THE A T I O N A L E O F CIRCULATING NUMBERS, WITH THE INVESTIGATIONS OP All the RULES and PECULIAR PROCESSES ufed in that Part of DECIMAL ARITHMETIC. TO WHICH ARE ADDEB, SEVERAL CURIOUS MATHEMATICAL Q^U E S T I O N S ; WITH SOME USEFUt REMARKS ON Adfeded ^Equations, and *the Dodrine of Fluxions, ADAPTED TO THE USE OF SCHOOLS. By H. CLARKE, €mnia quacunque a pr'imava rerum naturt conJlruBa funt, numerorum v'tdentur ra- tione format a. Hoc in'm fuit principaU in an'tmo tonditorh exemplar. BOETHIUS, L O N D O N t Printed for the Author; and fold by Mr. Murray, T-To. 32, in Fhet'Strtit, M DCC LXXVIIt 14' A CSSSr T O Thomas Butterzvorth Bayky^ Efq; OF HOPE, FELLOW OF THE ROYAL SOCIETY. S I R, IT gives me the highell: Satisfadlion and Pleafure, that you have condefcended to receive this my firft Effay under your Protedion. And all who are honoured with your Friendfliip, and are acquainted with your fuperlor Knowledge in polite and ufeful Learning, in which you have juftly included the Science of Numbers, will be fenfible of my Happinefs in be- ing thus permitted to addrefs you. Were [ iv ] Were my Abilities, Sir, equal to my Wifhes, I could with Pleafure dilate on thofe many excellent Qualifications, adorned with the utmoft Good-nature and Humanity, which have rendered your Charadler fo confpicuous. But, as I well know I Ihould fail in the Attempt, the only Ufe I can make of this Opportunity, is, to teftify my Regard to fo generous a Patron, by publicly acknowledging the many Favours which I, however unde- ferving, have received at your Hands ; and which I fhall always remember with the fincereft Gratitude. I am, SIR, Your moft obliged And obedient Servant, HENRY CLARKE. PREFACE. HEN we confider the Excellency and fuperior Ufefulnefs of Decimal Arithmetic above all o- ther Kinds of Computation, we fhall readily allow, that an Attempt to render the more intricate Parts thereof clear and intelligible, not only merits the peculiar Atten- tion of thofe concerned in the Inftruftion of Youth, but is particularly interefling to all others who defire Accu- racy in their Calculations ; and is therefore fo far from being an unneceflary Work, that it appears to be of the greateft Utility. But as I am fenfible how extremely difficult it is, even in the beft Performances of this Na- ture, to efcape the Malevolence of thofe, who fancy it their Intereft to keep others in a long Dependence on themfelvesi I (hall beg Leave to obferve, that I fhall be well pleafed, notwithftanding their Cenfure, if my De- fign meets with the Favour of the Candid and Ingenuous, who, I am perfuaded, upon a fufficicnt Perufal, will acknowledge, that the Method here purfued is not only- new, but that it is attended with all the Perfpicuity the Subjefl can admit of. The principal Defign then of this Treatife is, to re- trench all thofe Superfluities (as I may call them) which Cunn vi PREFACE. Cunn and others have loaded the Theory of Circulating Decimals with ; and to fliew, that the whole Bufmtrfs dc-pend"? upon, or may be deduced from, this one fingle Operation, That of finding a finite Vulgar Fra5iion equi' 'valent to an infinite Repeating Decimal This being once iinderftood, the Rationale of all the Rules will be obvi- ous ; and the Pupil will then go on, not only with Plea- fure, but with Speed : For as the Man, who is engaged in a Race with every Obftacle removed from his Courfe, has undoubtedly the Advantage of him who m.uft turn and wind to get clear of the Impediments •, fo not only in this, but in every Art and Science, when Difficulties are removed, as well as a concife general Rule pointed out, the Mind's chief Labour is accomplifhed. So Ho- race fays, ^icquid pr<£cipies^ ejio brevis : ut cito iiEla Percipiant animi dociles, teneantque fideles. Yet that I might not be thought to affecfi: an unintelligi- ble Concifenefs, I have confidered every Rule diflinftly,) illullrated them with proper Examples, and given the Invefligation lymbolically •, by which the Scholar may at one View comprehend the whole Proccfs. A Queftion may poffibly arife with fom.e, what Ad- vantage will refult from a farther Invefligation of the Nature of Decimals ? Have not the moft eminent Ma- thematicians written profeficdly on the Subjed:, and done all that was ufeful or curious therein ? In anfv/er to which, it will be neceflary to take a retrofpeflive View of the principal Authors who have treated upon Deci, mals ; PREFACE. Vll mals ; from whence it will clearly appear, that there is itill Room for farther Improvement. The firil Specimen of Decimal Arithmetic that we meet with, is in the Aftro- nomical Tables of Arzachel, a Moor, who was very emi- nent in Spain about the Beginning of the eleventh Cen- tury. They are adapted to the Meridian of Toledo; and as they are Calculated for the Arabian Year of the Hegira, were probably originally written in Arabic : The Perfians, Moors, Arabs, and Saracens, being about that Period very famous for their Knowledge in Agronomy. In thefe Ta- bles, the Places of the Heavenly Bodies are denoted by a centefmal Divifion of the great Circles of the Sphere, to which the Arabian Algorithm of Numbers was better ac- commodated than the Greek or Roman literal Notation which had been hitherto made Ufe of for the Egyptian Sexagefms in the Aftronomical Tables of Ptolemy, Alba- tegnius, Abenazra, and other ancient Writers. Gerard Voflius informs Us alfo of a Treatife entitled De Algorithm mo^ written by Johannes de Sacro Bofco, about the Mid- dle of the twelfth Century, v/ho made Ufe of a centefmal Notation for the Extradlions ot the Square and Cube Roots. About the Year 1460, John Muller, fometimes named Regiomontanus, publifhed his Book De Triangu- Us^ in which he had conftrucled a Table of Sines to the Radius 10,000,000 ; an Account of which may be feen in the Opus Palatimim de Triangtilis, by Otho and Rheti- cus. The next Improvement in this Part of Arithmetic, w^e find in a Treatife entitled Arithmetica Msmorativa, compofed in Latin Verfe, by William Buckley, about the Year 1530, wherein he has given a Rule for extract- ing the Square Root of a FracTdon •, the Operation being nearly the fame with the prefent Mode of extrading the Square Vlll P R E F A C E. Square Root of a Surd Number, excepting that it is limited to a certain Number of Cyphers : The Rule, as correded by Dr. Wallis is, ♦ Quadra to numsro*^ fen as pr^efigito Ciphras : Produolii ^iadri. Radix, per milk fecetur. Integra dat ^otiens ; 6f pars ita reota manehit^ Radiii ul Vtfre m pars millejfima defit. The Denominator being written under this Number, exprefics the Square Root of the Fradion. Afterwards Peter Ramus, in his Arithmetic, written about the Year 1570, and pnbhfhed by Schoner, (hews the Method of approximating to the Square and Cubic Roots of Surd Quantities, by adding Punctuations of Cyphers, exactly in the Manner we now pracftice. But the firft Treatife written profeffedly on this Subjeft, was publiihed at Ley- den, 1585, by Simon Stevens, entitled DISME, or De- cimals i which he tells us in his Geography, he believes to have been in Ufe among the Indians, and other Eaftem Nations, long before the SexagefTimal Notation was in- troduced by Ptolemy, in the Time of M. Aurelius. Af- ter this Time, Decimals began to be frequently ufed in Arithmetical Calculations, and were particularly much advanced by Briggs and Gellibrand, in their 'Trigo?iometria Britannica ; by Oughtred, in his Clavis Mathematics de^ nub limata \ alfo Wingate, Baker, Kerfey, and feveral other Authors of lefs Note, all contributed towards their Perfedion, in their different Treatifes of Arithmetic. Yet we do not find, that any Regard had been paid to the * Referring to the Produd of the Numerator and Denominator, mentioned in a former Rule. Nature PREFACE. ix Kature o^ Infini I e Circulating Decimals before Dr. Wallis's Time. He was, in all probability, the firlt who diftinft- ly confidered this curious Subje^l, as he himfelf informs us in his Treatifc of Infinites. But he has neither given the Demonftrations, nor fliewn their Application. The latter of thefe Defects, Mr. Brown, in his Decimal Arith- metic, and afterwards Mr. Cunn in !iis Treatife of Frac- tions, attempted to fupply, by giving Rules for their Operations. The former indeed has done this only in one fingie Cafe ; but the latter has extended it to all Cafes. But as thefe are alfo wanting in the main Point, namely, a Demonftration, and are moreover defignedly expreiTed in fuch a Manner, as to fet the Rationale of the Thing as far out of View as pofTible ; it is neceflary that either the Memory muft be loaded v/ith every Rukj or the Book be continually at Hand. Several other Authors have treated on Circulating Decimals. Martin, in his Decimal Arithmetic, has given fome pra6lical Rules, but hath not fufficiently demonftrated them. Emerfon, in his Cyclomathefis, is excellent in the Theory, but has Omitted the praftical Part. Pardon, Vyfe, Thompfon, and fome others, have alfo touched on this Subjedt ; but as they all feem to have borrowed from Cunn, they are in the fame Predicament. Malcolm and Donn are the only Authors I know of, who have treated the Dodrine of Circulates in an intelligible Manner. The principal Objedlion to the former is, that he is too concife, refer- ring conftantly to the Rules for Vulgar Fraflions, by which that Perfpicuity, which is the very Eflence of a Demonftration, is lofl j and the latter has omitted feveral Cafes in Multiplication and Divifion, which frequently occur in Praftice, and is alfo too curfory to afford the * Learner X PREFACE. Learner a proper Idea of the Subjeft. All our later Books of Arithmetic pafs by theDodtrine of Ilepetends unnoticed. I mufl here beg my Readers not haflily to impute Ar- rogance to me, as if I rejeded all that has been done on this Subjeft, or fuppofed myfelf capable of what fo many Men of ^reat Parts and g-reat Learning feem to have come Ihortof. Fori acknowledge myfelf indebted to moil of the Authors I have juft mentioned, particularly to Mal- colm and Emerfon, who have furnillied me with feveral ufeful Hints. And yet I can juftly make the fame Ob- fervation with the former Author, namely, That " the Rules I have given are chiefly the EfFedl of Speculation" made fome Years ago on this Subjed:, before I had feen his Syftem of Arithmetic, or even any of the before-men- tioned Authors. It is not impoflible but an Objedlion may be raifed by others, who have never adverted to the Subject propofed, as that this part of Arithmetic is of little or no no Ufe, iince all Decimal Operations may be performed fufficient- ly near the Truth without it. Thefe Perfons I fhall refer to the Praflical QueAions at the latter end of this Trea- tife, and only here obfervef that thofe Inllances, amono- innumerable others which might be produced, ferve to Ihew, that the common Way is very defeflive, thouf^h carried out to feven or eight Decimal Places, and that the Method of Circulates is abfolutely neceflary, where any Degree of Accuracy is required. The Want there- fore of a Treatile fomewhat ot the following Kind, which fhould at once fliew the particular Properties of Repeat- ing Decimals, and the Invefligations of the Rules for Ope- ration PREFACE XI ration from one certain general principle^ naturally gaveOc- cafion to the following Sheets. For as I had frequently obferved in the Courfe of Teaching, that Youth, by their not feeing the Reafon of the Thing, could not long retain the Rules in their Memory, I endeavoured to demonftrate them viva voce^ and then obliged the Scholar to give the Inveftigation himfelf ex fcripto j by which I could cafily difcern whether he fully