5x 2 as the first term of the root. The second term of the square,
- 30a⅛∕ gives - 3xy as the second term of the root. The last term 16y 4 gives ± 4y 2 ; which, since the last term but one is -, leads to the root 5ae 2 - 3xy + 4y 2 .
Ex. 3. Extract the cube root of
8z 6 - 36íc 5 + 66α4 - 63<ri 5 + 33ae 2 - 9x +1.
Since there are seven terms in the cube, there must be three terms in the root. The first is 2ae 2 , the second - 3x, the third 1, as will be seen at once by examining the cube of p - q +1, viz., ∕> 3 - 3p 2 q + ... - 3q +1.
These examples have been solved by the assumption that the root is capable of extraction without 1eaving a remainder. When this is not the case, or when there is no certainty that it is so, the only resource is to work the example through, abbreviating the process by the aid of detached coefficients.
Ex. 4. Extract the square root of 4λj 6 + 12x 5 y + 5^y 2 - 2x 3 y 3 + 7a¾∕ 4 - %xy b + y 6 . The work is written thus: [table]
We have [table]
Sect. III.— Feaotions.
25. In the operation of division, the divisor may be sometimes greater than the dividend, or may not be contained in it an exact number of times: in either case the quotient is expressed by means of a fraction. There can be no difficulty, however, in estimating the magnitude of such a quotient; if, for example, it were the fraction f, we may consider it as denoting either that some unit is divided into 7 equal parts, and that 5 of these are taken, or that 5 times the same unit is dividet into 7 equal parts, and one of them taken.
In any fraction the upper number, or the dividend, is called the numerator, and the lower number or divisor is cal1ed the denominator. Thus, in the fraction a is the o numerator, and b the denominator.
If the numerator be 1ess than the denominator, such a fraction is cal1ed a proper fraction; but if the numerator be either equal to or greater than the denominator, it is called an improper fraction; and if a quantity be made up of an integer and a fraction, it is called a mixed quantity.
Thus, is a proper fraction: - and -—- are both im- ’α+x i i 3 a a proper fractions; and b + is a mixed quantity.
The reciprocal of a fraction is another fraction, having its numerator and denominator respectively equal to the denominator and numerator of the former.
Thus, ~ið the reciprocal of the fraction ∣ ·
26. The following proposition is the foundation of the operations relating to fractions.
If the numerator and denominator of a fraction be either both multiplied or both divided by the same quantity, the value of the resulting fraction is the same as before.
To demonstrate this proposition we shall throw the definition of a fraction into a categorical form. We shall according1y define the fraction as such a magnitude, that when it is multiplied by b, the product is a. [table]
From this proposition, it is obvious that a fraction may be very differently expressed without changing its value, and that any integer may be reduced to the form of a fraction, by placing the product arising from its multiplication by any assumed quantity as the numerator, and the assumed quantity as the denominator of the fraction. It also appears that a fraction very complex in its form may often be reduced to another of the same value, but more simple, by finding a quantity which will divide both the numerator and denominator, without leaving a remainder. Such a common divisor, or common measure, may be either simple or compound; if it be simple, it is readily found by inspection, but if it be compound, it may be found as in the following problem.
27. Pkob. I.— To find the greatest common Measure of two Quantities. Rule 1. Range the quantities according to the power of some one of the letters, as in division, leaving out the simple divisors of each quantity.
2. Divide that quantity which is of most dimensions by the other one, and if there be a remainder, divide it by its greatest simple divisor; and then divide the last compound divisor by the resulting quantity, and if anything yet remain, divide it also by its greatest simple divisor, and the last compound divisor by the resulting quantity. Proceed in this way till nothing remain, and the last divisor shall be the common measure required.
Note.— It will sometimes be necessary to multiply the dividends by simple quantities in order to make the divisions succeed.
The demonstration of this proposition depends on the Axiom, that whatever divides a number divides any multiple of the number; and whatever divides two numbers divides their sum or difference. It was given by Euclid in Prop. 2, Book vii., very much as follows :—
Let a, b be the quantities, the smaller of which is b.
Let a be divided by b, with a remainder c, b by c, with a remainder d, c by d, with no remainder, d is the greatest common measure of a and b. We have a-pb = c, b-qc≈d, c = rd.
Now, (1.) d is a common measure of a and b ; for d divides c .∙. qc .’. qc + d .’. b .'. pb .’. pb + c .∙. a; i.e., d divides a and b.
(2.) It is the greatest common divisor. For if not, let e be the greatest; then, since e divides a and b, it divides a and pb, .∙. α-pb .∙. c .'. qc .'. b — qc .’. d ’ i i.e., e is less than d, and not greater.
Cor. Every other divisor of a and b divides their greatest common measure.
Observe that no fraction is in a form to be interpreted until it is reduced to its lowest terms.
Ex. 1. Required the greatest common measure of the quantities α¾-α> 3 and α 3 -2α¾ + αr 2 . The simple divisor x being taken out of the former of these quantities, and a out of the latter, they are reduced to α 2 — x 2 and α 2 - 2ax + x 2 ; and as the quantity a rises to the same dimensions in both, we may take either of them as the first ι divisor: let us take that which consists of fewest terms, and the operation will stand thus: [table]
Hence it appears that a — x is the greatest common meastιre required.
Ex. 2. Required the greatest common measure of 8α 2 Z> 2 — 10αδ 3 + 2δ 4 , and 9α 4 δ - 9α¾ 2 + 3α 2 δ 3 - 3αfr 4 .
It is evident, from inspection, that δ is a simple divisor of both quantities; it will therefore be a factor of the common measure required. Let the simple divisors be now left out of both quantities, and they are reduced to 4α 2 — 5aδ + 0 2 , and 3α 3 - 3α¾ + alf - b 3 ; but as the second of these is to be divided by the first, it must be multiplied by 4 to make the division succeed, and the operation will stand thus: [table]
This remainder is to be divided by b, and the new dividend multiplied by 3, to make the division again succeed, and the work will stand thus: [table]
This remainder is to be divided by -19δ, which being done, and the last divisor taken as a dividend as before, the rest of the operation will be as follows :— [table]
from which it appears that the common divisor sought is a-b, and remarking that the quantities proposed have also a simple divisor b, the greatest common measure which is required will be b(a - í).
It will be seen that the examples we have given are not on numbers, but on algebraic quantities. In fact, the axiom and the demonstration founded on it apply, with some restrictions and modifications, to such quantities. The most important of the modifications is this: that the divisor, instead of being a whole number, is an expression of the form x + m, where m is of the nature of a numerical quantity, and does not depend on x.
The application of this modified form of the axiom has a wide range in the higher analysis. We offer two additional examples for advanced students.
Ex. 1. If αx 2 + bx + c, α'x 2 + Px + c' have a common divisor of the form x + m, prove that (a ’ b — αb') (b'c — δc') = (α'c — αc') 2 .
Multiply the first expression by a, and the second by a, and subtract the products, the difference (ab - αb')x + ac — αc', is by the axiom divisible by x + τzι, α!c —α'c .
∙*∙ x +~lι Γ∕ wλγ + wi ∙ a o — ao
Again, multiply the first’ expression by c', and the second by c, and subtract them; the difference (α'c - ac')# 2 + (Pc - bc' y )x is divisible j by x + m, . ∙. x + is x + m. Qf C — (LC z - t 11 b ’ c — be' a'c — ac'
Consequently, 7 7 = —— 77, v αc-αc αo-αδ
the condition required.
Ex. 2. If ax 3 + 3δz 2 + d, ba? + 3cfø + e, have a common divisor;’then (ffbd - αe) 3 + 27(αd 2 + b 2 e) 2 = 0.
Treating this question exactly as the last, viz., multiplying first by b and a, and then by e and d, and subtracting, it appears (if u be written instead of bd - ae îor brevity) that the two following expressions have a common divisor,
3δ¾ 2 - 3adx + u and ux 2 - 3bex + 3d 2 , whence, by the last example, the condition is (3beu - 9ad 3 ) (3adu - 9δ⅞) = (u 2 - Qb 2 d 2 ) 2 , from which u divides out as a common factor, and the result reduces to that enunciated.
28. Prob. II.— To Reduce a Fraction to its Lowest Terms. Rule. Divide both numerator and denominator by their greatest· common measure, which may be found by Prob. I.
Ex. 1. Reduce to its lowest terms ∙
We have already found in the first example of Prob. I. that the greatest common measure of the numerator and denominator is a - x ; and dividing both by this quantity, 1 α¾-x 3 ax+x 2 we have , _ 9 9 = -,- τ ,
α 2 -2α¾+αz 2 a 2 -ax
In like manner we find 9α 4 δ-9α⅞ 2 +3α 2 δ 3 -3αδ 4 9α 3 +3αδ a 8α 2 δ 2 - 10αδ 3 +2δ 4 - 8ab - 2b 2 i the common measure being b(a - P), as was shown in Ex∙ ample 2, Problem I. ct^ 1) ’ ^ - ∣ - 2òc Ex. 2. Reduce ∙ 9 ι 9 —— to its lowest terms.
a 2 +b 2 -c 2 +2ab a 2 -b 2 -c 2 +2bc a 2 -(b-c) 2 (a+b — c) (a — b — c) a — b+e α 2 + δ 2 - c 2 + 2ab ~(a +δ) 2 -c 2 “(a+b + c)(α÷δ-c) - a+b + c Ex. 3. To find the value of fc+ 1 ) 2 ~ 3ig - 3 when χ = 2 ae 2 —2ae
Here the substitution of 2 in place of x renders the numerator and denominator separately equal to 0. This shows (Art. 20) that x - 2 is a divisor of each of them. We get, therefore, (as+l) 2 -3x-3 ae + l . 3
- 9 —õ = ’ which when x = 2 becomes -·
x i — 2a> x 2
Ex. 4. Find the value of -⅜~ ⅞ 3 4 f ^t 1 when x = 1.
ae 4 - 4x 3 +6x 2 - 4x +1
Dividing numerator and denominator by x-l, the -ae2 _ ß/ç
result is a Λ -3χ 2 + 3 a Γ Γ 1 , w ∙kich, when 1 is written in place 3
of x i becomes ð, or infinity.
29. Prob. III.— To Reduce a Mixed Quantity to an Improper Fraction. Rule. Multiply the integer by the denominator of the fraction, and to the product add the numerator; and the denominator being placed under this sum, the result will be the improper fraction required. 2j2 Ex. 1. Reduce a-x + ^-^ to an improper fraction. x 2 (a+x) (a-x)+x 2 a 2 . a+x a+x a+x
30. Prob. IV.— To Reduce an Improper Fraction to a Whole or Mixed Number. Rule. Divide the numerator by the denominator for the integral part, and place the remainder, if any, over the denominator; it will be the mixed quantity required. τ , n r > j ax+2x 2 , x 2 -y 2 . Ex. 1. Reduce —-— and to whole or mixed
α+ae x—y quantities.
τ ,. ax+2x 2 x 2 i l ιrst —-—- = x 4 ì the answer.
a+x a+x ae2 —- ?/$
And — — x + y a whole quantity, which is the χ ~ y answer.
31. Prob. V.— To Reduce Fractions having different Denominators to others of the same value which shall have a common Denominator. Rule. Multiply each numerator separately into all the denominators except its own for the new numerators, and all the denominators together for the common denominator.
Ex. 1. Reduce a ∏d tθ fractions of equal value, having a common denominator. ax(a + x) = a 2 x + ax 2 1 il
(α≡ - ic ≡) (a - x) = a’ - α¾ - ∞≈ + χ3 ∫ the new n≡≈≈>tors∙
(α — x)(a + x) = a 2 — x 2 , the common denominator, ττ ax a 2 x+ax 2 1 α 2 - x 2 a 3 — a 2 x-ax 2 +rf
Hence —- = ——- and —-— = 5—» ’
a—X a--χ 2 a+x a 2 -ar ’ 32. Prob. VI.— To Add or Subtract Fractions. Rule. Reduce the fractions to a common denominator, and add or subtract their numerators; and the sum or difference placed over the common denominator is the sum or remainder required.
In practice, however, it is generally better to separate the process into two or more parts analogous to the addition or subtraction of sums of money, where the pounds are added to the pounds, the shillings to the shillings, &c., and the result afterwards combined.
Ex. 1. Add together and r -^- · o a-b b-a
The latter fraction is ·
α —o , . ,. a b a—b -
.∙. the sum required is —.; = —; = 1.
j ∙ a —o a-b a-b
~. .. . 1 1 1 ι a∙ l-a∙ -
Similarly, . + — = -- —r +;=——τ = - 1.
j, a·-1 a~ n -l a n -l l-a" a*-l Ex. 2. Collect into a single fraction
1 1 2α
a — b α+δ^ a 2 -b 2 ςs . 1 1 2 ¾
Smce α -6 a+ b
1 1 2a 2(⅛+α) 2
a — b a+b^a 2 - ò 2 a 2 -b 2 a — b
11 7
Ex. 3. Collect -—- + -—- + —÷— ·
4x —8 3x-6 24 — 12x
We observe that x - 2 is common to all the denominators; the question may therefore be written, ii2. ⅛⅛-⅛θ∙
Ex. 4. Collect + + ∙
Here we commence by adding the 1st and 3d together, and the 2d and 4th together; which results in
6x 24x
9x 2 -4y 2 ^^ x 2 -16y 2 C ( 1 4 — 35x 2 = *∖9x 2 -4y 2 x 2 -*-16y 2 ∕ (9x 2 - 4y 2 )(x 2 - 16y 2 )
210x 3 fs (9x a - 4y 2 ) (x ? — 163∕ 2 )
_ * tv j xi. , l+x+x 2 +x 3 ι l-x+x 2 -x 3 vx. 5. Find the sum of 1 ., + ^ι,^+ 1 4, + ^ ·
The numerator will consist of the sum of two products, the one containing + x, exactly in the same way that th∈ other contains - x. If, then, we write down one of these products, and double the even powers of x in it, omitting the odd powers, we shall obtain the required result. The product of the denominators again may be readily obtainec by regarding it as that of the difference and sum of 1 + x· and x + x 3 . As such processes are of constant occurrence, we will indicate the work in full. [table]
Denominator, {1 + x 2 - (x + x 3 )} {1 + x 2 + (x + x 3 )} = (1 4 x 2 ) 2 -(x + x 3 ) 2 = 1 + 2x 2 + x i -x 2 - 2x*-x 6 = 1 + x 2 À , „ . 2÷6x 2 +6x 4 +2xθ
And the result is 1+aj2-a .l∑^-∙ Ex. 6. Collect into one fraction ———7-—— +
1 + x m-n + x"'~∙ ,
1 1
1 + x"-“ + x n ~P + 1 + χP- m +χP-∙*
Multiply numerator and denominator of the first fraction by x^^*, &c., and the given quantity becomes x~” χ-* x~^ p = J
x - "+x^^*+χ-* + χ-~+x~"+ x~ p + χ^^ m +or"+ x~ p
_ 1 tn nm 1 1 Ex ' lf i + l + ln + l+m+nd + l+n+nm~ 1, and
_J + wZ — + L_ =1 1 + l+ln l+ ι m+ml 1+n+nm ’ none of the denominators being zero, then I = m = n.
Multiply the first quantity by I, and subtract, thcro results I = n which, when substituted in the first quantity, gives m — n, whence the proposition.
33. The converse problem to collecting many fractions 1 into one is frequently as important as the direct—the pro- f blem, namely, of resolving a compound fraction into its components or partial fractions. For a first example, if it be required to find what simple fractions make up the compound fraction —- v we commence by observing that the denominator x 2 — α 2 is the product of x + a and x — a. 2x ·
Hence, -τ—0 is the sum of the fractions whose denomina- ’ x 2 -α 2 tors are x + a and x - a. 2x A. *β φ w
Let -7-—„ = 1 , where A and B are quantities x 2 -α 2 x+a x—a i which involve a only, not x, since x 2 does not appear in the numerator of the sum.
_ jj∙x∙ 2x A(x-α) + B(x+α)
By addition, - r ^- a
.∙. 2x = A(x - a) + B(x + a).
To obtain A and B from this equality, we remark that the equality is an identity, as in Art. 20. We may, therefore, deal with it in either of two ways: 1. Make the x’s on the left hand side to coincide with the x’s on the right, and the a’s in like manner. 2. As in Art. 20, write anything we please in place of x on both sides. We will in this example take the first method, and illustrate the second method by the subsequent examples. We get 2 = A + B, 0 = A - B; .∙. A = B = 1, and the result is
2x 1,1
—— ≈ — 1- — ·
x 2 -α 2 x-a x-a
P, a 1 a I - ?-
(x — α)(x- b) x — a x—b
.∙. l=A(x-δ) + B(x-α).
Write a for x i then 1 =A(αδ) .∙. A = ^-ð·
Write δforx, then l=B(δ-α)= -(α-δ). ’. B= -~ðî hence, (x-α)(x-δ) = 7 x^δ) ’
The reader will observe that we have treated —7 as if a—ö it were not itself a fraction. In fact, in the application of the subject before us, the letters a and b stand for arithmetical quantities, and the fraction is simply an arithmetical fraction, as contradistinguished from an algebraical fraction like ·
x-a Ex. 3. yg ^ t ^ g A i b + ° (x-a) (x-b) (x-c) χ — α x-b^ r x-c gives px + q = A(x -b)(x-c) + B(x - a) (x - c)
+ C(x -a)(x-b), .∙. pa + q = A(α - δ) (α - c), &c., and P a +i . J_ + (x — a) (x — b) (x — c) (a — b)(a- c) x — a pb+q . 1 + pc+q 1 (δ-α) (δ-c) x-δ^ t ^ (c-α)(c-δ) x-c Ex. 4. Find the sum of √ +-—¾ c - +
(δ-c)(c-α) (c-α)(α-δ)
c+a (a-b)(b-cV
Let α÷δ + c = sj and write in alphabetical order; it gives a-s b-s c—8 (a-b)(a-c) (b-a) (b - c) + (c - a) (a - b) ’ i.e. (Ex. 3), the A, B, C of the resolved fraction, x-s = A B C (x-α) (x-b) (x-c) x-a + x-b + x-c ’ and since x-s = Å(x -b) (x-c) + B(x -a)(x-c) + C(x-a) (≈-b), the sum required, being the coefficient of x 3 , is equal to 0.
The reader will easily extend this process to other cases, as, for instance, to prove r , k bed, eda Ex. 5. -∣ —— 4-
(α-δ) (a-c)(a-df r (b-c)(b-d) (δ-α) + dab dbc t l (c-d)(c-a)(c-b} (d-a)(d-b)(d-c)~ ’
34. Pkob. VII.— To Multiply Fractions. Pule. Multiply the numerators of the fractions for the numerator of the product, and the denominators for the denominator of the product.
The demonstration follows at once from the definition of a fraction given in Art. 26; thus sifice ^×b = a, ⅛×d = c, we have ∣×δ×∣×d<=αc, i.e., by the commutative law a c u r×-.×bd=ac. b d
T> X aC 7 7
But — × bd = ac bd • - ’ ∙ b × d~bd*
35. Pkob. VIII.— To Divide Fractions. Rule. Multiply the dividend by the reciprocal of the divisor, the product will be the quotient required.
This rule requires no demonstration.
Examples in Multiplication and Division of Fractions. Ex. 1. Multiply by -f f ,. Since 2- δ -=≤≠∖ 6 α * ’ α 2 -δ 2 ba ab
.∙. the product is a 2 -b 2 a 2 _ a 2 _ a ab × a 2 -b 2 ab~b ' Ex. 2. Multiply a ∖~ 3 *+ 2 by ≈ 2 -+ 2 *+ 1 . x 3 4-2x 2 4-2x4-1 y x 2 -5x4-4
Because the numerator of the first fraction, and the denominator of the second both become 0, when 1 is written for x, each is divisible by x - 1 (Art. 20). In the same way the denominator of the first fraction, and the numerator of the second are both divisible by x+l, Hence, x 3 -3x÷2 x 2 + 2x+1 _ x 3 -3x÷2 x 2 ÷2x4-l x 3 4-2x 2 4-2x 4-1 × x 2 -5x+4 - x 2 -5x+4 × x 3 +2x 2 +2x÷l ^ x 2 +x-2 x+1 x 3 +2x 2 -x-2
x-4 × x 2 +x÷l - x 3 -3x 2 -3x-4
r , „ τv ∙j a ð r α 2 ð 2 Ex. 3. Divide v - - by ττ - -z · δ a j δ 2 α 2 τr ∏ .∙ , · a 2 —δ 2 a 2 δ 2 ab
The quotient is —— × —.— ιi ≈ - s — 7 τ. ξl ab a i -b i a 2 4-δ 2 τ , . τ> 7 1 ∕δ 2 +c 2 -a 2 ∖2 .
Ex. 4. Reduce 1 - ( — ) to factorials.
