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THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 Education 
 
 GIFT OF 
 Snma T. Grimwood 
 
London, December 1, 1828. 
 JUST PUBLISHED 
 
 UY 
 
 JAMES DUNCAN, 
 
 37 PATERNOSTER ROW. 
 
 THE MODERN TRAVELLER ; containing a Popular Descrip- 
 tion, Geographical, Historical, and Topographical, of the various Countries 
 of the Globe, compiled from the latest and best Authorities. The Work 
 is published in Monthly Parts, closely and elegantly printed, price 2,9. Qd. each, 
 Two of which form a Volume. Each Part contains Two Engravings, and 
 (upon an average) five Sheets of Letter-press, comprising as much as an 
 ordinary octavo volume. The Publisher flatters himself, that it will be 
 considered as one of the cheapest Books published. 
 
 ** The various Countries, as completed, may be had in boards, price 5s. Gd. 
 per Volume; neatly half-bound and lettered, Gs. per Volume; in calf, 
 gilt extra, Is. per Volume. 
 
 Already published, 
 
 PALESTINE; OR, TEE HOLY 
 
 LAND, One Vol. 
 
 SYRIA and ASIA MINOR,TwoVols. 
 BRAZIL and BUENOS AYRES, 
 
 Two Vols. 
 MEXICO and GUATIMALA, Two 
 
 Vols ' NIA, Two Vols. 
 
 COLOMBIA, One Vol. 
 ARABIA, One Vol. 
 
 RUSSIA, One Vol. 
 
 SPAIN and PORTUGAL, Two Vols. 
 
 BIRMAN EMPIRE, One Vol. 
 
 GREECE, Two Vols. 
 
 TURKEY, One Vol. 
 
 EGYPT, NUBIA, and ABYSSI- 
 
 PERSIA and CHINA, Two Vols. 
 INDIA, Four Vols. 
 
 AFRICA, NORTH AMERICA, PERU, CHILI, &c. will follow in succession. 
 %* To be completed in 60 Parts. 
 
 The following Testimonies, selected from numerous notices recommendatory 
 of the Work, will prove the general estimation in which it is held : 
 
 " We speak within the most cautious bounds, when we say, that in any volume of this work 
 the traveller will find more of the actual material of which he stands hi need, the real, distinct, 
 matter-of-fact information, than in any ten ' voyages and travels 1 to the same region. Of the 
 minuteness of this history, India, the best judgment may be formed from its extent. In the 
 usual manner of printing, it would fill three large octavo volumes. The contents of the four 
 volumes exhibit a singular industry in the collection, and condensation in the quantity of valu- 
 able matter." Literary Gazette, Dec. 6, 1828. 
 
 " The portion on Turkey, in the Modern Traveller, contains the united excellencies of 
 every writer, past and present, on this important subject, and cannot be too frequently con- 
 sulted for correctness of information perspicuously delivered. This observation, indeed, applies 
 to the whole body of that work, which, taken altogether, is not exceeded by any similar publi- 
 cation throughout Europe; and reflects great credit on the spirit of the Publisher, and the 
 correct taste, deep reading, and patient industry of the Editor." Foreign Review, No. 3, p. 219. 
 
 " In our papers on Greece, we have looked occasionally to the works of the individuals on 
 whose fidelity and knowledge we had most reason to rely. But for the reader's purpose of 
 obtaining a view at once general and minute, animated and important, we can na-ne nothing 
 superior to the two little volumes of ' Greece,' contained in the Modern Traveller, a publication 
 which, amounting to fifteen pocket volumes, already contains more information of the actual 
 state of the world, than perhaps any other in existence. Its merit is, that it is not a compilation 
 of the writings of modern travellers, but a combination of their various knowledge, checked and 
 i often increased by the accuracy and information of the intelligent Editor. It deservf ,- a place in 
 ^e library of every inquiring person, who desires to become acquainted with the latf&t state of 
 nations, without the trouble of turning over a multitude of voyages and statistical works, 
 naturally imperfect and partial, and, of course, sometimes contradictory and untrue."' Black- 
 tvood's Magazine, Dec. 182(j. 
 
 " We feel ourselves justified in recommending this work to our readers, as promising to bo 
 the most judicious and interesting publication of the kind that has ever fallen under our notice." 
 Asiatic Journal, 
 
 " This is an excellent geographical work, compiled with much industry, learning, and intel- 
 ligence. Though printed with neatness, and forming an elegant stri: of little volumes, we 
 think that its intrinsic merit entitled it to a more dignified form. It is not the work it seems to 
 be but a better." Westminster Review. 
 
 - 
 
WORKS RECENTLY PUBLISHED. 
 I. 
 
 A SECOND COURSE of SERMONS for the YEAR; con- 
 taining Two for each Sunday, and one for each Holyday ; abridged from the 
 most eminent Divines of the Established Church, and adapted to the Service 
 of the Day : intended for the Use of Families and Schools. Dedicated, by 
 permission, to the Lord Bishop of London. By the Rev. J. R. PITMAN, 
 A.M., alternate Morning Preacher of Belgrave and Berkeley Chapels; and 
 alternate Evening Preacher of the Foundling and Magdalen Hospitals. In 
 Two Vols. 8vo, price 21 s. boards. 
 
 " There is no question which the Clergy are more frequently asked, and to which they find 
 it more difficult to give a satisfactory reply than this What Sermons would they recommend for 
 the use of a private family ? There are so many circumstances which render the greater part of 
 modern discourses totally unfit for the purposes of domestic instruction, and the old standards, 
 unmodernised, are so little intelligible to common ears, that it is no easy matter to point out any 
 set of discourses embracing a sufficient variety to excite attention, at the same time forcibly 
 inculcating the pure doctrines and practical precepts of Christianity, which is adapted in all 
 respects to the reader, and the usual circle of listeners met on fhe Sabbath evening for prayer 
 and edification. We really think that Mr. Pitman's work bids fair to supply the deficiency 
 which has been so much regretted." Quarterly Theological Review. 
 
 %* A Second Edition of the FIRST COURSE is just published, same size 
 and price as above. 
 
 II. 
 The WHOLE WORKS of the Ri;ht Rev. JEREMY TAYLOR, 
 
 D.D., Lord Bishop of Down, Connor, and Dromore, with a Life of the 
 Author, and a Critical Examination of his Writings. By the Right Rev. 
 REGINALD HEBER, D.D., late Lord Bishop of Calcutta. In 15 Volumes, 
 8vo, new edition, price 91. boards. 
 
 Also may be had, separate, by the same Author, 
 
 1. HOLY LIVING and DYING. 8vo. price 12s. boards. 
 
 2. A COURSE of SERMONS for all the SUNDAYS of the YEAR. Two 
 Vols. 8vo, price 24s. boards. 
 
 3. The LIFE of the Right Rev. JEREMY TAYLOR. In One Volume, 
 8vo, with a Portrait, price 10s. 6d. boards. 
 
 III. 
 
 The WORKS of the Right Rev. WILLIAM BEVERIDGE, 
 
 D.D., Lord Bishop of St. Asaph, now first collected : with a Memoir of the 
 Author, and a Critical Examination of his Writings, by the Rev. THOMAS 
 HARTWELL HORKE, M.A., of St. John's College, Cambridge; Author of the 
 " Introduction to the Holy Scriptures." In Nine Vols. 8vo, with a Portrait, 
 uniform with the Works of Bishop Taylor, price 51. 8s. boards. 
 
 Also may be had, by the same Author, 
 
 THESAURUS THEOLOGICUS ; or, a Complete System of 
 Divinity. In Two Volumes, 8vo. price 24s. boards. 
 
 IV. 
 
 A NEW FAMILY BIBLE, and Improved Version of the Holy 
 Scriptures ; from corrected Texts of the Originals, with Reflections on each 
 Chapter, suitable for the Family or the Closet. By B. BooTHROYD,D.D. Editor, 
 of the Biblia Hebraica, &c. Dedicated to his Majesty. In Three Vols. 4t(, 
 royal paper, 6/. Gs. ; demy paper, 41. 4s. boards. 
 
 ' The above important and interesting work has received the sanction of the highest literary 
 authorities; and has been introduced, and is constantly read, in the families of many respectable 
 Clergymen and Ministers, as well as of private Christians." 
 
 " We cannot, in concluding pur notice of Dr. Boothroyd's labours, withhold our cordial com- 
 mendation of his persevering diligence; and feeling, as we do, warmly interested in every well- 
 conducted attempt to promote Biblical studies and the intelligent use of the Scriptures, we con- 
 gratulate him on the completion of his arduous undertaking. We shall be glad to find that he 
 receives the patronage of the public to a gratifying extent, and recommend his Family Bible, and 
 Improved Version, as a highly meritorious publication." Eclectic Revieiv f Nov, 1826. 
 
Recently published by JAMES DUNCAN. 
 
 V. 
 
 THE WHOLE WORKS of the Most Reverend Father in God, 
 ROBERT LEIGHTON, D.D., Archbishop of Glasgow. To which is prefixed an 
 entire new Life of the Author, by the Rev. J. N. PEARSON, M.A., of Trinity 
 College, Cambridge, and Chaplain to the Most Noble the Marquess Wellesley. 
 In Four Vols. 8vo, with a Portrait, engraved by WARREN, price 36*. boards. 
 
 We have placed a new edition of Archbishop Leighton's Works at the head of this article ; 
 and as Mr. Coleridge has neglected to furnish the biographical notice he had promised, we shall 
 endeavour to supply its place by a few particulars of his life and writings, principally extracted 
 from a spirited and eloquent Memoir prefixed to the new edition, by the Rev. Norman Pearson. 
 It is a reproach to the present age, that his valuable writings, breathing as they do the sublimest 
 and purest spirit of piety, rich in beautiful images and classical learning, throughout abounding 
 in practical reflections, and all expressed with the sweetest and simplest eloquence, should have 
 been neglected among us." British Critic, October 1826. 
 
 %* The above may be had, printed in a small but neat type, and compressed 
 into Two Volumes, price 21s. boards. 
 
 VI. 
 
 A PRACTICAL COMMENTARY upon the FIRST EPISTLE 
 of ST. PETER, and other Expository Works. By ROBERT LEIGHTON, D.D* 
 Archbishop of Glasgow. To which is prefixed, an entire new Life of the 
 Author, by the Rev. J. N. PEARSON, M.A., Trinity College, Cambridge, &c. 
 In Two Volumes, 8vo., with Portrait, price 18s. boards. 
 
 %* Compressed into One Volume, price 10s. 6d. boards. 
 
 VII. 
 
 A SHORT HISTORY of the CHRISTIAN CHURCH, from 
 its Erection at Jerusalem down to the Present Time. Designed for the Use of 
 Schools, families, &c. By the Rev. JOHN FRY, B.A. Rector of Desford, 
 in Leicestershire, Author of " A New Translation and Exposition of the very 
 ancient Book of Job," " Lectures on the Epistle of St. Paul to the Romans," 
 " Present for the Convalescent," &c. &c. In One Vol. 8vo, 12s. boards. 
 
 " His matter is unquestionably selected with judgment, and luminously arranged; his lan- 
 guage is clear and concise, and not deficient in elegance ; and we rise from the perusal of his 
 work with very favourable impressions of his character, with which otherwise we are unac- 
 quainted." Theological Review. 
 
 " Mr. Fry's Compendium of Church History is an instructive and interesting survey of the 
 various changes in the Christian profession, in its direct relations to the Christian doctrine, and 
 as influenced by the secular associations with which it has but too extensively been allied. To 
 such readers as wish for an Ecclesiastical History, written on the model of Milner's. and animated 
 by the same spirit, Mr. F.'s work will be highly acceptable, particularly as it is complete, and 
 comprised within a single volume." Eclectic Review. 
 
 VIII. 
 
 LECTURES, Explanatory and Practical, on the EPISTLE of 
 ST. PAUL to the ROMANS. By the Rev. JOHN FRY, B.A. Author of 
 " A Short History of the Christian Church," &c. &c. Second Edition, in 
 One Vol. 8vo, price 12s. boards. 
 
 IX. 
 
 A NEW TRANSLATION and EXPOSITION of the very 
 Ancient BOOK OF JOB ; with Notes, explanatory and philological. By 
 the Rev. JOHN FRY, B.A., Author of " A Short History of the Christian 
 Church," &c. &c. In One Vol. 8vo, price 12s. boards. 
 
Recently published by JAMES DUNCAN. 
 
 X. 
 
 THE SECOND ADVENT; or, the GLORIOUS EPIPHANY 
 of our LORD JESUS CHRIST : being an Attempt to Elucidate, in Chrono- 
 logical Order, the Prophecies, both of the Old and NBAV Testaments, which 
 relate to the approaching Appearance of the Redeemer, &c. By the Rev. 
 JOHN FRY, B.A., Author of " A Short History of the Christian Church," 
 &c. &c. In Two Vols. 8vo, price 28s. boards. 
 
 XI. 
 
 CANTICLES; or, SONG of SOLOMON: a new Translation, 
 with Notes, and an Attempt to Interpret the SACRED ALLEGORIES con- 
 tained in that Book ; to which is added, an Essay on the Name and Character 
 of the REDEEMER. By the Rev. JOHN FRY, B.A. Author of " A Short 
 History of the Christian Church," &c. &c. In One Vol. 8vo. Second Edi- 
 tion, price 6s. boards. 
 
 XII. 
 
 LYRA DAVIDIS; or, a NEW TRANSLATION and EXPO- 
 SITION of the PSALMS; grounded on the Principles adopted in the post- 
 humous Work of the late Bishop Horsley ; viz. that these Sacred Oracles have 
 for the most part an immediate Reference to Christ, and to the Events of his 
 First and Second Advent. By the Rev. JOHN FRY, B.A., Author of " A 
 Short History of the Christian Church," &c. &c. In One Vol. 8vo, price 18s. 
 boards. 
 
 XIII. 
 A PRESENT for the CONVALESCENT, or, for those to 
 
 whom, it is hoped, some Recent Affliction has been attended with a Divine 
 Blessing ; and for New Converts to Religion in general. Intended as a Sequel 
 to " The Sick Man's Friend," By the Rev. JOHN FRY, B.A., Author of 
 " A Short History of the Christian Church," &c. &c. In One Vol. 12mo, 
 price 4s. boards. 
 
 XIV. 
 
 SERMONS of HUGH LATIMER, sometime Bishop of Wor- 
 cester, now first arranged according to the order of time in which they were 
 preached : collated by the early impressions, and occasionally illustrated with 
 Notes, explanatory of Obsolete Phrases, Particular Customs, and Historical 
 Allusions. To which is prefixed a Memoir of the Bishop. By JOHN WATKINS, 
 LL.D. In Two Vols. 8vo, with Portrait, price 24s. boards. 
 
 " He, more than any other man, promoted the Reformation by his preaching. The straight- 
 forward honesty of his remarks, the liveliness of his illustrations, his homely wit, his racy manner, 
 his manly freedom, the playfulness of his temper, the simplicity of his heart, the sincerity of his 
 understanding, gave life and vigour to his sermons when they were delivered, and render them 
 now the most amusing productions of that age, and to us, perhaps, the most valuable." 
 Southey's Book of the Church. 
 
 XV. 
 
 SERMONS on SEVERAL OCCASIONS. By the late Rev. 
 JOHN HILL, Minister of the Gospel in London. Ninth Edition. In One 
 Vol. 8vo, price 10s. 6rf. boards. 
 
Recently published by JAMES DUNCAN. 5 
 
 XVI. 
 
 THE COMMUNICANT'S SPIRITUAL COMPANION ; or, 
 an EVANGELICAL PREPARATION for the LORD'S SUPPER: with 
 Meditations and Helps for Prayer suitable to the Subject. By the "Rev. 
 T. HAWEIS, LL.B. and M.D. Rector of All Saints, Aldwinkle, Northampton- 
 shire. Twelfth Edition, price 1*. Crf. bound. 
 
 XVII. 
 
 THE DOMESTIC ALTAR, a Six Weeks' Course of Morning 
 and Evening Prayers, for the Use of Families. To which are added, a few on 
 particular Occasions. By the late Rev. W. SMITH, A.M. In One Vol. 12mo, 
 Sixth Edition, price 5s. boards. 
 
 " We can give this volume our decided approbation ; and most sincerely hope that every pious 
 Family will avail themselves of this, as one of the most comprehensive volumes of Domestic 
 Prayer extant." Eclectic Review. 
 
 " Its principles are sound, its strain is pious and devout, its language is plain, perspicuous, and 
 scriptural, and we cheerfully recommend it to the use of all who may find occasion to call in the 
 aid of such auxiliaries." Edinburgh Christian Instructor. 
 
 A few Copies of the 5th Edition may be had in One Volume 8vo, price 8*. 
 boards. 
 
 XVIII. 
 
 MEMOIRS of EMINENTLY PIOUS WOMEN. By THOMAS 
 GIBBON, D.D., embellished with Eighteen fine Portraits; corrected and en- 
 larged, with the Addition of New Lives, by the Rev. SAMUEL BURDER, 
 M.A., Author of " Oriental Customs," &c. In Three Vols. 12mo, a New 
 Edition, price 24s. boards. 
 
 " The Memoirs which now appear for the first time in this work, or have been expressly 
 written for this Edition, are those of Mrs. Lucy Hutchinson, Mrs. Evelyn, Mrs. Savage, Mrs. 
 Hulton, the Viscountess Glenorchy, Lady Maxwell, Mrs. Berry, Miss Sinclair, Mrs. Fletcher, 
 and Mrs. Graham. These extensive additions, it is presumed, are of a character to give an en- 
 hanced value to the publication, which has long been a favourite with a large class of the Reli- 
 gious Public. The work in its present state forms the most interesting collection of Female Bio- 
 graphy extant." Eclectic Review. 
 
 XIX. 
 
 A THEOLOGICAL DICTIONARY, containing Definitions of 
 all Religious Terms ; a comprehensive View of every Article in the System of 
 Divinity ; an impartial Account of all the principal Denominations which have 
 subsisted in the Religious World, from the birth of Christ to the present Day: 
 together with an accurate Statement of the most remarkable Transactions and 
 Events recorded in Ecclesiastical History. By the Rev. CHARLES BUCK. Sixth 
 Edition, revised and corrected. Two Vols. in one, 8vo. price 15s. boards. 
 
 XX. 
 
 A HISTORY of BRITISH ANIMALS, exhibiting the descriptive 
 Characters and systematical Arrangement of the Genera and Species of Qua- 
 drupeds, Birds, Reptiles, Fishes, Mollusca, and Radiata, of the United King- 
 dom ; including the Indigenous, Extirpated, and Extinct Kinds ; together with 
 Periodical and Occasional Visitants. By JOHN FLEMING, D.D., F.R.S.E., 
 M.W.S. &c., and Author of the " Philosophy of Zoology." In One Volume, 
 8vo, price 18s. boards. 
 
 " This very important work, which has just appeared, we consider as infinitely superior to any 
 Natural History of British Animals hitherto published. It will become the standard book on 
 British Animals." Jamieson's Journal of Science, April. 
 
 See also Brewster's Journal of Science for April. 
 
 
t 
 
 6 Recently published by J A M ES D u x c A N . 
 
 XXI. 
 
 CHRISTIAN RECORDS ; or, a Short and Plain History of the 
 CHURCH of CHRIST : containing the Lives of the Apostles ; an Account of 
 the Sufferings of Martyrs ; the Rise of the Reformation, and the present State 
 of the Christian Church. By the Rev. THOMAS SIMS, M.A. Third edition, 
 corrected and enlarged, in one volume, 18mo,with a beautiful Frontispiece, price 
 3s. 6d. boards. 
 
 XXII. 
 
 A COMPARATIVE ESTIMATE of the MINERAL and 
 MOSAICAL GEOLOGIES, revised, and enlarged with relation to the recent 
 Publications of Messrs. Buckland, Conybeare, Cuvier, and Humboldt. Second 
 Edition. With an Introduction ; to which is now added a Postscript, on the 
 Strictures of the last BRITISH and WESTMINSTER Reviews. By GRANVILLE 
 PENN, Esq. In Two Vols. 8vo, Second Edition, price II. Is. boards. 
 
 This Edition contains Dissertatory Notes ; 1. On the Mosaic Days of Creation. 2. On the Ju- 
 bilean Chronology of Frank. 3. On M. Humboldt's Theory cTf Rocks. 4. On M. Cuvier's Nu- 
 merous Revolutions of the Earth. 5. On the recent Discovery of Fossil Human Remains at 
 Durfort and Kosritz. 6. On the Eastern Origination of Mankind. Also, a Supplement on 
 Caves in Limestone Formations, containing Fossil Animal Remains. 
 
 XXIII. 
 
 ELEMENTS of CONCHOLOGY, according; to the Linnaean 
 System; illustrated by Twenty-eight Plates, drawn from Nature. By the Rev. 
 E. I. BURROAV, A.M., &c. Third Edition. In 8vo, price 16s. boards; or 
 beautifully coloured by Sowerby, price 11. 11s. 6d. boards. 
 
 XXIV. 
 
 'H KAINH AIA0HKH, Novum Testamentum Manuale. Glasguse, 
 ex Prelo Academico. 32mo, price 8*. boards. 
 
 " This edition contains the Greek Text only; it follows the text of Aitton, except in a few 
 instances, in which the received readings are supported by the best authorities, and consequently 
 are most to be preferred. It is beautifully printed on the finest blue-tinted writing paper ; it was 
 read six TIMES, with the utmost care, in passing through the press, and will be found unusually 
 accurate. No contractions are used. In point of size, it is the smallest edition of the Greek 
 Testament ever printed in this country." HORNE'S Introduction to the Critical Study and Know- 
 ledge of the Holy Scriptures, vol. ii. p. 138, 4th edition. 
 
 XXV. 
 
 A GREEK and ENGLISH LEXICON; for the Greek Classics 
 in general, but especially for the Septuagint, Apocrypha, and New Testament. 
 By the Rev. GREVILLE EWING, Glasgow. In One large Vol. 8vo. Third 
 Edition, price 24s. boards ; or the Lexicon may be had separate, price 185.; 
 the Grammar, 6s. boards. 
 
 " From its size, cheapness, and laudable brevity (in most respects), this book is capable of 
 becoming generally useful." British Critic and Theological Review. 
 
 " The student who is not neglectful of his own benefit in the most essential respects, will 
 possess himself of the book, if in his power. Its cheapness is only equalled by the beauties and 
 clearness of its typography; and in the grand point of accuracy it is exemplary." Eclectic Review. 
 
 XXVI. 
 
 NOVUM LEXICON GR^ECUM,Etymologicum et Reale: cui 
 
 pro basi substantial sunt, Concordantiae et Elucidationes Homericae et Pindaricae. 
 Auctore CHRISTIANO TOBIA DABIM, Rectore Gymnasii Coloniensis Berolini. 
 Editio de novo instructa, Voces nempe omnes praastans, primo, ordine literarum 
 explicatas; deinde,familiis etymologicis dispositas, cura JOHANNISM. DUNCAN, 
 A.B. In One very large Volume, 4to, price 31. 35. boards. 
 
Recently published by JAMES DUNCAX. 7 
 
 XXVII. 
 
 THE CLASSICAL STUDENT'S MANUAL ; containing an 
 
 Index to every Page, Section, and Note, in Matthias's Greek Grammar ; Her- 
 mann's Annotations to Vigerus on Idioms; Bos on Ellipses ; Hoogeveen on the 
 Greek Particles ; and Kuster on the Middle Verb : in which Thucydides, 
 Herodotus, Pindar, jEschylus, Sophocles, and the Four Plays of Euripides 
 edited by Professor PORSON, are illustrated and explained. Second Edition ; 
 to which is now added, the First Twelve Books of the Iliad of Homer. 
 By the Rev. WILLIAM COLLIER SMITHERS. Intended for Students in the 
 Universities, and the Higher Classes in Schools. In One Vol. 8vo, price 9s. 
 boards. 
 
 XXVIII. 
 
 BIBLIA HEBRAICA, Editio longe Accuratissima.' Ab EVE- 
 RARDO VAN DER HoooHT, V.D.M. In One large Vol. 8vo, (1200 pages) 
 price U. 5s. boards, on fine vellum paper 36s. boards. 
 
 It has been the particular object of the Publisher to offer to the Public a neat and correct copy 
 of the Hebrew Scriptures at a moderate price ; and to ensure every attainable degree of accuracy, 
 every page has been (independent of the care previously bestowed upon it) revised four times 
 after the stereotype plates were cast, by persons familiar with the Hebrew language. The errors 
 which have been discovered in the edition of Van der Hooght have in this been carefully cor- 
 rected ; and the Publisher is determined to avail himself of that security which stereotype print- 
 ing alone affords, to guard against their recurrence in future. 
 
 XXIX. 
 
 A GRAM MAR of the HEBREW LANGUAGE; comprised in a 
 Series of Lectures, compiled from the best Authorities, and augmented with 
 much Original Matter, drawn principally from Oriental Sources ; designed for 
 the Use of Students in the Universities. Dedicated, by permission, to the 
 Right Rev. the Lord Bishop of Lincoln, Regius Professor of Divinity in 
 the University of Cambridge. By the Rev. S. LEE, A.M.; D.D. of the 
 University of Halle ; Honorary Member of the Asiatic Society of Paris ; 
 Honorary Associate and F.R.S.L. and M.R.A.S., &c. &c. ; and Professor of 
 Arabic in the University of Cambridge. In One Vol. 8vo, price 16*. boards. 
 
 XXX. 
 
 AN EASY METHOD of ACQUIRING the READING of 
 HEBREW with the VOWEL-POINTS, according to the Ancient Practice. 
 Price Is. 6d. on a sheet of drawing paper, hot-pressed. 
 
 This Table includes Three Lessons: containing 1. The different Alphabets in use among the 
 Jews; 2. The Vowel-points, and the Rules respecting them ; 3. The Letters and Points, with the 
 Pronunciation; and will be found of great utility, not only to parents who superi" tend the edu- 
 cation of their own children, but also to the Tutor and Young Student, to whom it opens at one 
 view a concise but comprehensive and systematic introduction to the Hebrew Language. And 
 even the man of letters, unacquainted with the Hebrew character and its readings, will not fail 
 duly to appreciate it as a very usefu^ Table of Reference. 
 
 TABLES in the SYRIAC and ARABIC LANGUAGES, on 
 the same plan as the above, price Is. fid. each. 
 
 XXXI. 
 
 TABLES of INTEREST, at 3, 4, 4|, and 5 per cent, from 
 II. to 10,000/., and from 1 to 365 days, in a regtilar progression of single days ; 
 with Tables at all the above rates from 1 to 12 months, and from 1 to 10 years. 
 By JOHN THOMSON, Accountant in Edinburgh. In One Volume, Ninth 
 Edition, 12mo, price 8*. bound. 
 
8 Recently published by JAMES DUNCAN. 
 
 XXXII. 
 
 CORPUS POETARUM LATINORUM. Edited by W. S. 
 WALKER, Esq. Fellow of Trinity College, Cambridge. In One large Vol. 8vo, 
 price 21. 2s. 
 
 The authors comprised in this Volume constitute THE WHOLE OF THE CLASSICAL LATIN 
 POETS, chronologically arranged, with brief notices of their Lives. 
 
 The Texts of the CORPUS POETARUM have not only been selected by the Editor from the best 
 editions; but the Orthography and Punctuation have been by him reduced to a uniform Stand- 
 ard. The greatest care has been taken to ensure correctness in the Printing. 
 
 The peculiar advantages of this Edition are, its portability and its cheapness. The whole body of 
 Latin Poetry may now lie/or reference on the table of the Student, in a single Volume, printed in a 
 type of great distinctness; to the Scholar who is travelling, this advantage becomes doubly 
 valuable. The same works cannot at present be obtained in less than twenty Volumes. The cost 
 of the Collection is below all example. The very lowest price of a pocket edition of those Authors, 
 who are here given entire, without the omission of a single line, is about Six GUINEAS. In the 
 common Delphin Editions, they amount to EIGHT GUINEAS. The CORPUS POETARUM is thus 
 two-thirds cheaper than any edition, even of the Text only, of the Latin Poets. 
 
 XXXIII. 
 
 RESEARCHES in SOUTH AFRICA ; illustrating the Civil, 
 Moral, and Religious Condition of the Native Tribes : including Journals of 
 the Author's Travels in the Interior ; together with detailed Accounts of the 
 Progress of the Christian Missions, exhibiting the Influence of Christianity in 
 promoting Civilisation. By the Rev. JOHN PHILIP, D.D., Superintendent of 
 the Missions of the London Missionary Society at the Cape of Good Hope, 
 &c. &c. In Two Vols. 8vo, illustrated with a Map and other Engravings, 
 price 21s. boards. 
 
 " A very interesting work, entitled ' Researches in South Africa,' &c. has just been given 
 to the public by Dr. Philip. It is full of valuable information respecting the progress made by 
 the Missionaries in instructing and civilising the Hottentots, Bushmen, and Caffres, and presents 
 a view of the characters of these people very different from that which those interested in degrad- 
 ing them have falsely attributed to them. Whosoever wishes to obtain accurate and authentic in- 
 formation on the latter point, may turn with confidence to this publication." Times, April 24. 
 
 " These interesting volumes are valuable on two accounts. In the first place, they contain 
 the narrative of a most intelligent traveller, among a people of whose real character and disposi- 
 tions we at present know very little." Athenaeum, April 18. 
 
 " This is the most important work connected with the colonial policy and coloured popu- 
 lation o? the British Empire which has come before us since we commenced our Review. It is 
 evidently the production of a man of superior talent and high principle, who, with a perfect 
 knowledge of his subject, is animated by a deep and fervid zeal for the cause to which he has 
 devoted nimself namely, the emancipation and improvement of the native tribes of Southern 
 Africa. This is one of the few books which we can safely recommend to such of our readers as can 
 afford to purchase it. It ought to be carefully perused by every friend of humanity, who desires 
 to promote the spread of liberty, civilisation, and true religion, over the world." London Weekly 
 Review, April 26. 
 
 " He (Mr. Buxton) could not sit down without calling the attention of the Right Honourable 
 Colonial Secretary (Sir George Murray) to a book published by Dr. Philip, which contained 
 more information on the subject of our colonies than any other work which he had ever read ; 
 and he hoped the Right Honourable Gentleman would have an opportunity of perusing it." 
 _Afr. Buxton' s Speech, House of Commons, July 15. 
 
 XXXIV. 
 
 DESCENT of the DANUBE, from RATISBON to VIENNA, 
 
 during the Autumn of 1827 ; with Anecdotes and Recollections, Historical and 
 Legendary, of the Towns, Castles, Monasteries, &c. on the Banks of the River; 
 and their Inhabitants and Proprietors, Ancient andModern. ByJ.R. PLANCHE, 
 Author of " Lays and Legends on the Rhine;" " Oberon," an Opera, &c. 
 In One Vol. 8vo, embellished with a Map, &c. price 10s. 6d. boards. 
 
 " His Descent of the Danube from Ratisbon to Vienna, is a volume of such varied merit and 
 interest, as to ensure its popular reception." Literary Gazette, July 5. 
 
 " Mr. Planche's recent publication of his journey from Ratisbon to Vienna, over the Danube, 
 contains much useful information, as well as elaborate historical notices of the most remarkable 
 places situated on the banks of that river." Times Newspaper, July 19. 
 
 See also the Athenaeum, July 16; Atlas, July 20; London Weekly Review; London Magazine, 
 August. 
 
 LONDON: j. MOVES, TOOK'S COURT, CHANCERY LANE. 
 

TREATISE 
 
 ON 
 
 THE ELE M ENTS 
 
 ALGEBRA. 
 
 BY THE 
 
 REV. B. (BRIDGE, B.D. F.R.S. 
 
 FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE; 
 
 AND LATE PROFESSOR OF MATHEMATICS IN THE EAST-INDIA 
 
 COLLEGE, HERTS. 
 
 SIXTH EDITION, ENLARGED AND CORRECTED. 
 
 LONDON: 
 
 PRINTED FOR T. CADELL, STRAND, LONDON ; 
 
 DKTGHTONS, STEVENSON, AND BARRETT, CAMBRIDGE ; 
 AND PARKER, OXFORD. 
 
 1826. 
 
Itontvon: 
 
 PRINTED BY RICHARD WATTS, 
 Crtfwn Court, Temple Bar. 
 

 ADVERTISEMENT. 
 
 THE favourable reception which this Treatise 
 has met with from the Public has induced the 
 Author, in this Sixth Edition, to make some con- 
 siderable additions and alterations. By contracting 
 the letter-press, more particularly in the early part 
 of the work, these improvements have been effected 
 in such a manner as to render it unnecessary to 
 enlarge the size, or increase the price, of the 
 volume. The whole has also been revised, and the 
 press corrected, by a Friend upon whose judgment 
 and accuracy the Author has the greatest reliance : 
 it is hoped, therefore, that it may still retain its 
 character, as a useful Elementary Work on this 
 branch of Mathematical Science. 
 
 LONDON, June, 1856. 
 
 176 
 
ERRATA. 
 
 P. 10, Note, for " at the end of Art. 39," read " in Art. 6 
 
 P. 73, dele the first five lines. 
 
 P. 144, 1. 3 from bottom, dele comma after ~. 
 
 P. 208, The Answer to Ex. 11. has been omitted ; via. 
 _log.(dg-c)-log. a 
 log. b 
 
CONTENTS. 
 
 SECT. INTRODUCTION. PAGE 
 
 I. Eocplanation of the Algebraic method of notation - 1 
 
 II. Exemplification of the Algebraic signs and symbols - 4 
 
 CHAP. I. 
 
 On the Addition, Subtraction, Multiplication, 
 and Division of Algebraic Quantities. 
 
 III. Addition - - 7 
 
 IV. Subtraction - 9 
 
 V. Multiplication - - 11 
 
 VI. Division - 15 
 
 VII. On the application of the foregoing rules to quan- 
 
 tities witJi lite.ral coefficients - - 21 
 
 VIII. Some general theorems, deduced by means of the 
 
 foregoing rules - -23 
 
 CHAP. II. 
 On Algebraic Fractions. 
 
 IX. On the reduction of fractions - - 25 
 
 X. On the addition, subtraction, multiplication, and 
 
 division of fractions - - 81 
 
 XI. On the method of finding the greatest common mea- 
 
 sure of two or more quantities - 36 
 
 CHAP. III. 
 
 On the Involution and Evolution of Numbers and 
 of Algebraic Quantities. 
 
 XII. On the involution of numbers and simple algebraic 
 
 quantities - 41 
 
 XIII. On the involution of compound algebraic quantities, 42 
 
TABLE OF CONTENTS. 
 
 SECT. 
 
 XIV. 
 XV. 
 
 XVI. 
 
 XVII. 
 
 XVIII. 
 
 XIX. 
 
 XX. 
 XXI. 
 
 XXII. 
 
 XXIII. 
 XXIV. 
 
 XXV. 
 XXVI. 
 
 PAGE 
 
 On the evolution of algebraic quantities - 48 
 
 On the investigation of the rules for the extraction 
 of the square and cube roots of numbers - - 52 
 
 On the general mode of expressing the powers and 
 roots of quantities by means of indices - - 57 
 
 CHAP. IV. 
 On Simple Equations. 
 
 On the solution of simple equations containing 
 only one unknown quantity - - 60 
 
 On the solution of simple equations containing two 
 or more unknown quantities - - - - 69 
 
 The solution of questions producing simple equations) 75 
 
 CHAP. V. 
 
 On Quadratic Equations. 
 
 On the solution of pure quadratic equations 88 
 
 On the solution of adfecttd quadratic equations - 90 
 
 On the solution of questions producing quadratic 
 equations - - - - - - - 95 
 
 On quadratic equations having impossible roots - 100 
 
 On the solution of quadratic equations of the form 
 x* n +px n = q - - 101 
 
 On the solution of quadratic equations containing 
 two unknown quantities - - - - 103 
 
 On the solution of certain equations, in which the 
 two unknown quantities (x and y) are similarly 
 involved - - - - - - -108 
 
 CHAP. VI. 
 
 On Ratios, Proportion, and Variation. 
 
 XXVII. Definitions - - 111 
 
 XXVIII. On the comparison and composition of ratios - - 114 
 
 XXIX. On proportion - 117 
 
 XXX. On variation 124 
 
TABLE OF CONTENTS. 
 
 CHAP. VII. 
 On Arithmetical and Geometrical Progression. 
 
 PAGE 
 
 XXXI. 
 
 XXXII. 
 XXXIII. 
 XXXIV. 
 
 XXXV. 
 XXXVI. 
 
 Definitions 130 
 
 On arithmetical progression - - 131 
 
 On geometrical progression - - 136 
 
 On the method of finding any number of arithmetic 
 or geometric means between two numbers - - 138 
 
 On the solution of equations relating to numbers in 
 arithmetical or geometrical progression - - 140 
 
 On the summation of an infinite series of fractions 
 in geometric progression ; and on the method of 
 finding the value of circulating decimals - 144 
 
 CHAP. VIII. 
 On Surds. , 
 
 XXXVII. On the reduction of surds - - 147 
 
 XXXVIII. On the application of the fundamental rules of 
 
 arithmetic to surd quantities - 151 
 
 XXXIX. On the method of finding multipliers which shall 
 
 render binomial surd quantities rational - 154 
 
 XL. On the method of extracting the square root of 
 
 binomial surds - 157 
 
 CHAP. IX. 
 On Miscellaneous Subjects. 
 
 XLI. On prime numbers and their relations ; and on 
 the method of finding the least common multi- 
 ple of two or more numbers - 160 
 
 XLII. Properties of numbers - - 163 
 
 XLIII. Permutations and combinations - - 166 
 
 XLIV. Unlimited Problems - - 168 
 
 XLV. Diophantine Problems - 171 
 
 XL VI. The solution of two questions relating to numbers 
 
 in geometrical progression - - - 178 
 
TABLE OF CONTENTS. 
 
 CHAP. X. 
 
 On the Binomial Theorem, and subjects connected 
 with it. 
 
 PAGE 
 
 XL VII. The general demonstration of this Theorem - - 176 
 XLVIII. Some observations arising out of the foregoing 
 
 theorem - - 179 
 
 XLIX. On the expansion of series - - 181 
 
 L. On the method of finding the approximate ratio 
 of the -powers and roots of numbers whose dif- 
 ference is small - - 187 
 
 LI. On the method of extracting the nth root of a bi- 
 
 nomial quadratic 'surd - ] 88 
 
 LII. On the method of reverting a series - - 194 
 
 CHAP. XI. 
 
 On Logarithms, and subjects connected with them. 
 LIIL Definition and properties of logarithms - - 196 
 LIV. On the method of finding the logarithm of any 
 
 given number - - 198 
 
 LV. On the method of constructing logarithmic tables, 200 
 
 LVI. On the application of logarithms to complex arith- 
 metical operations, and to the solution of expo- 
 nential equations - - 206 
 
 LVII. On the summation of geometric series - 208 
 
 LVIII. On compound interest - 211 
 
 LIX. On the method of finding the increase of popula- 
 tion in any country, under given circumstances 
 of births and mortality - - 215 
 
 LX. A Table, exhibiting the period in which the po- 
 pulation of a country has a tendency to DOUBLE 
 itself, from an estimate of its increase per cent. 
 at the end of every ten years - - 221 
 
ELEMENTS 
 
 OF 
 
 ALGEBRA. 
 
 INTRODUCTION. 
 
 ALGEBRA is that branch of Mathematical science, 
 in which number or quantity in general, and its seve- 
 ral relations,, are made the subject of calculation, by 
 means of certain signs and symbols, the nature and 
 meaning of which may be explained as follows. 
 
 I. 
 
 Explanation of the Algebraic Method of Notation. 
 
 1. Quantities whose values are known or determined, are 
 generally expressed by the fast letters of the Alphabet, 
 a, l t , d t &c. ; and unknown or undetermined quantities are 
 commonly represented by the last letters of the Alphabet, 
 x, y t x t &c. 
 
 2. The multiples of these quantities, such as, twice a, 
 three times I, five times x, &c. are expressed by placing 
 numbers before them thus, 2 a, 3b t 5x, &c. ; and the numbers 
 2, 3, 5, &c. thus prefixed are called the coefficients of a, b t x, 
 &c. in the several quantities 2 a, 3 1, 5 x } &c. 
 
 3. The sign -f (plus) placed between two or more quan- 
 tities means that those quantities should be added together ; 
 thus, a + b + x+ &c. means the sum of the quantities a, b, x, 
 &c. ; and the sign (minus) placed before any quantity 
 means that such quantity should be subtracted from the 
 
 B quantity 
 
 
INTRODUCTION. 
 
 quantity or quantities with which it is combined; thus, a b 
 means the difference between a and b ; and a + b c t the 
 difference between a + b and c. 
 
 4. In the general expression a + 2b 4x+3y 5z, &c. 
 such quantities as have the sign + prefixed to them are 
 called positive or affirmative quantities ; and such as have the 
 sign prefixed to ^them, are called negative quantities. If 
 no sign be prefixed to a quantity, then the sign + is under- 
 stood ; thus in the foregoing expression the positive quantities 
 aretf,-J-26, + 3y, and the negative ones 4 #, 5 z, 
 
 5. The general sign for the multiplication of quantities 
 is x ; but the manner of expressing the product of two or 
 more quantities is varied according to circumstances. The 
 product of quantities consisting of single letters is expressed by- 
 placing those letters one after another, and generally according 
 to the order in which they stand in the Alphabet $ thus, the 
 product of a and b is expressed by ab ; of a, b t and x } by abx ; 
 of 3a, x, andy> by 3axy; &c. &c. The product of a-\-b 
 and c + d is expressed by a 4- b x c -j- d, or a -j- b . c + d, or 
 (a-\-b) (c-\-d); in the two former cases, the line drawn over 
 a + b and c + d, to mark them as distinct quantities, is called 
 a vinculum. 
 
 6. The sign -~ placed between two quantities means that 
 the former of those quantities is to be divided by the latter ; 
 thus, G~- means that a is to be divided by b; a + b~-c + d t 
 that a + b is to be divided by c -j- d. But since every fraction re- 
 presents the quotient of the numerator divided by the denomina- 
 tor, this division is more simply expressed by making the former 
 quantity the numerator, and the latter the denominator of a 
 
 fraction ; thus, -^ expresses the quotient of a divided by b ; and 
 
 a + b 
 
 -7, the quotient of a+ I by c+d. 
 
 7. The 
 
INTRODUCTION. > 3 
 
 7. The powers of algebraic quantities are expressed by 
 placing a small figure (equivalent to the number of factors, 
 and called the index or exponent of the power) at the right- 
 hand of the letter; thus, 
 
 a x a or the square of a . . is expressed by a 9 , 
 
 bxbxb . . . . or the cube of b by 6 3 , 
 
 xxxxxxx . <n I\\Q fourth power oix by a?*, 
 
 (a + 6) (a -f 6) (a* 6) or tiiecube of a + b byoTJP, 
 
 and so on. 
 
 8. The roots of quantities are expressed by the sign \f t 
 with the proper index annexed ; thus, 
 
 or v/tf, expresses the square root of c> % 
 
 cube root of b, 
 
 fourth or biquadrate root of a+x, 
 and so on. The roots of quantities may also be expressed by 
 fractional indices; but this method of notation requires an 
 explanation, which will be given in Chap. III. 
 
 9. Like quantities are such as consist of the same letter, 
 or the same combination of letters ; thus, 5 a and 7 a ; 4ab 
 and 9 ab; 2 bx* and 6 bx*; &c. are called like quantities; 
 and unlike quantities are such as consist of different letters, 
 or of different combinations of letters ; thus, 4 a, 3b, Tax, 
 5 b x*, &c. are unlike quantities. 
 
 10. Algebraic quantities have also different denominations, 
 according to the number of terms (connected by the signs 
 + or ) of which they consist ; thus, 
 
 a, 2 b, Sax, &c. quantities consisting of one term, are called 
 
 simple quantities. 
 
 a -f#, a quantity consisting of two terms, is called a binomial, 
 b c (that particular species of binomial which expresses the 
 
 difference between two quantities) is called a residual. 
 
4 INTRODUCTION. 
 
 bx+y x, a quantity consisting of three terms, is called a tri- 
 nomial. 
 
 a*x + by3c+d t a quantity consisting of four terms, is called 
 a quadrinomiaL 
 
 a+bc+xy &c. a quantity consisting of an indefinite 
 number of terms, a multinomial. 
 
 11. The sign = placed between two or nlore quan- 
 tities, expresses the equality of such quantities; thus, 
 
 means that a + b is equal to c + d; and 
 \ex+fy" mean that the quantities 
 ax + ly, cx + dy, and ex+fy } are all equal to each other. 
 When quantities are thus connected together by. this sign of 
 equality, the expression is called an equation. 
 
 12. In algebraical operations, the word thei-efore, or con- 
 sequently^ often occurs. To express this word, the symbol .*. 
 is generally made use of; thus, the sentence " therefore a + b 
 is equal to c + d," is expressed by " . . a + b 
 
 II. 
 
 Exemplification of the Algebraic Signs and Symbols. 
 
 13. The use of these several signs, symlols, and allre- 
 viationSy may be exemplified in the following manner : 
 
 Ex. 1. In the algebraic expression a-f-& c, let a =9, 1=7, 
 and c=3 5 then 
 
 = 16 3=13. 
 
 Ex. 2. In the expression a #-f a?/ xy, let a =5, x=2,?/ = 7; 
 then, to find its value, we have 
 
 ax+ ay xy = 5x2+5x7 2x7 
 = 10 + 35-14 
 =45-14 = 31. 
 
 Ex.3. 
 
/ 
 
 INTRODUCTION. 5 
 
 Ex.3. What is the value of 7 -. where a = 5, A=3, 
 
 b + x 
 
 #=7, and ^ = 5 ? 
 
 Here ax-\>by=.b X7 + 3X5 = 35 + 15=50, 
 
 Ex.4. In the expression r- 9 , let a=3, b 5, c=2, 
 
 U 3C"^" CL ~~* C 
 
 x=zQ; What is its numerical value ? 
 
 Hereao; a H-6 a =3x6x6 + 5 X5= 108 + 25 = 133, 
 and 6#-a 3 -c=5 x6 3 x3-2 = 30-Q-2= 195 
 
 a^ + 6 3 133 
 *' A.r-a 9 -^ 19 "~ 7 * 
 
 Ex. 5. There is a certain algebraic expression consisting of 
 six terms connected together by the sign plus; the first term 
 of it arises from multiplying three times the square of a by 
 the quantity b ; the second term is the sum of the squares 
 of a and b divided by the quantity c 3 the third is the product 
 of a, &, and c ; the fourth is two-thirds of the product of a and 
 6 ; the ^/7A arises from dividing the square of a by the c&e 
 o/", # ; and the last term is a fraction, whose binomial numerator 
 is the difference between a and b t and whose trinomial deno- 
 minator is the sum of the cubes of a and b and the fourth 
 power of c. 
 
 All this is expressed, in one line of algebraic writing, thus ; 
 2ab a* a-b 
 
 Let a=4, 1 then the value of this quantity is, 
 
 6 = 3 I 16 + 9 15 4-3 
 
 | 2 + 27 + 64 + 274 
 
 c=2;J or 
 
 25 16 I 
 
CHAP. I. 
 
 ON THE 
 
 ADDITION, SUBTRACTION, MULTIPLICATION, 
 
 AND 
 
 DIVISION OF ALGEBRAIC QUANTITIES. 
 
 14. PREVIOUSLY to the application of the fundamental 
 rules of Arithmetic to Algebraic quantities, it may be proper 
 to observe, that, although the explanation of the sign minus in 
 Art. 3. does not, in strictness, extend beyond the subtraction 
 of a less quantity from a greater one, it is convenient to consi- 
 der negative quantities abstractedly, without any reference to 
 others from which they may be supposed to be subtracted. 
 For although, when we say that 2 i is equal to 3, we 
 mean nothing more than that the addition of 2, and subtrac- 
 tion of 5, is, on the whole, equivalent to the subtraction of 3 ; 
 yet, after the algebraic operation has been performed upon 
 it, the quantity 2 5 assumes the definite value of ~ 3. 
 
 It must be farther observed, that the word Addition is, in 
 Algebra, taken in a much more comprehensive sense than in 
 common Arithmetic; and as denoting the union of two or 
 more quantities, positive or negative. Thus, the union of 2 
 with 5, in the foregoing example, is called the addition of 
 those quantities. The same remark is to be extended to 
 Subtraction ; which is, properly, the finding such a quantity, 
 as, being algebraically added to the subtrahend, will give the 
 quantity from which the subtraction is made. 
 

 III. 
 
 ADDITION. 
 
 From the division of algebraic quantities into positive and 
 negative, like and unlike, there arise three cases of Addition. 
 
 CASE I. 
 
 To add like quantities with like signs. 
 15. In tins case, the rule is, " To add the coefficients of 
 " the several quantities together, and to the result annex the 
 (f common sign, and the common letter or letters;'* for it is 
 evident, from the common principles of Arithmetic, if +2 <?, 
 + 3 a, and +5 a be added together, their sum must be + 10 a; 
 and if 3& 2 , 4^ 3 , and 8& a be added together, their sum 
 must be 
 
 Ex.1. Ex.2. Ex.3. 
 
 3a 
 3x + 
 
 4x+ 8a ?b 11#' J + 5Jcy 4bc 5a 6 2a*+ 4 
 
 3a 3 7a*+ 3 
 
 5a 3 14a a + 35 
 
 
 
 Ex. 6, 
 
 Qx*y 3x+ 2 
 
 4x*y<2x+ 1 
 
 3x\j 5x+ 10 
 
 xy x-\-\5 
 
 ( a ) In these Examples, it may be observed that some of the quantities 
 have no coefficient. In this case, unity or 1 is always understood. Thus, 
 in adding up this column, we say, 1 + 1+11 + 9 +7=29; in the third, 
 2 + 1 4-4 + 7 + 5= 19 ; and so of the rest. 
 
8 ADDITION. 
 
 CASE II. 
 
 To add like quantities with unlike signs. 
 
 16. Since the compound quantity a +bc+d e &c. is posi- 
 tive or negative, according as the sum of the positive terms is 
 greater or less than the sum of the negative ones, the aggregate 
 or sum of the quantities 2 a 4a + 7a 3 a will be +2 a, and 
 that of the quantities 7 b* 5 & a -f-2 3 8 ft will be 4& 2 ; for 
 in the former case, the excess of the sum of the positive terms 
 above the negative ones is 2 a ; and in the latter, that of the 
 negative above the positive is 4 #*. Hence this general rule for 
 the addition of like quantities with unlike signs; *' Collect the 
 " coefficients of the positive terms into one sum, and also those 
 " of the negative ; subtract the lesser of these sums from 
 " the greater ; to this difference, annex the sign of the 
 " greater together with the common letter or letters, and the 
 " result will be the sum required." 
 
 If the aggregate of the positive terms be equal to that of 
 the negative ones, then this difference is equal to ; and con- 
 sequently the sum of the quantities will be equal to 0, as in 
 the second column of Ex. 2. following. 
 