comprehended the Nature of them. But as the Inveftigation and Rule given by the Pupil were generally too prolix and incorrefl to be inferted in their Cyphering-books, I drew up the follov/ing Sketch, as a general Form by which they might correft theirs, if neceffary. By this Method of Procedure, I have the Pleafure to find, that a Boy, who is tolerably acquaint- ed with the common Rules in Arithmetic, will readily ac- quire as clear a Knowledge of the Rules for Circulating- Decimals, as he has of any common Operation in Whole Numbers, But as the Operations of Circulates (as well as all other Arithmetical Calculations) are moft eafily performed by I^ogarithms, I have fhewn the Method of finding the Lo- garithm of any Repeating Decimal ; whereby the whole Bufmefs is greatly facilitated, and the Difficulty and Intri- cacy of the Rules by Common Arithmetic avoided. And, for the Amufement of fuch Pupils as have touched on the firft Principles of Algebra and Geometry, I have in- ferted a few Qiieftions, chiefly Originals, with their Solu- tions •, and fome are given without Solutions, which are intended for the Exercife of thofe that are farther advan- ced. I have alfo added feveral Remarks on thofe Parts of ■m PREFACE. of the Mathematics which feem to the young Reader to be rather obfcurc, namely, On Cardan's and Colfon's Theorems for Cubic Equations, wherein a very clear and concife Rule is given for extrafting the Cubic Root of an impoffible Binomial ; by which Cardan's Theorem is rendered generally ufeful, in finding the Roots of an Equation when they are all real, as well as when there is but one real and two imaginary — On the improbability of obtaining general Formula for the Surlolid and other higher Equations — On the Method of tabulating Literal Equations, illqftrated by Examples •, from whence the Reverfion of a Series, however affected with Radicals may be eafily performed — On the diredt and inverfe Me- thod of Fluxions, v/herein the Principles are fully ex- plained, and by avoiding all Metaphyiical Confiderations, rendered clear to the loweft Capacity. The whole Bufi- nefs of finding Fluxions is reduced to one general Rule -, and the particular Forms of fluxionary Expreflions are fo diltinguiOied, that the Learner may almoft immediately dtrtermine in what Manner the Fluent may be obtained — On the Correction of a Fluent, and the Reafon of it — On Trigonometrical Fluxions, with their great Importance in Aftronomy — On the Phsenomena of Saturn's Ring, being a new and curious Atialytical Solution of the Pro- blem refpecfiiing the Times of its appearance and difap- pearance ; whereby is alfo exhibited a new Species of Curves, &c. v/hich is extracted from a Treatife juft pub- lifiied, entitled, EJpii fur ks Phenomenes relatifs aux dif- paritions period/'ques de Vanneau de Saiurne. By M. Dionis du Sejour, Fellow of the Royal Societies of London and Paris. In the Regiffer of the Royal Academy of Sciences at Paris, for 1775, we have the following Encomium on this PREFACE. xiu this Work, T^els font les ohjets que M. Du Sejour, a traites dam Jon Owurage •, et I' on voit qu'il yia rien laijfe a defirer fur la theorie dcs fhafes de Vanneau de Saturne' V elegance ^ lafinejfe et lajimplicite des met bodes dont it a fait ufage, r en- dent cet Ouvrage tresintereffant pour les Geometres-, et la difci'.ffion des Phenomenes depuis 1600 jufqti'en 1900, Ic rend neceffaire aux Afironomes qui voudront da'as la fuite ch- ferver avec predfion ccs apparences \ ainfi notis croycns qii^il pjcrite d''etre imprime avec V Approbation et le Privilege de rAcademie, Signed by M. M. d'Alemberr, Le Cheva- lier Borda, Bezour, Vandermonde, M. de la Place, and Jean de Fouchi •, fome of the greateft Mathematicians now living *. — I have alfo added fome new and ufeful Geometrical Proportions ; and, iaftly, have given a Ca- talogue of the mofl approved Authors in the feveral Branches of Mathematics, Philofophy, and Aftronomy ; from which are fdefted thofe that are generally efteerned the moft ufefulj and ranged in the Order they may be read * In tlii? Treatife of Sejour's we meet with the following Para- graph ; which, as it may, perhaps, convince thofe who ftill doubt the Exidence of Saturn's Satellites, I have taken the Liberty to in- ferr. — Le quatrnme Satellite fut cCahord tlt-couvcrt par M. Huy^hers, le 25 Mai 1655. Les quatre aittres Satellites ont ete decowverts fucceffinie- ment par Dominique CaJ/liii ; le cinquieme fur la Jin d'OSohre 1 67 1, le troijieme le 25 Decembre 1672, le premier et le fee and en 1684. Les Aug'- lais conttfilrent long terns I'exiftence de ces Satellites. Ce ne fut quen 1718, que M. Pound ay ant ele^ve att-dejfus du clocher de fa Paroifjc, r ex- cellent objetlif de cent ijingt trois pieds de foyer, donne par M. Huygens a In Socieie royale, on fut affure pour la premiere fois en Anglcterre, que Saturne a'voit reellemcnt cinq Satellites. Lor: de leur deccu-verte, M. Cafp.ni les a'voit nommes Aftra Lodoicea, par allufion a Louis XIF, a Vexemple des Satellites de Jupiter, que Galilee a'voit nommes Aftra Me- dicea L'on fit frapper en France une Medaille pour confacrer cette de- couverte ; la Medaille reprefentoit Saturne acccmpagne de fes cinq Satel- lites, avec r exergue fui-v ante ; Satunii Satellites priinum cogniti. xiv PREFACE. read by the young Student to the moft Advantage. The known and received Terms in the Circulates I have re- tained, except in one or two Infiances, where others offered thernfelves which feemed much more fignificant-, as for the y/ords Pure Con7pound and Mixed Compound,yjh]ch evidently carry in them an Air of Abfurdity, I have fubfcituted Pure MuUiple and Mixed Multiple. And for the greater Ele- gance and Perfpicuity in the Operations, I have made ufe of the common accentual Dafli over the Repetend ; by which alfo that Illegibility of the Figures, often caufed by the other Method, is avoidf;d. The Succefs I have met with in my own School, by ufing the following Rules for Circulating Decimals, and their Inveftigations, together with the Utility of the Ma- thematical Remarks, in rendering thofc intricate Affairs extremely eafy and intelligible to the Learner, and I may with fome Truth add that trite Apology, the Solicitations of feveral of my Friends, have induced me to publiOi them. And as I can manifeftly have no lucrative Views by fo fmall an Affair ; fo neither have I any great Anxiety about its Reception. Should its Ufefulnefs to either Mailer or Scholar be a Reafon for its furviving the At- tacks of a carping Zoilus, I have acknowledged already, it would be no unpleafmg Event. Should the contrary happen, I mult (as Mr. Harris obferves) acquiefce in its Fate ; and let it peaceably pafs to thofe deftined Regions, whither our modern Produdlions are daily paffmg : /;; vicum vend en tern thus & odores. HoR. Lib. ii. epiff. i. THE CONTENTS. Page CT^// E 'Theory of Circulates — — * 1 7 Addition • — 37 Suhtra£iio7i ■ 42 Multiplication ' 45 T)ivifion '" — *- 62 Of the Logarithms of Repeating Decimals — • 77 Pra&ical ^lejlions adapted to the preceding Rules 8 1 Mifcellaneous ^e (lions — — 84 Some ufeful Remarks on adfe£ied Equations — 1 1 1 An Eafy Rule for finding the Cubic Root of an im- poffible Binomial — — — 114 'the Solutions of fc.ne Surfolid Equations with ge- neral Coefficients — ■ 119 An Eafy Converging Series for a Surfolid Equation 122 ^he Newtonian Method of approximating to the Roots of literal Equations ilhfira ted -— 124 An Eafy Method of reverting a Series — • 132 Seme ufeful Remarks on the Nature of Fluxions — 134 Tragical Qhfervations drawn therefrom — 141 ■ illujirating the Method of finding the Fluxions of one or more variable ^Mntities — 143 the Fluxions of Logarithms 146 the Fluxions of Exprefjions with a va- riable Index ibid. the- Fluxions of Exprefpions when both ^.antity and Index are variable ■ 147 Remarks CONTENTS Remarks on the Inverfe Method of Fluxions — J4S ^'he Method of reducing Surd Fluxionary Expref- Jions by Infinite Series — — 152 Of the Correction of a Fluent — — 162 Of Trigonometrical Fluxions — — 165 Xhe Solutions of fome inter efiing Problems depends ing thereon — — — 173 'The Solution of a Curious Algebraic Equation^ ex- hibiting a new Species of Curves, and applicable to the Phenomena of Saturn's Ring — 1/7 Geometrical Propofttions — — — ic^S A Sele£} Catalogue of the mofi approved Authors in the feveral Branches of Mathematics^ Phila- fophy^ and Afironomy «-,_—, 208 THE THE RATIONALE OF CIRCULATING NUMBERS. SECT. I. The Theory of Circulates, I. A Circulate, or recurring decimal, is that wherein one -iTlL or more figures are continually repeated : they arc diftinguifhed into fingle and multiple, and thefe again into pure and mixed, 2. A pure lingle circulate Is that which repeats a digit only; as -666 &c. and is marked thus •6. 3. A puie multiple circulate is that in which feycral figures repeat; as '642642 Sec. marked •642. 4. A mixed iingle circulate Is one which confifts of a terminate part, and a fingle repeating figure ; as 4*333j ^'c. pr 4*3- 5. A mixed multiple circulate is that which contains a ter- minate part and feveral repeating figures; as 86*325. 6. That i6 RATIONALE OF 6. That part of the circulate which repeats is termed the repeteiid; as in 2'6, and 34*643, 6 and 643 are the refpecllve repetends. 7. In any pure circulate, the whole repeating part, being continued ad infinitum^ is equal to a vulgar fradlion, of which the numerator is the repeating number, having the decimal point removed as many places to the right-hand as there are figures in the repetend, and the denominator an equal number of n'lnei. For. in the pure multiple circulate '325325, &c. fine fincy if we put rrr the repeating figures '325, and cz=. the whole circulating part, and from c take the part of it- *' lOCO felf, welliall have c — = -325 — r; that is, 1000 From -325325? ^^- = ^ Takd '000325, kc. ■=. 1000 Rem. '325 ^=: c — =: r J but c — ^ lodo 1000 ir /: X 1 lOOO I OQQ 1 =r ^ X = ^ X — -^— ; hence I coo 1000 1000 g zz. i222_^lllr=^^ = '325325, &c, ad infinitum, where 999 999 there are as many cyphers and nines as repeating figures ; which procefs evidently holds good for any pure circulating numbers. JExamples where the circulation begins in the integral part. / N ' ' 370 / s ' ^ 2060 . ' .' £^'- (i.) 37 -—-' (2.) ro6 - — -.. iz-) 42-63 ~ 4-^53^0 2cc ^xampie^ CIUCULATING i^tlMBERS. 17 Examples where the circulation begins with the prime deci?naL E^' (i.) -3 = -' (2.) -^74 - —' (3-) -4736 = 4786^ 9 ' ' ' • 999 ^'" " 9999' (4.) -064 = ~. (5.) -0083 = — ^, &c. ^ 999 9999 Examples where there are Cyphers, which do not repeat^ betwixt the decimal point arid the firji Jtgnificant jfigiire. Ex, (i.) '04 = ~. (2.) .006 = — . (3.) '00026 =: 9 9 . (4.) -00706 =r - — . (r.) -007145 = -^— ^, &c. 99 999 ^ 9999 8. Hence if the repetend (punfluated according to its places) be divided by as many nines as there sire repeating figures, the quotient will give the whole circulating part, and the fraftion is a finite expreffion for the feries infinitely con- tinue d. 9. Hence alfb if any number be multiplied by an unit wltli as m^ny Cyphers annexed as it contains places^ and then divided by as many nines, it becomes a circulate; which is fingle or multiple according to the places in the given number, and pure or mixed as we take the whole or part of the given number for a repetend. Pure Circulaiess Example i, 6 X 10 =: 60, and 9)60- (6,666, &c. i8 RATIONALE O T Example 2. •6 X 10 rr 6, and 9)6(-666, &c. Example 3. 24 X 100 = 2400, and 99)2400(24*24,24, &c. Example 4. •234 X 1000 = 234, and 999)234>(*234>234, &c. Mixed Circulates, Example 5. 