∖ 2δc ∕
, ∕b 2 + c 2 -a 2 ∖ 2 _ P δ 2 +c 2 -α 2 ∖ ∕ δ 2 +c 2 -a 2 ∖
∖ 2δc ) ~ V + 2δc A 2δc ) -(δ÷c) 2 - a 2 a 2 -(b-c) 2 ~~ 2bc × 2δc
(δ + c+a) (δ + c-a) (a+b-c) (a-b + c) . ~ 4δ 2 c 2 Ex. 5. Reduce 1 - (a j^∖ c ∙⅛√*^ ) to factorials. ∖ 2(ab+cd) ∕
1 _ ∕ α 2 +δ 2 -c a -d 2 ∖2 = (a 2 +b 2 -c 2 -d 2 ì
∖ 2(αδ÷cd) ) ~ (+ 2(aδ÷cd) J ×
í I _ a 2 + δ 2 -c 2 -d 2 ì = ( a+ δ) 2 -(c-d) 2 (c+d) 2 -(a-δ) 2 ( 2(aδ+cd) J 2(aδ+cd) × 2(aδ + cd) (a+b+c-d) (a+b-c+<P) (c+d+a-b) (c+d-a+b) ½(db+cdf t Miscellaneous Examples in Fractions. Ex. 1. Find the value of ι 1 , +, -., ab-ax bx — ab (x-a)(bc-cx) j when x = — · Writing down every term with x ab+ac+bc 0 j first, there results—
1 1 -+χ+--2
- . 1 1 X , _ a b c _ Q
a(x - b) δ(x - a) c(x -a)(x-b)~ ~ (χ-a)(x-δ) Ex. 2. Find the value of —1— 4—î— i 4——, when x-3a x-3δ x+3c’
- + _ _ _ an d x _ 2(a 4- b - c). a b c v '
Restore symmetry by writing - c for c ; the numerator of the sum is (x - 3δ) (x - 3c) + (x - 3α) (x - 3c) + (x - 3α) (x - 3δ) = 3{x 2 - 2(α + b 4- c)x + 3(ab + ac + δc)}. But x = 2(α + δ + c), whence the first and second terms make up 0; and - + v + - =0, is the third term divided by abc, a 0 c ’ .∙. the sum required is 0.
Ex. 3. Given that (α 2 4- be) (6 2 + ac) (c 2 4- do) + (α 2 - be) (b 2 -ac)(c 2 -ab) = 0, when multiplied out and reduced, may be written a 3 + b 3 4- c? + abc = 0, prove that (a 2 + bc) (δ 2 + αc) (c 2 + ab) — (a? — be) (δ 2 - ac) (c 2 - áb) = 0, may be reduced to + p∙ + + A _ θ e qq ιe l a tter given equality, by dividing it by α 2 δc × b 2 ac × c 2 αδ. becomes [table]
which is identical with the first given equality, but with -, V, 7, written in place of a, b t c. The result therefore of CL O G reducing the second equality will be identical with that of reducing the first, when -, ∣, -, are written in place of a. b, c. Now the former result is a 3 + b 3 + c 3 + abe = 0,. ’ . the latter is -7 + w + -3 + ~τ- = 0 .
a j b i c λ abe Ex. 4. Prove that . 77—- 1 - -. ∙. +
(α 1 - α 2 ) (α 1 - α 3 ) ... (α 1 - α»)
τ 7 2 - x 7 r+ &c., is equal to 0 if m>n, (¾- α ι)(α 2- α 3 ) . · · (¾-α-)
and equal to 1 if wι = n. This is easily proved by resolv- ' x w —*
ing , 77 7 7 . into partial fractions (Art. 33).
o (x-c⅛)(x-α 2 )... (x-α n ) 1 v '
We have — _A_ + _Al + &c
(x-α 1 ) ... (x-α n ) x-α 1 x-α 2 .∙. x n ~ 1 = A 1 (x - α 2 ) . . (ar-α β )+ &c. . . (1), whence, writing a 1 , a a , &c., successively for x, we get A 1 , A 2 , &c.
The given quantity is A 1 + A 2 + ... + A n , and the equation marked (1), gives, by equating coefficients of like powers of x, the result required.
ä5. π r-r, t hen2±s∑r±i.
q s’ p-q r-s τ , p , r , . p + q r+s 1 p , r 1
For - +1 = - +1 gives j ^—- = , and - -1 1
q s ° q s q s gives 71∑J = . Divide the former by the latter.
Ex. 6. If l + →l = -⅜-, then α 2 = δ 2 = c 2 ∙ abe a+b+c p or a+b 1 ∣- 1 ~ 1 1 ~ a+b ab ~ a^ r b aMb+c c c(fl+b+c)
.∙. either a + b = 0, or -~ = - — — v 4 ·
’ ab ac bç c l
In the latter case, 4^ = ^4 + ~ + 4^ ’ c i ab ac be which is not changed by interchanging c and b or c and a, 111
·’· c 2~ δ 2- α 2 ’ so that on either alternative the proposition is true. r , f , „. j , ' ad-bc ac-bd. , , Ex. 7. Given that —7 ∏∙—rτ 7, and a not a-b-c+d a-b+c-d equal to b, nor c equal to d∙, to prove that a + b = c + d , , and that either of the fractions equals .
Write the equality thus,—
ac-bd a-b+(c-d) ad-be a-b-(c-d)
Apply Example 5, and there results, ac — bd+ad — be a — b ac — bd — ad+bc c-d . (a-bj(e+d) a-b , 1 7 t , ι.e. 7 —τ( = —;, whence a + b = c + d
(c-α)(α+δ) c-d
If now a - c or d — b be written by a single symbol x, the first fraction becomes
(c+x) (δ+x)-cδ δ + c + x α÷δ α÷δ+c+d
2x “2 * ~2~ “ 4
Ex. 8. If χ≈- r r- 7 r ,y≈- i 7 - 1 .
√δ÷√c-√α j ∣ dc + *fa->jb i
1 1
2 ≡≡ —. 24 = * - ·
√α+√δ-√c ’ √α+√δ + √c ’ prove that
(y+⅞-x+ιz) (¾+x-y+¾) (x+y-¾+¾) ^ {x+y+z-ufi
(δ + c — α) (c÷α-δ)(α+δ-c)
8αδc
Deal with the reciprocals of x, y, z, u; thus, y + z-x + u = y + z-(x-u) -∞(i-j) = 2 √≈(y≈-∞) -2√⅜∞ 1 (1-1)
= 4 Jayzxu{c + b-aj.
Hence, by symmetry, the numerator of the left hand fraction becomes
64 Jabe y 3 z⅛⅛ 3 (δ + c - a) (c + à - δ) (α + b - c).
Also, x + ÿ+2 - u -∞√θ + i) + ∞(1 -1)
= 2 Jc{xy+m) = 2 √⅛jκκ(i + -,
= 8 *Jabcxyzu,
Hence the result Sect. IV.— Surds.
36. It has been already observed (Art. 23), that the root of any proposed quantity is found by dividing the exponent of the quantity by the index of the root; and the rule has been illustrated by examples, in all of which, however, the quotient expressing the exponent of the result is a whole number; but there may be cases in which the quotient is a fraction. Thus, if the cube root of α 2 were required, it might be expressed, agreeably to the method of notation 2
already explained, either thus, ^∕α 2 , or thus, a s .
Quantities which have fractional exponents are called surds, or imperfect powers, and are said to be irrational, in opposition to others with integral exponents, which are called rational.
Surds may be denoted by means of the radical sign, but it will be often more convenient to use the notation of fractional exponents. The following examples will show how they may be expressed either way.
≈ α÷, √4^δ 2 = 2δΛ = ah i ,
= (α 2 + b 2 )⅛, ^∕(^≡δF = (α - δ) l ,
A^Ξ = (α + δ)⅜α⅛⅜.
√αδ
The operations concerning surds depend on the following principles :—
1. If the numerator and denominator of a fractional exponent be either both multiplied or both divided by the same quantity, the value of the power is the same. Thus, w cm n cn α ≈α .
2. The product of like powers (integral or fractional) is the same power of the product. Thus, a⅛b⅛ = (abfi.
37. I.— Reduction of a Rational Quantity to the form of a Surd of any given denomination. Rule. Reduce the exponent of the quantity to the form of a fraction of the same denomination as the given surd.
Ex. Reduce α 2 to the form of the cube root.
Here the exponent 2 must be reduced to the form of a fraction having 3 for a denominator, which will be the fraction f; therefore α 2 = a? = t fa 6 .
38. II.— .Reduction of Surds of different denominations to others of the same value and of the same denomination. Rule. Reduce the fractional exponents to others of the same value, and having the same common denominator.
Ex. Reduce fa and fb 2 , or a? and l> 3 to other equivalent surds of the same denomination.
The exponents ∣, f, when reduced to a common denominator, are f and ⅜; therefore the surds required are
3 4 6 __ 6
a s and δ 6 , or √α 3 and √δ 4 .
39. III.— Reduction of Surds to their most simple terms. Rule. Reduce the surd into two factors, so that one of them may be a complete power, having its exponent divisible by the index of the surd. Extract the root of that power, and place it before the remaining quantities, with the proper radical sign between them.
Ex. 1. Reduce J 48 to its most simple terms.
The number 48 may be resolved into the two factors 16 and 3, of which the first is a complete square; therefore √48 = (4 2 x3)* = 4×3⅛ = 4√37
Ex. 2. Reduce x ∕98α⅛, and λ ∕24α⅛ + 40a 3 x 2 , each to its most simple terms.
First, √98⅛ = (7 2 α 4 × 2x)* = 7a 2 × (2x)* = 7a 2 f2^x.
Also λ ∕24α⅛ + 40α⅛ 2 = (2 3 α 3 (3x + 5x 2 )) ir = 2a f3x + 5α> 2 .
40. IV.— Addition and Subtraction of Surds. Rule. If the surds are of different denominations, reduce them to others of the same denomination, by prob. 2, and then reduce them to their simplest terms by last problem. Then, if the surd part be the same in them all, annex it to the sum or difference of the rational parts, with the sign of multiplication, and it will give the sum or difference required. But if the surd part be not the same in all the quantities, they can only be added or subtracted by placing the signs + or - between them.
Ex. 1. Required the sum of J27 and λ ∕48^.
By prob. 3 we find f27 = 3 λ ∕3 and a ∕48 = 4 f3, therefore √∑7 + √48 = 3 √3 + 4 √3 = 7 √3.
3 — 3 —-
Ex. 2. Required the sum of 3 √∣ and 5 √ s 1 2 -.
3 . 3 - 3 — 3 3 — 3,—
3 f⅛- 3 a∕⅛~⅜ *J% an ^ ð λ∕⅛ = ð aA⅞ = f
3 _ 3— 3 _ 3 _ 3 —
therefore 3 f⅛ + 5 f= f2 + £ f2 ≈ 1 j 1 f2 . Ex. 3. Required the difference between f30a i x and √200⅜ 3 ^.
fS0a i x = (4 2 α 4 × 5ir)^ 2 ^ = 4α 2 f5x, and 5 ∕20α¾ 3 =
(2 2 α⅛ 2 × 5xf 1 =2ax fbx; therefore s ∕80 a i x- x ∕20α⅛ 3 = (4α 2 - 2ax) f5x.
41. V.— Multiplication and Division of Surds. Rule. If they are surds of the same rational quantity, add or subtract their exponents.
But if they are surds of different rational quantities, let them be brought to others of the same denomination, by prob. 2. Then, by multiplying or dividing these rational quantities, their product or quotient may be set under the common radical sign.
Note. If the surds have any rational coefficients, their product or quotient must be prefixed.
3 β —
Ex. 1. Required the product of √α 2 and √α 3 . 3 5 . 2 3 2 >3 19 16 .
√α 2 × Ja 3 = α≈xα κ = α 3+ff = a ri = √α 19 , Ans.
3
Ex. 2. Divide √α 2 - δ a by fa + δ .
These surds, when reduced to the same denomination, are (α 2 -δ 2 )* and (α + δ)*. Hence √±jf = K j k ' 3 √α+δ ∖(*+δ) 2 ∕
= f'*' t (α⅛^ tr ) i ” C“+ Ì)( “ ^ δy )* = √< a ÷ 6 X αs -*> 3 ∙
42. VI.— Involution and Evolution of Surds.
The powers and roots of surds are found in the same manner as any other quantities, namely, by multiplying or dividing their exponents by the index of the power or root 3 _ 2
required. Thus, the square of 3 a ∕3 is 3 × 3 × (fif s = 3 1 ·
9 5 ∕9. The n th power of x β, is x m . The cube root of 1 s ∕2 is I (2) κ = ∣ ≡∕2, and the nth root of x i * is x™.
43. The reduction of quadratic surds is facilitated by the following considerations, which appear hardly to requiro demonstration:
1. fa cannot = δ + fc, when fc is a surd.
2. a+ fb cannot = c + fd when s ∕δ, fd arc unequal surds.
3. a cannot ≈ fb fc when fb, fc are surds not involving the same irrational part, λ ∕2and f3ioν example.
4. fa cannot equal fb + fc when all are surds not involving the same irrational part.
Note. The irrational part of s ∕8, for instance, is f2, for f3 = 2√2.
44. For example, we extract the square root of a binomial surd such as 28 +10 f3 in the following way:
Let f23 +10 f3 = x + y, where one or both of x and y must be a surd.
Then 28 +10 f3 = x 2 + y 2 + 2xy ,
28 = α 2 + y 2 , 10 f3 = 2xy, or No. 2 above would be violated.
Hence f23 - 10 f3 = fx 2 + y 2 - 2xy = x-y .
And √784 - 300 = ic 2 -y 2 , or a> 2 + y 2 = 28
x 2 -y 2 ≈22 « = 5, y= f3 and 5 + f3 is the root required.
Additional Examples in Surds. Ex. 1. Add together , _L_, -A_, and -A- , l + √2 1-√2 1-2
-1- + -2-≡2V3 √3 + l^√3-l 3-1 v ° , .∙. f3 - 2 is the sum required.
Ex. 2. Find the difference between α+¾+√(α+x) z ∕ (α+g)-√(α+g) ∖ * (α+x) + √(α+x) 3 ^* ∖(α+x) 2 -√(α+x) 3 ∕
rrq t .1 √(α+αι)+l 1
The former is , . . -- ,. ’ —- = — .
√(a+x) l + √(a+aj) √(α+χ)
The latter is the square root of 1 √(α+as)-1. 1
a+x ‘ √(α+χ)-l , t ' 6 ' ’ ° α+jc ’.∙. the difference reαuired is 0.
Ex. 3. Find the value »
√(α 2 +x 2 )-x
, (δ-c)α
when x = a , r -=- ·
2√δc
At the first reduction a divides out, and the fraction is reduced to
√ 1 ,(δ- c ) 2 ∖ δ- c
A 4bc f 2Vbc = δ÷c+(δ-c) __ b f(↑ Λ b ~ c ^ ∖ b ~ c b+c-(b-c}~c ’
A 4δc J 2Vbc Ex. 4. Find the value of (, x-√(2α-x) ’ when x = x ∕(δ 2 + 4αδ) - δ.
By the process explained in this article √2α-x = √(2α÷δ- √δ 2 + 4αδ) = √∣, 1 , -, . √2δ+l whence the fraction reduces to ~τ=—- .
√2o-1
45. In arithmetic the square root of a number is another number, which, when multiplied by itself, shall produce the first number. In algebra, where quantity takes the place of number, the definition leads to a less limited result than in arithmetic. In the latter science there cannot be two square roots of the same thing; in the former, there will necessarily be two. For both 2 × +2 gives 4, and - 2 × - 2 gives 4; hence the square root of 4 is - 2 as well as + 2.
And, further, as in algebra, - 2 is a quantity subject to all the operations and definitions of the science, it is clearly competent to express, in some form or other, the result of extracting its square root. That form must of necessity be something very different in character from ∣ J2, whether a ∕2 be + or -. For the definition requires that the square root of - 2 shall be such a quantity as when multiplied by itself shall produce - 2. It is then clearly no arithmetical quantity either + or - , but some quantity connected with numerical quantities by its properties, but not by its nature. It is termed an impossible or im aginary quantity, and may be written f - 2 or ,J2 ,J - 1, and the same notation applies to the square roots of all negative quantities.
The properties of imaginary quantities are almost identical with those of surds, and we need not stop to consider them. One example of their application will suffice. It affords strong confirmation of the safety of assuming the commutative law to exist in every branch of pure algebra.
Ex. The product of the sum of two squares by the sum of two squares can always be represented under the form of the sum of two squares.
For (α 2 + δ 2 ) (c 2 + d 2 )≡(α + δ - δ √~1)
×(c + d√-l)(c-d√÷T)
- (α + δ √^T) (c + d √- 1 )×(α- δ √ - 1) (c - d √ - 1) ≈(ac-bd + ad + bc f J -Γ) (ac-bd-ad + bc √ - 1) = (αc - δd) 2 + (ad + δc) 2 .
Cor. (α 2 + δ 2 ) (c 2 + <7 2 ) ≈ (ac + δ<7) 2 + (ad - δc) 2 , or the product may be represented in two different ways, under the form of the sum of two squares.
Sect. V.— Proportion and Progression.
46. In comparing together any two quantities of the same kind in respect of magnitude, we may consider how much the one is greater than the other, or else how many times the one contains either the whole or some part of the other; or, which ú the same thing, we may consider either what is the difference between the quantities, or what iε the quotient arising from the division of the one quantity by the other: the former of these is called their arithmetical ratio, and the latter their geometrical ratio. These denominations, however, have been assumed arbitrarily, and have little or no connection with the relations they are intended to express.
I. Arithmetical Proportion and Progression.
47. When of four quantities the difference between the first and second is equal to the difference between the third and fourth, the quantities are called arithmetical proportionals. Such, for example, are the numbers 2, 5, 9, 12; and, in general, the quantities a, a + d, b, b + d.
48. The principal property of four arithmetical proportionals is this :—If four quantities be arithmetically proportional, the sum of the extreme terms is equal to the sum of the means. Let the quantities be a, a + d, b, b + d', where d is the difference between the first and second, and also between the third and fourth, the sum of the extremes is a + δ + d, and that of the means a + d + δ; so that the truth of the proposition is evident.
49. If a series of quantities be such, that the difference between any two adjacent terms is always the same, these terms form an arithmetical progression. Thus, the numbers 2, 4, 6, 8, 10, &c., form a series in arithmetical progression, and, in general, such a series may be represented thus:
a, a + d, a + 2d, α+⅛d,α + 4d, a + 5cZ, a + Gd, &c., where a denotes the first term, and d the common difference.
By a little attention to this series, we readily discover that it has the following properties:
1. The last term of the series is equal to the first term, together with the common difference taken as often as there are terms after the first. Thus, when the number of terms is 7, the last term is a + 6<Z√ and so on. Hence if z denote the last term, n the number of terms, and a and d express the first term and common difference, we have z = a + (n- 1)<7.
2. The sum of the first and last term is equal to the sum of any two terms at the same distance from them. Thus, suppose the number of terms to be 7, then the last term is a + 6cζ and the sum of the first and last 2α + 6cZ; but the same is also the sum of the second and last but one, of the third and last but two, and so on till we come to the middle term, which, because it is equally distant from the extremes, must be added to itself.
3. To find the sum of the series, it is only necessary to observe that, if the progression is written down twice, 1° from the beginning, 2° from the end, the terms of the former increase by the same amount as that by which the terms of the latter diminish; so that the sum of any two terms which stand under each other is always the same, viz., the same as the sum of the first and last terms; hence the double series converts addition into multiplication; so that if s denote the sum of the series, we have 2s = n(α + z), and β=2(α+ 2 ).
Ex. The sum of the odd numbers 1,3, 5, 7, 9, &c., continued to n terms, is equal to the square of the number of terms. For in this case α = 1, d≈2, z≈l +(n- 1) d≈2η — 1, therefore s = ^×2n = n 2 .
II. Geometrical Proportion and Progression.
50. When, of four quantities, the quotient arising from the division of the first by the second is equal to that arising from the division of the third by the fourth, these quantities are said to bo in geometrical proportion, or are called eimply proportionals. Thus, 12, 4, 15, 5, are four numbers in geometrical proportion; and, in general, nα, a, nb, b, may express any four proportionals, for — = n, and Cb , ηb also — = n. o
To denote that any four quantities a, b, c, d, are proportionals, it is common to place them thus, a: b:: c: d; or thus, a: b=c: d∙, which notation, when expressed in words, is read thus, a is to b as c to d, or the ratio of a to b is equal to the ratio of c to d.
The first and third terms of a proportion are called the antecedents, and the second and fourth the consequents.
"When the two middle terms of a proportion are the same, the remaining terms, and that quantity, constitute three geometrical proportionals; such as 4, 6, 9, and in general nα, a,-. In this case the middle quantity is called 71 a mean proportional between the other two.
51. The principal properties of four proportionals are the following:
1. If four quantities be proportionals, the product of the extremes is equal to the product of the means. Let a, b, c, d, be four quantities, such that a: b:: c: d∙> then, from the nature of proportionals, j i =f ^ et these equal quotients be multiplied by b d, and we have ad = be. It follows, that if any three of four proportionals be given, the remaining one may be found. Thus, let a, b, c, the first three, be given, and 1et it be required to find x, the fourth term; because a: b:: c: x, ax = bc, and dividing , be by a, x = — .
The converse is obviously true, viz., if four quantities be such that the product of two of them is equal to the product of the other two, these quantities are proportionals.
2. If four quantities are proportional, that is, if a: b:: c: d, then will each of the following combinations or arrangements of the quantities be also four proportionals.
Isí, By inversion, b: a:: d: c. 2d, By alternation, a: c:: b: d. Note.— The quantities in the second case must be all of the same kind.
3d, By composition, a + b ∙.a∙.∙.c + d∙.c, or, a + b ∙.b '.∙.c + d'.d. ith, By division, a-b: a ∙.∙.c~d: c, or, a — b :b:: c - d: d. f>th, By mixing, α + δ : a — b ∙.∙.c + d :c-d. 6th, By taking any equimultiples of the antecedents, and also any equimultiples of the consequents, na: pb:: nc: pd. r ith, Or, by taking any parts of the antecedents and con- . abed sequents, η p n p i That the preceding combinations of the quantities a, b, c, d, are proportionals, may be readily proved, by taking the products of the extremes and means; for from each of them we derive this conclusion, that ad = bc, which is known to be true, from the original assumption of the quantities.
Sth, If foιrr quantities be proportional, and also other four, the product of the corresponding terms will be proportional.
Let a: b :; c : d, And e ∖ f ∖ ∙.g :h; Then ae: bf:: eg: dh.
For ad=bc, and eh∙=fg, as before, therefore, multiplying together these equal quantities, adeh = bcfg, or ae × dh = bf × eg ; therefore, by the converse of the first property, ae: bf:: eg: dh.