 Ex. 1. Ex.2. Ex.3. 
 
 4x* 3x+ 4 7 ab + 3bc xy 5x 3 + 
 
 3x*5x+ 1 3ab bc+2xy 7x 3 + 
 
 4 2ab + 4bc 3xy Qx 3 
 
 xy 
 
 9#-K 9 ~<2ab * +3xy 4x 3 
 
 Ex.4. Ex.5. 
 
 5 a 3 2 a b + b* 
 
 ;_ y G 3 -f- fl^4-2/> a 
 
 7jc 3 x+Qy 4a 3 3ab-i- I* 
 
 CASE 
 
ADDITION. * 
 
 CASE III. 
 
 V 
 
 1 7. There now only remains the case where unlike quan- 
 tities are to be added together, which must be done by col- 
 lecting- them together into one line, and annexing their 
 proper signs ; thus the sum of 3x, <2a, -\-bb, 4y, is 
 3x 2a + 5b 4 ?/; except when like and unlike quantities 
 are mixed together, as in the following examples, where the 
 expressions may be simplified, by collecting together such 
 quantities as will coalesce into one sum. 
 
 Ex. 1. 
 
 x y 
 4c 2 + x 
 
 Collecting together like quanti- 
 ties, and beginning with 3ab t we 
 
 5ab 3c + d have 3 ab-\- 5ab 8ab} -\-x-\-x= 
 
 4y 
 
 3 c = + c > besides which there 
 are the two quantities + d and 
 
 which do not coalesce with* any of the others ; the sum re- 
 quired therefore is 
 
 Ex.2. Here4.r-*'=3* 
 
 . + . 
 
 IV. 
 SUBTRACTION. 
 
 18. If it were required to subtract 5 2 (i.e. 3) from 9, 
 it is evident that the remainder would be greater by 2, than 
 if 5 were subtracted. For the same reason, if b c were 
 
 C subtracted 
 
10 
 
 SUBTRACTION. 
 
 subtracted from a, the remainder would be greater by c, than 
 if I were subtracted. Now, if I is subtracted from a, the 
 remainder is a l\ and consequently, if b c be subtracted 
 from a, the remainder will be ab + c. Hence this, -general 
 Rule for the subtraction of algebraic quantities ; " Change 
 " the signs of the quantities to be subtracted, and then place 
 " them one after another, as in Addition/* 
 
 Ex. 1. From 5a + 3x 2l, take 2c-4y. Thequantity to 
 be subtracted with its signs changed, is 2c-j-4j/ ; therefore 
 the remainder is 5a+3x <2b 
 
 , take 
 
 Ex.2. From lx 
 The remainder is 7 $? 
 or 7x* 
 
 But when like quantities are to be subtracted from eachf 
 other, as in Ex. 2., the better way is to set one row under 
 the other, and apply the following Rule ; " Conceive the signs 
 " of the quantities to be subtracted to be changed, and then 
 " proceed as in Addition/' 
 
 Ex. 3. 
 
 From 7^ 
 Subtract 3 x* -f5 x \ 
 
 Ex. 4. Ex. 5. 
 
 3c-f- I 1 5?/ 3 4y + 3a 
 6y*-4y- a 
 
 a 2 
 
 Remainder 4 x* 7 x + 6 
 
 4 
 
 +4a 
 
 Ex. 6. 
 
 From 7x2/4-22 3y 
 Subtract 2 xy x+ y 
 
 Remainder 
 
 Ex. 7. Ex. 8. 
 
 s 5 13x 3 2x 3 + 7 
 z 1 1 # 3 -f X Q 6 
 
 MULTI- 
 
II 
 
 V. 
 
 MULTIPLICATION. 
 
 19. In the multiplication of algebraic quantities, the four 
 following Rules must be observed. 
 
 i. When quantities having like signs are multiplied together, 
 the sign of the product will be -{-; and if their signs are un- 
 like, the sign of the product will be .* 
 
 u. The coefficients of the factors must be multiplied to- 
 gether, to form the coefficient of the product. 
 
 in. The 
 
 * This Rule for the multiplication of the Signs may be thus ex- 
 plained ; 
 
 To multiply a b by c c?, is to add a b to itself as often as there 
 are units in c d; now this is done by adding it c times, and subtracting 
 it d times ; 
 
 But a b , added c times . . =ac c, 
 and a 5, subtracted d times =r a d-\-bd, 
 
 .*. a bXc d =ac be ad+bd. 
 
 i.e. 4- aX -}-c=-}-ac 
 
 bX -f 0= be 
 
 + aX d= ad 
 
 bXd=z+bd. 
 
 Or thus ; 
 
 I. If + is to be multiplied by +6, it means, that -fa is to be added 
 to itself as often as there are units in b ; and consequently the product 
 will be + ab. 
 
 II. If a is to be multiplied by -j-&, it means, that a is 'to be 
 added to itself as often as there are units in b ; and therefore the product 
 is a b. 
 
 III. If +a is to be multiplied by 6, it means, that -fa is to be 
 subtracted as often as there are units in b, as appears from the foregoing 
 explanation ; and consequently the product is ab. 
 
 IV. If a is to be multiplied by 6, it means, that a is to be sub- 
 tracted as often as there are units in b ; and, since to subtract a negative 
 quantity is the same as to add a positive one, the prodxict will be + ab. 
 
12 MULTIPLICATION. 
 
 in. The letters of which they are composed must be set 
 down, one after another; and generally according to their or- 
 der in the Alphabet. 
 
 iv. If the same letter is found in both factors, the indices 
 of it must be added together, to form the index of it in the 
 product. This follows immediately from Art. 7, as will appear 
 by the following example ; a 3 xa*~Qaaxaa aaaaa=.Q s . 
 
 Thus, -\-a multiplied by + b is equal to -\-ab, and 
 a multiplied by bis also equal to +ab; + 3xx5y 
 = 15 #7/5 3abx -f 4cJ= l<2abcd; 4a*b'*x 3 aid* 
 = -f- I'2a 3 b 3 d*', &c. &c. < 
 
 From the division of algebraic quantities into simple and 
 compound^ there arise three cases of Multiplication. In per- 
 forming the operation, the Rule is, " To determine first the 
 sign, then the coefficient, and afterwards the letters." 
 
 CASE I. 
 
 20. When both factors are simple quantities ; for which the 
 Rule has been already given. 
 
 Ex.1. Ex.2. Ex. ,3. Ex.4. 
 
 4ab Vaxy 3abc 5a*bc 
 
 3 a 3 ii 5a*b 2/>V 
 
 Ex.5. Ex.6. Ex.7. Ex.8. 
 
 4atc 9#y 
 
 3ac 2t/ 2 c 
 
 CASE II. 
 
 21. When one factor is compound and the other simple; 
 "Then each term of the compound factor must be multiplied 
 
 "by 
 
MULTIPLICATION. IS 
 
 " by the simple factor, as in thn last Case ; and the result will 
 " be the product required." 
 
 Ex. 1. Ex.2. 
 
 Multiply 3al Qac-{-d 3x* 2 a;* -f- 4 
 
 by 4 a 
 
 Product 1 2 a*b -8a*c + 4ad 
 
 Ex. 3. Ex. 4. 
 
 Multiply 7.r* 2x +4 a 12a 3 2a a -f 4 a 1 
 
 by 3 a 3x 
 
 Product 2 1 ax* + 6ax 1 2a* 
 
 Ex. 5. Ex. 6. 
 
 Multiply Qa*x+3a x+ 1 
 
 by -x* 3xy 
 
 Product 
 
 r 
 
 CASH III. 
 
 22. When loth factors are compound quantities, each term 
 of the multiplicand must be multiplied by each term of the 
 multiplier; and then placing like quantities under each other, 
 the sum of all the terms will be the product required. 
 
 Ex. 1. 
 
 
 Ex. 2. 
 
 Multiply a -\- 
 by a + 
 
 I 
 
 a + b 
 a- b 
 
 1st, bye . . a* + 
 2d, by b . . 
 
 ab 
 ab + b* 
 
 a* + ab 
 -al-V 
 
 Product G' + ' 
 
 2ab + b* 
 
 a 3 * -b* 
 
 a- b 
 
 Ex.4. 
 
14 MULTIPLICATION. 
 
 Ex. 4. Ex. 5. 
 
 3x' + 2o; 3z 3 
 
 4x + 7 ' 6x 7 
 
 8x* 
 f 21x a + 14tf 21a,- 2 4-14x 35 
 
 29^+ 14 J? 18.T 3 33.r 2 -f 44-Z 35 
 
 Ex. 6. 14ac Sal + 2 
 ac ab + 1 
 
 3a 2 bc+ 2ac 
 
 c +30*1* 2 
 
 14aV \lc? 
 
 T. Q ! 2 
 rv 7 T T I - 
 
 IJA, / . 0> n U/ *f 
 
 
 1 3 11 2 7 ,4 
 
 Ex. 8. Multiply a' + Sa'b+Satf+b 3 . . by a + ^. 
 
 ANSWER, 4 -f 4a 3 />+6Q 2 2 +4a 3 + 4 . 
 
 Ex.9 4x 2 y + 3a:?/-l .... by 2x~ x. . 
 
 Ais sw . 8 a 4 // + 2 a; 3 ?/ 2 x* 3 x*y + x. 
 
 Ex. 10 a: 3 -a: a +x-5 .... by 2a?* + jc+l. 
 
 ANSW. 2x 5 j; 4 + 2.r'' 10x 2 4J7 5. 
 
 Ex. 11 3a 2 +2a^-^ .... by 3a a - 2aZ>+ Z; 2 . 
 
 ANSW. 9a 4 -4a a Z' 2 +4aZ' 3 Z> 4 . 
 
 Ex. 12, 
 
MULTIPLICATION. 15 
 
 Ex. 12. Multiply x* + x*y + xif + y 3 . . by x~ y. 
 A NSW. x 4 y*. 
 
 Ex. 13 or-|r-t-l . . . . .by a; 2 *x. 
 
 5 3 11 2 i 
 ANSW. x 4 -OP + -x -x. 
 
 VI. 
 DIVISION. 
 
 23. In the division of Algebraic quantities, the four follow- 
 ing Rules (which arise immediately out of the consideration 
 that the quotient multiplied by the divisor gives the dividend) 
 are to be observed. 
 
 i. That if the signs of the dividend and divisor be like, then 
 the sign of the quotient will be -f ; if unlike y then the sign of 
 the quotient will be , (a) 
 
 ii. That the coefficient of the dividend is to be divided by 
 the coefficient of the divisor , to obtain the coefficient of the 
 quotient. 
 
 in. That all the letters common to both the dividend and 
 the divisor must be rejected in the quotient. (b) 
 
 ( a ) The Rule for the signs follows immediately from that in Multipli- 
 cation ; thus, 
 
 -f5 +ab 
 
 Since -fax -f i=-f * -; - = !"& an d r = . 
 
 + a +6 i.e. like signs 
 
 ^ a 6 j~ fl& I produce 
 
 b= -aft, . . . = -6, and =-\-a.y 
 
 I and unlike 
 
 -ax -i= 
 
 ( b ) If any letter 'or letters are found in the divisor, which are not in the 
 dividend, they must remain in the denominator of the fraction by which 
 the division is expressed. See Art. 35, with which this case coincides, 
 and the examples there. 
 
16 
 
 DIVISION. 
 
 iv. That if the same letter be found iu both the dividend 
 and divisor with different indices, then the index of that letter 
 n the divisor must be subtracted from its index in the dividend, 
 to obtain its index in the quotient. Thus, 
 
 -f a b c 
 i. -\-abc divided by -f cic . . . or = +U. 
 
 j OL C 
 
 Gabc 
 
 n. -\-Qabc ga...or =3 Ac. 
 
 2a 
 
 in. IQxyz -|5y...or- 
 
 IT. SOa'x'y 3 4axy, or 
 
 \0xyz 
 
 4axy 
 
 Cft Division, also, there are three Cases; the same as in 
 Multiplication. 
 
 CASE I. 
 
 24. When the dividend and divisor are both simple terms. 
 Ex.1. Ex.2. 
 
 Divide 18 ax* by 3 ax. Divide 15a 2 2 by 5 a. 
 
 i 
 
 = 6x. 
 
 Sax 5 a 
 
 Ex.3. Ex.4. 
 
 Divide <28x*y 3 by 4=xy. Divide 25 a 3 c' by 5a\ 
 
 Ex. 5. 
 
 Divide I4a 3 b*c b lac. 
 
 Ex. 6. 
 
 Divide 20x 4 ?yV by 
 ' 
 
 CASK II. 
 
 ( a ) If the index of any letter in the divisor should be greater than 
 that of the same letter in the dividend, the index in the quotient will, by 
 the rule, be negative. The signification of this negative index will be 
 explained at the end of Art. 89. 
 
DIVISION. 17 
 
 CASE II. 
 
 25. When the dividend is a compound quantity, and the 
 divisor a simple one, then each term of the dividend must 
 be divided separately, and the resulting quantities will be the 
 quotient required. 
 
 Ex.1. Divide 42a 4-3a& + 12a a by 3a. 
 42a-f Sab + 
 
 3 a 
 Ex. 2. Divide 90a*x 3 ]8ax* + 4d*x 2ax by 2ax. 
 
 <2ax 
 Ex. 3 . Divide 4 x 3 2 x* + 2 .r by 
 
 Ex. 4. Divide 24aVy 3axy+'6x*y* by 3xy. 
 24 a Vfy 3 a .r ?/ + 6 x 3 7/ a 
 3xy 
 
 Ex. 5. Divide 14a6 3 -f 7T 21 a 2 i 3 -f35a 5 6 by Tab. 
 
 6' 2 1 fl 
 
 CASE III. 
 
 26. When the dividend and divisor are both compound quan- 
 tities. In this case, the Rule is, "to arrange both dividend and 
 " divisor according to the powers of the same letter, beginning 
 " with the highest; then find how often the first term of the 
 t( divisor is contained in the first term of the dividend, and 
 " place the result in the quotient; multiply each term of the 
 " divisor by this quantity, and subtract the product from the 
 "dividend; to the remainder bringdown as many terms of the 
 " dividend, as will make its number of terms equal to the 
 
 D " number 
 
18 DIVISION. 
 
 " number of those in the divisor ; and then proceed as before, 
 " till all the terms of the dividend are brought down, as in 
 " common arithmetic." 
 
 Divide cf- 
 
 In this Example, the dividend is arranged according to the 
 powers of a, the first term of the divisor. Having done this, 
 we proaeed by the following steps ; 
 
 i. a is contained in a 3 , a 2 times; put this in the quotient. 
 ii. Multiply a I by a 2 , and it gives a*(il. 
 
 in. Subtract a 3 a*b from a 3 3a*b, and the remainder 
 is -2a*&. 
 
 iv. Bring down the next term + 3ab*. 
 
 v. a is contained in -Vcfl, <2al times; put this in the 
 quotient. 
 
 vi. Multiply and subtract as before, and the remainder 
 isa a . 
 
 vn. Bring down the last term b 3 . 
 
 vm. a is contained in ab* 9 +6* times; put this in the 
 quotient. 
 
 ix. Multiply and subtract as before, and nothing remains ; 
 the quotient therefore is a%ab + b*. 
 
 Ex, 2. 
 
DIVISION. 
 
 Ex. 2. 
 
 '#"+10aV + 5< 
 
 19 
 
 flV+lOflV 
 
 + 3aV 
 
 # * * 
 
 Ex. 3. 
 
 13.r 4 34x^4- 
 
 -*X Q 7# 
 1<2# 5 21x 4 
 
 * 
 
 Ex. 1 4. 
 
 6x 4 -96 
 6.r 4 12 
 
 * + isar 1 -96 
 
 + 12^ 24.r 2 
 
 * + 24^ - 96 
 + 24JC 2 
 
 ( a ) When there is a remainder, it must be made the numerator of a 
 Fraction whose denominator is the divisor; this Fraction must then be 
 placed in the quotient (with its proper sign), the same as in common 
 Arithmetic. 
 
20 DIVISION. 
 
 Ex. a. 
 
 \ * * I * % x 
 
 ,y,2 _J_ ,yi I I fft fy ! '' >V* I I '>"'* ____ fl' I /V" ____ ,V I 1 ! _L 
 
 */~J *A* ~ x +x x " ac^-jc+i 
 
 x'+^-a; 4 
 
 a; 1 
 
 Ex. 6. 
 
 
 -x 3 
 
 In this last Example, the division may be continued to any 
 number of terms at pleasure, observing only to place the whole 
 divisor under the last remainder. 
 
 Ex. 7. Divide a 4 + 4 cfb + 6 aT -f 4 a Z> 3 + 4 by a -f 6. 
 
 ANSWER, a 3 + 3 a*/> + 3 a a + Z- 3 . 
 
 Ex. 8 a 5 5a 4 .r4-10aV lOaV + 5ajc 4 x 5 
 
 by a 3 3 a a jc + 3 a x' - x 3 . 
 
 ATS s w. a* 2 a x -f .r a . 
 
 Ex. 9. ' 
 
DIVISION. 
 
 Ex. 9. Divide 25 j; 6 -a; 4 -fix 3 8 a;* by 5,x 3 -4x\ 
 
 ANSWER, 5 x 3 4- 4 # a + 3x4- 2. 
 
 Ex. 10 ..... a 4 4-8a 3 x4-24aV4-32ax 3 4- I6x 4 by 
 
 ANSW. a j 4 6 a*x 4- 1 2 a x* + 8 a; 3 . 
 
 Ex. 1) ..... a 5 -x & by a-*. 
 
 Aw s w. a* + 3 x 4- a V 4~ a: 3 4- x 4 . 
 
 Ex. 12 ..... 6*x 4 49* 2 - 2 0x by 3x'~3x. 
 
 AN s w . 2 x* 4- Q.r 4-5 
 
 Ex. 13 ..... 9x 6 46x 5 4 -95x*+l5Qx by :* 4jp~5. 
 
 ANSW. 9 x 4 1 Ox 3 4- 5 x* 30 x. 
 
 Ex. 14. . a'bv l-x\ 
 
 VII. 
 
 On the application of the foregoing Rule.* to Quantities with 
 literal Coefficients. 
 
 27. In applying the foregoing Rules to quantities with 
 literal coeffipients 3 such as, mx, ?iy, qx* &c. (where m t 
 n, q^ &c. may be considered as the coefficients of x, y, x*, &e.) 
 a compound quantity may be expressed by placing the coeffi- 
 cients of like quantities one after another (with their proper 
 signs) in a parenthesis, and then annexing the common letter 
 or letters. "Thus, the sum of mx and ?ix, which is mx-\-^^9 
 may be expressed by (m + ii)x % 9 their. difference, which is mx 
 noc, by (m ?z)jc; the multinomial mx ~\-nx~ px*-\-qx* 9 
 by (ra + ft p + 9)x 2 ; and the mixed multinomial pxy+qif 
 rxy -\-rny* nxy, by (pr ii) xy -\-(q +iri)y*' 9 &c. &c. 
 According to this method of notation the operations are per- 
 formed in the following Examples. 
 
 Ex. 1. 
 
APPLICATION OF THESE RULES TO QUANTITIES 
 
 Ex. I. 
 
 m# 2 -f ny-\- ,z 
 py* ry + nz 
 < qy'+my-vz 
 + ry-qz 
 
 Ex. 2. 
 
 From px 3 + qx* rx + s 
 Subtract mx 3 nx* -\-tx- v 
 
 Remainder (p m)x 3 + (q +n)x* 
 
 Ex. 3. 
 
 Multiply p# 4 + qx r 
 by mx n 
 
 mpa? + mq x* mrx 
 
 npx* nqx + nr 
 
 Product mx* + m 
 
 Ex. 4. 
 
 Multiply ax* lx +c 
 
 by x a ex + I 
 
 Product ax 4 
 
 ( ll ) As the sign prefixed to quantities in a parenthesis affects them 
 all; when this sign is negative, the signs of all those quantities must be 
 changed in putting them into the parenthesis. Thus, when -\-tx is sub- 
 tracted from r<r, the result is r x tx ; and, as this means that the 
 sum ofro?and tx is to be subtracted, that negative sum is expressed by 
 r= (r-\-)x. For the same reason, any multinomial quantity 
 ya? a +rr a , when put into a parenthesis with a negative 
 sign prefixed, becomes (m n + q r)# 4 . 
 
WITH LITERAL COEFFICIENTS. 23 
 
 Ex. 5. (Division.) 
 
 + aX2_( c a 
 
 -\-ax* 
 
 * _ 
 
 bx* +bcx* bx 
 
 * _j_ cx * c*x 
 
 + ex* -c*x 
 
 Ex. 6. Multiply mx*nxr ... by nxr. 
 
 ANSWER, 
 Ex. 7- Multiply a: 3 p x a + 90: r . by x a. 
 
 ANSW. ^-(a+p> 3 
 Ex.8. Multiply p.r a rx + q . . . by x*rx g. 
 
 ANSW. pa; 4 -(l +p)r^ 3 + (^+r a -p9)^- 9 9 . 
 Ex. 9. Divide ao; 3 -.(a 2 + &X + 6 a . by'ax-A. 
 
 ANSW. x*ax b. 
 VIII. 
 Some general Theorems, deduced by means of the foregoing 
 
 Rules. 
 
 From the clear and distinct manner in which quantity and its 
 several relations are represented throughout every part of an 
 Algebraic operation, the exemplification of its most ordinary 
 rules affords the means of investigating certain general Theorems 
 relating to the sum, difference, product, &c. &c. of numbers, of 
 which the following are examples. 
 
 28. Let a and b be any two numbers of which a is the 
 greater and b the lesser, and let their sum be represented by s 
 and their difference by d, 
 
 Then 0+^ = 5 
 
 and a b = d 
 
 .'. by Addition, 2 a = 
 
 and a = - 4. - i 
 2^2 ^ 
 
 by 
 
4 GENERAL THEOREMS. 
 
 by Subtraction, 2# = s d } 
 
 / s d t 
 
 = 2~i 3 
 
 From which we deduce this general Theorem, that "if theswm 
 " and difference of any two numbers be given, the greater of 
 " them may be found by adding half the given sum to half the 
 " given difference; and the lesser, by subtracting half the given 
 " difference from half the given sum." 
 
 29. Let a, b, s, d have the same relation as before, then 
 s = a+b 
 d=a-b 
 
 Hence,by Multiplication, s x dc?l* (See Ex.2.CaseIII. p. U ) 
 
 *-/;* 
 '' S ~ d 
 
 *A a *- 1 * 
 and a=. -- 
 
 s 
 
 From which it appears, that " if the sum and difference of 
 " any two numbers be multiplied together, the product of that 
 " sum and difference gives the difference of the squares of the 
 " two numbers;" and, that "if the difference of the squares 
 " of the two numbers be divided by their difference, it gives 
 " their sum; and if by their sum, it gives their difference." 
 
 30. Let the number c be divided into any two parts a and b, 
 Then c a + b 
 
 .-.by Multiplication, c 3 =a 2 -f 2 a l+b*(SeeEx. l. Case III. p. 14.) 
 From which we infer, that "if a number be divided into two 
 " parts, the square of the number is equal to the sum of the 
 " squares of the two parts, together with twice the product of 
 " those parts." 
 
 31. Let a and b be any two numbers; then, 
 
 Their difference =a b 
 The difference of their cubes =a*~-b* 
 
 By 
 
ALGEBRAIC FRACTIONS. 25 
 
 By actual division, a &Ja 3 6 3 (a*-f ab + b* (quotient) 
 
 Hence it appears, that " if the difference of the cubes of any 
 " two numbers be divided by their difference, the quotient 
 (f arising will be equal to the sum of the squares of the two 
 " nuinbers together with their product." 
 
 CHAP. II. 
 
 ON ALGEBRAIC FRACTIONS. 
 
 THE Rules for the management of Algebraic Fractions are 
 the same with those in Common Arithmetic. 
 
 IX. 
 
 ON THE REDUCTION OF FRACTIONS. 
 32. To reduce a mixed Quantity to a Fraction. 
 RULE. " Multiply the integral part by the denominator 
 " of the fractional, and to the product annex the numerator 
 " with its proper sign ; under this sum place the former deno- 
 " minator, and the result is the fraction required." 
 
 Ex. 1. R.educe 3 a + T to a fraction. 
 
 5a 
 
 The integral part x the denominator of the fraction -f the 
 numerator = 3 ax5a 2 -j-2#=15a 3 -f2o:; 
 
 . . 
 Hence, - * is the traction required. 
 
 o cr \ 
 
 E Ex. 2. 
 
ALGEBRAIC FRACTION*. 
 
 Ex.2. 
 
 4b 
 
 Reduce 5x -r IT to a fraction. 
 6a~ 
 
 Here 5 rx 6a e = 30a*JC$ to this add the numerator with its 
 
 30crx 4b . 
 proper sign, viz. 4b- } then - r- is the fraction re- 
 
 quired. 
 
 Ex.3. 
 
 2JC 3 
 Reduce 5# -- - to a fraction. 
 
 Here 5x X 7 = 35#. In adding the numerator 2x 3 with 
 its proper sign, it is to be recollected, that the sign affixed to 
 
 2# - 3 
 the fraction - means that the whole of that fraction is to 
 
 be subtracted, and consequently the signs of each term of 
 the numerator must be changed when it is combined with 35 x; 
 
 . 35X2X + 3 33X + 3 
 hence the fraction required is - - - = - 
 
 Ex. 4. Reduce 4 a b -f to a fraction. 
 
 ANSWER. 
 
 a -- to a fraction. 
 
 5 x 
 
 ANSW. 
 
 3a 
 
 5X 
 
 a? ax 
 o ..... ax-\- -- to a fraction. 
 
 a'-V 
 ANSW. -- 
 
 x 
 
 Ex. / ..... 3x - 1Q to a fraction. 
 
 ANSW . 
 
 
ALGEBRAIC ENACTIONS. 27 
 
 33. To reduce a Fraction to a mixed Quantity. 
 
 RULE. " Observe which terms of die numeral or are divisible 
 " by the denominator without a remainder, the quotient will 
 " give the integral part ; to this annex (with their proper signs, 
 " and observing the caution given in Ex. 3. of the last Article,) 
 " the remaining terms of the numerator with the denominator 
 " under them, and the result will be the mixed quantity re- 
 " quired." 
 
 EXAMPLE 1. 
 
 Reduce " -- to a mixed quantity. 
 
 
 Here * =a + b is the integral part, 
 and is tire fractional part ; 
 
 14 
 
 is the mixed quantity required. 
 
 Ex. 2. 
 3c 
 
 O CL 
 
 3c 
 Reduce - - - to a mixed quantity. 
 
 15 a* 
 Here- = 3a is the integral part, 
 
 O T 1 Q 
 
 and is the fractional part; 
 
 O Y* .---L Q f 
 
 .'. 3 a 4- is the mixed quantity required. 
 
 Ex. 3. Reduce ~ to a mixed quantity. 
 
 2 X 
 
 5 a 
 ANSWER, 2x 
 
 1 2 a* -f 4 a 3 c 
 Ex. 4 to a mixed quantity". 
 
 3c 
 
 ANSW. 3 a + 1 - 
 4a 
 
 Ex. 5. 
 
28 ALGEBRAIC FRACTIONS. 
 
 Ex. 5 - ' ; ~ to a mixed quantity. 
 
 ANSWER, 5 x* 5^T~* 
 
 34. To reduce Fractions to a common Denominator. 
 RULE. " Multiply each numerator into every denominator 
 " but its own for the new numerators, and all the denominators 
 " together for the common denominators." 
 
 & 
 
 EXAMPLE 1. 
 
 2.x 5x , 4 a 
 
 Reduce > *r-, and -> to a common denominator. 
 3 5 
 
 2x x 6 x 5 = \obx ~| r Hence the frac- 
 
 new numerators ; J tions required are 
 
 i 
 
 > 
 J 
 
 3 x ^ X 5 s= 1 5 b common denominator ; I 156' 156' 156* 
 
 Ex. 2. 
 
 Reduce ^ > and ~? to a common denominator. 
 
 Hence the frac- 
 tions required 
 
 Here(2a:+l)x4^ 
 
 3 x X 5 = 1 5 x 
 
 5 x 4 = 20 common denominator ; 
 
 are 
 
 \5x 
 
 Ex. 3. 
 
 5x ax 1 
 
 Reduce , ,and -, to a common denominator. 
 
 (I ~j~ X 5 ^. J7 
 
 Here 5 a: x 3x2jc=30a: a 4 
 
 (a a:) X (a + x) x 2^:= 2 aV 2 or 5 !/, the newfractions are- 
 
 'V 1 i V ^? V ^ T* 
 
 4 4J^iO/V*-*A^ 
 
 Ex. 4. 
 
 
ALGEBRAIC FRACTIONS. 29 
 
 Ex. 4. 
 
 3.r 4bx 5x* 
 
 Reduce - > ~T:~J and - j to a common denominator. 
 
 Qcfx <2Qalx 7 5 ax* 
 
 ANSWER, - 
 
 Ex. 5. 
 
 . 5X+1 
 
 Reduce -- > and - > to a common denominator. 
 3 
 
 ANSVV, 
 
 Ex.6. 
 
 x" 2 , 2. 
 Reduce --- - , -- , and -^jt to a common denominator. 
 
 40 ax 
 AN.VT. - -- , -, and 
 
 Ex.7. 
 
 7.r'-l ,4a;*-a;+2 
 Reduce - 5 and -- ? - > to a common denominator. 
 
 * J 
 
 14n 2a 
 
 ANSW - " and 
 
 35. To reduce a Fraction to its lowest terms. 
 RULE, " Observe what quantity will divide all the terms 
 " both of the numerator and denominator without a remainder ; 
 " Divide them by this quantity, and the fraction is reduced to 
 " its lowest terms." A more general Rule will be given at 
 the end of this Chapter. 
 
 EXAMPLE 1. 
 
 Reduce -- 9 - - to its lowest terms. 
 
 o*3 3C 
 
 The coefficient of every term of the numerator and denominator 
 of this fraction is divisible by 7>and the letter .r also enters into 
 every term ; therefore fx will divide both numerator and deno- 
 minator without a remainder. 
 
 Now 
 
30 ALGEBRAIC FRACTIONS. 
 
 ax+zix 1 
 
 __ =2 a -fa + 3 .r, 
 
 Hence, the fraction in its lowest terms is ^ X . 
 
 5x 
 
 Ex. 2. 
 
 Reduce - r*-s ~ - to its lowest terms. 
 
 J Q, C 
 
 Here the quantity which divides both numerator and deno- 
 minator without a remainder is 5#; the fraction therefore in 
 
 . 4bc a-f2c 
 
 its lowest terms is - * 
 a c 
 
 a -b 
 Ex. 3. Reduce Q *__^ to its lowest terms. 
 
 Here, a I will divide both numerator and denominator, 
 for by Ex.2. Case III. page 13. a a -/; a = (a + Z') (a-b); 
 
 hence ri is the fraction in its lowest terms. 
 
 lOx 3 
 
 to its lowest terms 
 
 Qx 
 
 Ex. 4. Reduce rr~~5 to its lowest terms. 
 
 ANSWER, 
 
 vJ 
 
 Salx* 
 
 Ex. 5 ..... -^ - to its lowest terms. 
 6ax 
 
 bx 
 ANSW. 
 
 Ex. 6 ..... - ~-^ - L to its lowest terms. 
 
 ANSW. 
 
 5lx 3 - 17^ + 340: 
 Ex. 7 ..... - r~3 - to its lowest terms. 
 
 ANSW. 
 
 Ex. 8. 
 
ALGEBRAIC FRACTIONS. SI 
 
 a-b 
 Ex. 8. Reduce CT to * ts l wes t terms. (See Art. 31.) 
 
 ANSW - 
 
 ON THE ADDITION, SUBTRACTION, MULTIPLICA- 
 TION, AND DIVISION OF FRACTIONS. 
 
 36. To add Fractions together. 
 
 RULE. " Reduce the fractions to a common denominator, 
 " and then add their numerators together ; bring the re- 
 " suiting fraction to its lowest terms, and it will be the sum 
 " required." 
 
 EXAMPLE 1. 
 
 3 x & x x 
 Add , -r- and -> together. 
 
 Of O 
 
 3.TX7 X3 = 
 
 5X7 X3=1O5 J 
 
 3.r4- 30;r+3 507 128r . 
 
 '" ToT" = 105 "the fraction 
 
 required. 
 
 (i 2a 5 ^ 
 
 Ex. 2. Add 7? 7, and > together, 
 v 36> 4a 
 
 3 fl -f 8 cfb -f 1 5Z> 3 __20a Q /> + 1 5 b 3 
 2a x 
 5bx 
 
 IN - . 
 
 I = (dividing by b) - 7 - is the 
 * y } ^ ai} 
 
 sum required. 
 
 _ 3xl 4x 
 
 Lx. 3. Add , - Q ^- > and -> together. 
 
 XQXX7 = 
 l)x3 X? =105.r 35 
 
 4JCX5 X2o;=40z* 68*+ 147^-35 , 
 
 is the sum 
 
 5X2^X7 =70x . 
 
 required. Lx. 4. 
 
C ALGEBRAIC FRACTIONS. 
 
 3 x 5 X 4 X 
 Ex.4. Add y-j *-j-> and ~r-> together. 
 
 934 x 
 
 ANSWER, - 
 693 
 
 Ex. 5. ... -J-) -jand; > together. 
 
 * v 7 ft , , . i, 
 
 105 a +28 a 0-f 30 1> 
 
 ANSW. - 
 
 . 
 Ex. 6. . . . - : - >and- together. 
 
 Axsw. 
 
 105 
 
 _ 
 
 Lx. 7. ... ^- > and - ^ , together. 
 
 ' ANSW. 32 
 
 Ex. 8. ... - , and - > together. 
 
 ANSW. 
 
 6x 
 Ex. 9. ... ~~^> and ~T^ together. 
 
 a -\-b a b 
 
 Ex. 10. .. .- r> and 7, together. 
 a b a-f-o 
 
 2 
 ANSW. - ? -_ 
 
 37. To Subtract Fractional Quantities. 
 RULE. " Reduce the fractions to a common denominator ; 
 and then subtract the numerators from each other, and under 
 the difference write the common denominator." 
 
 EXAMPLE 1. 
 
 3x I4x 
 
 Subtract from - 
 
 3 1 3 
 
 3xx 15 = 45^1 70x45x Q5x x, 
 
 X 5 = 
 
 5 x!5==75 required. 
 
 Ex. 2. 
 
ALGEBRAIC FRACTIONS. 
 Ex. 2. 
 
 
 . f 
 bubtract - from 
 
 6 7 
 
 x7 = 14#-f 7] 15#+6 14a;~7_^~ 1 . 
 
 x3 = 15x + 6 I '*' 21 " 21 
 
 x i = g j J fraction required. 
 
 Ex.3. 
 
 9 3# 5 
 
 From - - subtract ' 
 
 7 
 
 lO#-9)x7 = 70a?-63~) ^ 70# 
 (3a; 5)x8 = 24a;-40 I'*' 
 
 8 X 7 = 56 is 
 
 - 23 
 
 56 56 
 
 the fraction required. 
 
 Ex.4. 
 
 a+b ab 
 
 r rom - subtract ry 
 
 a u a+u 
 
 I ^^ _ r-w-rmm-r r, " ******* ' 
 
 b* J -r^p is the fraction required. 
 
 (a b)(a + b) = a* I? 
 
 4 <TC Q x 37 
 
 Ex. 5. Subtract from ~ ' .... ANSWER, 
 
 3 1 1 
 
 Ex. 
 
 6. . . 
 
 5x + 
 
 1 . Qlxl 
 
 trom "'" ' 
 
 -3 
 
 1 
 
 
 7 
 
 4 
 
 127^+17 
 
 _ 
 
 Fv 
 
 7 
 
 3x + 
 
 1 4:X 
 
 ANSW. 2g 
 
 Ex 
 
 8 
 
 2x 
 
 I fr m 5 
 3 , 4j; + ' 
 
 ANSW. ' ' ' " " ' . 
 
 Ex. 
 
 9. . . 
 
 3x 
 1 
 
 o 
 from ' ' 
 
 2b 
 ANSW ~* 73* 
 
 Ex. 
 
 10. , 
 
 ' ' a+b 
 3x- 
 
 - from 
 
 11^ + 49 
 ANSW. *-T: 
 
ALGEBRAIC FRACTIONS. 
 
 
 38. To Multiply Fractional Quantities. 
 RULE. " Multiply their numerators together for a new nu- 
 " merator, and their denominators together for a new denomi- 
 " nator, and reduce the resulting fraction to its lowest terms." 
 
 EXAMPLE 1. 
 
 8*' 
 
 Here 
 and 
 
 Multiply ~ by 
 7x9 =63 J ""* the fraction rec l uired is ~^' 
 
 A nn _i I \ /v 
 
 Ex. 2. Multiply - by 
 
 <24x* + 6x 
 
 .'. ~- = (dividing the nu- 
 merator and denominator by 3) 
 
 3X7 =21 
 
 . 
 
 is the traction required. 
 
 
 Ex. 3. Multiply - _^ by 
 
 By Ex. 2. CASE III. page 13. (a s - 2 ) x3a 2 = (a + b) 
 : b)x3cf; hence the product is 
 
 (dividing the numerator and/denominator by a + b) - , 
 
 5b 
 
 Ex.4. Multiply 
 
 Here 
 and 
 
 14 
 
 by 
 
 2 a; 3 3.r 
 21 ox 2 
 
 ToV = (dividing 
 
 the numerator and denomi- 
 
 nator by 7#) 5 is the 
 
 TC^t/ "~" O 
 
 fraction required. 
 
 Ex.5. 
 
ALGEBRAIC FRACTIOUS. 
 
 <2x 3x 6$ 
 
 Ex.5. Multiply ;~[ b y .... ANSWER, jTgl.y 
 
 .r, 10 
 Ex - 6 ...... - b 
 
 Ex - 7 
 
 33? \5x~30 9x 
 
 Ex - 8 ...... -^To ?-^- ' ANSW - V 
 
 39. On the Division of Fractions. 
 RULE. " Invert the divisor, and proceed as in Multiplication." 
 
 r i TV -i I4x \ Qx 
 Ex. 1. Divide - by - 
 
 3 1 4 T 3 3 
 
 Invert the divisor, and it becomes - ; hence ~- x -r~ 
 
 2o?' 9 <2x 
 
 42 X 7x 
 ~7~j> = "J" (dividing the numerator and denominator by 
 
 is the fraction required. 
 
 I4x 3 \0x 4 
 Ex. 2. Divide - - by ~ 
 
 \4x 3 ^25 _(l4.r 3) x 5^70 jc 15 
 
 " * 
 
 5, lO.x-4 lOo; 4 I0x 4 
 
 5 a __56 a 
 
 Ex. 3. Divide - ~ - by 
 
 
 
 2 a 
 
 5 ft 
 
 66 66 
 
 i X 
 
 6b 
 
 2 a 4 x (a -j- b) 
 
 _ 30 A (a b)__l5ab 
 
 2a 
 
 "iir is 
 
 the fraction required. 
 
 4x . 9x 20 
 
 Lx. 4. Divide --- by r .... ANSWER, 
 7 ^ O>3 
 
 ANSW. 
 
 3 
 
 Ex. 6. 
 
86 GREATEST COMMON MEASURE. 
 
 12 
 
 Ex. 6. Divide -.- by 
 
 ~T 
 
 ANSW. 
 
 XL 
 
 Orc the Method of finding the Greatest Common Measure 
 of two or more Quantities. 
 
 40. One quantity is said to measure another, when it is 
 contained in that other a certain number of times, without a 
 remainder. 
 
 41. A quantity is said to be a multiple- of another, when it 
 contains that other quantity a certain number of times, without 
 a remainder, 
 
 42. A common measure of two or more quantities is any 
 quantity which measures them all ; and the greatest common 
 measure is the greatest quantity which will so measure them. 
 Thus, 2 a is a common measure of the quantities 24 a b* 9 1 6 a*b c, 
 and l<2abc?, and their greatest common measure is 4 a b. 
 
 43. If one quantity measures another, it will also measure 
 any multiple of that quantity. Thus, let b measure a by the 
 units in m, then a=mb, and let na be a multiple (denoted by 
 the units in n) of a; then na nmb', consequently I measures 
 71 a by the units in ?i?n. 
 
 44. If one quantity measures two others, it will also mea- 
 sure their sum and difference. For let c measure a by the 
 units in w,, and b by the units in n, then a = mc, and b nc 
 therefore a + b^ =mc+nc,= (mn)c] consequently c mea- 
 sures a + b (their sum} by the units in m-\-?i, and a b (their 
 difference) by the units in m n. 
 
 45. The 
 i 
 
 ( a ) The quantity ffib^ means a plus or minux h. 
 
GREATEST COMMON MEASURE. 37 
 
 45. The Rule for finding the greatest common measure of 
 two numbers may be thus investigated. Let a and b be any 
 two numbers, whereof a is the greater -, and let the following 
 operation be performed upon them ; viz. 
 b)'a(p 
 
 pb_ 
 
 c)b(q 
 qc_ 
 
 rd 
 
 Where a divided by I gives the quotient p, and 
 remainder c ; b divided by c, the quotient q, and 
 remainder d; c divided by d, the quotient r, and 
 remainder o. Then, since in each case the divi- 
 dend is equal to the divisor multiplied by the quo- 
 
 tient plus the remainder, we have, 
 
 c=rd 
 
 qrd)qrd+d(qr-{' \)d 
 
 . Hence, 
 
 since p, q, r are whole numbers, d is contained in b as many 
 times as there are units in qr+ l, and in a as many times as 
 there are units in pqr+p + r ; consequently the last divisor d 
 is a common measure of a and b ; and this is evidently the case, 
 whatever be the length of the operation, provided that it be 
 carried on till the remainder is nothing. 
 
 This last divisor d is also the greatest common measure of 
 a and b. For let x be any common measure of a and b, such that 
 a=mx f and b = nx, then 
 c=apb = mxpnx=(mpn)x 
 
 d=b qc=nx (qmpqri)x=(n qm'}-pqri)x; .'.x mea- 
 sures d by the units in nqm+pqn 9 that is, every common 
 measure of a and b measures d. Now it has been shewn that 
 d is a common measure of a and b; and the greatest measure of 
 d is evidently Itself ; consequently d is the greatest common 
 measure of a and b. Hence this Rule for finding the greatest 
 common measure of two numbers ; " Divide the greater by the 
 " lesser, and the preceding divisor by the last remainder, till 
 " nothing remains ; the last divisor is the greatest common 
 " measure." 
 
 To 
 
38 GREATEST COMMON MEASURE. 
 
 To find the greatest common measure of three numbers, 
 a, b, c ; let d be the greatest common measure of a and b, and 
 x the greatest common measure of d and c ; then x is the 
 greatest common measure of a, b, and c. For, let a = md, 
 b = nd) d=px'j then a~mpx, and L=npx } therefore x is a 
 common measure of a and 6 ; and, since it also measures c, 
 it will be a common measure of a, b, and c. But, as above, 
 every common measure of a and I measures d ; therefore every 
 common measure of a, b, and c, measures d and c; and conse- 
 quently the greatest common measure of d and c, or x, will also 
 be the greatest common measure of fl, I, and c. 
 
 In general, let there be any set of numbers, a, b c, d t e, &c. ; 
 and let x be the greatest common measure of a and 5 ; y the 
 greatest common measure of x and c ; z the greatest common 
 measure of y and c? j &c. &c. ; then will y be the greatest 
 common measure of a, b, c ; z the greatest common measure 
 of a, , c, d ', &c. &c. 
 
 46. To find Hhe greatest simple common measure of Alge- 
 braic quantities, the Rule is, " to find the greatest common 
 " measure of their coefficients, and then annex to it the letters 
 common to all the quantities ;" thus the greatest common 
 measure of 24 a x*y*, \6bxy, and Saxy*; is <2xy. 
 
 To find the greatest compound common measure of two 
 algebraic quantities, " first divide each of them by their greatest 
 " simple common measure (if they have one) ; arrange their 
 " terms according to the dimensions of the same letter, and 
 " divide either, or both of them, by the greatest simple factor 
 " which it may contain ; then perform on them the same ope- 
 " ration as that for finding the greatest common measure of two 
 " numbers, observing only, that the remainders which arise 
 " are to be divided by their greatest simple factors, and that 
 " the dividends may, if requisite, be multiplied by any simple 
 " quantity which will make the first term of the dividend a 
 " multiple of the first term of the divisor. Lastly, multiply 
 " the compound common measure thus obtained by the simple 
 
 " one 
 
GREATEST COMMON MEASURE. 39 
 
 " one originally taken out,- and the product will be the greatest 
 " common measure required. " (a) 
 
 EXAMPLE 1. 
 
 Find the greatest common measure of 6 a 3 -f 1 1 ax + 3 x* and 
 6a* + lax 3x*. 
 
 These quantities having no simple divisors, we immediately 
 proceed as follows ; 
 
 6a* + lax- 3x*\ 6a* -f- 1 1 ax + 3# 
 * * 
 
 -f 4ax+6x 
 Dividing 4ax + 6x 2 by its greatest simple divisor 2x, we have, 
 
 2a 4- 3x) 6a* -4- 7 ax 3x* ( 3 a x 
 6a a + Qax 
 
 Hence 2a-f 3x is the greatest common measure. 
 
 Ex. 2. 
 
 Find the greatest common measure of 8 cPb* 1 Oa b* -f 2 b 4 
 and 9 4 6 - 9 a 3 ^> 2 + 3a'6 3 3 rz i 4 . 
 
 The greatest simple common measure of these quantities is b ; 
 which being taken out from both, they become Ba*l \Qab' 1 
 + 2^ 3 and 9 a 4 9 a 3 & -f- 3 a 2 ^ 3a& 3 ; the former of these is 
 divisible by 2 , and the latter by 3 a ; which divisions being 
 
 made, 
 
 ( a ) The rejection of these simple factors from the original quantities, 
 and from the remainders which arise in the process, or the multiplication 
 of the dividends pointed out in the Rule, will not affect the compound 
 common measure sought ; which can have no simple factor, because the 
 original quantities have (by the Rule) their simple factors taken out, pre- 
 viously to this part of the process. 
 
40 GREATEST COMMON MEASURE. 
 
 made, the given quantities are reduced to 4 a 9 5ab + 1\ and 
 3a 3 3a*b-\-ab b*. Multiply this last by 4, to make the 
 operation succeed, and we have 
 
 3a *l>+ ab*-4b 3 
 
 Dividing the remainder by b, and multiplying the new divi- 
 dend by 3, we have 
 
 igab+ I9^ a 
 Lastly, Divide the remainder by IQb, and proceed thus ; 
 
 4ab4b* 
 
 Which gives a I for the compound common measure ; and 
 this being multiplied into the simple one I, we have ab ~-b* for 
 the greatest common measure sought. 
 
 
li 
 
 CHAP. III. 
 
 INVOLUTION AND EVOLUTION OF NUMBERS 
 AND OF ALGEBRAIC QUANTITIES. 
 
 
 XII. 
 
 On the Involution of Numbers and Simple Algebraic 
 Quantities. 
 
 47. Involution, or "the raising of a quantity to a given 
 power," is performed by the continued multiplication of that 
 quantity into itself, till the number of factors amounts to the 
 number of units in the index of that given power. Thus, the 
 square of a or a 2 = <z x a \ the cube of b or b 3 =b x b x b ; the 
 fourth power of 2 = 2x2x2x2=16; the fifth power of 
 3 = 3X3X3X3X3 = 24 3; &c. &c. This rule as applied to 
 numbers will be readily understood by the mere inspection of 
 the following Table. 
 
 ROOTS AND POWERS OF NUMBERS. 
 
 Roots 
 
 1 
 
 o 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 Square 
 
 1 
 
 4 
 
 9 
 
 16 
 
 25 
 
 36 
 
 49 
 
 64 
 
 81 
 
 100 
 
 Cube 
 
 1 
 
 8 
 
 27 
 
 64 
 
 125 
 
 216 
 
 343 
 
 5J2 
 
 729 
 
 1000 
 
 4th power 
 
 1 
 
 16 
 
 81 
 
 256 
 
 625 
 
 1296 
 7776 
 
 2401 
 
 4096 
 
 6561 
 
 10000 
 
 5th power 
 
 1 
 
 32 
 
 243 
 
 1024 
 
 3125 
 
 16807 
 
 3276$ 
 
 59049 
 
 100000 
 
 48. The operation is performed in the same manner for 
 simple algebraic quantities, except that in this case it must be 
 observed, that the powers of negative quantities are alternately 
 4- and ; the even powers being positive, and the odd powers 
 negative. Thus the square of -f2ais +2ax +2 a or -f-4a*; 
 
 G the 
 
INVOLUTION. 
 
 the square of 2 a is 2 a x 2 a or + 4 a* ; but the cz/fo of 
 -2a=-2ax 2ax 2a = -f 4a 2 x -2a= 8a 3 . 
 
 The several powers of - are, 
 
 bb 
 
 a a a a 3 
 
 =- X 7 X~X- 
 
 b b b 6 s 
 
 a a a a a 4 
 -X-x-X-=-, 
 
 &c.=&c. 
 
 And the several powers of 
 
 6 6 6 2 
 
 Squares X = H 
 
 2c 2c 4c 2 
 
 2c 2c 2c Sc 3 
 
 6666 
 = -X ~-X - X ~ C 7 = 
 2c 2c 2c 2c 
 
 Upon this principle the powers of the several roots in the 
 following Table are calculated. 
 
 ROOTS AND POWERS OF SIMPLE ALGEBRAIC QUANTITIES. 
 
 Roots 
 
 a 
 
 -b 
 
 26* 
 
 a 
 
 ~25 
 
 SW 
 
 y 
 
 2a 
 36 
 
 a*6 
 
 2 
 ~6 3 
 
 3^r 
 5 
 
 x 
 J0 
 
 Square 
 
 a" 1 
 
 + ' 
 
 4i* 
 
 a a 
 
 46 a 
 
 -?- 
 
 4 a 
 96 2 
 
 a46 2 
 
 4 
 
 + 6^ 
 
 9 t r> 2 
 + 25 
 
 ^ 
 
 1(J^2 
 
 Cube 
 
 a 3 
 
 63 
 
 8 
 
 a 3 
 863 
 
 27^ 
 
 " ^ 
 
 8 a 3 
 
 276 3 
 
 a 6 63 
 
 a 6 
 ""69 
 
 27a^ 
 ""l23~ 
 
 ^ 3 
 
 647^ 
 
 4th , 
 
 //4 
 
 -4-W 
 
 1 fi/1 8 
 
 CZ4 
 
 + 8U8 
 
 16a 4 
 
 ,,814 
 
 8 
 
 8.J,r4 
 
 ^4 
 
 Power 
 
 
 
 
 1664 
 
 2^ 
 
 8164 
 
 
 + i* 
 
 + 625 
 
 256^4 
 
 5th 
 
 a* 
 
 b 5 
 
 3<2jio 
 
 a 
 
 243.T 10 
 
 32s 
 
 10 65 
 
 a 10 
 
 243^5 
 
 *5 
 
 Power 
 
 
 
 
 3265 
 
 * 
 
 24365 
 
 
 6'5 
 
 3125 
 
 10'24y5 
 
 XIII. 
 