4*26 X 10 =: 42-6, and 9)42*6(4.7333, &c. Example 6, 327 X 100 zz 32700, and 99)32700(330,30,30, &c. Example 7. '2071 X 1000 = 207-1, and 999)207'i(*2O73,o73, &c. 6 234 10. Since - — '666, See. -^ — '234234, &c. it is evident, that -6665 &c. X 9 r: 6; '234234, Sec x 999 rr 234; from whence it appears, that if any pure circulate be multi- plied by as many nines (conlidered as decimals) as it contains places, the refult will be the fame number complete and ter- minate. 11. And fince -9 = 1 — -i, -99 =: i — "Oj, '999 =: i — *00i, &c. it follows, that if any circulate be divided by 10, 100, Sec. according to the places of the repetend, and the quotient fubtrailed from the given circulate, there will remain the " CIRCULATING NUMBERS. 19 the fame number, terminate and complete, that conflituted the repetend. Example 1. •674,674, &:c. -r- 1600 = '000674674, &c. Then from -674674674, &c. take '000674674. &c. there remains '674 the repetend complete. Example 2. 3'737, &c. 4- 100 rz -03737, &c. and 3-737, — -03737? &c. - 3-7. Example 3* •00666, &c. -r 10 =: -00066, &c. and -0066, &c, — •00066, &c. z=. -006. 12. A vulgar fraction, of which the denominator is any number of nines not lefs than the number of fignificant figures in the numerator, is equal to a pure circulate, its repetend being the fignificant part of the numerator. And for the punftuation obferve, when the numerator and denominator are integral, if the places of the former exceed thofe of the latter, the excefs is the number of integral places in the circulate ; but if the places of the latter exceed thofe of the former, the ex- cefs ftiews the number of cyphers to be prefixed to the numerator for the repetend; and if the places of both are equal, the circulation begins with the prime decimal. If the numerator ^onfift of integers and decimals, or decimals only, there will be as many terminate cyphers as decimal places, prefixed to the numerator for the circulate. B z Ex, 20 RATIO NALEOF 2060 , , 426300 , , Ex. (i.) = 2'o6. (2.) = 42-63. 999 9999 674 / . 83 (3.) = -674. (4.) -0083. 999 9999 .06 , 7-06 , , (5.) = -006. (6.) =: -00736. 9 999 The truth of which appears from its being plainly the re«» verfe of Art. 7. 13, If the numerator of a Vulgar fra£lion confifhs of com- plete repetends, and the denominator hath as many nines as the figures in the numerator, one period of the repeating figures is equivalent to the whole fradlion. 2626 , , ... '/ Ex» (i.) zz. '2626 (Art. 12.) which is evidently =: "26- 9999 r ^ 434*3 / / , / (2.) = -04343 = -043. 9999 14. If a vulgar fraction hath a repeating numerator, and the denominator as many nina as fignificant figures in the repe- tend, that fradion is equal to the fum of two or more other fra6lions, which may be turned into circulates by Art-. 12. 4646 4600 46 ^ , , , £x. (i.) =: ! = 46-46 4 -46. 99 99 99 320320 320000 320 , t I , (2.) = • + — = 3232-32 + -32. ■ 99 99 99 - 454*5 450 4*5 / / >i (3) = -* = 45 + *045' 99 99 99 15. A CIRCULATING NUMBERS. 21 15. A vulgar fra^Ion, of which the numerator is unlty^ and denominator any number o^ nines ^ is equal to an infinite feries of jV.ivi^ions, the numerator of the firft term being an unit, and denominator as many cyphers as there are nines in the given fradlion, with an unit annexed ; and the fucceeding terms in a geometrical progreflion, of which the common ratio is the iiift term. Thus ; = -IIIII,&C. = A + -r4.+-r-.W, &C. &C. :=^^^-^+Z^' +7^\^Q, ■5~5"S"-©Tr"5-> 16. From whence it follows, that any circulating decimal may be confidered as confifting of a feries of fractions, of which the numerators are the repeating number, and the denomi- nators an unit, with as many cyphers annexed as exprefs the local value of the repetend. Example i. •6 = -666, &c. rr 1 1- , Sic. =: -6 + '06 4- 10 lOQ ICQO * * •006, &c. Example 2. / i 46 46 •46 =—-; + ——-> &c, =: '46 -f* '0046 + '000046, &c. B 5 Example 22. . R A T I O N A L E O F Example 3, < •C85 = . 1 ^ — , he. = -oSc -f -oooSc + 1000 loooooo -^ -' ^ "00000085, Stc,. From whence it Is alfo evident that any circulate, as 2*3 will be equal 2 -f- "3 =^ 2 + -3 + -03 + '003, &:c. or, 4-621 — 4'6 + 'C2 1 = 4*6 -f '02 1 "^ '0002 1, &c. 17. The fum of an Infinite feries of nines is equal to unity in the next left-band place; thus, '999, &c. =z i, '0999, &c. — 'I, -00999, &c. = -01, &c. For it is evident -9 ( =— ) V lO'' wants only — of urJfy, '99 wants , and -999, wants 10 100 , and fo on. So that if the feries were infinitely continued, I coo the difference between that feries and unity would be equal to an unit divided by infinity; which, from the do(5lrine of in- finites, is known to be equal mthing. 18, Hence It appears, that any pure circulate, being mul- tiplied by as many nicies as there are repeating figures, will alvvays carry the repttend to the next left-hand place. Example i. •666 X 9 n 5*999> &c. — 6. — -6 x 9 + -6. Example 2. .2424, &c. X 99 =^ 23-999, ^^' = 24' = "24 X 99 + -24. 19, The CIRCULATING NUMBERS. 23 19. The circulating figures may be fuppofed to begin at any place in the repetend. For 2*342342, &c. is evidently z=. ?42 ^423 234 342 - 2 =: 2.3 = 2'34 — - = 2.^42 - — , &c. 999 999 999 999 20. The repeating figures of any circulating number may be confidered as confifting of twice, or thrice, that number of figures; or any multiple thereof. Thus in 5*2434343, ^c. the repetend 43 having 2 places, may be confidered as having 4, 6, &rc. places. For fince 43 is fuppofed to be repeated y^r evej'f it is obvious, that 5*243 = / / It 5*24343 = 5*2434343> ^^• 2 r. Any number multiplied by an w«/V, with any number of cyphers annexed, and once fubtracted, is manifeflly the? fame as being multiplied by as many nines as there are cyphers. For lOrt — a — ga, 1000^ — i>—gggb. 22. A mixed circulate, fingle or multiple, is equal to a vulgar fraftion, of which the numerator is the given number multiplied by lO, 100, lOco, &c. (according to the number of repeating figures) and lefTened by its terminate part, confidered in it's local value, and denominator as many nines as repeating figures. Example i, 4 S — • For 4-3'= 4 1, (Art, 7.) - ^^^ . ^ ; but 4x9 + 33: B 4 4-3x9 24 RATIONALEDF 4-3 X 9 (Art. 18.) = 4-3 X 10 — 4-3 (Art. 21.) =: 43-3 — * SO ' 4"3 = 43 — 4= 39-*-~ = 4-3- Example 2. 5:243=^ - 09 4"; For 5-243 ~ 5,2 + -043 (Art. 16.) = 5-2 ^^ ::^ 99 C2 X 0Q + 4.-Q , ' ' "99" — "' ^'^ ^ 99 + 4'3 = 5*^43 x 99 = 5-243 X ICO — 5-243 ~524*3— 5*2 = 5i9'ij-*- 5*243 = 99 Ur thu?j 5*243 X 10 r: 52-43 = =^— , (t-x. i.;;andoi' lie rQ'i v)ding by 10, we have 5 24.3 rr ^2 , Example 3. ' ' 23'jcoQ 2357 iz-^il— . ■ ^ 999 For 235-7 = 235-735 (Art. 19;) = -^ —^ T _235'7 X 1000 — 200_2355OO ~ 999 ~ 999 ' ' ' ' 22CC ,T- V ,, Or thus, 235-7 "^ I'^o = 2-357 = -^^ (Ex. I.) = (by multiplying by 100) -^^^^^ — j as bsfore. Or ) CIRCULATING NUMBER?. 25 .Or taus, 2357 = 200 + 35'7 = 200 i22i — — _£22rr . 99V 999 as betore. Example 4. // 23'3 •235 =: ~^^. ^ 99 lor '235 = '2 -{- '035 = '2 ^:^-^ rr ^ - -^ -^ ~ 99 99 •'235X 100— •23^_2^-5— •2_23-3 99 99 99 233 Or thus, "235 x 10 — 2.35 = (Ex. i.) -^j which being (divided by 10 gives •2 3c=:-^-^: as before. Example 5. // 2-72 •0274 = — . T7 '" , "' '74 '0274x100 — '02 vox '0274 = '02 + '0074 =: '02 -^-~— — 99 99 99 I' '^_272 . ,. Or, -0274 X 100 = 274 — TT"! this divided by 100 gives . /, ; as before, -99 23. If the Denominator of a vulgar frailion, (not repeating In the numerator) confifting of nines^ have lefs places than the fignificant figures of tke numerator; that fradion is equal to a mixed circulate, of whicK the repetend has the fame places a6 R A T I O N A L E O F as there are nicies in the denominator; and its circulate value may be thus found : Expunge the denominator, and remove the decimal point as many places to the left-hand as there are nines in the denominator; then dafh off the fame number of fignificant figures for a repetend ; and the remaining figures to the left will be the terminate part. Add the terminate part to the numerator of the given fraftion, obferving to en- creafe the right-hand place by i, if there be i carried in the addition, the fum being dafhed for a repetend, and the decimal point removed as before, will give the correft circulate equal to the given fraction. The reafon of which is evidc... aom the laft article. Example i. Let ^^-^ — be propofed, 99 Which being ordered according to the rule will ftand thus; gg J 5i9*i> 5*1915 S"^9h 5 i9-i» T" 5*^ = 524*3 •*• 5*243 _5i92 99 ' Example 2. 2355QQ ' ' ' / 999 » 235500, 235-5» 235500 + 200 = 235700 .-. 2357 - 999 ' Example 3. a'7s /, ,/ 2-72 • ' — , 272. -0272, 272 + -02 = 274 .♦. 0274 = — • 24. A'ny CIRCULATING NUMBERS. 27 24. Any circulating number being multiplied by a givea number, the produ£l will be a circulating number, of which the repetend will have the fame places as before. For every repetend being equally multiplied muft produce equal pro- duas. Example I. •3 X 2 r:: '6 hence 6 6 6 •666, &c. =: •& But, if the produ£l confift of more places than the given number, the overplus belongs to the firft place of the next repetend. Example 2. •6x4 = 2*4> hence 24 24 24 2 2*666 &c. Example 3. •234 X 13 = 3-042, 3042 3042 3042 3 therefore 3-045045045 &c. = 3'045' 2*6. «5. Any vulgar fra£lion, of which the numerator and dc-* nominator are prime to each other (neither 2 nor 5) being re- duced to a decimal, will be a circulate, and the places of the repetend a5. RATIONALE OF repetend always lefs than the number of units in the denomi- nator. For it is evident, the remainder is always lefs than the divifor, one of which muft therefore return a fecond time, when the quotient confifts of as many places as there arc uniu in the divifor, if not before. Example i. Let the vulgar fra6lion - be propofed. 7 / 7)3-0(428571,428571,4, &c. 28 26 14 Here all the remainders are i, 2, 3> 4» 5> 6, one of which (3) re- turns in the yth place; therefore the repetend confifts of 6 places. • 60 56 40 35 50 49 10 7 3 ^'c. Let — be propofed. Example 2, 1 1)9*0(81,81, ^c. Here the remainders are i, 2, ^^ ^, 4, 5, 6, 7, 8, 9, 10; and as one of them returns in the 20 ' 21 3d place, the repetend has but 2 places. 9 ^c. a6. Any CIRCULATING NUMBERS. 29 26. Any vulgar fradlion, of which the denominator is a prime, (except 2 and 5) being thrown into a decimal, will have as many circulating figures as the denominator would re-^ quire nines for a dividend till it terminates. For if we divide 1000 &c. by any number till i remains, the quotient will evidently begin to repeat; but 999 Sec. is only i lefs thaa 1000 &c. therefore nothing muft remain when the figures have once circulated. It 15 evident from the lafl article, that, whatever the dividend be, provided it be prime to the divilbr, the number of circu^ lating figures will be the fame. Example i. Let — be propofed. II 11)99(9 Here are 2 nines ufed in the 99 dividend, therefore the rcpetead has 2 places. ii)3'o(*2727 &:c. = '27. 22 Proof 80 77 3 &C. Example 2. Suppofe - be given. 7)999999(142857. Here are 6 nines required : hence the repetend confifts of 6 places. 7)60 30 RATIONALE OF Proof 7)6-o(.857i42 56 ,857 See. = -857142. 40 35 50 49 10 7 - ^0 28 20 6 &c. Example 3. Let ^— ^ be propofed, ' 37 37)999(27. Here are 3 nines made ufe of; therefore, the repetend has 3 places. Proof , , 37)483-0 Scc.( 1 3-054,054, &c. = 13-054. 27. If two numbers be prime to each other, the leaft com- mon dividend, or number which exa6lly, contains them, is cxprefled by their producSt; but if they be not primes, exprefs them fradlionally, and reduce the fra(5tion to its loweft terms, the reciprocal produdt of the numerator and denominator of the two fractions is the leaft common dividend, Example CIRCULATING NUMBERS. ^31 Example i. Let the numbers propofed be g and 7. 