Hence it follows, that if there be any number of proportions whatever, the products of the corresponding terms will still be proportional.
52. If a series of quantities be so related to each other, that the quotient arising from the division of any term by that which precedes it is al·ways the same quantity, these are said to be in geometrical progression; such are the numbers 2, 4, 8, 16, 32, &c., also ⅜, ∣, ⅜, τ ⅛, &c., and in general, a series of such quantities may be represented thus, a, ar, art, art, art, αr 5 , &c. Here a is the first term, and r the quotient of any two adjoining terms, which is also called the common ratio.
By inspecting this series, we find that it has the following properties:
1. The last term is equal to the first, multiplied by the common ratio raised to a power, the index of which is one less than the number of terms. Therefore, if z denote the last term, and n the number of terms, z = αrt~ 1 .
2. The product of the first and last term is equal to the product of any two terms equally distant from them: thus, supposing art the last term, it is evident that a x αr 5 ≈ar× art = art × αr 3 , &c.
The sum of n terms of a geometrical· series may be found thus:
Let s = α + αr + αr 2 + αr 3 ...+αr n ~ 1 .
Then rs = ar + ar 2 + ar 3 ... + ar n ^^ i + art.
Subtract, rs - s = art — a.
That is, (r - 1) s = α(r" - 1).
ττ r η - 1 1— r n
Hence s = -a, or- a. r —1 1— r
Cor. The sum to infinity = .
Additional Examples in Proportion and Progressioni Ex. 1. How many strokes does a clock strike in twelve hours?
If s denote the number
«= 1+ 2 + ...12
s=12 + 11 + ... 1
.∙. 2s=13 + 13 + ...13 = 13x12; s = 78.
Ex. 2. Find the number of shot 1ying close together in the shape of an equilateral triangle.
Let n be the number of shot in a side of the triangle. Counting from one angle, and taking in successive rows parallel to the opposite side, we get as the number required l + 2 + ...w= k J Ex. 3. To find the number of shot in a pile of the form of a triangular pyramid.
As each shot lies in the holl·ow formed by those below it, the number of shot in the successive sides from the base upwards will evidently be n- 1, n- 2,...1.
Hence the number of shot in the pile will be n(n÷1) (n-l)n (n-2)(n-l) 1.2
2 + 2 + '" 2 ’
To sum this series induction may be employed. Thö result is
,n(n+l)(n+2)
" 6 __
Ex. 4. A ratio of greater inequality is diminished, and of less inequality increased, by adding the same quantity to each of its terms. ~7↑
Let a>b: then
’ δ+x δ
By multiplying out, this is evident.
Ex. 5. Find the vulgar fraction which is equivalent to the recurring decimal.
•3142
Let ae=·3Ì42, then 10a=3142
10,000z = 3142·Ï42
.∙. subtracting 9990z = 3139
ðl39
X ~ 9ð9O
Ex. G. A sum of money doubles itself in fifteen years at a rate a little below 5 per cent. A noble Scotch family have retained in their possession gold coins of the value of £500 since the days of Mary Stuart (300 years); what have they lost by not allowing the money to accumulate at the above rate 1
Every pound would have amounted to £2 20 ; .∙. £500 (2 20 -l) is the loss. It amounts to upwards of £524,000,000.
Ex. 7. The sum of the mixed series
α + (α + δ)r + (α + 2δ)r 2 + .. . . .
a δr(l-r n ~ 1 ) (α+ra-lδ)r n is i-r + (l-r) 2 “1-r Sect. VI.— Resolution of Equations involving one Unknown Quantity.
53. The primary object of algebraic investigation is to discover certain unknown quantities, by comparing them with other quantities which are given, or supposed to be known. The relation between the known and unknown quantities is either that of equality, or else such as may be reduced to equality; and a proposition which affirms that certain combinations of quantities are equal to one another is called an equation. Such are the following :—
X x _24
2 + 3 ~ X , ' 2x + 3y = xy.
The first of these equations expresses the relation between an unknown quantity x and certain known numbers; and the second expresses the relation which the two indefinite quantities x and y have to each other.
The conditions of a problem may be such as to require several equations and symbols of unknown quantities for their complete expression. These, however, by rules hereafter to be explained, may be reduced to one equation, involving only one unknown quantity and its powers, besides the known quantities; and the method of expressing that quantity by means of the known quantities constitutes the theory of equations, one of the most important as well as most intricate branches of algebraic analysis.
An equation is said to be resolved when the unknown quantity is made to stand alone on one side, and only known quantities on the other side; and the value of the unknown quantity is called a root of the equation. The general definition of a root of an equation is, that it is a numerical quantity (i.e., some combination of numbers) which, when written in place of the unknown quantity, renders the equation a numerical ideality; thus 1 is the root of the equation x = l, 1 and -1 are both roots of the equation íc 2 = I; I, x ∕-1,- f-1 and -1 are all roots of the equation x i = l.
54. Equations containing only one unknown quantity and its powers, are divided into different orders, according to the highest power of that quantity contained in any one of its terms. The equation, however, is supposed to be reduced to such a form that the unknown quantity is found only in the numerators of the terms, and that the exponents of its powers are expressed by positive integers.
If an equation contains only tħc first power of the unknown quantity, it is called a simple equation, or an equation of the first order. Such is ax + b = c, where x denotes an unknown, and a, b, c, known quantities.
If the equation contains the second power of the unknown quantity, it is said to be of the second degree, or is called a quadratic equation; such is 4ic 2 + 3x = 12, and in general ax 2 + bx = c. If it contains the third power of the unknown quantity, it is of the third degree, or is a cubic equation; such are ae 3 + 2ae 2 + 4# = 10, and ax 3 + bx 2 + cx≈d ; and so on with respect to equations of the higher orders. A simple equation is sometimes said to be linear, or of one dimension. In 1ike manner, quadratic equations are said to be of two dimensions, and cubic equations of three dimensions.
When in the course of an algebraic investigation we arrive at an equation involving only one unknown quantity, that quantity will often be so entangled in the different terms as to render several previous reductions necessary before the equation can be expressed under its characteristic form, so as to be resolved by the rules which belong to that form.
These reductions depend upon the operations which have been explained in the former part of this treatise, and the application of a few self-evident principles, namely, that if equal quantities be added to or subtracted from equal quantities, the sums or remainders will be equal; if equal quantities be multiplied or divided by the same quantity, the products or quotients will be equal; and, lastly, if equal quantities be raised to the same power, or have the same root extracted out of each, the results will stili be equal.
From these considerations are derived the following rules, which apply alike to equations of all orders, and are alone sufficient for the resolution of simple equations.
55. Pule 1. Any quantity may be transposed from one side of an equation to the other, by changing its sign.
Thus, if 3x- 10 = 2ic + 5 ,
Then 3#-2# = 5 +10,
Or zc = 15.
Again, if ax + b = cx-dx + e,
Then ax —cx + dx≈e-b,
Or (α- c + d)x≈e-b.
The reason of this rule is evident, for the transposing of a quantity from one side of an equation to the other is nothing more than adding the same quantity to each side of the equation, if the sign of the quantity transposed was -; or subtracting it, if the sign was +.
From this rule we may infer, that if any quantity be found on each side of the equation with the same sign, it may be left out of both. Also, that the signs of all the terms of an equation may be changed into the contrary, without affecting the truth of the equation.
Thus, if a + x = b + a + c,
Then x = b + c∙,
And if a-x=b-d,
Then x - a = d — b . 56. Pule 2. If the unknown quantity in an equation be multiplied by any quantity, that quantity may be taken away, by dividing all the other terms of the équation by it.
If 3« = 24,
Then x = = 8.
If ax = b — c, ™ b-c b c
Then x≈ =---∙ a a a Here equal quantities are divided by the same quantity, and therefore the quotients are equal.
57. Pule 3. If any term of an equation be a fraction, its denominator may be taken away, by multiplying all the other terms of the equation by that denominator.
If
5
Then x = 35 .
If - = 5-c + <Z, a
Then x = ab-ac + ad. τ , δ
It α--=c, x ’
We have ax-b = cx and x≈· a-c
In these examples, equal quantities are multiplied by the same quantity, and therefore the products are equal. 58. Pule 4. If the unknown quantity is found in any term which is a surd, let that surd be made to stand alone on one side of the equation, and the remaining terms on the opposite side; then involve each side to a power denoted by the index of the surd, and thus the unknown quantity shall be freed from the surd expression.
If √* + 6 = 10, _
Then, by transposition, Jx =10-6 = 4;
And, squaring both sides, l fx× l fx = 4 × 4,
Or a j = 16.
Also, if ,Ja 2 + x 2 -b = x,
By trans. ja 2 + χ 2 = b + x,
And, squaring, a 2 + x 2 = (b + x) 2 = b 2 + 2bx + x 2 , n 2_7.2
Hence a 2 = b 2 -t∙ 2bx, x = ——— ’ 2δ
59. Pule 5. If the side of the equation which contains the unknown quantity be a perfect power, the equation may be reduced to another of a lower order, by extracting the root of that power out of each side of the equation.
Thus, if ic 3 =64α 3 ,
Then, by extracting the cube root, x = 4a;
And if (a + x) 2 = b 2 -a 2 ,
Then a + x ≈ Jb 2 — a 2 .
60. In these examples we have been able to determine the value of the unknown quantity by the rules already delivered, because in every case the first, or at most the second power of that quantity, has been made to stand alone on one side of the equation, while the other consisted only of known quantities; but the same methods of reduction serve to bring equations of all degrees to a proper form for solution. Thus, if 1 ~P+⅞+. ιr _ χ _ jt , _ χ + x+l
-; by proper reduction, we have x 3 + px 2 + qχ = r, a cubic equation which may be resolved by rules to be afterwards explained.
Sect. VII.— Reduction of Equations involving moke than one Unknown Quantitv.
61. Having shown in the last section in what manner an equation involving one unknown quantity may be resolved, or at least fitted for a final solution, we are next to explain the methods by which two or more equations, involving as many unknown quantities, may at last be reduced to one equation and one unknown quantity.
As the unknown quantities may be combined together in very different ways, so as to constitute an equation, the methods most proper for their elimination must therefore be various. The three following, however, are of general application, and the last of them may be used with advantage, not only when the unknown quantity to be eliminated rises to the same power in all the equations, but also when the equations contain different powers of that quantity.
62. Method 1. Observe which of the unknown quantities is the least involved, and let its value be found from each equation, by the rules of last section.
Let the values thus found be put equal to each other, and hence new equations will arise, from which that quantity is wholly excluded. Let this operation be now repeated with these equations, thus eliminating the unknown quantities one by one, till at last an equation be found which contains only one unknown quantity.
Ex. Let it be required to determine x and y from these two equations.
2x'l· 3y = 23 , 5iε-2y=10.
From the first equation, a? = 23 3y .
2
From the second equation, x = -- · 5
Let these values of x be now put equal to each other, And we have — j^ 2y ≈ 23 ~-° y ,
5 2
Or 20 + 4y = 115 - I5y;
19y=95,
And y = 5:
a i * io+2« , . 1 _
Ana since x≈ ——% or x = —— from either of Λ 5
these values we find x = 4.
63. Metlιod 2. Let the value of the unknown quantity which is to be eliminated be found from that equation wherein it is least involved. Let this value and its powers be substituted for that quantity, and its respective powers in the other equations; and with the new equations thus arising, let the operation be repeated till there remain only one equation and one unknown quantity.
Ex. Let the given equations, as in last method, be
2<r + 3y = 23, 5x- 2y- 10.
From the first equation, x = 23 ~ 3y ;
2
And this value of x being substituted in the second 23 — 3?/
equation, we have 5 × ——- - 2y = 10
Or 115 - I5y-4y = 20;
95 = 19y,
And y — 5
And hence χ = ' 3 ^ = 4, as before.
Λ
64. Method 3. Let the given equations be multiplied or divided by such numbers or quantities, whether known or unknown, that the term which involves the highest power of the unknown quantity may be the same in each equation.
Then, by adding or subtracting the equations, as occasion may require, that term will vanish, and a new equation emerge, wherein the number of dimensions of the unknown quantity in some cases, and in others the number of unknown quantities will be diminished; and by a repetition of the same or similar operations, a final equation may be at last obtained, involving only one unknown quantity.
Ex. Let the same example be taken, as in the illustration of the former methods, namely,
2a? + 3y = 23,
5x-2y = 10.
To eliminate x, let the first equation be multiplied by 5, and the second by 2; thus we have
10x+15y = 115, 10#- 4y= 20.
Here the term involving x is the same in both equations; and it is obvious, that by subtracting the one from the other, the resulting equation will contain only y, and known numbers; for by such subtraction we find 19y = 95, and therefore y = 5.
Having got the value of y, it is easy to see how x may be found from either of the given equations; but it may also be found in the same manner as we found y. For let the first of the given equations be multiplied by 2, and the second by 3, we have
4# + 6y = 46, 15#-6y = 30.
By adding these equations, we find
19# = 76, and x = 4. 65. The following examples will serve further to illustrate these different methods of eliminating the unknown quantities from equations.
Ex. 1. Given í a f ~ c I to determine x and y. ∖ dχ+fy=g)
To eliminate y, let the first equation be multiplied by f, and the second by δ, and we have afx + bfy = cf, bdx + bfy — bg.
Taking now the difference between these equations, afx — bdx = cf— bg,
Or (af-bdf)x = cf-bg,
And therefore x = ⅛-⅛ . aj—ba
In the same manner may y be determined, by multiplying the first of the given equations by d, and the second by a∙, for we find adx + bdy = cd, adx + afy = ag.
And taking the difference as before, we get bdy - afy ≈ cd-ag, . , il , cd-ag
And therefore y = γί -, · ba — aj
66. This example may be considered as a general solution of the following problem. Two equations expressing the relation between the first powers of two unknown quantities being given, to determine those quantities; for whatever be the number of terms in each equation, it will readily appear, as in Art. 55, that by proper reduction they may be brought to the same form as those given in the above example.
67. Let us next consider such equations as involve three unknown quantities. [table]
Here the given equations, when cleared from fractions, become [table]
To eliminate z by the third method, let the first equa tion be multiplied by 10, the second bv 5, and the third by 3, the results will be these:
120#+ 80y + 60z = 14880, 100#+ 75y + 60z = 14100, 90# +72y + 60z = 13680.
Let the second equation be now subtracted from the first, and the third from the second, and we have
20# + 5y = 780, 10# + 3y = 420.
Next, to eliminate y, let the first of these equations bo multiplied by 3, and the second by 5; hence, 60# + 15y = 2340, 50# + 15y = 2100 .
Subtracting now the latter equation from the former, 10# = 240 and # = 24, 420-10# Rn y≈ —3— = 60, a 1 1448 —12# —8y 19n
And z = — - =120.
Ό
Ex. 3. Given x 2 -yz = a 2 , y 2 -xz = b 2 , z 2 -xy-c 2 , to find #, y, and z.
By subtraction, we have (x - y) ( x + y + 2 ) = a2 ~ ð 2 G) > (z - x) (x + y + z) = c 2 - a 2 (2), (y-z) (x + y + z') = b 2 -t> 2 .
Squaring, adding, and dividing by 2, we get (x 2 + y 2 + z 2 -xy-xz-yz) (# + y + z) 2 = α 4 + δ 4 + c 4 - a 2 b 2 — a 2 c 2 — b 2 c 2 .
But x 2 + y 2 + z 2 - xy - #z - yz is the sum of the three given expressions, and . ∙. equal to a 2 + b 2 + c 2 .
Hence (α 2 + b 2 + c 2 ) (# + y + z) 2 = α 4 + δ 4 + c 4 - α 2 δ 2 - α 2 c 2 - δ 2 c 2 , which gives # + y + z.
Equations (1) and (2) are now two simple equations, which, combined with the value of # + y + z as determined, give #, y, and z. Ex. 4. Given r Jx = + +
√*-√H H+≈-) +=√⅛( ∣+∣-θ
Multiply the first by t fx, the second by s ∕y, and the third by λ ∕z, and add two and two. There results !— 2c # + y= √∙zyz — Λ ,— 2b x + z = f dxyz — y ,— 2a y + z = ∙fχyz-x i.e. χz + yz= Jxyz2c χ y+yz≈ Jχyz 2b xy+xz = Jxyz 2a .∙. yz= t Jxyz (f + c-a) xz = f fxyz (a + c-b) χ V~ Jxyz{a + b-c)
Multiplying any two of these we get one of the unknown quantities:
.’·. # = (a + e - δ) (α + b - c), &c.
Sect. VIIL— Questions producing Simple Equations.
68. When the conditions of a problem have been expressed by equations, or translated from the common language into that of algebra, we must consider whether the problem be properly limited; for in some cases the conditions may be such as to admit of innumerable solutions, and in others they may involve an absurdity, and thus render the problem altogether impossible.
Now, by considering the examples of last section, it will appear, that to determine any number of unknown quantities, there must be given as many equations as there are unknown quantities. These, however, must be such as cannot be derived from each other, and they must not involve any contradiction; for in the one case the problem would admit of an unlimited number of answers, and in the other case it would be impossible. For example, if it were required to determine x and y from these two equations, 2x - 3y = 13, 4# - 6y = 26; as the latter equation is a consequence of the former (for each term of the one is the half of the corresponding term of the other), it is evident that innumerable values of x and y might be found to satisfy both equations. Again, if x and y were to be determined from these equations, #+2y = 8, 3# + 6y = 26, it is easy to see that it is impossible to find such values of x and y as will satisfy both; for, from the first, we find 3x ≈ 24 - 6y ; and from the second, 3# = 26 - 6y; and therefore 24 - 6y = 26 - 6y, or 24 = 26, which is absurd; and so also must have been the conditions from which this conclusion is drawn.
69. But there is yet another case in which a problem may be impossible; and that is, when there are more equations than unknown quantities; for it appears, that in this case, by the rules of last section, we should at last find two equations, each involving the same unknown quantity. Now, unless these happened to agree, the problem would admit of no solution. On the whole, therefore, it appears that a problem is limited when the conditions furnish just as many independent equations as there are unknown quantities to be determined: if there be fewer, the problem is indeterminate; but if there be more, the problem in general admits of no solution whatever.
70. We shall now apply the preceding observations to some examples, which are so chosen as to admit of being resolved by simple equations.
Ex. 1. What is that number, to which if there be added its half, its third, and its fourth parts, the sum will be 50!
Let x denote the number sought; then its half will be
, its third , and its fourth
Z 3 4
X X X r* r ∖ ft + 2 + 3 + i = δ0 ∙
Hence 24#+12# +8#+ 6# = 1200,
Or 50# = 1200;
∖∙. # = 24.
Thus it appears that the number sought is 24, which upon trial wi∏ be found to answer the conditions of the question.
Ex. 2. A post is I of its length in the mud, ⅜ in the water, and 10 feet above the water; what is its whole length 1
Let its length be # feet, then the part in the mud is X · X
-, and that in the water -; therefore, from the nature of the question, XX , λ - + - + 10 = #.
From this equation we find 7# + 120 = 12#, and # = 24.
Ex. 3. A market-woman bought a certain number of eggs at 2 a penny, and as many at 3 a penny, and sold them all out again at 5 for 2d.; but, instead of getting her own money for them, as sne expected, she lost 4d.: what number of eggs did she buy!
Let # be the number of eggs of each sort; X
Then will - be the price of the first sort;
ïj X
And - = the price of the second sort.
Now, the whole number being 2#, we have
4⅛c
5: 2#:: 2: — = price of both sorts at 5 for 2d, · õ
x x 4x . , , 1 .
- + õ - T = θy tlιe Question. 2j 3 O
Hence 15#+ 10#-24# = 120,
And #=120, the number of each sort.
Ex. 4. A person at play lost ∣ of his money, and then won 3s.; after which he lost ⅜ of what he then had, and then won 2s.; lastly he lost 4 of what he then had, and, this done, found he had only 12s. left: what had he at first!
Suppose he began to play with # shillings.
x x 3*Z∕
He lost I of his money, or -, and had left x - - = —.
ττ 9 j j 4-i> Q 3#+12
He won 3s. and had then — + 3 = ———.
4 4
He lost 4 of or , and had left
≡ 4 ’ 4
3#+12 ¾+4 2#+8
4 ~^~4~~ 4
TT o j κ J 4.1, 2ae+8 2#+l
He won 2s. and had then —- H 2 = ——
4 4
2#+16 2x+16 1 .
He lost 4 of —-—, or —-—- , and had 1eft 7 4 28 ’
2#+16 2#+16 12ae+96
4 28 28
And because he had now 12s. left, we have this equation, 12ae+96
28 1 ·
Hence 12# = 240, and # = 20 .
[Ex. 5. To divide the number 90 into 4 such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product and quotient shall be all equal to each other.
In this question there are four quantities to be determined; but instead of introducing several letters, having put # to denote the first of them, we may find an expression for each of the remaining ones, as follows
Because # + 2 = second quantity - 2, # + 4 = the second quantity;
And because # + 2 = third × 2; ae+2
. ’. = the third quantity.
And in like manner 2(# + 2) = the fourth quantity. Now by the question, the sum of all the four = 90; x + # + 4 + + 2(# + 2) = 90,
Hence 9#= 162, and # = 18;
Therefore the numbers required are 18, 22, 10, and 40.
Ex. 6. A and B together can perform a piece of work in 12 hours, A and C in 20, and B and C in 15 hours; in what time will each be able to perform it when working separately!
