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ELEMENTS OF ALGEBRA,
LEONARD EULER,
TRANSLATED FKOM THE FRENCH;
WITH THE
NOTES OF M. BERNOULLI, &c.
AND THE
ADDITIONS OF M. DE LA GRANGE.
FOURTH EDITION,
CAREFULLY REVISED AND CORRECTED.
BY THE REV. JOHN HEWLETT, B.D. F.A.S. &c.
TO WHICH IS PREFIXED,
^ i^Tcmoir of ^z %x\z aniy (J^j^aractcr of iEuler,
BY THE LATE
FRANCIS HORNER, ESQ., M. P.
LONDON :
PRINTED FOR LONGMAN, REES, ORME, AND CO.
PATERNOSTER-ROW.
1828.
M; (i/jai
Sc. ,ices
THIRD EDITION. 1%^S
Having prefixed my name to the present edition of Euler's Algebra, it may be proper to give some account of the Translation ; which I shall do with the greater pleasure, because it furnishes a fa- vorable opportunity of associating my own labors, with those of my distinguished pupil, and most excellent friend, the late Francis Horner, M. P.
When first placed under my tuition, at the cri- tical and interesting age of seventeen, he soon discovered uncommon powers of intellect, and the most ardent thirst for knowledge, united with a docility of temper, and a sweetness of disposition, which rendered instruction, indeed, a " delightful task." His diligence and attention were such, as to require the frequent interposition of some ra- tional amusement, in order to prevent the in- tenseness of his apphcation from injuring a con- stitution, which, though not delicate, had never been robust.
Perceiving that the natural tendency of his mind led to the exercise of reason, rather than to the indulgence of fancy ; — that he was particularly interested in discussing the merits of some specious theory, in exposing fallacies, and in forming legi- timate inductions, from any premises, that were
a 2
459399
IV ADVERTISEMENT. \
supposed to rest on the basis of truth ; but findin) also, that, from imitation and habit, he had bee led to think too highly of those metaphysical speculations, which abound in terms to which we annex no distinct ideas, and which often require the admission of principles, that are either unintel- ligible, or incapable of proof; I recommended to his notice Euler's Algebra, as affording an ad- mirable exercise of his reasoning powers, and the best means of cultivating that talent for analysis, close investigation, and logical inference, which he possessed at an early period, and which he after- wards displayed in so eminent a degree. At the same time, I was of opinion, that to translate a part of that excellent work from the French into English, when he wished to vary his studies, would improve his knowledge of both languages, and be the best introduction for him to the mathematics.
He was soon delighted with this occasional em- ployment, which seemed to supply his mind with food, that was both solid and nutricious ; and he generally produced, two or three times a week, as much as I could find time to revise and correct. In the course of the first twelvemonth, he had translated so large a portion of the two volumes, that it was determined to complete the whole, and to publish it for the benefit of English students : but he returned to Scotland before the manuscript was ready for the press ; and, therefore, the labor of editing it necessarily devolved on me.
I wished to give this short history of the Trans-
1^*"'.
M; tiiomatical
My
THIRD EDITION. / r! <' ^^
Having prefixed my name to the present edition of Elder's Algebra, it may be proper to give some account of the Translation ; which I shall do with the greater pleasure, because it furnishes a fa- vorable opportimity of associating my own labors, with those of my distinguished pupil, and most excellent friend, the late Francis Horner, M. P. When first placed under my tuition, at the cri- tical and interesting age of seventeen, he soon discovered uncommon powers of intellect, and the most ardent thirst for knowledge, united with a docility of temper, and a sweetness of disposition, which rendered instruction, indeed, a " delightful task." His ddigence and attention were such, as ; to require the frequent interposition of some ra- il tional amusement, in order to prevent the in- tenseness of his application from injuring a con- stitution, which, though not delicate, had never been robust.
Perceiving that the natural tendency of his
> mind led to the exercise of reason, rather than to
the indulgence of fancy ; — that he was particularly
^ interested in discussing the merits of some specious
;^ theory, in exposing fallacies, and in forming legi-
l timate inductions, from any premises, that were
^ a 2
supposed to rest on tlie basis of Iriith ; but findin also, that, from imitation and liabit, he had bee led to think too higlily of tliose metaphysical speculations, which abound in terms to which we annex no distinct ideas, and which often require the admission of principles, that arc either unintel- ligible, or incapable of proof; I recommended to his notice Euler's Algebra, as affording an ad- mirable exercise of his reasoning powers, and the best means of cultivating that talent for analysis, close investigation, and logical inference, which he possessed at an early period, and which he after- wards displayed in so eminent a degree. At the same time, I was of opinion, that to translate a part of that excellent work from the French into English, when he wished to vary his studies, would improve his knowledge of both languages, and be the best introduction for him to the mathematics.
He was soon delighted with this occasional em- ployment, which seemed to supply his mind with food, thai was both solid and nutricious ; and he generally produced, two or three times a week, as niuch as I could find time to revise and correct. In the course of the first twelvemonth, he had translated so large a portion of the two volumes, that it was determined to complete the whole, and to publish it for the benefit of English students : but he returned to Scotland before the manuscript was ready for the press ; and, therefore, the labor of editing it necessarily devolved on me.
X wished to give this short history of the Trans-
ADVERTISEMENT. V
lation at first, without any eulogium on his cha- racter and talents, while living, of course ; but he modestly, though, at the same time, resolutely opposed it, saying that whatever merit or emolu- ment might be attached to the work, it belonged to me. The same proposal was made to him, on publishing the second edition * ; but he still persisted in his former determination.
From the pleasure and instruction which he re- ceived from Euler's Algebra, it was natural for him to wish to know something more of the life and character of that profound mathematician. Having therefore in some measure satisfied his curiosity, and collected the necessary materials, by consulting the ordinary sources of information, I advised him, by way of literary exercise, to draw up a biographical Memoir on the subject. He readily complied with my wishes ; and this may be considered as one of his earliest productions. Its merits would do credit, in my opinion, to any writer ; and therefore in appreciating them, the reader will not deem any apology necessary on account of the author's youth.
I have been led into this short detail of circum- stances, first, because 1 disdain the contemptible vanity of shining in what may be thought bor- rowed plumes, and because I feel a melancholy pleasure in speaking of my highly valued, and
* The care of correcting the press for this edition was en- trusted to Mr. P. Barlow, being engaged myself, at that lime, in the laborious employment of editing the Bible.
MEMOIR
OF THE
LIFE AND CHARACTER OF EULER,
BY THE LATE
FRANCIS HORNER, ESQ., M. P.
Leonard Euler was the son of a clergyman in the neighbourhood of Basil, and was born on the 15th of April, I707. His natural turn for mathe- matics soon appeared, from the eagerness and fa- cility with which he became master of the elements under the instructions of his father, by whom he was sent to the university of Basil at an early age. There, his abilities and his application were so distinguished, that he attracted the particular no- tice of John Bernoulli. That excellent mathe- matician seemed to look forward to the youth's future achievements in science, while his own kind care strengthened the powers by which they were to be accomplished. In order to superintend his studies, which far outstripped the usual routine of the public lecture, he gave him a private lesson regularly once a week ; when they conversed to- gether on the acquisitions, which the pupil had been making since their last interview, considered whatever difficulties might have occurred in his
EULER. IX
progress, and arranged the reading and exercises for the ensuing week.
Under such eminent advantages, the capacity of Euler did not fail to make rapid improvements ; and in his seventeenth year, the degree of Master of Arts was conferred on him. On this occasion, he received high applause for his probationary discourse, the subject of which was a comparison between the Cartesian and Newtonian systemg.
His father, having all along intended him for his successor, enjoined him new to relinquish his mathematical studies, and to prepare himself by those of theology, and general erudition, for the ministerial functions. After some time, however, had been consumed, this plan was given up. The father, himself a man of learning and liberality, ^abandoned his own views for those, to which the inclination and talents of his son were of them- selves so powerfully directed ; persuaded, that in thwarting the propensities of genius, there is a sort of impiety against nature, and that there would be real injustice to mankind in smothering those abilities, which were evidently destined to extend the boundaries of science. Leonard was permitted, therefore, to resume his favorite pur- suits;, and, at the age of nineteen, transmitting two dissertations to the Academy of Sciences at Paris, one on the masting of ships, and the other on the philosophy of sound, he commenced that splendid career, which continued, for so long a period, the admiration and the glory of Europe.
About the same time, he stood candidate for a
X EULER.
vacant professorship in the university of Basil ; but having lost the election, he resolved, in con- sequence of this disappointment, to leave his na- tive country ; and in 1727 he set out for Peters- burg, where his friends, the young Bernoullis, had settled about two years before, and where he flattered himself with prospects of literary success under the patronage of Catherine I. Those pro- spects, however, were not immediately realised ; nor was it till after he had been frequently and long disappointed, that he obtained any prefer- ment. His first appointment appears to have been to the chair of natural philosophy ; and when Daniel Bernoulli removed from Petersburg, Euler succeeded him as professor of mathematics.
In this situation he remained for several years, engaged in the most laborious researches, enrich- ing the academical collections of the continent with papers of the highest value, and producing almost daily improvements in the various branches of physical, and, more particularly, analytical science. In 1741, he complied with a very press- ing invitation from Frederic the Great, and re- sided at Berlin till I766. Throughout this pe- riod, he continued the same literary labors, di- rected by the same wonderful sagacity and com- prehension of intellect. As he advanced with his own discoveries and inventions, the field of know- ledge seemed to widen before his view, and new subjects still multiplied on him for further specula- tion. The toils of intense study, with him, seemed only to invigorate his future exertions. Nor did
EULER. XI
the energies of Euler's mind give way, even when the organs of the body were overpowered : for in the year 1735, having completed, in three days, certain astronomical calculations, which the aca- demy called for in haste ; but which several ma- thematicians of eminence had declared could not be performed within a shorter period than some months, the intense application threw him into a fever, in which he lost the sight of one eye.
Shortly after his return to Petersburg, in I766, he became totally blind. His passion for science, however, suffered no decline ; the powers of his mind were not impaired, and he continued as in- defatigable as ever. Though the distresses of age likewise were now crowding fast upon him, for he had passed his sixtieth year ; yet it was in this latter period of his life, under infirmity, bodily pain, and loss of sight, that he produced some of his most valuable works ; such as command our astonishment, independently of the situation of the author, from the labor and originality which they display. In fact, his habits of study and composition, his inventions and discoveries, closed only with his life. The very day on which he died, he had been engaged in calculating the orbit of Herschel's planet, and the motions of aerostatic machines. His death happened suddenly in Sep- tember 1783, from a fit of apoplexy, when he was in the seventy-sixth year of his age.
Such is the short history of this illustrious man. The incidents of his life, like that of most other
Xll EULER.
laborious students, afford very scanty materials for biography ; little more than a journal of studies and a catalogue of publications : but curiosity may find ample compensation in surveying the charac- ter of his mind. An object of such magnitude, so far elevated above the ordinary range of human intellect, cannot be approached without reverence, nor nearly inspected, perhaps, without some de- gree of presumption. Should an apology be ne- cessary, therefore, for attempting the following estimate of Euler's character, let it be considered, that we can neither feel that admiration, nor offer that homage, which is worthy of genius, unless, aiming at something more than the dazzled sensa- tions of mere wonder, we subject it to actual ex- amination, and compare it with the standards of human nature in general.
Whoever is acquainted with the memoirs of those great men, to whom the human race is in- debted for the progress of knowledge, must have perceived, that, while mathematical genius is di- stinct from the other departments of intellectual excellence, it likewise admits in itself of much di- versity. The subjects of its speculation are become so extensive and so various, especially in modern times, and present so many interesting aspects, that it is natural for a person, whose talents are of this cast, to devote his principal curiosity and attention to particular views of the science. When this hap- pens, the faculties of the mind acquire a superior facility of operation, with respect to the objects
EULER. XIU
towards which they are most frequently directed, and the invention becomes habitually most active and most acute in that channel of inquiry.
The truth of these observations is strikingly illustrated by the character of Euler. His studies and discoveries lay not among the lines and figures of geometry, those characters, to use an expres- sion of Galileo in which the great book of the universe is written ; nor does he appear to have had a turn for philosophising by experiment, and advancing to discovery through the rules of in- ductive investigation. The region, in which he dehghted to speculate, was that of pure intellect. He surveyed the properties and affections of quantity under their most abstracted forms. With the same rapidity of perception, as a geometrician ascertains the relative position of portions of exten- sion, Euler ranges among those of abstract quan- tity, unfolding their most involved combinations, and tracing their most intricate proportions. That admirable system of mathematical logic and lan- guage, which at once teaches the rules of just inference, and furnishes an instrument for prose- cuting deductions, free from the defects which obscure and often falsify our reasonings on other subjects ; the different species of quantity, whether formed in the understanding by its own abstrac- tions, or derived from modifications of the repre- sentative system of signs ; the investigation of the various properties of these, their laws of genesis, the limits of comparison among the different
XIV EULER.
species, and the method of applying all this to the solution of physical problems ; these were the re- searches on which the mind of Euler delighted to dwell, and in which he never engaged without finding new objects of curiosity, detecting sources of inquiry, which had passed unobserved, and ex- ploring fields of speculation and discovery, which before were unknown.
The subjects, which we have here slightly enu- merated, form, when taken together, what is called the Modern Analysis : a science eminent for the profound discoveries which it has revealed ; for the refined artifices that have been devised, in order to bring the most abstruse parts of mathe- matics within the compass of our reasoning powers, and for applying them to the solution of actual phfenomena, as well as for the remarkable degree of systematic simplicity, with which the various methods of investigation are employed and com- bined, so as to confirm and throw light on one another. The materials, indeed, had been col- lecting for years, from about the middle of the seventeenth century ; the foundations had been laid by Newton, Leibnitz, the elder BernouUis, and a few others ; but Euler raised the superstruc- ture : it was reserved for him to work upon the materials, and to arrange this noble monument of luiman industry and genius in its present sym- metry. Through the whole course of his scientific labors, the ultimate and the constant aim on which he set his mind, was the perfection of Calculus
EULER. XV
and Analysis. Whatever physical inquiry he be- gan with, this always came in view, and very fre- quently received more of his attention than that which was professedly the main subject. His ideas ran so naturally in this train, that even in the perusal of Virgil's poetry, he met with images that would recall the associations of his more fa- miliar studies, and lead him back, from the fairy scenes of fiction, to mathematical abstraction, as to the element, most congenial to his nature.
That the sources of analysis might be ascertained in their full extent, as well as the various modifica- tions of form and restrictions of rule that become necessary in applying it to different views of nature ; he appears to have nearly gone through a complete course of philosophy. The theory of rational mechanics, the whole range of physical astronomy, the vibrations of elastic fluids, as well as the movements of those which are incom- pressible, naval architecture and tactics, the doc- trine of chances, probabilities, and political arith- metic, were successively subjected to the analytical method ; and all these sciences received from him fresh confirmation and further improvement*.
It cannot be denied that, in general, his at- tention is more occupied with the analysis itself,
* A complete edition of his works,, comprising the numerous papers which he sent to the academies of St. Petersburg, Berlin, Paris, and other public societies, his separate Treatises on Curves, the Analysis of Infinites, the differential and integral Calculus, &c. would occupy, at least, forty quarto volumes.
XVI EULER.
than with the subject to which he is applying it; and that he seems more taken up with his instru- ments, than with the work, which they are to assist him in executing. But this can hardly be made a ground of censure, or regret, since it is the very circumstance to which we owe the present per- fection of those instruments ; a perfection to which he could never have brought them, but by the un- remitted attention and enthusiastic preference which he gave to his favorite object. If he now and then exercised his ingenuity on a physical, or perhaps metaphysical, hypothesis, he must have been aware, as well as any one, that his conclusions would of course perish with that from which they were derived. What he regarded, was the proper means of arriving at those conclusions ; the new views of analysis, which the investigation might open ; and the new expedients of calculus, to which it might eventually give birth. This was his uni- form pursuit ; all other inquiries were prosecuted with reference to it ; and in this consisted the peculiar character of his mathematical genius.
The faculties that are subservient to invention he possessed in a very remarkable degree. His memory was at once so retentive and so ready, that he had perfectly at command all those nu- merous and complex formulae, which enunciate the rules and more important theorems of analysis. As is reported of Leibnitz, he could also repeat the ^neid from beginning to end ; and could trust his recollection for the first and last lines in
EULER. XVll
every page of the edition, which he had been ac- customed to use. These are instances of a kind of memory, more frequently to be found where the capacity is inferior to the ordinary standard, than accompanying original, scientific genius. But in Euler, they seem to have been not so much the result of natural constitution, as of his most wonderful attention ; a faculty, which, if we con- sider the testimony of Newton * sufficient evi- dence, is the great constituent of inventive power. It is that complete retirement of the mind within itself, during which the senses are locked up ; that intense meditation, on which no extraneous idea can intrude ; that firm, straight-forward pro- gress of thought, deviating into no irregular sally, which can alone place mathematical objects in a light sufficiently strong to illuminate them fully, and preserve the perceptions of " the mind's eye" in the same order that it moves along.
Two of Euler's pupils (we are told by M. Fuss, a pupil himself) had calculated a converging series as far as the seventeenth term ; but found, on comparing the written results, that they dif- fered one unit at the fiftieth figure : they com- municated this difference to their master, who went over the whole calculation by head, and his decision was found to be the true one. — For the purpose of exercising his little grandson in the extraction of roots, he has been known to form to
* This opinion of Sir Isaac Newton is recorded by Dr. Pemberton.
b
XVIU EULER.
liimself the table of the six first powers of all num- bers, from 1 to 100, and to have preserved it actually in his memory.
The dexterity which he had acquired in analysis and calculation, is remarkably exemplified by the manner in which he manages formulas of the greatest length and intricacy. He perceives, almost at a glance, the factors from which they may have been composed ; the particular system of factors belonging to the question under present consideration ; the various artifices by which that system may be simplified' and reduced ; and the relation of the several factors to the conditions of the hypothesis. His expertness in this particular probably resulted, in a great measure, from the ease with which he performed mathematical in- vestigations by head. He had always accustomed himself to that exercise ; and having practised it with assiduity, even before the loss of sight, which afterwards rendered it a matter of necessity, he is an instance to what an astonishing degree of per- fection that talent may be cultivated, and how much it improves the intellectual powers. No other discipline is so effectual in strengthening the faculty of attention ; it gives a facility of ap- prehension, an accuracy and steadiness to the conceptions ; and, what is a still more valuable acquisition, it habituates the mind to arrangement in its reasonings and reflections.
If the reader wants a further commentary on its advantages, let him proceed to the work of
EULER. XIX
Eiiler, of which we here offer a Translation ; and if he has any taste for the beauties of method, and of what is properly called composition^ we venture to promise him the highest satisfaction and pleasure. The subject is so aptly divided, the order is so luminous, the connected parts seem so truly to grow one out of the other, and are disposed altogether in a manner so suitable to their relative importance, and so conducive to their mutual illustration, that, when added to the precision, as well as clearness with which every thing is explained, and the judicious selection of examples, we do not hesitate to consider it, next to Euclid's Geometry, the most perfect model of elementary writing, of which the scientific world is in possession.
When our reader shall have studied so much of these volumes as to relish their admirable style, he will be the better qualified to reflect on the circumstances under which they were composed. They were drawn up soon after our author was deprived of sight, and were dictated to his ser- vant, who had originally been a tailor's apprentice ; and, without being distinguished for more than ordinary parts, was completely ignorant of mathe- matics. But Euler, blind as he was, had a mind to teach his amanuensis, as he went on with the subject. Perhaps, he undertook this task by way of exercise, with the view of conforming the operation of his faculties to the change, which the loss of sight had produced. Whatever was the
b 2
XX EULER.
motive, his Treatise had the advantage of being composed under an immediate experience of the method best adapted to the natural progress of a learner's ideas : from the want of which, men of the most profound knowledge are often awkward and unsatisfactory, when they attempt elementary instruction. It is not improbable, that we may be farther indebted to the circumstance of our Author's blindness ; for the loss of this sense is generally succeeded by the improvement of other faculties. As the surviving organs, in particular, acquire a degree of sensibility, which they did not previously possess ; so the most charming visions of poetical fancy have been the oiFspring of minds, on which external scenes had long been closed. And perhaps a philosopher, familiarly acquainted with Euler's writings, might trace some improve- ment in perspicuity of method, and in the flowing progress of his deductions, after this calamity had befallen him ; which, leaving " an universal blank of nature's works," favors that entire seclusion of the mind, which concentrates attention, and gives liveliness and vigor to the conceptions.