 On ^Ae Involution of Compound Algebraic Quantities. 
 
 49. The powers of compound algebraic quantities are raised 
 by the mere application of the Rule for Compound Multipli- 
 cation (Art. 22.) Thus, 
 
 Ex. 1. 
 
 I 
 
INVOLUTION. 4-0 
 
 Ex. 1. What is the square Ex. 2. What is the cube of 
 
 ofa + 2^? a xl 
 
 a ~\-2b cfx 
 
 a +<2l cf x 
 
 a*+2ab a* a*x 
 
 Square =a a 4-4aZ>-f4/> a Square= a 4 2 cfx + x* 
 
 a*x 
 
 = 6 - 3 a*x + 3 aV -a; 3 
 
 Ex.3. 
 
 What is the 5 th power of a + b ? 
 a +b 
 a +b 
 
 a'+ ab 
 
 + ab +1>* 
 
 Q 4- 2a b + b* = Square 
 a 4- b 
 
 b+ ab 1 
 
 + a*b+ 
 
 b* = Cube 
 
 a -f b 
 
 6aT-f 4a^ 3 -f 6 4 = 4 th Power 
 a 4- b 
 
 5 th Power. 
 
 Ex.4. 
 
44 BINOMIAL THEOREM. 
 
 Ex. 4. The 4 th power of a + 3# is a 4 + 12a 3 + 54a^ 
 
 Ex. 5. The square of 3x 2 + 2 #4-5 is 9#* + 1'2 a; 3 + 34 x* 
 
 Ex. 6. The cube of 3# 5 is 27,r 3 135# 3 + 225z 125. 
 Ex. 7. The cw&e of x* 2x+l is a: 6 6# 5 + I5x* 20x* 3 
 
 50. In the involution of a binomial quantity of the form a + 1> 9 
 the several terms in each successive power are found to bear 
 a certain relation to each other, and observe a certain law, 
 which the following Table is intended to explain. 
 
 TABLE OF THE POWERS OF fl -f I. 
 
 Powers. 
 
 Mode of 
 
 Powers expanded. 
 
 Square 
 
 Cube 
 
 (a + 6)3 
 
 4 th Power 
 
 5 th Power 
 
 6 th Power 
 
 (a+ft) 6 
 
 The successive powers of a b are precisely the same as 
 those of ci + l, except that the signs of the terms will be 
 alternately + and . Thus, the 4 th power of a b is 
 & 4 and so of the rest. 
 
 In reviewing that column of the foregoing Table, which con- 
 tains the powers of a + l expanded, we may observe, 
 
 i. That in each case, thejirst term is a raised tathe given 
 power, and the last term is b raised to the same power ; thus, 
 in the square, the Jirst term is a Q , and the last I 11 ; in the cube, 
 the first term is a 3 , and the last V ; and so of the rest. 
 
 ir. That, with respect to the intermediate terms, the powers 
 
 of a 
 
BINOMIAL THEOREM. 45 
 
 of a decrease, and the powers of I increase, by unity in each 
 successive term. Thus, in the fifth power, we have 
 In the second term .... a*l ; 
 
 third ........ a 3 ^; 
 
 fourth ....... aV; 
 
 fifth ........ a> 4 ; 
 
 and so in the other powers. 
 
 in. That in each case, the coefficient of the second term is 
 the same with the index of the given power. Thus, in the 
 square it is 2 ; in the cube it is 3 ; in the fourth power .it is 4 ; 
 and so of the rest. 
 
 iv. That if the coefficient of a in any term be multiplied by 
 its index, and the product divided by the number of terms to 
 that place, the quotient will give the coefficient of the next 
 term. Thus, 
 
 coeff. of a in the 2 d term x its index 
 In fre fourth power, - ' num ber of terms to that place 
 
 4X3 12 
 = ^ = -~= 6 = coefficient of third term. 
 
 til l& 
 
 coeff. of a in the 4 th term x its index 
 In the sixth power, number of terms to that place 
 
 20X3 60 r ~ f .i j 
 
 = - = =15= coefficient 01 fifth, term. 
 
 We are thus furnished with a general Rule for raising the 
 binomial a + b to any power, without the process of actual 
 multiplication. For instance, let it be required to raise a + b 
 to the eighth power ; then, according to the Rule just laid down, 
 Theirs* term is ............ '. ......... a 8 . 
 
 The second ....................... 8a 7 b. 
 
 8x7 
 The third ............... . <z 8 &' 
 
 The/oar/^ 
 
 The fifth 
 and so on. 
 
46 BINOMIAL THEOREM. 
 
 And thus we have 
 
 -f 
 
 In the same manner it will be found, 
 Ex.2. That (a- b) 7 = a 7 -7 a*b + 21a 5 &*-35a 4 3 -f 35a 3 
 
 Ex.3. That (ff-y) 9 =^ 9afy-t-36j;y--84ffy + 126-rV 
 
 Ex.4. That (x + a) w =x w + ioa 9 a H- 45 aV -f is 
 2 1 OtfV + 25 2#V H- 2 1 OusV + 1 20x s a 7 -f 45#V -f- 1 Oxa 9 -fa 10 . 
 In reviewing these several Examples, it may be observed, that, 
 when the number of terms in the resulting quantity is even 9 the 
 coefficients of the two middle terms are the same ; and that 72 
 all cases the coefficients increase as far as the middle terms, and 
 then decrease precisely in the same manner until we come to 
 the last term. By attending to this law of the coefficicjits, it 
 will only be necessary to calculate them as far as the middle 
 term, and then set down the rest in an inverted order. Thus, 
 
 in Ex. 3. .(ac-yl) 9 
 
 The first five coefficients are 1, 9, 36, 84, 126. 
 The last five ........ 126, 84, 36, 9, 1. 
 
 51. But we are not yet arrived at the most general form in 
 which this Rule may be exhibited. Suppose it was required to 
 raise the binomial a + b to any power denoted by the number 
 (n). Proceeding with n as we have done with the several in- 
 dices in the preceding examples, it appears that 
 
 The first term would be a". 
 
 The second ........ na n ~ l b. 
 
 The third ~ s~ 
 
 The fourth T-T 
 
 The>JM " ( "~' ) 2^ 
 
 The 
 
BINOMIAL THEOREM. 47 
 
 The last . . b*. 
 
 2.3.4 
 By the same process, (a ->)" = a" ; 
 
 a*~ 3 b 3 -f &c. ; the signs of the terms being 
 alternately 4- and . 
 
 This general and compendious method of raising a binomial 
 quantity to any given power, is called, from the name of its 
 celebrated inventor, Sir I. NEWTON'S " Binomial Theorem." 
 Its use will appear from the following Examples. 
 
 EXAMPLE 1. 
 
 Raise x*-{-3y to thcjifth power. 
 In comparing (x* + 3y 2 ) 5 with (a-|-6) n , we have, a = x*, 
 
 Substituting these quantities for a, b, n in the foregoing 
 general formula, it appears, that 
 Theirs/? . , ~ 
 
 r (ft n \ is \X P A ' 
 
 tenn 3 ' ' v / ' 
 
 is 
 
 ^ ) . . is 5 X 
 
 4/7. ____ M aB - 3 p i3 5 X - X x * X 
 
 \ 
 
 5*, "-~- is 5 X X X X ^X (,V)<=405, V . 
 
 . . (&") ...................... is 
 
 So that (# 2 + 3/) 5 =a' 10 + 1 5*V+ 900V + 270^ + 405jpV -f 243y 10 . 
 
 In the application of this formula, it may be observed, that 
 the number of terms of which the binomial consists, is always 
 one more than iheindexof the given power ; after having calcu- 
 lated therefore as many terms as there are units in the index of 
 the given power, we may immediately proceed to the last term. 
 
 Ex. 2. 
 
48 BINOMIAL THEOREM. 
 
 Ex. 2. 
 
 Raise 3x+Qy to the 6 th power. 
 Here steaj^ ( 3x + 2 ?/) 6 = 
 
 2 ~ 
 
 w = 6 3 
 
 Ex. 3. 
 
 Raise x Vy* to the 7 th power. 
 Here x=.a~l and comparing (XT- 2 y 2 ) 7 with (a I)*, we have 
 
 12 1287/ 14 for the quantity 
 required. 
 
 52. By means of this Theorem, we are enahled to raise a 
 trinomial or quadrinomlal quantity to any power, without the 
 process of actual multiplication. Thus, suppose it was required 
 to square a -f I + c ; inclosing it in a parenthesis (a-\-b), and con- 
 sidering it as one quantity, we should have 
 
 In the same manner we have, 
 
 Ex.2. 
 
 a + c 3 = a 3 4-3a^ + 3aZ; 2 + Z' 3 + 3a'c4-6flk + 3/' 2 c 
 c 3 = a 3 + 3 + c 3 + 3(a 2 Z> + ^ 2 -f G 2 c + ac 2 + ^ 2 c 4- 
 
 Ex.3. 
 
 XIV. 
 
 On <Ae Evolution of Algebraic Quantities. 
 53. Evolution, " or the Rule for extracting the root of any 
 quantity," is just the reverse of Involution; and to perform 
 the operation, we must inquire what quantity multiplied into 
 
 itself, 
 
EVOLUTION. ) 
 
 itself, till the number of factors amount to the number of units 
 in the index of the given root, will generate the quantity whose 
 root is to be extracted. 
 
 54. This Rule, as applied to small numbers and simple 
 algebraic quantities, may be easily explained by reference to 
 the Tables in Art. 47, 48. Thus, 
 
 49 = 7 x 7; /. the square root of 49, or ^49= 7. 
 63= _5 X __j x _ . ... t he cube root of 6 3 = (%/ 63) = 6. 
 2a 2a 2a 16a* 
 
 e r00t f 
 
 32 = 2 X 2 X 2X2 X 2 ; .'. theZA root of 32 (-{/32)= 2. 
 &c. &c. 
 
 55. If the quantity under the radical sign does not admit of 
 resolution into the number of factors indicated by that sign, or, 
 in other words, if it be not a complete power ^ then its exact 
 root cannot be extracted, and the quantity itself, with the 
 radical Sign annexed, is called a Surd. Thusy/37, v/ a * 
 \/ b*, v/ 4 ?> & c * & c are ^ ur ^ quantities. The application of 
 the fundamental rules of arithmetic to quantities of this kind 
 will form the subject of Chap. VIII. 
 
 56. In the involution of negative quantities, it was observed 
 that the even powers were all + , and the odd powers ; there 
 is consequently no quantity which, multiplied into itself in such 
 manner that the number of factors shall be even, can generate 
 a negative quantity. Hence quantities of the form ->/o a , 
 
 / 10, /^a 3 , v/~5> \/ a *> & c - & c - llave no real root > 
 and are therefore called impossible. 
 
 57. In extracting the roots of compound quantities, we must 
 observe in what manner the terms of the root may be derived 
 from those of the power. For instance, (by Art. 50.) the square 
 of a + 1 is a 2 + 2a/; + 3 , where the terms are arranged according 
 to the powers of a. On comparing a + b with a* + 2 a b + b*, we 
 observe that the first term of the power (a 2 ) is the square of the 
 
 H first 
 
50 EVOLUTION. 
 
 first term of the root (a). Put a 
 therefore for the first term of the 
 root, square it, and subtract that 
 
 square from the first term of the 
 power. Bring down the other two 
 
 terms 2 a b + &*, and double the first 
 
 term of the root ; set down 2 a, and 
 
 having divided the first term of the remainder (<2ab) by it, it 
 
 gives b) the other term of the root ; and since <2al + b" = (<2a + b)b, 
 
 if to <2a the term b is added, and this sum multiplied by b, the 
 
 result is 2 a b + 1* ,* which being subtracted from the two terms 
 
 brought down, nothing remains. 
 
 58. Again, the square of a + 1 + c (Art. 52.) is a 2 + 2 a b + 1* 
 + 2ac+2#c + c 2 ; in this case the root may be derived from 
 the power, by con- 
 tinuing the pro- 
 cess in the last 
 Article. Thus, 
 having found the 
 two first terms 
 (# -j_ fy of the root 
 as before, we bring 
 down the remain- 
 ing three terms 2<zc+2&c+e of the power, and dividing 2 ac 
 by 2 a, it gives c, the third term of the root. Next, let the last 
 term (I) of the preceding divisor be doubled, and add c to the 
 divisor thus increased, and it becomes 2 a + 2& + c ; multiply 
 this new divisor by c, and it gives 2ac+2&c + c a , which' being 
 subtracted from the three terms last brought down, leaves no 
 remainder. In this manner the following Examples are solved. 
 
 89 , . ../_ , 3 
 
 Ex. 1. 4: 
 
 89 
 
 15 a: + 25 
 
EVOLUTION. 51 
 
 Ex. 2. tf + 4# s -|-2x 4 +9# I -4x'-f 4(# 3 + 2.r a + 2 
 a* 
 
 2 X s -f- 4# a # 2x >4 -f 9 #* 4 x 
 
 a 2.r-r-2) -f 4x 3 
 
 59. The process for extracting the Cz^eRoot of a compound 
 quantity may be explained in the following manner. By 
 Art. 50, the cube 
 of a + b is 
 
 the terms being 3 a 9 -f 3 a b + ^3 o^'+ 3 a b* + V 
 arranged accord- \3a*b-{- 3al* + b 3 
 
 ing to the powers * * * 
 
 Of a. The first a-5-aaaasHSESB 
 
 term of the root is a, which being cubed, and this cube subtracted 
 from the first term in the power (a 3 ), bring down the remaining 
 three terms $ a z b -f 3 a b* -f b 3 . Next square the first term (a) 
 of the root, and having multiplied it by 3 5 place 3a a in the divisor, 
 divide 3a*b by 3a Q , and it gives b the second term of the root ; 
 to 3 a 9 add 3 a b + b*, and it forms the divisor 3a*-f 3ab + l* 9 
 which being multiplied by Ogives Scfb + Safr + b 3 ; subtract, 
 and nothing remains. 
 
 60. The cube root of a compound quantity, if that root 
 consists of three terms^ is found by continuing the process in a 
 similar manner. 
 
 (a + b) 3 -f 3(a -f b)*c + 3 (a + by + c s (a -f b + c 
 
2 EVOLUTION. 
 
 Thus (by Art. 52) the cube of a -f b -f c is 
 (a-f ) 3 + 3(a + 6)V + 3 (a + #)" + * j supposing the first two 
 terms of the root to have been found as in the preceding article, 
 cube a -f b arid subtract(a-f b) 3 from the first term of the power ; and 
 then bring down the next three terms 3(a + b}*c -f 3 (a + b)c -f c 3 . 
 Square the two terms already found ; which square being mul- 
 tiplied by 3, gives 3(fl-l-&) 2 ; divide 3 (a + b) 2 c by 3(a + b) z , and 
 we have c, the third term of the root. To 3(a-f-6) 2 add 
 3(a + b}c -f c 2 , and it forms the divisor 3(a + b) 2 + 3 (a + b}c + c*, 
 which being multiplied by c, gives 3 (a-f-) 2 c+3(a-i-&)c 2 -f-c 3 ; 
 subtract, and nothing remains. 
 
 If the quantity whose root is required be not an exact power, 
 the operation will not terminate, as in the above instances; but 
 it may be continued to any number of terms at pleasure. 
 Ex. Find the square root of a? + x-. 
 
 x \ x 
 2a-f -) 
 
 X J' 4 
 
 2 + - i) :* 
 
 a* 
 
 Iii these cases, however, the root is in general much more 
 easily found by help of the -Binomial Theorem, as will be ex- 
 plained hereafter. 
 
 xv. 
 
 On the investigation of the Rules for the Extraction of 
 
 the Square and Cube Roots of Numbers. 
 Before we proceed to the investigation of these Rules, it will 
 
 be 
 
 I 
 
EVOLUTION, 53 
 
 be necessary to explain the nature of the common arithmetical 
 notation. 
 
 61. It is very well known that the value of the figures in the 
 common arithmetical scale increases in a tenfold proportion from 
 the right to the left ; a number, therefore, may be expressed by 
 the addition of the units, tens, hundreds, &c. of which it consists. 
 Thus the number437 1 maybe expressed in the following manner, 
 viz.4000 + 300 + 70+ I,orby4 X 1000 + 3 X 100 + 7 X 10+1; 
 hence, if the digits (a) of a number be represented by o, b, c, d, e &c. 
 beginning from the left-hand, then, 
 
 A number of 2 figures may be expressed by 1 Oa + b. 
 ...... 3 figures ......'.. by 100a+ lob + c. 
 
 ...... 4figures ........ by 10000+ 100 + lOc -\ d. 
 
 &c. &c. &c. 
 
 62. Let a number of three figures (viz. lOOa+106 + c) be 
 squared, and its root extracted according to the Rule in Art. 58. 
 and the operation will stand thus; 
 I.10000a+2000a& + 
 
 1 OOOOa* 
 
 2000r;/>+ 1006* 
 
 u. Let a -- 2 | an( j ^ operation is transformed into the 
 ~^ C following one; 
 
 40000 + 1 2000 + 900 + 400 + 60 + 1 (200 + 30 + 1 
 40000 
 
 400 + 30) 1 2000 + 900 + 400 
 12000+900 
 
 400 + 60+1)400 + 60+1 
 400 + 60 + 1 
 
 (') By the digits of a number are meant the figures which compose it, 
 considered independently of the value which they possess in the arithmetical 
 
 scale. 
 
54 SVOLUTION. 
 
 in. But it is evident that this operation would not be a: 
 by collecting the several numbers which stand in the sarac 
 into one sum, and leaving out the 
 ciphers which are to be subtracted in 
 the several parts of the operation. Let 
 this be done; and let two figures be 
 brought down at a time, after the 
 square of the. first figure in the root 
 has been subtracted; then the ope- 
 ration may be exhibited in the manner 
 annexed; from which it appears that 
 the square root of 53361 is 231. 
 
 63. To explain the division of the given number into periods 
 consisting of two figures each, by placing a dot over every second 
 figure beginning with the units (as exhibited in the foregoing 
 operation) , it must be observed, that, since the square root of 
 100 is IO; of 10000 is 1OO; of 10QQOOO is 100O; c. &c. it 
 follows, that the square root of a number te$$ then IOO must 
 consist of one figure; of a number between IOO and 1OOOO, of 
 too figures; of a number between lOOOOand 10OOOOO, of three 
 figures ; &c* c* and consequently the number of these dots 
 will shew the number of figures contained in the square root of 
 the given number. From hence it also follows, that the Jtrst 
 figure of the root will be the square root of the greatest square 
 number contained in the first of those periods, reckoning from 
 thel^K Thus, in the case of 53361 (whose square root is a 
 number consisting of three figures) ; since the square of the 
 figure standing in the JbuKfrecfr place cannot be found either in 
 the last period (61), or in the last bat em (33), it must be 
 found inthejSrsI period (5); consequently the first figure of the 
 root will be the square root of the greatest square number 
 contained in 5; and as this number is 4, the first fiure of the 
 
 
EVOLUTIOK. 55 
 
 root will be 2. The remainder of the operation will*be readiiy 
 understood by comparing the steps of it with the several steps of 
 the process for finding the square root of (a + 1 + c) 2 in Art. 58 ; 
 for having subtracted 4 for the first period (5), there remains 1; 
 bring down the next two figures (33), and the dividend is 133 ; 
 double the first figure of the root (2), and place the result 4 in 
 the divisor; 4 is contained in 13 three times, 3 is therefore the 
 second figure of the root ; place this both in the divisor and 
 quotient, and the former is 43 ; multiply by 3, and subtract 
 12Q, the remainder is 4 ; to which bring down the next two, 
 figures (61), which gives 461 for the next dividend. Lastly, 
 double the last figure of the former divisor, and it becomes 46 ; 
 place this in the next divisor, and since 4 is contained in 4 once, 
 1 is the third figure of the root; place 1 therefore both in the 
 divisor and quotient ; multiply and subtract as before, and 
 nothing remains. 
 
 64-. The rule for extracting the cube root of numbers may be 
 understood by comparing the process for extracting the cube 
 root of (a + b + c) 3 in Art*. 59 and 60, with the following 
 operations, in which is deduced the cube root of the number 
 13997521. < 
 
 13997321 (200 + 40+1 
 a 3 =( 200) 3 = 8000000 
 
 3a 2 = 120000) 1st Remainder 5997521 
 
 3a*fl = 3 X ( 200)* X 40 = 4800000 
 3 a 6 2 = 3 X 200 X (40) 2 = 96000O 
 10x40 = 64000 
 
 5824000 
 
 - = 172600) 2d Remainder 1J3521 
 
 = 3(200 + 40)*X 1= 172600 
 = 3(200 + 40) Xl= 720 
 = ixlxl= l 
 
 173521 
 3d Remainder OOOOOO 
 
 Omitting 
 
56 EVOLUTION. 
 
 Omitting the superfluous ciphers, and bringing down three 
 figures at a time, the operation would stand thus ; 
 
 1399752l(241 
 
 2 3 = 8 
 
 3 X2* = 12)5997 
 
 300X2 2 X4 =4800 
 
 30X2 X4 2 = 960 
 
 4 3 = 64 
 
 5824 
 
 173521 
 
 300 X (24)' X 1 = 172800 
 
 SOX 24 X l a = 720 
 
 1 3 = 1 
 
 173521 
 000000 
 
 These operations may be explained in the following manner. 
 
 i. Since the cube root of 1000 is 10, of 1000000 is 100, &c. 
 it follows, that the cube root of a number less than 1000 will 
 consist of one figure; of a number between 1000 and 1000000 
 of two figures, &c. &c. ; if therefore the given number be 
 divided into periods, each consisting of three figures, by placing 
 a dot over every third figure beginning with the units, the 
 number of those dots will shew the number of figures of which 
 the cube root consists ; and for the reason assigned in the 
 preceding article (respecting the first figure of the square root), 
 the jirst figure of the root will be the cube root of the greatest 
 cube number contained in the first period. 
 
 n. Having pointed the number, we find that its cube root 
 consists of three figures. Thejirst figure is the cube root of the 
 greatest cube number contained in 13 ; this being 2, the value 
 of this figure is 200, or a = 200; consequently a 3 =8000000; 
 
 subtract 
 
 

 EVOLUTION. 57 
 
 subtract this number from 13997521, and the remainder is 
 5997321. Find the value of 3a a , and divide this latter number 
 by it, and it gives 40 for the value of b the second member of 
 the root ; put this in the quotient, and then calculate the value 
 of 3ft 2 Z' + 3a6 a + Z> 3 and subtract it, and there remains 173521. 
 Find now the value of 3(a + by and divide 173521 by it, and 
 it gives 1 for the value of c the third member of the root ; put 
 this in the quotient, and then calculate the amount of 3(a-f)*c 
 -f 3(a-f6)c 2 + c 3 , which subtract, and nothing remains. 
 
 in. In reviewing theirs* of these two operations, it is evident 
 that six ciphers might have been rejected in the value of a 3 , and 
 three in the value of 3 a?b -f 3 a b* -f- b 3 , without affecting the 
 substance of the operation ; having therefore simplified the 
 process as in the second operation, we are furnished with the 
 following Rule for extracting the cube root of numbers. 
 
 IV. " Point off every third figure, beginning with the units ; 
 " find the greatest cube number contained in the first period, 
 " and place the cube root of it in the quotient ; cube it and 
 " subtract it from the first period, and then bring down the next 
 " three figures \ divide the number thus brought down by 300 
 " times the square of the first figure of the root, and it will givei 
 " the second figure j then calculate the value of 300 x square 
 " of first figure x second figure + 30 X first figure x square 
 " of second + cube of second, subtract it, and then bring down 
 " the next period, and so proceed till all the periods are brought 
 " down." The Rules for extracting the higher powers of 
 numbers and of compound algebraic quantities re very tedious, 
 and of no great practical utility. 
 
 XVI. 
 
 On the general mode of expressing the Powers and 
 
 Roots of Quantities by means of Indices. 
 65. The* management of Surd quantities, and the method* 
 extracting the roots of compound algebraic quantities by 
 means of the Binomial Theorem, will be treated of hereafter ; 
 
 I but 
 
58 EVOLUTION. 
 
 but before we conclude this Chapter, it may be proper to 
 make a few observations on the method of expressing the 
 powers and roots of quantities by means of indices. 
 
 i. Since a x a* = a 3 = a l+z ; a*xa 3 = a 5 =a 2+3 ; or, in general, 
 a m xa n = a m+ * 9 it follows, that the different powers of any 
 quantity are multiplied together by adding the indices. 
 
 a* a 5 a m 
 
 ii. Again, -=a = a*~ 1 ', -3 = a a =a s ~ 3 ;or, in general, n = 
 
 a m ~ n ; from which it appears, that one power of a is divided by 
 another, by subtracting the index of the divisor from that of 
 the dividend. 
 
 in. The square of a a xa = a 1X2 =a 8 , 
 
 Cube of a' = c 2 X a 2 x a* = a 2x3 =a 6 , 
 
 or, in general, m th power of a n ==a" x a* x a n to m factors = a mtt ; 
 from this it follows, that the powers of a are raised to other 
 powers by multiplying the index of the original power by that 
 
 of the power to which it is to be raised. 
 
 jt 
 iv. Square root of a* =. a 1 = a* ; 
 
 Square root of a 4 =a* = a T ; 
 
 6 
 
 Cube root of a =a a = a5, & c . &c., i.e. the roots of the 
 powers of a are found by dividing the index of the power by 
 the number expressing the degree of the root to be taken. 
 
 66. From this method of considering the formation of the 
 powers and roots of quantities, a new species of algebraic notation 
 arises, of which the following are Examples. 
 
 i . The roots of quantities may be expressed by fractional 
 indices. Thus, 
 
 The Square root of a = a l ~ z =a? ; 
 
 Cube root of a =a 1-K3 =a * ; 
 
 or, in general, m th root of a = a l ^ m = a" 1 . 
 
 Again, Cube root of a 2 =a^ 3 =al 
 Square root of a 3 = a 34 " a =a^; 
 
 or, in general, m* root of a" = a n4 = as. 
 
 Ji. The 
 
EVOLUTION. 
 
 59 
 
 ii. The signification of the negative indices arising from 
 Rule 4 of Division (Art. 23.) will easily appear by an example. 
 
 By that Rule, - 5 = a~ 5 =a- 3 . But - 5 =- 3 ; consequently a~ 3 
 
 and -} (and, in general, cT and j are equivalent expressions. 
 
 Hence it follows that a will always represent unity, whatever 
 
 .a m 
 be the value of a ; for, by the Rule, '^=a m ~ m ) or 1 =a. 
 
 A comparison of the following series, in the first of which 
 every succeeding term is the quotient of the preceding divided 
 by a, and, in the second, the index of a is continually dimi- 
 nished by 1, will shew that the above conclusions naturally fol- 
 low from the notation adopted in Art. 7. 
 
 
 
 1 1 1 
 
 aaa aa a l 
 
 a aa aaa 
 a 3 a" a 1 a or 1 a~ 2 a- 3 
 
 in. From this it follows, that any factor may be removed 
 from the numerator of a fraction into the denominator, or from 
 the denominator into the numerator, by changing the sign of 
 its index. 
 
 Ex. 1. Thus (since -^=.1"^)^ may be expressed by a y l~ 3 ; 
 
 i / M a* 1 1 1 
 
 and /since a =^2 j> we have r 3 = ru X p= a _ a ^ 3 * 
 
 aU 3 
 
 Ex.2. The quantity & 4 4 may be expressed by a *l 
 c d e s 
 
 or by _, _ 3 j4 t - 
 
CO 
 
 CHAP. IV. 
 
 ON SIMPLE EQUATIONS. 
 
 WHEN two algebraic quantities are connected together by the 
 sign of equality, the whole expression thus formed is called 
 (Art. 1 1 .) an Equation. Equations, as applied to the solution 
 of questions or problems, consist of quantities, some of which 
 are k?iown, and others unknown; and by the solution of an 
 equation is meant, the operation by which the value of the 
 unknown quantities are found in terms of the known ones. If 
 an equation contains no power of the unknown quantities, but 
 those quantities merely in their simplest form, it is called a 
 Simple Equation ; if it contains the square of the unknown 
 quantity, it is called a Quadratic Equation; if the cube of the 
 unknown quantity, a Cubic Equation ; &c. &c. The present 
 Chapter will be occupied entirely with the solution of Simple 
 Equations, and questions depending upon them. 
 
 XVII. 
 
 On the Solution of Simple Equations, containing only 
 one unknown quantity. 
 
 67. The Rules absolutely necessary for the solution of simple 
 equations containing only one unknown quantity may be reduced 
 to four, and may be arranged in the following order. 
 
 RULE I. 
 
 The first Rule is, that " any quantity may be transferred from 
 " one side of the equation to the other, by changing its sign;" 
 
 and 
 
SIMPLE EQUATIONS. 61 
 
 and it is founded upon the axiom, that "if equals be added to 
 " or subtracted from equals, the sums or remainders will be 
 " equal." 
 
 Ex. 1. Let # + 8 = 15 ; subtract 8 from each side of the 
 equation, and it becomes x + 8 8=15 8; but 8 8 = 0, 
 .'..r=15 8 = 7. 
 
 Ex.2, Let x 7 = 20; add 7 to each side of the equa- 
 tion, them; 7 + 7 = 20 + 7; but -7 + 7 = 0, .'. j;=20+7 
 
 = 27. 
 
 Ex.3. Let 3x 5 = 2# + 9; add 5 to each side of the 
 equation, and it becomes 3x 5 + 5 = 2 x-\- 9 + 5, or 3x=<2x 
 + 9 + 5, Subtract *2x from each side of this latter equation, 
 then 3x 2x=2x 2# + 9 + 5; but2# 2x=0, .'.3x 2x 
 = 9 + 5. Now 30720;=^, and 9 + 5= 14; hence x 14. 
 
 On reviewing the steps of these examples, it appears 
 
 I. That x+ 8= 15 is equivalent to #=15 8. 
 
 u. . . . # 7 = 20 to #=20 + 7. 
 
 III.. . 3x 5 = 2.r + 9 to, 3x <2x= 9 + 5. 
 
 Or, that " the equality of the quantities on each side of the 
 " equation, is not affected by removing a quantity from one 
 " side of the equation to the other and changing Its sign." 
 
 From this Rule also it appears, that if the same quantity 
 with the same sign be found on both sides of an equation, it 
 may be left out of the equation; thus, if x + a=c-\-a, then 
 #=c + a a; but a # = 0, .'. #=c. 
 
 It further appears, that the signs of all the terms of an equa- 
 tion may be changed from + to , or from to +, without 
 altering the value of the unknown quantity. For let x b 
 = ca; then, by the Rule, xc a + b; change the signs of 
 all the terms, then I a; = a c, in which case I a + c = x t or 
 x = c a + by as before. 
 
 RULE 
 
62 SIMPLE EQUATIONS. 
 
 RULE II. 
 
 " If the unknown quantity has a coefficient, then its value 
 " may be found by dividing each side of the equation by that 
 " coefficient;" and the foundation of the Rule is, that" if equals 
 " be divided by the same, the quotients arising will be equal." 
 
 Ex. 1. Let 2 x = 14; then dividing both sides of the 
 
 <2X 14 2x 14 
 
 equation by 2, we have -=--; but -=#, and = 7, 
 
 /. x = 7. 
 
 Ex. 2. Let 6x+ I0 = 3x+ 22 ; then, by RULE I, 6x 3 x 
 
 3x 1 2 
 = 2"2 10, or 3#= 12 ; divide each side by 3, then -7T=ir> 
 
 o o 
 
 or .r=4. 
 
 Ex.3. Let ax = b + c-, then = ; but ~=#j 
 + c 
 
 ' = ' 
 
 RULE III. 
 
 " An equation may be cleared of fractions, by multiplying 
 " each side of the equation by the denominators of the frac- 
 " tions in succession, or by their product." This Rule goes 
 upon the principle, that " if equals be multiplied by the same, 
 " the products arising will be equal/* 
 
 Ex. 1 . Let = 6; multiply each side of the equation by 3 ? 
 
 then (since, from what has been already shewn, the multipli- 
 
 x 
 cation of the fraction - by 3 9 just takes away its denominator, 
 
 \J 
 
 and gives x) we have #=6x3 = 18. 
 
 T 1 'T* 
 
 Ex, 2. Let -+- = 7 ; multiply each side of the equation 
 2 5 
 
 2 X 
 by 2, and we b re x+ -r-=14; now multiply each side by 5, 
 
 O 
 
 and it becomes 5 jc + 2 = 70, or 7 jc=70; hence, by RULE II, 
 
 70 
 r =T= =Ja Ex.3. 
 
 

 SIMPLE EQUATIONS. 63 
 
 E,3. Lf+f~l3-J 
 
 O y* O />* 
 
 Multiply each side of the equation by 2, then x + = 26 . 
 
 6x 
 ................ . by3, and3x+2x= 78 . 
 
 by 4, . 
 
 By RULE I, 12,r+83: + 6tf= 
 or 
 
 312 
 .'. by Rule II, #==12. 
 
 This Example might have been solved more simply, by mul- 
 tiplying each side of the equation by the product of the nunv 
 bers 2, 3, 4 3 which is 24. 
 
 Thus, | + |= 13-f- 
 
 243? 24# 24.r 
 
 Multiply each side by 24, then -- + - = 312 - 9 
 
 or 12x + 8o:=312 6x, as before. 
 
 RULE IV. 
 
 " If the equation contains the square root of the unknown 
 f( quantity, or the square root of the unknown quantity combined 
 " with some known quantity; then, let this surd. quantity be 
 " brought by itself to one side of the equation, and let both 
 " sides of the equation be squared; the value of the unknown 
 ce quantity may then be found by the preceding Rules." This 
 Rule goes upon the supposition, that " if the square root of 
 " a quantity be equal to any given quantity, then the quantity 
 (C itself will be equal to the square of that given quantity." 
 
 Ex. 1 . Let v/x- 5 = 3 ; then by RULE I, v/x= 5 + 3 = 8; 
 square both sides of the equation, then #=8 X 8 = 64. 
 
 Ex.2. Let v /2x+l +2 = 5; then, by RULE I, v/2.r+ 1 
 =52 = 3; square both sides of the equation, and we have 
 
 8 
 
 = 9, .'. 2 #=9 1 = 8, and #=-=4, 
 
 68. The 
 
64- SIMPLE EQUATIONS. 
 
 68. The following Examples will serve to exercise the learner. 
 in these several Rules. 
 
 In RULE I. 
 
 Ex. 1. 207 + 3 = 07+17 . . . ANSWER, 57=14. 
 Ex. 2. 5x 4 = 407+25 ......... 07=29- 
 
 Ex.3. 7x9 =6x-3 ......... 07 = 6. 
 
 Ex.4. 4x + 2a=3x + 9b ......... x=9b-Qa. 
 
 In RULES I, II. 
 Ex. 1. 10^7=150 .... ANSWER, 07=15. 
 
 Ex. 2. 15# + 4 = 34 ......... x 2. 
 
 Ex.3. 8o7 + 7 = 6 + 27 ....... 07=10. 
 
 Ex.4. 907 3 = 407 + 22 ....... 0:= 5. 
 
 Ex.5. 1707 407 + 9 = 307 + 39 ....... x= 3. 
 
 Ex. 6. GO; c= + 2c ........ 07= 
 
 , In RULES I, II, III. 
 
 2.T X 
 
 Ex. 1. + - = 22 ........ ANSWER, 07=24. 
 
 707 5o? 55 
 Ex.2. T -^ = T .............. *=10. 
 
 07 07 07 
 
 Ex.3. 2 + 3 ==31 ""5 ............ 07=30. 
 
 207 X 07 
 
 Ex.4. T - B + - = 44 .............. *=60. 
 
 In RULE IV. 
 Ex. 1. \/xl=4 ....... ANSWER, 07= 25. 
 
 Ex. 2. v / 3a7+i+5 = io ........... 07=8. 
 
 Ex.3. 15 + \/x+7 = l9 ........... 07=9. 
 
 69. In the application of these Rules to the solution of simple 
 equations in general containing only one unknown quantity, it 
 will be proper to observe the following method. 
 
 i. To clear the equation of fractions by RULE III. 
 
 ii. To 
 
SIMPLE EQUATIONS. 65 
 
 ii. To collect the unknown quantities on one side of the 
 equation, and the known on the other, by RULE I. 
 
 in. To find the value of the unknown quantity by dividing 
 each side of the equation by its coefficient, as in RULE II. 
 
 iv. If the equation contains a surd quantity, then RULE IV. 
 must be immediately applied. 
 
 EXAMPLE 1. 
 
 3x X 13 
 
 Find the value of x in the equation + 1 =-+- 
 
 7 3 O 
 
 Multiply by 7, then 3x+ 7 = -^+ ; 
 
 ..... by 5, . . 15# + 35 = 7# + 91. 
 
 Collect the unknown quantities"^ 
 
 .. I 15#-7#=91-35, 
 
 on one side, and the known > 
 
 [ or 8#= 56. 
 on the other-, J 
 
 56 
 Divide by the coefficient of #, r= = 7. 
 
 EX. 2. ; 
 
 x 4-3 x 
 
 Find the value of x in the equation 1=2 - 
 
 5 x 
 
 Multiply by 5, then x+ 3 5 = 10 -j 
 ..... by 7? - 7# + 21-35 =70 5x. 
 Collect the unknown quantities" 
 
 r _ 7n 
 
 O/ - /U 
 
 on o/ze side, and the known > i2x = 84- 
 on the oMer ; J 
 
 84 
 
 Ex.3. 
 
 Find the value of x in the equation 
 
 ^-1 2# 2 
 
 4X- = ^ + - +24. 
 
 K Multiply 
 
66 SIMPLE EQUATIONS. 
 
 Multiply by the ? 40r _ to+5=l0r+4je _ 
 product (10), 3 
 
 By transposition^ Ox 5x 1 Ox 4x= 2404 5 
 or 40r 19JC=231, 
 
 231 
 i.e. 2]:r=231; .vxss = 11. 
 
 Ex. 4. 
 
 x 
 Find the value of x in the equation 2x ^ + 1 =5 x 2. 
 
 Multiply by 2, then 4xx+2= lOx 4. 
 
 By transposition, 4-f 2= lOx 4x+x, 
 
 6 
 or 6= 
 
 Ex. 5. 
 What is the value of # in the equation 
 
 x .r = 
 Divide each side of the equation hy 3a + 2b, which is the 
 
 coefficient of x; then x== ^ a i_2b 
 
 Ex.6. 
 
 Find the value of # in the equation 3bx + a=2ax+4c. 
 Bring the unknown quantities to one side of the equation, and 
 the known to the other ; then, 
 
 3bx 2ax = 4c a 
 but 3lx 2ax = 3b 2a x# 
 
 Divide by 3b 2a, and ^=^7 g~ * 
 
 Ex.7. 
 
 ( a ) As this step involves the case " where the sign stands before 
 a Fraction," when the numerator of that fraction is brought down into the 
 same line with 40 r, the signs of both its terms must be changed, for the 
 reasons assigned in Ex. 3, page 26 ; and we therefore make it 5j?-f-5, and 
 not 5jr 5. 
 
SIMPLE EQUATIONS. 67 
 
 Ex.7. 
 
 Find the value of x in the equation bx + x = 2x + 3a. 
 Transpose 2x, then bx + x 2x=3a, 
 or bx x=3a, 
 but bx x=(bl)x; 
 
 .'. (b l)jc=3a, or X~T . 
 
 Ex.8. 
 
 3 X x 2 x 
 
 Find the value of # in the equation - c-f^=4x-f- T' 
 
 Multiply by abd, thensbdx abed-}- adx=4abdx + 2abx. 
 By transposition, 3 bdx + adx 4abdx 2abx=abcd, 
 
 abed 
 
 Ex.9. 
 
 Let V# + Vtf +3;==^-- to find the value of a:. 
 Multiply by /a + x, then */x x */a+x+ a+x= 2a. 
 
 By transposition, Vjc x Va + x = 2a a o: =: a or. 
 
 Square both sides, x x (a + x) a fl 
 
 or ax+x* =a* 
 
 .'. Sax =a 3 
 
 3a 3 
 Ex. 10. 
 
 Let a + x^^a+xx/tf+x* to find the value of x. 
 Square both sides, and we heve a* + 2ax + X > =a' 2 -\- 
 
 or 2aj:-}-^ = W^-f a; 
 
 Divide by x, 2 a -f- jp = */b*+x\ 
 
 Square again, 4a* 
 
 or 4ax=zl*4a' t . 
 
 ^ a -4a' ^ a 
 
 Hence, x=. - - = . 
 4a 4a 
 
68 
 
 SIMPLE EQUATIONS. 
 
 Ex.11. ^+2 + 3 =11 * ' * * ANSWER J X==6 - 
 
 Ex. 12. 
 Ex. 13. 
 
 # x x x 
 
 3x 1 1O 
 y+ 
 
 # # #_ 
 
 C*X% 1 4. 7v ~T~ .-. ~~ i ~ c\ 
 
 Ex. 15. 
 
 1 x + 3 
 
 3x 
 
 Ex.16. -r-5 = 29-2# 
 
 3# 
 Ex. 17. 6# - 9=. 
 
 4 
 
 # + 3 
 Ex.18. 2# ^-+15 = 
 
 Ex. 19. 
 
 3 
 #-2 # 
 
 x-'Q 
 
 Ex.20. 5#- 
 
 Ex.21. 
 
 2x-l 
 
 _ 
 
 #=10. 
 
 6 
 
 x ~7 
 
 X =3 
 
 x=36. 
 
 = 12. 
 
 4a 
 
 x = . 
 
 Ex. 22. 
 Ex. 23. 
 Ex. 24. 
 
 a+x lx 
 3 a 
 
 XVIII, 
 
SIMPLE EQUATIONS. 69 
 
 XVIII. 
 
 On the Solution of Simple Equations containing Two 
 
 or more unknown Quantities. 
 
 For the solution of equations containing two or more un- 
 known quantities, as many independent equations are required 
 as there are unknown quantities. The two equations necessary 
 for the solution of the case when two unknown quantities are 
 concerned may be expressed in the following manner, 
 ax + by=c 
 
 where a, I, c, a', b', c' represent known quantities, and x, y the 
 unknown quantities whose values are to he found in terms of 
 these known quantities. 
 
 70. There are three different Rules by which the value of ope 
 of these unknown quantities may be determined ; 
 
 RULE I. 
 
 Let ax + by = c (A) \ be the two equations 
 and a'x + b'y=c (B) J to be solved. 
 
 Multiply equation (A) by a, then aa'x + a'by=za'c (C) 
 by a, . . . aa'x + al'y=ac r (D) 
 
 Subtract equation ()) from (C), then (a'b ab')y=a'c.ac f 
 
 a'c ac' 
 ' ' ^ ~" a I ah' 
 
 From which we deduce the following Rule. " Multiply the 
 " first equation by the coefficient of x in the second equation, 
 " and the second equation by the coefficient of x in the first 
 " equation , subtract the last of these resulting equations from 
 " thejirst, and there will arise an equation which contains only 
 " y and known quantities, from which the value of y is de- 
 
 P" termined." 
 RULE II. 
 
 From equation (A),axc ly, ..#= 
 
 [-D\ > ' T c'b'y 
 
 ........ (B) ) ax=^c-'ly ) .'.x^ ~- 
 
 Putting 
 
70 SIMPLE EQUATIONS. 
 
 Putting these two values of x equal to each other, we have 
 
 c ' b'y c ly 
 
 7-^ = - . an( j % . 4 ac aly~ac auy ; 
 
 By transposition, (a I al')y dc a c 
 
 a'cac' 
 
 and y= -r, - / * 
 * abab 
 
 From which it appears, that " if the value of x in t\\e first 
 " equation be put equal to its value in the second, there will 
 " arise a new equation involving only y, from which the same 
 " value of y is found as before." 
 
 RULE III. 
 
 c Ity 
 From equation (4), x - ; substitute this value of x in 
 
 equation (B), then a x 
 
 or a'c -a'ly + a I'y =a c' 
 
 .. a'cac=)dbal')y 
 a'cac 
 
 ^y^i^ai' 
 
 From which we infer> that " if the value of x found from the 
 "Jirst equation be substituted for it in the second, there will 
 " arise an equation which gives the same value of y as in the 
 " two former instances." 
 
 71. Having determined the value of T/, the value of .r may 
 be found in each case, by substituting this value for y either in 
 the first or second equation. The value of x in the first equation 
 
 c by a'c ac' c b(a'c-ac') 
 
 is - -; but y=-r - n " x = - , ,, - r \ = fbv re- 
 a y a'bab a a(abab') 
 
 ducing these fractions to a common denominator) - ,. . /a 
 
 . c b'y c' b' (a'cac) 
 
 The value or x in the second euuation is = - ,, ., - TA 
 
 a a a (abab) 
 
 =. (by reducing these fractions to a common denominator) 
 
 bc' b'c 
 
 ~rr - r? as before. 
 
 abab 
 
 72. From 
 
SIMPLE EQUATIONS. 
 
 71 
 
 72. From hence it appears, that in finding the value of y, 
 either of the three Rules may be applied ; and that in finding 
 the value of x, the value of y so found may be substituted either 
 in ihejlrst or second equation. In the choice of the Rule which 
 may be most adapted to practical application, experience only 
 can be our guide. It may further be observed, that there are 
 cases in which RULE I. may be somewhat varied ; for instance, 
 if the given equations be, 
 
 ax + by=.c (A) 
 dx b'y = c' (B) 
 
 Multiply equation (A) by b', then, ab'x+bb'y = b'c (C) 
 (B) by b, . . . a'bx-bb'y = bc'(D) 
 
 Add equation (D) to (C), then (ab' -\- a b) x=b'c + bc 
 
 b'c + bc 
 ~~ab' + a'b 
 
 Having the value of x, the value of y may be found by some of 
 the preceding methods. 
 
 73. The following examples are intended to illustrate each 
 Rule separately ; 
 
 EXAMPLE 1. 
 
 Let 5x + 4y = 55 (A)? to find the values 
 = 31 (B) $ of x and y. 
 
 By RULE I, 
 
 [ultiply (A) by 3, then 1 5 x + 1 2 y = 1 65 
 .-. . . (B) by 5 . . 
 
 10 
 
 .*. by subtraction, we have 2y=lO, orT/ = -=5. 
 
 55 4u 
 Now from equation (A) we have x= *=(sincey = 
 
 5520 35 
 
 T^T^ 7 - 
 
 Ex. 2. 
 
SIMPLE EQUATIONS. 
 
 Ex.2. 
 
 =l6 (A) 
 y=34 (B) 
 
 From equation (A), we have x=l6 4y. 
 
 ..... (B).....x= 3 ^- 
 
 34 v 
 Hence, by Rule II, 4 * = i6 4y, 
 
 or 34 y = 64-. 
 
 30 
 
 .*. 15y=30 or y= = 2. 
 1 5 
 
 It has already been shewn that #=16 4y=r (since 
 and /. 4y=8) 16-8=8. 
 
 Ex. 3. 
 
 8y=31 (A) 
 10^=192 5. 
 
 From equation (C),#=91 24y; by RULE III, substitute this 
 value of x in equation (D) ; then we have, 
 
 # + 40(91 -24 1/)=763 
 ory-f- 3640 960?/=763 
 
 .'. 959^=3640-763 
 2877 
 
 By referring to equation (C), we have x=Ql 24y = (since 
 y=3, and/. 247/=72)91-7 = 19. 
 
 Ex. 4. 
 
 Let 3^ + 4y=29 (A). 
 3y*=z36 (B). 
 
 In this example, the Rule mentioned in Art. 72 may be 
 
 applied. 
 
 Multipl 
 
 1 
 
SIMPLE EQUATIONS. 
 
 Ex. 4. 
 
 Let 3 
 
 ]7#-3# = 36 (B). 
 
 In this example, the Rule mentioned in Art. 72. may be 
 applied. 
 
 Multiply equation (A) by 3, then 9x+12y= 37 (C) 
 .......... (5) by4 . . 68#-12y=144 (D) 
 
 23 1 
 
 ion (D) to (C), then 7 7 #=231, or #= =3. 
 
 From equation (^4) we have 4y = 29 3 #= (since # = 3, and 
 
 20 
 /.3JC=9) 29 9 = 20; hence j/ =s - = 5. 
 
 Ex.5. 4^ + 3^31> > .ANSWER, 5 X = 4 
 
 Ex.6. 
 
 353 ' c= 5. 
 
 
 Ex.7. 5JC 4^=19? 
 4.r4.2y = 36S ' 
 
 Ex.8. 3x-f7?/ = 79? Ca?=10 
 
 2 = 93 ' ly- 7. 
 
 ]x=n 
 !y= 4 - 
 
 y^-f-3 = 4 
 
 ........ 
 
 2JC 4w 23 Jt/= 1. 
 
 3 
 
 Ex. 12. 
 
74 SIMPLE EQUATIONS. 
 
 3X-7 
 
 EX ' 12 ' 3 5 I U=13 
 
 > . . . . ANSWER, 1 
 
 ^n ?/ 
 
 5 
 
 74. When three unknown quantities are concerned, the 
 most general form under which simple equations can be ex- 
 pressed, is ax+ by + cz=d (E) 
 alx+b e y+c'z=d'(F) 
 d'3C + b"y + c"z=zd"(G), and the mode of 
 solution may be conducted in the following manner. 
 
 i. Multiply eq n . (E) by a, then a ax + a'by-\- a'cz=a'd (H) 
 (.F)bya . . . aa'x + ab'y + ac'z-=ad' (K) 
 
 Subtract(lQfrorri(H),then(a b - ab'}y + (a'c- ac>= a d - ad'(L). 
 By multiplying (F) by a", and (G) by a, and subtracting the 
 latter result from the former, we obtain in the same manner 
 (a"b'-ab")y + (ac-ac) z=ad'-dd"(M). 
 
 II. Next, let the coefficients in equation (L) be repre- 
 sented by a, /3, y respectively ; and those in equation (M) 
 by ', /3', y, respectively ; then those equations may be reduced 
 to the following form 5 viz. 
 
 *3/ + /3% y. ; 
 
 *y + &'%=?' From which, by making 
 
 the proper substitutions in RULE I, and in Art. 7 1, we have 
 
 , y oey 
 
 d uii *4- c% 
 in. From equation (E), we have #= --- - - -5 in which 
 
 substituting the values of y and s just now found, we obtain 
 
 This mode of operation might be easily extended to equations 
 containing any number of unknown quantities. 
 
 EXAMPLE 1. 
 
SIMPLE EQUATIONS. 75 
 
 EXAMPLE 1. 
 Let 
 
 j.Multiply(E)by3,then6# + 9y-t- 
 ..... (F)by2. . 
 