That we may have a general solution, let us suppose A and B can perform the work in a hours, A and C in b hours, and B and C in c hours. Let #, y, and z, denote the times in which A, B, and C, could perform it respectively, if each worked alone; and 1et the whole work be represented by 1. The question gives at once—
l + l-i, 4÷1 * l + i=i.
x y a’ x z b y z c
If these be added, and their sum divided by 2, we find
1111 1 1
- 4 h~==~ + .
x y z 2α 2o 2c
From this equation let each of the three preceding be subtracted in its turn: thus we get
1 _ 1 1 1 +ab+ac-bc z 2a 2b 2c 2abo ’
1 _ 1 1 1 + αδ- ac+bc y 2a 2b 2c = 2aδc ’
1 _ 1 _1 1 _ —ab+ac+bc x 2a 2b 2c 2abc ’ rr 2abc 7200 * λ +ab+ac-bc 120 ’
2abc 7200 _ Λ
?/ = ≡≡ = 20
+ab —ac+bc 360
2abc 7200 _ __
X ~ -ab+ac + bc≈ 240 - Sect. IX.— Solution of Quadratic Equations.
71. We are next to explain the resolution of equations of the second degree, or quadratic equations. These involve the second power of the unknown quantity, and may be divided into two kinds, pure, and adfected.
I. Pure quadratic equations are such as after proper reduction have the square of the unknown quantity in one term, while the remaining terms contain only known quantities. Thus, x 2 = 64, and ax 2 + b = c, are examples of pure quadratics.
II. Adfected quadratic equations contain the square of the unknown quantity in one term, and its first or simple power in another; the remaining terms consisting entirely of known quantities. Such are the following, # 2 + 3# = 28, 2# 2 = 33- 5x, ax 2 + bx-c≈d.
The manner of resolving a pure quadratic equation is sufficiently evident. If the unknown quantity be made to stand alone on one side, with unity as a coefficient, while the other side consists entirely of known quantities, and the square root of each side be taken, we immediately obtain the value of the simple power of the unknown quantity as directed by rule 5th of Sect. VI.
In extracting the square root of any quantity, it is necessary to observe, that the sign of the root may be either + or —, and that consequently a quadratic must always have two solutions.
72. When an adfected quadratic equation is to be re- so1ved, it may always, by proper reduction, be brought to the following form:
x 2 +px = q∙ j where p and q are numerical quantities, + or —.
Let us compare the side of it which involves the unknown quantity x "with the square of a binomial x + a; that is, let us compare x 2 +px with x 2 + 2ax + a 2 = (x + a) 2 , and P it will presently appear, that if we suppose p = 2a, or = α, A the quantities x 2 +px and x 2 + 2ax will be equal; and as x 2 + 2ax is rendered a complete square, by adding to it a 2 , so also may x 2 +px be completed into a square by adding λj2 . λj2 to it , which is equal to a 2 ; therefore, let =^- be added to both sides of the equation x 2 +px = q, and we have p 2 p 2 ∕ p ∖ 2 p 2 x 2 +px + ^ = ^+q,or(x+^ ≈^ + q∙, and, extracting the square root of each side, + hence ir =-2 i ,√ , 4 + 2∙
73. From these observations we derive the following general rules for resolving adfected quadratic equations.
1. Bring all the terms involving the unknown quantity to one side, and the known quantities to the other side, and so that the term involving the square of the unknown quantity may be positive.
2. If the square of the unknown quantity be multiplied by a coefficient, let the other terms be divided by it, so that the coefficient of the square of the unknown quantity may be 1.
3. Add to both sides the square of half the coefficient of the unknown quantity itself, and the side of the equation involving the unknown quantity will now be a complete square.
4. Extract the square root of both sides of the equation, by which it becomes simple with respect to the unknown quantity; and, by transposition, that quantity may be made to stand alone on one side of the equation, while the other side consists of known quantities; and therefore the equation is resolved.
Ex. 1. Given x 2 + 2x = 35, to determine x.
Here the coefficient of the second term is 2; therefore^ adding the square of its half to each side, we have x 2 + 2x + 1 = 35 +1 = 36,
And, extracting the square root, x +1 = ¼/36 = ± 6 .
Hence x = ±6-1, that is, x = + 5, or x = - 7, and either of these numbers will be found to satisfy the equation, for 5×5 + 2×5 = 35, also - 7 × - 7 + 2× - 7 = 35.
a - 2 Ex. 2. Given — - 12 = x, to find x .
6
This equation, when reduced, becomes x 2 - 6# = 72 , And, by completing the square, # 2 -6# + 9 = 72 + 9 = 81. Hence, by extracting the square root, x - 3 = ±9, and x≈ ±9 + 3;
Therefore x — +12, or # = - 6; and upon trial we find that each of these values satisfies the original equation, for i⅛^-12 = 12, also≡⅛^-12 = -6.
6 b Ex. 3. Given x 2 + 28 = 11#, to find x.
Then # 2 -ll#=-28.
And, by completing the square,
4 4 4
11 3
Therefore, by extracting the root, x - —- = ± - · Z A
11 3
Hence x = — ± -; that is, x = + 7, or x = + 4.
A Ji
In the first two examples, we found one positive value for x in each, and also one negative value; but in this example both the values of x are positive, and, upon trial, each of them is found to satisfy the equation; for 7 × 7 + 28 = 11 × 7, also 4 × 4 + 28 = 11 × 4.
74. As at first sight it appears remarkable, that in every quadratic equation the unknown quantity admits always of two distinct values or roots, it will be proper to consider a little further the circumstances upon which this peculiarity depends.
To do this, let us re-examine the equation x 2 + 2x = 35. By bringing all the terms to one side, the equation may be expressed thus, x 2 + 2x - 35 = 0; so that we shall have determined x, when we have found such a number as, when substituted for it in the quantity x 2 + 2x - 35, will render the result equal to 0. But x 2 + 2x - 35 is the product of these two factors x - 5 and x + 7, as may be proved by actual multiplication; therefore, to find x, we have (# - 5) (#+7) = 0; and as a product can only become =0 when one of its factors is reduced to 0, it follows that either of the two factors x-5 and #+ 7 may be assumed =0. If x - 5 = 0, then x = 5; but if x + 7 = 0, then x — -7; so that the two values of x, or two roots of the equation x 2 + 2x - 35, are + 5 and - 7, as we have already found in a different manner.
75. What has been shown in a particular case is true of any quadratic equation whatever; that is, if x 2 +px = q, or, by bringing all the terms to one side, x 2 +px -q≈Q, it is always possible to find two factors x-a, and x + b, such, that x 2 +px-q = (x-d) (x + b), where a and b are known quantities, which depend only upon p and q, the given numbers in the equation; and since that to have (x -α)(x + b) = 0, we may either assume x-a = 0 or x + b = Q, it evidently follows that the conditions of the equation x 2 +px - q = 0, or x i +px = q f are alike satisfied by taking x≈+ a or x=≈-b.
From these considerations it follows, that x can have only two values in a quadratic equation; for if it could be supposed to have three or more values, then it would be possible to resolve x 2 +px — q into as many factors, x-c,x-d i &c.; but the product of more than two factors must necessarily contain the third or higher powers of x, and as x 2 +px - q contains no higher power than the second, therefore no such resolution can take place.
7 6. Solution of Questions which produce Quadratic Equations. Ex. 1. It is required to divide the number 10 into two such parts that the sum of their squares may be 58.
Let x be the one part;
Then, since their sum is 10, the other is 10 - x;
.∙. by the question, x 2 + (10 - x) 2 = 58;
That is, # 2 +100 - 20ir + ^ 2 = 58 ,
Or 2a; 2 -20a; = 58-100=-42;
Hence x 2 - 10a; = - 21.
And completing the square, x 2 - 10# + 25 = 25 - 21 = 4;
Hence, by extracting the root, x - 5 = ÷ J4 = ÷ 2,
And x = 5 ÷ 2 ,
That is, x = 7, or x = 3.
If we take the greater value of x, viz. 7, the other number 10 - x will be 3; and if we take the less value of x 1 viz. 3, then the other number is 7. Thus it appears, that the greater value of the one number corresponds to the less value of the other; and indeed this must necessarily be the case, seeing that both are alike concerned in the question. Hence, the only numbers that will answer the conditions of the question are 7 and 3.
Ex. 2. A grazier bought as many sheep as cost him £60, out of which he reserved 15, and sold the remainder for £54, gaining 2s. each upon them. How many sheep did he buy, and what did each cost him î
Suppose that he bought x sheep.
1200
Then each would cost him - shillings. x o
Therefore, since after reserving 15, he sold each of the 1200
remaining x — 15 for 1- 2 shillings, he would receive x o for them (x - 15) + shillings. And, because £54
=1080 shillings, we have by the question,
(x-15ae-^ + 2) = 1080; which, by proper reduction, becomes x 2 + 45 x = 9000 · , 1 195 45 . 1 whence x — ± — —. And, taking the positive root, a; = 75, the number of sheep; and consequently i f^=16 75 shillings, the price of each.
Ex. 3. It is required to find two numbers, of λvhich the product shall be 6, and the sum of their cubes 35.
Let x be the one number; then - will be the other.
x 216
Therefore, by the question, a? + = 35;
Hence x c + 216 = 35z 3 ,
Or a; 6 - 35a; 3 =-216.
This equation, by putting a? = y, becomes y 2 -M>y≈ -216; '
Hence we find y = 27, or y = 8.
And since a^ = y, . ∙. x = 3, or x = 2.
If x = 3, then the other number is 2, and if χ = 2, the other number is 3; so that 2 and 3 are the numbers required.
In general, if it be required to find two numbers which are exactly alike concerned in a question that produces a quadratic equation, they will be the roots of that equation. A similar observation applies to any number of quantities which require for their determination the resolution of an equation of any degree whatever.
7 7. On some Anomalies in the Solution of a Problem which results in an Equation.
From what has preceded, it will be evident that a root of an equation may be a very different thing from the solution of the problem on which the equation is based. It will be proper to give a few illustrations of this difference before passing on to consider equations in general.
(1.) A solution may be inapplicable to the problem as a problem of arithmetic, applying only to the algebraic problem.
Ex. Find a number such that if it be first increased by 10, and then diminished by 10, the difference of the square roots of the results shall be equal to 10.
Let x be the number; then the problem requires that ∖4c+10- Jx- 10 = 10.
Transposing and squaring, we get x + 10 = 100 + 20 χ4Γ≡Π) + x - 10.
Transposing and squaring again, there results x -10 = 16, a; = 26.
Now, it is obvious that 26 does not satisfy the conditions of the problem, but that it is the solution of another problem, viz., that which substitutes “sum” for “difference” in the enunciation. Generally we may remark that an algebraic statement is not definite like an arithmetical one. The algebraic square root of a quantity being + or -, algebra cannot, as arithmetic does, distinguish between the two. The equation va;+10- √ar- 10 = 10 is algebraically the same as Jx+ 10+ *Jx- 10 = 10, &c.
(2.) A solution may be inverted, or rather may invert the statement.
Ex. Divide 15 into two such parts that the greater shall exceed three times the less by as much as half the less exceeds 3.
Let x be the greater, and . ’. 15 - x the less. The statement produces the equation,
Ä-3 (15-ar) = ⅜(15-x)-3, which gives at once x = 11, so that 11 is the greater, 4 the less part. But, on trying the solution, we find it is not that of the problem given, but of another problem, in which “exceeds” is replaced by “falls short of.” Algebra cannot, in every case, as arithmetic does, distinguish the order of subtraction in stating a difference. Ex. 2. Find a number such that the square root of the difference between its fourth power and its square being found and increased by 1, the square root of the sum shall be equal to the given number diminished by 1.
Let x be the number, then
√(1 + √⅛ f -⅛ a ) = a-l, . ’ . ∣ Jx i - x 1 = a? - 2x, whence x = 0, x = -, neither of which solves the problem as stated.
(3.) A solution may be illusory—that is, it may assume . , 0
the form -.
’ ' Ex. 1. There are two pieces of cloth of a and a' yards respectively. The owner sells the same number of yards of each at b and b' shillings per yard respectively, he then sells the remainder at c and c' shillings a yard, and finds that the prices received for both pieces are the same. Required the number of yards first sold.
The number is ., αc , α ° ,. b—b+c—o
As a particular case, if a and a' are 60 and 80; b and b' 10 shillings and 9 shillings; and c and c 4 shillings and 3 shillings, the answer assumes the form -.
The answer is in this case indeterminate; in other words, the conditions of the problem are satisfied independently of the number of yards first sold; any number will do.
It is not, however, a necessary interpretation of the form ð, that it may be replaced by any number whatever. Most frequently this form results from the fact that some fraction is not in its lowest terms. Solutions of this kind frequently occur—in ordinary equations they maybe avoided; and we offer an example simply to show the method applicable to cases in which they cannot be avoided.
Ex. 2. Find two numbers such that the sum of their products by a and b respectively is c, and the difference of their squares d.
We have ax + by = c x i -y 2 = d „ fc-ax ∖ 2 , t.e.
.∙, (α 2 - δ 2 ) x 2 - 2acx = -(c 2 + b 2 d) v _ac ± ∣ f'a 2 c 2 — (a 2 — δ 2 ) (c 2 +δ 2 d) i— *
To find the solution when a = b, we observe that (taking the negative sign of the square root) x = ð.
This arises from the fact that some power or root of a - b is common to the numerator and denominator of the fraction.
To divide this out, we may put α 2 - b 2 =pa 2 , and we shall get X- /..< pa 2 , 1 c 2 +b 2 d . ac - ac + ~pac— —1- &a = pa 2
1 c 2 +b 2 d 1 c 2 +α 2 d
2 ac 2 ac ’ when a is written for b, and 0 for p.
(4.) A solution may be introduced by the operation.
In the example last given, the positive sign presents us with a solution introduced by the operation, which, when a = b, is not a solution of the problem at all.
For in that case the two equations become x + y = ~> χ 2 -y 2 = d; the latter of λvhich is at once reduced to the simple equation x - y = — by means ot tne former, Accord, ingly, both equations are in this case simple equations, and can admit of only one solution.
(5.) As a solution may be introduced by the operation, so may it be dropped out, even when the operation is a perfectly legitimate one.
Ex. √(2ic+l)- s ∕(z + 4)≈= √(4a + 4)- √(3oj + 7). Taking reciprocals, we have
1 1
√(2x+1) - √(a!+4) = √(4ic+4) - √(3x+7) √(2a+l)+√(¾+4) _ √(4a+4)+√(3⅞+7)
ae-3 ae-3
or √(2a>+l)+ √(a> + 4)= √(4α + 4)+ √(3a + 7), which either added to the original equation, or subtracted , .. , 3
from it, produces x = - -.
But x = 3 is a solution of the equation which has been dropped out by the omission of the common denominator x - 3.
It is not necessary to point out that a solution may appear under the form κj-a or ∞.
In neither case can the problem be solved arithmetically.
Sect. X.— Equations in general.
78. Before we proceed to the resolution of cubic and the higher orders of equations, it will be proper to explain some general properties which belong to equations of every degree, and also certain transformations which must frequently be performed upon equations in order to prepare them for solution.
In treating of equations in general, we shall suppose all the terms brought to one side, and put equal to 0; so that an equation of the fourth degree will stand thus:
x i +pa? + qx 2 + rx + « = 0, where x denotes an unknown quantity, and p, q, r, s, num· bers or fractions, either positive or negative. Here the coefficient of the highest power of x is unity, but had it been any other quantity, that quantity might have been taken away, and the equation reduced to the above form by rules already explained (Sect. VI).
The terms being thus arranged, if such a quantity be found as, when substituted for x, will render both sides = 0, and therefore satisfy the equation, that quantity, whether it be positive or negative, or even imaginary, is defined to be a root of the equation. But we have seen that every quadratic equation has always two roots, real or imaginary; we may therefore assume that a similar diversity will take place in all equations of a higher degree; and this assumption appears to be well founded, by the following proposition, which is of great importance in the theory of equations.
If a root of any equation, as x i +px s + qx 2 + rx + s = 0, be represented by a, the first side of that equation is divisible by x - a;
For since rc 4 +px?· + qx 2 + rx + s = 0,
And also a 4 +y>a 3 + 2a 2 + ra + s = 0;
Therefore, by subtraction, ic 4 -α 4 + p(tf 3 -af) + q(x 2 -α 2 ) + r(x-a) = 0 .
But any quantity of this form x* - α*, where n denotes a whole positive number, is divisible by x - a (Art. 20, Ex. I).
Hence, since every term contains a factor of this form, the equation may be written under the form (x - α) (ae 3 +p'x 2 + <fx + r') = 0.
i.e.. the expression .r 4 4- nx? 4. aχ2 + rx + s is divisible bv x-a·, and since the same mode of reasoning will apply to any equation whatever, the truth of the proposition is evident.
We have found that lx - a) (íc 8 +p'x l + q x + r') = 0; and as a product becomes = 0 when any one of its factors = 0, therefore the equation will have its conditions fulfilled, not only when x - a = 0, but also when x s +p'x 2 4- fx 4- r' = 0.
Let us now suppose that b is a root of this equation; then, by reasoning exactly as before, x 3 +p ’ x 2 + (fx + r' = lx — b) lx 2 +p"x + <∕').
By proceeding in the same manner with the quadratic equation x 2 +p"x 4- cf' ≡ 0, we shall find that if c denote one of its roots, then x 2 + p"x + <f'≈(x-c) (x + c +√ ’ ).
So that if we put d = - (c +∕>"), we at last find x i 4- px 3 4- qx 2 + rx + s = (x - a) lx-b~) (x - c) (x - d); α, b, c, d, being the roots of the equation, x i +px 3 + qx 2 + rx + s = 0 .
The mode of reasoning which has been just now employed in a particular case, may be applied to an equation of any order whatever; we may therefore conclude, that every equation may be considered as the product of as many simple factors as the number denoting its order contains unity, and therefore, that the number of roots in any equation is precisely equal to the exponent of the highest power of the unknown quantity contained in that equation.
79. By considering equations of all degrees as formed from the products of factors x - a, x - b, x -c, &c., we discover certain relations subsisting between the roots of any equation and its coefficients. Thus, if we limit the number of factors to four, and suppose that a, b, c, d, are the roots of this equation of the fourth degree, íc 4 +px 3 + qx 2 + rx + í = 0 , we shall also have lx — a) lx -b) lx — c) lx - d)≈0 ∙, and therefore, by actual multiplication, [table]
If we compare together the coefficients of the same powers of x, we find the following series of equations: [table]
and as similar results will be obtained for equations of all degrees, we hence derive the following propositions, which are of great importance in the theory of equations.
1. The coefficient of the second term of any equation, taken with a contrary sign, is equal to the sum of all the roots.
2. The coefficient of the third term is equal to the sum of the products of the roots multiplied together two and two.
3. The coefficient of the fourth term, taken with a contrary sign, is equal to the sum of the roots multiplied together three and three; and so on for the remaining coefficients, till we come to the last term of the equation, which is equal to the product of all the roots having their signs changed.
Instead of supposing an equation to be produced by multiplying together simple equations, we may consider it as formed by the product of equations of any degree, provided that the sum of their dimensions be equal to that of the proposed equation. Thus, an equation of the fourth degree may be formed either from a simple and cubic equation, or from two quadratic equations.
80. When the roots of an equation are all positive, its simple factors will have this form, x - a, x-b, x-c, <tc., and if, for the sake of brevity, we take only these three, the cubic equation which results from their product will have this form, x 3 -px 2 + gx — r = 0 , where ∕> = α + δ + c, q = ab + ac + bc, r = abc; and here it appears that the signs of the terms are + and — alternately.
Hence we infer, that when the roots of an equation are all positive, the signs of its terms are positive and negative alternately.
If again the roots of the equation be all negative, and therefore its factors x + a, x + b, x + c, then p, q, and r being as before, the resulting equation will stand thus:
x 3 +px 2 + qx + r = Q.
And hence we conclude, that when the roots are all negative, there is no change whatever in the signs.
In general, if the roots of an equation be all real, that equation will have as many positive roots as there are changes of the signs from + to -, or from - to +; and the remaining roots are negative. This rule, however, does not apply when the equation has impossible roots, unless such roots be considered as either positive or negative.
The connection between the signs of the roots and the signs of the terms of an equation can be deduced from the proposition, that the introduction of a new positive root introduces a new change of signs amongst the terms of the equation.
The demonstration of this proposition depends on the fact already established, that an equation may be resolved into the product of simple factors, so that, for example, every equation of the fifth degree may be derived from some equation of the fourth, by multiplying the latter by x - a where a is the additional root. We shall show that the introduction of a vιom positive root produces an equation with at least one more change of signs than the original, and the introduction of a new negative root produces an equation with at least one more continuation of the same sign. To save space, it will suffice if we write the signs without the letters; thus, x 2 + px - q may be written + 4—. Let, then, any equation be written down (of the sixth degree, for instance), +4 1 1-; multiply by x - a, and write the multiplication in the usual form, [table]
The signs of the product are all determinate except two, which we have marked with a (î). Now the changes of sign in the original equation are three—one between the 1st and 3d terms, one between the 3d and 5th, and one between the 5th and 6th; and it is evident that whatever be the signs marked (?), the produced equation has as many changes of sign as the original between the same limits, and one change beyond those limits, viz., between the 7th and 8th terms. This proposition is perfectly general, that the introduction of a positive root causes the introduction of at least all the original changes of sign within their limits, and one more change beyond those limits. In the same manner we may prove that the introduction of a negative root introduces at least one more continuation of the same sign. Hence the conclusion, that an equation cannot have more positive roots than it has changes of sign, nor more negative roots than it has continuations of the same sign. This proposition is known as Descartes’ Rule of Signs.
81. Surd and impossible roots enter equations by pairs.
Let a + fb be a root, where δ is a positive or negative number or fraction; then a - ∕∕, ⅛ also a root.
If a + √δ be written for x in the quantity x* +px" r ~ l + &c., the result is composed of a series of powers of a and Jb. Of these all but the odd powers of f/b are numerical, whilst odd powers of f Jb may be written as numerical products of t fb itself. The result of the substitution is therefore of the form A + B ffb. But since a + t fb is a root of the equation x n +pxT~ l + &c. = 0, we must have A + B√δ = 0, and .∙. A = 0, B = 0.