In men devoted to study, we are not to look for those strong, complicated passions, which are con- tracted amidst the vicissitudes and tumult of public life. To delineate the character of Euler, requires no contrasts of coloring. Sweetness of disposition, moderation in tife passions, and simplicity of man- ners, were his leading features. Susceptible of tlie domestic aftections, lie was open to all their amiable
EULER. XXI
impressions, and was remarkably fond of children. His manners were simple, without being singular, and seemed to flow naturally from a heart that could dispense with those habits, by which many must be trained to artificial mildness, and with the forms that are often necessary for concealment. Nor did the equability and calmness of his temper indicate any defect of energy, but the serenity of a soul that overlooked the frivolous provocations, the petulant caprices, and jarring humours of ordinary mortals.
Possessing a mind of such wonderful compre- hension, and dispositions so admirably formed to virtue and to happiness, Euler found no difficulty in being a Christian : accordingly, *' his faith was unfeigned," and his love " was that of a pure and undefiled heart." The advocates for the truth of revealed religion, therefore, may rejoice to add to the bright catalogue, which already claims a Bacon, a Newton, a Locke, and a Hale, the illustrious name of Euler. But, on this subject, we shall permit one of his learned and grateful pupils * to sum up the character of his venerable master. *' His piety was rational and sincere ; his devotion " was fervent. He was fully persuaded of the " truth of Christianity ; he felt its importance to " the dignity and happiness of human nature ; " and looked upon its detractors, and opposers, as " the most pernicious enemies of man."
The length to which this account has been ex- * M. Fuss, Eulogy of M. L. Euler.
XXU EULER.
tended may require some apology ; but the cha- racter of Euler is an object so interesting, that, when reflections are once indulged, it is difficult to prescribe limits to them. One is attracted by a sentiment of admiration, that rises almost to the emotion of sublimity ; and curiosity becomes eager to examine what talents and qualities and habits belonged to a mind of such superior power. We hope, therefore, the student will not deem this an improper introduction to the work which he is about to peruse ; as we trust he is prepared to enter on it with that temper and disposition, which will open his mind both to the perception of ex- cellence, and to the ambition of emulating what he cannot but admire.
ADVERTISEMENT BY THE EDITORS OF THE ORIGINAL, IN GERMAN.
Wis present to the lovers of Algebra a work, of which a Russian translation appeared two years ago. The object of the celebrated author was to compose an Elementary Treatise, by which a beginner, without any other assistance, might make himself complete master of Algebra. The loss of sight had suggested the idea to him, and his activity of mind did not suffer him to defer the execution of it. For this purpose M. Euler pitched on a young man, whom he had engaged as a servant on his departure from Berlin, suf- ficiently master of arithmetic, but in other respects without the least knowledge of mathematics. He had learned the trade of a tailor ; and, with regard to his capacity, was not above mediocrity. This young man, however, has not only retained what his illustrious master taught and dictated to him, but in a short time was able to perform the most difficult algebraic calculations, and to resolve with readiness whatever analytical questions were proposed to him.
This fact must be a strong recommendation of the man- ner in which this work is composed, as the young man who wrote it down, who performed the calculations, and whose proficiency was so striking, received no instructions whatever but from this master, a superior one indeed, but deprived of sight.
Independently of so great an advantage, men of science will perceive, with pleasure and admiration, the manner in which the doctrine of logarithms is explained, and its con- nexion with other branches of calculus pointed out, as well
ADVERTISEMENT.
as the methods which are given for resolving equations of the third and fourth degrees.
Lastly, those who are fond of Diophantine problems will be pleased to find, in the last Section of the Second Part, all these problems reduced to a system, and all the processes of calculation, which are necessary for the solution of them, fully explained.
ADVERTISEMENT BY M. BERNOULLI, THE FRENCH TRANSLATOR.
The Treatise of Algebra, which I have undertaken to translate, was published in German, 1770, by the Royal Academy of Sciences at Petersburg. To praise its merits, would almost be injurious to the celebrated name of its author ; it is sufficient to read a few pages, to perceive, from the perspicuity with which every thing is explained, what advantage beginners may derive from it. Other subjects are the purpose of this advertisement,
I have departed from the division which is followed in the original, by introducing, in the first volume of the French translation, the first Section of the Second Volume of the original, because it completes the analysis of de- terminate quantities. The reason for this change is obvious : it not only favors the natural division of Algebra into de- terminate and indeterminate analysis ; but it was necessary to preserve some equality in the size of the two volumes, on account of the additions which are subjoined to the Second Part.
The reader will easily perceive that those additions come from the pen of M. De la Grange ; indeed, they formed one of the principal reasons that engaged me in this translation. I am happy in being the first to shew more generally to mathematicians, to what a pitch of perfection two of our most illustrious mathematicians have lately carried a branch of analysis but little known, the researches of which are at- tended with many difficulties, and, on the confession even of these great men, present the most difficult problems that they have ever resolved.
XXVI ADVERTISEMENT.
I have endeavoured to translate this algebra in the style best suited to works of the kind. My chief anxiety was to enter into the sense of the original, and to render it with the greatest perspicuity. • Perhaps I may presume to give my translation some superiority over the original, because that work having been dictated, and admitting of no revision from the author himself, it is easy to conceive that in many pas- sages it would stand in need of correction. If I have not submitted to translate literally, I have not failed to follow my author step by step ; I have preserved the same divisions in the articles, and it is only in so few places that I have taken the liberty of suppressing some details of calculation, and inserting one or two lines of illustration in the text, that I believe it unnecessary to enter into an explanation of the reasons by which I was justified in doing so.
Nor shall I take any more notice of the notes which I have added to the first part. They are not so numerous as to make me fear the reproach of having unnecessarily in- creased the volume ; and they may throw light on several points of mathematical history, as well as make known a great number of Tables that are of subsidiary utility.
With respect to the correctness of the press, 1 believe it will not yield to that of the original. I have carefully com- pared all the calculations, and having i-epeatcd a great num- ber of them myself, have by those means been enabled to correct several faults beside those which are indicated in the Errata.
CONTENTS.
PART I. Containing- the Analysis o/" Determinate Quantities.
SECTION I.
Of the Different Methods of calculating Simple Quantities.
Page
Chap. I. Of Mathematics in general - - - 1
II. Explanation of the signs + plus and — minus - 3
III. Of the Multiplication of Simple Quantities ~ 6
IV. Of the nature of whole Numbers, or Integers with
respect to their Factors - - - ]0
V. Of the Division of Simple Quantities - - 13 VI. Of the properties of Integers, with respect to their
Divisors - - - - _ ] 6 VII. Of Fractions in general - - - 20 VIII. Of the Properties of Fractions - - 24 IX. Of the Addition and Subtraction of Fractions - 27 X. Of the Multiplication and Division of Fractions 30 XI. Of Square Numbers - _ _ _ 36 XII. Of Square Roots, and of Irrational Numbers re- sulting from them - - - - 38
XIII. Of Impossible, or Imaginary Quantities, which
arise from the same source - - - 42
XIV. Of Cubic Numbers - - -* -45 XV. Of Cube Roots, and of Irrational Numbers re- sulting from them - - - 46
XVI. Of Powers in general - - - 48 XVII. Of the Calculation of Powers - - - 52 XVIII. Of Roots with relation to Powers in general - 54 XIX. Of the Method of representing Irrational Num- bers by Fractional Exponents - - 56 XX. Of the different Methods of Calculation, and of
their Mutual Connexion - - - 60
XXI. Of Logarithms in general - - - 63
XXII. Of the Logarithmic Tables that are now in use 66
XXIII. Of the Method of expressing Logarithms - 69
SECTION II.
Of the different Methods of calculating Compound Quantities.
Chap. I, Of the Addition of Compound Quantities - 76
II. Of the Subtraction of Compound Quantities - 78
III. Of the MuItipHcation of Compound Quantities - 79
IV. Of the Division of Compound Quantities - 84 V. Of the Resolution of Fractions into Infinite Series 88
VI. Of the Squares of Compound Quantities - 97
XXVIH CONTENTS.
Page
Chap. VII. Of the Extraction of Roots applied to Com- pound Quantities - - - 100 VIII. Of the Calculation of Irrational Quantities - 104
IX. Of Cubes, and of the Extraction of Cube Roots 107 X. Of the higher Powers of Compound Quantities 110
XI. Of the Transposition of the Letters, on which the demonstration of the preceding Rule is founded - - - - 115
XII. Of the Expression of Irrational Powers by In- finite Series - - - - 120 XIII. Of the Resolution of Negative Powers - 123
SECTION III.
O/* Ratios and Proportions.
Chap. I. Of Arithmetical Ratio, or the Difference be- tween two numbers - - - 126 II. Of Arithmetical Proportion - - 129
III. Of Arithmetical Progressions - - - 131
IV. Of the Summation of Arithmetical Progressions 135 V. Of Figurate, or Polygonal Numbers - - 139
VI. Of Geometrical Ratio _ - _ 146
VII. Of the greatest Common Divisor of two given
Numbers - - - - 148
Vlll. Of Geometrical Proportions - - - 152
IX. Observations on the Rules of Proportion and
their Utility - - - - 155
X, Of Compound Relations - _ - - 159
XI. Of Geometrical Progressions - - 164
XII. Of Infinite Decimal Fractions - - 121- XIII. Of the Calculation of Interest - ""-177
SECTION IV.
O/" Algebraic Equations, and of the Resolution of those Equations.
Chap. I. Of the Solution of Problems in General - 186
II. Of the Resolution of Simple Equations, or
Equations of the First Degree - - 189
III. Of the Solution of Questions relating to the pre-
ceding Chapter - - - 194
IV. Of the Resolution of two or more Equations of
the First Degree - _ - 206
V. Of the Resolution of Pure Quadratic Equations 216 VI. Of the Resolution of Mixed Equations of the
Second Degree - - - - 222
VJI. Of the Extraction of the Roots of Polygonal
Numbers . . _ . 230
VIII. Of the Extraction of Square Roots of Bino- mials - - - - 234
CONTENTS. XXIX
Page
(^hap. IX, Of the Nature of Equations of the Second
Degree .... 244
X. Of Pure Equations of the Third Degree - 248
XI. Of the Resolution of Complete Equations of
the Third Degree - - - 253
XII. Of the Rule of Cardan, or that o^ Scipio Ferreo 262
XIII. Of the Resolution of Equations of the Fourth
Degree - - - - 272
XIV. Of the Rule of Bomhelli, for reducing the Re-
solution of Equations of the Fourth Degree to that of Equations of the Third Degree - 278 XV. Of a new Method of resolving Equations of
the Fourth Degree . - _ 282
XVI. Of the Resolution of Equations by Approxi- mation - - . - - 289
PART II.
PART II. Containing the hna\ys,\% o/" Indeterminate Quantities.
Chap. I. Of the Resolution of Equations of the First De- gree, which contain more than one unknown Quantity - - - •■ - 299
II. Of the Rule which is called Regu/a Cceci, for de- termining, by means of two Equations, three or more Unknown Quantities - - - 312
III. Of Compound Indeterminate Equations, in which
one of the Unknown Quantities does not ex- ceed the First Degree - - - 3 1 7
IV. Of the Method of rendering Surd Quantities, of
the form (^/a + ax + c/t^"-). Rational - 322
V. Of the Cases in which the Formula a -f- b.v -\- c.%^
can never become a Square . - - 335
VI. Of the Cases in Integer Numbers, in which the
Formula ax~ -\- b becomes a Square - - 342
VII. Of a particular Method, by which the Formula
an^ -\- 1 becomes a Square in Integers - 3;") I
VIII. Of the Method of rendering the Irrational Formula
(v/a + bx -f- cx^ -h dx^) Rational - - 361
IX. Of the Method of rendering rational the incom- mensurable Formula {\/ x-\- hx ■\-cx"-\-dji^-\- ex* ) 3 68 X. Of the Method of rendering rational the irrational
Formula (Va -|- bx +cx^ + da,^) - • 379
XI. Of the Resolution of the Formula o^^-f hxy + cy-
into its Factors . _ _ - 387
XII. Of the Transformation of the Formula ax- -j- c^-
into Squares and higher Powers - - 396
Xlll. Of some Expressions of the Form r/a* + by*^
which are not reducible to Squares - - 40.>
XXX CONTENTS.
Page Chap. XIV. Solution of some Questions that belong to this
Part of Algebra - - - - 413
XV. Solutions of some Questions in which Cubes
are required _ - _ - 449
ADDITIONS BY M. DE LA GRANGE.
Advertisement _ . _ - 463
Chap. I. Of Continued Fractions _ . _ 465
II. Solution of some New and Curious Arithmetical
Problems - - - - 495
III. Of the Resolution in Integer Numbers of Equa-
tions of the First Degree containing two Un- known Quantities - - - - 530
IV. General Method for resolving in Integer Equa-
tions of two Unknown Quantities, one of which does not exceed the First Degree - 534
V. A direct and general Method for finding the values of x, that will render Quantities of the form »y{a-\- bx + cx^) Rational, and for re- solving, in Rational Numbers, the indeter- minate Equations of the second Degree, which have two Unknown Quantities, when they admit of Solutions of this kind - - 537
Resolution of the Equation Ap^ + 'sq^ — z^ in Integer Numbers _ - _ 539
VI. Of Double and Triple Equalities - - 547
VII. A direct and general Method for finding all the values of 2/ expressed in Integer Numbers, by which we may rejider Quantities of the form ^/ {A.y^ + b), rational; a and b being given Integer Numbers; and also for finding all the possible Solutions, in Integer Numbers, of in- determinaie Quadratic Equations of two un- known Quantities - - - - 550
Resolution of the Equation Cj/^— 2«_y:2 + nz^=. 1 in Integer Numbers _ - . 552
First Method - - - - ib.
Second Method _ - - » 555
Of the Manner of finding all the possible So- lutions of the Equations cy- — 2nyz + Bz^ = 1, when we know only one of them - - 559
Of the Manner of finding all the possible So- lutions, in whole Numbers, of Indeterminate Quadratic Equations of two Unknov*n Quan- tities _ . _ - - 565 VIII. Remarks on Equations of the Form j3*= Aq--{- \, and on the common Method of resolving them in whole Numbers . _ - 57s IX. Of the Manner of finding Algebraic Functions of all Degrees, which, when multiplied to- gether, may always produce similar Functions 583
ELEMENTS
OP
ALGEBRA.
PART I.
Containing the Analysis of Determinate Quantities.
SECTION I.
Of the different Methods of calculating Simple Quantities.
, CHAP. I.
Of Mathematics in general,
ARTICLE I:
Whatever is capable of increase or diminution, is called magnitude, or quantity.
A sum of money therefore is a quantity, since we may increase it or diminish it. It is the same with a weight, and other things of this nature.
2. From this definition, it is evident, that the different kinds of magnitude must be so various, as to render it dif- ficult to enumerate them : and this is the origin of the dif- ferent branches of the Mathematics, each being employed on a particular kind of magnitude. Mathematics, in general, is the science of quantity ; or, the science which investigates the means of measuring quantity.
3. Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known piece of money,
2 ELEMENTS SECT. I.
as a louis, a crown, a ducat, or some other coin, and shew how many of these pieces are contained in the given sum. In the same manner, if it were proposed to determine the quantity of a weight, we should take a certain known weight; for example, a pound, an ounce, &c. and then shew how many times one of these weights is contained in that which we are endeavouring to ascertain. If we wished to measure any length or extension, we should make use of some known length, such as a foot.
4. So that the determination, or the measure of mag- nitude of all kinds, is reduced to this : fix at pleasure upon any one known magnitude of the same species with that which is to be determined, and consider it as the measure or iinit ; then, determine the proportion of the proposed mag- nitude to this known measure. This proportion is always expressed by numbers ; so that a number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit.
5. From this it appears, that all magnitudes may be ex- pressed by numbers; and that the foundation of all the Mathematical Sciences must be laid in a complete treatise on the science of Numbers, and in an accurate examination of the different possible methods of calculation.
This fundamental part of mathematics is called Analysis, or Algebra *.
6. In Algebra then we consider only numbers, which represent quantities, without regarding the different kinds of quantity. These are the subjects of other branches of the mathematics.
7. Arithmetic treats of numbers in particular, and is the science of numhers properly so called; but this science ex- tends only to certain methods of calculation, which occur in common practice : Algebra, on the contrary, comprehends in general all the cases that can exist in the doctrine and calculation of numbers.
* Several mathematical writers make a distinction between Analijx'is and Algebra. By the term Analysis, they understand the method of determining those general rules, which assist the understanding in all mathematical investigations; and hy Algebra, the instrument wliich this method employs for accomplishing that end. This is the definition given by M. Bezoiit in the preface to his Algebra. F. T.
OHAP. ir. OF ALGEBRA.
CHAP. II.
Explanation of the Signs + Plus and — Minus.
8. When we have to add one given number to another, this is indicated by the sign + , which is placed before the second number, and is read plus. Thus 5+3 signifies that we must add 3 to the number 5, in which case, every one knows that the result is 8 ; in the same manner 12 + '7 make 19 ; 25 + 16 make 41 ; the sum of 25 -1- 41 is QQ, Sic.
9. We also make use of the same sign + plus, to con- nect several numbers together; for example, 7+5 + 9 signifies that to the number 7 we must add 5, and also 9, which make 21. The reader will therefore understand what is meant by
8 + 5 + 13+11 + 1+3 + 10, viz. the sum of all these numbers, which is 51.
10. All this is evident; and we have only to mention, that in Algebra, in order to generalise numbers, we re- present them by letters, as a, b, c, d, &c. Thus, the ex- pression a -r b, signifies the sum of two numbers, which we express by a and b, and these numbers may be either very great, or very small. In the same manner, y + m + b -\- x, signifies the sum of the numbers represented by these four letters.
If we know therefore the numbers that are represented by letters, we shall at all times be able to find, by arithmetic, the sum or value of such expressions.
11. When it is required, on the contrary, to subtract one given number from another, this operation is denoted by the sign — , which signifies minus, and is placed before the number to be subtracted : thus, 8—5 signifies that the number 5 is to be taken from the number 8 ; which being done, there remain 3. In like manner 12 — 7 is the same as 5 ; and 20 — 14 is the same as 6, &c.
12. Sometimes also we may have several numbers to subtract from a single one ; as, for instance, 50 — 1 — 3 — 5 — 7 — 9. This signifies, first, take 1 from 50, and there remain 49 ; take 3 from that remainder, and there will re- main 46 ; take away 5, and 41 remain ; take away 7, and 34 remain ; lastly, from that take 9, and there remain 25 : this last remainder is the value of the expression. But as the numbers 1, 3, 5, 7, 9, are all to be subtracted, it is the
b2
4 ELEMENTS SECT. I.
same thing if we subtract tlieir sum, wliich is 25, at once from 50, and the remainder will be 25 as before.
13. It is also easy to determine the value of similar ex- pressions, in which both the signs + plus and — minus are found. For example ;
12 — 3 — 5 + 2 — 1 is the same as 5. We have only to collect separately the sum of the numbers that have the sign + before them, and subtract from it the sum of those that have the sign — . Thus, the sum of 12 and 2 is 14; and that of 3, 5, and 1, is 9; hence 9 being- taken from 14, there remain 5.
14. It will be perceived, from these examples, that the order in which we write the numbers is perfectly indifferent and arbitrary, provided the proper sign of each be pi-eserved. We might with equal propriety have arranged the expression in the preceding article thus; 12 + 2 — 5 — 3 — 1, or 2 _ 1 _ 3 _ 5 + 12, or 2 + 12 - 3 - 1 - 5, or in still different orders; where it must be observed, that in the ar- rangement first proposed, the sign -f is supposed to be placed before the number 12.
15. It will not be attended with any more difficulty if, in order to generalise these operations, we make use of letters instead of real numbers. It is evident, for example, that
a — b — c + d ~ e, signifies that we have numbers expressed by a and cZ, and that from these numbers, or from their sum, we must sub- tract the numbers expressed by the letters b, c, e, which have before them the sign — .