 Subtract (g)from(fl). . . 5y+ 2a = 23(L). 
 
 MuItiply(F) by4,then 1 2x + 8 y + 20 z = 1 28 
 .... (G)by3 . 
 
 Subtract ..... y + 14x=53 (M). 
 ir. Hence the given equations are reduced to, 
 
 5y-f 22 = 23 (L) 
 
 Again . , . 5?/+ 2s = 
 Multiply (M) by 5, then-5y + 7021 = 2 
 
 288 
 By addition .... 72z=288 ? or = =4. 
 
 From equation (M) ....... y= 142J. 53 = 56 53 = 3. 
 
 29 3V-43 29 25 
 in. From equation (E) . . x= ^ - = - - =2. 
 
 Ex.2. #-f#-fs = 90 1 Tx=35 
 
 2x -f 40 =3^ + 20 v ...... ANSWER, <y = 3O 
 
 2X + 40 =4+ 103 ( = 25. 
 
 Ex.3, as-f 2/+ = 531 Tx=24 
 
 = 105 > ........... <= 6 
 
 = 134 
 
 XIX. 
 
 TAe Solution of Questions producing Simple Equations. 
 
 In the reduction and management of equations, we have pro- 
 ceeded by fixed and stated rules; but in the solution of questions 
 we have no such rules to guide us. Every particular question 
 
 requires 
 
76 SIMPLE EQUATIONS. 
 
 requires a distinct process? of reasoning, to bring it into an alge- 
 braic form ; and nothing but practice and experience can produce 
 expertness and facility in conducting this process. All that can 
 be done for the learner in this case, is, to explain the manner in 
 which the principles of this science may be made to bear upon 
 questions in general ; for as soon as they can be brought into 
 the shape of equations, we have only to apply the foregoing 
 Rules for finding the value of the unknown quantity or quan- 
 tities. Before we proceed, therefore, to any actual examples, 
 it may be proper to shew the relation which arithmetical and 
 algebraic operations stand in to each other. 
 
 75. Suppose the following arithmetical question was proposed 
 for solution; viz. "To divide the number 35 into two such 
 " parts, that one part may exceed the other part by 9." 
 A person unacquainted with Algebra might with no great 
 difficulty solve this question in the following manner. 
 
 i. It appears, in the first place, that there must be a greater 
 and a lesser part. 
 
 ii. The greater part must exceed the lesser by 9. . 
 
 in. But it is evident that the greater and lesser parts added 
 together must be equal to the whole number 35. 
 
 ' IV. If then we substitute for the greater part its equivalent, 
 viz. "the lesser part increased by 9," it follows, that the lesser 
 part increased by 9, with the addition of the said lesser part, 
 is equal to 35. 
 
 v. Or, in other words, that twice the lesser part with the 
 addition of 9, is equal to 35. 
 
 vi. Therefore, twice the lesser part must be equal to 35 3 with 
 9 subtracted from it. 
 
 vn. Hence, twice the lesser part is equal to 26. 
 
 vin. From which we conclude, that the lesser part is equal 
 to 26 divided by 2-, i. e. to 13. 
 
 ix. And 
 

 
 SIMPLE EQUATIONS. 77 
 
 ix. And consequently, as the greater part exceeds the lesser 
 by 9, it must be equal to 22. 
 
 But by adopting the method of algebraic notation, the dif- 
 ferent steps of this solution may be much more briefly expressed 
 as follows. + 
 
 i. Let the lesser part =x. 
 
 ii. Then the greater part =#-f9. 
 
 in. But greater part -f lesser part =35. 
 
 iv. /. x + 94-a? = 35 - 
 
 v. or 2#-f 9 - = 35 - , 
 
 vi, .'. 2# =35 9. 
 
 vii. or 2x = 26. 
 
 26 
 
 viii. .*. x (lesser part) s= ~2" == 13 * 
 
 ix. and x + 9 (greater part) . =13 + 9 = 22. 
 
 76. Having thus explained the manner in which the several 
 steps in the solution of an arithmetical question may be expressed 
 in the language of Algebra, we now proceed to its exemplifi- 
 cation. 
 
 QUESTION I. 
 
 There are two numbers whose difference is 15, and their 
 sum 59. What are the numbers? 
 
 / 
 
 As their difference is 15, it is evident that the greater number 
 must exceed the lesser by 15. 
 
 Let, therefore, #=the lesser number 
 then will x+ 15 = the greater 
 But their sum = 59 
 .\x + x + 15 = 59 
 or 2x+ 15 = 59 
 
 and 2#=59 15 = 44 
 
 44 
 /. #= = 22 the lesser number 
 
 and x + 15 = 22+ 15 = 37 the greater. 
 
 Qu. 2. 
 
78 SIMPLE EQUATIONS. 
 
 QUESTION 2. 
 
 What two numbers are those whose difference is 9 j and if 
 three times the greater be added to five times the lesser, the 
 sum shall be 35 1 
 
 Let #=the lesser number; 
 then x'+ 9 = greater number. 
 And 3 times the greater = 3 x(# + 9) = 3#-f 27. 
 
 5 times the lesser = 5 x. 
 
 Bu t by the question, 3 times the greater + 5 times the lesser =35. 
 Hence, (3a; + 27) + (5x) ...... =35, 
 
 or 8#=35--27 = 8; .*. #= 1 lesser number, 
 
 and r-f9=l+9=10 the greater number. 
 
 QUESTION 3. 
 
 What number is that to which 10 being added, |-ths of the 
 sum shall be 66 ? 
 
 Let #=the number required ; 
 then x+ 10= the number, with 10 added to it. 
 
 3, 3(^+10) 
 
 Now |ths of (x+ !0)=-(#-f Io)rV-r 
 
 8 3 D 
 
 But, by the question, -|ths of (x+ 10) =66 ; 
 
 3x + 30 
 
 Hence, - - = 66. 
 o 
 
 Multiply by 5, then 3 x + 30 = 330; 
 
 300 
 .'. 3o;=330 30 = 300; or a:=-r-=100. 
 
 9 
 
 QUESTION 4 
 
 What number is that which being multiplied by 6, the 
 product increased by 18, and that sum divided by 9, the 
 quotient shall be 20 ? 
 
 Let #=the number required; 
 then 6 x the number multiplied by 6 ; 
 6x+ 18 = the product increased by 18, 
 
 6.r-f 18 
 and j =that sum divided by 9- 
 
 Hence, 
 
SIMPLE EQUATIONS.t 79 
 
 6X+18 
 
 Hence, by the question, - ~ = 20. 
 
 Multiply by 9, then 6x+ 18 = 180, 
 
 162 
 or6tf=180-18=l62; .'.#= =27. 
 
 QUESTION 5. 
 
 A post is |th in the earth, -|ths in the water, and 1 3 feet out of 
 the water. What is the length of the post ? 
 Let x = length of the "post; 
 
 then - = the part of it in the earth, 
 
 5 5 
 
 O f * 
 
 =the part of it in the water, 
 
 13 = the part of it out of the water. 
 But part in earth -f part in water -f part out of water = whole post ; . 
 
 \5x 
 
 i *1 
 
 Multiply by 5, then j?+ z -f 65 =5.T 5 
 
 7 
 ..... by 7 . . 7x+l5x + 455=35 x, 
 
 455 
 Hence #=- = 35 length of post. 
 
 QUESTION 6. A- Q 
 
 After paying away Jth and Ith of my money, I had 85 /. left in $ // ; 
 
 my purse ; What money had I at first ? '< 
 
 Let x= money in my purse at first; 
 
 ^\ /y* IT* 
 then j-4-- = money paid away. 
 
 But money at first money paid away = money remaining. 
 
 / x x\ 
 Hence x-( -+- = 85, 
 
 Multiply 
 
 x x 
 i.e. x- --- = 85. 
 
80 SIMPLE EQUATIONS. 
 
 4X 
 Multiply by 4, then 4x x -- r = 340 ; 
 
 ..... by 7 ... 2Sx-7x-4:X =2380, 
 
 2380 
 .M7 = 2380; 
 
 QUESTION 7. 
 
 Of a battalion of soldiers (the officers being included), |ths are 
 on duty, -^th are sick, |ths of the remainder are absent, and there 
 are 48 officers. What is the number of persons in the battalion ? 
 
 Let #=the number of persons in the battalion. 
 
 3 x 
 Then |ths of x, or , =men on duty, 
 
 ith of x, or :> = the sick ; 
 
 , 3x x 34x \7x 
 And + ~, or ,= = men on duty and sick. 
 
 I7x 3x 
 Hence # == remainder, 
 
 3 x Q x 
 And |ths of , or = i tns f remainder = the absent. 
 
 But the men on duty, the sick, the absent, and the officers, 
 together make up the whole battalion; 
 I7x gx 
 
 Qx 
 
 or 17-r+-r-*-9 6 =20^; 
 
 =100x. 
 Hence 100# 85x 9.r = 4SOO, 
 
 4800 
 
 or 6x = 4800; or x= g =800. 
 
 QUESTION 8. 
 
 There are two numbers, such, that 3 times the greater added 
 to Jd the lesser is equal to 36 ; and if twice the greater be sub- 
 
 tracted 
 
 i 
 
SIMPLE EQUATIONS. 81 
 
 tracted from 6 times the lesser, and the remainder divided by 8, 
 the quotient will be 4. What are the numbers ? 
 Let .r= the greater number, 
 y = the lesser number; 
 
 Then3* + f = 36J gx + 
 
 > or, 
 
 6y 2x 6y 2x = 32; 
 
 ~- = 4 
 
 Or, y + QX. 
 
 6y-<2x= 32(5). 
 
 Multiply equation (A) by 6, then 6y + 54 #=648 
 Subtract equation (B) , . . 6y 2x= 32; 
 
 then 56^=616; 
 
 616 
 
 *=^6 =11 ' 
 From equation A .... y=108 9^=108-99=9. 
 
 QUESTION 9. 
 
 There is a certain fraction, such, that if I add 3 to the nu- 
 merator, its value will be J; and if I subtract one from the 
 denominator, its value will be I. What is the fraction? 
 
 Let x its numerator) , . r . # 
 
 > then the rractionis - 
 
 y= denominator J y 
 
 Add 3 to the numerator, then = - I Q _, ft 
 
 y 3 ( 3x-\-Q=y 
 
 Subtract one from denom r ., and-^-r = - i 
 
 By transposition, y~3x=g (A), 
 ?j-5x=l (B). 
 Subtract equation ( B) from (A), and we have 
 
 8 
 .*. x = -^ 4: the numerator. 
 
 From equation (A) y = 9 + 3 x= 9 + 1 2 = 2 1 the denominator. 
 
 4 
 Hence the fraction required is* 
 
 M ' 
 
82 SIMPLE EQUATIONS. 
 
 QUESTION 10. 
 
 A and B have certain sums of money ; says A to B, give me 
 15/. of your monet, and I shall have 5 times as much as you 
 will have left ; says B to A, give me 5/. of your money, and 
 I shall have exactly as much as you will have left. What sum 
 of money had each ? 
 
 Let x=A's money | then x+ 15 = what A would have after 
 y=B's . . . . f receiving 1 5 1. from B. 
 
 y 15 = what B would have left. 
 Again, y+ 5 = what B would have after 
 
 receiving 5/. from A. 
 x 5 = what A would have left. 
 
 Hence, by the question, #+15=5 x (y \b)=5y 15,\ 
 and?/ + 5=x 5. /. 
 
 By transposition, 5 y #=90 (A) , \ 
 and?/- #=10 (B). i 
 Subtract (B) from (A), 4?/=100; 
 
 .'. T/= 25 = J3's money. 
 From equation (B), #=?/ + 10=25+ 10=35 ~A's money. 
 
 QUESTION 11. 
 
 A person bought a certain number of sheep for 94/. ; having 
 lost 7 of them, he sold th of the remainder of them at prime 
 cost for 2Ql. How many sheep had he at first ? 
 Let #= number of sheep he had at first. 
 
 94 whole sum , 
 
 Then = r FT =what each sheep cost. 
 
 x number of sheep 
 
 Now x 7=i number remaining when 7 were lost; 
 
 .-. ^^=the number sold for 201. 
 4 
 
 But the number sold x price of each = whole price of sheep sold. 
 
 x f g4 
 Hence, by substitution, ~j x =20, 
 
 or (x 7)X94 =80#, 
 
 i.e. 9 
 or 9 
 
 4 a? 80#=658, 
 .'. 14#=658; or 
 
 658 
 ,==47. 
 
 
SIMPLE EQUATIONS. 83 
 
 QUESTION 12. 
 
 A and B have the same income ; A is extravagant, and 
 contracts an annual debt amounting to l - th of it ; B lives upon 
 ^ths of it ; at the end of 10 years, B lends A money enough 
 to pay off his debts, and has then 1 6oL to spare. What is 
 their income ? 
 
 Let 07 = their income. 
 
 x 
 Then jth of x 9 or - A\ annual debt, 
 
 and 10 x~ or - - =^'s debt contracted in 10 years. 
 
 As B lives upon f ths of his income, he saves annually jth of it; 
 
 x 
 hence, - = B's annual saving, 
 
 X 
 
 and 10x-, or -,or 2 x = B s savings m 1O years. 
 
 But, by the question, B's savings = ^4's debt-f- 160 ; 
 
 lOx 
 .'. by substitution, 2 x = r 4-16O, 
 
 or 14x=10x+ 112O, 
 
 1120 
 and 4x=1120; or x= = 280/. 
 
 QUESTION 13. 
 
 A person was desirous of relieving a certain number of beggars 
 by giving them 25. 6d. each, but found that he had not money 
 enough in his pocket by 3 shillings; he then gave them 
 2 shillings each, and had four shillings to spare. What money 
 had he in his pocket ; and how many beggars did he relieve ? 
 
 Let x = money in his pocket (in shillings). 
 y = number of beggars. 
 
 Then 2^ x y, or = N. of skill*, which would have 
 
 [been given at 2s. 6d. each. 
 
 and 2 xy, or 2y= . ....... at 25. each. 
 
 Hence, 
 
S'i SIMPLE EQUATIONS. 
 
 5y 
 Hence, by the question, =x + 3 (A)> 
 
 and <2y = x 4 (B). 
 
 Subtract (B) from (A} y then | =7, or i/=14, the number of 
 
 [beggars, 
 From eq n .(B), x=2?/-f4 = 28-f-4 = 32 shillings in his pocket. 
 
 QUESTION 14. 
 
 A person passed th of his age in childhood, ^th in youth, 
 Jth -f 5 years in matrimony ; he had then a son whom he sur- 
 vived 4 years, and who reached only J the age of his father. At 
 what age did this person die j 
 
 Let x= age of the person at the time of his death. 
 
 Then = time spent in childhood. 
 
 in youth. 
 
 in matrimony. 
 
 /. ^ + TT> + 7 4- 5 = age of the person when the son 
 
 r i 
 
 [was born, 
 
 ** f T 
 
 and ^~(3~~Y5~"^ "5 = interval between birth of the son 
 
 [and the old man's death ; 
 
 XXX 
 
 /. X~Q - 5 4 =sage of the son when he died. 
 
 But, by the question, the son died at J the age of his father, 
 
 xxx x 
 
 Hence, ;_-_-_- -9=-- 
 
 j 2 x 
 Multiply by 12, then 12# <2x-x 108 = 6 a?, 
 
 12x 
 or 3x -=108, 
 
 and 21# 12jc= 
 
 756 
 
 or x= - = 84. 
 y 
 
SIMPLE EQUATIONS. 85 
 
 QUESTION 15. 
 
 To find a number, such, that whether it is divided into two 
 or three equal parts, the continued product of the parts shall be 
 equal to the same quantity. 
 
 Let # = the number required. 
 
 x x 
 Then - x - = continued product, when the number 
 
 [is divided into two parts, 
 
 XXX 
 
 and - X- X~ = continued product, when the number 
 
 [is divided into three parts. 
 Hence, by ) x x x x x x" #" 
 
 * if l _ vx __ v v or ~~~ ~~~ 
 the question,] 2 X 2~3 X 3*3 5t 4~27' 
 
 Divide by x*, then 2 7 = 4 x y 
 
 27 
 and x= =6J, the number required. 
 
 Qu. 16. There is a certain number, consisting of two digits. 
 The sum of those digits is 5 ; and if 9 be added to the number 
 itself, the digits will be inverted. What is the number? 
 Let x=left-hand digit. 
 
 y = right-hand digit. 
 Then by Art. 6 1 . 10,r-f 7/ = the number itself, 
 
 and lOy + x the number with its digits inverted. 
 Hence, by the question, x+y = 5 (4), 
 
 Subtract (B) from (A), then 2?/==6, and y = 3, 
 
 .'. the number is (lOx + y) 23. 
 
 Add 9 to this number, and it becomes 32, which is the number 
 with the digits inverted. 
 
 Qu. 17. What two numbers are those whose difference 
 is 10; and if 15 be added to their sum, the whole will be 43 ? 
 
 ANSWER, 9? and 19. 
 Qu. 18. 
 
86 SIMPLE EQUATIONS. 
 
 Qu. 18. There are two numbers whose difference is 14; 
 and if 9 times the lesser be subtracted from 6 times the greater, 
 the remainder will be 33. What are the numbers ? 
 
 ANSWER, 17, and 31. 
 
 Qu. 19. What number is that, to which if I add 20, and 
 from |ds of this sum I subtract 12, the remainder shall be 10? 
 
 ANSW. 13. 
 
 Qu. 20. What number is that, of which if I add -fd, th, 
 and fths together, the sum shall be 73 ? 
 
 ANSW. 84<. 
 
 Qu. 21. Two persons, A and B, lay out equal sums of 
 money in trade ; A gains 1 20/., and jB loses 80l. ; and now 
 A's money is treble of B's. What sum had each at first ? 
 
 ANSW. ISO/. 
 
 Qu. 22. What number is that whose ^d part exceeds its -th 
 by 72 ? ANSW. 540. 
 
 Qu. 23. There are two numbers whose sum is 37; and if 
 3 times the lesser be subtracted from 4 times the greater, and 
 this difference divided by 6, the quotient will be 6. What are 
 the numbers ? ANSW. 21, and 16. 
 
 
 Qu. 24. There are two numbers whose sum is 49 ; and if 
 fth of the lesser be subtracted from -Jth of the greater, the 
 remainder will be 5. What are the numbers ? 
 
 ANSW. 35, and 14. 
 
 Qu. 25. What two^ numbers are those, to one-third the 
 sum of which if I add 13, the result shall be 17 ; and if from 
 half their difference I subtract one, the remainder shall be two ? 
 
 ANSW. 9, and 3. 
 
 Qu. 26. There is a certain fraction, such, that if I add one 
 to its numerator, it becomes J ; if 3 be added to the denomi- 
 nator, it becomes . What is the fraction ? 
 
 ANSW. - 
 
 
SIMPLE EQUATIONS. 87 
 
 Qu. 27. A person has two horses, and a saddle worth 10/.; 
 if the saddle be put on thejlrst horse, his value becomes double 
 that of the second ; but if the saddle be put on the second horse, 
 his value will not amount to that of the first horse by 13/. 
 What is the value of each horse ? ANSWER, 56 and 33. 
 
 Qu. 28. To divide the number 72 into three parts, so that 
 \ thejlrst part shall be equal to the second, and -fths of the 
 second part equal to the third. ANSW. 40, 20, and 12. 
 
 Qu. 29. A person after spending -fin of his income plus 
 10/., had then remaining J of it plus 35/. Required his 
 income. ANSW. 150/. 
 
 Qu. 30. A gamester at one sitting lost yth of his money, 
 and then won 10 shillings; at a second he lost d of the re- 
 mainder, and then won 3 shillings; after which he had 3 guineas 
 left. What money had he at first ? ANSW. 5L 
 
 Qu. 31. There are two numbers, such, that J the greater 
 added to Jd the lesser is 13 ; and if J the lesser be taken from 
 Jd the greater, the remainder is nothing. What are the 
 numbers? ANSW. 18 and 12. 
 
 Qu. 32. There is a certain number, to the sum of whose 
 digits if you add 7, the result will be three times the left-hand 
 digit ; and if from the number itself you subtract 18, the digits 
 will be inverted. What is the number ? ANSW. 53. 
 
 Qu. 33. Divide the number 90 into four such parts, that 
 the first increased by 2, the second diminished by 2, the third 
 multiplied by 2, and the fourth divided by 2, may all be equal 
 to the same quantity. ANSW. 18, 22. 10,40. 
 
 Qu. 34. A merchant has two kinds of tea, one worth 95. 6d. 
 per pound, the other 13s. 6d. How many pounds of each must 
 he take to form a chest of 104lbs. which shall be worth 56l ? 
 
 ANSW. 33 at 135. 6d. 
 71 at Qs. 6d. 
 
88 - SIMPLE EQUATIONS. 
 
 Qu. 35. A vessel containing 120 gallons is filled in 
 10 minutes by two spouts running successively ; the one runs 
 14 gallons in a minute, the other 9 gallons in a minute. For 
 what time has each spout run ? 
 
 ANSWER, 14-gallon spout runs 6 minutes. 
 9-gallon spout ... 4 minutes. 
 
 Qu. 36. In the composition of a certain number of pounds 
 of gunpowder, ds the whole + 1 was nitre ; Jth the whole 4^ 
 was sulphur ; and the charcoal was -th of the nitre 2. How 
 many pounds of gunpowder were there ? ANSW. 69 pounds. 
 
 Qu. 37. To find three numbers, such, that the first with 
 i the sum of the second and third shall be 1 20 ; the second 
 with |th the difference of the third andjirst shall be 70 5 and 
 J the sum of the three numbers shall be 95. 
 
 ANSW. 50, 65, 75. 
 
 CHAP. V. 
 
 ON QUADRATIC EQUATIONS. 
 
 QUADRATIC Equations are divided into pure and adjected. 
 Pure quadratic equations are those which contain only the 
 square of the unknown quantity; such aso; a = 36; # a -j-5=54; 
 ax a b = c; &c. Adftcted quadratic equations are those which 
 involve both the square and simple power of the unknown 
 quantity, such as x* + 4 jc = 45; Sx 1 2# = 21; ax*-t-2bx 
 = e + d; &c.&c. 
 
 XX. 
 
 On the Solution of Pure Quadratic Equations. 
 77. The Rule for the solution of pure quadratic equations 
 is this; "Transpose the terms of the equation in such a manner, 
 " that those which contain a?* may be on one side of the equa- 
 tion, and the known quantities on the other; divide (if ne- 
 " cessary) by the coefficient of # a ; then extract the square root 
 " of each side of the equation, and it will give the value of x." 
 
QUADRATIC EQUATIONS, 
 
 EXAMPLE 1. 
 Let a -f 5 = 54, 
 
 By transposition, x* = 54 5 = 49- 
 Extract the square root^ 
 
 of both sides of the (then # = \/49 = 7. 
 equation ...... } 
 
 Ex.2. 
 
 Let 3^ 4 = 71- 
 By transposition, 3jc 2 = 71 -f-4 = 75. 
 
 75 
 Divide by 3, #*= = 25. 
 
 Extract the square root, x = */25 = 5. 
 Ex. 3. 
 
 Let5# 2 27 = 3x 2 + 215. 
 By transposition, 5# 2 3o; 2 = 21 
 
 or, 2^ = 242; 
 
 242 
 
 .'. a? a = -r- 
 a 
 
 Ex. 4. 
 
 Let ax 7 6 = c; 
 then aa: 3 = c + 
 
 , , . 
 
 and a; = , or x=y 
 
 Ex.5. 
 
 Let flx a -5c = 6x 2 ~ 
 
 Then a# 2 Z>x a =5c 
 
 or (a 6).r Q = 
 
 Ex.6. 5.r 2 l. =244 . . . ANSWER, a;=7. 
 Ex.7. 
 
 Ex. 8. =45 
 
 N 
 
90 QUADRATIC EQUATIONS. 
 
 Ex. 10. 2 
 
 3 a c 
 
 XXI. 
 
 On the Solution of Adfected Quadratic Equations. 
 
 78. The most general form under which an adfected quadratic 
 equation can be exhibited is ax* + bx=c; where 3 I, c, may 
 be any quantities whatever, positive or negative^ integral or 
 
 fractional. Divide each side of this equation by a, then 
 
 be be 
 
 x* + ~x-. Let-=p,~ = q: then this equation is reduced 
 a a a ^' a * ' 
 
 to the form .r*-f px=q, where p and q may be any quantities 
 whatever, positive or negative, integral or fractional. 
 
 79. From the twofold form under which adfected quadratic 
 equations may be expressed, there arise two Rules for their 
 solution. 
 
 RULE I. 
 Let. 
 
 Add 7 to each sidef , p 2 p 2 
 
 4 
 
 of the equation, then 
 
 ^ . -\ P -f Vp* + 4o' 
 Extract the square root J xJr '- = ^- _L_ 2 
 
 of each side of the > 
 equation, then 3 an d # = 
 
 Hence it appears, that " if to each side of the equation there 
 " be added the square of half the coefficient of the second term, 
 " there will arise, on the left-hand side of the equation, a 
 
 " quantity which is the square of j: + - ; and by extracting the 
 
 " square 
 
 (*) Since the square of -{- is -ra a , and of a is also -fa 2 , the square 
 
 oot of + a may be either -ha or a ; hence the square root of j? -j- 4 <?may 
 
 be expressed by i vV 
 
 . 
 
QUADRATIC EQUATIONS. *1 
 
 " square root of each side of the resulting equation, we obtain 
 " a simple equation, from which the value of x may be dc- 
 " termined." 
 
 RULE II. 
 
 Let ax* + bx = c. 
 
 equation by 4tf, \ 
 
 Add b 2 to each side,we have 4a V -f lalx + b* = 
 Extract the square root as before, <2ax-\- b= */4ac -f- /;* 
 
 -- b 
 
 From which we infer, that " if each side of the equation be 
 " multiplied by four times the coefficient of x*, and to each side 
 " there be added the square of the coefficient ofx, the quantity 
 " on the left-hand side of the equation will be the square of 
 " 2ax + b. Extract the square root of each side of the equa- 
 " tion, and there arises a simple equation^ from which the value 
 " of x may be determined ^." 
 
 Ifa=l, the equation is reduced to the form x*-\-px = q; 
 in this case, therefore, the Rule may be applied, by " mul- 
 (e tiplying each side of the equation by 4, and adding the square 
 66 of the coefficient of a. 1 ." 
 
 80. Either of these Rules may of course be applied to the 
 solution of adfected quadratic equations ; but it may be proper 
 to observe, that in equations of the form ax* + bx=c where a 
 is a small number, and in those of the form x*+px = q where 
 p is an odd number, RULE II. will be found by far the most 
 convenient. 
 
 81. From the form in which the value of x is exhibited in 
 each of these Pvules, it is evident that it will have two values ; 
 
 one 
 
 ( b ) The principle of this Rule will be found in thefiija Ganita, a Hindoo 
 Treatise on the Elements of Algebra. See Mr. COLEBROOKE'S Translation 
 of this very curious work. 
 
9 QUADRATIC EQUATIONS. 
 
 one corresponding to the sign -f , and the other to the sign , 
 of the radical quantity. In the following examples, the positive 
 values only of x are inserted at the end of the solution. 
 
 EXAMPLE I. 
 
 Let x* + 8o?= 65. 
 
 2 
 
 By RULB I. add the square of 4 (i. e. ) to each side of 
 
 the equation, then .... x* + 8x+'\6-65 + 16 = 81. 
 Extract the square root of each side of the equation, then 
 
 (x+^ or)# + 4=^81=9, and x = 5. 
 
 Ex.2. 
 
 Let x a 4 x = 45. 
 
 By RULE I. add the? j **- 4 * + 4=45+4 = 49. 
 square of 2, z. e. 4, then 3 
 
 Extract the square root,and [x -,or j Jc 2 = V49 = 7,and j:= 9, 
 
 Ex.3. 
 
 Let 3 a -f5a;=42. 
 By RULE II. multiply eachl 
 
 side of the equation by V36#*-f 60 #=5 04. 
 (4 a) 12; then ..... 3 
 
 Add (*) 25 toeachside)^ o =5() =5 _ 
 
 of the equation, we have ) 
 Extract the square root of each side of the equation, which 
 
 .'. 6x=18, and # = 3. 
 
 Ex. 4. 
 Let 7# 2 20#=32, 
 
 a 20o: 32 
 .:x-. = y- 
 
 Complete the Square by RULE I. 
 
 20x , 100_32 . 100 224 , 100 324 
 then.,- f - T + = _ + = - 
 
 10 /324 18 28 
 
 Hence x - = ^/ - = , and ^= =4. 
 
QUADRATIC EQUATIONS. 93 
 
 Ex. 5. 
 
 Let #' I5x= 54. 
 
 By RULE II. mul-j then 4a ,_ 6ox=a! _ fllfi- 
 tiplyby4 . . ) 
 Add (4>) 225 to I 
 each side . . J 
 Extract the square root, 2x 15 = 
 
 .*. 2#=153=18 or 12, and #=9 or 6, 
 
 Ex.6. 
 
 Let 4x a 3#=85. 
 By RULE II. muM 
 tiply by 16, and! 
 
 add square of 3 /64x a -48# + 9= 1360 + 9= 1369. 
 to each side of 
 the equation . ' i 
 
 Extract the } 40 
 
 square root ^^-3 = N/1369 = 37,orx =T= : 5 , 
 
 Ex. 7. 
 
 4x* x 
 
 Let -11=5- 
 
 Multiply by 3, then 4x* 33 =x, 
 By transposition . . . 4x* x =33. 
 Multiply by 16, and addi 
 
 1 to each side of the>64x* 16#+ 1=528+ 1=529. 
 equation (RULE II.) 3 
 
 24 
 Extract the square root, 8 x 1 = \/5 29 = 23, or x = = 3. 
 
 Ex. 8. 
 
 Let 5# a -f-4#=273; 
 
 , 4x 273 
 then x -f -=-r- 
 o o - 
 
 4a; 4 273 4 1369 
 and by RULE I. * + T +-= +- = ^- 
 
 2 /1369 37 
 
 5=V -iT=T 
 
 37 2 35 
 
94 QUADRATIC EQUATIONS. 
 
 Ex. 9. 
 
 Mult y . by jc + 1, then 7 4- - = 
 
 
 By transposition, 5# a 4 a: = 2. 
 
 By RULE II. I00ff* 80j;+ 16 = 40+ -16 = 56. 
 Extract the square root, \0x 4 = \/56 
 
 and 103C=s/56-f 4 = 7.48 + 4=11.48 
 
 11.48 
 *=-I5- 
 
 Ex. 10. 
 Let 
 
 2jc 60 
 Divide by 13, a? '+ f5 = YJ' 
 
 , r j 6Q t 
 
 add the :square^ +- + _=-+_= + _ = _. 
 
 of fs' 
 
 Extract the ^ J^ _ ^781 __ 27.Q4 
 square root / X+ }3~~ 13 13 
 
 27.94 1 26.94 
 
 Ex. 11. 
 
 Let 
 
 By RULE II multiplyj 
 by 8 6 3 and add c , J 
 
 Extract the \ 
 
 square root) 4&*-c 
 
 Ex. 12. #-f 12x=108 ....... x = 6. 
 
 Ex. 13. x 2 I4jc=5l ........ x=l7. 
 
 Ex. 14. o; a + 6^jc = c 2 ......... *=Vc* 
 
 Ex. 15. 3x 7 + 2o:=l6l ....... jc=7. 
 
 Ex. 16. 2x 2 -5jc=117 ....... x = 9. 
 
QUADRATIC EQUATIONS. 95 
 
 Ex. 17. 
 
 Ex. 18. 
 
 Ex. 19. 5 j; 2 +4#=273 ....... x=7. 
 
 Ex. 20. 4# 3 7^=492 ....... #=12. 
 
 # 2 
 Ex. 21. g !=#+ 11 . ....... #=12. 
 
 Q x 1 7 
 
 Ex ' 22 - T + i = 5 - ......... *= 3o 
 
 Ex.23. | ~f=9 ........... a: = G. 
 
 Ex.24, 
 
 Ex. 25. ar'-34=jx ......... a: = 6. 
 
 x 5 
 Ex. 26. -+- = 5f .......... a;=25 or I. 
 
 O X 
 
 Ex . 27 . _. 
 
 x 6 
 
 Ex - 28 - 
 
 Ex. 29. x 9 6x+ 19=13 # = 4.732 or 1.268. 
 
 Ex.30. 5# 2 -f-4#=25 #=1.871. 
 
 r o i 11 & i ^^ + 1 6 ac . 
 rjX. O I. 4<23? VX = C ........ #= 3 
 
 8 A 
 
 Ex - 32 - a + x = a #=iVi-a\ 
 
 XXII. 
 
 Oft /Ae solution of Questions producing Quadratic 
 
 Equations. 
 
 82. In the solution of Questions which involve Quadratic 
 Equations, sometimes both, and sometirrrs only one of the values 
 of the unknown quantity, will answer the conditions required. 
 This is a circumstance which may always be very readily deter- 
 mined by the nature of the question itself. 
 
90 QUADRATIC EQUATIONS. 
 
 QUESTION I. 
 
 To divide the number 56 into two such parts, that their 
 product shall be 640. 
 
 Let x=one part, 
 then 56 .r=the other part, 
 and x (56 x) product of the two parts. 
 Hence, by the question, a: (5 6 x) = 640, 
 or 56jc~x'=:640. 
 By transposition, x* 56x= 640. 
 
 By completing the square, > 2 __ 
 (RULE I.) 3 
 
 .'. ac 28 = 12, and x = 40 or 16. 
 
 In this case it appears that the two values of the unknown 
 quantity are the two parts into which the given number was 
 required to be divided. 
 
 QUESTION 2. 
 
 There are two numbers whose difference is 7 3 and half their 
 product plus 30 is equal to the square of the lesser number. 
 What are the numbers ? 
 
 Let a;=the lesser number, 
 then x+ 7 =the greater number, 
 
 x x (x -f- 7) 
 and - - -f 30 = half their product phis 30. 
 
 Hence, by the question, - -- +30=x 2 (square of lesser). 
 
 Multiply by 2 
 
 By transposition .... x* 7x=6Q. 
 Multiply by 4, and a*dl 4a .,_ 
 49 (Rule II.) 
 
 . 
 
 X 2^-7 = 
 
 2 x = 24, or x = 1 2 = lesser number , 
 hence x + 7 = 1 9 = greater number. 
 
QUADRATIC EQUATIONS. 97 
 
 QUESTION 3. 
 
 To divide the number 30 into two such parts, that their 
 product may be equal to eight times their difference. 
 
 Let #=the lesser part, 
 then 30 x= the greater part, 
 and 30 x x or 30 2 x = their difference. 
 Hence, by the question, #(30 #) = 8(30 2#), 
 
 or 30x # a =240 
 By transposition, #* 46# = 240. 
 
 .'.a? 23 = 17, 
 
 and a? = 23 17 = 40 or 6 = lesser part; 
 30~#=30 6 = 24= greater part. 
 
 In this case, the solution of the equation gives 40 and 6 for 
 the lesser part. Now as 40 cannot possibly be a part of 30, 
 we take 6 for the lesser part, which gives 24 for the greater 
 part; and the two numbers, 24 and 6, answer the conditions 
 required. 
 
 QUESTION 4. 
 
 A person bought cloth for 33/. 155. which he sold again at 
 2/. 85. per piece, and gained by the bargain as much as one 
 piece cost him ; Required the number of pieces. 
 
 Let # = the number of pieces. 
 
 /" *7 r 
 
 Then = number of shillings each piece cost, 
 oc 
 
 and 48 x = number of shillings he sold the whole for; 
 .'. 4Sx 675 = what he gained by the bargain. 
 
 675 
 Hence, by the question, 48 x -675 = - ' 
 
 By transposition ^ 3 225 225 
 and division, J x ~~l6 X ~ = ^W 
 
 O Complete 
 
98 QUADRATIC EQUATIONS. 
 
 Complete the] , 223 50625 _ 225 50625 _ 65025 
 square, (RULE I.) j* ~~\6 X+ 1024 ~~76 + 1024 " 1024 ' 
 
 225 255 480 
 
 QUESTION 5. 
 
 A and B set off at the same time to a place at the distance 
 of 150 miles. A travels 3 miles an hour faster than B, and 
 arrives at his journey's end 8 hours and 20 minutes before him. 
 At what rate did each person travel per hour ? 
 
 Let #=rate per hour at which B travels. 
 Then x + 3= ........... A . . . . 
 
 And -- = number of hours for which B travels. 
 
 x 
 
 150 
 
 But A is 8 hours 20 minutes (8y hours) sooner at his 
 journey's en-d than J5; 
 
 150 150 
 
 Hence + 8^= , 
 
 150 25 150 
 
 By reduction, x*-f 3#=54. 
 
 9 9 225 
 
 Complete the square, x* + 3^+- = 54 + -= (RULE L); 
 
 3_15 
 
 153 
 and x= - = 6 miles an hour for B 
 
 x 4- 3 = 9 for A. 
 
 QUESTION 6. 
 
 Some bees had alighted upon a tree ; at one flight the square 
 root of half of them went away; at another Jths of them; two 
 bees then remained. How many then alighted on the tree ? fa) 
 
 (*) This question, and the mode of solution, is taken from the Bija 
 Canita. 
 
 
QUADRATIC EQUATIONS. 99 
 
 Let 2 x 9 =e the N of bees ; 
 
 16 a; 2 
 
 a: + - 1 + 2 = 2 ; r 9 , 
 
 7 
 
 or Qx -r 1 6# 2 + 1 8 = 1 8# f ; 
 .'. 18a? 8 16** 9#=18, 
 or 2# 9 9#=18. 
 
 (RULE II.) Multiply by 8, 
 
 I6x a 72#=144. 
 
 Add 81; then I6x*- 72X+Q1 =225, 
 or 4# 9=15; 
 .*. 4#= 15-f 9 = 24, and #=6, .*. 2o:* = 72 = N of bees. 
 
 Qu. 7. To divide the number 33 into two such parts, that 
 their product shall be 162. ANSWER, 27 and 6. 
 
 Qu. 8. What two numbers are those whose sum is 29, and 
 product 100? ANSW. 25 and 4. 
 
 Qu. 9. The difference of two numbers is 5, and Jth part 
 of their product is 26. What are the numbers ? 
 
 ANSW. 13 and 8. 
 
 Qu. 10. The difference of two numbers is 6 ; and if 47 be 
 added to twice the square of the lesser, it will be equal to the 
 square of the greater. What are the numbers ? 
 
 ANSW. 17 and 11. 
 
 Qu. 11. There are two* numbers whose sum is. 30; and 
 Jd of their product plus 13 is equal to the square of the lesser 
 number. What are the numbers ? ANSW. 21 and 9. 
 
 Qu. 1 2. There are two numbers whose product is , 1 2O. If 
 2 be added to the lesser, and 3 subtracted from the greater, the 
 product of the sum and remainder will also be 120. What are 
 the numbers ? ANSW. 15 and 8. 
 
 Qu. 13. A and B distribute 1200^. each among a certain 
 number of persons. A relieves 40 persons more than J5, and 
 B gives 5/. apiece to each person more than A. How many 
 persons were relieved by A and J3 respectively ? 
 
 ANSW. 120 by A, 80 by B. 
 
100 QUADRATIC EQUATIONS. 
 
 Qu. 14. A person bought a certain number of sheep for 
 1 20/. If there had been 8 more, each sheep would have cost 
 him 10 shillings less. How many sheep were there ? 
 
 40. 
 
 Qu. 15. A person bought a certain number of sheep for 
 57 1, Having lost 8 of them, and sold the remainder at 8 shil- 
 lings a-head profit, he is no loser by the bargain. How many 
 sheep did he buy ? ANSW. 38. 
 
 Qu. 16. A and B set off at the same time to a place at 
 the distance of 300 mires. A travels at the rate of one mile 
 an hour faster than B, and arrives at his journey's end 10 hours 
 before him. At what rate did each person travel per hour ? 
 ANSW. A travelled 6 miles per hour. 
 B ..... 5 ........ 
 
 XXIII. 
 
 On Quadratic Equations having Impossible Roots. 
 
 83. In the solution of the adfected Quadratic Equation, 
 = 9 (Art. 79.) the two values of x were shewn to be 
 
 equal to - g If q be a negative quantity, and p* 
 
 less than 4q, then the quantity p 4q is negative, and con- 
 sequently the quantity VjD* 4<? comes under the description 
 of the radical quantities mentioned in Art. 56. In this case, 
 the two roots, or values of x, are said to be impossible. 
 
 EXAMPLE 1. 
 
 Let #* + 8#H-31=0, or# 2 + 8.r= 31. 
 Complete the square, (RULE I.) 
 
 then x*+8x-\-l6= 31 +16= 15, 
 
 and x + 4 = V 15, or x 4\/ 15. 
 
 Ex.2. 
 Let # ? -12ac -H 50 = 0, or x*- l<2x= -50. 
 
 Complete 
 
QUADRATIC EQUATIONS. 
 
 101 
 
 Complete the square, (RuLE I.) 
 
 #_ 1 20; 4- 36= - 50 4- 36= - 14, 
 
 Ex.3. 
 
 To divide the number 16 into two such parts, that their 
 product shall be equal to 70. 
 
 Let # = one part, 
 then 16 #=the other part. 
 Hence x(\6x) or 16# # 2 =70. 
 
 Transpose, and # 2 1 6#= 70. 
 Complete the square, 
 
 Ex. 6, To divide the number 20 into two such parts, that 
 their product shall be 105 .... #= 10N/ 5. 
 
 XXIV. 
 
 On the Solution of Quadratic Equations of .the form 
 x* n +px n = q. 
 
 84. Lety = x n , then (by CASE III. Art. 65.) y a = x' M ; and 
 substituting these values for x** and x" in the equation 
 a?* n +p x n = q , it is transformed into y* + py = g, where the value 
 of y may be determined by the foregoing Rules. Having the 
 value of y, the value of may be found; for x n =y, .:x=\/y. 
 We are thus enabled to solve equations in which the unknown 
 
 quantity 
 
 ( a ) It is very well known that the greatest product which can arise from 
 the multiplication of the two parts into which any given number may be 
 divided, is when these parts are equal: the greatest product, therefore, 
 which could arise from the division of the number 1 6 into two parts, is 
 when each of them is 8 ; hence, in requiring "to divide the number 16 
 into two such parts that their product should be 70," the solution of the 
 question is impossible. 
 
102 QUADRATIC EQUATIONS. 
 
 quantity is found only in two terms, and where the index of the 
 highest power is double the index of the lowest, like common 
 quadratics. 
 
 EXAMPLE 1. 
 
 then x 4 =y*, 
 By RULE I. y* 6y + 9 = 27 +9 = 36, 
 
 and y 3 = 6, or y = 9. 
 But since x*=y, #=Vy, .*. x = *>/9 = 3. 
 
 Ex. 2. 
 
 These equations are often solved by the common Rules, 
 without the formality of substitution; thus, 
 
 Complete the square, (RuLE I.) x 6 2x 3 + 1 =48+ 1 =49. 
 Extract the root, x 3 1 = 7, .:x 3 = 8 } and#=rv/8 = 2. 
 
 Ex.3. 
 
 = ^ i .*. 22/ 2 7^ = 99. 
 
 By RULE II. 16?/ 2 56?/ + 49 = 792 + 49 = 841, 
 and 4 1/ 7 = 29, 
 
 Ex.4. 
 
 To resolve the number a into two such factors, that the sum 
 of their wth powers shall be equal to b. 
 Let x= one factor, 
 
 then - = the other factor. 
 x 
 
 a n 
 
 Hence x n + n *= I, or x* n + a n = & # n , .". ar w ^ jc n = a n . 
 x 
 
 By RULE II. 
 and 
 
 The two values of are the fewo factors required. 
 
QUADRATIC EQUATIONS. 103 
 
 Ex. 5. # 4 -j-4.r a =12 #=V/2. 
 
 Ex. 6. x 6 8# 3 = 513 x=3. 
 
 Ex. 7. 2# 4 #' = 496 #=4. 
 
 Ex. 8. To resolve the number 18 into two such factors, that 
 the sum of their cubes shall be 243 (See Ex. 4.) 
 
 ANSWER, 6 and 3. 
 
 XXV. 
 
 On the Solution of Quadratic Equations containing 
 
 Two unknown Quantities. 
 
 The solution of equations with two unknown quantities, in 
 which one or both these quantities are found in a quadratic 
 form, can only, in particular cases { ^, be effected by means of 
 the preceding Rules. Of these cases the two following are 
 very well known. 
 
 CASE I. 
 
 85. " When one of the equations by which the values of 
 " the unknown quantities are to be determined, is a simple 
 " equation ;" in which case, the Rule is, " to find a value of 
 " one of the unknown quantities from that simple equation, and 
 " then substitute for it the value so found, in the other equa- 
 " tion. The resulting equation will be a quadratic, which 
 " may be solved by the ordinary Rules." Thus, 
 
 Let ax + ly c 2 be the two equations, in which the values 
 a'x*-\-l'xy -fc'2/ a =c?5 of a; and y-^tre to be determined. 
 
 c ly 
 From the 1st equation a?=* 
 
 Substitute 
 
 ( b ) The most complete form under which quadratic equations containing 
 two unknown quantities could be expressed, is this, 
 
 a X* + b y z + c x y + d x + e y=. m 
 
 a'lf+b'y't + c'xy + d'x + e'y^m' ', but the general 
 
 solution of these equations can only be effected by means of equations 
 of higher dimensions than quadratics. 
 
104< QUADRATIC EQUATIONS. 
 
 Substitute this 1 t . . cy _j v , 
 
 fora; in the V thena'( ^j +1' ( J q ' j+cy=d, 
 
 2d equation, J 
 
 
 which reduced is (a'b* all' + aV)i/ a -f (a'c 2a'bc]y = a 2 J a'c 2 , 
 a common quadratic equation, from which the value of y may 
 
 be found. 
 
 EXAMPLE 1. 
 
 Let x + 2 y = 7, to find the ya , ues of ^ ftnd 
 a 2 
 
 From 1st equation, x=7 2y, .'. x* = 49 
 vSubstitute these values for # and a; 2 in the 2d equation, 
 then 49 28y + 4>y* + 2l y 6y*y*= 23, 
 
 or 3y a +7?/ = 49 23 = 26. 
 By RULE II. 36?/ 2 -l-84?/ + 49 = 3i2-{-49 = 36l 
 = 19 9 or 6^=12, andy = 2; 
 /. ^=.72^ = 7 4 = 3. 
 
 Ex.2. 
 
 _ 
 
 3 > to find the values of x and y. 
 
 and 3#?/ = 21oJ 
 
 From 1st equation, 2#-f?/ = 27; 
 
 27 -y 
 .'. 2j:=27 ,V> an d Jc= ^~~ 
 
 27-w 
 
 Hence, 3o:y = 3 X 
 
 or 3 x (27 
 
 or 2/ #=- 
 By RULE II, 4?/ 2 I08?/-f 729 = 729-560=169; 
 
 27 + 13 
 ,\2i/-27=13, ory = - ^ 
 
 27-20 
 and 0:= ^ =31. 
 
QUADRATIC EQUATIONS. 105 
 
 Ex.3. 
 
 There is a certain number consisting of two digits. The left- 
 hand digit is equal to 3 times the right-hand digit ; and if 1 2 
 be subtracted from the number itself, the remainder will be 
 equal to the square of the left-hand digit. What is the number ? 
 Let x be the left-hand digit,} then, by Art. 61, 10x-fy is the 
 and y the other ; $ number. 
 
 Hence >*= 3 ^ by the question; 
 and Wx+y 12 = x a $ 
 
 .'.by sub- 30 - 12 = 9 (f riox=30 y, and a" = 9 2/ a ); 
 
 stituton 
 
 9/-31y=-12 
 31 12 
 
 31 961 961 12 961-432 529 
 By RULE I, fy + =~ = 334 ' 
 
 31 23 - 54 
 
 Hence, 7,-- = -; or T/ ==-=3, 
 
 x = 3y = 9; and consequently the number is 93. 
 
 Ex.4. Let2*-3y = l ? to find the values of ,, and 
 
 2 .r< *- 5 *= 2 oi 
 
 Ex. 5. There are two numbers, such, that if the lesser be 
 taken from three times the greater, the remainder will be 35; 
 and if four times the greater be divided by three times the lesser 
 plus one, the quotient will be equal to the lesser number. 
 What are the numbers? ANSW. 13 and 4. 
 
 Ex. 6. What number is that, the sum of whose digits is 
 15; and if 31 be added to their product, the digits will be in- 
 verted? ANSW. 78. 
 
 CASE II. 
 
 86. When jc a , y 2 , or zy, is found in every term of the two 
 equations, they assume the form of 
 
 *=d' ; and their solution 
 may be effected in the following manner. 
 
 P Assume 
 
106 QUADRATIC EQUATIONS* 
 
 Assume xvy, then x*=v*y*' 9 substitute these values for 
 x 9 and x in both equations, then we have 
 
 = 9 or = 
 d X 
 
 
 
 
 
 or (a'd ud')v* + (b'dld')v cd'cd ; which is a quadratic 
 equation, from whence the value of v may be determined. Having 
 the value of v, the value of y may be found from either of the 
 equations (A) or (B); and then the value of #, from the equa- 
 tion x=vy. 
 
 EXAMPLE 1. 
 
 20 
 
 41 
 and 5z/ 
 
 u 20 41 
 
 ' e ~ a 
 
 which reduced is, 6v a 41 v= 13; 
 41v 13 
 
 41i? 1681 1369 
 By RULE l,,_-^+_ = _ 
 
 41 +37 4137 13 1 
 
 *- t; "T5 = "Tr 5 or V= ^T?" =T or 5" 
 
 41 41 369 
 
 == =9, or y = 3 , 
 
 Ex.2. 
 
 What two numbers are those, whose sum multiplied by the 
 
 greater 
 
QUADRATIC EQUATIONS. 107 
 
 greater is 7 7 ; and whose difference multiplied by the lesser is 
 
 equal to 12? 
 
 Let x= greater number, 
 
 y=. lesser. 
 
 Then (x + y) xx=.x* + xy = TJ 9 
 and (xy) xy-=xy y fl = 12. 
 Assume x=vy; 
 
 Then vy + vy*=77, or y* 4 ; 
 
 12 
 and vy* 7/ 3 = 12, or y*= - ' 
 
 12 77 
 
 ^^=7Z^ 
 
 or 12^ + 12^=77^77; 
 
 65 77 
 
 which gives v z j-y?= - 
 
 65 4225 529 
 
 65 23 88or42 11 
 -^T- =-^4 = T 
 Either value oft' will answer the conditions of the question; 
 
 7 12 12 48 48 
 
 but take *=-; then 3/ 2 = ~r^[ = lf^l = -j-= 16, 
 
 an^d 7/=4 .*. x=vy=- X4 = 7. 
 Hence, the numbers are 4 and 7 
 
 Ex. 3. Find two numbers, such, that the square of the 
 greater minus the square of the lesser may be 56; and the 
 square of the lesser plus ^d their product may be 40. 
 