9 and 7 are prime to each other ; therefore, 9x7=63 is the leaft number which contains 9 and 7 without a re- mainder. Example 2. Let 8 and 12 be propofed. 8 2 Thefe numbers are not primes ; therefore, — — — Hn it's 12 3 ^ Jowefl: terms) and 2X 12, or 8x3:= 24, the leaft common dividend. 28. If there be three or more numbers, find the leaft num- ber divifible by any two of them, then the leaft number divifible by this and any other, and fo on j the laft number found will be the leaft common dividend. Example, What is the leaft common dividend of 2, 3, 4, and 8 ? 3 IS prime to 3, .'.2x3 = 6; 6 is not prime to 4, .♦. - = -, 4 2 12 2 and 2x6 = 12; 12 is not prime to 8, .*. o- — - » and 2x12 o 2 — 24, the leaft common dividend. The reafon of this is evident from Euc. 37, 38. 7. 29- If the denominator of a fra£lion be compounded of two or more different primes (neither 2 nor 5) the leaft common dividend of all the numbers, for each fingle prime (found by the two hft Art.) will exprefs the number of circulating figures. Thus 3J RAtlONALEOF Thus let the propofcd fraction be — = — ^ — ; then, fince -^ 77 7x11 7 =:*42857i repeats in 6 places, and ——'ay repeats in 2 places, and the leaft common dividend of 6 and 2 is 6 ; therefore, we may fuppofe them both to circulate in 6 places, (Art. 20.) But 99 is divifible by 1 1 ; therefore 99,99,99, muft alfo be divifible by 11: and fince 999999 is alfo divifible by 7, it muft therefore be divifible by 7 x 1 1 =77 j confequently, the rcpetend will confift of 6 places. Example^ 77)99999(12987, 77 229 Where thdre ar^ 6 nines ufed j,^ _£_ before it terminates* 759 603 669 616 539 539 Hence the vulgar fradion -jf— will have a repetend of (s 2549 places. For -^'=- ■ ; the rcpetend by n, 7, and ^ 2849 11x7x37 "^ ^ * '' 37, hath 2, 6, and 3 places refpedively ; and the leafl num- ber divifible by thefe is 6 ; which is the number of places in the repetend. Example.^ CIRCULATING NUM-BERS. Example; 35 2849)999999(351 8547 14529 Where it terminates With 6 ^4245 nineSo 2849 2849 30, And generally, if the denominator of ?. vulgar fraiflion N -- be compounded of tlie prime numbers ABC Sec. which circulate in a b c kc. places refpedively ; and P be the ieaft . ■ . N number which a b c &c, meafure : then will ; AxBxL&c. ~ N . . . =- thrown into a decimal, repeat m P places, which is obvious from the laft Article. 31. If any circulating nuinbers, cohfifting of pure or mixed vepetends, of equal places, be added together, the fum will have a repetend of the fame number of places. For every column of repetends mufl evidently give the fame fum. Example i. Example 2. i3'054j054, &c. 2-7134,7134, &c. •5428,5428, &c. 2-342,342, 6cc. 7 1-437 1,437 1, &c. Sum, 15-396,396, &c. Sum, 74-6934,6934, &c. C 32. K 34 RATIO NALEOF 32. If any multiple repctends contain a certain number of places, and the places of the other repetends are found in a Geometrical progreflion, of any ratio, the fum will have a repetend confifting of as many places as the greateft term of the progreflion. For fuppofe the greateft repetend to circulate in 8 places, the next lefs repetend can only circulate in 4, and the next in 2 places, to be conterminous (Art. 20.) ; and 2, 4, 8 are in Geometrical proportion. Example i. 8*72, 72, 72, 72, 72 &c. Here 8, 4, 2 are the places of 74*12 74,12 74,12 &c. j]^g repetends J each of v/hich may therefore be fuppofed to 2*19 36 42 57,19 &c. 85-04 83 28 04,04 &c. confift of 8 places. Example 2. •234234234234,234 &:c. Here the repetends confift of 3, •410725410725,410 &c. 5 and 12 places; which being •983210645271,082 cS:c. . ,. . , ^- \ ^ -rj / y J jj^ vjeometncal progreihon, may 1-628170290231,628 &.C. ^^e confidcredas confifting of 12 places each ; therefore the fum confifts of 12 places. 33. If the repetends to be added have unequal places, fup- pofe them all to begin together, (Art. 19.) then will the leaft common dividend of thofe numbers give the places of repeating figures in the fum. For it is evident that each period of repe- tends muft then terminate together. (Art. 20.) Example CIRCULATING NUMBERS. Example i. 35 1 1 72*32 . Thefe repetends have 2, 3, 4 places, and tlie ' * leafl number divilible by thefe is 12; therefore ^ each repetend may be fuppofed to coniift of 12 1'04 places; which evidently cannot begin together ' ' till all the numbers become circulates. 4-714 Example 2. 78*141414 &c. Here the repetends have 2 and 3 places, 3*817817 &c. ^nd the leafl number divifiblc by thefe is 737373 • 5^ which therefore denotes the places of the repetend in the fum. 34. The difFerehce of any twb circulating numbers, of which the repetends can be made conterminous, will alfo be a circulating number of the fame places as the greater repetend. This plainly follows from the laft Article. Example i, Frorii 82*8546,8546, &:c. The greater repetend con- Take 8*72,72,72>72, &c . f,ft3 ^^ ^ pj^^^g. ^^ ^^'gg Rem. 71*1274,1274, &c. the remainder. Example 2. From 7*382,382,382,382, &c. Where the remainder Take 4*125 368,125 368, &:c. has 6 places, the fame Rem. 3*257014,257014, &c. as the greater repetend. C 2 Examples 36 RATIONALEOF Examfles for the Learner^s Exercife, Uequirsd the finite values of the following infinite repeating decimals. Ex. (i.) -3. (2.) HS- (3-) -741'. (4.) -5686. (5.) -07. (6.) -0046. (7.) 'ooos- (8.) 2-6. (9-)4-07- (10.) 762-3. (11.) ioo'o6. (12.) 'Oj. (13) '003. (14.) '00307. (15.) '00632. (16.) '070004. (17.) 2*6. . (i8.) 2*347. // // It It (19.) 723 6. (20.) -613. (21.) '0364. (22.) '005632. lilt J I It (23.) 12I2'I2 + 1*2. (24.) 7*6 -}- '076. Required the law of the infinite feries, or circulate decimal, equivalent to each of the following vulgar fractions. 3. 45. 741. |686, _7_. 46 . 3 . 260. 4070. 9* 99' 999' 9999' 99' 9999' 9999' 99' 999' 7623000 _ 10006000 _ '7. 03. 3*07^ 6*32. 7000*4, 24. 9999 ' 99999 ' 9* 9 ' 9 99* 9 99* 9999 9* 9* 232*4. 722900. 6o*7 ^ 3'6i . 5*627. 120120. 767*6. 99 ' 999 ' 99 ' 99 ' 999 ' 99 ' 99 •' SECTION CIRCULATING NUMBERS. 37 SECTION 11. Addition of Circulates, CASE I. IVhen the ytumbsrs to he added are pure^ or mixed fingle circulates* RULE. 1^ .CAKE them all end together j (Art. 33.) then add Jl"*/.!. as in common numbers, only increafe the right-hand place of decimals by as many units as there are nines in that column, and the laft figure will be the repetend. Or, continue the repetends two or three places forward, and add as before, obferving to rejeft the fuperfluous Figures to the right-hand of the repetend. Example. 24-68' 24-6888 88 7*234 7*2344 44 1-6 I '6000 00 73 Reduced •7333 33 25-56 28-5666 66 y / / 4*7401 47401 II 19*6 Sum 19-6666 66 87-2^01. C3 For 38 RATIONALE OF For 19-6 - 19-6666; 28-56 = 28-5666, &c. (Art. 4.) 1/ ' ' ' I > ' ^i^i^i3,4,8 and 6 + 1+6 + 3 + + 4 + 8= 9+9+9+9+9+g / A ^ > 28 I , ' (Art. 7.) =_zz3-:r 3+1. 9 9 Examples for the Learner's Exercife. (I.) (2.) (30 (4-) 21-43 716-4 1468. 6 6 67-8 271-6 21-4 •9 2-14 •8 871-23 •7 66- 4*62 •001 •2 4*1 46. 41-2 4- 34-81 7-12 6-i8 .06 68-7 •082 129-614 •005 CASE 11. JJ" the numbers be pure^ or 7nixed multiple circulates, RULE. Make them all end together, (Art. 33.) then add as In common numbers, only to the right-hand figure add as many units as are carried from the column where the circulation begins, ^nd rnark off the repetend from the fame place. Or, continue each repetend two or three places forward as before, rejecting the figures to the right-hand of the place where they ^ecome conterminous. Example. CIRCULATING NUMBERS. ^9 Example, 4'875 4-8758758758758 75 16-32 16-3232323232323 23 428-76 428'7666666666665 66 31*216 Reduced 38-2162162162162 16 'I t I 14*^184 1 4-6 1 846 1 846 1 845 18 / / •04.1 •0414141414141 41 •4142 •4142414241424 14 9 r Sum 503-2561085937322, The places of repetends are i, 2, 3, 4, and theleaft number divlflble by thefe is j 2. (Art. 28.) Each repetend may therefore be fuppofed to confifl of 12 Places (Art. 20.) and the circu- lation muft evidently begin from the column where they all begin to repeat together. The reafon of adding the units to the right-hand figure of the fum, is evident from continuing the repetends a few places forward. Exainpin 40 RATIONALE OF Examples for the Leanicrs Pra£ilce. (l.) (2.) (3.) (4-) 1 J 1 1 / / > 74-12 4127 127- ..4 •0017 / / / ./ 4*5 J62 5- 1-71 •682 •412 - ^27 •14 .'4 4" 1 2 28*21 6-214 » f 321 •016 6»O02 •182 / / / / / / 18-4 17-41 5-i6 4-12 A General Rule, Having made t'ne repetends conterminous, divide their funi by as many nines as they contain places, and carry the quotient to the next left-hand column to where the circulation begins j then add as in common numbers, and put down the remainder for the repetend, obferving to prefix cyphers (if ijecclTary) %i% make it equal in places to the other repetends. 67-51 4*1 23-401 4-62 I 7 "02 Example, Reduced / y 4- 1 1 1 ' 23-401 y / 4*622 y / 17*022 Suni i20'3O5 Here CIRCULATING NUMBERS, 4? Here the places of the repetends are i, 2, and the leaft jiumber divifible by thefe is 2, hence each repetend will have two places to become conterminous ; and the fum of the repetends is plainly ii^ll + l.+^4.^2_^33_l^_ , ± 99 99 99 99 99 99 99 99 = 1+05, (Art. 12.). Examples /or the LeamsrU Exexfifi, (5-) (6.) (7.) (8.) 4-6178 ?7-4 5'i62 •0171 2*45 j6-8i7 12*OI 4*151 j8-2 •008 y I'2I / / 42*684 46' I 2*1 27- •08 4-164 2167 44*4 2-8 4-127 •0098 i8-i' 1-68 (9O •784 y / •48 / *7 •681 r24i6 -0007 (10.) / / 24-24 r8 17-81' 5-48/ 76-281 (II.) •01 41- ■682 iS-4 -6871 21-48 (12.) 21* y 2*4 / y i8-8i y y •014 y y •004 286^ SECTION 45t RATIO ^^ ALE OF SECTION III. SiihtraBion of Circulates, CASE I. Jf the numbers conjijl of pure, or mixed fingle circulates. R y LE. MAKE them conterminous, and fubtra6l as in com-r •mon numbers ; the right-hand figure will be the r epc- tend. But if the repetend of the fubducend be greater than that of the minuend, inftead of lo add only 9 to the lefs repe- tend and fubtradl as before. Or, continue the repetends two or three places forward, and fubtradl as in common members, rejeding the figures to the right-hand of the repetend. Ex. I. From 24*67 Take j8*43 Rem. 6*24 Ex. 2. 71-42 38-7 Reduced 71*42 % 3877 7 32*64 Fpr (Ex. 2.) 71-42 — 3877' = li>^- — 387^- ^ 32-6* = 32-64. (Art. 7.), Examples CIRCULATING NUMBERS, 43 Examples for the Learner's "PraSlice, (I.) (2.) (3.) From 68-432 Take 42*27 7-41' 2-8 54-27 31-62 271*4 39-87 CASE II. (4.) When the numbers are pure, or mixed multiple circulate. RULE. Make them both conterminous and fubtra£t as In common Slumbers; the figures in the remainder, which ftand under the given repctends, will be the repetend of the difference. If the repetend of the fubducend be greater than that of the minuend, leflen the laft figure in the remainder by one, and it will be the true repetend. Or, continue the repetends a few places for\vardj and fubtraft as before. Example i. From 27*283 Take 13* 122 Reduced 27-2832832 83 13-1232323 23 !9is Remains 14* 1600509. Example 44 fv-ATIONALE OF Example 2. 'From 127*46 127*4627462746 27 / y Reduced y / Take 48*6 48*6486486486 48 Remains 788x40976259. In Ex. 2. the places of the repetends are 3, 4; and the Jeafl number divllible by thefe is 12 ; which is therefore the number of repeating figures in the remainder. The reafon of taking one from the laft figure of the re- mainder is evident from continuing the figures a few place* forward. Examples for the Learner's PraSiice. (3.) (4.) (5-) From 24*17 Take 12-4 (7 .) From 1*46 Take •08 32*18 / / 8'4i (8.) 17- •412 178-1 24*6 (9.) 8-41 2*418 (6.) 271-64 41*6 (10.) •28 < SECTI ON CIRCULATING NUMBERS. 