16. Kence it is absolutely necessary to consider what sign is prefixed to each number: for in Algebra, simple quan- tities are numbers considered with regard to the signs which ])recede, or affect them. Farther, we call those positive quaMitiCS, before which the sign + is found; and those are called negative quantities, which are affected by the sign — .
17. The manner in which we generally calculate a per- son's property, is an apt illustration of what has just been said. For we denote what a man really possesses by positive numbers, using, or understanding the sign + ; whereas his debts arc represented by negative numbers, or by using the sign — . Thus, when it is said of any one that he has 100 crowns, but owes 50, this means that his real possession amounts to 100 — 50; or, which is the same thing, + 100
— 50, that is to say, 50.
18. Since negative numbers may be considered as debts, because positive numbers represent real possessions, we
CHAP. 11. OF ALGEBRA.
may say that negative numbers are less than nothing. Thus, when a man has nothing of his own, and owes 50 crowns, it is certain that he has 50 crowns less than nothing ; for if any one were to make him a present of 50 crowns to pay his debts, he would still be only at the point nothing, though really richer than before.
19. In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive numbers by adding 1 to 0, that is to say, 1 to nothing ; and by con- tinuing always to increase thus from unity. This is the origin of the series of numbers called natural numbers ; the following being the leading terms of this series :
0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10, and so on to infinity.
But if, instead of continuing this series by successive ad- ditions, we continued it in the opposite direction, by per- petually subtracting unity, we should have the following series of negative numbers :
0, -1, -2, -S, -4, -5, -6, -7, -8, -9, -10, and so on to infinity.
20. All these numbers, whether positive or negative, have the known appellation of whole numbers, or integers, which consequently are either greater or less than nothing. We call them integers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak. For instance, 50 being greater by an entire unit than 49, it is easy to comprehend that there may be, between 49 and 50, an infinity of intermediate numbers, all greater than 49, and yet all less than 50. We need only imagine two lines, one 50 feet, the other 49 feet long, and it is evident that an infinite number of lines may be drawn, all longer than 49 feet, and yet shorter than 50.
21. It. is of the utmost importance through the whole of Algebra, that a precise idea should be formed of those ne- gative quantities, about which we have been speaking. I shall, however, content myself with remarking here, that all such expressions as
+ 1 - 1, + 2 - 2, +3—3, + 4 - 4, &c. are equal to 0, or nothing. And that
+ 2 — 5 is equal to — 3 : for if a person has 2 crowns, and owes 5, he has not only nothing, but still owes 3 crowns. In the same manner, 7 — 12 is equal to - 5, and 25 — 40 is equal to — 15.
22. The same observations hold true, when, to make the expression more general, letters are used instead of numbers;
6 ELEMENTS
SECT. 1.
thus 0, or nothing, will always be the value of + t*^ — " '•> but if we wish to know the value o^ + a ~ b, two cases are to be considered.
The first is when a is greater than b ; b must then be subtracted from a, and the remainder (before which is placed, or understood to be placed, the sign -[- ) shews the value sought.
The second case is that in which a is less than b : here a is to be subtracted from b, and the remainder being made negative, by placing before it the sign — , will be the value sought.
CHAP. III.
Of the Multiplication o/^' Simple Quantities.
23. When there are two or more equal numbers to be added together, the expression of their sum may be abridged : for example,
a + a is the same with 2 x a,
a + a + a - 3x«,
a + a -\- a -\- a 4xa, and so on ; where x is the
sign of multiplication. In this manner we may form an idea of multiplication ; and it is to be observed that, 2 X a signifies 2 times, or twice a
S X a 3 times, or thrice a
4i X a 4 times a, &c.
24. If therefore a number expressed by a letter is to be multiplied by any other number, we simply put that number before tl .e letter, thus ;
a multiplied by 20 is expressed by 20.'/, and b multiplied by 30 is expressed by oOb, &c. It is evident, also, that c taken once, or Ic, is the same as c.
25. Farther, it is extremely easy to multiply such pro- ducts again by other numbers ; for example :
2 times, or twice 3a, makes 6a
3 times, or thrice 4i, makes 12b 5 times 7a; makes 35.r,
and these products may be still multiplied by other numbers at ])leasure.
26. When the number by which we are to multiply is also represented by a letter, we place it immediately before the other letter; thus, in multiplying b by a, the product is
CHAP. III. OF ALGEBRA. 7
written ab ; and pq will be the product of the multiplication of the number q by p. Also, if we multiply this -pq again by a, we shall obtain apq.
27. It may be farther remarked here, that the order in which the letters are joined together is indifferent; thus ab is the same thing as ba ; for b multiplied by a is the same as a multiplied by b. To understand this, we have only to substitute, for a and ft, known numbers, as 3 and 4 ; and the truth will be self-evident ; for S times 4 is the same as 4 times 3.
28. It will not be difficult to perceive, that when we sub- stitute numbers for letters joined together, in the manner we have described, they cannot be written in the same way by putting them one after the other. For, if we were to write 34 for 3 times 4, we should have 34, and not 12. When therefore it is required to multiply common numbers, we must separate them by the sign x, or by a point: thus, 3 X 4, or 3.4, signifies 3 times 4 ; that is, 12. So, 1 x 2 is equal to 2; and 1x2x3 makes 6. In like manner, Ix2x3x4x 56 makes 1344 ; and Ix2x3x4x 5x6x7x8x9x 10 is equal to 3628800, &c.
29. In the same manner, we may discover the value of an expression of this form, S.^.S.abcd. It shews that 5 must be multiplied by 7, and that this product is to be again multiplied by 8 ; that we are then to multiply this product of the three numbers by a, next by b, then by c, and lastly by d. It may be observed, also, that instead of 5.7.8, we may write its value, 280; for we obtain this number when we multiply 35, (the product of 5 by 7) by 8.
30. The results which arise from the multiplication of two or more numbers are called products ; and the numbers, or individual letters, are cdWedi factors.
31. Hitherto we have considered only positive numbers; and there can be no doubt, but that the products which we have seen arise are positive also : viz. -\- a hy -\- b must necessarily give + ab. But we must separately examine what the multiplication of + a by — &, and of — « by — &, will produce.
32. Let us begin by multiplying —a by 3 or H-3. Now, since — a may be considered as a debt, it is evident that if we take that debt three times, it must thus become three times greater, and consequently the required product is — 3tf. So if we multiply —a by +b, we shall obtain —ba, or, which is the same thing, — ab. Hence we conclude, that if a positive quantity be multiplied by a negative quan- tity, the product will be negative; and it may be laid down
l>Ai).
8 ELEMENTS SECT. 1.
as a rule, that + by + makes + or plus ; and that, on the contrary, + by — , or — by +, gives — , or viinus.
33. It remains to resolve the case in which — is mul- tiplied by — ; or, for example, — « by — ^. It is evident, at first sightj with regard to the letters, that the product will be ab; but it is doubtful whether the sign +, or the sign — , is to be placed before it; all we know is, that it must be one or the other of these signs. Now, I say that it cannot be the sign — : for — « by +6 gives — a6, and —a by —b can- not produce the same result as —a by +6; but must pro- duce a contrary result, that is to say, + ah ; consequently, we have the following rule: — multiplied by — produces + , that is, the same as + multiplied by -\- *.
* A farther illustration of this rule is generally given by algebraists as follows :
First, we know that -\-a multiplied by +5 gives the product -\-ab ; and if +« be multiplied by a quantity less than b, as b — c, the product must necessarily be less than ab ; in short, from ab we must subtract the product of a, multiplied by c; hence a y. [b — c) must be expiessed by ab — ac; therefore it follows that ax — c gives the product — ac.
If now we consider the product arising from the multiplication of the two quantities (a—b), and (c — d), we know that it is less than that of (a — b) x c, or of ac — be; in short, from this pro- duct we must subtract that o^ [a — b) x d : but the product (a — b) X (c — d) becomes ac — be — ad, together with the product of —h X —d annexed 5 and the question is only what sign we must employ for this purpose, whether -f or — . Now, we have seen that from the product ac — be we must subtract the product of (a—b) x d; that is, we must subtract a quantity less than ad. We have therefore subtracted already too much by the quantity bd ; this product must therefore be added ; that is, it must have the sign -|- prefixed ; hence we see that
— b X —d gives -\- bd for a product ; or — jnhws multiplied by
— minus gives + j^/us. See Art. 273, 27+. Multiplication has been erroneously called a compendious
method of performing addition : whereas it is the taking, or re- peating of one given number as many times, as the number by which it is to be multiplied, contains units. Thus, 9x3 means that 9 is to be taken 3 times; or that the measure of multiplica- tion is 3 ; again 9 X | means that 9 is to be taken half a time, or that the measure of multiplication is f. In multiplication there are two factors, which are sometimes called the mul- tiplicand and the multiplier. These, it is evident, may re- ciprocally change places, and the product will be still the same: for 9X3 = 3X9, and 9 X f = i x9. Hence it appears, that numbers may be diminished by nuiltiplication, as well as in- creased, in any given ratio; which is wholly inconsistent with
CHAP. III. OF ALGEBRA. 9
34. The rules which we have explained are expressed more briefly as follows :
Like signs, multiplied together, give -f ; unlike or con- trary signs give — . Thus, when it is required to multiply the following numbers ; + «, — 6, — c, + ^ ; we have first + a multiplied by — 6, which makes — ab; this by — c, gives 4- o,bc; and this by + d, gives + abed.
35. The difficulties with respect to the signs being re- moved, we have only to shew how to multiply numbers that are themselves products. If we were, for instance, to mul- tiply the number ab by the number cd, the product would be abed, and it is obtained by multiplying first ab by c, and then the result of that multiplication by d. Or, if we had to multiply 36 by 12; since 12 is equal to 3 times 4, we
the natm-e of Addition ; for 9 x f = 4f , 9 x t = 1 » 9 X t^= Tg-o-, &c. The same will be found true with respect to algebraic quantitiesj a X b =^ ab, —9 x 3 =— 27, that is, 9 negative in- tegers multiplied by 3, or taken 3 times, are equal to —27, be- cause the measure of multiplication is 3. In tlie same manner, by inverting the factors, 3 positive integers multiplied by —9, or taken 9 times negatively, must give the same result. There- fore a positive quantity taken negatively, or a negative quantity taken positively, gives a negative product.
From these considerations, we may illustrate the present sub- ject in a different way, and shew, that the product of two ne- gative quantities must be positive. First, algebraic quantities may be considered as a series of numbers increasing in any ratio, on each side of nothing, to infinity. [See Art. 19.] Let us assume a small part only of such a series for the present purpose, in which the ratio is unity, and let us multiply every term of it by —2.
5, 4, 3, 2, 1, 0,-1,-2,-3,-4,-5, -2, —2, —2, —2, -2, -2, -2,-2, —2, —2, -2,
-10, -«, -6, -4, -2, 0, +2, +4, +6, +8, +10. Here, of course, we find the series inverted, and the ratio dou- bled. Farther, in order to illustrate the subject, we may con- sider the ratio of a series of fractions between 1 and 0, as in- definitely small, till the last term being multiplied by —2, the product would be equal to 0. If, after this, the multiplier having passed the middle term, 0, be multiplied into any negative term, however small, between 0 and — 1 , on the other side of the series, the product, it is evident, must be positive, otherwise the series could not go on. Hence it appears, that the taking of a negative quantity negatively destroys the very property of ne- gation, and is the conversion of negative into positive numbers. So that if -f X — = — , it necessarily follows that — x — must give a contrary product, that is, +. See Art. 170, 177.
10 ELEMENTS SECT. I.
should only multiply 36 first by 3, and then the product 108 by 4, in order to have the whole product of the mul- tiplication of 12 by 36, which is consequently 432.
36. But if we wished to multiply ^ab by Serf, we might write ^cd x Bah. However, as in the present instance the order of the numbers to be multiplied is indifferent, it will be better, as is also the custom, to place the common num- bers before the letters, and to express the product thus: 5 X Sabcd, or I5abcd; since 5 times 3 is 15.
So if we had to multiply 12j}qr by Kxy, we should obtain 12 X Ipqrxy, or ^^pqrxy.
ipC
CHAP. IV.
Of the Nature of whole Numbers, or Integers, with respect to their Factors.
.37. We have observed that a product is generated by the multiplication of two or more numbers together, and that these numbers are called factors. Thus, the numbers a, h, c, d, are the factors of the product abed.
38. If, therefore, we consider all whole numbers as pro- ducts of two or more numbers multiplied together, we shall soon find that some of them cannot result from such a mul- tiplication, and consequently have not any factors; while others may be the products of two or more numbers mul- tiplied together, and may consequently have two or more factors. Thus 4 is produced by 2 x 2; 6 by 2 x o ; 8 by 2 X 2 X 2 ; 27 by 3 X 3 X 3 ; and 10 by 2 k 5, &c.
39. But on the other hand, the numbers 2, 3, 5, 7, 11, 13, 17, &c. cannot be represented in the same manner by factors, iniless for that purpose we make use of unity, and represent 2, for instance, by 1x2. But the numbers which are multiplied by 1 remaining the same, it is not proper to reckon unity as a factor.
AH numbers, therefore, such as 2, 3, 5, 7, 11, 13, 17, &c. which cannot be represented by factors, are called simple, or prime numheis ; whereas others, as 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, &c. which may be represented by factors, are called comj^osite numbei's.
40. Simple or prime members deserve therefore particular attention, since they do not result from the multi[)lication of
i"" y
, /, 1 /
CHAP. IV. OF ALGEBRA. 11
two or more numbers. It is also partlculariy worthy of ob- servation, that if we write these numbers in succession as they follow each other, thus,
5?, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, &c. * we can trace no regular order; their increments being some- times greater, sometimes less ; and hitherto no one has been able to discover whether they follow any certain law or not. 41. All composite numbers, which may be represented by factors, result from the prime numbers above mentioned ; that is to say, all their factors are prime numbers. For, if we find a factor which is not a prime number, it may always be decomposed and represented by two or more prime num- bers. When we have represented, for instance, the number
* All the prime numbers from 1 to lOOOOO are to be found in the Tables of divisors, which I shall speak of in a succeeding note. But particular Tables of the prime numbers from 1 to 101000 have been published at Halle, by M. Kruger, in a Ger- man work entitled " Thoughts on Algebra;'' M. Kruger had received them from a person called Peter Jaeger, who had cal- culated them. M. Lambert has continued these Tables as far as 102000j and republished them in his supplements to the loga- rithmic and trigonometrical Tables, printed at Berlin in 1 770 3 a work which contains likewise several Tables that are of great use in the different branches of mathematics, and explanations which it would be too long to enumerate here.
The Royal Parisian Academy of Sciences is in possession of Tables of prime numbers, presented to it by P. Mercastel de rOratoire, and i)}' M. du Tour ; but they have not been pub- lished. They are spoken of in Vol. V. of the Foreign Memoirs, with a reference to a memoir, contained in that volume, by M. Rallier des Ourmes, Honorary Counsellor of the Presidial Court at Rennes, in which the author explains an easy method of finding prime numbers.
In the same volume we find another method by M. Rallier des Ourmes, which is entitled, " A new Method for Division, wlien the Dividend is a Multiple of the Divisor, and may therefore be divided without a remainder ; and for the Extraction of Roots when the Power is perfect." This method, moi-e curious, in- deed, than useful, is almost totally different from the common one : it is very easy, and has this singularity, that, provided v.e know as many figures on the right of the dividend, or the power, as there are to be in the quotient, or the root, we may pass over the figures which precede them, and thus obtain the quotient. M. Rallier des Ourmes was led to this new method by reflecting on the numbers terminating the numerical expressions of pro- ducts or powers, a species of numbers which 1 have remarked also, ou other occasions, it would be useful to consider. F. T.
|
4 |
is the |
same as |
2x2, |
|
8 |
_ |
- 2x |
2x2, |
|
10 |
- - |
_ |
2x5, |
|
14 |
- |
- |
2x7, |
|
16 |
- - |
2x2x |
2x2, |
12 ELEMENTS SECT. I.
30 by 5 X 6, it is evident that 6 not being a prime number, but being produced by 2 x 3, we might have represented 30 by 5 x 2 X 3, or by 2 X 3 X 5 ; that is to say, by fac- tors which are all prime numbers.
42. If we now consider those composite numbers which may be resolved into prime factors, we shall observe a great difference among them ; thus we shall find that some have only two factors, that others have three, and others a still greater number. We have already seen, for example, that
6 is the same as 2 x 3,
9 . . . „ 3x3,
12 - - - 2x3x2,
15 - - - - 3x5,
and so on.
43. Hence, it is easy to find a method for analysing any number, or resolving it into its simple factors. Let there be proposed, for instance, the number 360 ; we shall represent it first by 2 X 180. Now 180 is equal to 2 x 90, and
90~| r2x45,
45 Y is the same as -] 3 x 15, and lastly
15J (3x5.
So that the number 360 may be represented by these simple factors, 2x2x2x3x3x5; since all these numbers multiplied together produce 360 *.
44. This shews, that prime numbers cannot be divided by other numbers ; and, on the other hand, that the simple factors of compound numbers are found most conveniently, and with the greatest certainty, by seeking the simple, or prime numbers, by which those compound numbers are divisible. But for this Division is necessary ; we shall there- fore explain the rules of that operation in the following chapter.
* There is a Table at the end of a German book of arithmetic, published at Lcipsic, by Poetius, in 1728, in which all the numbers from 1 to 10000 are represented in this manner by their simple factors. F. T.
CHAP. V. OF ALGEBRA. 13
CHAP. V.
Of the Division o/*Simple Quantities.
45. When a number is to be separated into two, three, or more equal parts, it is done by means of division^ which enables us to determine the magnitude of one of those parts. When we wish, for example, to separate the number 12 into three equal parts, we find by division that each of those parts is equal to 4.
The following terms are made use of in this operation. The number which is to be decompounded, or divided, is called the dividend ; the number of equal parts souo-ht is called the divisor ; the magnitude of one of those parts, determined by the division, is called the quotient: thus, in the above example,
12 is the dividend, " 3 is the divisor, and
4 is the quotient.
46. It follows from this, that if we divide a number l)y 2, or into two equal parts, one of those parts, or the quotient, taken twice, makes exactly tlie number proposed ; and, in the same manner, if we have a number to divide by 3, the quotient taken thrice must give the same number again. In general, the multiplication of the quotient by the divisor must always reproduce the dividend.
47. It is for this reason that division is said to be a rule, which teaches us to find a number or quotient, which, being multiplied by the divisor, will exactly produce the dividend. For example, if 35 is to be divided by 5, we seek for a number, which multiplied by 5, will produce ^5. Now, this number is 7, since 5 times 7 is 35. The manner of expression employed in this reasoning, is ; 5 in 35 goes 7 times ; and 5 times 7 makes 35.
- 48. The dividend therefore may be considered as a product, of which one of the factors is the divisor, and the other the quotient. Thus, supposing v/e have 63 to divide by 7, we endeavour to find such a product, that, taking 7 for one of its factors, the other factor multiplied by this may exactly give 63. Now 7 x 9is such a product; and consequently 9 is the quotient obtained when we divide 6S by 7-
49. In general, if we have to divide a number ab by a, it is evident that the quotient will be 6; for a multiplied by h
14 ' ELEMENTS SECT. I.
gives tlie dividend ah. It is clear also, that if we liad to divide ah by 6, the quotient would be a. And in all ex- amples of division that can be proposed, if we divide the dividend by the quotient, we shall again obtain the divisor ; for as 24 divided by 4 gives 6, so 24 divided by 6 will give 4.
50. As the whole operation consists in representing- the dividend by two factors, of which one may be equal to the divisor, and the other to the quotient, the following ex- amples will be easily understood. I say first that the di- vidend abc, divided by a, gives hc\ for «, multiplied by ic, produces abo: in the same manner abc, being divided by 5, we shall have ac; and abc, divided by ac, gives h. It is also plain, that \2mn, divided by 3»2, gives 4?i; for 3«/, multiplied by 4«, makes \9.mii. But if this same number \%mn had been divided by 12, we should have obtained the quotient mil.
51. Since every number a may be expressed by la, or a, it is evident that if we had to divide «, or \a, by 1, the quotient would be the same number a. And, on the con- trai-y, if the same number a, or Iff, is to be divided by a, the quotient will be 1.
52. It often happens that we cannot represent the di- vidend as the product of two factors, of which one is equal to the divisor ; hence, in this case, the division cannot be performed in the manner we have described.