Now if a - fb be substituted for x, the result will be A - B s ∕δ, because even powers of - fb are the same as those of + √δ. But A = 0, B = 0.∙. A - B 5 ∕δ = 0; consequently a — fb is a root of the equation.
From this proposition it appears that every equation whose degree is denoted by an odd number, must have at least one real root.
82. We shall now explain some transformations which are frequently necessary to prepare the higher orders of equations for solution.
Any equation may have its positive roots changed into negative roots of the same value, and its negative roots into such as are positive, by changing the signs of the terms alternately, beginning with the second. The truth of this remark will be evident if we take the equation (x - α) (x - b) (x + c)≈αfi +px 2 + qx + r = 0, and write — x in place of x, producing - (x+α). - (x + b).( -x + c) = - x 3 +px 2 - qx + r = 0, i,e., (x + a) (x + δ) (x - c) = x 3 -pχ 2 + qx-r = 0, where it appears that the signs of the first and third terms are the same as in the original equation, but the signs of the second and fourth are the opposite. And this will be found to hold true of all equations, to whatever order they belong.
83. It will sometimes be useful to transform an equation into another that shall have each of its roots greater or less than the corresponding roots of the other equation, by some given quantity.
Let (x - a) (x - b) (x + c) = 0 be any proposed equation which is to be transformed into another, having its roots greater or less than those of the proposed equation by the given quantity n; then, because the roots of the transformed equation are to be + a ±n, + b±n, and — c ±n, the equation itself will be
(y τ n - α) (y =f n - b) (y =τ n + c) = 0.
Hence the reason of the following rule is evident.
If the new equation is to have its roots greater than those of the proposed equation, for x and its powers substitute y - n and its powers; but if the roots are to be less, then, for x substitute y + n ; and, in either case, a new equation will be produced, the roots of which shall have the property required.
84. By the preceding rule, an equation may be changed into another, which has its roots either all positive or all negative; but it is chiefly used in preparing cubic and biquadratic equations for solution, by transforming them into others of the same degree, but which want their second term.
Let ae 3 + px 2 + qx + r = 0 be any cubic equation; if we substitute y + n for x, the equation is changed into the following: [table]
Now, that this equation may want its second term, it is evident that we have only to suppose 3n + p = 0, or n =
—for this assumption being made, and the value of n substituted in the remaining terms, the equation becomes
^+(*-⅜÷iM + ^ « 2 , , 2υ 3 φq , ,, or, putting - — + q = q , and + gγ — - + r = r , the same equation may stand thus, y 3 + q' y + r' = 0.
85. In general, any equation whatever may be transformed into another, which shall want its second term, by the following rule.
Divide the coefficient of the second term of the proposed equation by the exponent of the first term, and add the quotient, with its sign changed, to a new unknown quantity; the sum being substituted for the unknown quantity in the proposed equation, a new equation will be produced, which wifi want the second term, as required.
By this rule any adfected quadratic equation may be readily resolved; for by transforming it into another equation which wants the second term, we thus reduce its solution to that of a pure quadratic. Thus, if the quadratic equation x i - 5x + 6 = 0 be proposed; by substituting y +1 for x, we find [table]
Hence y = ÷ 1 and since x = y + f, therefore x = = fc ⅛ + f ≈ + 3, or + 2.
86. Instead of taking away the second term from an equation, any other term may be made to vanish, by an assumption similar to that which has been employed to take away the second term. Thus, if in Art. 84 we assume 3n 2 + 2pn + 2 = 0, by resolving this quadratic equation, a value of n will be found which, when substituted in the equation, will cause the third term to vanish; and, by the resolution of a cubic equation, the fourth term may be taken away; and so on.
87. Another species of transformation, of use in the resolution of equations, is that by which an equation, having the coefficients of some of its terms expressed by fractional quantities, is changed into another, the coefficients of which are all integers.
Let ir 3 + -a> a +‰ + ~= 0 denote an equation to be so (i b c transformed, and let us assume y≈abcx, and therefore x = ~- ·, then, by substitution, our equation becomes abc ’ jL + ⅛ + J- v+ U 0 j
α⅞ 3 c 3 + α 3 δ 5 c^ + db 2 cr c and multiplying the whole equation by α 3 δ 3 c 3 , we have y 3 + bcpy 2 + dl·b&qy + α 3 δ 3 c 2 r = 0 .
Thus we have an equation free from fractions, while at the same time the coefficient of the highest power of the unknown quantity is unity, as before.
Examples of the Transformation and Solution of Equations when certain relations amongst the roots are known. Ex. 1. If a, b, c arc the roots of the equation x 3 - x 2 + %x _ 3 = 0, to form the equation of which the roots are
(1.) α + δ ,b + c, c + a.
Let y be any one root of the required equation; put y = a + b + c-x=l -x (Art. 79), and the values of y will be the roots of the equation required, which is therefore
(l-y) 3 -(l-y) 2 +2(1-y)-3 = 0, or y 3 — 2y 2 + 3y +1 = 0.
∕ 2 x . °- ∖- _ c -.
' ’ ' b+c — a’ a+c-b ’ a+b—c
Let z z y a+b+c- 2x l-2x ’
, 29 . 16 3 λ y≈+ 1 - 7 r-<- i7 y+ r -= 0 is the equation required.
(3.) α 2 , b 2 , c 2 .
If y = x 2 , the values of y are a 2 , b 2 , c 2 . Accordingly wc require to throw the given equation into a form which involves no odd powers of x. This is done as follows:
a? - x 2 + 2a; - 3 = x(χ 2 + 2) - (x 2 + 3) = 0; . ’ . x(x 2 + 2) = x 2 + 3.
squaring, x ∖ x 2 + 2) 2 = (a; 2 + 3) 2 , or y(y + 2) 2 = (y + 3) 2 , or y 3 + 3y 2 - 2y - 9 = 0.
(A ∖ L L L ' *' be ’ ac ’ ab ' τ , x i x 2 j ο o 2 1 λ Let 2 ’ = Λ=3'> and ^ + ≈ , -9≡ ’-3~ 0 is the equation required.
Ex . 2. Two roots of the equation x* - 16αr 3 + 86a; 2 - 176a; +105 = 0
are 1 and 5. Find the other roots.
The quantity on the left hand side of the equation is (Art. 78) divisible by {x -l')(x- 5), or by x 2 - 6x + 5.
The quotient is x 2 — 10α> + 21, which, being put =0, gives 3 and 7, the roots required.
Ex. 3. The equation a; 3 - 4x 2 + χ + r = 0 has one root, 3; find r and the other roots.
Write 3 for x, then r=6, and the equation may be written (x - 3)(o; 2 - x - 2) = 0, which gives x = 2, x = - 1.
Ex. 4. The equation x 3 + x 2 - 16a; - 16 = 0 has two roots of the form + α, - a ; find them.
If we write - x for x, we get the equation a; 3 — ae 2 - 16o;+16 = 0, which has also two roots, -a, +a, .∙. x 2 - a 2 is a common measure of the two quantities. But a; 2 - 16 is easily found to be a common measure of the two quantities, . ∙. a = 4. Ex. 5. The roots of the equation ae 3 - 6x 2 +llo; - 6 = 0
are in arithmetical progression; find them.
If a, a + b, α + 2δ be the roots, their sum is 3(α + δ), i.e., three times the middle root. But (Art. 79) their s nτn is 6, . ∙. a + δ = 2, also α(α + δ) (α + 2δ) = 6, i-e., α(4-α) = 3, α 2 -4α + 3 = 0,
·’· α = l, α = 3.
Ex. 6. The three roots of the equation a; 3 - 7x 2 +16a; - 8 = 0 are in geometrical progression; find them.
Let a, ar, aτ⅛ be the roots; then their product is (αr) 3 , . ∙. (ar) 3 = 8, and ar = 2, . ∙. their sum a + ar + ar 2 = 7, which gives a=l, r=2, and 1 2, 4 are the roots.
. Sect. XI.— Solution of Cubic Equations.
88. Cubic equations, like all equations above the first degree, are divided into two classes: they are said to be pure when they contain only one power of the unknown quantity; and adfected when they contain two or more powers of that quantity.
Pure cubic equations are therefore of this form, x 3 = 125, or x 3 = -27, or, in general, x 3 = r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube root of each side of the equation; thus, from the first of the three preceding examples we find x= +5, from the second or x = - 3, and from the third, x = ∖ Jr^.
It would seem at first sight that the only value which x can have in the cubic equation x 3 = r, or putting r = c 3 , x 3 - c 3 = 0, is this one, x = c; but since x 3 -c 3 may be resolved into these two factors, x-c and x 2 + cx + c 2 , it follows, that besides the value of x already found, which results from making the factor x - c = 0, it has yet other two values, which may be found by making the other factor x 2 + ex + c 2 = 0; and accordingly, by resolving the quadratic equation x 2 + ex = - c 2 , we find these values to be -c + √≡3c 5 - c -√^3c 2 -1 + √≡3 -l-√≡3
2 and 2 ’ ° r 2 Cand 2 C · Thus it appears, that any cubic equation of this form, x 3 = <?, or ir 3 - c 3 = 0, has these three roots,
-l + √≡3 -1-√Z3
<ε = C, « = 2 c > ic = 2 C ’ the first of which is real, but the two last are imaginary. If, however, each of the imaginary values of x be raised to the third power, the same results will be obtained as from the real value of x; the original equation ae 3 - e 3 = 0 may also be reproduced, by multiplying together the three factors x-c, x —θc, and x - — 1 —θc.
2 2
89. Let us now consider such cubic equations as have all their terms, and which are therefore of this form, x 3 + Ao; 2 + Bo; + C = 0, where A, B, and C denote known quantities, either positive or negative.
It has been shown (Art. 84) how an equation having all its terms may be transformed into another which wants A
the second term; therefore, assume x=y- -χ-, as directed 3
in that article; then, by proper substitution, the above equation will be changed into another of this form, y 3 + gy + r=0, where q and r denote known quantities, whether positive or negative; now the roots of this equation being found, it is evident that those of the former may be readily A obtained by means of the assumed equation x = y - —.
3
Resuming, therefore, the equation y 3 + qy + r = Q, let us suppose y = v + z, and it becomes v 3 + 3v 2 z + 3vz 2 + z 3 j + qv + qz > = 0.
+ r J
Thus we have a new equation, which, as it involves two unknown quantities, υ and z, may be resolved into any two others, which will simplify the determination of those quantities.
Now, it appears, that the only way in which we can divide that equation into two others, so as to simplify the question, is the following:
3v 2 z + 3vz 2 + qv + qz = 0, ιfi + z 3 + r = 0 .
The first of these may also be expressed thus,
(3vz + q)(y + «) = 0.
Hence, we must either suppose that v + z = 0, or that 3vz + q = 0; but the former supposition cannot be admitted without supposing also that y = 0; therefore we must adopt the latter. So that to determine υ and z we have these two equations, 3υz + q = 0, w 3 + z 3 + r = 0.
From the first, we find vz = -1, and. ∙. v 3 z 3 = - . This
3 27
reduces the second equation to a quadratic in v 3 , viz. t> 6 + rv 3 - gy = θ> the solution of which equation is v s = - ’ z+ √⅞ r +¾⅜ * ^ = - V- √ i ⅛ 1 +k a ∙.
v-V- i .r + √⅛√> + ⅛r≈j ≈ = √-∣r- √⅛ s + Pi and y = υ + z
-√-∣r+ √⅛ 8 + ⅛r≈ + v'-∣r- √⅛∙ + JrS,
Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved.
90. But this is only one of three values which y may have. Let us, for the sake of brevity, put
A= -⅛r + √⅛ 3 + ⅛r 2 , B= -Jr- √A^ 3 + ∣r 2 ,
„ -l+√-3
2 ’ andPUt fl . -l-√÷3
then, from what has been shown (Art. 88), it is evident that V and z have each these three values, v = ^∕'A, v=a 3 ∕A, υ = β ^/A; z = ≡∕B, z = α^∕B, z = β ≡∕B.
To determine the corresponding values of v and z, we must consider that υz = -~= ¾∕AB. Now if we observe that aβ = l, it will immediately appear that v + z has these three values, v + z = ^∕A + v z B,
ι> + z= α^A + j β*∕B υ + z = β ^∕A + a ^B, which are therefore the three values of y.
The first of these formulae is commonly known by the name of Cardan’s rule; but it is well known that Cardan was not the inventor, and that it ought to be attributed to Nicholas Tartalea and Scipio Ferreus, who discovered it much about the same time, and independently of each other. (See the Historical Introduction.')
The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows :— Because vz≈ -∣, therefore z = - ∣×^ = “ > hence v + z = £/A - - s ⅛-: thus it appears that the three values √A
of y may also be expressed thus:
S'" V*- ⅛
3 ∕~Γ y = α√A-^ y = ∕3√A-∣=∙.
91. To show the manner of applying these formulae, let it be required to determine x from the cubic equation ot? + 3∙r 2 + ⅛x - 13 = 0.
As this equation has all its terms, the first step towards its resolution is to transform it into another which shall want the second term, by substituting y-1 for x as directed (Art. 84). The operation will stand thus: z3 = y 3-3 y 2 + 3y - i
+ 3√≡ = + 3y 2 -6y + 3
+ 9x = +9y- 9
-13= -13
.∙. adding these, the transformed equation is y 3 + Gy — 20 = 0, which being compared with the general equation, y 3 + gy + r = 0, gives 2 = 6, r = - 20; hence
A = √-⅛r + √⅛ 3 +P = V z 10+ √108^, therefore the second formula of last article gives y = s ∕ = 2
√10 + √108~ 3∕ iλ 1 ~.-≡; but as this expression in- <v 10 + V108
volves a radical quantity, let the square root of 108 be taken and added to 10, and the cube root of the sum found; thus we have £/10 + √108 = 2732 nearly, and 2 2
therefore - = = 732; hence we at last find
√10+√108 2732
one of the values of y to be 2732 - 732 = 2.
In finding the cube root of the radical quantity £/10 + √108, we have taken only its approximate value, so as to have the expression for the root under a rational form, and in this way we can always find, as near as we please, the cube root of any surd of the form a+ s ∕b, where b is a positive number. But it will sometimes happen that the cube root of such a surd can be expressed exactly by another surd of the same form; and accordingly, in the present case, it appears that the cube root of 10+ √T08 is 1+ √3, as may be proved by actually raising 1 + √3 to the third power. Hence we find
_ 2 2 2 (l-√3), 1 ,5r.
‰+√108 l+√3 (l-√3)(l + √3) * ' ’ that y=l+ √3 + l- √3 = 2,as before.
The other two values of y will be had by substituting i ∩
1 + √3 and 1 - √3 for ≡∕A and in the second and v A
third formulae of last article, and restoring the values of α and β. We thus have y = =±⅞≤≡ X (1+ √3) + s ⅛≡ 3 × (> - √3)= - 1 + √≡^9 . __
y = ^⅛^x(l + √3) + ^⅞^- 3 x(l- √3)--l - λ ∕-9∙
So that the three values of y are
+ 2, -l+√≡9, -l-√^9j and since x = y - 1, the corresponding values of x are
+ 1, -2+√^9, -2-√~9.
Thus it appears that one of the roots of the proposed equation is real, and the other two imaginary.
The two imaginary roots might have been found otherwise, by considering that since one root of the equation is 1, the equation must be divisible by x-1 (Art. 78). Accordingly, the division being actually performed, and the quotient put = 0, we have the quadratic equation, a 2 + 4x+13 = 0j which gives x= - 2 ± √-9, the same imaginary values as before.
92. In the application of the preceding formulae to the resolution of the equation y 3 + qy + r = 0, it is necessary to find the square root of ⅛⅛ 3 + ⅛r 2 ; now, when that quantity is positive, as in the equation y 3 + Gy - 20 = 0, which was resolved in last article, no difficulty occurs, for its root may be found either exactly or to as great a degree of accuracy as we please.
As, however, the coefficients q and r are independent of each other, it is evident that q may be negative, and such that ½ j f is greater than ∣r 2 . In this case, the expression _i_ ? 3 + ι r 2 w i∩ b e negative, and therefore its square root an imaginary quantity; so that all the roots appear under an imaginary form. But we are certain (Art. 81) that every cubic equation must have at least one real root. The truth is that roots are frequently real, though they appear under au imaginary form. Take, for instance, the equation y 3 - 6y + 4 = 0, of which the roots are found to be y≈ √ 2 + 2 y£I + 4/ 2-2 √^I y = λ √2 + 2√ - l 1 + β 4/ 2 - 2 √- y = jS4∕2 + 2√^l + α√2-2√-l
It will be found by actual involution that the imaginary expressions 2 + 2 √ - 1 and 2-2 √^T are the cubes of - 1 + √ - 1 and - 1 - √ - 1 respectively, whence by substitution we find
27 = 2, y=l+ √3, and y=l- √3^.
93. We shall now prove, that as often as the roots of the equation a? + qx + r = 0 are real, q is negative, and ⅛q a greater than ∣r 2 ; and, conversely, that if ^ 7 -7 3 be greater than {r 2 , the roots are all real.
Let us suppose a to be a real root of the proposed equation ·
Then a; 3 + qx + r = 0,
And α 3 + qa + r = 0.
And therefore, by subtraction, x 3 - α 3 + γ(z - a) = 0; hence, dividing by x - a, we have x 2 + ax + a 2 + q = 0.
This quadratic equation is formed from the two remaining roots of the proposed equation, and by resolving it we find a: = - ∣α ± J -½ a 2 -q.
And as, by hypothesis, all the roots are real, it is evident that q must necessarily be negative, and greater than ∣α 2 ; for otherwise the expression λ J - ~a 2 - q would be imaginary. Let us change the sign of q, and put q = ∣α 2 + d ; thus the roots of the equation x 3 - qx + r = 0 will be
α, -∣α + √<Z, -⅛α- √<Z, and here d is a positive quantity.
To find an expression for r in terms of a and d, let {α 2 + d be substituted for q in the equation α 3 - qa + r = 0; we thence find r = - ∣α 3 + ad: so that to compare together the quantities q and r, we have these equations,
^ = ∣α 2 + tZ, r= -iα 3 + αiZ.
In order to make this comparison, let the cube of ∖ q be taken, also the square of ∣r, the results are
⅛Z 3 = ⅛ 5 + ^d + ⅛α 2 d 2 + ⅛d 3 ,
^ 2 = ⅛α 6 -⅛αV + ∣α 2 cZ 2 ÷
and therefore, by subtraction,
= 3J(⅛α 4 -⅛AZ + ⅛d 2 )
= 3d(iα 2 -∣rf) 2 .
Now the square of any real quantity being always positive, it follows that 3rf(∣α 2 - ∖ d') 2 will be positive when d is positive; hence it is evident that in this case ~<β must be greater than ∣r 2 , and that cannot be less than ⅛ 7 ' 2 > unless d be negative, that is, unless - ∖ a + s ∕ t Z, - {a - y ∕d, the two other roots of the equation are imaginary. If we suppose d = 0, then y f g 3 = ∣r 2 ; and the roots of the equations, which in this case are also real, two of them being equal.
Upon the whole, therefore, we infer, that since a cubic equation has always one real root, its roots will be all real as often as q is negative, and ÷q 3 greater than |r 2 ; and consequently, that in this case the formulae for the roots must express real quantities, notwithstanding their imaginary form.
94. Let y 3 - qy + r = 0 denote any equation of the form which has been considered in last article, namely, that which has its roots all real; then, if we put α = - ∣r, i 2 =-L·-2 3 - ∣r 2 , one of the roots, as expressed by the first formula (Art. 90) will be y = 4/ <ι ÷ δ — 1 + 4/ λ - δ — 1 ·
This expression, although under an imaginary form, must (as we have shown in last article) represent a real quantity, although we cannot obtain it by the ordinary process of arithmetic.
The case of cubic equations, in which the roots are all real, is now called the irreducible case.
It is remarkable that the expression y∕a + b — 1 + ∖ ∕a — b J — 1, and in general,
4/« + δ √ - 1 + χ/« - δ √ - 1, where n is any power of 2, admits of being reduced to another form, in which no impossible quantity is found. Thus, x ∕α + δ 5 ∕ - 1 + a ∕<i-0√-1= √ f 2α + 2 √α 2 + δ 2 , and 4∕α + δ√~l + ⅛a-b √~1 =
√(√2α + 2√α 2 + δ 2 + 2 + , as is easily proved by first squaring and then taking the square root of the imaginary formulae. But when n is 3, it does not seem that such reduction can possibly take place.
If each of the surds be expanded into an infinite series, and their sum be taken, the imaginary quantity f J - 1 will vanish, and thus the root may be found by a direct process.
Sect. XII.— Solution of Biquadratic Equations.
95. When a biquadratic equation contains all its terms, it has this form, x i + Aλγ 3 + Bz 2 + Cχ + D = 0
where A, B, C, D denote any known quantities whatever.
We shall first consider pure biquadratics, or such as contain only the first and last terms, and therefore are of this fonn, x i = b i . In this case it is evident that x τaa,γ be readily had by two extractions of the square root; by the first we find x 2 = b 2 , and by the second x = b. This, hθλv-ever, is only one of the values which x may have; for since x 4 = 6 4 , therefore χ 4 -δ 4 = 0; but x 4 - b i may be resolved into two factors x 2 -b 2 and x 2 + b 2 , each of which admits of a similar resolution; for x 2 -b 2 = (x-b)(x + δ) and x 2 + δ 2 ≈{x-b ,J- l)(ar + δ J- 1). Hence it appears that the equation x 4 - b i = 0 may also be expressed thus
(x-δ) (^ + 0)(z-6√Ti)(z + yΓi) = o i so that x may have these four values,
+ δ, -b, + δ x ∕-l, -δ√-l, two of which are real, and the others imaginary.
96. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form, x i + qx 2 + s = 0 .