45 SECTION IV. MultipUcatiofi of Circulates, CASE I. When me fa5ior is a terminate, and the other a pure circulate* RULE. ULTIPLY the repetend and terminate together; then remove the decimal point as many places to the right-hand as there are repeating figures, and divide the pro- duft by the fame number of nines, continuing the divifion till it repeats or terminates, which will then be the true product. Or, inftead of dividing by the ninei, add the produ£l to itfelf, removing it as many places forward as exceed the number of places in the repetend by one ; and thus proceed till the refult laft added be carried beyond the firft : lailly, add thefe feveral products together, beginning under the right-hand place of the firft, the fum will be the true produ<5l, having a repetend tqual in places to the given circulate. I. A fingle terminate, and a pure fingle circulate. Example.. Multiply '8 by 7. -8 Or thus, -8 L ^ 9)56(6'2i, &c. =: 6'2 56 5 6-2 For 4& RATIONALEOF For '8 X 7 = - j< -^ (Art. 7.) = ^ = 56 x - =^ 9 9 9 56x:i-+-i-+-i-, &c. =5-6 + -re + -056, &c. 10 100 1000 Examples fbr the Learner's PraSiice. . > / / ■/ > Multiply '6 by 2. '5 by 4 j 7 by 3 j "8 by 5 ; '07 by -04; •003 by 60 J '005 by 400; '0006 by 20. 2. A terminate confifting of feveral figures, and a pure fingle circulate. Example. Multiply 23.4 by •7. 23'4 Or thus, 16*3 8 '7 1638 1 63 99)i63'8(i8*2 16 I i8'i 9 = i8*2, (Art. 17.) For •7^=^=7 xl=7x:JL+_I_+.J_,&c.(Art.i5.) 9 9 10 100 1000 •«• '7 X 234 = i63'8 X :;^ + -i- + ^2_ &Ci = 16*38 + ' "10 100 1000 ^ I -63 + -1638, &c. For the Learner's Pra^ice* Multiply 21 by -3 ; 643 by 7 ; '265 by -06 j -0281 by 2' 2'; 4*©oi by '004.. a 3- A CIRCULATING NUMBERS. 4^ 3. A fing!c terminate, and a pure multiple circulate. Example. / / Multiply 3'4i by '4 3*41 Or thus, 1*364 •4 13 999)i364(i'365>365j ^c. I I'S^S r, ' ' 3410 1064 , I For 3-41 X '4 = ^^^— X '4 = -^-^ = 1364 X = 999 999 999 1364 X : 1 , Sec. = 1*364 H- '001364, &c, 1000 lOOOOOO For the Learner'' s Exerclfe, Multiply 2-1 by 8; '7 14 by '6; 72*1 by '02; 46*8 by '004. II II •0048 by '4; 'iB by 'oi. 4. A terminate of feveral figures, and a pure multiple circulate. Example. I I Multiply 4876 by 32'4i. 4876 Or thus, 1580*31 16 32-41 15803 9999)15803116(1580-4696,46, &c. ——-._-. 1580*4696 For 48 RATIO NALEOP ■n '"■ ' o r 3^4.100 „ , 1580*3116 For 52'4i X 48-76 r= ^-^ — X 48-76 — — — = ^ ^ 9999 ^ ^ 9999 1580*3116 - '15803 1 16 + &c. For the Learnei-''s Exercife. Multiply i/O. by 2*1; 4*68 by •04; 'oyi by 4-84; 28*4 by 21 ; '004 by 108 ; 4.8 by 'ooS. CASE II. Jf one faSloY be a terminate^ and the other a mixed circulate, ~ RULE; Multiply the circulate by an unity with as many cyphers an- hexed as the repetend has places, from which fubtradl the terminate part ; then multiply the remainder by the given terminate, and divide the produdl by as many nines as the circulate has repeating figures; or, add the prcduiSs as directed in Cafe I. |. A fingle terminate and a mixed fiilgle circulate. - Example. Mul tiply 4-23 by 6 4*23 lb Or, thus, 4.23 42*3 2^-38 4-2 2 '38-1 6 25-40; 9)-228-6 2r4 For CIRCULATING NUMBERS. 49 * 4-23 X 10— 4-2X6 228-6 For 4-2'2x6 — (Art. 22.) rz ^ 9 9 = 25-4. / "i And 4*23 X 6 rr 4-2 + — X 6 (Art. 16.) — 25*2 + '2. For the Learner*s Exercife^ / / / / Multiply 4*2 by 7; 27*6 by "3; 'loS by -04; I4'8 by 'i; •271 by '005; '9 by -316. 2. A fingle terminate, and a mixed multiple circulate. Example. Multiply 4*264 by '3. 4*264 Or thus, 4*264,64, &c. 100 "I 426-4 i'2793.93> ^'c- 4*2 422*2 ^ 3_ 1*2666 1266 12 1*2793. „ ," 4-'264 X 100 — 4-2 X *? ' 126 66 For 4-264 X "2 r: -^^ -^^ — — 99 99 r: 1*2666 + -012666 + *oooi26, 2cc. The other method is obvious. D Fo- ' f 50: RATIONALE OF For the Learner s Exercife, Muftiply 2'4i6 by 4; -871 by -3; 0146 by -7; 2 by '47 ; •008 by 3"87i. 3. A terminate confifting of fcveral figures, and a mixed fingle circulate. ts* Example. Multiply 2- 16 by 4-65 2'i6 Or thus, 4-65 X2'i Or thus, 2*166, &c. 10 '0 21*6 2*1 19*5 4-65 9)2-790 •310 465 9 30 4 05 1083' 13000 190*675 10075 866 66_. 10075 10-0749. ^ J r 2*16x10 — 2*t X4-6i: 90-67C For 2-i6 X 4-6c — ^—^ — ^ ^ ,9 9 = 10.075. ' > -6 And 2*16 =: 2*1 + "06 (Art. 16.) = 2*1 + — , .*. 2*i X 4*65 + - X 4*05 = I0075- The other proccfs is evidcAt, For CIRCULATING NUMBERS. 51 For the Learner s PraSlice. I J i Multiply 487 by 874; 2*83 by '21; 48-2 by '084; .48*76 by 21.8 J '016 by '0087. 4. A terminate of feveral figures, and a mixed multiple circulate. H^xample. Ml iltiply '467 by 32 / •4- 32'4 Or thus. 32-4,2, &c. 100 •467 3240 1 1 3 2269 3210 467 19454 129696 14-9907 1499 / / 14 151421 / / 15*1421 // 32-4x100 — - 2 CAS£ 52 RATIONALEOF CASE III. The faSiorSy a pure circulate, and a mixed circulate. RULE. Multiply the mixed circulate by an unit with as many cyphers annexed as the repetend has places, and from the product fubtradl the terminate part; then multiply the re- mainder by the other repetend, having its decimal point re- moved as far to the right as it contains places, and referve the refult. Write down as many nif:es as there arc figures in the greater repetend, and annex as many cyphers as the places in the lefs repetend, from which fubtracl as many nines, and divide the referved refult by the remainder j the quotient, con- tinued till it terminates or repeats, is the true product:. Or, inftead of finding a divifor, place the refalt under itfelf accord- ing to the places of either repetend as in Cafe I; and then place the fum again under itfelf as before, obferving the places of the other repetend ; or divide it by as many nifies. I. A pure fingle circulate, and a mixed fingle circulate. Example, Multi ply 2-3 by •7- 23 Or Uius, •14-7 10 147 14 23 i 2 90 21 9)i6'3 9 7 81)147(1 • * • •814,8; &c i-8i4,8,&c. For CIRCULATING NUMBERS. S3 ^ ' ' '7 X 10 a-'^xio — 2,. , 147 For "] X 2*3 = X -^ (Art. 9.) = -li- 9 9 ^ ^^ 9x9 =—^2—. (Art. 12.) 9 X 10—9 A J 147 ^^1 I 14*7 + 1*47 + -147, &c. And --^^= 147 X - X -=:--!^-i ^^^^ -iLi! 9x9 99 9 For the Learner's Exercife. t If t ) t I I Multiply '4 by 2*6; '8 by -014; 487 by '\\ 4*68 by 9; / II I •2A|lby 2i 487 by -04. 2. A pure fingle circul ate, and a mi xed multiple circulate. Example. Multiply 424-3 by •03. 424*3 1000 Or thus. 9) 424*3 •3 424 300 4 i 1 127*29 423 900 •3 14-14 127*17 127 12-729 1 272 127 12 I I * 14*14 D 3 For 54 RATIONALEOF r' " ' ' 424.3 X icoo— 400 '3 127170 For 424*3 x*03 = -^ -t — X - = — ' 999 9 9><999 = i27*i7 + 'i27i7 + *oooi27, &c. x - = 12*729 + 1*2729 -f '12729 + '0127, ^^^■• The reafon of the other procefs is evident. For the Learner' i Exerc'tfe* I I / // y // til / Multiply '276 by '6; 48-51 by -3; 7*i04by 'oaj •68by •! ; •Oi by 2*461 ; '06 by 4 002. ' • 3' A pure multiple circulate, and a mixed fmgle circulate. Example. Multiply 26*8 by -026. •026 Or thus, 268 X '02 10 •06 '26 •02 9)1*609 '24 26800 •I788455I2 6*432 -536 64. 9) 6-438 •/I5382048 •715382048. 'or CIRCULATING NUMBERS. 55 ^ / / / 26-8 X 1000 '026 X lO' — '02 6452 For 26-8 X -026 = X =-_l^__ 999 9 9x999 ^6432 + '006 4, &c. "~ 9 / i '06 lit And -0261= '02 4- '006=1 02 H (Art. 16. 7.).*. •026x26*8 9 ^ \ o' , 06 X 26-8 = '02 X 20-0 + . 9 For the Learner's Pra^ice. 'I t I i til ^ Multiply 468 by 4-1 ; -41 by 687; '681 by •0046; / i I / t I •46 by 2*46; "278 by 4-6. 4. A pure mujtiple circulate, and a mixed multiple cir- culate. Example. Multiply '0423 by '038. •0423 100 Or thus. •0423 3-8 9900 99 4*23 •04 4-19 3-8 338's 1269 980 I )i5*922(-ooi624528-|- '0016082, 82, 82, he. 160 82, 82, &c. I 60, 82, ^c. I 60, &:c. •0016245 28 -J- D^ For 56 RATIONALEOF y/ // "0423 X 100— -04^ 3 8_ I5'Q22 For-0423X-038 = — ^ -^ ^^^ -^ ^ 99 9 9 99 X 100 — 99 / / I I ' I / /' And '038 = 3 8 X — .*. '0423 X 3-8 X — ■=z '0016082 4- •000016082 + Uc, For the Learner' i Exercife. Multiply 2V8'' by 684'; 68-7 by 46-8 j 2V82 by 4-6V; ' 682-1 by 4-6; 2-19 by -416. CASE IV. When the favors are both pure circulates. RULE. Multiply the numbers as terminates, and remove the decimal point as many places to the right-hand as there are figures in both repetends ; then find a divifor, agreeable to the places of the repetend ; or, add the products as before. J, Two pure fingle circulates. Example. > Multiply '8 by •03' •8 Or thus, •8 90 -03 •3 9 "— - 8i)2-4( 0256 V 9)-26 •0296. For CIRCULATING NUMBERS. 57 _ ' ' •8x10 -03x10 'OX -03x100 For -8 X -03 = X -^^ =: ^ , 9 9 9x9 / "J I ' '8 X '^ And 'OX — — .'. '8 X -03 = -' ^9 9 For the Learner's Exercife. t J t J J i / Multiply '8 by '4; '07 by -oi j '4 by '2 j -06 by m; •004 by '5. 2. A pure fingle circulate, and a pure multiple circulate. Example. Multiply 421*421 by "ooa. 421* •002 8-42 84 &c. 9)8-42 •936492047603. T^ ' ' ' 421X1000 '002X10 8420 X -5-1-^ For 421*421 X '002=-^^ X =— J^ 2_E_2 999 9 9 _ 8-42 + '00842, &c. 9 For 58 R A T I O N A L E O F For the Learney's PraSlice. Multiply 68'2by2j i7-6by8; .762 byo4; '68 byoo6; •04 by -683. 3. Two pure multiple circulates. Example. Mu ikiply •324 by 4-6. '324 4-6 Or thus. •324 460 149-04 149.. / / 194-5 3-491891 14918 249 3 99) 1297 149^18 1-50696. 1-50696 -, ' ' '' -324 X 1000 ^4*6 X 100 , For -^24 X 4-6 = •^— I X =:-224X4*6 •" ^ 999 99 ^ II ' ^ * ^ looooo X — ■ x — =:i-49i84-'oi49i84- '000x4, ^c. 999 99 f / a6o -^24x460 ' , / And 4-6 = - — .'. ^^- • = 1-50696. ^99 99 For the Learner** Exercife. I •48 by "048 Viulliply -871 by 4-1; 68'2 by -014; -601 by -002; CASE CIRCULATING NUMBERS. 59 CASE V. Jf both faHors are mixed circulates, RULE. Multiply the numbers as terminates, and remove the de- cimal point as many places to the right hand as the number of figures in both repetends ; to which add the produdl of the ter- minate parts, and referve the fum. Multiply the terminate parts by the circalates reciprocally, having their decimal points brought forward according to their places, and fubtradl the fum of thele produ£ls from the referved fum : then proceed as before, by finding a divifor; or adding the remainders. I. Two mixed fingle circulates. Example. Multiply 2*14 by 4*3. 2-14 4-3 Or thus. 2-14x4 3 920*2 4X2'i= 8-4 43 21 90-3 + 21-4 4 85-6 9)6-43 928-6 175*9 = •7148 9)752-7 S-5777 9) ^3-63 9-2925 9*29 5 r ' ' 2-14x10 — 2*1 4'^xio — 4 For 2- 14 X A- 2 — — X —^ — "^ __9 9 2-14 X 4-3 X 100+ 2-1 X4 2-1 X 4-3 X 1044X2-1 X 10 9X9 • And 6o RATIONALEOF And 4-3 =4 + ^.-. 2-14x4 3 = 4x2-144-' ^* For the Learner i Exercife, I ^ J ' i I I Multiply 2-8 by 416; 687 by -021; -6? by 4-87; 4*89 by 6.4 J -46 by 017. 2. A mixed fingle circulate, and a mixed multiple circulate. Example. Multiply 'oi) by 1*003. 1-003 Or tl '017 lusj, 1-003 x*o I •07 170-51 •OJ 9) -0702 1 170-52 lO'Z •007801 13446 Sec, •01 003003003 •16032 160 •01 7831 16449 9)- 16048 •01783116449 ^ * ' -017x10 — '01 I'OO^XlOOO— I For -017 X 1-003 =: x = = t7o--5i 4--OI — iO'2 X -9-0-9- 9 •07 999 And *oi7 :=: 'oi + — « ^ 9 CIRCULATING NUMBERS. 61 For the Learner's Exercife, Multiply 46*8 by 46-8; '218 by 4-68; 26-4 by '9024; 7 I '6 by •46 ; 48-91 by 6-827. 3, Two mixed raultipl e circulates. Example. Multiply -0132 by 20' / J. •OT32 Or thus, i 1 20-1 X-OI 20'I .32 9900 99 2653-2 •2 •06432,32,32, 64' 321 32. 64>33i 64 2653-4 46-5 9801 )26o6-9(-265983c6 + -06497296 -20101010 "26598306 -F ,, '■' ' ' •0112 X ICO — 'OI 20*1XI00' ror -01^2 X 20*i rz — X — 99 99 _ 2606*9 99 X 100—99- 32 And -0132 = 'CI + '0032 — 'oi -f - 99 For the Learner*i Exercife* I* 4 ^ * /' 4 4 4 / I 4 Multiply i4-68bya*6B J '168 by -01463 6^7*12 by -4871 ; 4 / * f •687 1 -96 by 68719*5. SECTION 62 RATIONALEOF SECTION V. Divifion of Circulates. CASE I. fVhen the dividend is a circulate (pure or mixed) and the dlvifar a terminate. RULE. ROC E ED as in common divifion, continuing the re- peating figures in the dividend (if neceffary) till the quotient repeats or terminates. Example i. Divide '427 by 8. 