When we have, for example, 24 to divide by 7, it is at first sight obvious, that the number 7 is not a factor of 24; for the product of 7 x 3 is only 21, and consequently too small ; and 7x4 makes 28, which is greater than 24. We discover, however, from this, that the quotient must be greater than 3, and less than 4. In order therefore to de- termine it exactly, we employ another species of numbei's, which are called fractions, and which we shall consider in one of the following chapters.
53. Before we pi-oceed to the use of fractions, it is usual to be satisfied with the whole number which approaches nearest to the true quotient, but at the same time paying attention to the remainder which is left ; thus we say, 7 in 24 goes 3 times, and the remainder is 3, because 3 times 7 produces only 21, which is 3 less than 24. We may also consider the following examples in the same manner :
6)34(5, that is to say, the divisor is 6, the
30 dividend 34, the quotient 5, and tlie
remainder 4.
4
CHAP. V. OF ALGEBRA. 15
9)41(4, here the divisor is 9, the dividend 36 41, the quotient 4, and the remain-
der 5.
5 The following rule is to be observed in examples where there is a remainder.
54. Multiply the divisor by the quotient, and to the pro- duct add the remainder, and the result will be the dividend. This is the method of proving the division, and of dis- covering whether the calculation is right or not. Thus, in the first of the two last examples, if we multiply 6 by 5, and to the product 30 add the remainder 4, we obtain 34, or the dividend. And in the last example, if we multiply the divisor 9 by the quotient 4, and to the product 36 add the remainder 5, we obtain the dividend 41.
55. Lastly, it is necessary to remark here, with regard to the signs + pZz^^ and — minus, that if we divide + ah by
+ rt, the quotient will be +6, which is evident. But if we divide -\- abhy — cr, the quotient will be — 6 ; because —a
X — b gives + ab. If the dividend is — ab, and is to be divided by the divisor +a, the quotient will he —b; because it is —b which, multiphed by +a, makes —ab. Lastly, if we have to divide the dividend —ab by the divisor —a, the quotient will be + 6 ; for the dividend — ab is the product of — a by -\- b.
56. With regard, therefore, to the signs + and — , di- vision requires the same rules to be observed that we have seen take place in multiplication ; viz.
-h by 4- makes + ; + by — makes — ;
— by 4- makes — ; — by — makes + :
or, in few words, like signs give plus, and unlike signs give viinus.
57. Thus when we divide 18pq by — 3p, the quotient is — 6q. Farther ;
— SOxi/ divided by -j- 6y gives — 5x, and
— 54a6c divided by — 9b gives + 6ac ;
for, in this last example, — 9b multiplied by 4- 6ac makes —6 X 9abc, or — 54fl6c. But enough has been said on the division of simple quantities ; we shall therefore hasten to, the explanation of fractions, after having added some further remarks on the nature of numbers, with respect to their divisors.
16 ELEMENTS SECT. I.
CHAP. VI.
Of the Properties o/" Integers, with respect to their Divisors.
58. As we have seen that some numbers are divisible by certain divisors, while others are not; it will be proper, in order to obtain a more particular knowledge of numbers, that this difference should be carefully observed, both by distinguishing the numbers that are divisible by divisors from those which are not, and by considering the remainder that is left in the division of the latter. For this purpose, let us examine the divisors ;
2, 3, 4, .5, 6, 7, 8, 9, 10, &c. 59- First let the divisor be 2 ; the numbers divisible by it are, 2, 4, 6, 8, 10, \% 14, 16, 18, 20, &c. which, it appears, increase always by two. These numbers, as far as they can be continued, are called even numbers. But there are other numbers, viz.
1,3, 5, 7,9, 11, 13, 15, 17, 19, &c. which are uniformly less or greater than the former by unity, and which cannot be divided by 2, without the remainder 1 ; these are called odd numbers.
The even numbers may all be comprehended in the ge- neral expression 2a ; for they are all obtained by successively substituting for a the integers 1, 2, 3, 4, 5, 6, 7, &c. and hence it follows that the odd numbers are all comprehended in the expression 2a + 1, because 2a + 1 is greater by unity than the even number 2fl.
60. In the second pla<;e, let the number 3 be the divisor ; the numbers divisible by it are,
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on ; which numbers may be represented by the expression 3a ; for 3a, divided by 3, gives the quotient a without a re- mainder. All other numbers which we would divide by 3, will give 1 or 2 for a remainder, and are consequently of two Kinds. Those which after the division leave the re- mainder 1, are,
1, 4, 7, 10, 13, 16, 19, &c. and are contained in the expression 3a + 1 ; but the other kind, where the numbers give the remainder 2, are,
2,5,8, 11, 14, 17, 20, &c. which may be generally represented by 3a + 2 ; so that all numbers may be expressed either by 3a, or by 3a -f 1, or by 3a + 2.
CHAP. VI. OF ALGEBRA. 17
61. Let US now suppose that 4 is the divisor under con- sideration ; then the numbers which it divides are,
4, 8, L?, 16, 20, 24, &c. which increase uniformly by 4, and are comprehended in the expression 4a. All other numbers, that is, those which are not divisible by 4, may either leave the remainder 1, or be greater than the former by 1 ; as,
1, 5, 9, 13, 17, 21, 25, &c.
and consequently may be comprehended in the expression 4a + 1 : or thev may give the remainder 2 ; as,
2, 6, 10, 14, 18, 22, 26, &c.
and be expressed by 4a + 2 ; or, lastly, they may give the remainder 3 ; as,
3,7, 11, 15,19,23,27, &c. and may then be represented by the expression 4a + 3.
All possible integer numbers are contained therefore in one or other of these four expressions :
4a, 4a + 1, 4a + 2, 4a + 3.
62. It is also nearly the same when the divisor is 5; for all numbers which can be divided by it are compre- hended in the expression 5a, and those which cannot be divided by 5, are reducible to one of the following ex- pressions :
5a + 1, 5a + 2, 5a + 3, 5a + 4 ; and in the same manner we may continue, and consider any greater divisor.
63. It is here proper to recollect what has been already said on the resolution of numbers into their simple factors; for every number, among the factors of which is found
2, or 3, or 4, or 5, or 7, or any other number, will be divisible by those numbers. For example; 60 being equal to 2 x 2 x 3 y 5, it is evident that 60 is divisible by 2, and by 3, and by 5 *'.
* There are some numbers which it is easy to perceive whether they are divisors of a given number or not.
1. A given number is divisible by 2, if the last digit is even ; it is divisible by 4, if the two last digits are divisible by 4 ; it is divisible by 8, if the three last digits are divisible by 8 ; and, in general, it is divisible by 2", if the n last digits are divisible by 2".
2. A number is divisible by 3, if the sum of the digits is di- visible by 3 ; it may be divided by (3, if, beside this, the last digit is even ; it is divisible by 9, if the sum of the digits may be divided by 9.
3. Every number that has the last digit O or 5, is divisible by 5.
c
^*
18 ELEMENTS SECT. I,
64. Fartlier, as the general expression alicd is not only divisible by o, and b, and c, and d, but also by
ah, ac, nd, be, bd, cd, and by abc, add, acd, bed, and lastly by abed, that is to say, its own value ; it follows that 60, or 2 x 2 x 3 x 5, may be divided not only by these simple numbers, but also by those which are composed of any two of them; that is to say, by 4, 6, 10, 15 : and also by those which are composed of any three of its simple factors; that is to say, by 12, 20, 30, and lastly also, by 60 itself.
65. When, therefore, we have represented any number assumed at pleasure, by its simple factors, it will be very easy to exhibit all the numbers by which it is divisible. For we have only, first, to take the simple factors one by one, and then to multiply them together two by two,
4. A number is divisible by 11^ when the sum of the first, third, fifths &c. digits is equal to the sum of the second, fourth, sixth, &c. digits.
It would be easy to explain the reason of these rules, and to extend them to the products of the divisors which we have just now considered. Rules might be devised likewise for some other numbers, but the application of them would in general be longer than an actual trial of the division.
For example, I say that the number 53704689213 is divisible by 7, because I find that the sum of the digits of the number 64004245433 is divisible by 7; and this second number is formed, according to a very simple rule, from the remainders found after dividing the component parts of the former number by 7-
Thus, 53704689213 = 50000000000 + 3000000000 + 700000000 + 0 4- 4000000 + 600000 + 80000 + 9000 + 200 + 10 + 3; which being, each of them, divided by 7, will leave the remainders 6, 4, 0, 0, 4, 2, 4, 5, 4, 3, 3', the num.ber here given. Bernoidli.
If a, b, c, dy e, &c. be the digits composing any number, the number itself may be expressed universally thus; a -\- ]0b + 10*c + 10^0? H- \0*c, Sec. to 10"^; where it is easy to perceive that, if each of the terms a, \0b, lOV, &c. be divisible by 7i, the number itself a + lOb + 10%, &c. will also be divisible by n.
^ , .- « 106 \0"'C ^ . , . , „ . .
And, it — , — , , &c. leave the reraamders p, q, r, &c. it is
n H 71 ' ^
obvious, that a 4- 106 + lO'c, &c. will be divisible by n, when p ■{• q + r, is divisible by n ; which renders the principle of the rule sufficiently clear.
The reader is indebted to that excellent mathematician, the late Professor Bonnycastle, for this satisfactory illustration of M. Bernoulli's note.
CHAP. VT.
OF ALGEBRA.
19
three by three, four by four, &c. till we arrive at the number proposed.
66. It must here be particular!}^ observed, that every number is divisible by 1 ; and also, that every number is divisible by itself; so that every number has at least two factors, or divisors, the number itself, and unity : but every number which has no other divisor than these two, belongs to the class of numbers, which we have before called simple, or prime numbers.
Except these simple numbers, all other numbers have, beside unity and themselves, other divisors, as may be seen from the following Table, in which are placed under each number all its divisors *.
TABLE.
|
^ |
■^ |
^ |
-*- |
-v-~ |
'*' |
||||||||||||||
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
I |
1 |
1 |
1 |
1 |
1 |
1 |
! |
1 |
] |
1 |
1 |
|
2 |
3 |
2 |
5 |
2 |
7 |
2 |
3 |
2 |
11 |
2 |
13 |
2 |
3 |
2 |
17 |
2 |
19 |
2 |
|
|
4 |
3 6 |
4 8 |
9 |
5 10 |
3 4 6 12 |
7 14 |
5 15 |
4 8 IG |
3 6 9 18 |
4 5 10 20 |
|||||||||
|
1 |
2 |
2 |
3 |
2 |
4 |
2 |
4 |
3 |
4 |
2 |
6 |
2 |
4 |
4 |
5 |
2 |
6 |
2 |
6 |
|
p. |
P. |
P. |
P. |
P. |
P. |
P. |
P. |
P. |
67. Lastly, it ought to be observed that 0, or nothing, may be considered as a number which has the property of being divisible by all possible numbers; because by what- ever number a we divide 0, the quotient is always 0 ; for it must be remarked, that the multiplication of any number by nothing produces nothing, and therefore 0 times «, or Oa, is 0.
* A similar Table for all the divisors of the natural numbers, from 1 to 10000, was published at Ley den, in 1767, by M. Henry Anjema. We have likewise another table of divisors, which goes as far as 100000, but it gives only the least divisor of each number. It is to be found in Harris's Lexicon Tech- nicum, the Encyclopedic, and in M. Lambert's Recueil, which we have quoted in the note to p. 11. In this lavSt work, it is continued as far as 102000. F. T.
c£
20 ELEMENTS SECT. I.
CHAP. VII.
Of Fractions in general.
68. When a number, as 7, for instance, is said not to be divisible by another number, let us suppose by 3, this only means, that the quotient cannot be expressed by an integer number ; but it must not by any means be thought that it is impossible to form an idea of that quotient. Only imagine a line of 7 feet in length ; nobody can doubt the possibility of dividing this line into 3 equal parts, and of forming a notion of the length of one of those parts.
69. Since therefore we may form a precise idea of the quotient obtained in similar cases, though that quotient may not be an integer number, this leads us to consider a par- ticular species of numbers, caWedi fractions, or broken num.' hers ; of which the instance adduced furnishes an illustration. For if we have to divide 7 by 3, we easily conceive the quotient which should result, and express it by \- ; placing the divisor under the dividend, and separating the two numbers by a stroke, or line.
70. So, in general, when the number a is to be divided by
the number b, we represent the quotient by y-, and call this form of expression a fraction. We cannot therefore give a better idea of a fraction -^-j than by saying that it ex- presses the quotient resulting from the division of the upper number by the lower. We must remember also, that in all fractions the lower number is called the denominator, and that above the line the numerator.
71. In the above fraction ^, which we read seven tliirds, 7 is the numerator, and 3 the denominator. We must also read y, two thirds; |, three fourths; |-, three eighths; -x-o^;;, twelve hundredths; and 4, one half, &c.
72. In order to obtain a more perfect knowledge of the nature of fractions, we shall begin by considering the case in which the numerator is equal to the denominator, as in
— . Now, since this expresses the quotient obtained by
dividing a by o, it is evident that this quotient is exactly
unity, and that consequently the fraction — is of the same
CHAP. VII. OF ALGEBllA. 21
value as 1, or one integer; for the same reason, all the fol- lowing fractions, ^
a 3 4- 5 6 7 8 f^f,
o-J T» "4' T' "6' T» T' ^
are equal to one another, each being equal to 1, or one integer.
73. We have seen that a fraction whose numerator is equal to the denominator, is equal to unity. All fractions therefore whose numerators are less than the denominators, have a value less than unity : for if I have a number to divide by another, which is greater than itself, the result must necessarily be less than 1. If we cut a line, for ex- ample, two feet long, into three equal parts, one of those parts will undoubtedly be shorter than a foot : it is evident then, that ^ is less than 1, for the same reason ; that is, the numerator 2 is less than the denominator 3.
74. If tlie numerator, on the contrary, be greater than the denominator, the value of the fraction is greater than unity. Thus \ is greater than 1, for ^ is equal to ^ together with i. Now ^ is exactly 1 ; consequently 1 is equal to 1 + 4» ^^^^ is, to an integer and a half. In the same manner, ^ is equal to ly, 4" to 1|., and |- to 2|. And, in general, it is sufficient in such cases to divide the upper number by the lower, and to add to the quotient a fraction, having the remainder for the numerator, and the divisor for the denominator. If the given fraction, for example, were ^|^, we should have for the quotient 3, and 7 for the remainder ; whence we should conclude that 41 is the same as S-^^.
75. Thus we see how fractions, whose numerators are greater than the denominators, are resolved into two mem- bers ; one of which is an integer, and the other a fractional number, having the numerator less than the denominator. Such fractions as contain one or more integers, are called improper Jr actions, to distinguish them from fractions pro- perly so called, which having the numerator less than the denominator, are less than uviity, or than an integer.
76. The nature of fractions is frequently considered in another way, which may throw additional light on the sub- ject. If, for example, we consider the fraction |, it is evident that it is three times greater than \. Now, this fraction \ means, that if we divide 1 into 4 equal part?, this will be the value of one of those parts; it is obvious then, that by taking 3 of those parts we shall have the value of the fraction |.
In the same manner we may consider every other fraction ; for example, -J^; if we divide unity into 12 equal parts, 7 of those parts will be equal to the fraction proposed.
22 ELEMENTS SECT. I.
77. From this manner of considering fractions, the ex- pressions numerator and denominator are derived. For, as in the preceding fraction -/^, the number under the line shews that 1 2 is the number of parts into which unity is to be divided ; and as it may be said to denote, or name, the parts, it has not improperly been called the denominator.
Farther, as the upper number, viz. 7, shews that, in order to have the value of the fraction, we must take, or collect, 7 of those parts, and therefore may be said to reckon or num- ber them, it has been thought proper to call the number above the line the numerator.
78. As it is easy to understand what | is, when we know the signification of |, we may consider the fractions whose numerator is unity, as the foundation of all others. Such are the fractions,
I I I I I I I I J I I Crp
'ii T» T' T» 6"> 7' T' 9"' ~o> TTTJ XTJ *-^^*
and it is observable that these fractions go on continually diminishing : for the more you divide an integer, or the greater the number of parts into which you distribute it, the less does each of those parts become. Thus, -^^-o is less than -rV ; -ToW is less than ^^ ; and -rohro is less than
79- As we have seen that the more we increase the de- nominator of such fractions the less their values become, it may be asked, whether it is not possible to make the de- nominator so gi'eat that the fraction shall be reduced to nothing? I answer, no; for into whatever number of parts unity (tiie length of a foot, for instance) is divided ; let those parts be ever so small, they will still preserve a certain magnitude, and therefore can never be absolutely reduced to nothing.
SO. It is true, if we divide the length of a foot into 1000 parts, those })art5 will not easily fall under the cognisance of our senses ; but view them through a good microscope, and cacii of them will appear large enough to be still subdivided into 100 parts, and more.
At present, however, we have nothing to do with what depends on ourselves, or with what we are really capable of performing, and what our eyes can perceive; the question is rather what is possible in itself: and, in this sense, it is certain, that however great we suppose the denominator, the fraction will never entirely vanish, or become equal to 0.
81. We can never therefore arrive completely at 0, or nothing, however great the denominator may be ; and, con- se([uentlv, as those fractions nuist always preserve a cer- tain quantity, we may continue the series of fractions in the
CHAP. VII. OF ALGEBKA. 23
78th article without interruption. This circumstance has in- troduced the expression, that the denominator must be in- finite, or infinitely great, in order that the fraction may be reduced to 0, or to nothing; hence the word infinite in reality signifies here, that we can never arrive at the end of the series of the above-men tionedjTraci^iowi-.
82. To express this idea, according to the sense of it above-mentioned, we make use of the sign x , which con- sequently indicates a number infinitely great ; and we may therefore say, that this fraction ^ is in reality nothing ; be- cause a fraction cannot be reduced to nothing, until the denominator has been increased to injinity.
83. 1 1 is the more necessary to pay attention to this idea of infinity, as it is derived from the first elements of our know- ledge, and as it will be of the greatest importance in the following part of this treatise.
We may here deduce from it a few consequences that are extremely curious, and worthy of attention. The fraction ^ represents the quotient resulting from the division of the dividend 1 by the divisor co . Now, we know, that if we divide the dividend 1 by the quotient ^, which is equal to nothing, we obtain again the divisor oo : hence we acquire a new idea of infinity ; and learn that it arises from the division of 1 by 0; so that we are thence authorised in saying, that 1 divided by 0 expresses a number infinitely great, or oo .
84. It may be necessary also, in this place, to correct the mistake of those who assert, that a number infinitely great is not susceptible of increase. This opinion is inconsistent with the just principles which we have laid down ; for ^^ signifying a number infinitely great, and ~ being incon- testably the double of ^, it is evident that a number, though infinitely great, may still become twice, thrice,- or any num- ber of times greater *.
* There appears to be a fallacy in this reasoning, which con- sists in taking the sign of infinity for infinity itself; and applying the property of fractious in general to a fractional expression, whose denominator bears no, assignable relation to unity. It is certain, that infinity may be represented by a series of units (that
is, by ^ = = I -f- 1 +1, &c.) or by a series of numbers
increasing in any given ratio. Now, though any definite part of one infinite series may be the half, the third, &c. of a definite part of another, yet still that part bears no proportion to the whole, and the series can only be said, in that case, to go on to infinity in a diffeitnt lalio. But, farther, -^j or any other nu-
24 ELEMENTS SECT. I.
CHAP. VIII.
Of the Properties of Fractions. 85. We have already seen, that each of the fractions,
2 3 4 S 6 7 a C,„
T> 3» T> T» ■g' T> "3> *^'-*
makes an integer, and that consequently they are all equal to one another. The same equality prevails in the following fractions,
T» •;• ) T*
each of them making two integers ; for the numerator of each, divided by its denominator, gives 2. So all the fractions
3 6 9 la IJ 1_8 ^p
are equal to one another, since 3 is their common value.