These may be resolved in the manner of quadratic equations; for if we put y = x 2 , we have y 2 + 5 ’ y + β=0, from which we find y = — ~ 4 *, and therefore
-' √≡≡≡.
97. When a biquadratic equation has all its terms, tho manner of resolving it is not so obvious as in the two former cases, but its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following, which wo select as one of the most ingenious, was first given by Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra.
We have already explained, Art. 92, how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form, y 4 +py 2 + qy + r =0, where the second term is wanting, and where p, q, r, denote any known quantities whatever.
That we may form an equation similar to the above, let us assume y = Ja + Jb + √c, and also suppose that the letters a, b, c denote the roots of the cubic equation z 3 + Pz 2 + Qz - R = 0; then, from the theory of equations we have
α + δ + c= -P, αδ + αc + δc = Q, αδc = R.
We square the assumed formula y= Ja,+ Jb + Jc,_ and obtain y 2 = α + δ + c + 2 (s ∕αδ+ v z αc + Jbc), or, substituting - P for a + b + c, and transposing; y 2 + P = 2(Jab + Jac + J be).
Let this equation be also squared, and we have y 4 + 2Py 2 +P 2 = 4(αδ + ac + be) + 8(√α⅛c + √αδ 2 c + √aδc 2 ); and since ab + ac + bc≈ Q, and ∖ fa 2 bc + √αδ 2 c+√ αδc 2 = *∕abc(J a+*Jb+ √c) = √R. y, the same equation may be expressed thus:
y 4 + 2Py 2 + P 2 = 4Q + 8 √R.y.
Thus we have the biquadratic equation
∕ + 2Py 2 -8√R.y + P 2 -4Q = 0, one of the roots of which is y = Ja+ Jb+ Jc, while a, b, c are the roots of the cubic equation z 3 + Pz 2 + Qz - R = 0.
98. In order to apply this resolution to the proposed equation y 4 +py i j rqy j rτ=⅛, we must express the assumed coefficients P, Q, R by means of p, q, r, the coefficients of that equation. For this purpose, let us compare the equations y 4 +py 2 +gy + r = 0, y< + 2Py 2 -8√Ry + P 2 --4Q≈0, and it immediately appears that
2P=p, -8√R = i , P 2 -4Q = r and from these equations we find p_P Q-P 2 ~ 4r R = f 1- 2 ’ 16 ’ K 64*
Hence it follows that the roots of the proposed equation ire generally expressed by the formula y= Ja+ Jb + Jc∙, vhere α, δ, c denote the roots of this cubic equation, i 3 + ⅛ + t⅛-i,0.
2 16 64
But to find each particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities Ja, Jb, Jc may have either the sign + or - prefixed to it; and. hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities Ja, Jb, Jc must be equal to x ∕R or to - ∣; when q is positive, their product must be a negative 8
quantity, and this can only be effected by making either one or three of them negative; again, when q is negative, their product must be a positive quantity; so that in this case they must either be all positive, or two of them must be negative. These considerations enable us to determine, that four of the eight expressions for the root belong to the case in which q is positive, and the other four to that in which it is negative.
99. We shall now give the result of the preceding investigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall transform it into another, in which the coefficients are integers, by supposing z = ⅛∙ Thus the equation + 2 + 16 64
becomes, after reduction, υ 3 + 2∕w 2 + (p 2 - 4r)w - q 2 = 0, it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are , 4 , 4 , so that our rule may now be expressed thus:
Let y i +py 2 + qy + r = 0 be any biquadratic equation wanting its second term. Form this cubic equation v 3 + 2pυ 2 + (y 2 - 4r)v - q 2 = 0, and find its roots, which, let us denote by a, b, ci
Then the roots of the proposed biquadratic equation are, [table]
100. As an example 01 tne metnoα 01 resolving a oι-quadratic equation, let it be required to determine the roots of the following, x i - 25z 2 + 60z - 36 = 0.
By comparing this equation with the general formula, we have ρ = - 25, q = + 60, r = - 36; hence 2p = - 50, p 2 - 4r = 769, j 2 = 3600, and the cubic equation to be resolved is v 3 -50v 2 + 769v-3600 = 0; the roots of which are found, by the. rules for cubics, to be 9, 16, and 25, so that Ja = 3>, Jb = i, c = 5. Now in this case a is positive, therefore [table]
101. We have now explained the particular rules by which the roots of equations belonging to each of the first four orders may be determined; and this is the greatest length mathematicians have been able to go in the direct resolution of equations; for as to those of the fifth, and all higher degrees, no general method has hitherto been found, either for resolving them directly, or reducing them to others of an inferior degree.
It even appears that the formulae which express the roots of cubic equations are not of universal application; for in one case, that is, when the roots are all real, they become illusory, so that no conclusion can be drawn from them. The same observation will also apply to the formulae for the roots of biquadratic equations, because, before they can be applied, it is always necessary to find the roots of a cubic equation. But both in cubics and in biquadratic equations, even when the formulae involve no imaginary quantities, and therefore can be always applied, it is more convenient in practice to employ other methods, λvhich we are hereafter to explain.
Sect. XIII.— Solution of Equations in which certain RELATIONS ARE KNOWN TO EXIST AMONGST THE ROOTS.
102. When the coefficients of the terms of an equation form the same numerical series, whether taken in a direct or an inverted order, as in this example, x i +px 3 + qx 2 +px +1 = 0, it may always be transformed into another of a degree denoted by half the exponent of the highest power of the unknown quantity, if that exponent be an even number; or half the exponent diminished by unity, if it be an odd number.
The same observation will also apply to any equation of this form, x i +pax 3 + qcpχ 2 +pa 3 x + α 4 = 0.
103. That we may effect the proposed transformation upon the equation x i +px 3 + qx 2 +px + 1 = 0, let every two terms which are equally distant from the extremes be collected into one, and the whole be divided by x 2 , then
* 2 + ⅛+7 5 (* + ^)+!7 = 0 ∙
Let us assume 1 1
x + - = z; X then x 2 + 2 + = z 2 , and x 2 + = z 2 - 2.
Thus the equation x 2 + +∕> (x + - ) + g , = 0, X δ X becomes z 2 +pz + q — 2 = 0
and since x + ~ = z, therefore x 2 -zx+l = 0.
X
Hence, to determine the roots of the biquadratic equation x i +px? + qx 2 +px +1 = 0, we form the quadratic z 2 +pz + q-2 = Q, and find its roots, which, let us suppose denoted by z z and ∕'; then the four roots of the proposed equation will be found by resolving two quadratic equations, viz. x 2 -zx + 1 = 0, x 2 -z'x +1 = 0. .
104. may be observed, respecting these two quadratic equations, that since the last term of each is unity, if we put a, a' to denote the roots of the one, and b, b' those of the other, we have from the theory of equations, aa' = 1, and therefore a' = -; also bb , = 1, and δ ’ = τ: now a, a', b, Cl 0 b' are also the roots of the equation x i + px 3 + qx 2 +px +1 = 0.
Hence it appears that the proposed equation has this property, that one-half of its roots are the reciprocals of the other half; and to that circumstance we are indebted for the simplicity of its resolution.
105. If the greatest exponent of the unknown quantity in a reciprocal equation is an odd number, as in this example, x 5 +px 4 + qx 3 + qx 2 +px + 1 = 0, the equation will always be satisfied by substituting - 1 for x; hence, — 1 must be a root of the equation, and therefore the equation must be divisible byx+l. Accordingly, if the division be actually performed, we shall have in the present case xl· + (p - l)x 3 - - 7 - l)x 2 + (p - l)x + 1 = 0, another reciprocal equation, in which the greatest exponent of X is an even number, and therefore resolvable in the manner we have already explained.
106. As an application of thê theory of reciprocal equations, let it be proposed to find x from this equation,
≈ 5 +L-
(x+l) 5 σ, where a denotes a given number.
Every expression of the form x n + 1 is divisible by x +1 when n is an odd number. In the present case, the numerator and denominator being divided by x + 1, the equation becomes x 4 — x 3 +x 2 -x+l x 4 +4x 3 +6x 2 +4x+l a ' ’ and this again, by proper reduction, becomes
(a - l)x 4 + (4a + l)x 3 + (6a- l)x 2 + (4a + l)x + a- 1=0;
, . 4a + l 6a —1
and, putting p = , q = i , x i +pa? + qx 2 +px +1=0; a reciprocal equation, resolvable into two quadratics.
Equations which have Equal Roots.
107. When an equation has two or more equal roots, these may always be discovered, and the equation reduced to another of an inferior degree, by a method of resolution which is peculiar to this class of equations.
Although the method of resolution we are to employ will apply alike to equations of every degree, having equal roots, yet, for the sake of brevity, we shall take a biquadratic equation, x 4 +px 3 + qx 2 + rx + s = 0, the roots of which may be generally denoted by α, b i c, and d. Thus we have, from the theory of equations, (x — a)(x — b){x - c)(x -d) = x i +px 3 + qx 2 + rx + s.
Let us put
A = (x - a)(x - b}(x — c), A" = (x - a)(x - c)(x.- d), A! = (x-a)(x - b)(x - d), A'" = ∖ x - b)(x - c)(x - d): then, by actual multiplication, we have
A = x 3 - a + ab
- b x 2 +ac>x- abc , -c ) + bc )
A′ =x 3 -aj +ab i
- b x 2 + ad > x - abd,
-d) + bd)
A″ —a? — aj +ac i -c > x 2 +ad ∖ x-axd t -d ) +cd )
A‴ = x 3 -δ) +δc)
- c x 2 +δd>x - bed :
- d) + cd ) and taking the sum of these four equations;
A + A ’ +A"+A w = 4x 3 -3α∖ + 2aδ Ί
- 3b 2 ÷2ac — abc
- 3c ( + 2ad I —abd
- 3d) +2bc Γ x -acd + 2bd — bed. + 2cd ,
But since a, b, c, d are the roots of the equation x 4 +px 3 +g ι x 2 + rx+s = 0, we have -3(α + δ + c + d) = 3p, 2(ab + ac + ad+bc+bd+cd) = 2q, — (abc+abd+acd+bed) = r; A+A'+A"+A w = 4x 3 + 3px 2 + 2qx+r.
This result expressed in its most general, form is as follows :—Let A represent the product of all the differences x- a, &c., except one, SA, the sum of all such products; then SA = nx n ~ l + (n - 1 )px n ~ 2 + (n - 2)qx n ~ 3 - ∣ &c.
108. Let us now suppose that the proposed biquadratic equation has two equal roots, or a = b; then x-a = x-b, and since one or other of these equal factors enters each of the four products A, A', A", A"', it is evident that A + A' + A" + A'", or 4x 3 + 3px 2 + 2qx + r must be divisible by x - a, or x - b. Thus it appears that if the proposed equation χ i +paP + qx 2 + rx + s-Q have two equal roots, each of them must also be a root of this equation,
4∙r 3 + 3p# 2 + 2qx + r = 0; for when the first of these equations is divisible by (x — α) 2 , the latter 13 necessarily divisible by x - a.
Let us next suppose that the proposed equation has three equal roots, or a = b — c; then, two at least of the three equal factors x-a, x-b, x-c, must enter each of the four products A, A', A", A'", it is evident that A + A' + A" + A' ’ ', or 4∞ 3 + 3px 2 + 2qx + r must be twice divisible by x - a. Hence it follows that as often as the proposed equation has three equal roots, two of them must also be equal roots of the equation
4<r 3 + 3/># 2 + 2qx + r = 0.
106. Proceeding in the same manner, it may be shown, that whatever number of equal roots are in the proposed equation
#4 +7%c 3 + qx 2 + rx + s = 0, they will remain, except one, in this equation
4x 3 + 3p# 2 + 2qx + r = 0, which may be derived from the former, by multiplying each of its terms by the exponent of x in that term, and then diminishing the exponent by unity.
110. If we suppose that the proposed equation has two equal roots, or a = b, and also two other equal roots, or c = d, then, by reasoning as before, it will appear that the equation derived from it must have one root equal to a or 7, and another equal to c or d; so that when the former is divisible both by (x - a) 2 and (x - <∙) 2 , the latter will be divisible by (x-a) (x — c).
111. As a particular example, let us take this equation, # 5 -13# 4 + 67# 3 -171# 2 +216#-108 = 0, and apply to it the method we have explained, in order to discover whether it has equal roots, and if so, what they are. We must seek the greatest common measure of the proposed equation and this other equation, which is formed agreeably to what has been shown (Art. 109), 5x i - 52x 3 + 201# 2 - 342z + 21G = 0; and the operation being performed, we find that they have a common divisor, x i — 8x 2 + 21# - 18, which is of the third degree, and consequently may have several factors. Let us therefore try whether the last equation, and the following,
20# 8 - 156x 2 + 402# - 342 = 0, which is derived from it by the same process, have any common divisor; and, by proceeding as before, we find that they admit of this divisor, # - 3, which is also a factor of the last divisor, x i - 8# 2 + 21# - 18; and therefore the product of the remaining factors is immediately found by division to be x i — 5# + G, which is evidently resolvable into # - 2 and # — 3.
Thus it appears that the common divisor of the original equation, and that which is immediately derived from it, is (#-2) (#-3) 2 ; and that the common divisor of the second and third equations is # - 3. Hence it follows that the proposed equation has (# - 2) 2 for one factor, and (# - 3) 3 for another factor, and may therefore be expressed thus, (# - 2) 2 (# - 3) 3 = 0. The truth of this conclusion may be easily verified by multiplication. The five roots are 2, 2, 3, 3, and 3.
112. The property proved in Art. 107 enables us to establish numerous relations between the coefficients and roots of an equation, in addition to the fundamental one established in Art. 79, such as the following:—
Since #" + px κ ~^ x + gx"^ 2 + <fcc. = (#-«)(#- 7>) (# - c) &c.
and nx*~ 1 + (n - 1 )7xr*^^ 2 + (n - 2)g'χ∙~ , + <kc. =
(# - &)(# - c) .. . + (# - a)(# - c) ... + ..., by division nx*~ 1 + (x - l)px n ^ 2 + (n - 2)¾x*~ 3 + &c. x n +px** -1 + gx n ^ 2 + &c.
= — + j -τ + - + &c.
x — a x — b x — c
1 S 1 S 2 S, ,
= →4+4+4+ ιtc ∙
x x 2 x i x i where S 1 , S 2 , S 3 , &c., are the sums of the first, second, third, &c., powers of the roots of the equation.
Multiplying out and equating coefficients, we get— 6ι-l)p = S 1 ÷np
(n-2)2 = S 2 +pS 1 +nQ
(n - 3)r = S 3 +pS 2 +⅛S 1 +
&c. =&c.
Or S 1 +p = 0
S 2 +pS 1 + 2g = O
S3+pS 2 +{S 1 +3r — 0
&c. &c.
A⅛. 1. As a particular case, take the cubic equation a?+ qx + r = 0.
Here S 1 = 0
S 2 +2g = 0
S 3 +3r=0
S 4 +gS 2 = 0 S s +jS3+rS 2 = O
2S 4 =⅛, S δ = 5 2 r.
The last may be written—
S 5 S 3 S 2
5 qr 3’2*
i.e., if a + b + c = 0, then will α 6 + fcδ +c 5 a 3 + ^3 +c 3 a 2 + J2 + 0 2^
5 - 3 2
From S 7 + g ’ S 5 + rS 4 = 0, we get S,≈¾.⅛+¾∙⅜∙
Or ⅛-⅜.⅜,<⅛c. te Ex. 2. Take the biquadratic equation sc*+<μs 2 +rχ+i=O
Here
S l =0
S 2 +2<∕=0
S 3 +3r = 0
S 4 -∣- jSjj -f-4s = 0 S 6 +2S3+rS 2 = O s 0 -s,.⅛+s, .⅛
or ⅛=⅛.S>i
5 3 2
t.e., if α + δ + c + <Z = 0, 1 α 6 +⅛ 5 + c 5 + d 5 a 3 +δ 3 +c 3 +d ’ a 2 ÷δ 2 + c ’ ÷d , then — _
Equations whose Roots are Rational
113. It has been shown in Art. 79, that the last term of any equation is always the product of its roots taken with contrary signs. Hence, when the roots are rational, they may be discovered by the following rule:
Bring all the terms of the equation to one side; find all the divisors of the last term, and substitute them successively for the unknown quantity. Then each divisor, which produces a result equal to 0, is a root of the equation.
Ex. Let 3? - 4# a - 7# +10 = 0.
The divisors of 10, the last term, are 1, 2, 5, 10, each of which may be taken either positively or negatively; and these being substituted successivelv for x we obtain the following results:
By putting +1 for x, 1 - 4 - 7 +10 = 0,
-1, -1- 4+ 7 + 10= 12,
+2, 8- 16-14+10=-12,
— 2, -8- 16 + 14 + 10= 0,
+ 5, 125-100-35 + 10= 0.
Here the divisors which produce results equal to 0 are + 1, -2, and +5; therefore these are the three roots of the proposed equation Sect. XIV. Solution of Equations by Approximation
114. When the roots of an equation cannot be accurately expressed by rational numbers, it is necessary to have recourse to methods of approximation; and by these we can always determine the numerical values of the roots to as great a degree of accuracy as we please.
The application of methods of approximation is rendered easy by means of the following proposition:
If two numbers, either whole or fractional, be found, which, when substituted for the unknown quantity in any equation, produce results with contrary signs, we may conclude that at least one root of the proposed equation is between those numbers, and is consequently real.
Let the proposed equation be ^-5λ 2 + 10x-15 = 0, which, by collecting the positive terms into one sum, and the negative into another, may also be expressed thus,
Λ 3 + 10x-(5Λ 2 + 15) = 0j then, to determine a root of the equation, we must find such a number as, when substituted for x, will render r 3 +10x = 5x 2 +15.
Let us suppose x to increase and to have every degree of magnitude from 0 upwards in the scale of number; then a? + 10z and 5.r 2 +15 will both continually increase, but with different degrees of quickness, as appears from the following table :—
Successive values of x; 0, 1, 2, 3, 4, 5, 6, &c.
of rr 3 + l(kj 0, 11, 28, 57, 104, 175, 276, &c. of 5z 2 + 15; 15, 20, 35, 60, 95, 140, 195, &c.
By inspecting this table, it appears that while x increases from 0 to a certain numerical value, which exceeds 3, the positive part of the equation, or x 2 ∙ + 10∙r, is always less than the negative part, or 5x 2 + 15; so that the expression ic 3 + 10∙r- (5x 2 +15) or it 3 - 5x 2 + 10a; — 15 must necessarily be negative.
It also appears, that when x has increased beyond that numerical value, and which is evidently less than 4, the positive part of the equation, instead of being less than the negative part, is now greater, and therefore the expression ar ’-5τ 2 +10ΛJ-15
is changed from a negative to a positive quantity.
Hence we may conclude that there is some real and determinate value of x, which is greater than 3, but less than 4, and which will render the positive and negative parts of the equation equal to one another; therefore that value of x must be a root of the proposed equation; and as what has been just now shown in a particular case will readily apply to any equation whatever, the truth of the proposition is obvious.
115. From the preceding proposition it will not be difficult to discover, by means of a few trials, the nearest integers to the roots of any proposed numerical equation; and those being found, we may approximate to the roots continually, as in the following example:
z 4 -4^-3x + 27 = 0.
To determine the limits of the roots, let 0, 1, 2, 3, 4, be substituted successively for x ; thus we obtain the following corresponding results:
Substitutions for x, 0, 1, 2, 3, 4,
Results, +27,+ 21,+ 5,-9,+ 15.
Hence it appears that the equation has two real roots, one between 2 and 3, and another between 3 and 4.
That we may approximate to the first root, let us suppose x = 2 + y, where y is a fraction less than unity, and therefore its second and higher powers small in comparison to its first power: hence, in finding an approximate value of y, they may be rejected. Thus we have ic 4 = +16 + 32y, &c. -4x 3 = -32-48y, &c. — 3x = — 6 — 3y + 27 =+27
Hence 0 = 5 — 19y nearly, g and y = — = *26; therefore, for a first approximation we have x = 2'26.
Let us next suppose ic = 2 - 26 + y , ; then, rejecting as before the second and higher powers of y' on account of their smallness, and retaining three decimal places, we have y' = — 1 ⅛ = ’ 0075, and x≈ 2∙26 + y' = 2∙2675. This value lo’iiy of x is true to the last figure, but a more accurate value maybe obtained by supposing x = 2 ’ 2675 +y", and proceeding as before.
116. The method we have hitherto employed for approximating to the roots of equations is known by the name of the method of successive substitutions, and was first proposed by Newton. It has been since improved by Lagrange, who has given it a form which has the advantage of showing the progress made in the approximation by each operation. This improved form we now proceed to explain.
Let a denote the whole number next less to the root sought, and - the fraction, which, when added to a, com· 2/
pletes the root; then x = a + -. If this value of x be substituted in the proposed equation, a new equation involving y will be had, which, when cleared of fractions, will necessarily have a root greater than unity.
Let b be the whole number which is next lese than that root; then, for the first approximation, we have x = a + ⅛. But b being only an approximate value of y, in the same manner as a is an approximate value of x, we may suppose y = b +~,', then, by substituting b + -, for y, we shall have 2/ . 2/
a new equation, involving only y', which must be greater than unity. Putting therefore b' to denote the next whole number less than the root of the equation involving y', we have y = b + ÿ‘, and substituting this value in that of x t the result is
Λ = α-t —
i + s∙
for a second approximate value of x.
To find a third value, we may take y' = ⅛' + -⅛, and so on, to obtain more accurate approximations.
Sect. XV.— Indeterminate Problems.