 ANSWER, 9 and 5. 
 
 Ex. 4. There are two numbers, such, that 3 times the 
 square of the greater plus twice the square of the lesser is 1 10; 
 and half their product plus the square of the lesser is 4. What 
 are the numbers? ANSW. 6 and l/ a) 
 
 ( a ) For a great variety of questions relating to quadratic equations which 
 contain two unknown quantities, see ELAND'S Algebraical Problems^ 1812, 
 
 
108 QUADRATIC EQUATIONS. 
 
 XXVI. 
 
 On the Solution of certain Equations, in which the Two 
 unknown Quantities (x and y) are similarly involved. 
 87. Let x and y be any two numbers, of which x is the 
 greater, and y the lesser; let #-f-?/ = 2s, x ?/ = 2z; then, by 
 Art. 28, x=s + z, and y = s z. Nowlet# a -f 2/ a = 0,x 3 +2/ 3 = 6, 
 x* + y*=c, andx 5 + y s =d; then the values of x and y may be 
 found in terms of the known quantities s, a, b } c, d, in the fol- 
 lowing manner. 
 
 .*. by addition, 
 
 a-2*' A^2? 
 
 x z + y'\a) = 2/4- 2 " 2 ? and x = - or % = ' 
 
 Hence j:=, , m , , 
 
 II. a= 
 
 x 3 + ?/ 3 ^) = 25 3 -f 65 2 > and *'= ^J*j or ^=Y/ 
 
 65 
 Hence a7= 1 s-r-\/ ^"^^ , and?/ = 5--' 
 
 in. # 4 = 
 
 x 4 4-7/ 4 (c) = 25 4 -f 125V-f 2z 4 is a quadratic equation from 
 which the value of z may be found. 
 
 IV. X 5 = 
 
 -f- 1052;* is a quadratic equation 
 from which the value of z may be found (b) . 
 
 ( b ) In reviewing these operations, it may be observed, that those terms 
 where the index of* is an odd number destroy each other in the successive 
 
 series : 
 
QUADRATIC EQUATIONS. 109 
 
 88. Let x+y = 2s and x y = 2z as before, and let 
 
 y x y - y x y x 
 
 means of the equations in the preceding Article (87), the 
 values of x and ?/ may be found in terms of the known quan- 
 tities, s, a, b' } c, d'. 
 
 By CASK II. (87). x s -fy 3 =2i 3 -f 65^; 
 
 , -255 7(6' 
 
 and z = ,/ , ., .or z = \/ ^-p 
 6 -f-65 ' V 6 
 
 Hence x = 
 
 By CASK III. (87). 
 
 Hence c'(5 a - z a ) = 2 5 4 -f 1 2 5V -f 2 z* is a quadratic 
 
 equation, by which the value of x ntay be found. 
 
 series ; hence if the operations had been continued to i^+y 6 and <"-J-y 7 , 
 the resulting equations would have been equations of suf dimensions in a 
 cubic form; if they had been carried on to a^ + y 8 and J? 9 + 9 , the resulting 
 equations would have been equations of eight dimensions in a biquadratic 
 form. Hence the Problem of " Given the sum of two numbers, and the 
 sum of their th powers, to find the numbers themselves/* may be solved 
 as far as the 9lh power, by means either of quadratic, cubtCy or biquadratic 
 equations. 
 
110 QUADRATIC EQUATIONS. 
 
 iv. gr&> . 
 
 By CASE IV. (87). # 5 +2/ 5 =2s 5 ~l-20sV-f lOsz* -, and by 
 equating these two values of x s + y 5 , there arises a quadratic 
 equation by which the value of z may be determined. 
 
 89. Let x + y = s, and xy=.p ; then the sums of the several 
 powers of x and y may be found in terms of the known quan- 
 tities p and s, in the following manner. 
 
 ii. 
 
 or 
 
 .e. 
 
 in. 
 
 or 
 
 i. e. a*+y 4 +p(s*-2p) =s 4 - 3ps* ; 
 
 iv 
 
 or 
 i.e. 
 
 or in general # M + y "=5" nps*~* + n -p 2 ^"" 4 &c. 
 
 EXAMPLE 1. 
 
 The sum of two numbers is 6, and the sum of their fifth 
 powers is 1056. What are the numbers ? 
 
 This Example belongs to CASE IV. Art. 87, where 5 = 3, and 
 
 The equation to find the value of z is 
 
 25 5 -f 20.5V+10sz 4 = d, 
 or 486 -f 540 z a 4- 30z 4 = 1056 ; 
 
 Divide 
 
 
QUADRATIC EQUATIONS. Ill 
 
 Divide by 6, 81 4- 90z a -f 5# 4 =176 ; 
 
 By RULE I. z 4 + 18z* + 8l = 100, 
 
 or z* + 9= 10; .'. z*=l, and z= 1. 
 Hence x=s+z=3 + 1=4, 
 
 y=.s z=3 1 = 2. 
 
 Ex. 2. There are two numbers whose sum is 18, and the 
 square of the greater divided by the lesser plus the square of the 
 lesser divided by the greater is 27 ; What are the numbers ? 
 In CASE II. (88). 5 = 9, and &'=27; hence z= 
 
 9X81 __ 
 
 7=9 + 3 = 12, and y=s~-z = 9 3 = 6; and the two num- 
 bers are 1 2 and 6. 
 
 Ex. 3. The sum of two numbers is 5 (.?), and their product 
 6 (p) ; What is the sum of their 4th powers ? 
 
 By CASE III. (Art. 89.) x 4 + ?/ 4 = j 4 - 
 + 72 = 25+72 = 97. 
 
 CHAP. VI. 
 
 ON RATIOS, PROPORTION, AND VARIATION. 
 
 XXVII. 
 
 Definitions. 
 
 90. By RATIO is meant the relation which one quantity bears 
 to another, with respect to magnitude. It is evident that this 
 relation can exist rnly between quantities of a similar kind ; 
 thus, a number must be compared with a number; a line with 
 a line; &e. &c.; and it would be absurd to compare a certain 
 number of feet with a certain number of pounds; &c. &c. 
 
 91. There are two ways in which the magnitude of quantities 
 
 may 
 
RATIOS. 
 
 may be compared. In the first place, they may be compared 
 with regard to their difference; and then the question asked, 
 is, " How much one quantity is greater or less than another." 
 The relation which quantities bear to each other in this respect, 
 is called their Arithmetical Ratio. The other way in which they 
 may be compared, is, by inquiring, " How often one quantity is 
 contained in the other." This relation between quantities is cal- 
 led their Geometrical Ratio. The term ratio, when simply ap- 
 plied, is generally understood in the latter sense; and it is in this 
 sense that the word will be made use of in the present Chapter. 
 
 92. In considering how often one quantity is contained in 
 another, the natural process is to divide the one by the other. 
 Thus, in comparing the number 12 with the numbers 4 and 3, 
 we know that 4 is contained in 12 three times, and that 3 is 
 contained in the same number four times; from which we infer 
 that the ratio of 12 : 3 is greater than the ratio of 12 : 4, the 
 magnitude of a ratio being measured by the number of times 
 one quantity is contained in another. For the same reason, 
 the ratio of 1 1 : 7 is said to be less than the ratio of 1 1 : 5. 
 When a ratio is thus expressed, the first term of it is called 
 the antecedent, the last term the consequent, of that ratio. 
 
 93. From this mode of estimating the magnitude of a ratio, 
 it appears that when the consequent of a ratio is not an aliquot 
 part of the antecedent, the value of the ratio must be expres- 
 sed by a fraction, whose numerator is the antecedent, and de- 
 nominator the consequent of that ratio. Thus the magnitude 
 
 of the ratio of 15 : 7 is expressed by the fraction -, and of 
 
 4 ' 
 
 the ratio 4 : 13 by the fraction When the antecedent of a 
 
 ratio is greater than the consequent, it is called a ratio of greater 
 inequality ; when the antecedent is less than the consequent, 
 
 a ratio 
 
 ( a ) In expressing the ratio of two quantities, the word " to", is generally 
 supplied by two dots; thus, the ratio of " a to 6" is expressed by " : 6." 
 
RATIOS. 
 
 a ratio of Lesser inequality ; and if the two terms of a ratio be 
 the same, then it is said to be a ratio of equality. 
 
 94. The foregoing definitions evidently apply only to those 
 instances, in which the consequent of a ratio is contained a 
 certain number of times in the antecedent, or to those in which 
 
 .the magnitude of the ratio may be expressed by some definite 
 fraction. It does not therefore comprehend such ratios as 
 V/2 : 5; ^/3 : v/7,' 4 : v/ 1 5 & c - & C -S where the values of 
 the quantities v/2, \/3, ^/7 &c. can only be expressed in 
 decimal fractions which do not terminate. The ratio which 
 exists between quantities of this latter kind, when the radical 
 quantity is expressed by a decimal fraction^ is called their ap- 
 proximate ratio. i 
 
 95. Proportion consists in the equality of ratios; thus, 'since 
 4 is contained in 12, the same number of times that 6 is in 18, 
 the ratio of 12 : 4 is said to be equal to the ratio of 18 : 6, or, 
 in other words, that 12:4:: 18 : 6/ b) Of the four terms of 
 which every proportion consists, the first and last terms are 
 called the extremes, and the second and third the means of that 
 proportion. 
 
 96. If there be a set of quantities related together in the 
 following manner, viz. a : I : : b : c : : c : d : : d : e, &c. where the 
 consequent of every preceding ratio is the antecedent of the 
 following one, then the quantities , /;, c, d, e, &c. are said to 
 be in continued proportion ; and if only three quantities be 
 concerned, as in the proportion a : I : : b : c, then b is said to 
 be a mean proportional between the two extremes a and c. 
 
 97. Since the proportion a : I : : c : d expresses the equality 
 of the ratios a : b and c : d; u and since the magnitude of the 
 
 ratio a : b is measured by the fraction 7, and that of the ratio 
 
 c : d by 
 
 ( b ) In stating a proportion, the words "is to" and "to" are generally 
 supplied by two dots, and the word "as" by four dots; thus, the propor- 
 tion " a, is to b as c to rf, M is expressed by " a : b : : c : d" 
 
 Q 
 
114 RATIO S. 
 
 c a, c 
 
 c:d by the fraction -7, it follows that l=j ? or that "when 
 
 " four quantities are proportional, the quotient of the first 
 " divided by the second is equal to the quotient of the third 
 " divided by the fourth 5" and vice versa^ "if there be four 
 
 Cl (* 
 
 " quantities a, b, c, d, such, that 7= -3, then those four quan- 
 " tities are proportional, or a : I : : c : d." 
 
 XXVIII. 
 
 On the Comparison and Composition of Ratios. 
 
 98. On the comparison of Ratios. 
 
 I. Since the ratio of a : b may be expressed by the fraction 
 a 
 
 T? let the numerator and denominator of this fraction be raul- 
 b 
 
 tiplied by any quantity m (in being either integral o? fractional), 
 
 ma. a, 
 then 7~ = I J an d ' the ratio of ma : mb is the same with the 
 
 ratio of a:b; from which we infer, that "if the terms of a 
 " ratio be multiplied or divided by the same quantity, it 
 " does not alter the value of the ratio." From hence also it 
 appears, that a ratio is reduced to its lowest terms by dividing 
 its antecedent and consequent by their greatest common measure. 
 
 IT. " Ratios are compared together by reducing the fractions 
 " by which their values are respectively represented, to a common 
 
 "denominator." Thus, the ratio of 8 : 5 is represented by the 
 
 8 9 
 
 fraction -? and the ratio of 9 : 6 by the fraction ^5 reduce these 
 
 fractions to others of the same value having a common deno- 
 minator, and they become and respectively ; and since 
 * 30 30 J 
 
 48 . 45 
 
 is greater than ? the ratio 8 : 5 is greater than the ratio 
 
 of 9 : 6. 
 
 in. "A ratio of greater inequality is diminished, and a ratio 
 " of lesser inequality is increased, by adding the same quantity 
 
 " to 
 
RATIOS. 
 
 " to both its terms." Let a + b : a represent a ratio of greater 
 inequality, and let x be added to each of its terms, and it 
 becomes the ratio of a-\-l + x:a-\-x. Now the ratio of 
 
 a + b 
 
 a + b:a = -- > and that of a+b+x: a-\- x- -- 5 let 
 a a + x 
 
 these fractions be reduced to others of the same value having 
 
 a* + ab + ax + bx 
 a common denominator, and they become - , x 
 
 a a -f-a-fa# 
 and - , v respectively; and since a -\-ab-\- ax + bx is 
 
 evidently greater than d* + ab + ax, the ratio of a -\-b:a is 
 greater than the ratio of a + b + x:a+x', i.e. the ratio of 
 a + b: a has been diminished by adding x to each of its terms. 
 Next, let a I: a represent a ratio of lesser inequality; then 
 
 a I) a b + x 
 
 proceeding with the fractions - and - T~ - > as in the 
 
 ct a ~7"" oo 
 
 
 former instance, the resulting fractions are -- a (a-\- ') 
 
 and - -, r^ 5 and since a 1 a b -\-ax-bx is less than 
 a((i + ) 
 
 a*~ab + ax, the ratio of a I: a is less than the ratio of 
 a b+x : a + x, and consequently the ratio of a b : a has been 
 increased by addin'g x to each of its terms. In the same man- 
 ner it might be shewn that " a ratio of greater inequality is in- 
 " creased, and a ratio of lesser inequality is diminished, bv 
 " subtracting the same quantity from each of its terms." 
 
 99. On the composition of Ratios. 
 
 i. Ratios are compounded together by multiplying their an- 
 tecedents together for a new antecedent, and their consequents 
 together for a new consequent. Thus, if the ratio of a : b be 
 compounded with the ratio of c : d, the resulting ratio is that 
 ofacibd; or if the ratios 4 : 3; 5:2; and 7 : 1, be com- 
 pounded together, there results the ratio of 4x5x7:3x2X1, 
 or of 140 : 6, or (dividing each term by 2) of 70 : 3. 
 
 ii. If 
 
116 RATIOS. 
 
 ii. If the same ratio be compounded with itself once, twice, 
 thrice, &c. the resulting ratios are those of fl a : Q ; a*:b 3 ; 
 c 4 : l\ &c. &c. The ratio of a a : I? is called the duplicate 
 ratio of a : I ; a 3 : b 3 the triplicate ; a*: I 4 the quadruplicate ; 
 &c. &c. ; and as these ratios receive their denominations from 
 the indices of the several powers of a and b, the ratio of 
 */a: \/^ is called the sul duplicate ratio of a. : b; the ratio of 
 I/ a : /, the subtriplicate; &c. &c. 
 
 in. "If a set of ratios, whereof the consequent of the pre- 
 " ceding ratio is the same with the antecedent of the succeeding 
 i( one, be compounded together, the resulting ratio is that of 
 t( thejlrst antecedent to the last consequent." Thus, when the 
 ratios of a:b; l\c\ cid-j d;e\ are compounded together, 
 the resulting ratio is that of abed : bcde, or (dividing by bed) 
 that of a : e, or of the^rs^ antecedent : the last consequent ; and 
 the same will be the case whatever be the number of ratios. 
 
 iv. "A ratio of greater inequality compounded with another 
 " ratio, increases it; and a ratio of lesser inequality compounded 
 " with another ratio, diminishes it." Thus, let 1 -\-n : 1 re- 
 present a ratio of greater inequality, and let it be compounded 
 with the ratio a : b, the resulting ratio is that of a + na:b, 
 which is evidently greater than the ratio of a : b; on the other 
 hand, let 1 n : 1 represent a ratio of lesser inequality, and 
 let it be compounded with the ratio of a : b, then the resulting 
 ratio is that of ana:b, which is evidently less than the 
 ratio of a : b. 
 
 EXAMPLES. 
 
 Ex. 1. Reduce the ratio of 360:315, and 1595:667, to 
 their lowest terms. 
 
 Ex. 2. Reduce the ratio of a s + 2 a*x : a 9 to its lowest terms. 
 
 Ex. 3. Which is the greatest, the ratio of 16 : 15, or that 
 of 17 : 14? 
 
 Ex. 4. Which is the least of the three ratios, 20 : 1 7, 22 : 18, 
 
 or 
 
 I 
 
RATIOS. 117 
 
 or 25 : 23 ? and which is the greatest of the three ratios 
 8:7; 6:5; and 10:9? 
 
 Ex. 5. Which is the greatest, the- ratio of a+ 2 : ^a + 4, or 
 that of a + 4: 10 + 5? 
 
 ANSWER, The ratio of a 4 4 : Ja + 5. 
 
 Ex. 6. Compound together the ratios of 1 1 : 3, 7:2, and 
 5 : Q. ANSW. 385 : 54. 
 
 Ex. 7. Compound together the ratios of 15 : 12, 6 : 7? and 
 9:4; and then reduce the resulting ratio to its lowest terms. 
 
 ANSW. 135 : 56. 
 
 Ex. 8. Express in the simplest terms the ratio compounded 
 of a a x^'.a*, a + x: I, and 6 : a x. ANSW. (a + x)* : a*. 
 
 x 2 y* 
 Ex. 9. If the ratios of x + y : a, xy : I, and I : , 
 
 be compounded together, shew that the resulting ratio is a ratio 
 of equality. 
 
 Ex. 10. I f the ratios of 3 a -f- 2: 6a-fl, and of 2a-f 3 : a + 2, 
 be compounded together, is the resulting ratio a ratio of greater 
 or lesser inequality ? ANSW. A ratio of greater inequality. 
 
 Ex. 1 1 . What are the least numbers in the ratio compounded 
 of the three following ratios, viz. the ratio of 7 : 5, the dupli- 
 cate ratio of 4 : 9, and the triplicate ratio of 3 : 2 ? 
 
 ANSW. 14 and 15. 
 
 Ex. 12. Compound the sulduplicate ratio of x* : y*, with 
 the quadruplicate ratio of >J x : */y. ANSW. x 3 : y 3 . 
 
 XXIX. 
 
 On Proportion. 
 
 100. The most useful Theorems relating to proportional 
 quantities are the following. 
 
 TH. 1. " If four quantities be proportional, the product of 
 " the extremes will be equal to the product of the means }" for 
 
 let 
 
118 PROPORTION. 
 
 let a : I : : c : d, then, by Art. 97, 1 = j> " ad = lc. From 
 
 hence also it follows, fe that if any three terms of a proportion 
 be known, the fourth may be found;" for, from the equation 
 
 lc ad ad be 
 
 aa = t'c, we have a=-7- ; 1= c~r-- and d= 
 d J c 3 b ' a 
 
 TH. 2. The converse of the foregoing Theorem is also true ; 
 yiz. " If the product of any two quantities be equal to the 
 " product of two others, those four quantities will constitute 
 " a proportion, provided that the terms of one product be made 
 " the means, and the terms of the other product be made the 
 " extremes) of such proportion." Thus, if the four quantities 
 
 CL C 
 
 a, I, c, d, be such that ad=bc, then (dividing by Id) y= -%5 
 /. by Art. 97, a : I : : c : d. 
 
 TH. 3. " If three quantities be proportional, the product of 
 " the two extremes is equal to the square of the mean 5" for, if 
 a : b : : I : c, then, by THEOR. 2, ac = b*. From hence also it 
 follows, that " a mean proportional between any two quantities 
 is equal to the square root of their product;" for let x be 
 a mean proportional between a and c, then a : x : : x : c, 
 
 TH. 4. " If four quantities be proportional, they will also be 
 proportional, when taken inversely or alternately ;" thus, if 
 
 a : I :: c : d> then -/=-}; invert the fractions, then -=- ; 
 
 b d ' a c ' 
 
 .*. b : a : d : c. Again, since ad be, then (dividing by cd) 
 
 ad be a b 
 
 we have -} = - 7, or- = -> ; .*. a : c : b : a. 
 cd cd' c d 3 
 
 TH. 5. "If there be six proportional quantities, and the 
 " first be to the second as the third to the fourth; and the third 
 u to the fourth as the fifth to the sixth ; then will the first 
 " be to the second as the fifth to the sixth." For let a : b : : c : d, 
 
 and 
 
PROPORTION. 
 
 c 
 
 , 
 
 and c:a::?:y; then 7 = j and -y T.> .*. 7=7-' or by 
 
 Art.97,a:b::e:f. 
 
 TH. 6. " If four quantities be proportional, then the ^w or 
 " difference of the first and second will be to the second as 
 " the sum or difference of the third and fourth is to the fourth/' 
 
 CL C 
 
 For let a : I : : c : d, then T= jj add or subtract 1 from each 
 
 a c al cd 
 
 side of the equation; then 71=->1, .*. 7- = 7-, con- 
 
 sequently, by Art. 97, a b : I : : c d : d. 
 
 TH. 7. " If four quantities be proportional, the first is to the 
 " sum or difference of the first and second, as the third to 
 " the sum or difference of the third and fourth." For by 
 THEOR. 6, ab : b: : cd : d, and alternately ab:cd 
 :: I : d', but by THEOR. 4, b : d : : a : c ; hence, by THEOR. 5. 
 a-^.1 : cc?: : a: c, and alternately a : a : : c(2: c, .'.in- 
 versely a : a & : : c : c d. 
 
 TH. 8. "If four quantities be proportional, then the sum 
 " of the first and second is to theif difference, as the sum of 
 " the third and fourth is to their difference." For by 
 
 a + b c + d ab cd 
 IHEOR. 6, = j-) and -y = -r- > zwver the two last 
 
 b d a + b b c + d d 
 
 fractions, then - r = - -jj hence , x - 7 = ~ j~ X - ^> 
 a b cd b ab d cd 
 
 or - 7 = - j5 .*. by Art, 97, a + l : a b : : c + d: c d. 
 
 TH. 9. " If four quantities be proportional, and any equi- 
 " multiples or equal parts whatever be taken of the first and 
 " second, and also of the third and fourth; then will the result- 
 " ing quantities, taken in the same order, be still proportional." 
 For let a: b : : c:d; then, by Case I. Art. 98, the ratio of 
 in a : m b is the same with the ratio of a : b ; and for the same 
 reason, the ratio of nc:nd is the same with the ratio of c : d; 
 
 hence 
 
PROPORTION. 
 
 hence (Art. 95.) ma : ml : : nc : nd t where m and n may be 
 any quantities whatever, either integral or fractional. 
 
 TH. 10. The same theorem is true "if any equimultiples or 
 " equal parts whatever be taken of the first and third, and also 
 
 a c 
 "of the second and fourth-" for since 7 = 3' multiply each 
 
 m ma me 
 side of the equation by- then^- = ^, .'. ma:nb : : mcind, 
 
 where m and n may be any quantities whatever, either integral 
 or fractional. 
 
 TH. 11. "If four quantities be proportional, any powers or 
 roots of those quantities will also be proportional." P'or since 
 
 r=*-f we have ^ = ^ ' a * : n ' ' c " : ^ where n may be any 
 number either integral or fractional. 
 
 TH. 12. " If the corresponding terms of two sets of propor- 
 " tionals be multiplied together, or divided, by each other, the 
 " resulting quantities taken in order will still be proportional." 
 Thus let 
 
 " a 
 
 Again, by TH. 1, ad=bc, and eh=fg-, .: =--', hence, by 
 
 a b c d 
 TH. 2, - : j: -: T* The same will evidently be true of any 
 
 number of proportions. 
 
 TH. 13. " If there be two rows of proportional quantities, 
 " whereof the second and fourth of the first row are the same 
 " with the first and third of the second row, then will the 
 " remaining quantities, taken in order, be proportional ;" 
 thus, let a : b : : c : d 
 
 and I : e : : d if, then by THEOR. 1 2, ab : be : : cd : df, 
 or (reducing each tatio to its lowest terms) a : e : : c if. 
 
 TH. 14. 
 
PROPORTION. 121 
 
 TH. 14. " If there be a set of proportional quantities, 
 " a : b : : c : d : : e :f : : g : h &c. &c., then will the frst be 
 " to the second as the sum of all the antecedents to the sum of 
 " all the consequents.' 9 
 
 For,sincea = 6tf, and (byTHEOR 8 . 1 and 5) ad*=bc, af=be, 
 ah=bg, &c. we have ab-{-ad + af-{-ah + &Q-. = ba-}-bc + be 
 , or 
 
 . *.(by THEOR. 2.)a:b : 
 
 TH. 15. " If a : b : : b : c : : c : d : : d : e &c. as in Art. 96. 
 " then a : c : : a 2 : b 2 , or in the duplicate ratio of a : b ; 
 " a : d : : a 3 : b 3 , or in the triplicate ratio of a : b ; 
 " a : e : : a* : 4 , or in the quadruplicate ratio of a : b\" 
 &c. c. &c. &c. 
 
 For a : b : : a : b ; 
 and b : c : : a : b ; 
 .-. byTHEOR. 12, a : c : : a? : b*. 
 
 Again, a : c : : a a : & 2 , 
 
 but c : d : : a : b ; 
 
 .-. by THEOR. 12, a : d : : a 3 : b\ 
 
 Moreover, a : d : : a 3 : b 3 , 
 
 but d : e : : a : b ; 
 
 .*. by THEOR. 12, a : e : : a 4 : b\ 
 
 &c. &c. &c. &c. 
 
 101. The following Examples are intended to illustrate the 
 use of the foregoing Theorems. 
 
 EXAMPLE 1. 
 
 To divide the number 60 into two such parts, that the pro- 
 duct shall be to the sum of the squares : : 2 : 5. 
 
 R Let 
 
122 PROPORTION. 
 
 Let x=one part; 
 then 60 x the other part, 
 (60 x) xx=60x # a =the product, 
 
 and jc a -{-(60 #) 2 =2# a -f3600 l<20x=sum of the squares. 
 
 Hence, by the question, 60xx*i 2x a + 3600 120o; : : 2 : 5 \ 
 
 .'. by THEOR. I, (60 x x*) x 5 = (2# 2 -f 3600 120x) x 2, 
 
 or 300.r-5# 2 =4 l r 2 + 7200 240ff; 
 by transposition & division,!? 60 x= 800 ; 
 
 .'. tf a 60 # + 900 = 900 800=100, 
 and# 30 = 10; 
 
 or #=30 10 = 40 or 20 the 
 [parts required. 
 
 Ex.2. 
 
 The number 20 is divided into two parts, which are to each 
 other in the duplicate ratio of 3 : 1 . Find a mean proportional 
 between those parts. 
 
 Let x= greater part, 
 then 20 ~x= lesser part ; 
 .'. by the question, x: 20 x:: 3 2 : l a : :9: 1. 
 Hence, by THEOR. 1, x 180 g#, 
 or 10^=180; 
 
 .*. x= 18= greater part, 
 and 20 #=20 1 8 = 2=: lesser part. 
 By THEOR. 3, a mean proportional between 18 and 2 is 
 equal to V 18 X 2= /36 = 6, the number required. 
 
 Ex.3.' 
 
 If (a + x)* : (a x) 2 : : x + y : x y, shew that a : x : : */ 2 a y : A/y. 
 By expansion, a* + 2ax + x 2 : a*2ax + x 2 ::x+y:xy. 
 
 By THEOR. 8, 2 a a + 2 # a : 4ax :: 2x:2y. 
 Divide by 2, then d'+x* : <2ax :: x:y; 
 .: by THEOR. 1, (a'+# 2 ) xy = 2axxx=2axx*. 
 Hence, by THEO R . 2, a a + x 9 : x 9 : : 2 a :y. 
 By THEOR. 6, a* : #* : : 2a-y:y; 
 and by THEOR, 1 1, (w being )a : x : : N/2a y : >/y. 
 
 Ex.4. 
 
PROPORTION. 
 
 123 
 
 Ex.4. 
 
 lfx:y in the triplicate ratio of a:b, and a:b::\/c + x 
 \/a+y, shew that dx=cy. 
 
 Since x : y a 3 : b 3 , 
 
 and by THEOR. 11, a 3 : 1 3 c + x: d+y, 
 
 .*. by THEOR. 5, x:y c+x:d + y t 
 
 or c+x: d+y x :y, 
 
 and by THEOR. 4, c + x: x d + y:y; 
 
 .: by THEOR. 6, c : x d :y; 
 
 (and by THEOR. 1, dx=cy. 
 Ex.5. 
 
 There are two numbers whose product is 24, and the difference 
 of their cubes : cube of their difference :: 19 : 1. What are the 
 numbers ? 
 
 Let xzc.gr eater number, 
 and y = lesser number. 
 Then, by the question, xy = 24, 
 andx 3 y 3 :(xy) 3 : : 19 : 1. 
 
 By expansion, x* y 3 : x 3 3x*y + 3xy*y 3 OT(xyy : : 19 : 1. 
 ByTHEOR.6, 3x*y3xy*:(x-y) 3 :: 18: 1, 
 or 3xyx (xy) : (xy) 3 :: 18:1. 
 Divide by xy, then 3xy: (xy)*: : 18 : 1; 
 
 Hence, by THEOR. 1, 18 x (x y)*=72, 
 
 .*. xy = 2. 
 
 Again, x* 2 xy + y* = 4, 
 and 4xy =96. 
 
 12 
 
 or x+y=lO,i' 
 
 y= - =4. 
 
 Ex.6. 
 
PROPORTION. 
 
 Ex. 6. To divide the number 24 into two such parts, that 
 their product shall be to the sum of their squares : : 3 : 10. 
 
 ANSWER, 18 and 6. 
 
 Ex. 7. There are two numbers which are to each other as 
 3:2. If 6 be added to the greater) and subtracted from the 
 lesser, the sum and remainder will be to each other : : 3:1. 
 What are the numbers? ANSW. 24 and 16. 
 
 Ex. 8. There are two numbers which are to each other in the 
 duplicate ratio of 4 : 3, and 24 is a mean proportional between 
 them. What are the numbers? ANSW. 32 and 18. 
 
 a*-x* 
 Ex. 9. If ~, = 4 a; shew that +.T : 2 a : : 2b : a jr. 
 
 Ex. 10. If x 9 : y* : : 36 : 25, and 2x+y : x + 2 in a ratio 
 compounded of the ratios of 1 7 : : 2 and 2:7; what are the 
 numbers? ANSW. 12 and 10. 
 
 Ex. 11. There are two numbers whose product is 135, and 
 the difference of their squares is to the square of their dif- 
 ference : : 4 : 1. What are the numbers ? 
 
 ANSW. ] 5 and 9. 
 
 xxx. 
 
 On Variation. 
 
 102. If the quantities under consideration be of a variable 
 naturej then their relation to each other may be expressed in 
 the following manner. 
 
 i. Let A and B be two variable quantities so related to 
 each other, that whilst the value of A is changed to a, the 
 value of B is changed to I ; then if these two quantities 
 A and B always bear the same ratio to each other, i. e. if 
 A : B : : a : b (or by Theor. 4. of Proportion, A : a : : B : I) 
 throughout the whole period of their variation, they are said to 
 vary directly as each other. 
 
 EXAM. Suppose a body to move uniformly along at the rate 
 
 of 3 
 
VARIATION. J 25 
 
 of 3 feet in one second of time ; then in the Jirst second it 
 would describe 3 feet, in two seconds 6 feet, in three seconds 
 9 feet, &c. &c. ; hence, whilst the time varies through 1, 2, 3, 
 4, &c. seconds, the space varies through 3, 6, 9, 12, &c. feet ; 
 but the numbers 3, 6, 9, &c. are respectively in the same ratio 
 with the numbers 1, 2, 3, &c. When a body moves uniformly, 
 therefore, " the space varies directly a k s the time." 
 
 ii. If the relation between A and B be such, that whilst 
 A by increasing is changed to , and B by decreasing is 
 
 changed to b, in such manner, that A : a : : 5 : : 7 (or) : : b : B 
 
 throughout the whole period of their variation, then A is said 
 to vary inversely as B. 
 
 EXAM. The area of a triangle is equal to half the rectangle 
 contained by its base and perpendicular altitude; if, therefore, 
 the form of the triangle be changed whilst its area remains 
 the same, it is evident that as its altitude increases its base 
 must decrease. Let A and- B represent its altitude and base at 
 any one period of its variation, and a and I its altitude and 
 
 AxB axb 
 base at any other period, then o~ r=z ~~o~~ or ^x$ = ax&, 
 
 .'. (by TH. 2. Prop n .) A : a :: b : B : : 5 : -,, i.e. " the alti- 
 
 " tude of a triangle whose area is given varies inversely as its 
 " base, and vice versa." 
 
 in. If there be three variable quantities A, B, C, whose re- 
 lation to each other is such, that whilst B is changed to I, and 
 C to c, A is changed in the compound ratio of the change of B 
 and C; i. e. if A : a in the ratio compounded of the ratios of 
 B : I and C : c, or (Art. 99, I.) A: a :: BC : be, then A is 
 said to vary as B and C conjointly. 
 
 EXAM. Let A represent the area, B the base, and C the 
 perpendicular altitude of a triangle ; and when these are 
 changed, let a represent the area, b the base, and c the 
 
 altitude 
 
VARIATION. 
 
 TO / 
 
 altitude at any period of their variation ; then A = c 
 
 lc BC be 
 
 and a = j .'.A: a : : r- : :: BC : be, or "the area of a 
 
 " triangle varies as its base and perpendicular altitude con- 
 "jointly." 
 
 iv. If the relation between the three quantities A, B, C be 
 such, that when A is changed to a, B to b, and C to c, B : b 
 
 in the ratio compounded of the ratios of A: a and p, : -, or 
 
 A a 
 (Art. 99, 1.) B : b :: ~ : -> then B is said to vary directly as A, 
 
 and inversely as C. 
 
 EXAM. Let A, B, C, a, &, c represent the same quantities 
 
 BC QA 
 as in the last Example, then since A=- ~~> B= -~ > and since 
 
 be <2a QA Qa A a , 
 
 a > # = * Hence Bib:: -77-: :: 7,1-9 i. e. "the base 
 2 c u c C c 
 
 " will vary as the area directly, and as the perpendicular 
 " altitude inversely " 
 
 103. These several relations of variable quantities are often 
 more briefly expressed by placing the mark oc between them ; 
 thus 
 A : a : : B : b, or "A varies as 5," is expressed by . . A<x B. 
 
 A : a : : T> : > or "A varies inversely as 5," by .... ^oc-jv 
 
 Aia-.'.BC: be, or "Ovaries as 5 and C conjointly," byAocBC. 
 A ator "B varies directly as ^4, and 
 
 This notation is made use of in the following Theorems. 
 
 TH. 1. "If one quantity varies as another, it will also vary 
 as any multiple , part, power, or root of the other." Thus, 
 let^ocB, then A:a::B:b; multiply the last ratio by m, 
 then (Art. 98, 1.) A:a::mB:mb, .*. (Art. i02,I.)^ocmJ5 ? 
 
 where 
 
VARI4TION. 127 
 
 where m may be any number either integral or fractional. 
 Again, since A:a::B:l, (by TH. 1 1 of Proportion,) A n : a n 
 :: B n : b n ; ,:A n( ^B n , where n may be any number whatever, 
 
 integral m fractional. 
 
 TH. 2. " If one quantity varies as another, and each of 
 " them be multiplied or divided by any quantity variable or 
 " invariable, then will the products or quotients, thus 'arising, 
 "vary as each other." Thus, let Ac B, then^f: a:: B: b; 
 let m be an invariable quantity, an.d multiply all the terms of 
 the proportion by it, then mA:ma::mB:mb, .'. mA c m B. 
 Let C be a variable quantity, then we have 
 A:a::Bib\ rAC:ac::BC:bc,orAC<x;BC; 
 
 - V- b / P TH ; 12 | and 
 
 f of Prop". \AaBb A B 
 C:o::C:c) ( g! - :: g : -, or - Q oc g - 
 
 COR. 1. From hence it follows, that " if one quantity varies 
 " as two others jointly, then either of those quantities varies 
 " as the first directly and the other inversely." Thus let 
 
 A 
 
 Aoc B C, then, dividing each by C,Boc ~ or "as A directly 
 
 v A 
 
 and C inversely;" divide by B, then Coc -g or "as A directly 
 
 and B inversely." 
 
 COR. 2. If the product of two quantities be invariable, then 
 those quantities vary inversely as each other." For let 
 
 in 1 m 
 
 AxB m, then A=-r> which varies as5> and jB=~; which 
 
 ri Jj jfl 
 
 varies as -j? m being a constant quantity. 
 
 TH. 3. "If one quantity varies as a second, and the second 
 as a third, then will the first quantity vary as the third." For 
 let^oc B, then^ta ::B : 6; andlet .Boc C, then B : b::C:c- 9 
 .'. by TH. 5. of Prop n . A : a : : C: c', hence AOC C. 
 
 TH. 4. "If any two quantities vary as a third, then will 
 
 " their 
 
 
128 VARIATION. 
 
 " their sum or difference or the square root of their product 
 " vary as the third." Thus let Aac Cand Boc C, then, by 
 Th. 3, Ac JB; /. A : a : : B : b or A : B : : a : I ; arid, by 
 Th. 6 of Proportion, AB : B :: ab : b or AB : ab 
 :: B : b; but since Boc C, B : b :: C: c, hence AB :ab 
 :: C : c, or ABoz C. 
 Again, since 
 
 A', a:: Crc^byTh. 12 of Prop . AB : ab :: <7 : c\ 
 and: b:: C:c$and,Th. 11 of Prop". JAB: Jab:: C:c. 
 Hence 
 
 TH.'5. " If the square of the sum of two quantities varies 
 " as the square of their difference, then the sum of their 
 tf squares varies as their product." For let (A-\- JB) 9 oc (AB)*, 
 then (A + BY : (a + *) ::(A- )' : (a - b)*, 
 
 or (A+B)* : (A-Bf :: (a-b)* : (a + b}\ 
 By Expansion, andl 2^ a -f 25 a : 4>AB :: 2a a +26 a : 4ab, 
 byTH.8ofProp n .J orA' + B*: 2AB :: a* + b* : 2ab; 
 .'.A* + B*: a* + b*::2AB : 2ab :: AB : ab. 
 Hence 
 
 TH. 6. " If there be two sets of quantities, A, J5, C, ), 
 " &c. and Pi Q, R, S, &c. which vary as each other re- 
 " spectively, viz. A&z P, BQZ g, then will the products of 
 " those quantities vary as each other." For, let a, b, c, &c. 
 p, q, r, &e. be corresponding values of A, B, C, &c. P, Q, R, &c. 
 then, since A cc P, A : a : : P : p 
 
 . . . BxQ,B:b::Q:g 
 . . . C oc R, C : c : : R : r 
 &c. &c. 
 
 .-. By THEOR. 12 of Proportion, 
 
 ABC&c.: abc&c. :: PQR&c.:pqr 
 Hence ABC &c. 
 
 TH. 7. "If any quantity A depends upon a set of quanti- 
 " ties, P, Q, R, S, in such a manner, that if Q, R, S, are con- 
 
 " stant. 
 
VARIATION. 
 
 129 
 
 slant, AocP; ifP,R t S are constant, ^ocg;&c. &c. 
 
 " then if they all vary, ^ will vary as their product.'' 
 
 For let ^ be changed 
 
 to x, by the variation of P to p, the rest being constant, 
 
 from xtoy of Q to 9, 
 
 from y to z of J? to r, 
 
 from z to a of S to s 
 
 then, when all vary, we have A : a; :: P : p\ Hence, by com- 
 
 x : y :; Q : <? f position of ratios, 
 y:z s;^:f M^i^&^S 
 z : a:: : 5 ) : p^rs or ^4 ex: 
 
 PQRS-, and the Theorem would evidently be true, whatever 
 
 be the number of quantities P,g,#,S, &c. 
 
 TH. 8. " If one quantity varies as another, it is equal to 
 " that quantity multiplied into some constant quantity; and 
 <f the value of this constant quantity will be known, if the 
 " actual relation between the two variable quantities at some 
 " given period of their increase or decrease be known/' For 
 let AocBy then A:a::B:b or AiBnail, i.e. the ratio 
 of A : B is always the same through the whole period of 
 
 their variation; let this ratio be that of m-. 1, then AiB 
 
 ji 
 
 : : m : 1, and A=mB, or m = -5' If therefore the correspond- 
 ing values of A and B at any period of their variation be known, 
 the value of m will be known. 
 
 EXAM. The space described by a body descending perpen- 
 dicularly near the surface of the Earth varies as the square of 
 the time; let the space =S, the corresponding time=T 9 then 
 by this Theorem S^mT*', now it is known by experiment, 
 that a body falls through a space of about 1 6 feet in the first 
 second of its fall; hence, when S= 16, T= 1 , .\m=l6, and 
 the general relation between the space and time of a body thus 
 falling is S=l6T*. 
 
 COR. Since ^=wz, it follows, " that if one quantity varies 
 
 " as another, the fraction arising from dividing the one 
 " quantity by the other, is a constant quantity." 
 
 S 
 
130 
 
 CHAP. VII. 
 
 ON ARITHMETICAL AND GEOMETRICAL 
 PROGRESSION. 
 
 XXXI. 
 
 DEFINITIONS. 
 
 104. If a series of quantities increase or decrease by the 
 continual addition or subtraction of the same quantity, then 
 those quantities are said to be in Arithmetical Progression. 
 Thus the numbers 1, 2, 3, 4, 5, 6, &c. (which increase by the 
 addition of 1 to each successive term), and the numbers 21,19, 
 1 7, 15, 13, 1 1, &cc. (which decrease by the subtraction of 2 from 
 each successive term), are in arithmetical progression. 
 
 105. In general, if a represents thejirst term of any arith- 
 metical progression, and b the common difference, then may the 
 series itself be expressed by a, a + b, a + 2 b } a + 3 b, a + 4 b, &c. 
 which will evidently be an increasing or a decreasing one, 
 according as b is positive or negative. In the foregoing series, 
 the coefficient of b in the second term is one; in the third term 
 is two ; in the fourth is three, &c. ; i. e. the coefficient of b in 
 any term is always less by unity than the number which denotes 
 the place of that term in the series. Hence, if the number of 
 terms in the series be denoted by (n), the nth or last term in 
 the progression will be a + (n l)&. 
 
 106. If a series of quantities increase or decrease by the con- 
 tinual multiplication or division of the same quantity, then those 
 quantities are said to be in Geometrical Progression. Thus 
 the numbers 1,2,4,8,16, &c. (which increase by continual 
 
 multiplication by 2), and the numbers !,->-> rr> &c. (which 
 
 decrease by continued division by 3, or multiplication by -J> 
 are in Geometrical Progression. 
 
ARITHMETICAL PROGRESSION. 131 
 
 1 07. In general, if a represents thejirst term of such a series, 
 and r the common multiple or ratio, then may the series itself 
 be represented by a, ar, ar\ ar 3 , ar 4 , &c. which will evidently 
 be an increasing or decreasing series, according as r is a whole 
 number or a proper fraction. In the foregoing series, the index 
 o/Y in any term is less by unity than the number which denotes 
 the place of that term in the series. Hence, if the number of 
 terms in the series be denoted by (n), the last term will be ar*" 1 . 
 
 XXXII. 
 
 On Arithmetical Progression. 
 
 1 08. Let S be the sum of the .series a, a + 1, a + 2 b t a + 3 b, 
 &c.; then 
 
 &c. ... + + (-)+ a 'K n - 
 
 where the /ow/er series is the same as the upper one, except 
 that the order of the terms is inverted. 
 
 Add the two series together, and we have, 
 
 l)6) + &c. to n terms=25, 
 
 109. From the equation (2a + (w l)6j. 72=2 S, it appears 
 that if any three of the four quantities a, b, n t S are given, the 
 fourth may be found. For we have, 
 
 i. By Art. 108 ............. S=(Qa + (n- 
 
 ii. By actual multiplication, <Zan + ln*-*-bn = 2 S; 
 
 and a= 
 
 ( a ) Since the sum of any two terms=:(a+a-i-( w l)6j=sum of first 
 
 and last term, and since SzzfQa-^-tn l)&-, it appears that the sum of 
 
 the series is equal to the sum of the first and last terms (or of any two terms 
 equally distant from the first and last terms\ multiplied into half the number 
 of terms. 
 
ARITHMETICAL PROGRESSION. 
 
 in 
 
 Again, Irf ln^ 
 or (w 72)6 = 
 
 iv. To find 72, we have, 2 9 
 by transposition, ) 
 
 or &72 a + (2a 6)72=2$; 
 2^-6 2S 
 
 W+ ~T~~- 7Z= T 
 
 Solve this quadratic equation, and it gives the value of n. 
 
 EXAMPLE 1. 
 
 Find the sum of the series 1, 3, 5, 7, 9, 1 1, &c. continued to 
 1 20 terms. 
 Herea=l, 1 -, 
 
 = (2 + 1 19 X 2)60 = 240 X 60= 14400. 
 
 Ex.2. 
 
 Find the sum of the series 15, 11, 7, 3, I, 5, &c. to 
 20 terms. 
 
 Here a = 15,J c 
 ' 
 
 = (30-19X4)X 10. 
 
 = (30-76) X 10=-46x 10= 460. 
 
 Ex. 3. 
 
 12 45 7 
 Find the sum of 150 terms of the series - -> 1, -> -> 2, -, &c. 
 
 a o o o A 
 
 Here a = |, 
 
 1 
 
 6 = ~, 
 
 150 
 
 2 149 
 
 151 
 
 = ^-+j75=-~X 75 = 3775. 
 
ARITHMETICAL PROGRESSION. 
 
 Ex.4. 
 
 The sum of an arithmetic series is 1240, common difference 
 4, and number of terms 20. What is theirs* term? 
 Here S= 1240, J. gS-fln' + fl^ 
 
 = -4, 
 
 __ I 2480+1600-80 4000 
 
 ;z = 20; I ss - - = - =100 
 40 40 
 
 Hence the series is 100, 96, 92, 88, &c. 
 
 Ex. 5. 
 
 The sum of an arithmetic series is 1455, the first term 5, and 
 the number of terms 30. What is the common difference ? 
 Here S= 1455,} QS-Qan 
 
 G = 5, Y n*-n 
 
 n = 30; 5 2910-300 2610 
 
 900 30 = " 870 ~" 3 * 
 Hence the series is 5, 8, 1 \, 14, &c. 
 
 Ex.6. 
 
 The sum of an arithmetic series is 567, I\\Q first term 7, the 
 common difference 2. What are the number of terms? 
 HereS=567,} 2a-b 2S 
 
 fl=7 , [ 7z +-r~* w== T 
 
 ^ = 2; ) is rc'-f f)72 =567; 
 
 andw a -f 6^ + 9 = 567 -f 9 = 576;' 
 .-. n -f 3 =24,or;z = 21. 
 
 Ex. 7. 
 
 How much ground does a person pass over in gathering up 
 200 stones placed in a straight line, at intervals of 2 feet from 
 each other; supposing that he brings each stone singly to a 
 basket standing at the distance of 20 yards from the first 
 stone, and that he starts from the spot where the basket stands ? 
 
 It is evident that the space passed over by this person will 
 be twice the sum of an arithmetic series, whose first term is 
 20 yards (i.e. 60 feet), common difference 2 feet, and number 
 
 of terms 200. 
 
 Here 
 
134 ARITHMETICAL PROGRESSION. 
 
 Here a = 60, J / x n 
 
 = ( 120 + 398)lOO. 
 
 = 5 18 X 100 = 51800 feet. 
 
 feet, miles, furlongs, feet, 
 
 Hence the distance required =103600= 19 . 4 . 640. 
 
 Ex. 8. 
 
 A traveller bound to a place at the distance of 198 miles, 
 goes 30 miles the first day, 28 the second, 26 the third, and 
 so on. In how many days will he arrive at his journey's end ? 
 Here is given a = 30, 
 
 to find the number of terms. 
 
 _ ^ 2X198 
 
 Now n + ji=-y .'. n 31n= - = 198, 
 
 961 961 169 
 
 * 3172 + = 198 + = ' 
 
 31 13 31 13 
 ; Hence TZ = , and 72 = = 22 or 9. 
 
 To explain the apparent difficulty arising from the two po- 
 sitive values of w, which give us two different periods of the 
 traveller's arrival at his journey's end, we must observe, that 
 if the proposed series, 30, 28, 26, &c. be carried to 22 terms, 
 the 16 th term will be nothing, and the remaining six negative; 
 by which is indicated the rest of the traveller on the 16 th day, 
 and his return in the opposite direction during the six days 
 following; and this will bring him again, at the end of the 22 d 
 day, to the same point at which he was at the end of the 9 th , 
 viz. 198 miles from the place whence he set out. 
 
 Ex. 9. 
 
 There are a certain number of quantities in arithmetic pro- 
 gression, whose common difference is 2, and whose sum is equal 
 to eight times their number ; moreover, if 1 3 be added to the 
 second term, and this sum be divided by the number of terms, 
 
 the 
 
ARITHMETICAL PROGRESSION. 135 
 
 the quotient will l>e equal to thejirs/ term. What are the 
 numbers ? 
 
 Let the first term~x, "I then the second term will be ,r+ 2, 
 and the number ofierms=y; / . . . the last term ..... # + (#- 1) x 2. 
 
 In the expression (2 a + (n 1 )#;; SUDst i tute #fc> r a, 2 for, 
 and y for n, and it becomes tzx + ty l)2Y-(= xy + if y) 9 
 
 for the sum of the series. 
 
 By the question, xy + y*-y = 8y, or y = Q x y 
 
 , # + 2+13 
 and y=-. 
 
 # + 2+13 
 
 Hence, . = #, or # 2 8# = 15 ; 
 y jc 
 
 .-. #* 8#+l6=l6-15=l, 
 
 and x 4= 1 j * #=5 or 3, 
 y = Qx =4 or 6. 
 
 From which it appears that there are two sets of numbers 
 which will answer the conditions required; viz. 5, 7, 9, 11, or 
 3, 5, 7, 9,11, 13. 
 
 Ex. 10. Find the sum of 25 terms of the series, 
 
 2, 5, 8, 11, 14, &c. ANSWER, 950. 
 
 Ex. 11. Find the sum of 36 terms of the series, 
 
 40, 38, 36, 34, &c. ANSW. 180. 
 
 Ex. 12. Find the sum of 32 terms of the series, 
 
 1, Ij, 2, 2^, 3, &c. ANSW. 280. 
 
 Ex. 13. The sum of an arithmetic series is 950, the common 
 difference 3, and number of terms 25. What is the first term ? 
 
 ANSW. 2. 
 
 Ex. 14. The sum of an arithmetic series is 165, the Jlrst 
 term 3, and the number of terms 10. What is the common 
 difference? ANSW. 3. 
 
136 ARITHMETICAL PROGRESSION. 
 
 Ex. 15. The sum of an arithmetic series is 440, first term 3, 
 and common difference 2. What is the number of terms ? 
 
 ANSW. 20. 
 
 Ex. 16. The sum of an arithmetic series is 54, first term 
 14, and common difference -2. What is the number of 
 terms ? ANSW. 9, or 6. 
 