86. We may likewise represent the value of any fraction in an infinite variety of ways. For if we multiply both the numerator and the denominator of a fraction by the same number, which may be assumed at pleasure, this fraction will still preserve the same value. For this reason, all the fractions
1.234 5 6 7 8 910 0^^
2' 4» 6> "»> "ns"; TT' TT> TT> T?' ''"O'' "'*^'
are equal, the value of each being i. Also,
13 3 4 s 6 7 8 9 10 0,^
3! TJ TJ TT> -rsi T1b» ITTJ TT> 'JT> TTVy tX^ •
are equal fractions, the value of each being ^. The fractions
84 3 10 12 14 16 0,„
T> T' TT5 TTJ T7» "iTTJ TT' "■*'•
have likewise all the same value. Hence we may conclude, in general, that the fraction -7- may be represented by any
of the following expressions, each of which is equal to -^; viz.
merator, having 0 for its denominator, is, when expanded, pre- cisely the same as -i^.
2 Thu3j ^ = 7z — -i by division becomes A— ^
2—2)2 (1 + 1 + 1, &c- ad infinitum
2-2
2
2-2
2 2-2
2, &c.
CHAP. VIll. OF ALGEBRA. 25
a 2a 3a 4a 5a 6a 7a T' 2b' 3b' W 5// W W ^'''
87. To be convinced of this, we have only to write for the
value of the fraction -7- a certain letter c, representing by
this letter c the quotient of the division of « by 6 ; and to recollect that the multiplication of the quotient c by the divisor b must give the dividend. For since c multiplied by b gives a, it is evident that c multiplied by 2b will give 2a, that c multiplied by 3b will give Sa, and that, in general, c multiplied by mb will give ma. Now, changing this into an example of division, and dividing the product ma by 7nb, one of the factors, the quotient must be equal to the other factor c; but ma divided by ?nb gives also the fraction
—7, which is consequently equal to c ; and this is what was
to be proved : for c having been assumed as the value of the
fraction -y-, it is evident that this fraction is equal to the
fraction — r, whatever be the value of m. mb
88. We have seen that every fraction may be represented in an infinite number of forms, each of which contains the same value ; and it is evident that of all these forms, that which is composed of the least numbers, will be most easily understood. For example, we may substitute, instead of y, the following fractions,
4 6 8 I 0 I 2 Sj-f,
6> J> TT' TJ) TY'
but of all these expressions ^ is that of which it is easiest to form an idea. Here therefore a problem arises, how a fraction, such as —^ which is not expressed by the least possible numbers, may be reduced to its simplest form, or to its least terms; that is to ?ay, in our present example, to ^.
89. It will be easy to resolve this problem, if we consider
that a fraction still preserves its value, when we multiply
both its terms, or its numerator and denominator, by the
same number. For from this it also follows, that if we
divide the numerator and denominator of a fraction by the
same number, the fraction will still preserve the same value.
This is made more evident by means of the general ex-
ma pression — 7 ; for if we divide both the numerator tna and mb
the denominator mb by the number m, we obtain the fraction a .. . . ma
-7-, whicli, as was before proved, is e(jual to — r.
^'•^
Xb ELEMENTS SECT. I.
90. In order therefore to reduce a given fraction to its least termS) it is required to find a number, by which both the numerator and denominator may be divided. Such a number is called a common divisor ; and as long as we can find a common divisor to the numerator and the denominator, it is certain that the fraction may be reduced to a lower form ; but, on the contrary, when we see that, except vmity, no other common divisor can be found, this shews that the fraction is already in its simplest form.
91. To make this more clear, let us consider the fraction ^-. We see immediately that both the terms are divisible by 2, and that there results the fraction -|^ ; which may also be divided by 2, and reduced to ~- ; and as this likewise has 2 for a common divisor, it is evident that it may be re- duced to -^^. But now we easily perceive, that the nume- rator and denominator are still divisible by 3; performing this division, therefore, we obtain the fraction -I-, which is equal to the fraction proposed, and gives the simplest ex- l^ression to which it can be reduced ; for 2 and 5 have no common divisor but 1, which cannot diminish these numbers any farther.
92. This property of fractions preserving an invariable value, whether we divide or multiply the numerator and denominator by the same number, is of the greatest import- ance, and is the principal foundation of the doctrine of fractions. For example, we can seldom add together two fracticnis, or subtract the one from the other, before we have, by means of this property, reduced them to other forms; that is to say, to expressions whose denominators are equal. Of this we shall treat in the following chapter.
93. We will conclude the present, however, by remarking, that all whole numbers may also be represented by fractions. For example, 6 is the same as ~, because 6 divided by 1 makes 6 ; we may also, in the same manner, express the number 6 by the fractions '^^, 'j?, ^^*, \^, and an infinite number of others, which have the same value.
QUESTIONS FOR PRACTICE.
1. Reduce — -- — -— to its lowest terms. Ans. — r.
ca^ + a-x a-
% Reduce — — ^rr r to its lowest terms. Ans. — — r-.
a:2 + 2Zi.r + Z»- x-\-b
^i J4 x"+b-
Ci. Reduce -,- — , — :, to its lowest terms. Ans. :r— .
ijj^
CHAP. IX. OF ALGEBUA. 27
X^—l/^ 1
4. Reduce — r r to its lowest terms. Ans.
X*—7J* ' * X-+I/"'
Q* ^*
5. Reduce -r :; . to its lowest terms.
a-+x'^
o. Reduce -^ — , ^ , ^ , ,. — . 2 to its lowest terms.
r-ta^ + ^a^x Ans.
a-x + ax" + a;'*
CHAP. IX.
Of the Addition and Subtraction q/" Fractions.
94. When fractions have equal denominators, there is no difficulty in adding and subtracting them ; for ~ + ^ is equal to 4, and ^ — ^ is equal to ~. In this case, therefore, either for addition or subtraction, we alter only the nume- rators, and place the common denominator under the line, thus;
7 _|_ 9 I a I s I 2 o I'c pniinl tfi 9 .
""00^ i^ To^' joo 100 '^ 100'^'' tv^uaj. vj -To^o- ,
|4°- ^ - i4 + i4 is equal to f^, or if
1 o
+ 44isequal to44, ori:; also ^ + |. is equal to -I, or 1, that is to say, an integer ; and ^ — f 4- i is equal to ^, that is to say, nothing, or 0.
95. But when fractions have not equal denominators, we can always change them into other fractions that have tlie same denominator. For example, when it is proposed to add together the fractions i and g, we must consider that i is the same as ^^ and that g is equivalent to |^ ; we have therefore, instead of the two fractions proposed, |. + |:, the sum of which is ~. And if the two fractions were united by the sign m'mus, as i — 3> we should have |- — -|, or i.
As another example, let the fractions proposed be | + |. Here, since | is the same as |-, this value may be substituted for i, and we may then say f + |- makes -g-, or If.
Suppose farther, that the sum of ^ and | were required, I say that it is -J^ ; for -J- = /^, and i = -^^ : therefore ^^- 4- -^- = -^-.
96. We may have a greater number of fractions to reduce
28 ELEMENTS SECT. I.
to a common denominator ; for example, ^, ~, |^, ^, 4- I" this case, the whole depends on finding a number that shall be divisible by all the denominators of those fractions. In this instance, 60 is the number which has that property, and which consequently becomes the common denominator. We shall therefore have ||, instead of ~ ; |-°, instead of -f ; ^|, instead of | ; ||, instead of ^^ ; and ^, instead of |. If now it be required to add together all these fractions, |4, 4^, 4-1, 4_8-, and -g-l ; we have only to add all the numerators, and under the sum place the common denominator 60 ; that is to say, we shall have y^^- , or 3 integers, and the fractional remainder, ||-, or 44- .
97. The whole of this operation consists, as we before stated, in changing fractions, whose denominators are un- equal, into others whose denominators are equal. In order,
therefore, to perform it generally, let -r- and —j- be the frac- tions proposed. First, multiply the two terms of the first fraction by d, and we shall have the fraction y-^ equal
to -J- ; next multiply the two terms of the second fraction
by 6, and we shall have an equivalent value of it expressed
be by j-^ ; thus the two denominators are become equal. Now,
if the sum of the two proposed fractions be required, we
VI I • • o,d-\-bc , . . , . -.
may immediately answer that it is — 7-7— ; and il their dif- ference be asked, we say that it is — tt~- If the fractions
I and ^, for example, were proposed, we should obtain, in their stead, A| andff; of which the sum is '-^' and the difference — *.
98. To this part of the subject belongs also the question, Of two proposed fractions which is the greater or the less ?
* The rule for reducing fractions to a common denominator may be concisely expressed thus. Multiply each numerator into every denominator except its own, for a new numerator, and all the denominators together for the common denomi- nator. When this operation has been performed, it will appear that the numerator and denominator of each fraction have been nuiltiplied by the same quantity, and consequently that the iVactions retain the same value.
CHAP. IX. OF ALGEBRA. 29
for, to resolve this, we have only to reduce the two fractions to the same denominator. Let us take, for example, the two fractions -| and ^ ; when reduced to the same denominator, the first becomes 4-r> ^"^ the second i4> where it is evident that the second, or i^, is the greater, and exceeds the former
Again, if the fractions -f and | be proposed, we shall have to substitute for them 1 J- and ^ ; whence we may conclude that ^ exceeds ^, but only by ^.
99. When it is required to subtract a fraction from an integer, it is sufficient to change one of the units of that integer into a fraction, which has the same denominator as that which is to be subtracted ; then in the rest of the opera- tion there is no difficulty. If it be required, for example, to subti'act ~ from 1, we write |- instead of 1, and say that |- taken from 4 leaves the remainder j-. So — , subtracted from 1, leaves -^.
If it were required to subtract f from 2, we should write 1 and 1^ instead of 2, and should then immediately see that after the subtraction there must remain li.
100. It happens also sometimes, that having added two or more fractions together, we obtain more than an integer; that is to say, a numerator greater than the denominator : this is a case which has already occurred, and deserves attention.
We found, for example [Article 96], that the sum of the five fractions i, f, ~, -f, and ^ was %'--J, and remarked that the value of this sum was 3|4 or S^-. Likewise, ^-\-^, or ^ 4- _?_., makes ^, or l-j-'^^. We have therefore only to perform the actual division of the numerator by the deno- minator, to see how many integers there are for the quotient, and to set down the remainder.
Nearly the same must be done to add together numbers compounded of integers and fractions; w^e first add the fractions, and if the sum produces one or more integers, these are added to the other integers. If it be proposed, for ex- ample, to add 3{- and 2y; we first take the sum of i and |, or of 1^ and |, which is |^, or 1- ; and thus we find the total sum to be 6|.
QUESTIONS FOU PRACTICE.
Qx b
1. Reduce — and — to a common denominator. a c
9.CX ah
Ans. — and — . ac ac
30 ELEMENTS SECT. I,
- _^ - a , a + Z)
2. Reduce -v- sincl to a common denominator.
o c
flc ah-{-h"
Am. -J- and — -, .
be be
„ „ , 3^ 25 , , ^ . , .
3. Reduce TT", rz-. and fZ to tractions havma; a common
2a' 3c '^
9ca; 4«& Qacd
denominator. Ans. y, — , 7: — , and ;^ — .
bac oac oac
, „ , 3 2.r , 2a;
4. Reduce 7, -tti and a H to a common denominator.
4 3' a
9a 8a■^' 12a2+24a:
^??s. 77:-, tttj and 75— -•
12ft 12ft 12ft
^ „ . 1 ft- , a;- + ft' -
5. Reduce -, -tt-, and to a common denominator.
2' 3' a; + fl
^ 3ar + 3ft 2a2^ + 2fl^ 6.r'- + 6fl2 * 6a; + 6ft' 6^ + 6ft ' 6^ + 6ft '
6. Reduce 7i — , -tt-, and — to a common denominator.
2ft- 2ft' a
. ^.a^b ^a?c , ^aH b ac . 2aJ
^^"- 1^' 4^' ""^ -4^' °^' 2^' 2^=' ^"^^ 2^-
CHAP. X.
Of the Multiplication and Division of Fractions.
101. The rule for tlie multiplication of a fraction by an integer, or whole number, is to multiply the numerator only by the given number, and not to change the deno- minator : thus,
2 times, or twice ^ makes ^, or 1 integer ;
2 times, or twice ^ makes ~ ; and
3 times, or thrice ^ makes -|) or -|^ ;
4 times -^^ makes 44? or l-i^, or 1|..
But, instead of this rule, we may use that of dividing the denominator by the given integer, which is preferable, when it can be done, because it shortens the operation. Let it be required, for example, to multiply |. by 3 ; if we multiply the numerator by the given integer we obtain Y» ^vhich
CHAP. X. OF ALGEBRA. 31
product we must reduce to y. But if we do not c'aangc the numerator, and divide the denominator by the integer, we find immediately ^, or 2|-, for the given product ; and, in the same manner, 44 multiphed by 6 gives y , or 3^^.
102. In general, therefore, the product of the multiplica-
tion of a fraction -j- by c is -j- ; and here it may be re- marked, when the integer is exactly equal to t!ie denominator, that the product must be equal to the numerator.
( i taken twice, gives 1 ; So that< ^ taken thrice, gives 2 ;
( i taken four times, gives 3.
And, in general, if we multiply the fraction -j- by the number b, the product must be a, as we have already shewn ; for since -j- expresses the quotient resulting from the di- vision of the dividend a by the divisor b, and because it has been demonstrated that the quotient multiplied by the divisor
will give the dividend, it is evident that -j- multiplied by b
must produce a.
103. Having thus shewn how a fraction is to be mul- tiplied by an integer ; let us now consider also how a fraction is to be divided by an integer. This inquiry is necessary, before we proceed to the multiplication of fractions by frac- tions. It is evident, if we have to divide the fraction ~ by 2, that the result must be^; and that the quotient of|- divided by 3 is y. The rule therefore is, to divide the numerator by the integer without changing the denominator. Thus:
i-i divided by 2 gives -^ ; ^ divided by 3 gives -— ; and i|- divided by 4 gives ^ ; &c.
104. This rule may be easily practised, provided the numerator be divisible by the number proposed ; but very often it is not : it must therefore be observed, that a fraction may be transformed into an infinite number of other ex- pressions, and in that number there must be some, by which the numerator might be divided by the given integer. If it were required, for example, to divide ^ by 2, we should change the fraction into |, and then dividing the numerator by 2, we should immediately have |- for the quotient sought.
32 ELEMENTS SECT. I.
, a In general, if it be proposed to divide the fraction -j-
ctc by c, we change it into -,— , and then dividing- the nume- rator ac by c, write -j— for the quotient sought.
105. When therefore a fraction -j- is to be divided by an
integer c, we have only to multiply the denominator by that number, and leave the numerator as it is. Thus |^ divided by 3 gives -—, and -^ divided by 5 gives ^^^-.
This operation becomes easier, when the numerator itself is divisible by the integer, as we have supposed in article 103. For example, -f'-^ divided by 3 would give, according to our last rule, ^^; but by the first rule, which is applica- ble here, we obtain -^-^^ an expression equivalent to ^y, but more simple.
106. We shall now be able to understand how one fraction
d c
■J- may be multiplied by another fraction -j. For this pur-
pose, we have only to consider that — means that c is di- vided by d; and on this principle we shall first multiply the fraction -j- by c, which produces the result -j- ; after which
ttC
we shall divide by d, which gives y-v.
Hence the following rule for multiplying fractions. Mul- tiply the numerators together for a numerator, and the de- nominators together for a denominator.
Thus ~ by i- gives the product ^, or ^ ; ■f- by A makes -—- ; 4 by it: produces i|-, or -^-^ ; &c.
107. It now remains to shew how one fraction may be divided by another. Here we remark first, that if the two fractions have the same number for a denominator, the division takes place only with respect to the numerators ; for it is evident, that -^ are contained as many times in ^ as 3 is contained in 9, that is to say, three times ; and, in the same manner, in order to divide -^ by -j^, we have only to divide 8 by 9, which gives ^. We shall also have -^^ in 44, 3 times; -^l^ in -±%, 7 times; ^ in -i-j-, |-, &c.
108. But when the fractions have not equal denominators,
CHAP. \. OF ALGF.BRA. 85
we must have recourse to the method ah-eady mentioned for reducing them to a common denominator. Let there be,
for example, the fraction — to be divided by the fraction
c -7-. We first reduce them to the same denominator, and
there results 7-^ to be divided hy -rr;\t is now evident that
bd ■^ do
the quotient must be represented simply by the division of
ad by be ; which gives -j — .
Hence the following rule : Multiply the numerator of the dividend by the denominator of the divisor, and the de- nominator of the dividend by the numerator of the divisor ; then the first product will be the numerator of the quotient, and tlie second will be its denominator.
109. Applying this rule to the division of |- by i, we shall have the quotient i^ ' ^^^^ the division of i by f will give I, or A, or If ; and |4 by l- will give 44°, or f."'
110. This rule for division is often expressed in a manner that is more easily remembered, as follows : Invert the terms of the divisor, so that tlie denominator may be in the place of the numerator, and the latter be written under the line ; then multiply the fraction, which is the dividend bv this inverted fraction, and the product will be the quo- tient sought. Thus, I divided by t is the same as | mul- tiplied by ^ , which makes 1, or \\. Also |- divided by i is the same as |- multiplied by 4, which is \t ; or 1|- divided by -i gives the same as i|- multiplied by ~, the product of which is ^4^°, or |.
We see then, in general, that to divide by the fraction | is the same as to multiply by 3, or 2; and that dividing by i amounts to multiplying by \, or by 3, &c.
111. The number 100 divided'by f will give 200; and 1000 divided by \ will give 3000. Farther, if it were re- quired to divide 1 ^y -reooi the quotient would be 1000; and dividing 1 by -q-oWo^j the quotient is 100000. This enables us to conceive that, when any number is divided by 0, the result must be a number indefinitely great ; for even the division of 1 by the small fraction -to^tto^ c^-oo o gives for the quotient the very great number 1000000000.
112. Every number, when divided by itself, producing unity, it is evident that a fraction divided'by itself must also give 1 for the quotient ,• and the same follows from our rule : for, in order to divide | by |, we must multiply i by 4, iii
34 ELEMENTS SECT. I.
which case we obtain 44, or 1 ; and if it be required to divide -f-hy ~r~, we multiply -t" by — ; where the product
ah .
—r IS also equal to 1.
113. We have still to explain an expression which is frequently used. It may be asked, for example, what is the half of I? This means, that we must multiply i by f. So likewise, if the value of ~ of |- were required, we should multiply I by f , which produces i^ ; and | of -^-^ is the same as -j^^. multiplied by f, which produces ^.
114. Lastly, we must here observe, with respect to the signs + and — , the same rules that we before laid down for integers. Thus + f multiplied by — 4, makes — -g ; and — y multiplied by — *, gives + J,-. Farther — A divided by + I, gives — 4-^; and — i divided by — |, gives + 4-|, or + 1.
QUESTIOXS FOR PRACTICE.
1. Required the product of _- and — . ^ns. -^^.
« -n -11 1 „ X 4>x _ 10.r . 4a,''
S. Required the product or —, --, and -^y-. Ans. ■^'
o. Required the product 01 — and . Ans.
* *^ a a+c a- + ac
4. Required the product of -^ and -j-. Ans. -^j--
2a7 3j.» 3-J.3
5. Required the product of — and ^— . Ans. -^'
o. Required the product ot — , — , and -^j-. A7i\. 9ax.
7. Required the product of b \ and — .
ab + bx
Ans.
8. Required the product of —. — and
x^-b'' Ans.
b"c + bc^
CHAP. X. OF ALGETiliA. 35
9. Required the product of .r, , and r-
* ^ a a+o
x" — :i'
Am. -^r-, — i.. a--\-ao
10. Required the quotient of -^ divided by -^. Ans. \~.
^ ft 4c
11. Required the quotient of y- divided by -y.
Ans. ---. 2bc
12. Required the quotient of ^ divided by - .
. 5.V- + 6ax + a-
9.x" -W-
X
y>^X X
13. Required the quotient of -z divided by .
^ ^ a^^x' •' x+a
^x- + 9,ax
Ans. 3 , , .
7.C 12 91a;
14. Required the quotient of-^ divided by ^77. Ans. -^.
15. Required the quotient of-=- divided by 5x. Ans. ^z.
X -\-\ 2,r
16. Required the quotient of .. divided by — .
. 07-1-1
Ans. —. — -. 4.r
17. Required the quotient of , divided by -jy .
. X — b
Ans. ri .
bc'x ^4. _ n
18. Required the quotient of -;;; — r— ^^ divided by - -
3^" ^ /COX "T" O"
x^-\-bx b-
b2
36
ELEMENTS SECT. I.
CHAP. XI.
O/^^"^''^ Numbers.