117. When the conditions of a question are such that the number of unknown quantities exceeds the number of equations, that question will admit of innumerable solutions, and is therefore said to be indeterminate. Thus, if it be required to find two numbers subject to no other limitation than that their sum be 10, we have two unknown quantities x and y, and only one equation, viz. xiy = 10, which may evidently be satisfied by innumerable different values of x and y, if fractional solutions be admitted. It is, however, usual, in such questions as this, to restrict values of the numbers sought to positive integers, and therefore, in this case, we can have only these nine solutions, ar=l, 2, 3, 4, 5, 6, 7, 8,9; y = 9, 8, 7, 6, 5, 4, 3, 2, 1; which indeed may be reduced to five; for the first four become the same as the last four, by simply changing x into y, and the contrary.
118. Indeterminate problems are of different orders, according to the dimensions of the equation which is obtained after all the unknown quantities but two have been eliminated by means of the given equations. Those of the first order lead always to equations of this form, ax + by — c, where a, δ, c, denote given whole numbers, and x, y, two numbers to be found, so that both may be integers. That this condition may be fulfilled, it is necessary that the coefficients α, δ, have no common divisor which is not also a divisor of c; for if a = md and b = me, then ax + by = mdx + mey = c, and dx + ey = but d, e, x, y, are supposed to be whole numbers, therefore — is a whole number; hence ’m . m must be a divisor of c.
We proceed to illustrate the manner of resolving indeterminate equations of the first order, by some numerical examples.
Ex. 1. Given 2x + 3y = 25, to determine x and y in whole positive numbers.
25 3™
From the given equation we have x = —r—= 12 — y - 2i
. Now, since x must be a whole number, it follows that must be a whole number. Let us assume * 2» 2
= z, then y = 1 + 2z; and x — 11 - 3z, where z might be any whole number whatever, if there were no limitation as to the signs of x and y. But since these quantities are required to be positive, it is evident, from the value of y, that z must be either 0 or positive, and from the value of x, that it must be less than 4; hence z may have these three values, 0, 1, 2, 3.
If z = 0, z=l, z=2, z=3;
Then∫* = 11 ' * = 8 ’ * = 5 ’ ® = 2 ’
τhen U= 1, y = 3, y-5, y = 7.
Ex. 2. It is required to find all the possible ways in which £60 can be paid in guineas and moidores only.
Let x be the number of guineas, and y the number of moidores. Then the value of the guineas, expressed in shillings, is 21a?, and that of the moidores 27y; therefore, fτom the nature of the question, 21x + 27y = 1200, or, dividing the equation by 3, 7x + 9y — 400; hence, proceeding as before, we obtain y = 7ι>-3, x= 61 - 9v.
From the value of x, it appears that υ cannot exceed 6, and from the value of y, that it cannot be less than 1.
Hence if v= 1, 2, 3, 4, 5, 6, we have a; = 52, 43, 34, 25, 16, 7, y= 4, 11, 18, 25, 32, 39.
119. In the foregoing examples the unknown quantities x and y have each a determinate number of positive values; and this will evidently be the case as often as the proposed equation is of this form, ax + by = c. If, however, b be negative, that is, if the equation be of this form, ax - by = c, or ax = by + c, we shall have questions of a different kind, admitting each of an infinite number of solutions; these, however, may be resolved in the same manner as the preceding.
120. If an equation were proposed involving three unknown quantities, as ax + by+ cz = d, by transposition we have ax + by = d-cz, and, putting d-cz = c', ax + by = c'. From this last equation we may find values of x and y of this form, x = mr + nc, y-mr + n'c', or x = mr + n(d - cz), y = m'r + n'(d — cz); where z and τ may be taken at pleasure, except in so far as the values of x, y, z, may be required to be all positive; for from such restriction the values of z and r may be confined within certain limits to be determined from the given equation.
121. We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y 2 — a + bx + ex 2 , where x is to be found, so that y may be a rational quantity. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature.
Case 1. Let a be a square number; then, putting g 2 for a, we have y 2 = g 2 + bx + cx 2 . Suppose Jg 2 + bx + cx 2 =g + mx; then g 2 + bx + cx 2 = g 2 + 2gmx + m 2 x 2 , or bx + cx 2 = 2gmx + τn⅛ 2 , that is, b + ex = 2gm + rø 2 #; hence
⅛-δ , < i cg-bm+gm* x ~ c ~ m z >V~ +bx+cx ~ c . m 2 Case 2. Let c be a square number =g 2 ; then, putting x∕<z + bx+g 2 x 2 = m +gx, we find a + bx +g 2 x 2 = m 2 + 2mgx + g 2 x 2 , or a- ∖ -bx-m 2 + 2mgχ∙, hence we find r m2 ~ a υ - J q + bx+σ 2 x 2 - im ~ gmi ~- 9 . χ -b-2wf y ~ + +9 b-2mg Case 3. When neither a nor c is a square number, yet if the expression a + bx + cx 2 can be resolved into two simple factors, as f+gx and h + kx, the irrationality may be taken away as follows:
Assume <Ja + bx + cx 2 = V(∕ + gx) (Λ + kχ) = m(f+ gx), then (∕+ gx)(h + kx) = m 2 (f+gx) 2 , or h + kx = m 2 (f+gx); hence we find
∕m 2 - h ∏- r -. x / i--,--- - (ft ~ 9 h ') m . and in all these formulae m may be taken at pleasure.
Case 4. The expression a + bx + cx 2 may be transformed into a square as often as. it can be resolved into two parts, one of which is a complete square, and the other a product of two simple factors; for then it has this form, p 2 + qr, where p, q, and r are quantities which cont ain no power of x higher than the first. Let us assume ^Jp 2 + qr=p + mq; thus we have p 2 + qr = p 2 + 2mpq + m 2 q 2 and r = 2mp + m 2 q, and as this equation involves only the first power of x, we may by proper reduction obtain from it rational values of x and y, as in the three foregoing cases.
The application of the preceding general methods of resolution to any particular case is very easy; we shall there fore conclude with a single example.
Ex. It is required to find two square numbers whose sum is a given square number.
Let α 2 be the given square number, and x 2 , y 2 , the numbers required; then, by the question, x 2 + y 2 == a 2 , and y = <ja 2 - x 2 . This equation is evidently of such a form as to be resolvable by the method employed in case 1. Accordingly, by comparing >Ja 2 - x 2 with the general expression >Jg 2 + ‰c + cx 2 , we have g = a, b = ⅛, c— - 1, and substituting these values in the formulae, and also - n for + m, we find 2an a(n 2 - 1)
ar ~n 2 + l , ^ n 2 +l
If α = Λ 2 4-lj there results a; = 2n, y = n 2 -l, α = n 2 +1. Hence if r be an even number, the three sides of a rational right-angled triangle are r, - 1, +1. If τ be an
2*2 __ 1^∙2 _|_ odd number, they become (dividing by 2) r, - , —- ·
For example, if r = 4; 4, 4 - 1, 4 +1, or 4, 3, 5 are the sides of a right-angled triangle; if r=7, 7, 24, 25 are the sides of a right-angled triangle.
Sect. XVI.— Theorems of Expansion.
1. Binomial Theorem.
122. To demonstrate this theorem, which has for its object the expansion of (α + a;)" in the form P + Qx + Kx 2 + Hx 3 + &c., we shall first find P and Q; and then determine the other coefficients A, B, &c. in terms of P and Q.
(1.) (a + x)* = {<≈(l+ξ) }"=α∙(l+≡)^, it being assumed that the power of a product is the product of the powers of the factors, whatever be the index.
(2.) Let n be a whole number. Since
(1 +ar) 1 = 1 + a;
(1+z) 2 = 1 + 2λ + λj 2
(1 +x) 3 = 1 + 3x+ &c.; if we assume (1 + x) n ~ l = 1 + (n - 1) a; + &c., and multiply both sides by 1 + x, we shall obtain ( 1 + x}* = 1 + nx + &c.; whence our induction is complete to prove that the numerical of coefficient x is the same as the index.
(3.) Let n be a positive fraction^ · We may take
(1+ ar)⅜ = 1 + Qλ +&c.
. ’. (1+ αr)p = (l +Q# +&c.)’, or, 1 +px + &c. = 1 + çQz + &c. (Case 2.)
q =s.
(4.) If n be negative = - m α+ ^≈(1+ ≈)-=<1⅛
= ι + J+⅛ c . < caae3 ∙>
= 1 — mx + &c. by division.
Hence, generally the numerical coefficient of x is the same as the index.
To obtain A, B, &c., in terms of the first and second terms, we break up x into two parts, y, z, which enables us to write the expression 1 + x in two different ways: lsi, retaining the parts of x in connection; 2d, disseveri.ιg them. In the first form we simply multiply out, and thus exhibit a result not dependent on the properties of an index, except in so far as relates to the first and second terms. In the second form we apply the properties of an index to every term. The consequence is, that the latter form, bearing a more intimate connection with the property of an index than the former, is more determinate than the other. The comparison of the two results completes the demonstration.
I. (l+x)∙ = (l + y-∣-z) n '
= l + n(τ∕+z) +A(y+») 2 +B(y+a) 3 +&c = l + iu∕ + Aι∕ 2 +B?/ 3 +&c.
+ nz + ζ 2kyz + 3Bj∕⅜+&c.
4- &c. + &c.
II. (1 + x)* = (l+y+z)" = (l +z+y) κ
=(ι+ i, )"(1 + ⅛)
= (1 + z)" + n(l+z)"~ 1 y+A(l+ z) n ~ 2 y 2 +&c.
= l + ny +Ay 2 + &c.
+ nz+n(n — V)zy + A(n — 2i)zy 2 +&c.
+ &c. 4- &c. + &c.
Now, as these two expansions are the expansions of the same thing in the same form, the coefficients of z, zy, zy 2 , &c., must be the same in both. Comparing them, we get n = n, 2A = n(n - 1), 3B = A(n - 2) &c.
. n(n-1) p A(n-2) n(n-l)(n-2)
A= 1.2 ,B= 3 - 1.2.3 '
&c. = &c.; and finally, whatever n be, (l+ rι ∙.l^+⅛ 1 ≈ + ⅛ r ⅛⅛ + ⅛ Cor. 1. If n is a positive whole number, the series is finite, since every term after the (n + l)th will involve n - n as a factor.
Cor. 2. Since the coefficients, when the index is a whole number, are the results of simple multiplication, they are necessarily whole numbers, i.e., any such expression as —∖ is a whole number when n is such
1.2.3
Cor. 3. The sum of the numerical coefficients is 2*, for it is equal to (14-1)”, as will appear if we write 1 for x.
Cor. 4. The sum of the coefficients in the even places is equal to the sum of the coefficients in the odd. This will appear if we write - 1 in place of x.
Cor. 5. If the index is a whole number, the coefficients from the end are the same as those from the beginning; for they occur at the beginning of (x + l) n in the same positions as at the end of (1 4-a?)».
Cor. 6. The product 1.2.3.... r is sometimes expressed by the abbreviated form ∣ With this notation the coeffi-I n ci ent of af in (1 +x)" may be written^ — ~ ·
Cor. 7. The sum of the squares of the coefficients of (1 +x)∙ is the coefficient of x* in the expansion of (1 + a?) 2 ", I2n and is equal to τ¼=⅛ ·
(∣n) 2 Examples. ExΛ, (1 + ^)- 2 =1 + ^ λj + ^≡x2 + ^z3^4 λ , + &c.
= 1 — 2a? + 3x 2 - 4x 3 + &c. generally (l+aj) - ∙ = l- τ⅛r + w fo+ 1 ) 3 a _ & c and (1 - x)~ * = 1 + wa: + 3 a +
1.2
Ex. 2. Find the coefficient of x l in fa; + x 2 + 0c s 1 a? 4 + x 5 + a: 6 ) 2 .
The expression may be written -+ ’ = x 2 (l - 2x 6 + λj 12 )(1 + 2x + 3x 2 + 4x 3 + &c.)
The coefficient required is therefore that of x 5 in the last factor, viz. 6.
Ex. 3. Find the sum of 1 + 2n + 3 n ^-4.4 ⅛- 2 -
1.2 1·2.3
+ &C.
By writing 1 + 1 for 2, 1+2 for 3, &c., this series may be broken up into the sum of 1 + n + n ^ ∖ + &c.
1 · 2
and n + 2 1 ~~ l ~ + &c.
1 · 2
The latter is to ∣ 1 + —j-* + ———— + &c. ∣ .∙. the sum required is 2" + to2^~ 1 . Ex. 4. Find the sum of 1 + ∣ n + ∣ ~γ~+ &c.
2 3 1.2
Multiply by to + 1; the product is n+ ι + ⅛t⅛ + ("+ ι 1 ⅛- 1 ) + ⅛ e .
.1 J 2∙+ ’-l
. ’. the sum required is ∑—
1 n+1
Ex. 5. If x r denote the product x(x - 1)(# - 2) . . . (x - r +1) whatever be r, and a similar notation be applied to y, and (x + y), then
(x + y) r = x r + r∙r p -, 1 y 1 + +p a⅛- 8 y a + &c.
We have (1 + α)' = 1 + x l a + -p⅛ a 2 + r ^- 3 a 3 + &c.
(1 +a) y =l +y 1 a + p^ a 2 + Ac.
,∙. (1 + a)* 4s = their product.
But (1 + a)* + * = 1 + (x + y) l a + -γ 7 g- a 2 + &c.
Equating coefficients of a" in the two expressions for (l + a)* +y , and multiplying by 1 . 2 . . . to, the required result is obtained.
Ex. 6. If x and to be less than 1, then (1 + a:)“< (1 + nx).
For (1+x)" = 1+tojj + , ⅜⅛-^ 2 +. . . .
1.2
1 1 n(t -to) 9 (λ 2-n ∖ _ = l+nx- v 1 - l x 2 il —αj - &c.
Ex. 7. On the same hypothesis (1 + x) n <
Prove that (1 + x)~ n > 1 - nx exactly as in the last example.
Ex. 8. If x<l; TO>r<r+lj then (l+ar)">the sum of the first r + 1 terms of the expansion; and < the sum of the first r + 2 terms.
Ex. 9. The difference between the sums of the squares of the even coefficients of the expansion of (1 + #)", when to is an even whole number, and the sum of the squares
• J‰ υf the odd coefficients is ( - 1) 2 ∕ I n ∖ 2 .
∖∣2∕
2. Logarithmic Theorem.
123. The definition of a logarithm is precisely the same as that of an index or exponent (Art. 21) viz.— the logarithm of a product is equal to the sum of the logarithms of the factors. Such being the case, we are at liberty to employ the definition, either in the form first given, or in the algebraic form a* = y. In this last form x is called the logarithm of y to the index or base a. The base of the common or tabular logarithms is 10.
124. Before proceeding to the demonstration of the theorem by which a logarithm is expressed in the form of a series, it may be as well to illustrate the definition as applied to common logarithms.
l st, Since 1 is the logarithm of 10, we may inquire of what is ⅜ the logarithm; if we resume the form 10* = y, and write ⅜ for x, we have to inquire what is y.
Since 10* = y, 10* × 10* = y 2 .
But 10⅜ × 10* = 10* + * (def.) = 10, y 2 = 10 and y= √10 = 3.1622777, so that the number of which ⅜ is the logarithm is not a whole number, but a fraction lying between 3 and 3⅜.
In the same way, we may, but with great labour, ascertain the numbers of which any given fraction is the logarithm.
2d, The definition will evidently enable us to obtain· a large number of logarithms, when a few have become known. For example: Given log 2 = . 30103 to find log 4 and log 5.
Log 4 = log (2 × 2) = log 2 + log 2 (def.)
= 2 log 2 = . 60206;
Log 5 = log j = log 10 -log 2
= 1-log 2 = . 69897.
If in addition to log 2, log 3 be known, we can find a vast number of others. For example: Given-
Log 3 =. 47712 to find log 6 and log 72.
Log 6 = log 2 × 3 = log 2 +log 3 = . 77815,
Log 72 = log 8×9 = 3 log 2 + 2 log 3 = 1 . 85733.
125. To expand log (1 +x) in terms of x.
Since log 1=0; the expansion must commence with the first power of x, the coefficient of which will depend on the radix or base. This coefficient we shall determine afterwards for the common logarithms. In the meantime we shall denote it by A.
Let then log ( 1 + a?) = Ax + Bx 2 + Cx 3 + &c.
Put y + z for x ; then
I. Log (l + y + z) = A(j∕ + z) + B(y+z) 2 +Ac∙
-Ay + By 2 + C⅛ 3 +Ac, + Az +2B2/z + 3Cÿ 2 z+&c.
+ &c. + &c.
II. Log (I + y +z) = log (1 + y) (1 +
=log(l + 2∕)+log(l + j^) =Aj/+By 2 +&c.
+—
1+y
+ &c.
=Ay+By 2 +<fcc,
+ Az (1 — y + y 2 -&c.)
+ &c.
Equating coefficients of z, yz, y 2 z, <tc., in the two expansions, there results
A = A, 2B= -A, 3U = - A log(l +i r)-A(x-≤ + ≤-*c)
126. Cor. If x≈a-l, where a is the base of the system, v. e have 1 = A ∣ a - 1 - ∣ (a - 1) 2 + &c. ∣
This expansion of log (l+a ’) is not convergent, i.e., the terms do not diminish as we advance, but the contrary, when x is any whole number greater than 1. We can, however, readily obtain from it a converging series for the difference between the logarithms of the consecutive numbers.
For log (1 + x) = a(x- y + <tc.J
log (1 —ar) = A^-ar-γ-&ο.) .∙. log (1 + x) - log (1 - x) = 2A^x + + y + & c ·)
Now log (1 + x) - log (1 - x) = log ,
1 — X
14" ic and x has to be found, so that j—shall be the quotient of consecutive numbers = ·
u
This gives ic =2⅛i∙
and i og l±ï_ 2A |_l_ + |_l_ j+&c .J
127. To apply this formula to the calculation of common logarithms, we will commence by finding from it a few logarithms of the system for which A= 1. In this system
1. Ifu=l, log 2=2{| + |.i+&c.]·
= . 693, 147, 2.
2. If u = 4, log 5 = log 4 + 2∣∣ + ∣∙ ^+&c. ∣
= 1.609, 437, 9 .
Hence log 10 = log 2 + log 5
= 2.302, 585, 1.
This system, for which A = 1 is the so-called Napierian system, which assumes no base, but defines a logarithm to be such that the increment of the number shall be the product of the number by the increment of the logarithm. In this system the number of which the logarithm is 1 is 2.718 .. .. and is generally designated by the letter e.
To pass from Napierian logarithms to common, we observe that if e , = 10 y = nj ae is the logarithm of n to the base e, and y to the base 10. Now, taking the Napierian logarithm of each side of this equation, we obtain x = y Nap. log 10.
q j , ® x_ y ~ Nap. log 10 - 2.302, 585, 1 ’
=x× . 4342944819
= a; × . 4343 very nearly.
This multiplier, which was previously denoted by A, is called the modulus of the common system of logarithms. A celebrated calculator of the last century, Mr A. Sharp, found it to be
•43429448190325182765112891891660508229439700 5803666566114454.
For further details on the construction and use of logarithmic tables, the reader is referred to the Article od Logarithms.
3. Exponential Theorem.
128. It is now required to expand a* in terms of x.
1. Write 1 + a - 1 for a, and apply the binomial theorem; the result is
{l + (α-l)}-≈=i +α (α -l) + ⅛±)(α -l)2 + ife c.
Hero the only term which does not contain x is 1; and the coefficient of x being traced through the different terms, is easily seen to be
α-l -∣(α-l) 2 + ∣(α-l) 3 -<fec,
Thus will seem (Art. 126) to be the reciprocal of the modulus ot the system of logarithms whose base is α: call it r. We have now to determine B, C, &c., in terms of r, from the form of expansion a^ = l + rx + Bz 2 + Car 3 + &c.
Write y + 2 is place of χ∙ i then
I. a»+* = l+r(ÿ+«) +Btø+2) 2 4-&c.
= l + ry +By 2 +&c.
+ rz +2BT/Z + &C.
+ &c.
IL aP +, = av ×a ’ =(l+ry+Bi/ 2 +&c.)
×(1 + rz +Bz 2 +&c.)
= 1 +ry + Bτ∕ 2 +&c.
+rz + r 2 yz+rBy 2 z +&c.
+ &c.
Equating coefficients of 2, yz, y i z, &c., in I. and 11., we get r = r, 2B = r 2 , 3C = rB, &c.
b "⅛ σ -E⅛3-
and o -.l +ra + 2⅛ + r r⅛L + , fe
129. Now, since e is such (Art. 127) that e - 1 - I (e - 1) 2 + &c. = 1, and r = a - 1 — ∣ (α - 1) 2 + &c., whatever a be, it follows that when e takes the place of α, τ becomes 1.
e '- 1 +≈ + ⅛ + i43÷* 0 ∙
and putting x≈ 1, e= 1 + 1 + + &c.
= 2.718281828459045...
• 7*2
Again e r =l+r + —+ &c.
but since, a" = 1 + rx + + &c.
1.2
r 2 we have a = 1 + r 4- —- + &c.
1.2
= e r
From this equation we have r = Nap. log α, a result obtained before.
130. We may approximate directly to the value of τ when α = 10, thus
∕1∖* κ 1 r⅛ 2 p
U -<>≈-' = l- κr+ --&O.
so that the coefficient of xinQ)'⅛ -r.
Now = ^1 + ^- 1^ , whence (Art. 128) the coeffi cient of x in is
·· '→∣H) ’ ÷∣HA*
9 1∕9∖2 1∕9∖3 p - IO + 2∖1O∕ + 3∖lθ) +&C ·
= 2.302...
Additional Examples. Ex. 1. To find the value to which (1 approaches as m becomes larger and larger.
By the binomial theorem Λ ®λ* 1 ae τrfm —1)∕ a ∖ 2 .
0 + ™) = 1+ ^÷ K2 +&C ·
ι-l Λ-lVι-A∖ =i +a . + _^*2-∣Λ 3 <^+&c.