 Ex. 17. A person bought 47 sheep, and gave 1 shilling for 
 the first sheep, 3 for the second t 5 for the third, and so on. 
 What did all the sheep cost him ? ANSW. .110. Qs. 
 
 Ex. 1 S. A person began the year by giving away a. farthing 
 the first day, a halfpenny the second, three far things the third, 
 and so on. What money had he disposed of in charity at the 
 end of the year ? ANSW. .69. 1 Is. 6fd. 
 
 Ex. 19. A travels uniformly at the rate of 6 miles an hour, 
 and sets off upon his journey 3 hours and 20 minutes before jB ; 
 B follows him at the rate of 5 miles the first hour, 6 the second, 
 1 the third, and so on. In how many hours will B overtake A? 
 
 ANSW. In 8 hours. 
 
 Ex. 20. There are a certain number of quantities in arith- 
 metic progression, whose first term is 2, and whose sum is equal 
 to 8 times their number ; if 7 be added to the third term, and 
 that sum be divided by the number of terms, the quotient will 
 be equal to the common difference. What are the numbers ? 
 
 ANSW. 2, 5, 8, 11, 14. 
 
 XXXIII. 
 
 On Geometrical Progression. 
 
 110. Let S be the sum of the series a, ar, ar*, ar 3 , &c. 
 (Art. 107), then 
 
 a + ar-f ar' + ar'-f &c. . . . ar n -* + ar*" 1 =S. 
 Multiply the equation by r, and it becomes 
 
 ar-f ar*-far 3 -f &c. . . . ar"- 8 + ar*-' +ar*=rS. 
 
 Subtract 
 
GEOMETRICAL PROGRESSION. 137 
 
 Subtract the upper equation from the lower, and we have, 
 ar*a=rS S, or (r l)S = ar n a; 
 
 and therefore, iS= 
 
 If r is a proper fraction, then r and its powers are less than 1. 
 For the convenience of calculation, therefore, it is better in this 
 
 case to transform the equation into = ? by multiplying 
 the numerator and denominator of the fraction - by 1 . 
 111. If I be the last terrjj of a series of this kind, then 
 l=ar r -\ .*. rl=ar n ; hence S= (~~^) =7~* From 
 this equation, therefore, if any three of the four quantities 
 S) a } r,l, be given, the fourth may be found. For S= 5 
 
 S-a ( r _i)S + 
 
 'c_l> an d 1= ~ * The value 
 
 of n cannot be found from the equation S= ^ except by 
 means of Logarithms, as will be shewn in a future chapter. 
 
 EXAMPLE 1. 
 
 Find the sum of the series 1, 3, 9, 27, &c. to 12 terms. 
 Herea=l, ^ n ar n a 1 x 3 12 1 
 
 3 - 1 
 81 s 1 
 2 
 
 = ^ =265720. 
 
 2 
 Ex. 2. 
 
 Find the sum of ten terms of the series 1 -f- - + ~-\ 
 
 32 
 
138 GEOMETRICAL PROGRESSION. 
 
 10 l 1024 
 
 /2\ 
 N W U = 3 ry = 
 
 /2v 10 _ 1024 58025 
 
 '^""Vaj =1 "" 59049 ""59049' 
 
 3X58025 174075 
 59049 = 59049 ' 
 
 Ex. 3. Find the sum of 1, 2, 4, 8, 16, &c. to 14 terms. 
 
 ANSWER, 16383. 
 
 1 1 1 
 Ex. 4 ......... *' 3* a' 27 3 to 8 terms - 
 
 3280 
 
 XXXIV. 
 
 On the method of finding any number of Arithmetic or 
 Geometric Means between two numbers. 
 
 112. Let I be the last term of an arithmetic series, whose 
 first term is (a), common difference (b), and number of terms 
 
 I a 
 (w); then Z=a+(7z l)6; .*. (n l) b = la f or #=^ [' 
 
 Now the number of intermediate terms between the first and 
 the last 'is n 2j let n 2 = m, then TZ l=m + l. Hence 
 
 Z a 
 Z> = T~Tj which gives the following Rule for finding any num- 
 
 ler of arithmetic means between two numbers ; " Divide the 
 " difference of the two numbers by the given number of means 
 " increased by unity, and the quotient will be the common 
 " difference" Having the common difference, the means 
 themselves will be known. 
 
 113. Let I be the last term of a geometric series, then I=ar n ~ l 9 
 
 l *-yT 
 
 and r n ~ l =~, .'. r=\/ -. The number of intermediate terms 
 
 as before is n 2; let TZ 2 = m, then n *ls=w+l, and 
 
 m+1 /!T 
 r =. Y/ -, which gives the following rule for finding any number 
 
 
ARITHMETIC AND GEOMETRIC MEANS. 139 
 
 of geometric means between two numbers; viz. "Divide 
 " one number ly the other, and take that root of the quotient 
 " which is denoted by m + I ; the result will be the common 
 " ratio." Having the common ratio, the means are found by 
 common multiplication. 
 
 EXAMPLE I. 
 Find six arithmetic means between 1 and 43. 
 
 Here /=43. J 
 
 ' l-a 43-1 42 
 
 By adding this common difference continually to the lesser 
 number (l), we have 7, 13, 19* 25, 31, 37, for the six means 
 required. 
 
 Ex.2. 
 
 Find three geometric means between 2 and 32. 
 Herea = 2, 1 
 
 1=39,5 /.r= 
 771 = 3 ; 3 
 
 and the means required are, 4, 8, 16. 
 
 Ex. 3. 
 
 16 
 
 Find two geometric means between ~ and 2. 
 
 16 +r 3 
 
 /=2, 8 4 
 
 _ I .*. the two means are ~ and - 
 771=2 5j 9 3 
 
 Ex. 4. Find seven arithmetic means between 3 and 59. 
 
 ANSWER, 10, 17, 24, 31, 38, 45, 52. 
 Ex. 5. Find eight arithmetic means between 4 and 67. 
 
 Ex. 6. Find nine arithmetic means between 9 and 109. 
 
 Ex. 7. Find two geometric means between 4 and 256. 
 
 ANSW. 16 and 64. 
 
 Ex. 8. Find three geometric means between - and 9. 
 
 y 
 
 ANSW. -, l, 3. 
 
140 EQUATIONS RELATING TO NUMBERS 
 
 114. Let a, a -f I, a -f- 2 1 be three quantities in arithmetic 
 progression, then the sum of the first and last =2a + 2& = 
 2(a + l) ; .*. a + b=:halftl\e sum of the first and last ; hence 
 " an arithmetic mean between any two quantities is found, by 
 " taking half their sum." Again, let a, ar, ar* be any three 
 quantities in geometric progression, then the product of the first 
 and last =aV = the square of the mean term, from which it 
 appears that " a geometric mean between any two quantities is 
 found by taking the square root of their product." From 
 hence also it appears, that an arithmetic mean between any two 
 numbers is greater than a geometric mean; for let the two 
 numbers be a + x and a x, then the arithmetic mean is a and 
 the geometric is Va a #% which is evidently less than a. 
 
 XXXV. 
 
 On the solution of Equations relating to Numbers in 
 Arithmetical or Geometrical Progression. 
 
 115. As the several terms of any arithmetic or geometric 
 series may be expressed by means of hvo unknown quantities^ 
 it is not difficult to find the value of quantities of this kind, 
 which shall bear such relations to each other as may be deter- 
 midedby *M/O equations-^ of which the following are Examples. 
 
 EXAMPLE 1. 
 
 Find four numbers in arithmetical progression, such, that 
 their sum shall be 56, and the sum of their squares 864. 
 
 Let # = the second of these four numbers, 
 and y = their common difference. 
 
 Then 
 
 ( a ) It may be proper here to observe, that quantities which are in geometric 
 progression are also in continued proportion; for a : ar :: ar : ar* :: ar 2 : ar 3 :: 
 &c. The differences of quantities in geometric progression are also in con- 
 tinued proportion ; for the successive differences of the terms of the series 
 a, ar, ar*, ar^ar 4 , &c. arear a, ar 2 ar, ar 3 ar 2 , &c. or ar a,(ar a)r, 
 (ar a)r 2 ,&c. which is a geometric progression whose first term is ar a. 
 and common ratio r- 
 
 ien 
 
 f* 
 
IN ARITHMETIC AND GEOMETRIC PROGRESSION. 141 
 
 Then the four numbers may be represented by xy, x, 
 
 Hence, by the question, 
 (x-y) +x + (x + y)' + (x + Sy) -4x + <2y = 56, 
 
 From 1st equation, 
 Square this equation, then 4 x 9 + 4 x y -f- y* = 7 84 (A) , 
 ' ' -y = 864(B). 
 
 
 Subtract (A] from (5), and we have 5?/*=80, 
 or y*= 16, and y = 4; 
 28 - 24 
 
 Hence 8, 12, 16, 20 are the four numbers required. 
 
 JEx.2. 
 ' The sum of three numbers in arithmetic progression is 9, and 
 the sum of their cubes is 153. What are the numbers? 
 
 Let xy,x,x + y, be the numbers. 
 Then(jc y) +x + (x+y) =3x =9, 
 
 (x yf + x 3 + (x+ yY = 3 x 3 + 6 xy*= 153. 
 
 From 1st equation, x=- = 3; 
 
 .\bysulstitution,in 2d equation, 81 + I8y*= 153, 
 
 or 182/ a =153 81 = 72; 
 
 72 
 
 Hence, the numbers are 1, 3, 5. 
 
 Ex.3. 
 
 Find three numbers in geometric progression, such, that their 
 sum shall be equal to 7, and the sum of their squares to 21. 
 
 Let x, y, z, be the numbers. 
 
 Then, by the question, x +y +z = 7, 1st equation. ^ 
 And ff Q -fi/ a -f;z a =21, 2d equation. $ 
 
 By 
 
142 EQUATIONS RELATING TO NUMBERS 
 
 By Note( a ) Art. 1 14, x : y : : y : z; .-.y'=xz. 
 
 From 1st equation, x + z=7 y 
 Square this equation, and # 2 -f 2#2+z 4 =49 
 
 but <2xz = Qy 9 
 
 Subtract (B) from (A), then # 3 -f2; 2 = 49 14yy z 
 But, from second equation, x* + z 9 =21 ?/ a . 
 Hence, 49 14y y 2 = 21- ; - y 2 , 
 or 49 147/ = 21; 
 
 /. 14y = 49 21 = 28. 
 
 28 
 .'. 2/ = 
 
 Again, since a; + z = 7^=7 -2 = 5, 
 we have #* + 2.r 2-1-2* =25; 
 
 but 4xz =16, for xz y*-, 
 .'. by subtraction, a? a 2 0:2; + ^ 2 = 25 16 = 9, 
 and x z=3. 
 
 Hence, x + z =5, ) .'. 2x=8, or a;=4 
 a; =3; 3 2s; = 2, or 2=1, 
 and the three numbers are 1, 2, 4. 
 
 Ex. 4. 
 
 The sum of four numbers in geometric progression is 30, and 
 
 4. 
 
 the last term divided by the sum of the mean terms is - > 
 
 3 
 
 What are the numbers ? 
 
 Let#=nrst term, > then the numbers themselves will 
 
 y = the common ratio; 3 be a?, xy, xy*, xy 3 . 
 Hence, by the question, x 4- xy -f xy* + o;^ 3 = 30, 1 st equation,^ 
 
 xy 3 4 }, 
 
 and * = ' 2d e q uation - 
 
 xy+xy* = i' e q uaon - 
 From 1st? 30 
 
 = 30 Or *= 
 
 From 2d ^ xy x y* 4 y* 4 
 equation, Ja;y X (1 +y) ^s' (T T + y == 3^ '" 
 
 By 
 
IN ARITHMETIC AND GEOMETRIC PROGRESSION. 143 
 
 By reaction of >, 
 equation (B), } * 
 
 4__4 4_16 
 2 4. 6__ 
 
 30 30 
 
 _ = _ 
 
 Hence from equation , 1+2+4 + 8 
 
 The four numbers are therefore 2, 4, 8, 16. 
 
 Ex.5. 
 
 There are three numbers in geometric progression, whose 
 product is 64, and sum of their cubes 584; What are the 
 numbers? 
 
 Let the numbers be x, xy, xy*. 
 
 Then, by the question, x X xy x 2?y,ora?y = 64, 1st equation. 
 And a; 3 -f x 3 y 3 + x 3 y 6 = 5 84, 2d equation. 
 
 64 4096 
 
 r rom 1 st equation, y = ^y> and ?/ = r"* 
 
 By substitution, in ? 4096 
 
 
 
 adequation, 5 *' ' x 3 "' 
 
 Hence, X G + 64x 3 + 4096 = 584*% 
 
 or a; 6 520 x 3 = 4096. 
 
 Solve this equation by ( 
 
 J = 8; or x=2. 
 
 the Rule in Art. 84. \ 
 
 64 64 
 
 And the three numbers are 2, 4, 8. 
 
 Ex. 6. The sum of three numbers in arithmetic progression 
 is 15; and the sum of the squares of the two extremes is 58. 
 What are the numbers? ANSWER, 3, 5, 7. 
 
 Ex. 7 There are four numbers in arithmetic progression ; 
 the sum of the two extremes is 8, and the product of the 
 means is 15. What are the numbers? ANSW. 1, 3, 5, 7. 
 
 
144 INFINITE SERIES OF FRACTIONS AND 
 
 Ex. 8. There are four numbers in arithmetic progression; 
 the sum of the squares of the two means is 2, and the sum of 
 the squares of the two extremes is 18. What are the numbers? 
 
 ANSW. 3, 1, 1, 3. 
 
 Ex. 9. There are three numbers in geometric progression, 
 whose sum is 21, and sum of their squares 189. What are 
 the numbers? ANWS. 3, 6, 12. 
 
 Ex. 10. There are three numbers in geometric progression ; 
 the sum of theirs/ and last is 52, and the square of the mean 
 is 100. What are the numbers? ANSW. 2,10,50. 
 
 Ex. 11. There are three numbers in geometric progression, 
 whose sum is 3 1, and the sum of thejirst and last is 26. What 
 are the numbers ? ANSW. 1, 5, 25. (a> 
 
 XXXVI 
 
 On the Summation of an infinite Series of Fractions in 
 Geometric Progression; and on the method of finding 
 the value of Circulating Decimals. 
 
 \ 16, The general expression for the sum of a geometric series 
 
 a ar n 
 whose common ratio (r) is a fraction, is (Art. 1 10) S= -j"^ 
 
 Suppose now n to increase indefinitely, then r n (r being a 
 proper fraction) will decrease indefinitely^; therefore ar* will 
 decrease indefinitely with respect to a, or a will be the limit of 
 
 a aar n 
 
 aar n , and __ the limit of __ or S; and consequently 
 
 , will express the value of the series when the number of 
 
 its terms is supposed to be indefinitely increased, or (as it is 
 commonly called) the sum of the series ad infinitum. 
 
 ( a ) Some curious Theorems relating to numbers in Geometrical Progression 
 will be found in "Siemens d'Algebre, par L'Huilier," Vol. II. p. 177.. .208. 
 Ed. 1812. A great variety of questions, both in Arithmetical and Geometrical 
 Progression, will also be found in Eland's "Algebraical Problems." 
 
 (b) Letr = ^, for instance; then r*= ^^ 1*=-^, &c. 
 
 from which it is evident, that if there be no limit to the increase of tl 
 index , there will be none to the decrease of the fraction r n . 
 
CIRCULATING DECIMALS IN GEOMETRIC PROGRESSION. 145 
 
 EXAMPLE 1. 
 Find the sum of the series 1 + o "J~ 2 ""*" Q' & c ' ac ^ wfinitum. 
 
 Here a= 1, 
 
 Ex.2. 
 
 5 ' QS 4 "" 
 
 Find the value of - + +7^1 ~*~ & c > afi ^ infinitum. 
 
 1 5 1~4" 
 
 Ex.3. 
 Find the value of ~ + ~ + g 4- - + i + &c. ad wfinltum. 
 
 Ex. 4. Find the value of 1 +3+5 + ^+ ^ c ' adwfinitum. 
 
 ANSW. - 
 Ex. 5 ........... -+ l + + + & c ' adinfinitum. 
 
 
 ANSW. 4 . 
 
 117. These operations furnish us with an expeditious method 
 of finding the value of circulating decimals, the numbers com- 
 posing which are geometric progressions, whose common ratios 
 
 i 
 
 are 77? V7^ ? i7m^> & c< according to the number of factors 
 
 IU 1UU JLUUU 
 
 contained in the repeating decimal. 
 
 U 
 
146 CIRCULATING DECIMALS IN GEOMETRIC PROGRESSION. 
 
 EXAMPLE 1. 
 
 Find the value of the circulating decimal .33333, &c. 
 This decimal is represented by the geometric series 
 333 3 
 
 10 + 100 + 1000 + SC f rst term 1S 10' common 
 
 ratio-' 
 
 Hencea=^ 3 3 
 
 r =T5' 
 
 1""9""3 
 
 Ex.2. 
 
 Find the value of .32323232, &c. ad wjinitum. 
 
 32 ) 32 
 
 Herea= , I ^^ 32 32 
 
 JL-1 " l ~~ r 1 "100 1~ 99 
 
 ^ : 1 1 '"ioo 
 
 Ex.3. 
 
 Find the value of .713333, &c. ad wfinitum. 
 The series of fractions representing the value of this decimal 
 
 e + (geometric series) 
 
 3 
 
 Here a== Tooo > I 1000 
 
 1000-100 900~300 
 
 r =To ; I " 10 
 
 Hence the value of the decimal =( 
 _107 
 
 ""1^0* 
 
 Ex.4. 
 
 Find the value of .81343434, &c. ad iiifinitum. 
 34 34 
 
 lOOOO 34 34 
 
 ~~T = 1 0000 1 OO """ 99OO * 
 
ON THE REDUCTION OF SURDS. 147 
 
 . , 81 81 34 8053 
 
 And the value of the decimal + S = 
 
 Ex. 5. Find the value of .77777, &c. ad infinitum. 
 
 ANSWER, - 
 y 
 
 Ex. 6. Find the values of .232323 &c. ; .83333 &c. ; 
 .7141414 &c. ; and .Q56666 &c. ad infinitum. 
 
 23 5 707 , 287 
 
 ANSW - ; ~ ; d respectively. 
 
 CHAP. VIII. 
 
 ON SURDS. 
 
 SURD Quantities have already been defined in Art. 55. and 
 may be expressed either by the radical sign, or by their frac- 
 tional indices (as in Art. 66.) ; thus the square root of 2, 
 the cube root of 3, thewth root of a -f 1, the cube root of(a + #)% 
 &c. &c. may be expressed either by v/2, \/3, %/ a + b, \/(a+#) a , 
 &c. or by 2*, 3*, (a + b)\ (a+xf, &c. 
 
 The precise value of these quantities cannot be ascertained ; 
 it can only be expressed by means of decimals or series which 
 do not terminate 5 and in this sense they are called irrational, 
 to distinguish them from all other quantities whatever, integral 
 or fractional^ whose values are determinate, and which are 
 therefore denominated rational. 
 
 XXXVII. 
 
 ON THE REDUCTION OF SURDS. 
 
 CASE I. 
 
 1 IS. A RATIONAL quantity may be reduced to the form of 
 a surd, ly raising it to the power denoted ly the root of the 
 surd, and then annexing the radical sign. 
 
148 ON THE REDUCTION OF SUROS. 
 
 EXAMPLE 1. 
 
 Reduce 3 to the form of the square root) and it becomes 
 /s/3* or */9. 
 
 * Ex. 2. 
 
 2 3 /2 3 3/ 8 
 
 Reduce - ..... cube root, ..... V ~3* OT V 27* 
 
 Ex.3. 
 
 Reduce a -f b . . . square root, .... Y/(a + &) 2 . 
 
 Ex. 4. 
 Reduce 4&* . . . . cube root, ..... v/64& a . 
 
 CASE II. 
 
 119. Surds of different indices are reduced to equivalent 
 ones having the same radical sign, by bringing their fractional 
 indices to a common denominator, 
 
 Ex. 1. Reduce a* and a* to surds of the same radical sign. 
 
 The fractions - and - reduced to a common denominator, 
 
 3 j 2 . 
 are ~ and -5 > 
 6 o 
 
 .*. a* = a^= v/ a3 > ? which are surds with the same 
 and tt^ = a* = v/a 9 ; ^ radical sign . 
 
 Ex. 2. Reduce 3^ and 5% to surds of the same radical sign. 
 
 2 
 - 
 
 2 1 
 
 The fractions - and -> reduced to a common denominator, 
 
 4 ,3 
 are ^ and ~- 
 o o 
 
 Now 3'c-= v/3*= v/81 $ and 5^ = v/5 3 = 
 Ex. 3. Reduced and b* \ (Ays. v 
 
 Ex. 4. , . c* and d^ f wifh'u.e j . . 
 
 s same / 
 
 Ex. 5 3</2 & 2v/5 ( ^! j . . . 3< 
 
 Ex. 6 4^ and 15^ J ' . . . 
 
 * CASE 
 
ON THE REDUCTION OF SURDS. 149 
 
 CASE III. 
 
 1 20. Surds are reduced to their simplest form, by observing 
 whether the quantity under the radical sign contains, as a fac- 
 tor, a power corresponding to the given surd root, and then 
 extracting the root. 
 
 EXAMPLES. 
 Ex. 1. v/^=>A 
 Ex. 2. %/a m x=%/a 
 
 Ex.3, v/72 =v / 36x2=v/36x N/ 2 = 
 
 Ex. 4. ^/108=v / 27 X4=\/27 
 Ex. 5. 
 
 Ex. 6. Reduce y g 4 6c &^/ Q Sofa 
 Ex. 7 ..... / 
 
 Ex.8 ..... V56 and 
 Ex.9 ..... J/243and/96 
 
 The quantity without the radical sign is called the coefficient 
 of the surd; and it is evident, that this quantity may always 
 be put under the radical sign, by raising it to the power denoted 
 by the index of the surd. 
 
 Thus, 7 Cv/2 x= (by Case I.)v/7 a x 7 a X v/2 #. 
 
 = 
 Also, xv/2a #= 
 
 CASE IV. 
 
 121. If the quantity under the radical sign be a fraction, it 
 may be reduced to an integral form by the following process. 
 
 Multiply the numerator and denominator of the fraction 
 by such a quantity as will make the denominator a complete 
 power, corresponding to the root; and then proceed as in 
 CASE III. 
 
150 
 Ex 
 
 ON THE REDUCTION OF SURDS. 
 
 c /a" c /(fb 
 
 - L d x V b=d*V V 
 
 C l(f 
 
 =2 X V b' x ^ b - 
 
 c a 
 
 ac 
 
 3 /2 3 /2X7 
 
 Ex.2. |*V V ?*?*V?W 
 
 = jXyXv/14 = 2 ^/14. 
 
 Ex.3. 
 
 Reduced 
 
 V/ 16 V/ 
 
 ' 8 X 2 
 
 1 2 3 /2 
 
 5 x i x Vi' 
 
 3V 81~~3V 
 
 27 X 3" 
 
 
 = 
 
 - x \-y? 
 
 9 X V 3 
 
 i 
 
 
 : 
 
 2 3/1 
 
 9 X V 
 
 2 y 
 
 9 X Vs 
 
 ' X 3* 
 
 3 3 
 
 ~X18. 
 
 
 _. 
 
 -x-x^/ 
 
 18=- 
 
 Ex. 4. 
 
 
 
 
 /* 3/c' 
 /- and a^/ -] 
 
 to integral 
 Surds in 
 
 ANs.^yand^ 
 
 Ex.5. 
 /50 3 /3 
 
 1 their sim- ' 
 plest form. 
 
 5 
 
 
 /- and2V/ : 
 
 
 ' ' ' 21 V 
 
 6 and v 
 
ADDITION AND SUBTRACTION OF SURDS. 151 
 
 XXXVIII. 
 
 On the application of the Fundamental Rules of 
 Arithmetic to Surd Quantities. 
 
 1 22. On the Addition and Subtraction of Surds. 
 
 RULE. Reduce them to their simplest form ; and if the 
 surd part be the same in both, then their sum or difference 
 will be found by taking the sum or difference of their co- 
 efficients. 
 
 EXAMPLE 1. 
 
 Find the sum and difference of^/l6a*x and^/4a 9 x. 
 By ART. 120. </l6a*x=:4a</x, 
 
 and \/4a*x=2a\/x; 
 
 .'. the sum = 4 a\/x + 2 a\/x (4 a + 2 a) x \/x= 6 a ^/x. 
 the differ ence = 4 a\/x 2 a ^/x=(4a2 a] x \/x~2a\/x. 
 
 Ex.2. 
 
 Find the sum and difference of v/ 192 and v/24. 
 By ART. 120. v/192=\/64 x 3 =4^/3, 
 = 1 / 8x3 =2</3; 
 6^/3 or 
 
 Ex.3. 
 
 Find the sum and difference of I/ and \/ ^* 
 
 Q 1 
 
 The two fractions ~ and ^, reduced to a common deno- 
 
 48 27 
 
 minatory are 
 
 N W 
 
 / 48 / 16 x 3 4/3 
 
 > V lT2 = V ^TIT^ =9V 2* 
 
 27 / 
 
 T65 = V 
 
 9 X 3 3 
 
 i' 
 
 3 
 
 Hence x- -- -, or 
 
152 MULTIPLICATION AND DIVISION OF SURDS. 
 
 If the surd part be not the same in the quantities to be 
 added or subtracted from each other, it is evident that such 
 addition or subtraction can only be performed by placing the 
 signs + or between them. 
 
 Ex. 4. Add,v/27a 4 #and \/3a*x together . . ANSW. 4aV3#. 
 Ex. 5. . . . -v/128 and V72 ............... 14^2. 
 
 Ex. 6. . . .JS135 and ^/ 40 ............... 5^5. 
 
 Ex. 7. Subtract 3\ / from 4\/ 7 .......... N/TJT 
 
 V 07 V 5 15 
 
 Ex. 8 ...... v/108 from 9\/ 4 ........... 6</4. 
 
 123. On *Ae Multiplication and Division of Surds. 
 
 RULE. Reduce them, if necessary, to equivalent ones with 
 the same index, and then multiply or divide loth the rational 
 and irrational parts respectively. 
 
 EXAMPLE 1. 
 Multiply *J a by $ I, or a* by $. 
 
 The fractions - and - reduced to common denominators, 
 
 3 2 
 
 are ^> and ^ > 
 6 6 
 
 .'.a* = a*=ya 3 ; and i*=fl*= 
 Hence ^/a x ^ ^= x 6 / a 3 x </^ = ^/a 3 6 2 . 
 
 Ex.2. 
 
 Multiply 5 -y/ 5 by 3 A/ 8. 
 5-V/5 X 3 \/8= 15 v x 40= 15 v/4 X 10. 
 = 15 X 2 X X /10 = 30 V / 10. 
 
 Ex. 3. 
 
 Multiply 2^3 by 3 v/4. 
 By reduction, 2>y/3 = 2 X 3^=2 x v/3 3 =2 v/27, 
 
 and 3 v V4 = 3X4^=3X 
 Hence V3X 3^/4 = 2^/27 X 3 
 
MULTIPLICATION AND DIVISION OF SURDS. 153 
 
 Ex. 4. 
 
 Divide 2 \/bc by 3 \/ac. 
 
 and 3 v/ac=3 x(ac)*=3v/a 3 c 3 ; 
 
 2ij/6c 2 6 /W_2 */&_ 
 '* 3 v/ ac~*3 X V aV~~3 V a 3 e 
 
 Ex.5. 
 
 Divide 10^/108 by 5^4. 
 10v/108=10v/27 X 4= 10X3 X v/4 = 
 
 rO^/108 
 
 = 2X3 = 6. 
 
 Ex. 6. Multiply v/ 15 by \/lo . . . ANSWER, v/225000. 
 Ex. 7 ...... l -f/6 by 5^/18 ....... 4/4. 
 
 Ex. 8, Divide 10^27 by 2 \/3 ....... 15. 
 
 2 
 Ex. 9. .... 10v/108 by 5</84 ...... -</441. 
 
 124. On tfAe Involution and Evolution of Surds. 
 
 RULE. Raise the rational part to the power or root^required^ 
 and then multiply the fractional index of the surd part by the 
 index of that power or root. 
 
 Ex. I. 'Yhesquar 
 Ex. 2. Cule o 
 
 Ex. 3. 4^ pother of 2</2= 16 X 2^ x 4 = 16 X 2^= 16^/16 
 
 [=32^2. 
 Ex. 4. Square root of a*6* = a* x * i* x * = a^b* . 
 
 Ex. 5. Cube root of L/2=i x2* Xi =i X2 
 
 82 2 
 
 1 3 
 
 Ex.6. Cube-V3 ....... ANSWER, g 
 
 Ex. 7. Find fourth power of V6 ..... 
 
 X 
 
154 INVOLUTION AND EVOLUTION OF SURDS. 
 
 Ex. 8. Find square root of 9 f/3 ANSW. 3v/3. 
 
 ~ 
 
 O A 
 
 Ex. 9. ... fourth root of ~ </a* o 
 
 1 /*V v/* 6 
 
 Ex. 10.. .. fifth rootof-x(-) ..... -f 
 
 125. From the preceding rules we easily deduce the method 
 of converting fractions whose denominators are surd quantities, 
 into others whose denominators shall be rational. Thus, let 
 
 a 
 both the numerator and denominator of the fraction , be 
 
 multiplied by /x, and it becomes 5 and by multiplying 
 
 b 
 
 the numerator and denominator of the fraction s , by 
 
 a ~T~X 
 
 Or 
 
 in general, if both the numerator and denominator of a fraction 
 of the form / be multiplied by /x/tf*"" 1 , it becomes > 
 
 ^^/ ul/ 
 
 a fraction whose denominator is a rational quantity. 
 
 XXXIX. 
 
 On the method of finding Multipliers which shall render 
 Binomial Surd quantities Rational. 
 
 1 26. Compound surd quantities are such as consist of two or 
 more terms, some or all of which are irrational; and if a quan- 
 tity of this kind consist only of two terms, it is called a binomial 
 surd. The rule for finding a multiplier which shall render a 
 binomial surd quantity rational, is derived from observing the 
 quotient which arises from the actual division of the numerator 
 of the following fractions by the denominator. Thus. 
 
 x n y n 
 I. y =zx n ~ l +x n ~*y + x n ~Y + &c... -f y""" 1 to n terms, 
 
 whether n be even or odd. 
 
ON FINDING MULTIPLIERS &C. 155 
 
 x n -y n 
 n. =x fl - 1 -tf n -^+a^y-&c...-y n - 1 to n terms, 
 
 .when n is an even number. 
 
 x n +v n 
 m. -^^ = x n - l --x*-*y + x n ~y-&c...+y n - 1 to n terms, 
 
 when n is an odfJ aumber 00 . 
 127. Now let # a =a, y=&, then x\/a, y=f/b, and 
 
 these fractions severally become M/ fl _ /^> /a 4- V6 
 
 a-\-b 
 n/ a , nsjj', and by the application of the foregoing rules we 
 
 haveo^s^/a*- 1 ; a; n -^=-C/a n - 2 ; ^ n - 3 =-C/a n - 3 , &c.; also, 
 y*=3 b *> y*=3V; &^; hence, a?~*y = ?a*-*x?l = 
 Va n -*b', x n -y=Va n - 3 xyb*=t/a n - 3 b*;&c. By substitu- 
 ting these values of x n ~ l , x n ~*y, x"~ 9 y*, &c. in the several 
 quotients, we have 
 
 1 to 
 
 nn , 
 
 v/fl \/ 
 
 TZ terms; where n may be any whole number whatever 
 
 And 
 
 ab n ..C/a-' -C/a- a & +</a"- 3 ^ + &c ..... 
 v/a+\X^ 
 
 to TZ terms ; where the terms I and v/6 71 " 1 have the sign +, 
 when n is an odd number; and the sign , when n is an even 
 number. 
 
 128. Since the divisor multiplied by the quotient gives the 
 dividend, it appears from the foregoing operations that "if 
 
 " a binomial 
 
 For i. 
 
 &c. 
 
156 ON FINDING MULTIPLIERS TO RENDER 
 
 " a binomial surd of the form $/a \/b be multiplied by 
 "S/a n ~ 1 +t/a n - 2 b + ya n - 9 b 9 + &c....+Z/b n - 1 (n being any 
 " whole number whatever), the product will be a b, a rational 
 " quantity; and if a binomial surd of the form v/a-f-^X b be 
 multiplied by ^/a"- 1 -^a^b + ^a n -^*-&c.... ^ 6 1 , 
 " the product will be a + b or a b, according as the index n is 
 te an odd or an even number." The great use of this rule is, 
 (( to convert fractions having surd denominators, into others 
 " which shall have rational ones;" of which the following are 
 examples, 
 
 EXAMPLE 1. 
 
 Reduce - 7- and ; , TT to fractions having rational 
 a 
 
 denominators. 
 
 Since "the sum into the difference of two quantities gives 
 the difference of their squares" it is evident that these fractions 
 may be reduced to others having rational denominators, by 
 multiplying their numerators and denominators by a -f \/x and 
 v'S \/3 respectively, without the formal application of the 
 rule. 
 
 and (a-^)(a + V^)=a'-# re duced to 
 
 Again v/6(-/8- i/3) = v /48- -/18= (Art. 120.)4\/3-3i/2, 
 and 
 
 and the fraction is reduced to 
 
 Ex.2. 
 
 Reduce T/ r/r to a fraction with a rational denominator. 
 
 > v/2 
 
 To find the multiplier which shall make #3<X2 rational, 
 
 l ^ = 
 Now 
 
 ( a ) The number of terms of the general series to be taken, is always 
 equal to n; in the present instance, therefore, the number to be taken is 3; 
 and so in all other cases j recollecting that the last term is always &b* 1 . 
 
BINOMIAL SURD QUANTITIES RATIONAL. 157 
 
 Now 
 
 and (/3 -v/2) ($/g + v/6 + /4) = (a-fl) 3 -2 = 1 j 
 .*. the denominator is 1 , and the fraction is reduced to 2y/9 
 4-2^/6 + 2^/4. 
 
 Ex.3. 
 
 Reduce -5 3 to a fraction with a rational denominator. 
 
 Here w=3, 0=0:, ^ = 2/> the sign of \/b is +, and ?z an 
 number ; .'. the multiplier is ^/a 51 " 1 C/a n -*5 -f-^//> B ~ l 
 
 *) Hence the fraction required is f 3 , " */ y / 
 
 Ex. 4. 
 3 
 Reduce */ 4 y to a fraction with a rational denominator. 
 
 Here w=4, a = 5, 6 = 3, the sign of \/b is -f> and TZ an 
 number, .-. the multiplier is ^/a n ~ l t/a n ~*b +%/a n - 3 b* 
 yi n ~ l = ^125 ^75 + ^45 ^27. Hence the fraction 
 
 / _3 __ \ / y/125-^ 
 required is ( 5 +3 j I /125- 
 
 XL. 
 
 On ^e method of extracting the Square Root of 
 
 Binomial Surds. 
 
 129. Let +J~x and x/jtjTbe two quadratic surds, which are 
 not reducible to the same irrational part ; their product will be 
 
 _ _ _ m m _ _ 
 
 irrational. For, if *Jx x *Jy = m, *Jx=-== ~~\/y ', that 
 
 is, v^~i s reducible to the irrational part ^/ y, contrary to the 
 supposition. 
 
 130. Next 
 
158 METHOD OF EXTRACTING 
 
 130. Next, let \/x+ <*Jy be a binomial, both whose terms 
 are quadratic surds, not reducible to the same irrational part. 
 If this binomial be squared, the result is x+y + 2\/xy, a 
 quantity of which one part is rational, and the other (Art. 129.) 
 irrational. Let x+y = a and 2*J xy = \/b, then it appears 
 that every Binomial surd whose square root can be exhibited 
 under the form Vx+t/if must be of the form a -f ^/ b ; 
 a being a rational quantity and J b a quadratic surd. The 
 same will evidently be true, if one of the terms, as ^/ a?, be sup- 
 posed rational. 
 
 131. The square root of a rational quantity cannot be partly 
 rational and partly a quadratic surd. For, if possible, let 
 
 then x=a* + l2a*>/b 9 and ^&= 
 
 a rational quantity. But, by the supposition, \/b is a surd; 
 hence \/x cannot be expressed under the form a\/b. In the 
 same manner it may be proved, that the square root of a ra- 
 tional quantity cannot be equal to the sum or difference of two 
 quadratic surds not reducible to the same irrational part. 
 For, if possible, let >J x-=. */ a.<Jb, then x=a + b2</ab, 
 
 and \/ab= rr-^ , which is impossible by Art. 129. 
 
 132. In any equation x + */y = a + \/L 9 consisting of rational 
 quantities and quadratic surds, the rational parts on each side 
 are equal, and also the irrational. For if x be not equal to 
 a, let x^a+m, then am + ^y~a+ */b, or m + ^/y 
 = \/&, i.e.^/b is partly rational and partly irrational, which 
 has already been proved to be impossible. In a similar man- 
 ner it may be shewn, that in any equation m*/x + n*/y=p*/x 
 + q*/y 9 where N/JC &\fy cannot be reduced to the same irrational 
 part, m^xp^X) and n*/y = q*/y. For, if q be not equal to n, 
 by transposition, m'Jx=p*/x + q*/y n*/y=.p*/x + (q 
 contrary to Art. 131. .*. q^y = n^y } .and consequently 
 
THE SQUARE ROOT OF BINOMIAL SURDS. 
 
 159 
 
 133. To find the square root of the binomial quadratic surd 
 . Assume */x + vfy = *,/a + */b, then x + y + ^xy = 
 ; .'. (by Art. 132.) x+y = a, and 2+Jxy=z*/b; hence 
 a*(^), and4#2/ = &(.B); subtract (5) from (^4), 
 then a; 2 2#y +y a =a 2 6, and xy=\fa* l- t we have, 
 therefore, 
 
 ? ^ 2;r=a-fVa a , and a;= 
 " 2y = a*/cf b, and y = 
 
 Hence Vjc+V^ = N ia + i^a 2 ~6 + N ia-jVa 2 6, an ex- 
 pression which can evidently be of the form V,r-fVy, only 
 when /^/a 3 6 is a rational quantity. The square root of 
 the binomial surd quantity a + *Jb can therefore be exhibited 
 under the form Vx + V^ only when a a b is a square number. 
 By a similar process it might be shewn that the square root of 
 
 4>/a 3 , subject to the 
 
 a \ s x a -f 
 same limitation. 
 
 EXAMPLE 1. 
 What is the square root of 19 + 8N/3? 
 
 a- Va a -6= + 
 
 Ex. 2. 
 Find the square root of 12 N/140. 
 
 ' *'- 6 =144- 140=4, and 
 
 Hence 
 
 Ex. 3. 
 Find the square root of 3 1 + 1 2 */ 5. 
 
 Here 
 
160 ON PRIME NUMBERS &C< 
 
 s 1681, and Jtf 
 
 or b= 720 
 Hence V^a -f */a* b + V a-^vV-= V T+T 
 
 Ex. 4. Find the square root of 7-f 4N/3 . ANSW. 2-f Vs. 
 Ex. 5 ...... ....... 7-sVlO .... V5-V2. 
 
 Ex.6 ..... . ...... 18-10V-7. . . 5 V^7. 
 
 CHAP. IX. 
 
 ON MISCELLANEOUS SUBJECTS. 
 
 WE now proceed to apply the principles laid down in the 
 preceding Chapters to the investigation of questions of a mis- 
 cellaneous nature, beginning with some observations upon 
 prime numbers and their several relations. 
 
 XLI. 
 
 On Prime Numbers and their Relations; and on the 
 Method of finding the least Common Multiple of two 
 or more numbers. 
 
 134. Numbers which admit of no exact divisor, or which 
 have no measure (Art. 40), except themselves and unity, are 
 called Prime Numbers, as 2, 3, 5, 7, &c. ; and two or more 
 numbers, which have no common divisor, or measure, greater 
 than unity, are said to be prime to each other, as 8 and 9 ; 
 11, 14, and 15; &c. 
 
 135. " Let a b, the product of any two numbers, be divisible 
 " by e; then, if c be prime to I, it will be a divisor of a." For 
 
 suppose 
 
ON PRIME NUMBERS &C. 161 
 
 suppose b to be greater thane; then, if the operation in Art. 45 
 be performed on them, the last divisor, or greatest common 
 measure, will be unity, because b and c are prime to each other. 
 Let the operation stand as follows ; 
 
 c)l(p 
 
 then we have these equations. ; 
 
 b-cp = d, 1 i abacp = ad 
 cdq = e > or < ac adqae 
 der=l 1 fad aer=za. 
 
 Consequently, since c, by supposition, measures ab, it will 
 measure ab acp, or ad; zndacadq, or ae; and adaer, 
 or a. (Art 8 . 43,44.) 
 
 If c be supposed greater than b, we shall, by a similar process, 
 arrive at the same conclusion; which will be equally true, what- 
 ever be the number of divisions in the operation. 
 
 ^\ 
 136. Hence it follows, that if the numerator and denominator 
 
 of a fraction be prime to each other, there can exist no other 
 equal fraction having its numerator and denominator respectively 
 less than those of the first. 
 
 a m 
 
 In the fraction T let a be prime to b ; and let be an equal 
 
 u 7 n 
 
 fraction; then, since T = > m=z-r' Consequently b will be a 
 
 divisor of an; and since, by supposition, it is prime to a, it 
 must (Art. 135) be a divisor of n, and therefore less than n. 
 In the same manner it may be proved that a is less than m, 
 
 and the fraction 7 is therefore in its least possible terms. 
 
 Again, since b is a divisor of n, let T=p ; then n=pb, and 
 
 pa a m 
 consequently, since 7=y = -,m will pa-, that is, " if 
 
 Y " two 
 
ON PRIME NUMBERS &C. 
 
 " two fractions, of which the former is in its least terms, be 
 " equal, the numerator and denominator of the latter will be 
 " equimultiples of the numerator' and denominator of the 
 " former, respectively/' 
 
 137. If a and b ar x e both prime to c, al will be prime to c. 
 For if not, suppose ab and c to have a common measure ?w, and 
 let ab=mp, and c~mq. Then, since a is prime to c, or my, 
 it is prime to m ; for if a and m had a common measure, 
 this would (Art. 43) be a common measure of a and m q. For 
 the same reason, b is prime to m. But, since ab=>mp, 
 
 - = T, and (Art. 13(>) is in its lowest terms ; therefore b is 
 
 either equal to m, or (Art. 136) a multiple of m, which is 
 absurd, because b has been proved to be prime to T; .'. ab 
 and c can have no common measure, and consequently a b must 
 be prime to c. In the same way, if Q, b, c are all prime to d, 
 ale is prime to d, and so on. Hence, if a be prime to d, a* 9 
 a 3 , a 4 , &c. will all be prime to d. 
 
 Again, if a, b } c, &c. are each of them prime to each of 
 d, e,f, &c. ale &c. will be prime to def See. For, since 
 a, I, c, &c. are prime to d, abc &c. will be prime to d. For 
 the same reason, abc &ic. is prime to e,f, &c. 5 and conse- 
 quently todef &c* Hence, if a be prime to d, a 2 will be 
 prime to d 2 , a 3 to d 3 , and so on. 
 
 138. A common multiple of two or more numbers is any 
 number which is measured by each of them ; and their least 
 common multiple is the least number which is so measured. 
 
 Let c be the greatest common measure of a and b, and let 
 
 ab 
 
 a mCy b nc. Then ab = mnc\ and mnc=na=mb; 
 
 c 
 
 therefore is a common multiple of a and I. It is also 
 
 their least common multiple; for let d be any other common 
 
 q a m 
 multiple of a and b, and let d=pa = (]b; then - = ^ = -> 
 
 where 
 
PROPERTIES OF CUMBERS. 
 
 where is in its least terms, because (c being the greatest 
 
 common measure of a and I) m and n are prime to each other; 
 therefore q and p are equimultiples (Art. 136) of m and n 
 respectively, and q is greater than/ft; hence, qb is greater 
 
 ab 
 
 than mb, or d greater than * Hence, "the least common 
 
 c 
 
 " 'multiple of two numbers is equal to their product divided 
 " by their greatest common measure/' It may be farther 
 observed, that "every other common multiple of a and b 
 " is a multiple of their least common multiple;" for since q 
 
 ab 
 is a multiple of m, qb or d is a. multiple of mb, or 7. 
 
 To find the least common multiple of three numbers, a, b, c; 
 " let m be the least common multiple of a and b, and n the 
 " least common multiple of m and c; then n will be the least 
 t( common multiple required." For since m is a common 
 multiple of a and b } and n a common multiple of m and c, 
 n will obviously be a common multiple of a, b, c. It will 
 also be their least common multiple; for let d be any other 
 multiple of a, b, c, then d will be a multiple of m, as has just 
 been shewn ; and since it is also a multiple of c, it will be a 
 multiple of n, and therefore must be greater than n ; hence n 
 is the least common multiple of a, b, c. 
 
 XLII. 
 
 Properties of Numbers. 
 
 139. Let a } I, c, d, &c. represent the digits of a number, a 
 being the digit in the unit's place, b the digit in the ten's place, 
 c the digit in the hundred's place, &c. &c., and let r= 10, then 
 the general value of any number may be represented bya + br + 
 cr'+dr s +&c.; thus, 357 = 7 + 50 + 300 = 7 + 5 X 10 + 3x10*; 
 and 4213 = 3 + 1 x 10 + 2 x 10* + 4 x 10 3 ; &c. &c. From this 
 mode of representing a number, the following properties are 
 very readily deduced. 
 
 i. " If from any number the sum of its digits be subtracted, the remainder 
 " is divisible by 9." 
 
 For let a -f b r + c r 1 + dr* + &c. = the number 
 Subtract + *+ c + d +&c. 
 
 Then we have &(r-l) + c(r a -l) +</(r 3 - l)rj-&c. for the value of the 
 
 number 
 
16 i PROPERTIES OF NUMBERS* 
 
 number whenits digits are subtracted from it; but by Art. 126, this quantity 
 is divisible by r 1 or 9. Take, for instance, the number 37591, subtract 
 the sumofits digits, and the remainder is 37566=9 X 4174. 
 
 ii. " If the sum of the digits of any number be divisible by 9, the num. 
 " ber itself is divisible by 9." For let the number be N, and the sum of 
 its digits S, and let S= 9 m. Then (by Property i.) N S is divisible by 9 ; 
 Jet XTS=9p, and we have JV-9m=9p, '. tf=9p+9m=9 (p + m), 
 which is divisible by 9 ; consequently JV is divisible by 9. Thus the 
 numbers 171, 387, 51489, &c., the sum of whose digits is divisible by 9, 
 are themselves divisible by 9. 
 
 in. " If the sum of the digits of any number be divisible % 3, then the 
 " number itself is divisible by 3." Let N and S represent the number 
 and sum of its die/its as before, and let Szz 3m. Now N S=9p, 
 .*. N 3m=9p, or N=9p + 3m, which is evidently divisible by 3. Thus 
 the numbers 111, 123, 258, 1713, &c. are all divisible by 3. 
 
 iv. " If from any number the sum of the digits standing in the ODD places 
 " be SUBTRACTED, and to it the sum of the digits standing in the EVEN places 
 u be ADDED, then the result is divisible by 11." 
 
 For let the number be a+ir+cr'-MrS + e/H&c. 
 Adda+b c +d e -f&e. 
 
 il^Q result is &.r+l +c.r*-l + d.r*-\-\ +e.r*- 1 -f &c. ; 
 but by Art. 126, the quantities r+1, r 2 1, r*+ 1, ?4 i, &c. are all di- 
 visible by r-H ; therefore b.r-l -fc.r 2 1 +rf.r 3 +l + e.r* 1 + &c. 
 is divisible by r+1, or =11. Take, for instance, the number 57937; 
 subtract 5 + 9 + 7 = 21, and add 7 +3 = 10, or, in other words, subtract 11, 
 then the remainder 57926= 1 1 X 5266. 
 
 v. "If the sum of the digits standing in the EVEN places, be equal to the sum 
 " of the digits standing in the ODD places, then the number is divisible by 1 1." 
 Let N = the number, tfr^the sum of the even digits, 5= the sum of 
 the odd digits ; then (iv.) 2V+ S s is divisible by 1 1 ; but if S=:s, then 
 Ss=Q, .*. N is divisible by 11. Thus the numbers 121, 363, 12133, 
 48422, &c. are all divisible by 1 1. 
 
 The number r (which is called the root of the scale) has 
 here been supposed = 10, that being its value in the common 
 system of notation; but the above properties of numbers are 
 true for any other system. For instance, if the system of 
 notation be such that the value of the digits increase only in a 
 sixfold instead of a tenfold proportion from the right to the 
 left, then (since f==6, and consequently r 1 =5, r-f 1=7) 
 
 what 
 
PROPERTIES OF NUMBERS. 
 
 what has just been proved with respect to the numbers 9 and 
 11, is equally true with respect to the numbers 5 and 7 iw the 
 system the root of whose scale is 6. 
 
 140. Suppose, now, that it was required to transform a 
 number of the common arithmetical scale into another of the 
 name value, where the root of the scale shall be r ; let the 
 given number be N 9 and let the Digits of the number where 
 the root of the scale is r, be a, b, c, d, &c. ; then we have 
 
 an equation in which N and r are given, to find the values of 
 a, b, c, d, &c. Divide JVby r, then the quotient is b + cr + 
 dr*-+-&zc. and the remainder a; divide l + cr-{-dr' i -\'&c. by r, 
 the quotient is c-fc2r-J-&c., and the remainder I ; divide 
 c-f-dr-f&c. by r, the quotient is d &c. and the remainder c ; 
 so that the rule is, " to divide the given number continually 
 " by r till the last quotient is less than r, then this last quo- 
 " tientj together with the several remainders taken in the re- 
 " verse order, will be the digits of the number required." For 
 instance, let it be required to convert the number 3714 into 
 another number of the same value, wherein the value of each 
 digit shall increase in a fourfold proportion from the right hand 
 to the left. Here ?-=4; and the operation will stand thus; 
 
 4)3714(2= 1 st remainder ^j Hence 322002, where the value 
 
 4)928(0 = 2 d D. I of each digit increases in a fourfold 
 
 4)232(0 = 3 d D. ^proportion, is a Number of the 
 
 4)58(2 = 4 th D. / same value with 37 14, where the 
 
 4)l4(2=5 t D. I value of each digit increases in 
 
 ) a tenfold proportion. 
 
 141. The foregoing properties of Numbers have been deduced 
 from the manner in which they are represented by means of 
 the series a + &r-fcr a -fc?r 3 + &c. But numbers may also be 
 considered as arising from the continued multiplication of cer- 
 tain factors. The most general form under which numbers 
 may be thus represented is a n b m c r d s &c, where a, b, c,d &c. 
 are prime numbers, and 77, m, r, s, &c. any whole numbers 
 
 whatever. 
 