115. Tho product of a number, when multiplied by itself, is called a square ; and, for this reason, the number, considered in relation to such a product, is called a square root. For example, when we multiply 12 by 12, the product 144 is a square, of which the root is 12.
The origin of this term is borrowed from geometry, which teaches us that the contents of a square are found by mul- tiplying its side by itself.
116. Square numbers are found therefore by multiplica- tion; that is to say, by multiplying the root by itself: thus, 1 is the square of 1, since 1 multiplied by 1 makes 1 ; like- wise, 4 is the square of 2; and 9 the square of tJ; 2 also is the root of 4, and 3 is the root of 9.
We shall begin by considering the squares of natural numbers ; and for this purpose shall give the following small Table, on the first line of which several numbers, or roots, are ranged, and on the second their squares *.
|
Numbers. Squares. |
1 2 1 4 |
3 9 |
4 5 16 25 |
6 7 36|49 |
8 64 |
9 81 |
10 100 |
11 121 |
12 144 |
13 169 |
117. Here it will be readily perceived that the series of square numbers thus arranged has a singular property; namely, that if each of them be subtracted from that which immediately follows, the remainders alwa3's increase by 2, and form this series ;
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c. which is that of the odd numbers.
118. The squares of fractions are found in the same manner, by multiplying any given fraction by itself. For example, the square of \ is ^,
* We have very complete tables for the squares of natural numbers, published under the title " Tctragonometria Tabularia, &c. Auct. J. Jobo Ludolfo, Amstelodami, 1690, in 4to." These Tables are continued from 1 to 100000, not only for finding those squares, but also the products of any two numbers less than 100000; not to mention several otiier uses, which are explained in the introduction to the work. F. T.
CHAT. Xr. OF ALGEBRA. b I
The square of -
3 1 9 I
We have only therefore to divide the square of the numerator by the square of the denominator, and tb.e friaction which expresses tliat division will be the square of the given fraction; thus, ||- is the square of ^; and re- ciprocally, ^ is the root of |^|^.
119- When the square of a mixed number, or a number comjKJsed of an integer and a fraction, is required, we have only to reduce it to a single fraction, and then take tlie square of that fraction. Let it be required, for example, to find the square of ^\ ; we first express this mixed number by i^, and taking the square of that fraction, we have y , or 6i, for the value of the square of 2;. Also to obtain the square of 3i, we say 3;^ is equal to y ; therefore its square is equal to ^-^ , or to 10/^. The squares of the numbers between 3 and 4, supposing them to increase by one fourth, arc as follow :
|
Numbers. | 3 | .S-| | 3^ j 3| | ^1 1 |
|
Squares. | 9 1 [0^\\ 12i | 14^-'^ | u\ |
From this small Table we may infer, that if a root cojitain a fraction, its square also contains one. Let the root, for cxavnple, be 1~; its square is ~?^, or ^ri? 5 that is to say, a little greater than the integer 2.
120. Let us now proceed to general expressions. First, when the root is a, the square must be aa ; if the root be *la, the square is 4a«; which shews that by doubling the root, the square becomes 4 times greater ; also, if the i"oot be oa, the square is ^3aa ; and if the root be 4«, the h(|uare is iQaa. Farther, if the root be ah, the square is aabb ; and if the root be ahc, the square is aabbcc; or ab-c^.
12L Thus, when the root is composed of two, or more factors, we multiply their squares together ; and, reciprocally, if a square be composed of two, or more factors, of which each is a square, we have only to multiply together the roots of those squares, to obtain the complete root of the square proposed. Thus, 2304 is equal to 4 x 16 x 36, the square root of which is 2 x 4 x 6, or 48 ; and 48 is found to be the true square root of 2304, because 48 x 48 gives .^304.
122. Let us now consider what must be observed on this subject with regard to the oigns + and -. First, it ib
459399
38 ELEMENTS SECT. I.
evident that if tlie root have the sign +, that is to say, if it be a positive number, its square must necessarily be a positive number also, because + multiplied by + makes + : hence the square of + a will be + an : but if the root be a negative number, as — a, the square is still positive, for it is + aa. We may therefoi-e conclude, that + aa is the square both of -i- a and of - a, and that consequently every square has two roots, one positive, and the other negative. The square root of 25, for example, is both -f- 5 and — 5, because — 5 mul- tiplied by — 5 gives 25, as well as + 5 by -f 5.
CHAP. XII.
Of Square Roots, and of Irrational Numbers re suiting from
them.
\2S. What we have said in the preceding chapter amounts to this ; that the square root of a given number is that num- ber whose square is equal to the given number ; and that we may juit before those roots either the positive, or the negative sio-n.
121. So that when a square number is given, provided Ave retain in our memory a sufficient number of square num- bers, it is easy to find its root. If 196, for example, be the given nmnber, we know that its square root is 14.
Fractions, likewise, are easily managed in the same way. It is evident, for example, that ■?- is the square root of i|-; to be convinced of which, we have only to take the square root of the numerator and that of the denominator.
If the number proposed be a mixed number, as 121, ^ve reduce it to a single fraction, which, in this case, will be *^ ; and from this we immediately perceive that ^-, or 3i, must be the square root of 12|.
125. But when the given number is not a square, as 12, for example, it is not ])ossible to extract its square root ; or to find a number, which, multiplied by itself, will give the product 12. Ave know, however, that "the square root of 12 must be greater than 3, because 3x3 produces only 9; and less than 4, because 4 x 4 produces 16, which is more than 12; we know also, that this root is less than 3^, for we have seen that the square of 3[, or ^, is 12'-; and we may approach still nearer to this root, by comparing it with 3/. ; for the square of 3/3, or of ]-],, is Vz°yj t)r 12^^ 5; so that this
CHAP. XII. OF ALGEBRA. 39
fraction is still greater than tlie root rec[uired, though but very little so, as the difference of the two squares is only -^-f 3^.
126. We may suppose that as 3^ and 3^^ are numbers greater than the root of 12, it might be possible to add to 3 a fraction a little less than ~, and precisely such, that the square of the sum would be equal to 12.
Let us therefore try with 3^-, since ^'isa little less than ^^y. Now 3f is equal to y-, the square of which is %^^ , and con- sequently less by il than 12, which may be expressed by 5_?^. It is, therefore, proved that 3} is less, and that 3/5- is greater than the root required. Let us then try a num- ber a little greater than 3^, but yet less than S/^-; for ex- ample, 3-j^,- ; tills number, which is equal to ~^, has for its square '^/Z ; and by reducing 12 to this denominator, we obtain V^rr^ which sheAvs that 3 j^,- is still less than the root of 12, viz. by -rlx^ let us thereibre substitute for ■^\- the fraction -i^y, which is a little greater, and see what will be the result of the comparison of the square of 3,-3-, with the proposed num- ber 12. Here the square of 3/3- is \°g^/ ; and 12 reduced to the same denominator is VeV 5 ^^ ^-'^^^ ^rr ^^ ^^'^^ ^'^^ small, though only by -y-f-^, whilst 3,^ has been found too great.
127. It is evident, therefore, that whatever fraction is joined to 3, the square of that sum must always contain a fraction, and can never be exactly equal to the integer 12. Thus, although Ave know that the square root of 12 is greater than 3 ^^3-, and less than o-^\, yet we are unable to assign an intermediate fraction between these two, which, at the same time, if added to 3, would express exactly the square root of 12; but notwithstanding this, we are not to assert that the square root of 12 is absolutely and in itself indeterminate : it only follows from what has been said, that this root, though it necessarily has a determinate magnitude, cannot be ex- pressed by fractions.
128. There is therefore a sort of numbers, which carmot be assigned by fractions, but which are nevei'theless determinate quantities; as, for instance, the square root of 12 : and we call this new species of numbers, irrational numbers. They occur whenever we endeavour to find the square root of a number which is not a square; thus, 2 not being a perfect square, the square root of 2, or the number which multiplied by itself would produce 2, is an irrational quantity. These numbers are also called surd quanUtieSj or incoimnen- surahles.
129. These irrational quantities, though they cannot be expressed by fraction;>, are nevertheless magnitudes ol which we may form an accurate idea ; since, however concealed
40 ELEMENTS SECT. I.
the square root of 12, for example, may appear, we are not Ignorant that it must be a number, which, when multiphed by itself, would exactly produce 12; and this property is sufficient to give us an idea of the number, because it is in our power to appi'oximate towards its value continually.
130. As we are therefore sufficiently acquainted with the nature of irrational numbers, under our present con- sideration, a particular sign has been agreed on to express the square roots of all numbers that are not perfect squares ; which sign is written thus ^^5 and is read square root. Thus, ^/12 represents the square root of 12, or the number which, multiplied by itself, produces 12 ; and a/2 represents the square root of 2 ; ^/.S the square root of 3 ; ^/^ that of ■J; and, in general, ^^a represents tlie square root of the number a. AVhenever, therefore, we would expi'ess the square root of a number, which is not a square, we need only make use of the mark V by placing it before the number,
131. The explanation which we have given of irrational numbers will readily enable us to apply to them the known methods of calculation. For knowing that the square root of 2, multiplied by itself, must produce 2 ; we know also, that the multiplication of ^'2 by V2 must necessarily pro- duce 2 ; that, in the same manner, the multiplication of v/3 by a/3 must give 3; that v'S by ./5 makes 5; that V^ by \/y makes f; and, in general, that x^a multiplied by Va produces a.
132. But when it is required to multiply \/ftby ^/Z>, the product is ^/ab ; for we have already shewn, that if a square has two or more factors, its root must be composed of the roots of those factors ; we therefore find the square root of the product ab, which is ^^ab, by multiplying the square root of a, or x/«, by the square root of b, or A/b ; &c. It is evident from this, that if b were equal to a, we should have ^/aa for the product of a^« by Vb. But ^/aa is evidently a, since aa is the square of «.
133. In division, if it were recjuired, for example, to
divide \/ a hy ^^b, we obtain A/y; and, in this instance,
the irrationality may vanish in the quotient. Thus, having to divide VlS by v/8, the quotient is v/'/, which is re- duced to ^/.^-, and consequently to |-, because ?^ is the square of|.
131". When the number before which we have pkiccd the radical sign a/, is itself a s(|uarc, its root is expressed in the
CHAP. XII.
OF ALGEBRA.
41
usual way; thus, -v^4 is the same as 2 ; v9 is the same as 3; a/36 the same as 6; and v/12j, the same as |^, or 3i. In these instances, the irrationality is only apparent, and vanishes of course.
J 35. It is easy also to multiply irrational numbers by or- dinary numbers; thus, for example, 2 multiplied by ^/5 makes 2 x/5; and 3 times v/2 makes 3 \/2. In the second example, however, as 3 is equal to \/9, we may also express 3 times ^2 by VO multiplied by v'2, orby a/18; also2v/« is the same as A/4a, and 3 \/a the same as V9a ; and, in general, b ^/rt has the same value as the square root of 66a, or \/bba: whence we infer reciprocally, that when the num- ber which is preceded by the radical sign contains a square, we may take the root of that square, and put it before the sign, as we should do in writing bVa instead of ^/bba, After this, the following reductions will be easily under- stood :
a/8, or a/(2.4)*1 f2A/2
Vl2, or a/(3.4) I 2VS
./IS, or a/(2.9) ( is equal to J ^^?. a/24, or a/(6.4) p^ equal to g ^g
a/32, or a/ (2. 16) | 4 a/2
a/75, or a/ (3.25) J {^5 V^
and so on.
136. Division is founded on the same principles ; as \/ii
divided by ^/b eives — r, or v/t-- In the same manner,
•' ^ a/6 6
a/8 ~72 a/18
8
a/^, or a/ 4, or 2
18
-Tq~ } is equal to ^ V-^i or a/9, or 3
Farther,
a/12
^" o
V2 3
12
a/itj or a/^, or 2.
rA/4
/2' a/9
or a/-^, or a/ 2, 9
—^ )-is equal to ^ — ^, or a/^, or a/3,
x/3 12
V6
v/3
a/144 144
^g— , or ^-g-, or a/24,
or
a/(6 X 4), or lastly 2 a/6. 137. There is nothing in particular to be observed in ad-
* The point between 2.4, 3.4, &.c. indicates multiplication.
42 ELEMENTS
S£( T. T.
(iilion .'iiilI sublracliDn, because we oi)ly connect the nunibeis by the signs + and — : for example, V2 added to a/8 is written ^/^ + y^ ; and v^o subtracted from ^5 is written Vo — v3.
138. We may observe, lastly, that in order to distinguish the irrational numbers, we call ;)ll other numbers, both in- tegral and fractional, rational numbers; so that, whenever we speak of rational numbers, we understand integers, or fractions.
CHAP. XIII.
O/" Impossible, or Imaginary Quantities, loltich arise from the same source.
139- We have already seen that the squares of nurhbers, negative as well as positive, are always positive, or affected by the sign -|- ; having shewn that — a multiplied by — a gives + ««, the same as the product of + a by + a : where- fore, in the preceding chapter, we supposed that all the numbers, of which it was required to extract the square roots, were positive.
140. When it is required, therefore, to extract the root of a negative number, a great difficulty arises ; since there is no assignable number, the square of which would be a nega- tive quantity. Suppose, for example, that we wished to
iir^ extract the root of — 4 ; we here require such a number as, ^ when multiplied by itself, would produce —4: now, this ^ number is neither + 2 nor — 2, because the square both of -}- 2 and of — 2, is + 4, and not — 4.
141. We must therefore conclude, that the square root of a negative number cannot be either a positive number or a negative number, since the squares of negative luunbers also
/- take the sign plus : consequently, the root in question must
^1 belong to an entirely distmct species of numbers; since it
cannot be ranked either among positive or negative numbers.
142. Now, we before remarked, that positive numbers are all greater than nothing, or 0, and that negative numbers are all less than nothing, or 0 ; so that whatever exceeds 0 is expressed by positive numbers, and whatever is less than 0 is expressed by negative numbers. The square roots of negative numbers, therefore, arc neither greater nor less than nothing; yet wc cannot say, that they are 0; for 0
CIIAI'. XIII.
OF ALGEUUA. 4{3
iimltiplied by 0 produces 0, and consequently does not give a negative number.
143. And, since all numbers which it is possible to con- ceive, are either greater or less than 0, or are 0 itself, it is evident that wo cannot rank the square root of a negative number amongst possible numbers, and we must therefore say that it is an impossible quantity. In this manner we are
led to the idea of numbers, which from their nature are im- ^ possible ; and therefore they are usually called imaginary ) / quantities, because they exist merely in "the imagination.
144. All such expressions, as a,/— 1, v — 2, \/ —o, s/— 4, &c. are consequently impossible, or imaginary numbers, since they represent roots of negative quantities ; and of such numbers we may truly assert that they are neither nothing, nor greater than noticing, nor less than nothing ; which ne- cessarily constitutes them imaginary, or impossible.
145. But notwithstanding this, these numbers present themselves to the mind ; they exist in our imagination, and we still have a sufficient idea of them ; since we know that by a/ — 4 is meant a number which, multiplied by itself, produces — 4 ; for this reason also, nothing prevents us from making use of these imaginary numbers, and employ- ing them in calculation.
146. The first idea that occurs on the present subject is, that the square of ^/ — o, for example, or the product of
a/ — 3 by A,/ — 3, must be — 3 ; that the product of \/ — 1 by V— l,is ~ 1; and, in general, that by multiplying
a/ — a by .%/ — a, or by taking the square of ^/ — a we ob- tain — a.
147. Now, as — G is equal to +« multiplied by — 1, and as the square root of a product is found by multiplying to- gether the roots of its factors, it follows that the root of a times — 1, or >y — «, is equal to ^Ui multiplied by ^Z — 1 ; hnV A/a is a possible or real number, consequently the whole impossibility of an imaginary quantity may be always re- duced to \/ — 1 ; for this reason, -/ — 4 is equal to \/^ mul- tiplied by a/ —1, or equal to 2 a/ — 1, because a/4 is equal to 2; likewise — 9 is reduced to V^ X V— 1, or 3 a/— 1 ; and a/ — 16 is equal to 4^/ — 1.
148. Moreover, as a^^ multiplied by V 6 makes s/ah,wc shall have v^6 for the value of \/ — 2 multiplied by a./ — 3; and v/^, or 2, for the value of the product of a/ — 1 by a/ — 4. Thus we see that two imaginary numbers, mul- tiplied together, produce a real, or possible one.
But, on the contrary, a possible number, multiplied by an
■f
44 ELEMExNTS SECT. I.
impossible mimbor, gives always an imaginary product : thus, v/— '3 by ^/ + 5, gives a/ - 15.
149. It is the same with regard to division ; for ^.^a
divided by a/6 making V-j-j it is evident that -v/ — 4 di- vided by x/ — 1 will make ^/ + 4, or 2 ; that a/ + 3 divided by \/ — o will give V — 1 ; and that 1 divided by -v' — 1
gives ^/ — r, or ^,/ — 1 ; because 1 is equal to ^/ + 1.
150. We have before observed, that the square root of any number has always two values, one positive and the other negative; that a/4, for example, is both +2 and —2, and that, in general, we may take — Va as well as + Vff for the square root of a. This remark applies also to ima- ginary numbers ; the square root of — a is both + \/ — a and — V — a; but we must not confound the signs -j- and — , which are before the radical sign y- , with the sign which comes after it.
151. It remains for us to remove any doubt, which may be entertained concerning the utility of the numbers of which we have been speaking ; for those numbers being im- possible, it would not be surprising if they were thought entirely useless, and the object only of an unfounded specu- lation. This, however, would be a mistake ; for the cal- culation of imaginary quantities is of the greatest importance, as questions frequently arise, of which we cannot imme- diately say whether they include any thing real and possible, or not; but when the solution of such a question leads to imaginary numbers, we are certain that what is required is impossible.
In order to illustrate what Ave have said by an example, suppose it were proposed to divide the number 12 into two such parts, that the product of those parts may be 40. If we resolve this question by the ordinary rules, we find for the parts sought 6 + a/— 4 and 6 — \/ —4 ; but these num- bers being imaginary, we conclude, that it is impossible to resolve the question.
The difference will be easily perceived, if we suppose tlie question had been to divide 12 into two parts which nud- tiplied together would produce 35 ; for it is evident that those parts must be 7 and 5.
^^-/z^ =--*- ^-^'^ ^/^^-4.*
CHAP. XIV.
OF ALCIEBUA.
in
CHAP. XIV
Of Cubic Numbers.
15^. Wlien a number has been multiplied twice by itself, or, which is the same thing, when the square of a number has been multiplied once more by that number, we obtain a product which is called a cubcj or a cubic number. Thus, the cube of a is aaa, since it is the product obtained by multiplying a by itself, or by a, and that square aa again by a.
The cubes of the natural numbers, therefore, succeed each other in the following order * :
|
Numbers. Cubes. |
1 2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
1 8 |
27 |
64 |
125 |
216|343 |
512 |
729 |
1000 |
^-
yP
153. If we consider the differences of those cubes, as we did of the squares, by subtracting each cube from that which comes after it, we obtain the following series of numbers :
7, 19, 37, 61, 91, 127, 169, 217, 271. Where we do not at first observe any regularity in them ; but if we take the respective differences of these numbers, we find the following series :
12, 18, 24, 30, 36, 42, 48, 54, 60 ; in which the terms, it is evident, increase always by 6.
154. After the definition we have given of a cube, it will not be difficult to find the cubes of fractional numbers; thus, i- is the cube of ^ ; -j^ is the cube of i ; and ^y is the cube of i. In the same manner, we have only to take the cube of the numerator and that of the denominator sepa- rately, and we shall have ^^ for the cube of -|.
155. If it be required to find the cube of a mixed num- ber, we must first reduce it to a single fraction, and then proceed in the manner that has been described. To find, for example; the cube of 1^, we must take that of |^, which
* We are indebted to a mathematician of the name of .J. Paul Buchner, for Tables published at Nuremberg in 1701, in which are to be found the cubes^ as well as the squares, of all numbers from 1 to 1 2000. F. T.
4(> ELEMENTS SF.CT. 1.
is y , or Qj- ; also the cube of 1'^, or of the single fraction ^, is 'eV* or l|l ; and the cube of 3a, or of y , is ^;|.% or 84^4.