8)*427,427,4 &c.(.o53428,4 &c. Example 2. Divide 2*i68-;. by 68*7. 68'7)2*i684,684j6 Uc.{'oii^(i\i he. Examples for the Learner'* s Exercife, Divide 6*87 by 27; '241 by 8*i; 68'4i by '091; o-4i by •06; •0214 by 48. CASE CIRCULATING NUMBERS. 6j; CASE 11. JVhsn the dividend is a terminate, and the divifor a pure circulate, RULE. Multiply the terminate by as many nines as there are re- peating figures in the circulate for a new dividend, and remove the decimal point in the circulate the fame places to the right for a new dlvifor ; then proceed as in common divifion, con- tinuing the quotient (if ncceflary) till it repeats or terminates^ Example. Divide 7 by -6. 7 9 6)63(10-5 f- / 6 , 6 9 7 63 For -S = -, and 7 ~ - =: r X - = -- = lo's- 9' '9616 For the Learnst^s PraSlice. I t t I I Divide 4 by*6 J '8 by 2'^ ; '5 by 'oS > -5 by.'oS j *c6 by *4 j •007 by 2. 2. A terminate confjfting of feyeral figures, and. a pure flngle circulate. Example. Divide 43-2 by -7. 43*2 9 7)388-8(55-542857i For 64 RATIONALEOF For ./ = 2, and 43*2 ^ I = ^- ^ ^ = ^J^ = 5S'54> &c- /^or the Learner's PraSiice. Divide 764 by3; -68 by 8-8 i '2816 by -oob ; 2*2 by '007 > •087 by -5. 3. A fingle terminate, and a pure multiple circulate. ♦ Example. Divide '6 by 2 '31. *6 999 23io)599*4(*25948, &c. -^ ' ' 2310 . 2310 oQe , For 2-31 — -^^ — , and -6 ~ -^ — — --^-^^ x '6 999 999 2310 r= '25948, Sec. For the Learner s Exercife. t / / t > I y/ Divide 8 by 23*1 j "] by 87'i6 j *o8 by '0873 4 bj 6i9'6i9; •CO9 by •0017. 4. A terminate of feveral figures, and a pure multiple circulate. Example. Divide 48-76 by 32*4i. 487600 4876 524rioo)48755i-24( 1-5043, &c. For CIRCULATING NUMBERS. . 65 _ * I 324100 JO/:. 324100 9999x4876 For ^2*41 —^—^ , and 4876 — ^- — Z22J. — 3L_i^ 9999 9999 Z^\lo^ ^ 4876 X loooo — 4876 "~ 324100 For the Learner's Exerclfe* It I I I I Divide, 3871 by 46*8; 4871 by 71-6} '0087 by 2*687; 487*1 by '6871} 271.271. CASE III. tf the dividend he a terminate y dnd the divifor a mixed circuldte* RULE. Multiply the terminate by as many nines as there are places in the repetend for a new dividend ; then remove the decimal point in the circulate the fame places to the right-hand, from which fubtradl the given circulate, and the remainder will be the new divifor, with which proceed as in common divifion^ 1. A fingle terminate, and a mixed fingle circulate. Example. Divide 8 by 3*27* 3*27 3277 8 3-27 9 29-5 ) 72(2-44067, &Ci / / _ 3-27x10 -3 -27 29-5 , Q . 29'S For 3*27 = — = -2^ and 8 -f -2-^ — 999 ^x8=-Il, 29-5 29-5 E rsr 66 RATIONALEOF For the Learner's Pra6ike, f Divide, 7 by 487; •4by48*2i -02 by 27 ; •Oo8byo87j It / •007 by 687; 8 by 46.9. 2. A lingle terminate, and a mixed multiple circulate. Example. Divide '6 by 3*?74. 3-274 327-474 &c. 6 3-274 he. 99 324-2 )59*4(*i832i &c. -, " 324*2 , 324*2 For 3-274 = ^ ^ , .'. -6 ~- ■ — 99x6 99 99 324*^ For the Learner's PraSiice, // lit It. * t Divide 4 by 2-i68 \ -8 by 66-8i j 7 by -487 j 6 by -219 j •04 by 581-901. 3, A terminate of feveral figures, and a mixed fingle circulate. Example. Divide 4-56 by 3*14. 3*14 31-4 4*56 31 9 28-3 )4i-04( 1-45017, &c. For CIRCULATING NUMBERS. 67 r ' 28*3 , . 28-3 4*56x9 For 3-14 = — ^, ••• 4-56 -, ^ - ^ — 2. •* ^ 9 9 28-3 i^or //;i. E2 CASE 68 RATIONALEOF C A S E IV. If the dividend he a pure circulate, and the divifor a mixed circulate, R U L E» Remove the decimal point of the dividend as many places to the right-hand as repeating figures in both numbers, from which fubtradl the faid dividend, punctuated according to its place, for a new dividend. Then multiply the divifor by an an unit v/ith as many cyphers annexed as its repetend hath places ; this produdl leflened by its terminate part, and the decimal point removed to the right, according to the places of the firft dividend, and again lelTencd by the laft remainder, will give the new divifor. 5. A pure fingle circulate, and a mixed fingle circulate. Example, Divide 't by 2*3. Or thus, 2i)6.(-28S7i, &c. 2-3 10 23 2 — f 21 •6 a 10 60 21 6 189 )s4C28S7i>^c. For CIRCULATING NUMBERS. 69 -, ' 6 , / 2-3x10-2^3 ai 6 21 For 'o =: — , and 2*3 = — — = — , .•,--:: 9- 9 999 9 ^ _ 6x 10—6 6 ^^ 9 "" a*3xio — 2x10 — 2-3x10 — 2 ~" 21' For the Learner's Exercife. i * t it It » Divide '4 by 2*6} '8 by 46*8 ; 'oS by '687; '7 by 68'4i 5 •08 by 46'98. a, A pure finglc circulate, and mixed multiple circulate. Example. Divide '07 by 03 1*4. 631*4 100 63140 6 62540 '07 625400 700 62540 '7 562860 ) 699*3('ooi24&c, t , , 'O'] X loooo — '07 X 10 For '07 -r 63r4 = 631*4 X 100—6 X 10— 631*4 X joo — Q ^ 56286^* ^o RATIONALEOF For the Learner' i PraSiUe. Divide -4 by 27*14; 77 by '4871; 'ooS by 3826; 6'& by '216; ii'i by 4-8187. 3. A pure multiple circulate, ^nd a mixed fingle circulate. Example. . . ' ' ' Divide 34'6 by '04.3. •043 10 , , •43 •04 •39 54*6 39Q 346000 •39 34600 389*61 ) 31 1400(799*26079, &c. / / / 34'6 X loooo — 34'6 x 1000 For 34-6^*043: •043 X 10 — '.04 X 1000 — -043 X 10— '04 38961° For the Learner'' s PraSlice. t I lit » i 7 I Divide 2'4i by 71*4; 2*48 by '2713 41 68 by '064; •487 by^68-9; 46-8 by 48-8, 4. A pure multiple circulate, and a mixed multiple circulate. Example. CIRCULATING NUMBERS. 71 Example. Divide '042 by .0326, •0326 Or thus, 3*23)4*2( 1*3003, &c. 100 326 •03 3*23 '042 3 23 420 3*23 4*51 3^9*77 ) 4i5*8(i*3003, &c. For 'oLl ^ -0.26 - '^^^ ^ ^""^ ^ -0326xroo-'03 _ 99 * 99 4-2 3*23* For the Learner's Exerclfe, Divide 4-26 by 42-76; 41-6 by '0714; '087 by '00568} 26*8 by 4*17; 6o*oi by .01756. Remark i. It appears from the firft and laft Examples, that when the places of repeating figures in both numbers are equal, the work, may be contracted, by rejeding the denominators in the finite values of the given circulates. 2. It is alfo evident that this rule holds good (mutath mu- tandis) when the dividend is a mixed circulate, and the divifor a pure circulate. E 4 Example. It R A T I O N A L E O F Example, Divide 4*2076 by 34*12. 4*2076 1 00 420*76 4*2 34*12 416*56 34120000 4165600 341200 416*56 33778800 )4i65i83*44(-i233, &c. ,, , , 4'2076xioo — 4'20 Xi 0000— 4-2076X100— 4*x For 4-2076 ~ 34*12 = 34-I2X icooooo — 34"i2X 1000. For the Learner's PraSike, it II Divide 4-27 by 2*16; 84*71 by 501*6; 4*871 by40-i7; / / / / / / / II II •52ic; by 5 0763 '00176 by '0146; '00743 by 'OOiyio. CASE V. When both the dividend and divifor are pure circulates. RULE. Find a new dividend and a new divifor, by removing the decimal points as many places to the right-hand as there are repeating figures in both numbers, and then fubtradting the numerators of their reipedive terminate values. I. Two pure fingle circulates. Example, CIRCULATING NUMBERS. 73 Example. Divide '6 by '04. / ?04 "6 Or, *4)6(i5 4 60 •4 ^ 3-6) 54(15- „ / 6 ' '4 6 , M '6x100 — 6 For 'o ■=. -J and '04 =:-,.•,-—•-=: • 9 9 9 9 '04x100 — 4 jP<7r /Z?^ Learner^ s Pra^ice, > / / y / / / i Divide '7 by 'B ; '04 by V2; 7*7 by •& ; 'Oi by 4*4 j lit I •8 by '07 J '008 by •0004. 2. Two pure multiple circulates. Example. Divide '432 by 2*3. 2*3 '432 2 30000 43200 230 432 229770 )42768(-i8526, &c. For the Learner's Exercife, pivide 46*8 by 67; '504 by i"68; '014 by 7*65; 21*8 by '0014; '0819 by '071. 3. A pure fmgle circulate, and a pure multiple circulate. Example. 74 RATIONALEOF Example i. I f Divide '004 by 341*341 341 -004 34 I 00 CO 40 341000 '04 3069000 ) 39'96(*ooooi302, &c. Example 2. Divide 70*046 by 4*4. 4*4 4 000000 7^046000 40 7004600 3999960 ) 6304x400(15-7605, &c. For the Learner's FraSilce, II II Divide '4 by •6825 'oS by '427; 'ooi by 5*04; '0871 by -7 ; t I i 4-81 by '04, CASE VI. Jf the dividend and divifor are both mixed circulates, RULE. Find the new divifor and dividend by removing the djcimal point of the terminate value of each numerator as many places to the right-hand as there are repeating figures in the other number, and then fubtra<5ling their refpedliYC numerators. 1. Two mixed finsflc circulates. O" Example. CIRCULATING NUMBERS. 75 Example. Divide 21*4 by 3*4. 3*4 21*4 Or thus, 31(193(6*2258, Sec. 34 214 3 21 310 1930 31 193 279 ) 1737(6*2258, &c, XT ' . ' 9 2I'4X 10—21 tor 2i'4 -r- 3*4 = X — -^ — ^ = 34 X 10— 3 9 193x10— -193 _ 193 31x10—31 31* For the Learner's PraSlice. Divide 41*7 by 8'i7; 98*4 by •418; '127 by '046; / i I lit i'04 by '087 J '714 by -0084; -687 by 99'g. 2. Two mixed multiple circulates. Example, Divide -0132 by 32*0 1, 31980 1-31 3198000 I 310 3198Q 1-31 3166020 )i3o8*69(*ooo4i33, &c. F$r 76 RATIONALEOF For the Learner's Exercife, ft II II II II f i Divide 4*871 by*6i8; 4-109 by 7 181 ; '4271 by '0714; // // // // I I . I i 5*071 by '007 14 J '4876 by '01436} '098714 by 48*71659, 3. A mixed fmgle circulate and a mixed multiple circulate. Example i. Divide 'oiS by i'oi4. 1013 -15 10130 150. 1013 -15 91 17 ) i49'S5(*oi643> &c. Example 2» II I Divide 21*07 ^y 4*2 ?» 37-9 21050 37 900 210500 37.*9 21050 37862*1 ) 189450(5*0034, &c. For the Learner's Exercife, I It I II I It Divide 4* 1 6 by 2*184; 21*87 by 4-106; 187*19 by '0176 j * I til I •4871 by 50*1; 201*47 by '4071. SECTION CIRCULATING NUMBERS. 77 SECTION VI. Of the Logarithms of Repeating Decimals* HAVING, in the foregoing Seftions, fupplied thepupli with rules for managing the whole dodrine of circu- lating numbers by common Arithmetic, and given the theory and reafons for the fame, we fhall therefore now proceed to Ihew how the whole bufinefs may be eafily performed loga* rithmlcallyp having firft premifed the following LEMMA. When one number is to be divided by another, the quotient will be the fame as if unity were divided by the latter number, and the quotient multiplied by the former. r, A I X A I . From hence, and the nature of Logarithms, it is evident, that L. ^ r: L.A — L.B = L.A + L.i— L.B; and fmce B in the terminate value of any circulating number, the denomi- nator confifts of as many nines as figures in the repetend, we (hall evidently have this general rule for the logarithm of any circulate, pure or mixed. RULE. To the log. cf the numerator of the terminate valiic of the given circulate add the arithmetical complement of as many nines as the repetend has places, and the fum will be the logarithm of the given circulate. Example 78 R A T I O N A L E O F Example i. Required the logarithm of the repetend '2, * 2 " = ? To the tabular logarithm of 2 - - - r: o'30lo'30o Add the arithmetical eomplement of the logarithm of 9 j - - •9'0457S74 The fum is the logarithm of the circulate '2 •=. •9*3467874 Example 2. What is the logarithm of 4*64 ? ' , ' 4640 4*04 = , .*. 999 To the log. of 4640 ------ — 3'6665 1 80 Add the arithmetical complement of 999 - — •7-0004344 The fum is the logarithm of - 4*64 - ::: o'6669524 Example 3. Required the logarithm of '007145. ' ' 71*45 •007145 = -^-^, .*. 9999 To the log. of ----- 71*45- — 1-8540022 Add the arith. corap. of - - - 9999 - = •6-0000433 The fum Is the logarithm of - "007145 '■■=. *7*8540455 Example CIRCULATING NUMBERS. 79 Example 4. Required the logarithm of 4'3. ' 39 • 4-3 = f.--. To the log. of------ 39"=: i'59io646 Add the arlth. comp. of - - - g • zz *9'0457574 The fum is the logarithm of - - 4*3 - — 06358220 Example 5. What is the logarithm of 235'7 ? ' ' 231:500 2357 = ^^^ , .*. ^^ 999 To the log. of - - - _ 235500 - = 5*3719909 Add the arith. comp. of - - - 999 - = •7*0004344 ' ' — — — ^— The fum is the logarithm of - 235*7 - = 2*3724253 For the Learner's Exercife, Required the logarithms of the following repeating decimals. I i I J J ill III I 21*4; 76-8; •0764; 46*6; 28*19; 466*9; '187 J 68*214; * I // // // // It 4*687; 716*498; 24*24; *oo987; '698; 187*1649; / J I I II 48716*8714; 2876'8i43; •ooooo67» A Tabic %o RATIONALE OF A Table of the nine Digits perpetually circulating, and of the Arithmetical Complements of the De- nominators. Number, Logarithm. Number. Logarithm. # I = 0*0457574 I 9 = •9*0457574 2 = 0-3467874 I 99 I 999 =r •8-0043647 3 4 — 0-5228786 o'6478i74 == •7-0004344 5 — 0-7447274 I 9999 - — •6*0000433 6 7 — o'8239o86 0-8908554 0-9488474 I 99999 I = •5-0000043 8 999999 I 4 0000004 •3'ooooooo 9 — I '0000000 9999999 PR AC- CIRCULATING NUMBERS. 