PROPERTIES OF NUMBERS. 
 
 whatever. One of the simplest cases of this kind is when the 
 number comes under the form a n b; and under this form we 
 are enabled to investigate the expression for what is called a 
 perfect number, i. e, " a number which is equal to the sum of 
 all its divisors." 
 
 The process ia this. The divisors of a n b are 1, a, a a , a 3 , &c. . . . n and 
 6, ab, a?b, 3 6, &c. . . . a n ~ l b ; hence by the supposition 
 
 a n+\ _ i 
 
 but since b is a 
 
 , . _ . 
 
 a n+1 2a n -fl 
 
 whole number, suppose n+l 2 a w -f 1 equal to unity ', and consequently 
 a*+i 2=0, and a 2=0, or=2; hence i=2* +l 1 ; and the ex- 
 pression a n b becomes 2 w ('2 7l+l 1), where 2 n+1 1 must be a prime num- 
 ber. Let ?i= I, 2, 3, 4, 5, 6, &c., then 2 n+1 -l = 3, 7, 15, 31, 63, 127, 
 255, &c. ; of which the prime numbers are 3, 7, 31, 127, c., and the 
 corresponding values of n are 1,2,4, 6, &c. ; hence a system of perfect 
 numbers may be generated in the following manner ; 
 
 2 (2 a -l) = 2X 3 =6 } 
 
 2*(23-l)= 4X 7 =28 f proceeding in this manner, 
 
 31 =496 ( the next P erfect numl3er is f und 
 
 ~1) =64 X 127 = 8128 to be 33550336. 
 
 XLIII. 
 
 Permutations and Combinations. 
 
 142. By Permutations are meant the number of changes 
 which any quantities a, I, c, d, &c. may undergo with respect 
 to their order, when taken two and two together, three and 
 three, &c. &c. Thus ab, ac, ad, la, be, Id, ca, cl, cd, da, 
 db, dc, are the different permutations of the Jour quantities 
 a, b, c, d, when taken two and two together; a be, acb, Lac, 
 be a, cab, cba, of the three quantities a, b, c, when taken three 
 and three together; &c. &c. 
 
 143. By Combinations are meant the number of collections 
 which may be formed out of the quantities a, b, c, d, e, c. 
 taken two and two together, >three and three together, &c. &c. 
 without having regard to the order in which the quantities arc 
 
 arranged 
 
PERMUTATIONS AND COMBINATIONS. 107 
 
 arranged in each collection. Thus al>, ac, ad, be. Id, cd, 
 are the combinations .\\\i\c\\ can be formed out of the four 
 quantities a, b, c, d, taken two and two together; abc, abd, 
 acd, bcd t the combinations which may be formed out of the 
 same quantities, when taken three and three together; c. &c. 
 
 144. Let there be n quantities, a, b, c, d, e, &c., taken two 
 and two together; then, by Art. 142, it appears that there 
 will be (n [) permutations in which a stands first; for the 
 same reason there will be (72 l) permutations in which I stands 
 first ; and so of c, d, e, Sec, Hence there will be n times (n 1 ) 
 permutations of the form a I, ac, ad, ae, &c.; la, be, bd, 
 be, &c.; ca, cb, cd, ce, e. ; i. e. " the number of permutations 
 ofn things taken two and two is n(n l)." 
 
 145. If these n quantities be taken three and three together, 
 then there will be n(n 1 )(/i 2) permutations. For if ( 1 ) 
 be substituted for n in the last article, then the number of per- 
 mutations of 7i 1 things taken two and two together will be 
 (n l)(/i 2); hence the number of permutations of b, c, d, e, &c. 
 taken two and two together, are (n l)(w 2), and conse- 
 quently there are ( !)(;/ 2) permutations of the quantities 
 a, b, c, d, e, &c. taken three and three together, in which a may 
 stand first; for the same reason there are (n l)(w 2) permu- 
 tations in which b may stand first; and so of c, d, e, &c. The 
 number of permutations of this kind will therefore amount 
 to n(n !)(. 2). 
 
 146. In the same way it appears, that if the number of 
 quantities be n, and they are taken m and m together, the 
 
 number of permutations will be n(n ]) (n 2) &c 
 
 (nrn-r-l); and if m = ?i, i.e. if the permutations respect all 
 the quantities at once, then (since TW 72 = 0) the 7iumber of 
 
 them will be TZ(TI l)(n 2) &c 2.1. Thus, the number of 
 
 permutations which might be formed from the letters com- 
 posing the word "virtue" are 6x5x4x3X2x1 = 720. 
 
 147. But if in this latter case the same letter should occur 
 
 any 
 
168 PERMUTATIONS AND COMBINATIONS. 
 
 any number of times, then it is evident that we must divide 
 the whole number of permutations, by the number of times the 
 permutations are multiplied by having different letters instead 
 of the repetition of the same letter. Thus if the same letter 
 should occur twice, then we must divide by 2 x 1 ; if it should 
 occur thrice, we must divide by 3 x 2 x i; ifp times, by 1.2.3...p ; 
 and so for any other letter which may occur more than once. 
 Hence the general expression for the number of permutations 
 of n things, of which there are p of one kind; r of another ; 
 
 n(n l) (n 2)(n- 3)....2. 1 
 
 q of another; &c. &c. is ^..px 1.2.3..rx 1.2.3.^ Th " S 
 the permutations which may be formed by the letters composing 
 the word " easiness" (since s occurs thrice, e twice) are 
 8.7.6.5.4.3.2.1 
 
 1. 8.3. XI. 9 
 
 148. From the expression (in Art. 146) for finding the 
 number of permutations of n things taken m and m together, 
 we immediately deduce the theorem for finding the number of 
 combinations of n things taken in the same manner. For the 
 permutations of n things taken two a?id two together being 
 n(n l), and each combination admitting of as many permit - 
 tations as may be made by two things (which is 2 x l), the 
 number of combinations must be equal to the number of per- 
 mutations divided by 2; i.e. the number of combinations of n 
 
 n(n l) 
 things taken two and two together is - For the same 
 
 reason, the combinations of n things, taken three and three 
 
 n(n l)(n 2) 
 together, must be equal to - T^T^ - ' an " m general, the 
 
 1 ^i ,o 
 
 combinations of n things taken m and m together must be equal 
 n(n l)(n 2)....(nm,+ l) 
 1.2.3....ro 
 
 XLIV. 
 
 Unlimited Problems. 
 
 149. It has already been observed (Art. 69), that in order 
 
 to 
 
^ UNLIMITED PROBLEMS. 169 
 
 to obtain the solution of equations containing any number of 
 unknown quantities, it is necessary that there should be as many 
 equations as there are unknown quantities. If the number of 
 equations be less than that of the unknown quantities, then the 
 number of values of the unknown quantities will be unlimited, 
 unless the problem be limited by circumstances. This will be 
 readily understood by taking the simple case of j; -f ?/ = 1 0, 
 where it is evident that the values of x and y may vary through 
 all degrees of fractional and integral magnitude between 
 and 10; forif#=l, then^=9j; if x= 1, then y = 9; if 
 x l, then #=8f ; &c. &c. ; but if the hypothesis be limited 
 to the integral and positive values of x and y 9 then the num- 
 ber of answers is limited to nine, for if #=1, 2, 3, 4, 5, 6, 7> 
 8, or 9, then the corresponding values of y are 9, 8, 7, 6, 5, 
 4, 3, 2, or 1. 
 
 150. Suppose now it was required to find all the integral 
 and positive values of x and y in the equation 2.r+3y=17. 
 
 y (y~~l\ 
 
 =8 + ^ y -5 =8 y \ )j and since 
 
 y\ 
 x and y are whole numbers, it is evident that " must be 
 
 y 1 
 also a whole number. Let =p, then y = 2/) + l ? and 
 
 # = (8 ?/ p = )8 2p 1 p=7 3 p. To make x a positive 
 number, p cannot be taken greater than 2 ; let p = 0, 1, or 2, 
 then jc = 7, 4 or 1, and the corresponding values of y (2p-f l) 
 are 1, 3, and 5; so that the number of positive and integral 
 values of x and y are limited to three. 
 
 151. Next let it be required to find the same in the equation 
 
 5y = 7. Here T/= - - = -^ --- ; and since 5 is 
 
 2 . _ i 
 not a divisor of 7, ^ - must be a whole number (Art. 135). 
 
 3 
 
 Z Let 
 
170 UNLIMITED PROBLEMS. 
 
 Let =/>, then 2x=5/> + l, & x = 2p-f 
 
 then p = 2q 1; hence x=(<2p-\-q=) 4>q 2-t-q = 5q 2 r 
 
 and t = 
 
 Let 9=1, 2, 3, 4, 5, &c. 1 In this case the positive 
 then #=3, 8, 13, 18,23, &c. > and integral valuesofaandy 
 y = 7) 21,35, 49, 63, &c. ) are unlimited. 
 
 By attending to the several parts of the process in the two 
 last Articles, the solution of the following Questions will be 
 readily understood. 
 
 T. In how many ways may the sum of .b be paid, in crowns and seven* 
 shilling-pieces'* Let #= the N. of seven-shilling-pieces, y=the N. of 
 
 crowns ; then 1 x + 5 y = 1 00, y r= = 20 - x (where x must 
 
 5 5 
 
 be divisible by 5). Let -=J> then a-=5jo, and #=^20 x -- - = J20 
 
 5p 2/)=:20 1p (where p must evidently be less than 3). Let p = l 
 or 2, then #=5 or 10, and 2/=13 or 6, so that a payment of this sort can 
 only be effected in two ways. 
 
 ii. What is the least number of pieces in which a bill of .7 can be paid 
 in half-guineas and seven-shilling-pieces? Let #=N. of half-guineas,, 
 y=N. of seven-shilling-pieces, then 21 x-}- 14/=280, or 3<z i +2# == 40, 
 
 and y= - = 20 x - (where x must be divisible by 2). Let -=jo, 
 22 2 
 
 then #=2j9, and y = - ^ = 20-3jt? (where p must be less than 7). 
 
 Let j>=xl, 2, 3, 4, 5, or 6, "1 so that the number of ways in which thia 
 then a?=2, 4, 6, 8, 10, or 12, payment may be made is six; and the 
 and yr=17, 14,11,8,5, or 2, ) feos/ N. of pieces is 14, 
 
 in. A person owes me seven shillings ; he has no other money about him 
 but half -guineas, and I no other but crown -pieces ; what is the least number 
 of pieces by which this debt may be settled? Let #=N. of half-guineas, 
 
 y=rN. of crowns, then 21 x 10^=14, and y=. =2#-l -t . 
 
 Let a ^=p, then x= Wp+ 4,andy=(20p+8 1 -f p=)21 p + 7(where 
 10 
 
 p may be 0, or any whole number whatever). 
 
 Let 
 
UNLIMITED PROBLEMS. 171 
 
 T A 1 s0 that the least N- of pieces is 11, viz. 
 
 Mt/>=0, 1, 2, 3, 4, &c. / 4half _ guineasand7crowns . butthentwii- 
 
 then * = 4, 14, 24, 34,44,&c. S^ tf - n which tfae t be 
 
 *=7, SB, 49, 70, 91,4* } efffe ^ ^ wM 
 
 iv. JHf '* required to find the least number which when divided by 19 shall 
 leave the remainder ^ ; and when divided by 28, the remainder 13. Let 
 * and y be the quotients arising respectively from such division, then 
 
 3 y + 2 must be divisible by 1 9). Let ~ = p, then y = -~ = 6 j> 
 
 H -- ; ; put - 7, then p=3g-{- 2 ; and as it is required to find tho 
 3 3 
 
 least number which will answer the conditions required, let <?r=0, then 
 
 - =18, in which case 19^ + 7 
 19 
 
 and 28 y + 13 are each equal to 349, which is the N. required. 
 
 v. What is the least whole number which-, when divided by 5, 6, 7> re- 
 spectively, shall leave remainders 1 , 2, 3 ? Let or, y, * be the quotients 
 arising from this division, then 5 #+ l=6# + 2=7# -4-3- Now ,r= 
 
 = P , then y=5p-l, and 6y + 2= 30/-4 
 
 == 7 5f _i_3 ; hence *= - -- = 4 p 1 + ( where p must be divisible 
 
 by 7). Let -= 7, then p 1 q, and z=(4p- 1 + ~ = J28?- 1 -f 2</= 
 
 30 <?!. 
 
 Let q= 1, 2, 3, &c. Iso that the feas/ number which 
 
 then #= 29, 59, 89, &c. ? will answer the conditions required 
 
 and 7 * + 3 = 206, 4 1 6, 626, &c. 3 is 206. 
 
 XLV. 
 
 Diophantine Problems. 
 
 152. These are a species of unlimited Problems, principally 
 respecting square and cube numbers. No general rules can be 
 laid down for the solution of them ; but the following examples 
 may serve to give the learner an insight into their nature, and 
 the manner of solving them* 
 
112 DIOPIIANTINE PROBLEMS. 
 
 i. To find ttvo square numbers whose sum shall also be a square number. Let 
 * a and a* represent the two square numbers required ; then the values of <r" 
 and a* must be such, that a? -f* a m ay be a square number. Now a? -+ a* 
 is greater than # a] 4 (for # a] 2 =r # 2 -f- a * 2 a #) ; we may therefore as- 
 sume x* -J-a a =m# a] 2 , where w is some number greater tlian unity ; but 
 =7j,- a*=:nV 2zna.r -f a% then ** = V 2a#, or 
 
 2 TO a 2 
 
 'ar *= 2wa; .'. d? = 3 _ ; hence the two numbers required are fl 
 
 and a 2 , where TO and a may be any whole numbers whatever ; but that 
 ^ - may be an integer, it is necessary that 2 ma be some multiple of 
 
 m s 1. Let TO=2, a=3, then the two numbers are 16 and 9, and their 
 
 225 
 
 sum 25. Letm=3, a=r5, then the two numbers are and 25, whose 
 
 16 
 
 gOg 
 
 sum -V is also a square number. Let 7/1= 3, a=8, then the numbers 
 ID 
 
 are 36 and 64 and their sum 1 00 ; &c. &c. 
 
 11. To find a number ( t c) such that x-\-a and x a shall both be square 
 numbers. Let x + a= m*, then x a=w a 2 a; assume w a 2a=m a 2 
 a% then 2a= 2wa+a' i or 2ma = a 2 
 
 a + 2 . , a + 4a-f-4 a a +4 a + 4 
 
 - , andm r= - ; hence ^=w r= -- = 
 
 2 4 4 4 
 
 where a may be any number whatever ; and if it be an even number, then 
 (and consequently x 4- a and x a) will be a whole number. 
 
 Let =1, then ^zzp=|; J? 4- = -^+ 1=: ? ; ^-a^- l = i, 
 
 9 + 4 13 13, . 25 13 
 
 =, ^ = - ? -= I ; * + fl =-+3=-; *-a = --.3 = i, 
 
 16+4 
 o=4, . . . #=: = 5; <r + a=5+4=:9 ; J?-ar=54=;l ; 
 
 & &c. &c. &c. 
 
 and this is a general property of square numbers, viz. that if we 
 take any number^ square it, add 4 to that square, and then divide the result 
 by 4, it will give such a number, that the sum and difference of it, and the 
 original number, shall be a square number. 
 
 in. To find three square numbers which shall be in arithmetic progression. 
 Let the numbers be x\y*,y\ then <r'-f-* 3 = 2/. Put x\=p+ q, 
 and * = ;>-?, then **+ * 2 = 2;>'-H 9 2 = 2y a > '> P*+q* = #*, and the 
 
 question 
 
/ DIOPHANTINE PROBLEMS. 
 
 question resolves itself into the finding p and q t such that p a +? fl shall 
 be a square number. Let, therefore (Ex. I.) p = - r, ? = a, then 
 
 771 1 
 
 A where a and m may be any numbers 
 
 ,r=r p -f- q = - :: | i j- a I whatever. For instance, let a = 3, w = 2, 
 
 " - = 7, y = 5, z=\, and the square 
 
 * = p q - -- numbers in arithmetic progression are 49, 
 25, 1. Let a = 8, m = 3, then o?=14, 
 
 __ /-TTi _._ 2?L+l y=10, z= 2, .'.the square numbers 
 rn a 1 ] j n Arithmetic Progression are 196, 100,4. 
 
 XLVL 
 
 The Solution of two Questions relating to Numbers in 
 Geometrical Progression. 
 
 153. Let a be the first term, r the common ratio, n the 
 number of terms, and S the sum of a Geometric Series; then 
 
 (by Art. 110), S=^- -j andifa=l, 5=1-^1. Now let 
 r 1 r 1 
 
 S be the sum of the series arising from the successive addition 
 of 1, 2, 3, 4, &c. . . . n terms of the geometric series ; then we 
 shall have, 
 
 __ 9 
 
 r-l r 2 -! r 3 -! r 4 - 1 r w 1 
 
 H--T + I T+&C + 
 
 1 ' r I ' r l ' r 1 ' ' r 1 
 
 1 
 
 ... to 72 terms) 
 
 r~i\ r 
 
 of which the following are examples. 
 
 Let r==2, then S=sl-f 2+ 4+ 8 -f l6 + &c...2 n ~ l =2" 1. 
 
 31+&c...2"-l = 2 n+1 - 
 
174 QUESTIONS RELATING TO NUMBERS 
 
 ii. Letr = 3, then S=l+3+ 9 + 27+ 81 + &c...3 n ~ l = 
 2= 1+4+ 13 +40+ 121 +&c... ^-= : 
 
 HI. Letr = 4, then S=l +4 
 
 + &c... 
 
 4, n -l 
 
 &c. &c. 
 
 &C. 
 
 154. Let- + + 
 
 &c. 
 
 . be an 
 
 infinite series of fractions whose numerators are in Arithmetical 
 and their denominators in Geometrical Progression. For find- 
 ing its sum (Sj, this series may be resolved into the following ; 
 
 a a 
 
 ,. 
 
 ++ *+~ + '+ &c - ad l 
 
 Art. U6) 
 
 ar 
 
 (a) 
 
 b 
 
 ^ + c/ + 
 b b 
 
 ,(,-!) 
 
 cr* 
 
 b b 
 
 cr(r 1) 
 
 b 
 
 
 cr* + cr* + 
 
 cr\r-l) 
 b 
 
 
 &c 
 
 c^(r-l) 
 .. -& c . 
 
 
 
 Hence 
 
IN GEOMETRICAL PROGRESSION. 175 
 
 Hence S =^^ + ,-^(1 +; + J+p + &c.adlnfinitu m } 
 
 ar I r ar Lr 
 
 r-J) + c(r-l) X 7 T 
 which the following are examples. 
 
 i. Let fl= 1, 6= 1, c= 1, r = 2, then 
 
 ii. Let o=l, 6=2, c = 3, r=2, then 
 
 13579 24 
 
 in. Let a = 2, 6a=3, c=5, r=3, then 
 
 2 5 8 11 14 6 9 21 
 
 CHAP. X. 
 
 ON THE BINOMIAL THEOREM, 
 AND SUBJECTS CONNECTED WITH IT. 
 
 SIR ISAAC NEWTON'S theorem for raising a binomial to any 
 power was given in Chap. III. The index (n) was there sup- 
 posed to be an integral and positive number ; but the great 
 value and importance of this theorem is derived from its being 
 equally true, whether the index be integral or fractional, 
 positive or negative; for this circumstance enables us not 
 only to obtain the roots, as well as powers, of Algebraic quantities 
 in a much more easy manner than by the common processes, but 
 to apply the theorem itself to many very useful and important 
 investigations in the higher branches of analysis. 
 
176 DEMONSTRATION OF BINOMIAL THEOREM. 
 
 XLVII. 
 
 The general Demonstration of this Theorem. 
 155. Previously to the investigation of this Theorem, it will 
 be necessary to ascertain the two first terms and the general 
 form of the series which expresses the value of (l + ax + bx* + 
 ca? 3 -f-&c.) M , whether n be integral or fractional, positive or 
 negative/ 50 
 
 i. If n be a positive whole number, then, by the ordinary 
 process of involution exhibited in Art. 49, we have 
 
 from which it appears that 
 in finding the value of 
 
 1+ ax + &c. 
 1+ ax + &c. 
 
 * .) n , the 
 
 H-aax+fcc-forthcS^nre. i two first terms will be 
 
 1-f- ax + &c. ) 
 
 1 + n a x ; ana from the na- 
 ture of the process it is 
 
 -f- 
 
 c. for the Cube. 
 
 &c. &c. 
 
 evident that the powers of 
 x will increase regularly. 
 
 II. If TZ=-, then since the indices of x in the quantity 
 .1 -fajc4'&z 4 -t-cj; 3 - r -&:c. are all supposed to be integral and 
 positive, it is evident that the indices of x in the series which 
 expresses the r th root of this quantity will be integral and po- 
 sitive also ; for if any of the indices in the root were fractional 
 or negative, we should, in the re-composition of the power 
 from the root, have fractional or negative indices also in the 
 power ; which is contrary to the supposition. 
 
 With 
 _ 
 
 (*) The general form of a multinomial quantity in which the powers 
 of x regularly ascend is ^4 + J5<r-f-C<z >7 -f-Z)^4-&c. ; but this is easily 
 reduced to a form much more simple, yet equally general, by dividing the 
 
 B CD 
 
 whole by A, in which case it becomes 1 -f ^r + ~^4*-^ 3 -f &c- or 
 
 A A A 
 
 BCD \ 
 
 making - = , ~ l ' =b '> 7 ==c &c - r*^K"* 
 
DEMONSTRATION OF BINOMIAL THEOREM. 177 
 
 With respect to the two first terms of the root, it is mani- 
 fest that the first of them will he unity, and that the second 
 will he such a quantity as, in the recomposition of the power 
 from the root, will give ax for the second term of the power ; 
 now, by Case L, this must be such a quantity as when multiplied 
 
 by r will produce ax, i.e. it must be -ax* Hence we have 
 
 " 
 
 = 1 -f - 
 
 = 1 + n a x -f &c., since - = * 
 r 
 
 in. Now let n z , then 
 involution (Case I.), (l-f #-f />.r 2 rf &c,) w = 
 ttracting the ?- th root, (l -fas -f- # 2 4- &c.)?ss(l 4- a# + &c.) a>: 
 
 = 1 -f - (max) -f &c. (by Case 1 1.) 
 
 = l-f -(ajr-)+&c* 
 = 1 -f ?z :r -i- &c. as in Cases L 1 1* 
 iv. If = 5, where 5 is either integral or fractional, then 
 
 = 1 ifl.r4-&c. by actual division. 
 = 1 -f- n a x -f- &c. as in former cases. 
 
 Hence it appears, that whether n be integral QT fmctioval, 
 positive or negative, the first two terms of the series expressing 
 the value of (l -f a #-}-/' #* + &c.) w will be 1 -\-nax t and that in 
 the subsequent terms the powers of at will be integral and positive. 
 Now, suppose a= l ; ^ = 0; cz=0; &c. then the multino- 
 mial quantity 1 +ax+lx* + &c. is reduced to the binomial 
 1 -fjj; and we are evidently at liberty to assume 
 
 (l-f #)*=! 4. nx + qx' i + rx* + sx 4 + c. 
 where (7, r, 5, &c. are quantities whose values are hereafter to 
 
 A A be 
 
178 DEMONSTRATION OF BINOMIAL THEOREM. 
 
 x\ n 
 be determined. Hence, also, since (a + x) n = a n { 1 -f^j , we 
 
 nx qx z rjj s 
 
 = a* 4- n a n ~ } x -f q a n ~ V -f r a n ~ V -f- &c. 
 156. Now let the trinomial quantity (l +x-f-^) n be expanded, 
 first by considering x + h. as o?ze quantity, and secondly by con- 
 sidering 1 4-.r as one quantity, and there will arise two scries, 
 from the comparison of which the values of q, r, S 9 &c. may 
 be obtained. Thus 
 
 = 1 -f nx+qx* + r* 4- ** + &c. + n*-h2y^r+3rAaf* -f 4*for* H- &c. 
 omitting the higher powers of A, as unnecessary for our purpose. 
 
 = 1 4- a? -f ?* 2 + r,r3 4. S x* -f &c. -f nA -!- n(n - 
 
 Since the series (A) and (J5) arise from the expansion of the same quantity \-\-x 
 they are evidently equal ; rejecting therefore the part common to both, we have 
 
 SqJw + 3rhx* -f 4shx 3 -f &c. = n(n 1 }lix -f nq'hx* + nr ^ -f &c. 
 and equating the coefficients^ 5 , we have 
 
 t n(n-\) (w-l)(-2)( c ) 
 
 Syrsw^n 1), or 9 - j and by parity of reasoning,*? - - - 
 2 2 
 
 '*(*--')(-8) _*(n-l)(n-8). . (n- l)(n~8)(n- 
 
 2 2.3 2-3 
 
 / n(n~l)(n-8)(n~3) w(n~ l)(w-2)(w-3) , (n-.l)(n-r2)(n 3)(n- 
 
 ~T3~~ 2.3.4 2.3.4 
 
 &C.=:&C. = &C. 
 
 By 
 
 ( a ) In assuming a series for the value of ( 1 -|-#) T '~ l , the first two terms 
 (by Art. 155) will be 1 +(n 1> ; and the other coefficients will also be 
 different from those of the series which expresses the value of ( 1 -fa') 71 . 
 To preserve an uniformity of notation, we have made them 9', r , *', &c. 
 
 ( b ) This process of equating coefficients requires explanation ; for which 
 purpose, let us suppose a + &# + c<z >2 -fd 1 a,' 3 +&c. and a4-/8d?4-7<r c + S^-f 
 &c. to be two series arising from the different modes of expansion (by 
 division, evolution, &c.) of the same quantity, or of equal quantities, in 
 ivhioh a, b, c, d, &c. and o, j8, 7, 5, &c. are constant quantities, but x a 
 quantity varying through all decrees of magnitude. Since the two series 
 
OBSERVATIONS ON THE FOREGOING THEOREM. 171J 
 
 By substituting the values ofp, q, r, &c. thus found, we have 
 
 --" . and 
 
 so that the series expressing the value of (a -}-#)", w being any 
 number whatever, either integral or fractional, positive or we- 
 gative, observes the same law as that which was deduced in 
 Chap. III., on the supposition of n being a positive whole 
 number. 
 
 XLVIII. 
 
 Some Observations arising out of the fore going Theorem. 
 157. Resuming the notation adopted in Chap. III. we have 
 
 n(n I)(TZ 2) (n m + <2) 
 
 &c. the m ih term of the series being \~v~3 ( ITTS 
 
 a n ~ m+l l m ~ l . Hence if n be a positive whole number , the 
 
 series 
 
 are equal to one another, whatever be the value of'*, let us suppose # = 0, 
 and we have a = a ; and these two quantities being invariable, a, will always 
 be equal to a for every value of a?. Now since a=za, bx+cx^ + dx* + &c. 
 must be equal to 3J? +7^-f5r 3 -f-&c. ; divide by *, and we have 
 b-\-cx-\- rfa >2 + &c. = /3-r - 7a?+5^-t-&c. ; suppose again o?=o, then 6 = ^3, 
 and so on; hence o= a, 6=/5, 0=7, rf S, &c. The same is also true in 
 the equation (a + 6* + c* 2 -f &c.)y + p ^ + Q^ 3 + &e - = ( + ^ + 7 ** +&c.) 
 . ; for divide by y, then a-j- 6,1? + cx* + &. 
 
 +Q'/-r- & c. ; Iet2/ = 0, then a-f 6 
 . and a, 6, c, &c. may be proved equal to 
 a, j8, 7, &c. respectively, as before. 
 
 ( c ) For if the coefficient of the third term of the series which expresses 
 the value of (l + #)" be ~ l \ the coefficient of the third term of the 
 series which expresses the value of (l H- .r) n1 W M1 (7 substituting n -* J 
 ftr w) be ^ ; and so of the rest, r', a', &c. 
 
180 OBSERVATIONS ON THE FOREGOING THEOREM. 
 
 series will terminate after w-fl terms; for let m= 
 then n m + 2 0, and consequently the coefficient 
 which involves the factor (w wi+2) vanishes. Let 
 mz=n+l, then n w + 2=l, ?* ra + 1=0, and ? l=w; 
 
 the (ra+lth) (or /a*/) term is 
 
 a*b n OT l n . If n be fractional or negative, the series will not 
 terminate, and in this case the value of any expanded binomial 
 can only be expressed in the form of an infinite series. 
 
 158. If in the series expressing the value of (a + b) 9 , for I 
 we put b, then those terms which involve the odd powers of b 
 will be changed from -f to ; Hence we have, 
 
 and (a-by> = a*-n<r* 
 
 /. by addition, (a + b) n + (a-l) n = <2a" + n(n-\} a"-W + &c, 
 or 
 
 - ~^ - t 
 
 o 
 
 159. Let fl=i, Z=l, then (a-}-6) w = (l 4- l) n = 2 n ; and 
 sirice the several powers of a and b are, in this case, each of 
 
 them equal to 1, we have 1 + rc + " fn ~ l) 
 
 2 2.3 
 
 &c. = 2 n , i. e. the sum of the coefficients of the nth power 
 of any binomial is equal to that power of 2 whose index 
 is n. Thus, for the square, 1+2 + 1=4 = 2 2 ; for the cube 
 1 +3 + 3 + 1 = 8 = 2 3 ; for the fourth power, 1+4 + 6 + 4+1 
 = 16=2 4 ; &c. &c. If a=l, #=1, in the expression 
 (a b) n , then (i -i) n =:0, which shews that the sum of the 
 positive coefficients of (ab) n is equal to the sum of the 
 negative ones. 
 
EXPANSION OF SERIES. 181 
 
 XLIX. 
 
 On the Expansion of Series. 
 160. It has already been observed (Art. 157) that when n is 
 
 a negative number or & fraction, then the series expressing the 
 
 m 
 value of (a-f- &) n does not terminate. Let ?i = > and substitute 
 
 m 
 
 for n in the series (Art. 156); then 
 
 m /m \ in /m \ /m \ 
 
 m m ^ m , -I -- I ) (- l) - 2) ro 
 
 -UV".i=*^ _ a r >+ rVr r _ -fl- 3 *' + &c. 
 
 ma r (l\^m(m r)a T fl\ ^ 
 = r + V+ 2? UV 
 
 2.3.r 3 
 
 m(m r) (?w 2 
 
 which is a general expression for finding the value of any bino- 
 mial surd quantity in a series, being either positive or nega- 
 tive, and m and r any whole numbers whatever. 
 
 EXAMPLE 1. 
 Find the value of 
 
 Herea = c 3 
 
 l = x*\ .'.a'=!/c* = c>, 
 
 m\ 
 
 ( a ) This series is derived from the preceding one, by resolving the powers 
 
 m 
 ?-l ~ _j - ] r --9 - _2 
 
 of a into two factors; thus a r rr a r X a = r X = a ? r F= r X a 
 
182 
 
 EXPANSION OF SERIES. 
 
 m(m-r)lt 
 
 2.3 2 V 
 
 x* 
 
 2.3r 3 \aV~- 2.3.3 3 
 
 &c. = &c. 
 
 tyjfl 
 
 Ex.2. 
 
 Find the value of - or ;.. -w.. or 
 
 Here a=e 2 "^ ^ ? __/_* _ A | 
 m(m-r}(b z \ - l(~l-2)/^ 4 
 
 w ( m _ r )(w - 2 r) I 3 \ - 1 ( - 1 - 2 ) ( - 1 - 4 ) /.f 6 
 2.3r 3 l a 3 /" 2.3.2 3 
 
 5 x 
 
 and 
 
 Vc' + a;' 
 
 Find the value of 
 Here a=c 
 
 m= 2 
 
 Ex.3. 
 
 or 
 
 2r *a 
 m(mr) (m <2t 
 2.3 r* 
 
 2.3 
 
 Hence 
 
 
v 
 EXPANSION OF SERIES, 183 
 
 <2x 3x* 4x 3 
 
 -T + --^ 
 
 This series is easily verified by the division of 1 by c'-f 2c#-f x*. 
 
 Ex. 4. 
 
 Find the value of (e'-jc 1 )*. 
 Herea=a<? 
 
 - 
 
 m = 3 /_/_* =_.:. 
 
 r==4 J T\ a/ 4V ?'/ 2*.c*> 
 
 m(w-r)//A_3(3-4)/J; 4 \ 3 a; 4 
 
 2r 2 VaV 2.4" \c 4 / 2 5 .c 4 5 
 
 m(m-r)(m-2r) ( b\ 3(3-4)(3~8)/ a?\ Sx 6 . 
 
 2.3r 3 \a a /~" 2.3.4 3 "" ^ ~ c 6 ^ ~~ ~ 2 7 .C 6 * 
 
 &c. = &c. 
 
 Hence c c -x a = 
 
 __^ _ 
 
 161. Nowletw=l, then (a + i)' = (a-f ^) r = ^ a-f //; 
 
 
 and "?= >y a ; hence the series in Art, 160 is transformed into 
 
 73 J + 
 
 Let a 1, b=sl, then 
 
 H 3 
 
 2.3 r 3 2.3.4.r 4 
 
 -f &c. (B.) Thus 
 
 Ifr = 8, then ^2= i+i- +l. -+l 5 - 
 
 3 3 2 3 4 3 5 3 6 3" 
 
 c. = &c. 
 
 By means of the series marked (A) 9 the ?^th root of many 
 other numbers may be found, if a and b be so assumed, that 
 
 I is 
 
184 EXPANSION OP SERIES. 
 
 I is a small number with respect to a t and JJ a. a whole 
 number; thus, 
 
 EXAMPLE 1. 
 Let a=4, b\, r=2, then v / a = v / 4 = 2, and we have 
 
 Ex. 2. 
 
 > =1, r = 3, then ^0 = ^/8 = 2, and we ol)tain 
 
 Ex.3. 
 
 1) 2 1 
 
 Let fli=8, 6= 2, r = 3, then -=-=-, and we have 
 
 / 1 1 5 2.5 \ 
 
 s/ Q Q _ v/ _ ofl _ - _ - _ &c. 1 1 
 /8-2-V6-2U ^ 3 - u , 3 . U3 35<44 c.i, 
 
 The several terms of these series are found by substituting for 
 o, by and r their values in the general series marked (A) or (B), 
 and then rejecting the factors common to both the numerators 
 and denominators of the fractions. Thus, for instance, to find 
 the seventh term of the series exhibiting the value of ^/2, 
 we take the 7th term of the series marked B, which is 
 
 (\ _ r )(l 2r)(l 3r)(l 4r)(l 5?-) , . 
 
 -L- .: V. A M g 2 j and since r=2, the 
 
 r t . 3.5.7.9 7-9 f . 93 
 
 fraction is r = _ = since ..=- 
 
 2.3.4.5.6.2 2.4.6.2 6 t 62 
 
 37 37 
 
 ~ 2 ~ e "jib* To find the 5th term of the series express- 
 ing the approximate value of y'Q, we take the 5th term of the 
 general series marked (A\ which is -f 
 
 -V O T* 7* 
 
 -A where a=:8, b=\ > and r=3; .*. the value of the 
 
 2.5.8 / 1\ 2.5 2.5 
 
 fraction is 4 l ^j )= 4 4 = ~3^* ^ n tn ' s man " 
 
 ner each term of the several series is calculated. 
 
 1C2. These 
 
 
EXPANSION OF SERIES. 185 
 
 162. These series converge very fast, so that a few terms 
 would give the rth root of certain numbers with a great degree 
 of accuracy. But a more practical method of finding the 
 higher roots of such numbers, is, by making the number whose 
 root is to be extracted equal toa r + b, and then assuming 
 a + x=.*ya T + b) x being some decimal fraction; for in this 
 case (a + jr) r =a r -f-6, and by expanding (a-\-x) r and neglecting 
 all the powers of x after x 9 (being very small compared with 
 the preceding ones) we have 
 
 a r + ? a r ~'x + r ( ~^~ J a r ~ V = a r -f I ; 
 
 .'.ra r ~ } x + r ( -Ja r ~'V=6 (^4) an equation 
 from which the value of x may be found in two ways. 
 
 I. By arranging the terms, and dividing by r( l r ~% we 
 
 2ax 
 have 
 
 -r r -l-r (r _y 
 and by solving the quadratic, 
 
 a / 2b 
 
 ~^~1 + V r(r-l 
 
 Hence yf + b = a+x= 
 
 which is HALLEY'S Rule, (Philosophical Transactions, 1694). 
 II. From equation (A) we have x ( r a r ~ { -{- r ^ ) a r ~ 2 jrj = 6, 
 
 r 1 
 
 "Bya-Jlrst approximation, neglecting the term which involves^, we 
 = rfl f^ 1 ; substitute this value for x in the fraction TT=5 
 
 - 
 
 _ 1 J , and we obtain a second approximation which 
 x / BB gives 
 
180 EXPANSION OF SERIES. 
 
 and 
 
 which is the Rule given by LA CROIX (Complement d'Algebre), 
 and ascribed to LAMBERT. 
 
 EXAMPLE 1. 
 
 Find an approximate value of the cube root of 67. 
 Here 67 = 64 + 3 = 4 3 + 3; .'. Q = 4, 6 = 3, r = 3; hence, 
 
 + -, or 
 
 = 2 2.0615 = 4.0615. 
 
 Ex.2. 
 
 Find an approximate value of the fifth root of 30. 
 
 Here 30 = 32 2 = 2 s 2 ; .*. a = 2, b= 2, r=5 ; hence, 
 
 /, /l__x 
 by the second method, a + x=a-{- s( ' 4 ^ \, or 
 
 /, S IT. .v 
 
 = a }- 3 f 4 ^ V 
 
 v a+ ib^y 
 
 20 
 
 The method of finding the rth root of certain numbers as 
 exhibited in this and the foregoing Article, is a matter rather of 
 curiosity than practical utility, as the rth root of any number what- 
 ever may be found with great facility by means of Logarithms. 
 This method would be useful, however, in an operation where it 
 was required to express this root in the form of a vulgar fraction ; 
 as in the last Example, where we obtained the approximate 
 
 value of the 5th root of 30 in the shape of the fraction ~ 
 
APPROXIMATION OF RATIOS. 187 
 
 L. 
 
 On the method of finding the approximate Ratio of 
 the Powers and Roots of Numbers whose Difference 
 is small. 
 
 163. Let a + x and a be two numbers whose difference is x, 
 
 n(n\) n(n \}(n 2) 
 
 { '- 
 
 a w ~ 3 .r 3 -f &c. : a*:: (dividing each term of the ratio by a*" 1 ) 
 
 164. Suppose now that n is not a large number, and that x 
 
 x* x 3 
 is very small when compared with a, then the fractions > &c. 
 
 will be small also, and those terms in which they are involved 
 will be very small when compared with the integral part a-f nx 
 of the series; in this case, therefore, the ratio of ( + #)": a* 
 approximates to the ratio of a + nx:a. Thus the ratio of 
 (a -f- x)* : a* approximates to the ratio of a -f- 2 x : a ; of (a -f x) 3 : a 3 
 to the ratio of a + 3x:a; &c. &c.; or if n = J, ^, &c. 
 then the ratio of v / a -\-x\a approximates to the ratio of 
 a + J# : a ; of ^/ a -{-x : a to the ratio of a -f %x : a ; &c. &c. 
 For instance, the ratio of the square of 501 to the square of 500 
 (in which case, a = 500, x= 1, 72 = 2) is 502 : 500 very nearly; 
 the ratio of the cube of 62 to the cube of 61, is 64 : 61 very 
 nearly; &c. &c. Again, the ratio of the square root of 501 
 to the square root of 500 is 500^: 500; and of the cube root 
 of 103 to the cube root of 100, is 101 : 100, very nearly. 
 
 165. If the difference between the two numbers is not very 
 small when compared with the numbers themselves, then the 
 three first terms of the series must be taken instead of two, in 
 which case the approximate ratio of (a + x) n : a* becomes that of 
 
 I : a. For instance, let it be required to 
 
 a I 
 
 find 
 
188 EXTRACTION OF THE nth ROOT 
 
 find a near approximation to the ratio of \/\\ : \/lO, then 
 a=10, x= I, w=-, and the approximate ratio becomes that 
 
 1 1 900-f 30-1 
 
 of 10 + g- : 10, or of - ^ -- : 10, or of 929 : 900. 
 
 By the Theorem in Art. 164, this approximation would be 
 10J: 10, or 31 : 30, i.e. 930:900. 
 
 Another method, which gives a much nearer approximation, 
 is as follows. Let S = half the sum of the given numbers, 
 and JD = half their difference; then (Art. 28) the numbers 
 themselves will be S -f D and S D. Hence the ratio of their Tzth 
 powers is that of S n + nS n ~ l D + &c. : S n nS*~ l D + &c. or of 
 S -f n D -f- &c. : S nD + &c. and their approximate ratio that of 
 S + nD: SnD. If this method be applied to the last 
 
 21 1 
 
 Example, $=>)=-> and the approximate ratio is that of 
 
 21 1 21 1 
 
 +g : g? or of 64 : 62, or of 32 : 31, which is nearer 
 
 the truth than that of 929 : 900, given by the last method. 
 
 LI. 
 
 On the method of extracting the nth Root of a 
 Binomial Quadratic Surd. 
 
 166. In the expression x + \f'y, let x be a rational quantity 
 and >^y a quadratic surd, then (x-{- */y) n = x n + nx n ~ l \/y -f 
 
 c. ( P). Let the sum 
 
 t 
 
 2 23 
 
 of the rational terms in the series (P) be equal to fl, and of the 
 Irrational p^/y^^/p^y^ which maybe expressed in the form 
 kjb, \/b being a quadratic surd containing the surd </y. 
 Hence (x-f-v/ ?/)*= + A/ &, and $ a + \/b x+^y\ if, there- 
 fore, the nth root of a quadratic surd of the form a-\~\/b can 
 be extracted, it may be expressed under the form x-r^/y, 
 whether n be an odd or even number. 
 
 167. Let 
 
OF A BINOMIAL SURD. 189 
 
 167. Let v/x-f */y be a binomial quadratic surd, in which 
 *Sx and */y are surds not reducible to the same irrational 
 
 n n 1 n _ j 772 
 
 part, then (\/x+ v /i/) n =a? z + nx * \/ y + n --- x 2 y + n. 
 
 fl J 77 2 "~ 3 
 
 --- x 2 2/A/7/ + &C. (Q). If TZ be an evew number, then 
 
 the 1st, 3d, 5th, &c. terms of the series (Q) are rational, and 
 the 2d, 4th, 6th, &c. irrational^ (Art. 129.) and (</ x +^y) n 
 may (as before) be expressed under the form a+ *Jb 9 where 
 s/& is a quadratic surd containing the surd */xy. Hence 
 ( v /x+ v /2/) w =fl+ Jb, or ^/a + ^ v'tf + */y\ and, if the 
 wth root of a + \/b can be extracted, it may be expressed under 
 the form Vx + \/y. 
 
 168. If TZ be an odd number, then - - &c. refractions, 
 and ^ > - j &c. whole numbers; hence the 1st, 3d, 5th, 
 
 ** 4* 
 
 &c. terms of the series (Q) will be surd quantities involving <Sx, 
 and the 2d, 4th, 6th, &c. terms, surd quantities involving Vy ; 
 the series may therefore be expressed under the formp/x 
 = Vf*x + Vq*y) or under the more general form 
 where \/a and \/b are quadratic surds involving the surds 
 and //// respectively. In this case, then, (Vx + Vy)*= 
 
 or $ N/a -h N//' = *Jx -|- N/y ; and^onsequently, if the wth root 
 O f Va 4- A/^ can be extracted, it may be expressed under the 
 form 
 
 1 69. From hence it appears that the 7^th root of a + Vb may 
 be expressed under the form x + */y, whether n be an odd or 
 an even number ; that the w.th root of a -}- */b may a/jo be ex- 
 pressed under the form >/a? + >A/, when TZ is an even number; 
 but that the ?/th root of N/a-fV6 can be expressed in the form 
 of a binomial quadratic surd only when n is an odd number, and 
 then under the form sAr + N///. 
 
 170. Sup- 
 
190 EXTRACTION OF THE nth ROOT 
 
 170. Suppose now that ^/ a-\-^b = x + */y, then a-fv7; = 
 ( x + VyY =x n + n x- Vy + ?*. x^y + n. 7 !! ^~ 2 jcs 
 
 ^ .-w O 
 
 .; .-. by Art. 132, a = a; M + ?z.!^lix n - 2 ^ + &c. and 
 
 &c.; hence a-V 
 
 - 3 
 
 2 23 
 
 = (x N/^/) W , or *ya */b=x*/y; from which it appears that 
 i 
 
 In the same manner, if vM- V^ = \/r-f Vy, where TZ is an 
 number, it may be shewn that \sa 
 
 171. Let ^/ *Ja-\-^l = ^x- 3 r\/y (n being an oc?c? number), 
 
 then v/a-h */b = (*/x + /y) n = x* +7ix~*/y+nJ?^x ~^ y + 
 
 2 
 
 y?> yz "" 1 .^'~ 2 a?Vi/ N /y + &c.; hence by Art. 132. (since Va is 
 2 3 
 
 a quadratic surd involving */x, and v7> a quadratic surd in- 
 
 n ~j _ i w 2 M 1 
 
 volving N/Z/) Va = o:2 +?z. a? * y-f&c. and ^/b=nx * \/y 
 
 ; from which it follows, that if 
 , then <y*/a*/b = */x*/y. 
 
 172. Suppose now that A + B is a binomial quadratic surd, 
 one or both of whose terms is irrational, then, from what has 
 been shewn, it appears that if A and B are both irrational, the 
 rath root of A + B can be extracted only when n is an odd 
 number ; but if A be rational, then the wth root of A+ B may 
 be extracted, whether n be an odd or an even number. In the 
 
 following 
 
OF A BINOMIAL SURD. 191 
 
 following Rule for extracting the nth root of A+B, the two 
 terms are supposed to be so arranged that ^4 is greater than B ; 
 and it consists of two cases, depending upon the value of 
 
 A*-B\ 
 
 CASE I. 
 
 When A* B* is a complete nih power ; i.e. when A* B* 
 = ee, n or r^/ ^L B* = a, being some whole number. 
 
 Assume $' A + B = */x -f */y(R) 9 
 then (by Art. 1 70 or 1 7 1 
 
 By squaring equation (R) , %/A* -f B J H- 2 A B x + y -f 2 Vx y, 
 ...... equation (S 
 
 whole number. 
 
 Now letv/^ 2 -f B*+2AB=p+f y where pis the nearest 
 whole number less than the true root, and consequently/ a 
 proper fraction; 
 
 and let iyA* + B* <2AB = q /', where q is the nearest 
 whole number greater than the true root, and consequently f a 
 proper fraction. 
 
 Then /A* + B* + 2AB+ t/A*+B*-2AB=l)+q+f-f 
 = SL whole number, 
 -/ = 0, to or/=f; hence 
 
 = t}.-. 2x = t + *> and V f x = 
 = *-, and V f = 
 
 Hence 
 
 (*) Since /and /' are both proper fractions, it is evident that//' cannot 
 be a whole number, and consequently ja-J- q+ff cannot be a whole 
 number, unless/ /'Q, or/=/ / . 
 
 ( b ) For4a? = ^-i-3o, /. 2v^^=^+^, and ^*=i^+2a; in the 
 same manner it may be shewn that -Sy=.\ -/ilo.. 
 
EXTRACTION OF THE tttll ROOT 
 
 CASE II. 
 
 When A 11 B* is not a complete 7zth power. 
 In this case let Cbe so assumed as to make (A* B*)Ca complete 
 nth power, i.e. let (A*-B*)C= n , OTf/(A*-B')C=* ; then 
 
 */X -f 
 
 assume ( + =*x + * 9 or 
 
 and, 
 
 From which we deduce, as before, 
 
 EXAMPLE 1. 
 Find the cube root of 26 + 1 5>/3. 
 
 e ~ 
 
 780^3 = 13 -f/, 
 
 -780^/3= 1 -/; 
 
 .*. /= 13 + 1 = 14. 
 
 Hence 
 
 12 
 
 Ex. 2. 
 
 Find the cube r.oot of QN/3 1 1*/2. 
 ^,_ si= 
 
 = y 243 + 242 + 198 v / 6 = 9 +/, 
 
 Hence t = 9 + 1 = 10, and 
 
 = y 243 + 242 -198^ 6= 1 -/. 
 ~2*=i 12 ~ 
 
 Ex. 3. 
 
OF A BINOMIAL SURD. 193 
 
 Ex.3. 
 
 Find the cube root of 8 + 4V5, or 4 A /5 +8. 
 
 =4x/52.\ ^ 8 -jB a = 80-64= 16, which is not a 
 J3 = 8 \cule number, and the least number which 
 multiplied into it will produce a cube number is 4, (a) .*. C=4, 
 and (>4 2 -~J3 a )C=l6x4 = 64; hence * 3 = (54, and = 4. 
 Now J/(A* + B* + 2AB)C=<y (80 + 64: + 64</5)4 = 10 f/, 
 
 Hence 
 
 ( a ) In finding the feas* number by which a given number (a) must be 
 multiplied so as to give a product which shall be a complete wth power, it 
 may be observed, that if a be a prime number, it must always be multiplied 
 by a"" 1 ; thus, there is no other number by which 3 can be multiplied to 
 make it a cube number, but 3* or 9, which gives the product 27 ; nor is 
 there any other number by which 5 can be multiplied to make it a biquadrate 
 number, but 5 3 or 125, which gives the product 625. But if the given 
 number is resolvable into factors, one or more of which are square, cube, &c. 
 numbers, then a less number than a n ~' will answer the purpose. Thus 
 1 2 = 3 X 4 = 3 X 2 a ; and if 3 X 2 2 be multiplied by 3* X 2, it gives 3 3 X 2 3 , 
 which is the cube of 8 X 2 ; i.e. if 12 be multiplied by 18 it gives 216 the 
 cube of 6. Or in general, if the given number (a) be resolvable into factors 
 a, j3, 7, &c. such that a=o m j8 p 7' &c., then if this number be multiplied 
 by a""" 1 jS""^ 7"-* &c. it gives a n 0" 7" &c. which is the nth power of 
 ajS7&c. Thus 360 = 8X9X5=2 3 X3*X5; here m=3, J?=2, 3= 1 ; 
 and if it be required to find a multiplier which should make it a biquadrate 
 number, then w=4, .'. n m=l, n p~2, n g> = 3 ; hence the mul- 
 tiplier is 2 X 3^ X 5 3 : =2250, and we have 360 X 3250= 810000, which is the 
 fourth power of 2 x 3 X 5 or 30. If one or more of the indices ?n, />, #, &c. 
 be greater than w, then, in finding the multiplier, such a multiple of n must 
 be taken as to make the indices of all the factors in the multiplier positive; 
 thus if m be greater than n but less than 2w, then the multiplier to be taken 
 is cr 2 "- j8 n ~^ 7"-', which gives for the product of it and a'" &> 7' the quan- 
 tity a 2M jff 1 7", which is the nth power of a? j8 7. 
 