156. Since aaa is the cube of a, that of ab will be aaabbb ; whence we see, that if a number has two or more factors, we may find its cube by multiplying together the cubes of those factors. For example, as 12 is equal to 3 x 4, we multiply the cube of 3, which is 27, by the cube of 4, which is 64, and we obtain 1728, the cube of 12; and farther, the cube of 2rt is Saaa, and consequently 8 times greater than the cube of a : likewise, the cube of 3a is 9!7aaa ; that is to say, 27 times greater than the cube of a.
157. Let us attend here also to the signs -|- and — . It is evident that the cube of a positive number +a must also be positive, that is + ciaa ; but if it be required to cube a negative number —a, it is found by first taking the square, which is -jraa, and then multiplying, according to the rule, this square by —a, which gives for the cube required —aaa. In this respect, therefore, it is not the same with cubic num- bers as with squares, since the latter are always positive : Avhereas the cube of —1 is —1, that of —2 is ~8, that of — 3 is —27, and so on.
CHAP. XV
Of Cube Roots, and o/" Irrational Numbers resulting ft-om
iliem,
158. As we can, in the manner already explained, find the cube of a given number, so, when a number is proposed, we may also reciprocally find a number, which, multiplied twice by itself, will produce that number. The number here sought is called, with relation to the other, the cube root; so that the cube root of a given number is the number whose cube is equal to that given number.
159. It is easy therefore to determine the cube root, when the number proposed is a real cube ; such as in the examples in the last chapter ; for we easily perceive that the cube root of 1 is 1 ; that of 8 is 2 ; that of 27 is 3 ; that of 64 is 4, and so on. And, in the same manner, the cube root of —27 is —3; and that of —125 is —5.
Farther, if the proposed number be a fraction, as ^^y, the
CHAP. XV. OV ALGEBRA. 47
cube root of it must be i- ; and that of J-^y is -^, Lastly, the cube root of a mixed number, such as Si-5. must be ^, or 14- ; because 2— is equal to f^.
160. But if the proposed number be not a cube, its cube root cannot be expressed either in integers, or in fractional numbers. For example, 43 is not a cubic number; there- fore it is impossible to assign any number, either integer or fractional, whose cube shall be exactly 43. We may how- ever affirm, that the cube root of that number is greater than 3, since the cube of 3 is only 27; and less than 4, because the cube of 4 is 64 : we know, therefore, that the cube root required is necessarily contained between the numbers 3 and 4.
161. Since the cube root of 43 is greater than 3, if we add a fraction to 3, it is certain that we may approximate still nearer and nearer to the true value of this root : but we can never assign the number which expresses the value ex- actly ; because the cube of a mixed number can never be perfectly equal to an integer, such as 43. If we were to suppose, for example, 3i, or -^ to be the cube root required, the error would be ^; for the cube of ^^ is only ^^^, or 421.
162. This therefore shews, that the cube root of 43 can- not be expressed in any way, either by integers or by frac- tions. However, we have a distinct idea of the magnitude of this root ; and therefore we use, in order to represent it, the sign i/, which we place before the proposed number, and which is read cube root, to distinguish it from the square root, which is often called simply the root ; thus it/ 43 means the cube root of 43 ; that is to say, the number whose cube is 43, or which, multiplied by itself, and then by itself again, produces 43.
163. Now, it is evident that such expressions cannot belong to rational quantities, but that they rather form a particular species of irrational quantities. They have no- thing in common with square roots, and it is not possible to express such a cube root by a square root ; as, for ex- ample, by ^12; for the square of a/ 12 being 12, its cube will be 12 a/12, consequently still irrational, and therefore it cannot be equal to 43.
164. If the proposed number be a real cube, our ex- pressions become rational. Thus, X/\ is equal to 1 ; a/8 is equal to 2 ; -^27 is equal to 3 ; and, generally, l/aaa is equal to a,
165. If it were proposed to multiply one cube root, X/a, by another, l/h^ the product must be y/nh; for we know that
48 ELEMENTS SECT. I.
the cube root of a product ah is found by uuilllplyino; to^ gether the cube roots of the factors. Hence, also, if we
divide X/a by X/h, the quotient will be 1/-t-
166. We farther perceive, that ^i/a is equal to VSfl!, because 2 is equivalent to ^8 ; that 3^/« is equal to ^/27cf, hl/a is equal to X/ahbb ; and, reciprocally, if the number under the radical sign has a factor which is a cube, w^e may make it disappear by placing its cube root before the sign ; for example, instead of ^/64« we may write Vi/a ; and 5\/a instead of iyi25« : hence \/\ 6 is equal to 2^/2, because 16 is equal to S x 2.
167. When a number proposed is negative, its cube root is not subject to the same difficulties that occurred in treating of square roots ; for, since the cubes of negative numbers are negative, it follows that the cube roots of negative num- bers are also negative; thus ^/ — 8 is equal to —2, and ^/ — 27 to — o. It follows also, that ^ — 12 is the same as
— yi2, and thaty/— a maybe expressed by —l/a. Whence Ave see that the sign — , when it is found after the sign of the cube root, might also have been placed before it. We are not therefore led here to impossible, or imaginary num- bers, which happened in considering the square roots of negative numbers.
CHAP. XVI.
Of Powers in general.
168. The product which we obtain by multiplying a number once, or several times by itself, is called a power. Thus, a square which arises from the multiplication of a number by itself, and u cube which we obtain by mul- tiplying a number twice by itself, arc powers. We say also in the former case, that the number is raised to the second degree, or to the second power ; and in the latter, that the number is raised to the third degree, or to the third power.
169- We distinguish these powers from one another" by the number of times that the given number has been mul- tiplied by itself. For example, a square is called the second
CHAP. XVI.
OF ALGEBRA.
49
power, because a certain given number has been multiplied by itself; and if a number has been multiplied twice by itself we call the product the third power, which therefore means the same as the cube; also if we multiply a number three times by itself we obtain its fourth power, or what is commonly called the h'lquadrate : and thus it will be easy to understand what is meant by the fifth, sixth, seventh, &c. power of a number. I shall only add, that powers, after the fourth degree, cease to have any other but these numeral distinctions.
170. To illustrate this still better, we may observe, in the first place, that the powers of 1 remain alwa3's the same ; because, Avhatever number of times we multiply I by itself, the product is found to be always 1. We shall therefore begin by representing the powers of 2 and of 3, which succeed each other as in the following order :
|
Powers. |
Of the number 2. |
Of the number 3. |
|
1st |
2 |
3 |
|
M |
4 |
9 |
|
3d |
8 |
27 |
|
4th |
16 |
81 |
|
5th |
32 |
243 |
|
6th |
64 |
729 |
|
7th |
128 |
2187 |
|
8th |
256 |
6561 |
|
9th |
512 |
19683 |
|
10th |
1024 |
59049 |
|
11th |
2048 |
177147 |
|
12th |
4096 |
531441 |
|
13th |
8192 |
1594323 |
|
14th |
16384 |
4782969 |
|
loth |
32768 |
14348907 |
|
16th |
Q55'6Q |
43046721 |
|
17th |
131072 |
129140163 |
|
18th |
262144 |
387420489 |
But the powers of the number 10 ai*e the most remark- able : for on these powers the system of our arithmetic is founded. A few of them ranged in order, and beginning with the first power, are as follow :
1st 2d 3d 4th 5th 6th
10, 100, 1000, 10000, 100000, 1000000, &c. 171. In, order to Illustrate this subject, and to consider it in a more general manner, w^e may observe, that the
£
\\
.50 ELEMENTS SECT. I.
powers of any number, a, succeed each other in the fol- lowing order :
1st 5d 3d 4th 5th 6th
o, aa, aaa, aaaa^ aaaaa^ aaaaaa, &c.
But we soon feel the inconvenience attending this manner of writing the powers, which consists in the necessity of re- peating the same letter very often, to express high powers ; and the reader also would have no less trouble, if he were obliged to count all the letters, to know what power is in- tended to be represented. The hundredth power, for ex- ample, could not be conveniently written in this manner ; and it would be equally difficult to read it.
172. To avoid this inconvenience, a much more com- modious method of expressing such powers has been devised, which, from its extensive use, deserves to be carefully ex- plained. Thus, for example, to express the hundredth power, we simply write the number 100 above the quantity, whose hundredth power we would express, and a little to- wards the right-hand; thus, a'"° represents a raised to the 100th power, or the hundredth power of a. It must be observed, also, that the name exponent is given to the num- ber written above that whose power, or degree, it represents, which, in the present instance, is 100.
173. In the same manner, a- signifies a raised to the 2d power, or the second power of a, which we represent some- times also by aa, because both these expressions are written and understood with equal facility ; but to express the cube, or the third power aaa, we write a^, according to the rule, that we may occupy less room ; so a* signifies the fourth, a^ the fifth, and a^ the sixth power of «.
174. In a word, the different powers of a will be re- presented by a, Or, a^, ft*, «'', d\ a''^ a^, a?, ft'", &c. Hence we see that in this manner we might very properly have written ft' instead of a for the first term, to shew the order of the series more clearly. In fact, «' is no more than a, as this unit shews that the letter a is to be written only once. Such a series of powers is called also a geometrical pro- gression, because each term is greater by one-time, or term, than the preceding.
175. As in this series of powers each term is found by multiplying the preceding term by a, which increases the exponent by 1 ; so when any term is given, we may also find the preceding term, if we divide by o, because this diminishes the exponent by 1. This shews that the term which precedes the first term ft^ must necessarily be
CHAP, XVI.
OF ALGEBRA.
51
— , or 1 ; and, if we proceed according to the exponents, we
Of
immediately conclude, that the term which precedes the first must be a° ; and hence we deduce this remarkable property, that a° is always equal to 1, however great or small the value of the number a may be, and even when a is^ nothing; that is to say, a" is equal to 1.
176. We may also continue our series of powers in a retro- grade order, and that in two different ways ; first, by dividing always by a ; and secondly, by diminishing the exponent by unity ; and it is evident that, whether we follow the one or the other, the terms are still perfectly equal. This decreasing series is represented in both forms in the fol- lowing Table, which must be read backwards, or from right to left.
1st. 2cl.
177. We are now come to the knowledge of powers
whose exponents are negative, and are enabled to assign
the precise value of those powers. Thus from what has been said, it appears that
|
I 1 I |
1 |
, |
I aa |
X a |
1 |
a |
|
aaaaaa aaaaa |
aaaa |
aaa |
||||
|
1 1 I a8 1 a> |
a* |
I 0-3 |
i a" a-2 |
a-' |
a° |
a^ |
|
1 a-6 1 a-5 |
a-* |
J 1
1- is equal to .{—oi'^
a'
74'
&c.
178. It will also be easy, from the foregoing notation, to find the powers of a product, ah ; for they must evidently be a6, or a'6% a"h\ aW, a'^b*, a^b^, &c. and the powers of fractions will be found in the same manner ; for example,
those of -r- are
a"
or
a'
a"
^7'
&c.
EV7
52 ELEMENTS SECT. I*
179- Lastly, we have to consider tlie powers of negative numbers. Suppose the given number to be ~a; then its powers will form the following series :
— a, +«-, —a", +tt\ — «^5 +0,'^, &c. Where we may observe, that those powers only become negative, whose exponents are odd numbers, and that, on the contrary, all the powers, which have an even number for the exponent, are positive. So that the third, fifth, seventh, ninth, &c. powers have all the sign — ; and the second, fourth, sixth, eighth, &c. powers are affected by the sign +•
CHAP. XVII.
Of the Calculation of Powers.
180. We have nothing particular to observe with regard to the Addition and Subtraction of powers ; for we only represent those operations by means of the signs \- and — , when the powers are different. For example, a^ + a- is the sum of the second and third powers of a ; and a' — a* is what remains when we subtract the fourth power of a from the fifth ; and neither of these results can be abridged : but when we have powers of the same kind or degree, it is evidently' unnecessary to connect them by signs ; as a^ + <3f' becomes 2fi^, &c.
181. But in the Mtdti plication of powers, several circum- stances require attention.
First, when it is required to multiply any power of « by «, we obtain the succeeding power ; that is to say, the power whose exponent is greater by an unit. Thus, a^, multiplied by a, produces a^ ; and a\ multiplied by a, produces «■*. In the same manner, when it is required to multiply by a the powers of an}' number represented by a, having negative exponents, we have only to add 1 to the exponent. Thus, a~' multiplied by a produces a.°, or 1 ; which is made more
evident by considering that a~^ is equal to — , and that the
I . a . .
product of — by a being , it is consequently equal to 1 ;
likewise a'"^ multiplied by <?, produces a~^, or — ; and
CHAP. XVII. or ALGEURA. 53
«~"^ multiplied by Oy gives a~^, and so on. [See Art. 175, 17(3.]
182. Next, if it be required to multiply any power of « by a'^, or the second power, I say that the exponent becomes greater by '■2. Thus, the product of a- by a^ is a*; that of a"^ by ci' is a^\ that of a" by a^ is a^ \ and more generally, a" multiplied by a- makes a"+-. With regard to negative exponents, we shall have a^, or a, for the product of a~^ by
«-; for a~^ being equal to — , it is the same as if we had ^ ^ a
divided aa by a; consequently, the product required is
— , or a ; also a~^, multiplied by a-, produces a°, or 1 ; and
a~^, multiplied by a-, produces a~'.
188. It is no less evident, that to multiply any power of a by «^ we must increase its exponent by three units; con- sequently, the product of «" by a^ is a"''"^ And whenever it is required to multiply together two powers of a, the pro- duct will be also a power of a, and such that its exponent will be the sum of those of the two given powers. For example, a* multiplied by a^ will make uP, and d^~ multiplied by a' will produce a'^, &c.
184. From these considerations we may easily determine the highest powers. To find, for instance, the twenty-fourth power of 2, I multiply the twelfth power by the twelfth power, because 2-* is equal to 2^' x 2'". Now, we have already seen [Table, p. 49] that 2'- is 4096 ; I say there- fore that the number 16777216, or the product of 4096 by 4096, expresses the power required, namely, 2'-^.
185. Let us now proceed to division. We shall remark, in the first place, that to divide a power of a by a, we must subtract 1 from the exponent, or diminish it by unity ; thus, a^ divided by a gives a*; and a°, or 1, divided by a, is equal
to a~' or — : also a~^ divided by a, iiives a~*.
186. If we have to divide a given power of a by «% we must diminish the exponent by 2; and if by a% we must subtract 3 units from the exponent of the power proposed ; and, in general, whatever power of a it is required to divide by any other power of a, the rule is always to subtract the exponent of the second from the exponent of the first of those powers: thus a^^ divided by a'' will give a^; «" divided by a^ will give a"' ; and «-=* divided by «* will give ar''.
187. From what has been said, it is easy to understand
54 ELEMENTS SECT. 1.
the method of finding the powers of powers, this being- done by nuiltiphcation. When we seek, for example, the square, or tlie second power of a^, we find a^'; and in the same manner we find a^- for the third power^ or the cube, of a*. To obtain the square of a power, we have only to double its exponent ; for its cube, we must triple the exponent ; and so on. Thus, the square of a" is a-" ; the cube of «" is a^" ; the seventh power of a" is a'", &c.
188. The square of a", or the square of the square of fl, being a*, we see why the fourth power is called the bigua- drate: also, the square of a? being a^^ the sixth power has received the name of tlie square-cubed.
Lastly, the cube of a^ being «9, we call the ninth power the cubo~C7ibe: after this, no other denominations of this kind have been introduced for powers; and, indeed, the two last are very little used.
CHAP. XV III.
()/ Hoots, zoith relation to Powers in general.
181). Since the square root of a given number is a num- ber, whose square is equal to that given number; and since the cube root of a given number is a number, whose cube is equal to that given number; it follows that any number whatever being given, we may always suppose such roots of it, that the fourth, or the fifth, or any other power of them, respectively, may be equal to the given number. To distin- guish these different kinds of roots better, we shall call the square root, the second root ; and the cube root, the third root ; because, according to this denomination, we may call the fourth root^ that whose biquadrate is equal to a given number; and the fifth root, that whose fifth power is equal to a given number, &c.
190. As the square, or second root, is marked by the sign v^, and the cubic, or third root, by the sign ^, so the fourth root is represented by the sign i/ ; the fifth root by the sign ^/ ; and so on. It is evident that, according to this method of expression, the sign of the square root ought to be ^: but as of all roots this occurs most frequently, it has been agreed, for the sake of brevity, to omit the number 2 as the sign ol" this root. So that when the radical siau lias no num-
CHAP. Will. OF ALGEJiUA. OO
ber prefixed to it, this always shews that the square ^-oot is meant.
191. To explain this matter still better, we shall here exhibit the different roots of the number a, with their re- spective values :
i/af i/a\- is the
i/aj {^6th ) ^fl, and so on.
So that, conversely, Tlie 2d The 3d
The 4th V power of -K t/a V is equal to The 5th The 6th
192. Whether the number a therefore be great or small, we know what value to affix to all these roots of different degrees.
It must be remarked also, that if we substitute unity for a, all those roots remain constantly 1 ; because all the powers of 1 have unity for their value. If the number a be greater than 1 , all its roots will also exceed unity. Lastly, if that number be less than 1, all its roots will also be less than unity.
193. When the number a is positive, we know from what was before said of the square and cube roots, that all the other roots may also be determined, and will be real and possible numbers
But if the number a be negative, its second, fourth, sixth, and all its even roots, become impossible, or imaginary num- bers ; because all the powers of an even order, whether of positive or of negative numbers, are affected by the sign -i- : whereas the third, fifth, seventh, and all its odd roots, become negative, but rational ; because the odd powers of negative numbers are also negative.
194. We have here also an inexhaustible source of new kinds of surds, or irrational quantities; for whenever the number a is not really such a power, as some one of the foregoing indices represents, or seems to require, it is im- possible to express that root either in whole numbers or in fractions; and, consequently, it must be classed among the numbers which are called irrational.
56 ELEMENTS SECT. I.
CHAP. XIX.
Of the Method of representing Irrational Numbers by Fractional Exponents.
195. We have shewn in the preceding chapter, that the square of any power is found by doubling the exponent of that power ; or that, in general, tlie square, or the second power, of o", is or" ; and the converse also follows, viz. that the square root of the power a-"- is a", whicli is found bv taking half the exponent of that power, or dividing it by 2.
196. Thus, the square root of a'^ is a\ or a\ that of «* is a- ; tliat of c^'' is a^ ; and so on : and, as this is general,
the square root of a^ must necessarily be a^, and that of a^
must be a^; consequently, we shall in the same manner
have a^ for the square root of a'. Whence we see that d^ is equal to ^/«; which new method of representing the square root demands particular attention.
197. We have also shewn, that, to find the cube of a power, as «", we must multiply its exponent by 3, and con- sequently that cube is a^".
Hence, conversely, when it is required to find the third, or cube root, of the power «^", we have onl}^ to divide that exponent by 3, and may therefore with certainty conclude, that the root required is a : consequently a\ or a, is the cube root of a^; a- is the cube root of a*^; a^of a^; and so on.
198. There is nothing to prevcr': us from applying the same reasoning to those cases, in which the exponent is not divisible by 3, or from concluding that the cube root of a*
is a^, and that the cube root of rt* is a% or a^T; conse-
quently, the third, or cube root of «, or a^, must be a^ :
I >
whence also, it appears, that aJ is the same as v/o. i
199. It is tlie same with roots of a higher degree : ?t'hus,
the fourth root of a will be ci*, which expression has the
same value as^/rt ; the fifth root of a will be a% which is consequently equivalent to Vrt; and the same observation may be extended to all roots of a higher degree.
CHAP. XIX. OF ALGEBRA. 57
200. We may 'therefore entirely reject the radical signs at present made use of, and employ in their stead the fractional exponents which we have just explained: but as we have been long accustomed to those signs, and meet with them in most books of Algebra, it might be wrong to banish them entirely from calculation ; there is, however, sufficient reason also to employ, as is now frequently done, the other method
of notation, because it manifestly corresponds with the nature
I of the thing. In fact, we see immediately that a^ is the
I square root of a, because we know that the square of a^, that
I I
is to say, a/^ multiplied by a^, is equal to «\ or a.
201. What has been now said is sufficient to shew how Ave are to understand all other fractional exponents that may
occur. If we have, for example, «% this means, that we must first take the fourth power of a, and then extract its
4-
cube, or third root ; so that aJ is the same as the et»mmon
expressions/a*. Hence, to find tlie value of flf*^, we must
first take the cube, or the third power of «, which is o^, and
i . then extract the fourth root of that power; so that a+ is the
* . same as Va^y and aJ is equal to s/a'', &c. .'