8i PRACTICAL QUESTIONS, ADAPTED TO THE PRECEDING RtTL^S. Qucftion I. WHAT Is the Produd of 14 feet 8 inches, by 11 feet 10 inches ? L , / / io6*c 17.2 * That IS 14-6 X 11*83 = X -^ = 173-5 = 173 f. 6i. 8p. Queftion 11. Required the Area of i8 f. 9 I. 10 p. by ii i. 9 p. ? That is l^nn X ^'^'^^ = 18 f. 5 i. I p. 6 th. 6 fo. 9 9 Queftion III. There is a Hall, the length of which is 96 f. 7 | i. and the breadth 68 f. 8 f i. which is to be paved with marble fquaresj cack if. I I i. fqUi How many fuch fquares will it take, and what does it come to at as 6d f per foot ? Queftion IV. If I C^^ 2 q'-. 18 lb. of Sugar coft 5I is 8d. what will I a C^^^ 3 q". coft at the fame rate ? F Queftion 82 RATIONALEOF . Queftion V. What muft I give for 8 lb of Tobacco, when | C'-. coft 4I i 7s 8d ? Queftion VI. If the Penny-loaf weigh 7 | Ounces, when Wheat is at 6s 4d per Buftiel, what muft be the weight of the Penny-loaf when Wheat is at 3s lod per Bufliel ? Queftion VIT. Suppofs 4 Hh''% 3 Firkins, and 5 Gallons of Beer, coft 61 14s 8d. How much is that per Hh'', and per G". ? Queftion VIII. Apiece of Land 4 rods broad and 40 long, being a Statute- Acre; it is required to know what lengtli, with 10 rods and 2 yards breadth, will make an Acre ? Queftion IX. A Bill of Exchange was accepted at London for the Pay- ment of / 847*53, for the fame value delivered at Lifbon in Millrees; Exchange at 5s 4d /)fr piece. How many Millrees were paid at Lifbon ? Queftion X. A Merchant is defirous to know how much of each of the following Wines he muft take, fo that the whole Quantit)-- may be 84 | G"-. at 5s loi per G". Malaga at 7s 6d. Canary at 6s gd. Sherry at 5s. and White Wine at 4s 3d. />^r G". Queftion XL In what time will £>] per Annwu pay a Debt of 120I 8s at 6 per Cent, Simple Intereft ? Queftion CIRCULATING NUMBERS. 83 Queftion XII. Stippofe a Freehold Eftate of ,^40 - per Ann. Is to be fold ; what is it worth, allowing the Buyer ^5 per Ceni> Comp. Int. for his Money ? Queftion XIII. What is the Difcount of 87I 13s 4d for 233 day^, at £il, per Cent, per Ann. f Qiiefilon XIV. A challenges B to run a Peace with him, if he will give Kirii 3 rods in 10 ; now the velocity of B's running to that of A, is as 31 to 27,. Which of the two beat? Qi..eftion XV. In the Year of our Lord 1775, the Cycle of the Sun was 9, and the Cycle of the Moon 20. Required from hence the Year of the Dionyfian Period, Queftion XVi. Required the Area of the Parabola, whofe Ordinate and AbfcifTa are 60 '42 and 41*24 refpedively. Queftion XVIL Required the Solidity of a Spheroid, of which fiie greater Axis is 12 |, and the lefs 6*427. Queftion XVIlI. Required the Quadrature of Hippocrates' Luries, C. Di (Fig. I.) the Diameter A B being 18 i, and A E i\\. F 2 . Queftion 84 MISCELLANEOUS Qucftion XIX. What is the Solidity of a Parabolic Conoid, the Diameter of the Bafc being 9*75, and the Height ii'25 ? Queftion XX. How many Solid Inches arc contained in an Icofahedron, the fide of which is 2-4 Inches ? MISCELLANEOUS Q^UESTIONS. Queftion XXI. ** There arc five whole numbers, the three firft of which are in geometrical progreffion, and the three laft in arith- metical, the fecond being the common difference in the laft three. Now the fum of the four laft r: 102, and the produft of the fecond and laft number is 504. Required the numbers." Anfwer, Let z;, ey, a*, ^, z, reprefent the five numbers; then, per ^efl. vx ■=: w% x •{■ w rz. y^ x -\- 'i w zz %^ w + x -\- y •\- "z zz 102, and w % ■= 504. For j; and % fubftitute their values in the 4th. Equat. and it becomes 4 zt -f 3 *" ^^ 102. And from the 3d and 5th Equat. we find x — =— ^ — 2W, this value of X being fubftituted in the laft expreiHon, gives 4.W + — 6 w — 102, reduced, iv' •\- ^i w zz 756, hence iv rr 12, and the required numbers are 8, i2, 18, 30, and 42. Queftion XXI f. ** To find the leaft whole number, which, being divided by 19, fhall leave 17; if divided by 28, fhall leave 21 re- mainder; and if the fum of the two refulting quotients be fubtrad.cd (QUESTIONS. 8s fubtra£led from the number fought, the remainder, being divided by 3, fhall leave 2 remaining."' Anfwer. The leaft whole numb?r that can poflibly fatisfy the two firft conditions of the Queflion is 245. per Simpfon's Algebra, /». 289. From whence likewifc it is plain that the number fought may be reprefented by 532 x + 245. But in order to obtain a general expreflion for the fum of the refult- ing quotients, we murt obfcrve, that as 532 is a multiple of the Divifors 19 and 28, any multiple {x) of the faid number will likewife contain a like multiple of thefe divifors; and 245 being conftant, the quotients 12 and 8 will be fo too. There- fore 19-I-28 xr-f 20 will truly reprefcnt the fum of the quo- tients, which being fubtrafled from the number fought leaves 485*'4-225. Now 2 being taken from this number, and the • 1 1- • I 1 » 1 • 48^ A -f 9 2^ remainder divided by 3, the quotient will be a 3 whole number by the queftion, from whence the leaft value of X will be found — i, and confequently that of 532A-f-245 — 777; which is the number required. ■Note. As 47^+20 Is a general cxprcfllrn for the fum of the quotients, any other divifor and remainder befides 3 and 2 may be propofed, and the number anfwerlng the conditions of ;he que ;{ ion fovxad as ab.yve, Queftion XXIII, '* Require] the dimenfions of the greateft cylinder that ran be iufcribed in a folid, formed by the rotation of a curve round its axis, whofe equation is a>^ — <*==>'% abfcifs 40, and femiordinate 28 inches ?" Anpiver. The greai-eft cylinder that can be infcribed in a folid, generated by a curve revolving about its axis, is when F 3 the 86 MISCELLANEOUS the height of the cylinder is equal to half the fubtangent (per Emcrfon's Conies, p. 53). From the equation of the curve we have — the fubtangent ; therefore Zax — I ^ax- /i6o« + 6 ~ 40 — X, reduced x zz. . J9*9988. Therefore 40 — 19 9988 =: 20'ooii = the height of the greateft infcribed cylinder; from whence the diameter is eafily found by the equation of the curve. Note, J-Iad the equation of the curve been ax' — 2*=;'% 2 then tl'.c expreffion for the fubtangent would be 2ox — 2 evidently zz zx (which is the property of the common or Apollonian Parabola) and the height of the cylinder — 20 exaflly. Queftion XXIV^. *' If in a plane triangle a right line be drawn from the vertical angle to the bare, forming an angle at the fame equal to the coniplement of half that at the vertex ; the line fo drawn will divide the difference of the fegments of the bafe, in the ratio of the fides, including the vertical angle: required the Pemonftration." Anfiver. Draw BD (fig. B) making AD rr the difference of the fegments ; and make BEC ~ the complement of half the vertical angle ABC. Let BF be taken == BC; and CF be drawn, and like wife AH j| to DB. Then it is evident becaufe of parallels, that the angle HAB = ABDn:BCA--BAC zz. the Difference of the angles at the bafe. And in the tri- angles EGC,FGB, the angle FGB = EGC; and BED = BFG per conftruc. therefore GCErrFBG. But {^pcr Simpfjn's Trig. /». 62.) the angle GCE zzz half the difference of the angles Q^ a E S T I O N S. S^ angles at the bafe ; confequently BE bifefts the angle ABD, whence AB : BD (=BC) :: AE : ED (per Euc. 3, 6.) ^E.D. Queflion XXV. *' In a plane triangle ARC are given two fides, AC,BC, and the line CD = BC drawn from the vertex C to terminate in and bifeft the bafe AB, to conftruit the triangle geo- metrically." Anfwer. Let CA (fig. C.) zr one of the given fides; with the other given fide CB, defcribe BD ; from A, draw the tangent AE, on which defcribe the fniicircle AFE ; with AF =z FE defcribe FD, draw ADB and it is done. For AF^ + FE^ = 2 AF^ r: AE" = AD x AB ; but AD =: AF per conftruc. •.• AD X AB z= AF x AF + DB = AF^ + AF X DB ; hence 2 AF" = AF" + AF x DB, Gonfequently AF (AD) = DB. ^E,D. Queftion XXVI. " In lat. 53" N. ftands a Tower the fl'iade of whofe fummit on Tuefday ^une 9, 1772, defcribed a curve on the plane of the horizon whofe tranverfe axis was \ 50 yards : required the height of the faid Tower geoniitricaliv." Jnfwer. It is too well known to need demonfl:rating here, that when the fun's declination is lefs than the complement of latitude, the curve formed by tlie fetflion of the horizontal plane with the Cone of R;iys is an hyperbola. Therefore on the given line AB (fig. D. ) n the tranfverfe axis, defcribe the fegment of a circle to contain an angle of twice the fun's given declination. Make the angle BAD — the difference of the colatitude and the faid declination; and draw DC J_ AE, and it win be tht height of the tower required. For let DE bire(^ F 4. the 88 MISCELLANEOUS the angle BDF, and AG be perpend, to ED produced ; then will the angle GDA — EDF := the codeclin. hence the z. GAD — the fun's declination. And becaufe GE is evidently- parallel to the earth's axis, the angle GAE = the colatitude; therefore the Z. DAB =: the difference of the colat. and the given declination. Moi'cover becaufe BDF is the cone of rays defer ibed by the fun in his parallel, whofe vertex is D, and AC the plane of the horizon cutting it in the hyperbola BH, AB is rr.anifeftly the tranfverfe axis ; confequently DC is the true height of the fummit from the horizontal plane. The calculatiori is very eafy ; for in the triangle ADB, we have AB and ail the angles to find another fide, fuppofe AD ; from whence, and the angle DAC, we find DC = 43.588. N. B. I have taken the fun's declination — 23*^. Queftion XXVU. *' The fnorteft fide of a right-angled triangle is given, and a perpendicular let fall from the right-angle cuts the hy- pothenufe in extreme and mean proportion. Jt is required to cOMftrud the triangle geometrically." jinfwer. On the line AB (fig. E) drawn at pleafure, make AD ■=. the given fide of the triangle; raife the indefinite X DC, and take BD to AD in the given proporcion. Then make BC zz DA, and join CA, and ABC will be the require4 triangle. For BA is divided in the given ratio by the JL DC, and BC rr the given fide ^^r conftruc. and fince BA X BD (ziDA'^) rr BC% the ii at C is a right one, ^er Simpfon's Geom. p. 4. Theo. J 9. Queftion (QUESTIONS. 