 Cc 
 
194 METHOD OF REVERTING A SERIES. 
 
 LII. 
 
 On the Method of reverting a Series. 
 
 Let x=ay + ly* + cy 3 +dy* + &c., where the value of x is 
 expressed in a series containing the powers of y ; by the re- 
 version of the series is meant such an operation as shall exhibit 
 the value of y in a series containing the powers of x. 
 
 173. Previously to the reversion of a series, it will be ne- 
 cessary to shew the manner in which it may be raised to any 
 power (n). This is done by separating the first term from the 
 rest, and then applying the binomial theorem to the involution 
 of the series so transformed ; thus 
 
 = a V -f- na*- 1 *"- 1 ^ * 7 -f ctf +dx* -f &c.) + 
 
 2 3 
 
 ^ 
 
 2 . 3 
 
 174. Let us now suppose the following equation to be true, 
 whatever be the value of x 9 
 
 then, by transposition, we have 
 
 Now whatever is true in the original equation, must also be 
 true in the transposed equation ; but it has already been 
 proved with respect to the former equation (Note ( b ), p. 17 8), 
 that a = ; 6 = /3; c=y; d=3', &c.; hence a =0; b /3 = 0; 
 e y=0; d ^=0; &c. ; from which it follows that if an 
 equation of the form (B) be true for any value of x } its coeffi- 
 cients will all become equal to at the same time. 
 
 175. Resuming the equation x = ay + ly* + cy* -f-cty 4 -f&c. 
 
 let 
 
METHOD OF REVERTING A SERIES. 195 
 
 let it be required to find the value of y in terms of x. Trans- 
 
 pose x to the other side of the equation, then 
 
 + dy* + &c.x=0. Assume y = 
 
 and finding the value of the successive powers of y, by Art. 173, 
 
 we have 
 
 ay = 
 
 &c.= ......... + &c. 
 
 X= X 
 
 Hence, by Art. 1 74, a* 1 =0, or *=- ; 
 
 x =0, or/3=- =~,- a ; 
 
 Substitute these values for #, /3, y, 5 5 &c. then 
 
 x _lx* (2b*-ac)x 3 ^(5L 3 -5abc + a*d)x 4 &c 
 
 and if a=l, or ^=2/ + % 2 + c?/ 3 + 4 + &c. then 
 
 176. In the following chapter it will be shewn, that if / be 
 the logarithm of the number 1 +7i,l=n \rf + Jrc 3 Jw 4 -f &c.; 
 suppose therefore it was required to find the number in terms 
 of the logarithm, i.e. to find n in terms of /, then comparing 
 the equation l=n in* + J 3 -ra 4 + &c. with the equation 
 x=y + l)if + c?/ 3 + c??/ 4 +&c. and substituting / for x and n for y 
 in the equation (^f), we should have 
 
 where 
 
196 METHOD OF REVERTING A SERIES. 
 
 where = j, c=~, d= - ; &c. ; 
 
 CHAP. XI. 
 
 ON LOGARITHMS, 
 
 AND SUBJECTS CONNECTED WITH THEM. 
 
 LIII. 
 
 Definition and Properties of Logarithms. 
 
 177. IN the two following series of quantities, a x , a*', a- r " t 
 a'"' 9 &c. (A) ; x, x', x", x" 9 &c. (E) ; where a is some given 
 number, and x, x', x", x", &c. any variable quantities whatever, 
 the several terms of the series (B) are called the logarithms of 
 the several terms corresponding to them in the series (A). 
 Thus tfa x =y,a*=y', a*'~y", &c. then x=log.i/; x'=\og.y'; 
 x" = \og.y"; c. 
 
 178. In adapting the series (A) to the numbers 1, 2, 3, 4, 
 5, 6, &c. the given number a must be greater than unity, the 
 
 first 
 
LOGARITHMS. 197 
 
 first index x must be equal to 0, and the several indices x, x", 
 x", &c. must keep continually increasing. For in this case, since 
 (by Art. 66.) a= 1, this series will increase from 1 to infinity ; 
 and by properly adjusting the values of x', x\ x", &c. it is 
 evident that the several quantities a*, a*", a*'", &e. may be 
 made to coincide with the numbers 2, 3, 4, 5, 6, &c. For 
 instance, let a=10; then (since 10= 1 and 10'= 10), the 
 indices of 10 which would give lO 1 ', 10*", 10*"', &c. equal to 
 the numbers 2, 3, 4, 5, &c. must be fractions between and 1. 
 
 Take for example the number 5. Now 10^=v/10 a =v/100 
 = 4.64; from which we infer, that a fraction (x') somewhat 
 
 2 
 greater than -( = .666666, &c.) being made the index of 10, 
 
 iJ 
 
 would give 10*' = 5 ; this fraction is found by calculation to be 
 .6989700; hence 1 >69B97 = 5 ; i.e. when a=10, the loga- 
 rithm of 5 is .6989700. 
 
 179. From hence it appears that the logarithm of any given 
 number will depend upon the value of a, and that different 
 systems of logarithms would be formed by assuming it equal to 
 different numbers, but that (since a= l) in every system the 
 logarithm of one would be 0. This constant quantity a, from 
 whose powers the natural numbers are formed, is called the base 
 of the system to which it belongs. But before we proceed to 
 calculate a system of logarithms, it will be proper to explain 
 some of their properties. 
 
 1 80. Let N and n be any two numbers belonging to the 
 series (A) ; let N (for instance) =a x , and n (f""- 9 then 
 Nn a*xa*""=:a* +x ""', but by Art. 177, the logarithm of 
 0*+*"" i s x + x"" 9 .'. the logarithm of Nni=x + x""=]og. a* + 
 log. a*""=Iog. IV -I- log. n. In the same manner, if n, n', 
 n", ri", &c. be any set of numbers belonging to the series (A], 
 it might be shewn that the logarithm of n TZ W, &c. = log. n 
 
 4- log. Tz' + log. ?/' + log. ri" + &c. ; i. e. " the logarithm of the 
 "product of any number of factors is equal to the sum of their 
 <( logarithms." 
 
198 LOGARITHMS. 
 
 N a* 
 
 181. Again, =z-x^=za x *""$ but the logarithm of a**"" 
 
 =x-x""; /. the logarithm of ^=#-*""=log. a*~log.a*"" 
 
 = log. AT log. TZ; from henoe it appears that " the logarithm of 
 " the quotient of any two numbers is equal to the difference of 
 
 /N\ 
 " their logarithms ; and that the logarithm of & fraction f 1 
 
 " is equal to the logarithm of its numerator minus the logarithm 
 " of its denominator." If N be less than n, then log. IV 
 log. n is negative ; consequently the logarithms of all proper 
 fractions are negative quantities. 
 
 182. Let N=a x be raised to the wth power, then N m = a mx ; 
 but the logarithm of a m ' = mx; hence the logarithm ofN m =mx 
 
 = w.log. a*=7W. log. N; for the same reason, since jyN=N m 
 
 x x loff 7V" 
 
 = a", the logarithm of v/ N=- = - : from which we 
 
 m m ' 
 
 infer that "the logarithm of the rath power of any number is 
 " found by multiplying its logarithm by m ; and of the mth 
 " root of any number, by dividing its logarithm by m." 
 
 183. If the series (A) consists of quantities of the form a x , 
 a% a 31 , a 4x , &c ...... a"*, then the corresponding terms of the 
 
 series (J5) are x, 2x, 3x, 4x, &c ..... nx; i.e. "if a series of 
 
 " quantities be in geometrical progression, their logarithms will 
 " be in arithmetical progression." 
 
 LIV. 
 
 On the Method of finding the Logarithm of any given 
 Number. 
 
 184. Let 1 +n be any number in the common arithmetical 
 scale, and x its logarithm, then, Art. 177, a x = 1 +n ; and let 
 a 1 +b ; then, to find the logarithm of 1 +TZ, we have only 
 to solve the equation (1 + l) x = 1 -f n, where x is the unknown 
 quantity. 
 
 Let 
 
LOGARITHMS. 199 
 
 Let both sides of this equation be raised to the power h, then 
 (l+l) h *=(l+n) h , or 
 
 J>.6 2 2.3 
 
 rejecting 1 from each side of the equation and dividing by h, we have 
 * + & , =n+M * + <* 
 
 Now let h = Q, and we have 
 
 n 
 
 .), if W e make 
 to M 
 
 185. But the series which thus expresses the value of # in 
 terms of n y will not converge so quickly as to make the sum- 
 mation of a few terms of it a sufficient approximation to that 
 
 value, unless n be a. proper fraction. Let, therefore, Ji=-^-_ lf 
 where JVmay be any number greater than 2, then 
 
 1+n i+jV^Tl JV 
 \-n~ i 
 
 and log. (1 -H?i)-log.(l -w) =log. JV-log.(IV- 2). 
 
 Now log.(l +n) =M(w-i7z a + K-K + T^ 
 
 Hence, 
 
200 LOGARITHMS. 
 
 Hence, by subtraction, log.(l + ?i) log.(l -w) = 2M(?i 
 
 or lag-^log.(^)^ 
 from which we have 
 log. N 
 
 which is a very commodious series for constructing a lalle of 
 logarithms, when some value has been assigned to A/. 
 
 LV. 
 
 On *Ae Method of constructing Logarithmic Tables. 
 186. Since a may be arbitrarily assumed, let us first suppose 
 
 in which case the equation in the foregoing Article becomes 
 
 But since J^must be some number greater than 2, we must 
 find the logarithm of 2, before we can proceed to the actual 
 calculation of a table of logarithms. Now this may be done 
 by making N=4 in the first instance, for then we have 
 
 log. 4 = log. 2* = 2 log. 2 = 2 (j + ^3 + ~s + &c.) + lg- 2 - 
 
 and by subtracting log. 2 from each side of the equation, we have 
 
 log. 2 = 2-+ + 3 + &c - to 7 terms =0.693 1472. 
 
 Having thus obtained the logarithm of 2, we are enabled to 
 construct a Table of logarithms, by substituting in the fore- 
 going series all the prime numbers for IV" in succession, and 
 availing ourselves of the properties of logarithms for finding 
 the logarithms of all other numbers. Thus, 
 
 log. 
 
 
LOGARITHMS. # 
 
 log. 
 
 1 = ......................... 0.0000000 
 
 2 = ......................... 0.6931472 
 
 \ 2 . 3 . 2 <5. y 
 
 4 =2 log. 2 =1.3862944 
 
 5 (^H-^p + J-p-f&c. to 6 terms) + log. 3. =1.6094379 
 
 6 = log. 3 + log. 2 =1.7917595 
 
 . =1.9459101 
 
 8 = log. 4 -J- log. 2, or log. 2 :{ = 3 log. 2 ..... =2.0794415 
 
 9 =log. 3 3 =2 log. 3 ............... =2.1972246 
 
 10 =log. 5+ log. 2 ................ =2.3025851 
 
 &c. = &c ....................... = &c. 
 
 A sufficient number of terms has here b^en made use of to 
 make the logarithms true to 7 places of decimals. This par- 
 ticular system of logarithms (viz. where M = 1 ) are called 
 Napier's logarithms, from their inventor ; and they are also 
 called Hyperbolic logarithms, from their connection with the 
 quadrature of the equilateral hyperbola. 
 
 187. To find the base of this system of logarithms, let log. 
 (l+w) = , then (since M= i), J=7z-iw a +w 3 J 4 -f &c., 
 and reverting the series by Art. 176, we obtain 
 
 but since a 1 = a, the base of any system of logarithms is that 
 number whose logarithm is 1 ; if therefore in this series, which 
 expresses the value of the number in terms of the logarithm, 
 we substitute 1 for I, we shall immediately obtain, for the base 
 of this particular system, the series 
 
 = 2.7182818, by actual calculation. 
 
 DD The 
 
LOGARITHMS. 
 
 The constant multiplier M is called the Modulus ; hence, in 
 that particular system of logarithms whose Modulus is 1, the 
 base is 2. 7 1828 18. Call this number e, and the logarithms of 
 the several powers of e (viz. e^e 2 , e 3 , e 4 , &c.) being 1, 2, 3, 4, &c. 
 we might have interposed in the preceding Table 
 Log. 2.7182818 =1.0000000 
 
 Log. 7.3890559 (being the square of 2.7 182 8 18) = 2. 0000000 
 &c. =&c. 
 
 The numbers whose logarithms are 1,2, 3, 4, &c. in this 
 system are, therefore, decimal numbers. 
 
 188. In the common system of logarithms, which are much 
 more convenient for ordinary arithmetical operations than the 
 Napierian or Hyperbolic logarithms, the base a= 10; hence 
 a a =dOO, a 3 =1000, a 4 = 10000, &c., and the numbers whose 
 logarithms are 1, 2, 3, 4, c. in this system, are 10, 100, 100O, 
 10000, &c. To find the logarithms of the intermediate num- 
 bers, i.e. to construct a table of logarithms of this kind, 
 we must find the value of M when a = 1 0. Which is done thus, 
 
 In a system whose Modulus is M, log. 
 
 In the Napierian system, Iog.(l +ri)z=n 
 
 Hence log.(l +ri) to Modulus M=M x Nap. log. (1 +n) 
 In the common system, let 1 4-72=10, then 
 
 log. 1 = M x Nap. log. 1 
 
 or 1 = MX 2.3025851, see Art,l 86. 
 
 202585 
 
 For the actual construction of a Table of common logarithms, 
 we must therefore substitute this value of M in the equation 
 at the end of Art. 185, which then becomes 
 
 and it is by the substitution of all the prime Numbers in suc- 
 cession for j!V in this expression, that the following Table is 
 calculated. 
 
LOGARITHMS. 203 
 
 k* /111 \ , 
 
 2 = .86858S96(-+ i + 7-rs + &c. to7 terms/ 10 =0.3010300 
 
 \<3 oo O.o / 
 
 111 X 
 
 - + 3 + ir^5 + &c. tolOtermsJ =0.4771213 
 
 4=2 log. 2 .................... =0.6020600 
 
 5 =log. =Jog. 10 log. 2 = 1 log. 2 .... =0.6989700 
 
 6 =log. 3-f log. 2 ............ . . . . =0.7781513 
 
 7 =.86858896(g-f ^ + ^5+y^- 7 ) + log. 5. . =0.8450980 
 
 8 =log. 2 3 = 3log. 2 ............... =0.9030900 
 
 9 =log. 3* = 2 log. 3 ............... =0.9542425 
 
 10 = ........................ = 1.0000000 
 
 5 )+log.9. . . =1.0413927 
 
 3 
 
 12 = log. 3 -flog. 4 ................ =1.0791812 
 
 (I 1 1 \ 
 
 13 =.86858896\--f^j^3rf^Y 5 ) +log. 11 . . =1.1139434 
 
 14 = log. 7+ log. 2 ................ =1.1461280 
 
 15 =log. 5+log.3 ................ =1.1760913 
 
 16 =log. 4*= 2 log. 4 ............... =1.2041200 
 
 17 =.86858896^-f-g^g3 + ^jg5 + og. 15 . . =1.2304489 
 
 18 =log. 9 -Hog. 2 ................ =1.2552725 
 
 19 = 8685 88 96 (- 8 -f 5^3 + ^Y^ 5 ) -flog. 17 . . =1.2787536 
 
 20 =log. 10 + log. 2 ................ =1.3010300 
 
 21 =log. 7 + log. 3 ................. =1.3222193 
 
 22 =log. 11 + log. 2 ................ =1.3424227 
 
 23 =-86858896(^4- 5^3 + ^5) +log. 21 . . =1.3617278 
 
 The next number which requires calculation by means of the 
 series, is 29 ; and from this number to 400 inclusive, two terms 
 of the series are sufficient to make the logarithms true to 7 places 
 
 of 
 
 ( a ) See Art. 1 86. 
 
204 LOGARITHMS. 
 
 of decimals. ' After 400, one term is sufficient ; thus log. 401 
 .86858896 
 
 = .0021714724 -f 2.6009729 = 
 
 2.6031444 (very nearly) ; and in this manner the table might 
 be continued with great facility to any extent, by means of tiie 
 logarithms previously calculated. For the most expeditious 
 manner of dividing "the number .86858896 by the denominators 
 of the several fractions composing the series, and for the manner 
 of using logarithmic tables, the reader is referred to the Preface 
 annexed to Dr, BUTTON'S Tables. 
 
 1S9. Since log. 1 =0, log. 10=1, log. 100 = 2, log. 1000 
 = 3, &c., it follows that the logarithms of all numbers between 
 1 and 10 will be some decimal number less than unity ; between 
 10 and 100, some decimal number between 1 and 2 ; between 100 
 and 1000, some decimal number between 2 and 3 ; &c. &c. The 
 whole number annexed to the decimal is called the index or 
 characteristic of the logarithm ; and consequently for all numbers 
 between 10 and 100 the index is 1 ; between 100 and 1000, 
 the index is 2; between 1000 and 10000, the index is 3 ; &c. &c. 
 From the circumstance of log. 10=1, it also follows that the 
 logarithms of all numbers in decuple proportion involve the 
 same decimal number, and differ only by their index. 
 
 Thus, Log. 1132 =3.0538464. 
 
 1132 
 Log.. 113.2:=log. =iog. 11321=2.0538464. 
 
 1132 
 Log. 1 1.3-2 =-log: j-=log. 113.2-1 = 1.0538464. 
 
 11.32 
 Log. 1.132=log/ -*y^- =log. 11.32-1=0.0538464. 
 
 1.132 , - 
 
 Log, .1132 = log. j^-=!og. 1.132-1 = 1.0538464. 
 
 .1132 - 
 
 Log. ,01132 = log. j^- = log. .1132 1 = 2.0538464. 
 
 .01132 (> 
 
 Log. .00 11 32 = log. -=log.. 01 132-1 = 3.0538464; 
 
 1Q 
 
 where 
 
 (*) The index of a logarithm may in all cases be determined by the 
 following simple rules ; 
 
 i. If 
 
LOGARITHMS. 
 
 where the negative sign is placed above the index of the last 
 three logarithms, to shew that it does not extend to the deci- 
 mals, which are .supposed positive. Thus 3.0538464 means 
 3 + .0538464, or 2.9461536. 
 
 190. The foregoing property, belonging to that particular 
 system of logarithms arising out of the supposition of the base 
 =z 10, is not only of great practical utility in their application 
 to arithmetical purposes, but also very much facilitates the 
 construction and use of the tables founded upon that system. 
 Since the same decimal logarithm always applies to a number 
 consisting of the game digits, it follows that in the construction 
 of a table of common logarithms it is only necessary to register 
 the digits of the number and the decimal logarithm in parallel 
 columns; for the index of the logarithm may always be deter- 
 mined from the actual value of the number; and, vice versa, the 
 actual value of the number may always be determined from the 
 index of the logarithm. For instance, in the common tables . 
 where the logarithms are registered for all numbers consisting 
 of five figures, the decimal logarithm belonging to the number 
 98637 is .9940399; if this number be a whole number, then 
 since it consists of 5 integral digits, we know that its logarithm 
 is 4. 9940399; ifadecimal point be placed before the last figure, 
 then the value of the number is 9863.7, which has four inte- 
 gral digits, and therefore its logarithm is 3.9940399; if a de- 
 cimal point be placed before the last figure but one, then the 
 number is 986.37, and its logarithm 2.9940399 ; &c. &c. On 
 the other hand, if the logarithm 1.9940399 was given to find 
 the corresponding number, then since the decimal part of it 
 belongs to the digits 98637, and since from the index of the 
 
 logarithm 
 
 i. If the number be integral, with or without decimals annexed, the 
 index of the logarithm will be one less than the number of digits in the 
 integer. 
 
 n. If the number be a proper decimal fraction, the negative index will 
 be equal to the place of the first significant digit after the decimal point. 
 
206 APPLICATION OF LOGARITHMS 
 
 logarithm we know that the numher has two integral digits, 
 the figures 98637 must he pointed 98.637; &c. &c. The 
 utility of this system was so obvious, that the tables for ordi- 
 nary purposes were founded upon it very soon after the inven- 
 tion of logarithms. 
 
 LVI. 
 
 On the application of Logarithms to Complex Arith- 
 metical Operations, and to the solution of Exponential 
 Equations. 
 
 191. Logarithms are of considerable use in the ordinary 
 operations of multiplying or dividing one large number by 
 another; but it is in the raising of powers, and the extraction 
 of roots, and in their application to complicated numerical 
 expressions, that their utility most plainly appears. 
 
 EXAMPLE 1. 
 
 Find the 5th root of 2593. 
 By Art. 182, the logarithm of the 5th root of 2593 = 
 
 log. 2593 3.4138025 
 
 -= = .6827605 = log. 4.8168; /.the 5th 
 
 o o 
 
 root of 2593 = 4.8168. 
 
 Ex.2. 
 
 2 20 X3 7 X2.013 
 Find the value of the fraction 
 
 / A / xs y .** O \J 
 
 By Art. 181, the logarithm of this fraction is equal to the 
 log. of its numerator minus log. of its denominator. 
 By Art 5 . 180, 182,log.2 c 'x 3 7 X 2.013 = 20log.2 + 7log.3 + log.2.013, 
 
 and, log. 17 X 9350 =log.!7+log. 9350. 
 
 Now 20 X log. 2 = 6.0206000 . . log. 17=1.2304489. 
 7 X log. 3 = 3.3398491 . . log. 9350 = 3.9708116. 
 log. 2.013=0.3038438 
 
 By addition = 9. 6642929 (A.} 5.2012605 (J5). 
 
 Subtract (B) from (A), and we have 4.4630324, which is 
 the logarithm of 29042, the number required. 
 
TO COMPLEX ARITHMETICAL OPERATIONS. 207 
 
 Ex.3. 
 
 r" j u t 
 
 Find the value of 
 
 Call the numerator of this fraction (JV), and its denomi- 
 nator (n) ; 
 Then,byArtUSl,lS2>g.of\/~= - ] S- n t 
 
 V /Z O 
 
 Now log. (317)' = 2 x log. 317 = 5.0021156. 
 log. x/S^xlog. 3 = 0.2385606. 
 5=0.2329900. 
 
 5.4736692 = log. AT. 
 
 log. 251 = 2.3996737; 
 
 b 
 
 .*. 3.0739955=log.A 1 '-log.n. 
 
 log.IV-log.72 3.0739955 
 Hence - = =0.6147991, which is the 
 
 O O 
 
 logarithm of 4.1 19, the number required. 
 
 Ex. 4. 
 
 Find a fourth proportional to the 6th power of 9, the 4th 
 power of 7, and the 5th power of 5. 
 
 7 4 X 5 s 
 Let x= the number required, then 9 6 : 7 4 :: 5 s :a;= -g ; 
 
 .'. log. x = 4 log. 7-f 5 log. 5 6 log. 9 = 3.3803920 + 
 3.4948500 5.7254550= 1.1497870 = log. 14.118; hence 
 #=14.118. 
 
 192. Equations into which the unknown quantity enters in 
 the form of an index, are called Exponential Equations ; and 
 are solved by means of Logarithms, as in the following ex- 
 amples. * 
 
 Ex. 5. 
 
 Find the value of x in the equation a x = b. 
 Taking the logarithm of the equation a* =6, we have 
 
 x. log. a = log. I, ..#== p^- ; thus, let a = 5, =100, then 
 
 loff. 100 2.0000000 
 in the equation 5*. 100, x -;- = ; 
 
 log;. 5 0.< 
 
203 APPLICATION tfF LOGARITHMS &C. 
 
 Ex. 6. 
 
 To Jin d the value ofx in the equation G &X =C. 
 Assume (a) b x y, then aV c, and y. log. a = log. c, .'. y = 
 
 loi^. c l^g c 
 
 ; hence ^ ar = , - ('which let) =6/. Take the logarithm 
 log. a ' log. a v 
 
 of the equation l x = d, then (by Ex. 5.) , = ; ~ i ; thus, let 
 
 x log. c 
 
 # = 9, 1 = 3, c= 1000, then in the equation 9 s = 1000, T = 
 
 log. 1000 IOJT. d log. 3.14 .4969^96 
 
 1.04. 
 
 31 X33 X 255 X315 
 Lx. 7. Find the value of - 35x357 - 
 
 ANSWER, 6576.4. 
 
 Ex. 8. Divide the 20th power of 2 by the 12th power of 3 
 
 ANSVV. 1.973. 
 
 Ex. 9- Find a third proportional to ^117 
 
 ANSVV. 10.252. 
 
 r rv i i ,^935 X V 14xJXlOO 
 
 Ex. 10. Find the value oP' - ~~^T~ 
 
 <v 
 ANSVV. 3. 3593. 
 
 a l x -f- c 
 Ex. 1 1. Find the value of x in the equation -j =e. 
 
 LVII. 
 
 On the Summation of Geometric Series. 
 
 193. Logarithms are found very useful in ascertaining the 
 
 ar tt a aar* 
 
 value of S in the equation S= - or > where n is 
 
 r 1 1 r 
 
 not a very small number. 
 
 EXAMPLE 1. 
 
 3 9 27 
 Find the sum of 20 terms of the series 1, -^ -> -> &c. 
 
 ( a ) In considering the nature of an exponential of the form a 6 *, it must 
 be recollected that i t means a to the power of A*, and r.ot a 6 to Hie power ofj?. 
 
SUMMATION OF GEOMETRIC SERIES. 209 
 
 ar-a ' 
 
 Q 20 o 
 
 Now log. (-) = 20 X log. - 
 
 2 
 
 = SOX (log. 3 log. 2.) 
 = 3.5 2 18260 = log. 3325.263 ; 
 3 v 20 
 
 .-.(-) =3325.263. 
 
 ~ol-0 
 
 Hence S=2 X (-J l) =2 X 3324.263 = 6648.526. 
 
 Ex. 2. 
 
 Find the sum of 1O terms of the series l, - -p ^> &c. 
 
 O oO 210 
 
 aar" 
 
 10 
 
 1-rl X 1-1 
 
 6 
 
 /5\ 10 5 
 
 Now log. (^) =lOxlog. ^ 
 
 = lOx(log. 5 log. 6.) 
 = 1OX .0791813. 
 = -.7918130. 
 = .2081870 1.0000000. 
 = log. 1.6150 log. 10. 
 1.6150 
 
 Hence S = 6l-g =:6(l -.1615) = 5.031. 
 
 194. If the sum of the series, the common ratio, and the 
 first term be given ; the number of terms may be found thus 
 (See Art. Ill); 
 
 E E Since 
 
210 SUMMATION OF GEOMETRIC SERIES. 
 
 Since r S S = a r* a ; 
 By transposition, a?-* rSS + a, 
 
 rS-S + a 
 and r n 3= ; 
 
 .* . log. r * or w x log. r = log. (rSS + a) log. fl, 
 
 Hence M= 
 
 ; 
 log. r. 
 
 Ex.3. 
 
 The sum of a geometric series is 6560, its first term 2, and 
 common ratio 3, What is the number of terms ? 
 
 Here S= 6560, 1 log. (rS-S + a) log, a 
 
 r 3; 3 log. 13122 log. 2 
 
 log. 3 
 3.8169700 
 
 .4771213 
 
 = 8. 
 
 Ex. 4. A servant agreed to serve his master for one year 
 (13- months), at the rate of sixpence for the first month, a 
 shilling for the second, two shillings for the third, and so on ; 
 What had he to receive at the end of the year ? 
 
 ANSWER, 2O4/. 155. 6d. 
 
 5 2 5 
 
 Ex. 5. Find the sum of 1 1 terms of the series. i,- 5 ~, &c. 
 
 4 lo 
 
 ANSW. 42.568. 
 
 Ex.6. The sum of a geometric series is 1023, the^ro/ 
 term 1, and common ratio 2 ; Find the number of terms. 
 
 ANSW. 10. 
 
 Ex. 7. A person undertakes a journey of 364 miles, going 
 one mile the^T5^ day, three the second, nine the third, and so 
 on ; When will he arrive at his journey's end ? 
 
 ANSW. in 6 days. 
 
COMPOUND INTEREST. 
 
 . LVIII. 
 
 On Compound Interest. 
 
 Let (P) be the principal, or sum put out to compound in- 
 terest; (r) the fraction which expresses the rate of interest per 
 cent. 00 ; (A) the amount at the end of (ri) years, the interest 
 heing paid yearly ; Then the following Theorems may be 
 established, by means of logarithms. 
 
 THEOREM I. 
 
 195. " Log. Alog.P + n x log. (l +r)." 
 For since .1, at the end of the first year, becomes 1-fr, 
 and that the amount is increased each year in the same ratio, 
 we have, by the rule of proportion, 
 
 ]-: 1 + r : : P : P ( 1 + r] = amount of P at end of first year. 
 
 1 :1 +r::P(l +r) :P(l-f r)'= . . . . . .... . second year. 
 
 l:l+r::P(l+r)':P(l + r) 8 = ..... ..... third year. 
 
 &c. &c. 
 
 So that, at the end of n years, the amount is P(l -f r)*. 
 
 Hence A=P(l+r) n ', 
 
 and, taking the logarithm, log. A log. P + rax log. (l+r). 
 From which we deduce, 
 
 Log. P=log. Anx log. (l + r), 
 
 log. A -log. P 
 Log.(l+r) = -2 - - -- ! 
 
 log. A- log. P 
 
 Any /^ree of the quantities A, P, r, w, being given, theybwr//i 
 may therefore be found. 
 
 THEOR. 2. 
 
 log. ?7i 
 
 " 
 
 196. "Let A=mP, then n = 
 
 For, in this case, m P=P(1 + r} n . 
 'Divide by P, then w=(l-fr) n , 
 _J Take 
 
 ( a ) That is, the fraction which expresses the ratio of the interest to the 
 principal. Let the interest, for example, be 5 per cent. ; then this frac- 
 tion (r) will be or 
 
 100 20 
 
COMPOUND INTEREST. 
 
 Take the logarithm, log. m c= n x log.( 1 -f- r) j .'. n = - ""Tj~r-y 
 
 By means of this Theorem, we ascertain the period or number 
 of years in which a sum of money would double 9 treble, &c. or 
 amount to m times itself, when put out at compound interest, 
 at r rate per cent. 
 
 THEOR. 3. 
 
 197. "Suppose the interest to be paid half' yearly, and at the 
 " same time converted into principal, then will log. A log. P 
 
 For in this case, 2 n must be substituted for n, and |r for r. 
 Hence, at the end of n years, A P(l +ir) Sfl ; 
 and, taking the logarithm, log. ^4= log. P-f 2 n x log. (l -f Jr). 
 
 THEOR. 4. 
 
 1 98. " Suppose 7iow, that besides the interest being converted 
 " into principal at the end of every year, the sum P is at the 
 " same time invested in capital; then the amount (A), at the 
 
 P R(R n _ i^ 
 " end of n years, will be - ^_ 1 '(\fR=l+r)." 
 
 In this case, the principal (P) is put out for n, nl, n <2, 
 &c. years, in succession ; the amount therefore is the sum of 
 the several amounts of ( P) put out for n,nl,n <2, &c. years ; 
 
 **). 
 
 p( ft*+ 1 p\ 
 
 = P x (Geo.Prog.firsttermR,comwonratwR)=-^-n~-^ -. 
 
 PR(R*-l) 
 
 R ~ l EXAMPLE 1. 
 
 What would be the amount of 200/. placed out for 7 years, 
 at 4 per cent, compound interest? 
 Here P= 200, 
 
 ' .-.byTH.l.iog.^=log.P + nxlog.(l -fr). 
 = log. 200 + 7 X log. 1 .04. 
 
 r =i? 
 
 = 1 .04, 
 
 = 2.4202631. 
 = log. 263.18.- 
 
 Hcnce, ^/=263/. 3s. 
 
COMPOUND INTEREST. 
 
 213 
 
 Ex. 2. 
 
 How much money must be placed out at compound interest, 
 to amount to 500/. in 12 years, at 5 per cent. ? 
 
 Here A = 500, > 
 
 l By Th. 1. log. P = log. An x log.(l -fr). 
 ~20' = log. 500 12 X log. 1.05. 
 
 1 / =2.4446984. ' 
 
 20* = log. 278.41. 
 
 = 1.05, Hence, / } =278/. 8s. 2^d. 
 
 Ex.3. 
 
 At what rate of interest must 400/. be placed out, that it 
 may amount to 569/. 6s. 8d. in 9 yeats, at compound interest? 
 
 ByTh.l.log.(l+r) = 
 
 log.^-log.P 
 
 log.569.33 log.400 
 
 9 
 = .0170338. 
 
 Hence 1 --f r= 1 -f j 
 
 /. r= or the rate of interest 4 per cent. 
 
 Ex.4. 
 
 In how many years will 500/. amount to 900/., at 5 per 
 cent, compound interest? 
 
 Here^f=900, ^ T ^_ , , _ log.^-Iojr. P 
 P=500, 
 
 1 
 
 1 -f-r==1.05. 
 
 log.(i +7") 
 
 log. 900 log. 50O 
 log. 
 
 ,2552725 
 
COMPOUND INTEREST. 
 
 Ex.5. 
 
 In what time will a sum of money double and treble itself, 
 at 5 per cent, compound interest? 
 
 By Theor. 2. ( since r = J 
 
 log. 2. .3010300 
 
 ; i^Ko3 = .^rT^ 3 "= 14 ' 2 >' ears - 
 
 log. 3 .4771213 
 77t = 3 of treuling'zz] 
 
 Ex. 6. 
 
 Supposing the interest to be paid half yearly, what will be 
 theamountof500/.in 8 years, at 5 per cent, compound interest? 
 Here P = 500, 
 
 By Th.3. log. A\o{ 
 r= = log. 5004- 16 xlog.(1.025). 
 
 IJ-JU- 109^ =2.8705524 = log. 742.25. 
 
 A ~y~ "o"' " ~ *-*\J&3m 
 
 Hence^=742t. 55, 
 
 72 = 8. 
 
 Ex. 7. 
 
 Suppose a person to place out annually 100/. for 10 suc- 
 cessive years, and suffer the whole to accumulate at the rate of 
 5 per cent, compound interest ; What sum would lie have to 
 receive at the end of the tenth year ? 
 
 Here P= 100, ^ .'. by Theor. 4. 
 
 #=1.05A A PR(R*-l) 105(1. 05| 10 -l) 
 
 7z=10; ) R~~ l -O5 
 
 = 2100(T051 10 -1). 
 Now log. (l.05) to =10xlog. 1.05. 
 = .2118930. 
 
 = log. 1.6289; .*. (1.05) 10 1 = .6289. 
 Henoe ^ = 2100 X .6289. 
 
 = 1320/. 135. 9|^ 
 
 EXAMPLES FOR PRACTICE. 
 
 Ex. 8. What would be the amount of 1000/. placed out 
 at compound interest of 5 per cent, for 10 years ? 
 
 ANSWER, 1628/. 18s. 
 
COMPOUND INTEREST. 215 
 
 Ex. 9. What sum must be placed out at compound in- 
 terest, at 4 per cent., to amount to 2000A in 15 years ? 
 
 ANSW. l.llO/. 105. 
 
 Ex. 10. At what rate of compound interest must 518/. 6s. 
 be placed out, to amount to 600/. in 3 years ? 
 
 ANSW. 5 per cent. 
 
 Ex. 11. In how many years wili 200/. amount to 318/. 165. 
 at 6 per cent, compound interest? 
 
 ANSW. 8 years. 
 
 Ex. 12. In how many years will a sum of money double 
 itself, at 4 per cent, compound interest ? 
 
 ANSW. 17. 6 years. 
 
 Ex. 13. Find the amount of 1200/. put out to compound 
 interest at 6 per cent, for 10 years, the interest being con- 
 verted into principal every half year. 
 
 ANSW. 2167/. 65. 
 
 Ex. 14. Suppose a person to place out annually the sum of 
 20l. for 40 successive years, and suffer the whole to accumulate, 
 at the rate of 5 per cent, compound interest ; What would he 
 have to receive at the end of 40 years ? 
 
 ANSW. 2536Z. 165. 
 
 LIX. 
 
 On the method ofjlnding the Increase of Population 
 in any Country , under given circumstances of Births 
 and Mortality. 
 199. "Let (P) represent the population of a country at 
 
 " any given period ; (- J the fractional part of the population 
 
 " wlrich die in a year (or ratio of mortality) j ( 7 ) the propor- 
 
 " tion of births in a year; then, if (^f) represents the state 
 " of the population at the end of (n) years, log. A = log. P 
 
 The 
 
INCREASE OF POPULATION. 
 
 The rate of increase of population in one year = r -- = - -. ; 
 
 b m rub 
 
 m b m b\ 
 
 : : : ^ * ~*~ ' ~ state ^ t ' ie PP a ^ at ' on at 
 
 ' * "" nib ~~ mb 
 
 the end of theflrst year. 
 
 But it is increased every year in the same proportion ; 
 mb / m b 
 
 population at the end of the second year. 
 
 In the same manner we may prove, that the state of the 
 population at" the end of (//.) years will be P N -f - 
 
 (ill b 
 
 From which we deduce, 
 
 Log. P=rlog. A n x log. 
 , log. A- log. P 
 
 Of the quantities A, P, ra, b, n, any four being given, the 
 fifth may therefore be found. 
 
 EXAMPLE 1. 
 
 Suppose the population of Great Britain in the year 1800 to * 
 have been ten millions ; that i () th part die annually ; that the 
 births are to the deaths as 40 : 30 ; and that no emigration 
 takes place during the present century ; What will be the state 
 of its population in the year 1900 ? 
 Here P= 10000000," 
 
 12] 
 = log. 10000000 + 100 x log. , 
 
 1=7.3604200 
 = log. 22931000. 
 Hence y/ = 22931000. 
 
INCREASE OF POPULATION. 
 
 217 
 
 Ex. 2. 
 
 Suppose the population of France, in the year 1792, to have 
 been 27000000; the ratio of mortality during the 18th cen- 
 tury tq have been ^th, and the number of births jfih*, What 
 was the state of its population in the year 1700? 
 
 = 27000000, 
 
 1 + 
 
 S m -b\ 
 = log.^-?z x log. ( i+- j^y J 
 
 196 
 = 26, = lo g- 27000000 - 9 2 X log. 
 
 m b IQ6 =7.2269858 
 
 mb = T95 =log. 16864980, nearly; 
 
 .'.P= 16864980. 
 Ex. 3. 
 
 Suppose the population of North America to have been five 
 millions in the year 1800; In how many years will it amount 
 to 16 millions, taking the ratio of mortality at ^th, and the 
 annual proportion of births at J^th ? 
 Here A 16000000, ^ 
 
 P= 5000000, ' . ~ f m-b 
 m = 45, 
 6 = 24; 
 
 _ 
 mb "~360* 
 
 log. 16000000 -log. 5000000 
 ~ 367 
 
 .5051506 
 
 ' .0083636 - 
 
 Ex.4. 
 
 The population of a province in the year 1760, was estimated 
 at 500000 persons; in the year 1800, it amounted to 720000; 
 from the bills of mortality it appeared, that, upon an average, 
 55th part of the population had died annually; no register had 
 been kept of the births; What was the annual proportion of 
 them during this period ? 
 
 Here 
 
 ^=720000, 
 P = 500000, 
 ra=50, 
 H=40. 
 
 mb\ log. A log. P 
 
 ' -. I r^ 5 
 
 mu I n 
 
 50 M log. 720000 - log. 500000 
 ______ 
 
 = .0039590 = log. 1.009. 
 F F Hence 
 
INCREASE OF POPULATION. 
 
 50-b 9 
 
 Hence] + -^-=,.009= 1-f , 
 
 50-b 9 
 
 /. 50000 1000 /> = 450 . 
 50000 
 
 The annual proportion of births, therefore, was about ^tb. 
 
 200. But " in any country, under given circumstances of 
 " births and mortality, the fraction 7ir is always a given quan- 
 " tity ; Let it be represented by -Jj ', then the relation between 
 " the four quantities A,P,p,n, is expressed by A~P (l + ^) n - 
 "If^=?wP, we have mP = P (!+)", or wz=(i4-J)"j 
 " and taking the logarithm, log. m = nx log. ( 1 4- p). Hence 
 " we deduce the six following formulae." 
 
 II. 
 III. 
 
 IV. 
 
 log. m 
 V. n= -- f -- pr, for finding the period in which 
 
 the population would be increased m times. 
 
 VI. Log. ( 1 + -) = -~j^ for finding the rate (- ) at which 
 the population would be increased m times in n years. 
 
 The following Questions are intended to illustrate the use of 
 these formulae, in the order in which they stand. 
 
 QUESTION 1. 
 
 Suppose the population of a country to begin with six persons, 
 
 and 
 
INCREASE OP POPULATION, 
 
 219 
 
 and to increase annually by jUh of the whole ; What will 
 be the state of its population, at the end of 200 years ? 
 
 ANSWER, 1106448 persons. 
 
 QUESTION 2. 
 
 If (as stated in the 3d Example) the population of North 
 America was five millions in the year 1800, and the rate of increase 
 had been J^th for 50 years previous ; What was the state of 
 its' population in the year 1750 ? 
 
 ANSW. 1908930 persons. 
 
 QUESTION 3. 
 
 Suppose the population of an empire to be 40 millions, and 
 the annual increase ^.th ; How long will it be before it amounts 
 to 50 millions ? ANSW. 43.6 years. 
 
 QUESTION 4. 
 
 What must be the rate ofmcrease, that the population of a 
 country may be changed from 1106400 persons to five millions, 
 in 100 years ? ANSW. about ^,th annually. 
 
 QUESTION 5. 
 
 log. m 
 
 uy means 01 me loimuia // = , verijy me louow- 
 ing Table. Io &''( 1+ p) 
 
 1 
 P 
 
 Period of doubling 
 
 Period of trelliTts 
 
 o 
 
 Period of being increased 
 10 times. 
 
 1 
 120 
 
 83.5 years 
 
 132.3 years 
 
 27 7 A years 
 
 52 
 
 36.3 years 
 
 57.6 years 
 
 120.8 years 
 
 QUESTION 6. 
 
 What must be the annual increase of population in any 
 country, that it may douMe itself every century ? 
 
 ANSW. BetweenTT^d and TTltli. 
 143 144 
 
INCREASE OF POPULATION. 
 
 201. Supposing that a census of the whole population of a 
 country is taken every n years, and that it is found to have 
 increased * per cent, during that interval, then if P repre- 
 sents the amount of the population at the commencement of 
 
 it? 
 the 72 years, -P-r-jQQ will represent the amount of the popu- 
 
 lation at the end, of the n years. 
 
 If the annual increase be -, then, (by Art. 200), the amount 
 of the population at the end of n years is P(l +-J ; hence 
 
 T 
 
 l\ n <* 
 
 ) : l + = 
 
 oo 
 
 ^) = log.(100 + ,r)-log. 100 
 
 =log.(100-r-*-)-2,sincelog.lOO=2, 
 
 and log. '(l + -) = -(log.(100 + *)-2). 
 
 Substitute this value of log. (l -f--j in the expression 
 
 
 ( Form ulaV, Art. 200), and we have r 
 
 for the number of years in which the population of a country 
 will be increased m times, if it goes on increasing at the same 
 rate as it has done for the last n years preceding the period at 
 which the census is taken. 
 
 202. If the census be taken every ten years, and the period 
 of doubling be required, then n 10, m = 2, and the foregoing 
 
 Log. 2 
 expression becomes - - Bv substituting in 
 
 it for 9r the particular value of the per centage, the following 
 Table exhibits the corresponding period of doubling. 
 
INCREASE OF POPULATION. 
 
 LX. 
 
 A TABLE, exhibiting the Period in which the population of a Coun- 
 try has a tendency to DOUBLE itself, from an estimate of its increase 
 per cent, taken at the end of every Ten Years. 
 
 I. 
 
 II. 
 
 III. 
 
 Per Centage 
 increase in ten 
 years. 
 
 Numerical Value of 
 Tg(log.(lOO + )-s). 
 
 Period of doubling. 
 Log. 2, or .3010300 
 
 Tj(log.(lOO + )-s) 
 
 *= I.Q 
 1.5 
 2-0 
 2-5 
 3-0 
 3-5 
 4-0 
 4-5 
 5-0 
 
 OOO43914, . 
 
 696-60 years 
 465-55 
 350.02 
 280-70 
 23449 
 201-48 
 176-73 
 157-47 
 142-06 
 
 
 oooftOO9 
 
 nOlfY^^Q 
 
 0019^372 . 
 
 OO14Q4O3 . 
 
 .OO 170333 . . 
 
 OOlQllft3 . 
 
 OO9 1 1 ftQ3 . 
 
 
 w= 5-5 
 6-0 
 6-5 
 7-0 
 7-5 
 8-0 
 8-5 
 9-0 
 9.5 
 ,10-0 
 
 OO93' 7 ^2 f i . 
 
 129-46 years 
 118-95 
 110-06 
 102-44 
 95-84 
 90-06 
 84-96 
 8043 
 7637 
 72-72 
 
 OO9f>3O^Q . 
 
 O0273496 . 
 
 OO9Q3838 
 
 OO3 1 40*3^ . 
 
 
 003^4997 . . . 
 
 00^749^^ . 
 
 00004141 
 
 O04 13997 . 
 
 
 *-=10.5 
 11.0 
 1L5 
 12.0 
 12-5 
 ISO 
 13-5 
 14-0 
 14-5 
 15-0 
 
 0043<3fi93 . 
 
 69-42 years 
 66-41 
 6367 
 61-16 
 58-84 
 5671 
 54-73 
 52-90 
 51-19 
 49-59 
 
 004^0030 . 
 
 OO47974Q 
 
 00492180 > 
 
 OO^l 1^9*1 . 
 
 .00^30784 . 
 
 00^499^9 . 
 
 OO^fiQO4Q 
 
 OO^SfiOT'i . 
 
 OOfiOQ7ft 
 
 
222 
 
 INCREASE OF POPULATION. 
 
 A TABLE, exhibiting the Period in which the population of a Coun- 
 try has a tendency to DOUBLE itself, from an estimate of its increase 
 per cent, taken at the end of every Ten Years. 
 
 I. 
 
 II. 
 
 III. 
 
 Per Centage 
 increase in ten 
 years. 
 
 Numerical Value of 
 ^(log,(lOO-{-7r)-2). 
 
 Period of doubling. 
 Log. 2, or .301 0300 
 
 Sj(log.(lOO+)- S ) 
 
 *-=15-5 
 16-0 
 16-5 
 17-0 
 17-5 
 18-0 
 18-5 
 19-0 
 19-5 
 200 
 
 .OOfi'25820 . 
 
 48-10 years 
 46-70 
 45-38 
 44-14 
 42-98 
 41-87 
 40-83 
 39-84 
 38-91 
 38-01 
 
 .00fvl4^fift 
 
 
 00681859 
 
 00700S79 . 
 
 007188^0 
 
 00737184 
 
 00755470 
 .0077 %'7Q . 
 
 .O0791812 . 
 
 
 ^=20.5 
 2LO 
 21-5 
 220 
 22-5 
 23.0 
 23-5 
 24-0 
 24-5 
 25-0 
 
 00809870 
 
 37-17 years 
 36-36 
 35-59 
 34-85 
 34-15 
 33-48 
 32-83 
 32-22 
 31-63 
 31-06 
 
 .OOfi l >7R ( i4 . 
 
 00845763 
 
 .OORfft'Wft . 
 
 .nnoc iq/^i 
 
 00899051 
 
 00916670 
 
 .009S4217 . 
 
 00951694 
 
 00969100 
 
 
 *-=255 
 260 
 265 
 27-0 
 27-5 
 28.0 
 28-5 
 29-0 
 29.5 
 30-0 
 
 .0098^4^7 . 
 
 30-51 years 
 29-99 
 
 01003705 
 
 01020905 
 
 29.48 
 
 .mo^fto^7 . 
 
 28-99 
 28-53 
 28-07 
 27-64 
 27-22 
 26-81 
 26-41 
 
 0105510'? . 
 
 .01O7'210O . 
 
 01089031 
 
 O11O5S97 . 
 
 01122698 
 
 1 1 H94'?4 
 
 
INCREASE OF POPULATION. 
 
 A TABLE, exhibiting the Period in which the population of a Coun- 
 try has a tendency to DOUBLE itself, from an estimate of its increase 
 per cent- taken at the end of every Ten Years. 
 
 I. 
 
 II. 
 
 III. 
 
 Per Centage 
 increase in ten 
 years. 
 
 Numerical Value of 
 l(log.(lOO-K)-2). 
 
 Period of doubling. 
 Log. 2, or .3010300 
 
 4(log.(lOO-f)7r-2) 
 
 *-=30-5 
 31-0 
 31-5 
 32-0 
 325 
 33-0 
 33-5 
 34-0 
 34-5 
 35-0 
 
 01156105 
 
 26 03 years 
 25-67 
 25-31 
 24-96 
 2463 
 24-30 
 23-99 
 23-68 
 23-38 
 23-09 
 
 .0117<>713 . 
 
 .01 1^Q958 
 
 Ol 9O^73Q 
 
 Ol '?*?>> 1 P>O 
 
 01238516 
 
 .019*4813 . 
 
 .Ol -771048 . 
 
 .01287223 . 
 
 .Ol 303338 
 
 
 T=35-5 
 36.0 
 36-5 
 37-0 
 37-5 
 38-0 
 38.5 
 39-0 
 39.5 
 40.0 
 
 .O131Q3Q3 
 
 22-81 years 
 22.54 
 22-27 
 22-01 
 21-76 
 21-52 
 21-28 
 21-04 
 20-82 
 20-59 
 
 01335389 
 
 .01351327 . 
 
 .01 367906 
 
 .01383027 . . 
 
 O13Q87Q1 
 
 .Ol 41 4498 . 
 
 .Ol 4301 48 
 
 .01445742 .'. 
 
 01461820 . 
 
 
 "=41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 
 01492191 
 
 20-17 years 
 19-76 
 19-37 
 19-00 
 18-65 
 18-31 
 17-99 
 17-68 
 17-38 
 17-09 
 
 01522883 
 O155336O 
 
 01583625 
 
 .01613680 . 
 
 01643529 
 
 01673173 
 
 0170-2617 . 
 
 .01731863 
 
 01760913 . 
 
 
INCREASE OF POPULATION. 
 
 This is the Table of which the jirst and third Columns have 
 been inserted by Mr. Malthus, at page 498, Vol. I. of the sixth 
 edition of his Essay on Population. 
 
 From the Parliamentary Report of the population of England 
 and Wales, it appears 
 
 \ which gives an increase 
 That in 1800 lt amounted to 9168000 
 
 1810 ........ 1050250 from 1800 to 1810, and 
 
 1820 ........ I 2218500 of about 16.3 per cent. 
 
 [persons. 
 
 From hence, by referring to the Table, we infer that, taking 
 the average rate of increase from 1800 to 1810, the population 
 of England and Wales had in 1810 a tendency to double itself 
 in about 51 years ; and, taking the average rate of increase 
 from 1810 to 1820, it had in 1820 a tendency to double itself 
 in about 46 years. 
 
 THE END. 
 
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