≡ 1 ÷*÷⅛⅛÷^
+—+ ¾ + <fec.
m m i bo that ( 1 + ~) approaches the value 1 + ® + —- + &c. or e*, as m becomes larger and larger.
Ex. 2. + approaches to e* as m increases.
Ex. 3. n∙-ψ(n-l)∙ + ⅛(n-2)∙-⅛c.
1 ι. z
= 1.2.. .n when n is a whole number.
, t 2 χ 2 e nz = 1 + nx + —— &c.
1.2 e (->> = l + (n - l> + cw ~ 1 ^ + <fec.
Now I. (e β - 1)" = e"* - ne<- 1 >* + e <-*> - &c.
But e*-l = l+ ar + γ-% + &c. - 1 = x + -γ-% + &c.
.∙. II. (e* - 1)" = x n +px'+ 1 + &c.
Equating coefficients of x n in I. and II., we get ¾" j _ n(w-l)* n(⅞-l) (¾-2)" _ & 1
1.2...n 1.2...»» 1.2 1.2...n which is the required result.
Cor. When r is less than n, n r -n[n-l) r + - 2) r - &c. = 0 .
Ex. 4. The logarithm of a number to the base a n is a mean proportional between its logarithms to the bases a and α" 2 .
Jf «, y, z, are the logarithms to the three bases in order, we have
(α∙)' = α* = (α∙ 2 )*
. ’ . nx = y nx = n 2 z consequently x 2 = yz. Ex. 5. e* >1 + x, whatever be x.
If x be positive, or if it be negative and less than unity, the expansion may be thrown into the form e *= 1+ir+ ]⅞ (1+ l) + * fec ∙ every term of which after 1 + x is positive.
Ex. 6. e">^⅛^∙
For e κ > 1 +x .’. e∙ >1 +-
n . ’ . en n >(l+n) n
Hence e.l 1 >2 1 e.2 2 >3 2 en">(l + n) r and, by multiplication, e n ∣n >(1 + n)*. Ex. 7. If n be a whole number > e, n* +1 > (n +1)*. By the demonstration of Ex. 6, en">(l +n)*
But n>e .∙. n"+ 1 >(l+n)∙.
Ex. 8. If n∙ +1 = (n +1 )", then n > 1 < e.
For n is evidently >1. If then we suppose n>l<2, = i+i+ ⅛G + ⅛) +4c ∙
= 2 + a series of positive terms by the hypothesis n< 2, which is absurd, .∙. n>2.
Taking the Napierian logarithms of each side of the equation n = + , we get lθgn-l-l + i-fe, --l(⅛p∙ <1
.. n<e, Ex. 9. Nap. log x>l --<x-l.
1 11/ 1∖ 2
Because log x = .- log -=1-- + -(1--J + &°·
.∙. log x> ∖ - - · t5 x
And because when x > 1, x< 1 + (x -1) + (x -1) 2 + &c.
<e- 1
.∙. log »<«-1; when x< 1, log x = log (1 - 1 - x)
= - 11 - ® + | (1 -aj) 2 + <fcc. I
= ®-l-|(l-®) 2 -&c.
<x-l. Ex. 10. Nap. log x approaches to 2* (a^ 5, - 1) as n increases.
Sect. XVII.— Continued Fractions.
131. Every quantity which admits of being expressed by a common fraction may also be expressed in the form of what is called a continued fraction. The nature of such fractions will be easily understood by the following example:
314159
Let the fraction be .- azvvv c , or, which is the same, 3 + lUVVvv 1 ⅛⅛⅛∙ Since 100000 = 7 × 14159 + 887, therefore 100000
14159 14159 1 , 314159 _ „
100000 - 7 × 14159+887 , 887 100000
7+ 14159
1
14159
887
By treating the fraction in the same way, and continuing the process, we readily obtain
3141591 1
100000 + 7 + ~. 1 1 lδ + l + - 1 1 25 ÷i + Ul 7 4
By an operation in all respects the same as has been just now performed, may any proper fraction whatever be reduced to the form
— + 1 + 1
and it is then called a continued fraction.
132. When the root of any equation is found by the method explained in Art. 116, the value of the unknown quantity is evidently expressed by a continued fraction. For if X be the root sought, we have x = a + -, y≈.b + ~, y'≈b' + -ζ, y" = b" + -‰ &c. where a, b, b', b", &c. denote the whole numbers, which are next less than the true values of x, y, y', y", &c. If, therefore, in the value of x we substitute b + Λ for y, it becomes y
1
x = a- ∖ τ-
δ +4∙
Again, if in this second value of x we substitute b' + for y', it become
1
a ∙ = α + τ 1
δ + i -
δ '÷7'∙
And so on continually,
133. It is easy to see in what manner the inverse of the preceding operation is to be performed, or a continued fraction reduced to a common fraction.
The fractions which result from omitting portions of a continued fraction are termed the convergents to that frac-
— 1 1
tion. Thus, if the fraction be a, -I— , 1: — is the first ¾+- α l convergent, — c ⅛-- the second, &c.
134. The principal practical application of the properties of continued fractions is to approximate to the value of a given fraction. The proposition on which this application depends is the following :—
Vo fraction in terms equally low can give so good an approximation to the vahιe of a fraction as a convergent to the continued fraction which expresses it does.
To demonstrate this proposition, it is requisite to establish three preliminary propositions, which we shall do very briefly.
135. (1.) If denote the nth convergent, or the reduced fraction which results from stopping at α n , and reducing, then p n ± l = a n+1 p n +p n ~ 1 , g, + ι = a n+1 q n + q^ 1 .
Since no denominator can be multiplied by itself, the reduced fraction must give p n = a n A, + B.
Now ∕> n+1 is obtained by writing a n + —— for a n , and . <⅛+ι
reducing, P»+i = ⅞ +1 (α,A + B) + A = a κ+1 p n + A; i,e., the multiplier of any a is the previous p, and the other term is the multiplier of a in the previous convergent, hence the proposition.
136. (2.) p tl+l q n - q n+1 p n = ( - 1)∙.
This is at once obtained by eliminating a tt+1 from the two equations of last article.
137. (3.) The successive convergents are alternately greater and less than the complete fraction, and each convergent approaches nearer in value to it than the preceding.
If A denote the complete denominator α β+1 + &c. ; u the complete fraction; then ¾ f =A <⅛*~ι ; and by s ∏btract- - ft -2*÷ 2«—ι
ing successively and P — from w in this form, it will be 2« 2»-i seen at once that the results have different signs, and that the latter difference is the larger.
t 138. We are now able to prove the proposition enunciated.
Let - bea fraction nearer to u than: then since 2 2»+i the convergents are alternately too great and too small; ^ n - + - 1 > ~ i ~ i must be in order of magnitude.
2.+1 2 2« o
. ∙. If the first be the greatest,
?«+l P* >P Pn 2«+i 2» 2 2« ’
Reducing and applying Prop. 2, there results q>q n + ∖ *
Similarly by inverting the fractions, it may be proved that p >p n+1 . Ex. 1. To determine when a transit of Venus may be expected.
The relative sidereal periods of Venus and the earth are 224,700 days and 365,256 days. The continued fraction which expresses the quotient of these numbers is
H+ i+1 1 1-4-1 ,
8 " + ^ + &C 235
The fifth convergent is —; the sixth — ·
13 382
On account of the smallness of the formeris a very close approximation, i.e., 8 years and 13 sidereal periods of Venus are very nearly equal.
In consequence of this, a transit occurs after one period of 8 years, and then again not till after 235 years have been completed.
The last pair of transits at the descending node occurred in 1769, 1777; and at the ascending node in 1639, 1647.
The next pair will accordingly occur at the latter node in 1874 and 1882. The days of transit will be December 8 and December 6, respectively.
Ex. 2. To find the periods of probable recurrence of eclipses of the sun.
An eclipse of the sun will occur whenever the place of the new moon is within about 13° of the line of nodes. Now, the interval between two new moons is 29 - 5306 days; and the mean synodic period of the earth and thé line of nodes is 346 ’ 6196 days. The proportion of the latter of these numbers to the former, reduced to a con- 47 223 tιnued fraction, gives as convergents —, —, &c.
Hence, after 47 lunar months, things have come nearly to their original position, and after 223 lunar months, very nearly. This latter period, termed the saros, has been known from the remotest antiquity. It enabled the Chaldean shepherds to predict the return of eclipses. It amounts to 18 years and 10 or 11 days. Thus, there was a total eclipse on the 18th July 1860; adding 18 years 11 days, we get for an eclipse 29th July 1878. If we add 47 lunations or 1388 days, we get 6th March 1864, on which day there was an eclipse.
This period of 1388 days, multiplied by 5, makes exactly 19 years—a period which is designated as the cycle of Meton, giving ecEpses which occur on the same day of the month. Thus, echpses happened 18th July 1841 and 18th July 1860, and another will happen 18th July 1879.
Ex. 3. The fraction given (Art. 131) represents the ratio of the circumference of a circle to its diameter. By taking the first two terms we have 7γ=3 + ∣ = 2 τ 2 nearly: and this is the proportion which was found by Archimedes.
Again, by taking the first three terms, we have + 7 + ⅛ + 106 106 ’
15
which is nearer the truth than the former.
And, by taking the first four terms, we have , 1 i _ae5
5 ⅛ + 113 which is the proportion assigned by Metius.
Ex. 4. The mean tropical year consists of 365·2422642 days.
The fraction ’ 2422642, reduced to a continued fraction, gives as successive convergents
1 7 8 39 47 321 v r
4 ’ 29 ’ 33 ’ 161 ’ 194 ’ 1325 ’ &C ‘
To make the civil year approximate to the tropical, 1 leap year in 4 (the Julian Calendar) serves but imperfectly. 7 leap years in 29 would be inconvenient. The Gregorian Calendar, now in use, is based on combining the fractions and -, by doubling the numerator and denominator of the former, and trebling those of the latter, and adding 97 them respectively. The resulting fraction is —, giving 97 leap years in 400 years, instead of 100 as the Julian does. This diminution of 3 leap years in 400 years is produced periodically, by causing years which indicate the completion of centuries not to be leap years unless the number of centuries is divisible by 4. Thus, 1900 will not be a leap year.
Sect. XVIII.— Permutations, Combinations, and Probabilities.
139. Hitherto we have supposed the letters of the alphabet, a, b, c, &c., to stand for arithmetical quantities of some kind or other. Now we have to employ them, as in geometry, to represent magnitudes or objects, such as pens, pencils, &c., and to investigate the numbers of different ways in which a given set of them can be grouped according to a certain law.
Permutations are their arrangements in a line, reference being had to the order of sequence; thus ab and ba are the two permutations of a and b; combinations are their arrangements in groups, without reference to the order of sequence; thus abc is a combination involving a, b, and c; and bac is the same combination, both consisting simply of a, b, and c grouped together.
Prop. 1. To find the number of permutations of n things (1), two and two (2), three and three, &c., together.
Set aside a , and lay down the other things in a line; place a before each of them in succession, and you obtain ab, ac, ad, &c., i.e., n - 1 arrangements, each containing two things, with a first.
In the same way you can form n - 1 arrangements, each containing two things, with b first. The same is true of each of the other letters, and as there are n of them, the total number of arrangements of the n things, two together, is n(n -1).
Again, lay aside a, and group the other n - 1 things, two together; as we have just shown, there are(n - 1) (n — 2) such groups. Place a before each of them, and there will be formed (n-l)(n-2) arrangements, each containing three things, with a first; and there can be no more arrangements with a first.
Treat b, c, &c., in the same manner, and it will appear that there are (n-l)(n-2) groups of things, three together, in which every separate thing in succession stands first. Hence, the total number of arrangements, three and three, is n(n - 1) (n - 2).
By proceeding in the same manner we shall find the total number of permutations of n things, r together, to be n(n- 1) (n-r + 1).
Cor. The number of permutations of n things, all together, is n(n - 1). ...3.2.1.
Prop. 2. To find the number of combinations of n things, r together.
Let x be the number required.
Take any one of the x groups of r things. The number of permutations which can be formed with it will be (Prop. 1. Cor.) r(r - 1) 1, or 1.2 .. . r.
Now, since each of the x groups is different from all the others, if we treat each of the x groups separately in this way, we shall form 1.2 r × x permutations, all different. Also, since the x groups contain every possible combination of the n things, r together, we shall thus have formed all the permutations which can be formed; and consequently (Prop. 1) the number is n(n - 1)... (n - r + 1) .
n(n- l)....(n-r + l) x ~ 1.2....r
Prop. 3. To find the number of combinations which can be formed of n sets of things, containing respectively r, s, t, &c. things, by taking one from each set to form a combination.
1. Let there be two sets, one containing r and the other s things.
Any one (say a ) of the r things may be placed successively with each of the s things, and thus form s groups, in each of which a appears. The same is true of b, c, &c.; i.e., each of the r things gives rise to s groups, .’. the number required is rs.
2. Any one of the t things may be placed in succession with each of the groups of two things referred to in Case 1, so that every one of the t things will give rise to rs combinations of three things; .∙. the number required is rst. The same may be indefinitely extended.
140. The first and most obvious application of the theory of combinations is to the doctrine of chances. As, however, this application will form the subject of a separate article, all that is requisite for us now to do is to indicate the connecting link between the two subjects.
If we agree to designate certainty by unity, then the chance of an event happening, when it is less than certainty, will be designated by a proper fraction. Thus, if the average number of wet days and of dry is the same, the chance of any day named at random turning out wet will be represented by the fraction -; that is, if the number of days under consideration be 100, the chance is 50 number of wet days . ,. ,
, or ——- r - ,- · Chance is accordingly de- 100 , total number of days j n . . , , number of favourable events fined by the fraction —-—;; 7 τ ·
total number of events.
The only proposition by which chances are combined that we shall offer is this.
If there are two events, and the probability of one of them happening to be , and of the other; then the probability that both will happen is ·
For a and c may be taken to represent the favourable events respectively, and be combined (Art. 139, Prop. 3) so as to give ac ways in which they may happen together. And in the same way b and d may be combined to give the total number of events.
Ex. A bag contains 3 white and 4 black balls. Find the chance of drawing (1) two white balls; (2) a white and a black; (3) one white at least, when two balls are drawn.
The chance of drawing two white balls is the fraction, Number of combinations of 3 things, 2 together Number of combinations of 7 things, 2 together.
3.2
1.2 = 1
7.6 7
1.2
The chance of drawing a white and a black is (Art. 139, Prop 3),
3.4 _4
7÷6 ~7 ’
1.2
To find the chance of drawing at least a white ball, we may remark that it is the same as the chance of not drawing two black balls, i.e., certainty—the chance of drawing two black balls.
Now the chance of drawing two black balls is
4.3
1.2 _ 2
7.6 “7
1.2
.∙. the chance of drawing at least one white ball is 1-≡=Λ.
7 7
Sect. XIX.— On Series in General; their Summation and Convergence.
141. Certain series, from their very appearance, indicate that they are really the sums or differences of two other scries. From this circumstance their sum may frequently be determined, as in the following examples :—
Aer. 1. 1 .2 + 2.3 + n(n+l)
τ i 11 1
12 n
-111 1 x ~ 1 -⅛ - 2 + 3 + «+Ί
.∙. by subtraction, n+l ∖1 2∕ τ ∖2 3√ ∖n n+l) 1 ι 1 ι 1 “1.2 + 2.3 + * * * n(n+l)
and the sum is 1 1 -- , that is, .
n+l ’ ’ n + l
142. The sum of a series may often be easily found by the method of increments or differences, and this method is especially adapted to the summation of integral series, such as the squares of the natural numbers. We shall exhibit one or two illustrations only.
If we write S b = n(n +1), we have S< + ,≡(n+l)(n+2), .∙. S, +i -S, = 2( b + 1).
Hence conversely, and dividing by 2; if
S n+1 - S n = n+1, then will S n = n (n + 1 ).
Similarly, if S m+1 -S,≡(λ+1)(λ + 2) . . . (n+r-l), thon will g - i ,÷+l) · · O+r-l)
r
This last conclusion, of course, assumes that S n is 0 when n is 0. If it be otherwise, some numerical constant, easy of determination, will have to be added.
Ex. 2. l 2 + 2 2 + 3 2 + . . .n 2 .
Here S m+1 - S b = (n + 1) 2 = (n +1) (n + 2) - (n + 1);
. Q n (n + l)(n + 2) n(n + l) n(n+l)(2n + l)
’ * 3 2 “6
Ex. 3. l l f2 4 + 3 4 + . . . n 4 .
S, w -S B = (n + l)<
Let
(n+1) 4 = (n +1) (n+2) (n + 3) (n+4)+.A(n +1)(n + 2) + 3)
+ B(n +1) (n + 2) + C(n +1) .
Dividing by n + l, and proceeding as in Art. 33, we get A= -6, B = 7, C= -1.
S n = ∣n (n + 1) (n + 2) (n + 3) (n + 4) -
3 7 1
2?i(n +1) (n + 2) (n+3) + 4 n(n +1) (n+2) - g« (n +1) _ΐί&+ΐ).(2»±υρ Β(Β+1) _ 1 ρ
On the Convergency and Divergency of Infinite Series.
143. Def. If the limit to which the sum of a series approaches, as the number of terms increases, is finite, the series is a converging series; if otherwise, diverging. For example, the sum of the series 1 + r + 1 a + . . . to n 1 — r n terms is -—— (Art. 52), which, when r is less than 1, approaches to , in which case the scries is a converging series.
Prop. 1. It is necessary and sufficient for convergency that the remaining terms after the nth have zero for their limit, both individually and collectively, as n increases.
It is obviously necessary and sufficient for convergency that the sum of the series after the nth term shall have 0 as its limit; and consequently, when all the terms of the series are positive, the same must be true of each individual term. But when the terms are alternately positive and negative, though it is necessary for convergency that the sum of the consecutive terms with their proper signe should have 0 as its limit, this is not sufficient; for, were it so, the sum to n terms would depend on whether n is even or odd.
Ex. 1. l+∣ + ^+ . · · is not a converging series; for Λ o although each term after the nth tends to 0 as its limit— the sum of n terms aftér the nth, viz., —-4 + . . .
n+l n+2
, which is greater than -÷- + · . . ~ to n terms, i.e., greater than ∣, does not tend to 0 as its limit.
Ex. 2. 1 +1 + + 1 ~ g 3 + . . . the expression for e (Art. 129), is convergent.
The sum of the terms after the nth is f( 1+ ⅛i + * c ∙) <⅛(1 ÷⅛∙∙∙)
- 1 1 - 1 < k ' 1 _ 1 ' i ^(n-l) ∣ n-l n the limit of which as n increases is 0.
Prop. 2. If the limit of the nth term is 0, and the terms continually diminish; then when the signs of the terms are alternately + and - , the series is convergent.
Let u 1 -u 2 + u 3 - dre., be the series; the terms after the nth ( + or — ) make up the series of positive groups
(n β+1 - n. +2 ) + (n. +3 - u n+i ) + &c.
But these terms may also be written n, +1 - (n <+2 - n w+s ) - &c., which, since the whole group is positive, must be less than u, +1 , the only positive term in it. But w. +l has 0 for its limit, therefore the series is convergent.
Ex. 3. 1 — è +1 ~ · . · is convergent, for the sum of 2 o the series after the nth term is less than —, which has 0 as its limit.
Prop. 3. If the terms of the series are all positive, and the limit of the nth term is 0; then if the limit of the quotient of the (n+ l)th term by the nth be less than 1, the series is convergent; but if the limit be greater than 1, the sum is divergent.
1. Let k be the greatest value of , after a certain value of n, and k < 1; then,
%«+i + ‰+2 + · · · <¼ι(l+H½ 2 +∙ . .), which has 0 for its limit. Hence the series is convergent (Prop. 1).
2. Let k, the least value of —after a certain finite value of n , be greater than 1; then n n+1 = or >ku n w«+2 = 0r >k 2 u n &c. &c.
.∙. n n+1 + w n+2 + &c. = or >kufl +I' + F + &c.), which is infinite. Hence the series is divergent.
Prop. 4. If be less than 1; then the two series u 1 ÷ n 2 ÷ n 3 ÷ n 4 + . . . . (11
u 1 + 2n a + 4n 4 + 8n 8 + . . . . (2)
are both convergent, or both divergent together.
Series (2) + n 1 = 2(n 1 + n 2 + 2n 4 + 4n 8 + . . .), which is equal to or less than the following, term by term, viz. :—
2{w 1 + u a + (u 9 + w 4 ) + (u b + u 6 + u 7 + u a ) + . . .} , i.e., twice series (l).
Hence if the one series be convergent, the other will be also convergent; and if series (2) be divergent, series (1) is also divergent,
Again, series (2) is equal to or greater than the following, term by term, w 1 + (⅝ + ⅝) + (w 4 + ⅝ + ⅝ + m 7)+ · · · which is series (l).
Hence if series (l) be divergent, series (2) is also divergent.
Ex. 1. The series -⅞ r + + ,, 1 + . . . is convergent if l r 2 r o r r>l, but divergent if r= or < 1.
The two series (1) and (2) now become
1<∙ + 2 r + 3 r + ' ’ ’
12 4
F + F + F + · · ·
the latter of which is the geometric series
1 + g7→ + 4r z 1 + · · · »
which is convergent or divergent according as r> 1 or the contrary. Hence the same is true of the given series.
Ex. 2. The binomial series l+nx + <fcc., is convergent when x< 1, divergent when x>l. Ex. 3. To find when the binomial series 1 - n + —
1 · 2 &c., is convergent.
Let n< 1; the (r +1) term may be written n τ-n-^ ∖ , r—n—2 r r-1 r— 2
<"(ι+ ⅛Γ (1+ ⅛Γ (Art · 122 ∙ Ex · 7 >
<X÷F(⅛F∙∙
n
< r »+l ’
whence (Prop. 4, Ex. 1) the series is convergent. Similarly in other cases. (p. k.)