202. When the fraction which represents the exponent exceeds unity, we may express the value of the given quan-
tity in another way : for instance, suppose it to be a^ ; this
quantity is equivalent to a-^, which is the product of a" by
i -L . ... i_ .
a"" : now a- being equal to ^/a, it is evident that a^ is
\ '_? I .
equal to a-A/a\'. also a^ , or aV, is equal to a^^/a; and
a * , that is, a^T, expresses a^y^a^. These examples are suf- ficient to illustrate the great utility of fractional exponents.
203. Their use extends also to fractional numbers : for if
1 there be given -— , we know that this quantity is equal to
1
— ; and we have seen already that a fraction of the Ibrm
1 , 1
-;7- may be expressed by a~" ; so that instead of — ;;- wc
may use the expression a ^; and, in the .•aunic man-
58 ELEMENTS SECT. I.
i . _i . . . a^ .
ner, ■^—- is equal to a t. Again, if the quantity
proposed; let it be transformed into this, — ^, which is the
^_
product of «- by a -^^ ; now this product is equivalent to
5 ,
rt'+, or to aH, or lastly, to a'X/a. Practice will render similar reductions easy.
204. We shall observe, in the last place, that each root may be represented in a variety of ways; for >^ a being the
same as a^, and ~ being transformable into the fractions, |, -|, *, -E^o> -Ta? ^c- it is evident that >y a\% equal to X/a"^ or to Va^, or to ^a% and so on. In the same manner, %/a^ which
I
is equal to «^, will be equal to ^/a/^, or to Xfa^^ or to ^^a*. Hence also we see that the number «, or a\ might be repre- sented by the following radical expressions : Va\ l/a\ Va\ ^«^ &c.
205. Tills property is of great use in multiplication and division ; for if we have, for example, to multiply 1/a by l/a, we v%'iite Va^ for ^a, and ^a^ instead of i^a ; so that in this manner we obtain the same radical sign for both, and the multiplication being now performed, gives the product y^a^.
The same result is also deduced from a^ ', which is the
-L . . i.
product of a^ multiplied by a^ ; for | 4- ^ is |, and conse-
1. quently the product required is a^, or ^a^.
On the contrary, if it were required to divide ^a, or «^, hy l/a, or d^, we should have for the quotient a* ^, ova^ ^, that is to say, a^y or ^/a.
UUESTIONS FOR PllACTICE RESPECTING SURDS.
1. lleduce 0 to the form of \/5. Ans. ^/o6.
2. Reduce a ■]- b to the form of Vbc.
Ans. \^{aa -\- ^ab -r bb).
y. Reduce 7 — — to the form of \^d. Ans. k/tT'
b ^/c• bbc
i. Reduce a- and 6- to the common index \.
Ans. (I 1%
and A^P.
CHAP. XIX. OF ALGEBRA. 59
5. Reduce 'v/48 to its simplest form. Ans. ^s/^.
6. Reduce ^/{a^x — a'-x-) to its simplest form.
t Ans. aV{ax — xx^).
- *< 27a^b^
7. Reduce i^pr; — tt- to its simplest form.
So—Sa ^
. ' A7is.—'l/- .
^'^ 8. Add v/6 to 2v/6; and V8 to ^/50.
^«s. S-v/G; and 7^/2. 9. Add -v/4a and t/a^ together. Ans, {a + 2) v'a.
10. Add— 1^ and 4 c I 0
A ^' + C=
together. Ans. , , .
11. Subtract ^/4a from t^a^. J/zs. (a — 2) a/a.
"cli. ~^- h- — c'^ 1
12. Subtract -rr from — "". J.7zs. — j — V-i-.
o\ c o , be
13. Multiply x^-^ by a/-^. Ans. .
14. Multiply a/c/ by l/ub. Ans. ^(a"'b-d^).
15. Multiply v/(4a - 3a:) by 2a.
^7is. a/(16«^ - 12a=a:).
16. Multiply ^ -/(« — x) by (c — <i) s/ax.
. ac—ad Ans. — 7^7 — V(a-x — ax"), yib
17. Divide a^ by a+ ; and a" by a'".
Ans. a ' ^ ; and a mn ,
lo. Divide — oX~~ V\C''X — ax-) by ^ \/{ci — x).
Ans. (c — d) \/ax.
19. Divide a- — ad — b + d \/b by a — -^/b.
Ans. a -f ^/b — d.
20. What is the cube of -\/2 ? Ans. a/8.
21. What is the square of 3yZ>c'' ? Ans. 9cl/b-c.
22. What is the fourth power of -^ V 7 .?
•^'"' 4&V-2icTF)-
23. What is the square of 3 + a/5 .? ^7i5. 14+6 ^/5.
3
24. What is the square root of a"' .'' A716. a^ ; or \/a\
25. What is the cube root of x'(a'- — x-)f
Am. ^{a- — a-'-).
60 ELEMENTS SECT. I.
26. What multi|)lier will render a + v^3 rational ?
Ans. a — -v/3.
27. What multiplier will render -v^ff — ^/b rational ?
Ans. v''« + \^b.
28. What multiplier will render the denominator of the
fraction — -;:^ j rational.? Ans. a/7 — \/3.
CHAP. XX.
Of the different Methods of Calculation, and of their mutual Connexion.
206. Hitherto we have only explained the different me- thods of calculation : namely, addition, subtraction, mul- tiplication, and division; the involution of powers, and the extraction of roots. It will not be improper, therefore, in this place, to trace back the origin of these different methods, and to explain the connexion which subsists among them ; in order that we may satisfy ourselves whether it be possible or not for other operations of the same kind to exist. This inquiry will throw new light on the subjects which we have considered.
In prosecuting this design, we shall make use of a new character, which may be employed instead of the expression that has been so often repeated, is equal to ; this sign is =, which is read is equal to: thus, when I write a = b, this means that a is equal to b: so, for example, 3x5 = 15.
207. The first mode of calculation that presents itself to the mind, is undoubtedly addition, by which we add two numbers together and find their sum : let therefore a and b be the two given numbers, and let their sum be expressed by the letter c, then we shall have a + b == c; so that when we know the two numbers a and b, addition teaches us to find the number c.
208. Preserving this comparison a -j- 6 = c, let us reverse the question by asking, how we are to find the number b, when we know the numbers a and c.
It is here required therefore to know what number must be added to «, in order that the sum may be the number c: su))posc, for example, « = 3 and c = 8; so that we must have o -f 6 = 8; then b will evidently be found by sub-
CHAP. XX.
OF ALGEBRA. (>1
tracting 3 from 8 : and, in general, to find Z>, we must sub- tract a from c, whence arises b — c — a\ for, by adding a to both sides again, we have 6 + a = c — a + «, that is to say, — c, as we supposed.
209. Subtraction therefore takes place, when we invert the question which gives rise to addition. But the number which it is i-equired to subtract may happen to be greater than that from which it is to be subtracted ; as, for example, if it were required to subtract 9 from 5 : this instance there- fore furnishes us with the idea of a new kind of numbers, which we call negative numbers, because 5 — 9 = — 4.
210. When several numbers are to be added together, which are all equal, their sum is found by multiplication, and is called a product. Thus, ah means the product arising from the multiplication of a by b, or from the addition of the number «, h number of times; and if we represent this pro- duct by the letter c, we shall have ab — c\ thus multiplica- tion teaches us how to determine the number c, when the numbers a and h are known.
211. Let us now propose the following question: the numbers a and c being known, to find the number b. Sup- pose, for example, a = S, and c = 15., so that Sb = 15, and let us inquire by what number '^ must be multiplied, in order that the product may be 1-5 ; for the question pro- posed is reduced to this. This is a case of division ; and the number required is found by dividing 15 by 3; and, in general, the number b is found by dividing c by a ; from
c
which results the equation b = — . ^ a
212. Now, as it frequently happens that the number c cannot be really divided by the number a, while the letter b must however have a determinate value, another new kind of numbers present themselves, which are called fractions. For example, suppose a — 4, and c — 3, so that 45 = 3 ; then it is evident that b cannot be an integer, but a fraction, and that we shall have 6 = ^.
213. We have seen that multiplication arises from ad- dition ; that is to say, from the addition of several equal quantities : and if we now proceed farther, we shall perceive that, from the multiplication of several equal quantities to- gether, powers are derived ; which powers are represented in a general manner b}- the expression a''. This signifies that the number a must be multiplied as many times by itself, minus 1, as is indicated by the number b. And we know from wl at has been already said, that, in the present in-
62 ELEMENTS SECT. T.
stance, a is called the root, h the exponent, and d' the power.
214. Farther, if we represent this power also by the letter c, we have «'' = c, an equation in which three letters a, Z», c, are found ; and we have shewn in treating of powers, how to find the power itself, that is, the letter c, when a root a and its exponent h are given. Suppose, for example, rt = 5, and & = 3, so that c = 5^: then it is evident that we must take the third power of 5, which is 1^5, so that in this case c = 125.
215. We have now seen how to determine the power c, by means of the root a and the exponent 6; but if we wish to reverse the question, we shall find that this may be done in tv/o Vv'ays, and that there are two different cases to be con- sidered : for if two of these three numbers a, 6, c, were given, and it were required to find the third, we should immediately perceive that this question would admit of three different suppositions, and consequently of three solutions. We have considered the case in which a and h were the given num- bers ; we may therefore suppose farther that c and a, or c and 6, are known, and that it is required to determine the third letter. But, before we proceed any farther, let us point out a very essential distinction between involution and the two operations which lead to it. When, in addition, we re- versed the question, it could be done only in one way; it was a matter of indifference whether we took c and «, or c and 6, for the given numbers, because we might indifferently write a ■\- ^, or 6 + a ; and it was also the same with mul- tiplication ; we could at pleasure take the letters a and h for each other, the equation ah — c being exactly the same as ba = c: but in the calculation of powers, the same thing does not take place, and we can by no means write b" in- stead of a'' ; as a single example will be sufficient to il- lustrate : for let a — 5, and b = 3; then we shall have «* = 5^ = 125; but Z>'' = 3^ = 243: which are two very different results.
216. It is evident then, that we may propose two ques- tions more: one, to find the loot a by means of the given power c, and the exponent b ; the other, to find the ex- ponent b, supposing the power c and the root a to be known.
217. It may be said, indeed, that the former of these questions has been resolved in the chapter on the extraction of roots; since if 6 = 2, for example, and a- = c, we know by this means, that a is a number whose square is equal to c, and consequently that a = ^/c In the same manner, if
CHAP. XXI. OF ALOEBHA. 03
b = 3 and a^ = c, we know that the cube of a must be equal to the given number c, and consequently that a = \/c. It is therefore easy to conclude, generally, from this, how to determine the letter a by means of the letters c and b ; for we must necessarily have a = \/c.
218. We have already remarked also the consequence which follows, when the given number is not a real power ; a case which very frequently occurs ; nraiiely, that then the required root, a, can neither be expressed by integers, nor by fractions ; yet since this root must necessarUy have a de- terminate value, the same consideration led us to a new kind of numbers, which, as we observed, are called surds, or ir7-a- ^«or?«/ numbers ; and which we have seen are divisible into an infinite number of different sorts, on account of the great variety of roots. Lastly, by the same inquiry, we wejie led to the knowledge of another particular kind of numbers, which have been called imagmari/ numbers.
219. It remains now to consider the second question, which was to determine the exponent, the power c, and the root a, both being known. On this question, which has not yet occurred, is founded the important theory of Logarithms, the use of which is so extensive through the whole compass of mathematics, that scarcely any long calculation can be carried on without their assistance ; and we shall find, in the following chapter, for which we reserve this theory, that it will lead us to another kind of numbers entirely new, as they cannot be ranked among the irrational numbers before mentioned.
CHAP. XXI
Of Logarithms in general.
220. Resuming the equation d' — f, we shall begin by remarking that, in the doctrine of Logarithms, we assume for the root a, a certain number taken at pleasure, and sup- pose this root to preserve invariably its assumed value. This being laid down, we take the exponent b such, that the power a!' becomes equal to a given number c ; in which case this exponent b is said to be the logarithm of the number c. To express this, v/e shall use the letter L. or the initial letters log. Thus, by A = L. r, or b = log. c,
.64 ELEMENTS SECT. I.
we mean that b is equal to the logarithm of the number r, or that the logarithm of c is h.
221. We see then, that the value of the root a being once established, the logarithm of any number, c, is nothing more than the exponent of that power of «, which is equal to c : so that c being = a^, b is the logarithm of the power a^. If, for the present, we suppose 6 = 1, we have 1 for the logarithm of «', and consequently log. a = 1 ; but if we suppose b = 2, we have 2 for the logarithm of a- ; that is to say, log. a- = % and we may, in the same manner, obtain log. a^ — 3 ; log. «•* = 4 ; log. a-' — 5, and so on.
222. If we make Z> = 0, it is evident that 0 will be the logarithm of a"; but a° = l; consequently log. 1=0, what- ever be the value of the root a.
Suppose 6 = — 1, then — 1 will be the logarithm of
1 1
a ' ; but a ^ = — : so that we have log. — = — 1, and in
the same manner, we sliall have lo^. — ~ = — 2 ; log. —r
= -3; % ^V= - 4',&c.
223. It is evident, then, how we may represent the loga- rithms of all the powers of «, and even those of fractions, which have unity for the numerator, and for the denominator a power of a. We see also, that in all those cases the loga- rithms are integers; but it must be observed, that if 6 were a fraction, it would be the logarithm of an irrational num- ber : if we suppose, for example, 6 = |, it follows, that k is
the logarithm of «^, or of \/« ; consequently we have also log. \/a — \ ; and we shall find, in the same manner, that log. Va = i, log. %/a = i, &c.
224. But if it be required to find the logarithm of another number c, it will be I'eadily perceived, that it can neither be an integer, nor a fraction ; yet there must be such an ex- ponent h, that the power a'' may become equal to the nuni- ber proposed ; we have therefore b — lo^. c ; and generally, a' •'• = C-.
225. Let us now consider another number <7, Avhose loga- rithm has been represented in a similar manner by log. d ; so that a'"' = d. Here if we multiply this expression by the preceding one a^" = c, we shall have a^-'-"^^-" = cd ; hence, the exponent is always the logarithm of the poxaer ; consequently, log. c + log. d = log. cd. But if, instead of multiplying, we divide the former expression by the latter,
CHAP. XX r. OF ALGEBRA. 65
C
we shall obtain a' -'"^ ' = -y ; and, consequently, log. c —
log. d = log. — . ^
226. This leads us to the two principal properties of loga- rithms, which are contained in the equations log. c + log. d
Q
= log. c(/, and log. c — log. d = log. —r. The former of
these equations teaches us, that the logairthm of a product, as cd, is found by adding together the logarithms of the factors ; and the latter shews us tliis property, namely, that the logarithm of a fraction may be determined by sub- tracting the logarithm of the denominator from that of the numerator.
327. It also follows from this, that when it is required to multiply, or divide, two numbers by one another, we have only to add, or subtract, their logarithms ; and this is what constitutes the singular utility ot" logarithms in calculation : for it is evidently much easier to add, or subtract, tlian to multip]}'^, or divide, particularly when the question involves large numbers.
228. Logarithms are attended with still greater advan- tages, in the involution of powers, and in the extraction of roots ; for if d = c, we have, by the first property, log. c + log. c =■ log. cc, or C-; consequently, log. cc =2 log. c ; and, in the same manner, we obtain log. c^ ■— 3 log. c; log. c* = 4 log. c; and, generally, log. 0^=- n log. c. If we now sub- stitute fractional numbers for ??, we shall have, for example,
I log.c^, that is to say, log. \/c, = ^log. c; and lastly, if we suppose n to represent negative numbers, we shall have log.
c-\ or log. — , =i — log. c ; log. c~^, or log. —7, = —2 log.
c, and so on ; which follows not only from the equation log. c'* = n log. c, but also from log. 1 = 0, as we have already seen.
229. If therefore we had Tables, in which logarithms were calculated for all numbers, we might certainly derive from them very great assistance in performing the most prolix calculations ; such, for instance, as require frequent multiplications, divisions, involutions, and extractions of roots : for, in such Tables, we should have not only the logarithms of all numbers, but also the numbers answering to all logarithms. If it were required, for example, to find the square root of the number c, we must first find the loga-
F
f>(> ELEMENTS SECT. T.
rithm of c, that is, log. c, and next taking the half of that logarithm, or ^og. c, we should have the logarithm of the square root required : we have therefore only to look in the Tables fo)- the number answering to that logarithm, in order to obtain the root required.
230. We have alreadyseen, that the numbers, 1, 2, 3, 4, 5, 6, &c. that is to say, all positive numbers, are logarithms of the root a, and of its positive powers; consequently, logarithms of numbers greater than unity : and, on the con- trary, that the negative numbers, as —1, —2, &c. are loga-
1 1 nthms of the fractions — , — -, &c. which are less than unity, a a"
but yet greater than nothing.
Hence, it follows, that, if the logarithm be positive, the number is always greater than unity : but if the logarithm be negative, the number is always less than unity, and yet greater than 0 ; consequently, we cannot express the loga- rithms ("'"^negative numbers : we must therefore conclude, that the logarith'ns of negative numbers are impossible, and that they belong to the class of imaginary quantities.
231. In order to illustrate this moi*e fully, it will be proper to fix on a determinate number for the root a. Let us make choice of that, on Avhich the common Logarithmic Tables are formed, that is, the number 10, which has been preferrec, because it is the foundation of our Arithmetic. But it if vident that any other number, provided it wei'e greater ' . .lan unity, would answer the same purpose : and the reason why we cannot suppose a = unity, or 1, is manifest ; because all the powers a^' would then be con- stantly equal to unity, and could never become equal to another given number, c.
CHAP. XXII.
Of the Logarithmic Tables now in use.
232. In those Tables, as we have already mentioned, we begin with the supposition, that the root a is = 10; so that the logarithm of any number, c, is the exponent to which we must raise the number 10, in order that the power resulting from it may be equal to the number c; or if we denote the logarithm of c by L.c,^.'Ave shall always have lO""' = c.
CPIAl'. XXII.
OF ALGKBRA. 67
233. We liave already observed, that the logarithm of the number 1 is always 0; and w,e have also 10" = 1 ; con- sequently, log. 1=0; log. 10 ^^ 1; log. TOO ~ 2; los,. 1000 = 3; log. 10000 -= 4; log. 100000 =^ 5; log. lOOOCOO = 6. Farther, log. -V = - 1 ; log. -l^ = ~ 2 ; log. -^g- = - 3; log. j^^^- = - 4; log. too'o^ = - 5; log.
I o o o o o o ^° _
234. The logarithms of the principal numbers, therefore, are easily determined ; but it is much more difficult to find the logarithms of all the other intervening numbers; and yet they must be inserted in the Tables. This however is not the place to lay down all the rules that are necessary for such an inquiry ; we shall therefore at present content our- selves with a general view only of the subject.
235. First, since log. 1 — 0, and log. 10 = 1, it is evident that the logarithms of all numbers between 1 and 10 must be included between 0 and unity ; and, consequently, be greater than 0, and less than 1. It will therefore be sufirient to consider the single number 2; the logarithm of which is certainly greater than 0, but less than unity : and if we repre- sent this logarithm by the letter x, so that log, 2 — x, the value of that letter must be such as to give exactly 10 = 2.
We easily perceive, also, that x must be considerably
I less than i, or which amounts to the same thing, 10^ is greater than 2; for if we square both sides, th-^ square of
10^ = 10, and the square of 2 := 4. Now, thi: 'atter is
much less than the former; and, in the same manner, we
I see that x is also less than ~; that is to say, 10^ is greater
I than 2: for the cube of 10^ is 10, and that of 2 is only 8. But, on the contrary, by makings: = i, we give it too small
a value; because the fourth power of 10"* being 10, and
I that of 2 being 16, it is evident that 10*^ is less than 2. Thus, we see that x, or the log. 2, is less than 4-, but greater than -■ : and, in the same manner, we may determine, with respect to every fraction contained between } and |, whether it be too great or too small.
In making trial, for example, with ~, which is less than |,
and greater than l, 10 , or 10^, ought to be = 2; or the
seventh power of 10^, that is to