89 Qiieftion XXVIir. " In the common experiment of the double cone rolling (apparently) upwards, how tar will it move between the inde- finite right lines, which fupport it, fuppofing they meet in an angle of 12 degrees, and the plane of fituatioii elevated 4 de- grees above the horizon, the common bafe of the double cone being 6 inches and diftance between the vertices 13 inches?" AnftViV. It is well known, in the experiment of the double cone moving apparently upwards, that the center of gravity thereof does actually defcend; hence, flao-uld the right lines or rulers which fupport the cone be fo pofited, as to caufe the center of gravity (on its bting rolled upwards) either to move horizon- tally, or to recede from the bafe or horizontal line, it mani- feftly can have no motion but what is caufed by an external power. Tt alfo appears from experiment, that the cone will move with a grr-ater or lefs ceKrity, according as the right lines or rulers are inclined to one another in a greater or lefs angle, and that the angle may be made fo acute, or the rulers biought fo near together, as to caufe the cone to roll downwards ; con- fequently there muft be a certain angle, in which the cone will reft in any part of the inclined plan; now this angle is very eafily determined as follows; Let ABCD (fig. F.) be an horlzop.tal plan-', and AIHD a plane inclined to it in the angle FMG. Let alfo MH, MI, reprefent the indefinite right lines or rulers, and KMLE a vertical feiStion of the double cone. Then becaufe FG is parallel and equal to NM, and HI parallel and equal to KL, it is evident, that while the plane KMLE 90 xMISCELLANEOUS KMLE Aides vertically betwixt the lines MH, MI, (which is the fame thing 2s the cone rolling) the point N will move horizontally on the line NF, and when it arrives at F, the lines KL, HI, will coincide. And moreover, from this equality and p::ralielifm of the lines, we have this Analogv, as NiVI:NL, or as EM : KL :: FG : FI. But if wc call FM radius, FG will be the fine of the angle FMG, and FI the tani>ent of the angle FMI ; therefore as the diameter of the common bafe of two cones, is to the diftance of their vertices, fo is the the fine of the plane's elevation above the horizon, to the tangent of half the an:^le included hy the two right lines or rulers. Hence, {hould the given angle be lefs than tl^at found by this proportion, it is plain, that the cone cannot afcend, fmce on its being rolled upwards, the center of gravity muft move upwards likewife. Let this now be applied to the prefent cafe, and we fliall find that the angle, or inclination, of the indefinite right lines, ought to be fomething more than 17° 12', to caufe the leaft apparent afcent of the cone ; but this exceeds the given angle in the Queflion (viz. 12") confequently the cone can in this cafe have no motion of itfelf. This v/ill appear ftill plainer, if we fufpofe the cone to be rolled along the in- clined plane, to any given diftance (fuppo'e 10 inches) from the point of contaft of the rulers j for its center of gravity in that fituation would be found by calculation to have receded from the horizontal bafe line 6*212 inches, and confequently the center of gravity would have afcended '212 of an inch, which is contrary to both theory and experiment. We mi2;ht purfue this matter fiill farther, by fliewing the grcateft angle of elevation of the inclined plane, as well as the greateft or leafl dimenfions of the cone, whereby it could pofTibly afcend. Rut as all this mud be very obvious to any one (QUESTIONS. 91 one v^ho duly attends to the principle of gravitation, it is jneedlefs to fay any thing more. Qiieftion XXIX. Required a general rule for the infcribing of regular Poly- gons in a given circle. The rule given by Mr. Malton in his Royal Road to Geo- metry, Prob. 25. whiv.h is alfo the fame with that in Ward's Mathematician's Guide, Prob. 20. is this, *' Draw a diameter AB, on which conftrucl: an equilateral triangle ADB ; or draw the two arks only, interfering at D, (this preparation is the f.ane for any polygon whatever;) then divide the diameter into as many equal parts as the polygon, required, has fides; and through the fccond divifion, from either extreme, draw a right line from D to the oppofite fide of the concave circumference." Upon this Prob. Mr. Malton makes the following remarks : '^ Thus may a fidi* of any polygon whatever, contained in a circle, be obtained; by obferving the rule given above. And it is truly worthy of notice, that any right line drawn from D, cutting the diameter and the concave circumference, will cut them both in the fame proportion; or in whatever iPitio one of them is divided, a right line being diawn, from D, through the point of divifion, will alfo cut the other in the ume ratio." " Of this conftruclion or equal divifion of the diameter and Uie circumference, no demonil:ration can be given, having cqnfulted feveral able Geometricians concerning it; who fay, that it ib only an approximation, and not mathematically true. Yet 92 MISCELLANEOUS Yet I muft own, that I do believe it to be perfe6lly true, or it could never anTwer fo very accurately, as it does, in ail divilions whatever." It feems to me fomething very extraordinary to fee a pro- fefled Geometrician reafon fo very ungecmetricnlly. I always thought that not even a uiere reader, much lefs a reformer of Euclid, could give his aflent to the truth of a geometrical con- ftruiflion barely from a feeming concurrence of points, or coincidence of lines; but from an obvious re;jular ded j£\:ion fropi firft principles. For I am very clear, that there can be nothing efFe<5led by lines (at leaft in p!ane Geometry) but a demonftration may be given, directly or indirecTtly, of its truth or falfity. If Geometry were founded on no better a bafis than the bare teftimony of external fenfc, 1 am afraid we fliould foon view the whole fabric in ruins. Mr. Malton, through his whole performance, feems to lay a great ftrefs on an ocular demonftration. From whence it fhould fcem, that in order to become a proficient in geometry, it is indifpenfably necf fTiry to be furnifhed with the whole apparatus of a good microfcope, which ftiould be the criterion of every linear conftruftion. It is to be hoped, however, that the following invefligation will fully convince this Gentleman (without relying wholly on our optic faculty) that this rule is fo far from being *' perfec'^iy true" for all regular polygons, that it anfwcrs in one cafe only, when the coline of the angle at the center fubtcnded by the fide of the polygon is equal to half the radius, which is eafily fhewn to be the property of an arch of 60 or 120 degrees, anfwering to the trigon or hex^Tgon. It is fomcwhat furprizing that fo many able Mathematicians Ih'^uld be confulted, in order to be fatisfied of the truth or falfity of this rule, which may be fo eafily denionftrated in the following manner. Bifea (QUESTIONS* 93 Bifecl AB (fig. G.) the diam. of the given circle In C, through which perpendicular to AB, draw indefinitely HD, Take Al to AB as AG is to AGB, and draw GID. Then muft D evidently be the true point, from whence a right line being drawn through the given point I, will divide the diam, and concave circumference in the fame ratio. Draw GF perpendicular to AB ; then will CD be a fourth pro- portional to IF, IC, and FG. But FG is the fine, and FC equal the cofine of the arch AG, and CI is known from AB and AI being given; hence we have this analogy for findin^^ the point D, fo that DG fhall divide AB and AHB in the fame ratio. As the cofine of AG — CI : fine of AG :: CI : CD. T^'hus for the trigon, if we call AC, i, AI will be equal -, and 3 thence IC equal - .*. As Cof. 60" : S. 60" :: - : i'7220C 3 3 3 "^ ^ r:CD. After the fame manner we hnd the diftance CD for the regular polygons as follow : Pentagon, as Cof. 72° : S. 72° :: -: i-74478,=:CD. Hexagon, as Cof. 60^ : S. Co'' :: - : i -73205, =:CD. Heptagon, as Cof. 51°, 21 -— -:S. 51°, 21- ::- : i*7l903» = CD. '^ ^ ' Odagon, as Cof. 45"* : S. 45^:: - : 1*707 106, =:CD, Nonagon, as Cof. 40°—- : S. 40° :: - : 1 -60654, =:CD. 9 9 Decagon, as Cof. 36°— ^ : S. 36^ :: J : 1-68728, =:CD. '11 ' 1 1 Undecagon, as Cof. 32°, 43 — - 77 '• S. 32°:» 43 77 •' ~[^'^ 1-679165 =CD. o 2 Duodecagon, as Cof, 30' — -: S. 30':: -: 1-67202, r=CD. Now, 94 MISCELLANEOUS Now, by Maltoii's (Ward's) rule CD is equal v/aIT'^— -aO^ rr \/'3 zz 173205; which correfponds only with the trio-on or hexagon. The reafon why this conftru£tion anfwers to the arch of 60 or 120° will be evident if we confidcr, that in this cafe, CD is double of GF, and thence FI is - of EC ; but 3 FC is — -y and therefore Al is =z - of AB : which is the fame part as the arch AG is of AHB. Hence it plainly ap- pears, that the point D found by Ward's method is not true for any polygon whatever, excepting in one fingle cafe ; for it is evident that the fide of the tetragon cannot be faid to be found by this conftrudlion ; for by the above analogy it will be, asCof. 00° — 1 : S» 90° :: i : -> which fhews that CD in this o cafe is infinite. The latter part of Mr. Malton's 47th Prob. (which is alfo Ward's) is in the fame predicament with the other, being proved to be falfe as follows : The conftru6tion being made accordingjto the rule (fio-. H.) it is evident from the like fituation of the circles, kc. that wherever the point B is taken in GK the lines EF, DC, will be always parallel to each other, and the z. DCF = CDE, as alfo DEF = CFE. The fides CF, FE, and ED, are likewifc equal, being each a radius of the fame or equal circles. What remains then to be proved is, whether the z. DCF be rr the greater angle in a regular pentagon, formed by the diagonal and a fide. The moft direft (if not the only) method of invefti- gating which appears to be by a calculation of that angle. Having drawn fuch lines as appear by the fig. EF will be. the fide of a regular hexagon, and FB the fide of a regular dodecagon, infcribed in the fame circle, and they are therefore in e (QUESTIONS. 95 in the ratio of i to V2 — \/ ■^. And fince FG is ± IK, and KC II FG, the z. GKC is aright z. ; and the z. KBCrzABI is evidently zz - 3. right z.. Alio, as BF is the fide of a do- decagon, the z. FIB r: - of a right z, and confequently the zFBI =: ^ of a right z ; but the z. CBF is the fup. of KBC + FBI •.' the z FBC = -ofa riffht z =6o'\ Then as the fines of the zs FBC, FCB are in the ratio of their op- pofite fides FC, FB, we have, as i : Va — v'3 :: S. 6o° : S. 26% 38',2" nearly; hence the zDCFzrDCB + BCFrryi", 38', 2'^ But the z DCF in a regular pentagon is 72°. Therefore this conftruclion is falfe aifo. N. B. Since the above was written, the inveftigation of the veracity of thefe rules has been propofed in the Ladies Diary, to which the above anfwers were fent with feme alterations. Queflion XXX. To find the fum of a feries confilling of looo furfolid num- bers whofe roots are -, -, -, -, X &€. 63236 Jnfwer. The general expreflion for the fum of a feries of powersas2" + 3"-{-4'', &c. tor-i terms is 1 V , &t, «+i 2 3-4 I I « «-}-« — I X n — 2 5 A J r ..1, 1 — ! -, &c. And lince the roots «+i 2 3.4 2.3.45.6. of the propofed feries are evidently in Arith. Progrefs. and the common difference equal to the lirft term, wc may eafily adapt this general equation to the given feries by adding unity to MISCELLANEOUS to both fides thereof, and multiplying the whole by ~\ . For then it becomes ^j + ^l "^ ^ + 61 V -61 + 3I j._ 4- - the given fenes i = - x : [— J -^ , occ. , &c. 3.4 2.3.4.5.6. «+i 2 3.4 icoof . loool' f X 1005* loodi* X :— 7— + +- 62 12 12 (till it terminates) zr - zz 21497824502732767 the fum required. Quefllon XXXr. In a geometrical progreffion beginning from unity, having the common ratio and number of terms given, the fum of all the changes in the feries will be cxpreffed by this general theorem r" — I XI. 2.^- ft a ; — ^, X : I + 10+ 100+ 1000, &c. br — to b terms, where each term muft be diftinilly confidered, and eftimated according to its variable local value in the feries. Required the demonikation or invefligation ? Queftion XXXIL If A be any given number, P the places of figures, and C the number of changes, then will the fum of all thofe changes be r 1 u u- /• • ^C , loAC looAC , ^ reprefented by this leries -q- i ^ — I 5 — > ^c. - r r r continued to P terms. Required the invefligation. Queftion XXXIIL Required to find a fra£lion fuch, that being taken from its reciprocal the remainder (hall be a fquarc. This ^/tf.D ^ Q^ U E S T I O N S. ' ^^ This queftion was propofed in the Ladies' Diary for 17C8 ; but as the poflibility, or impoffibility, of fuch a fradion exit- ing has never yet been fliewn, and therefore the queiVion not anfwered, I thought it not improper to do it here. If we put - for the required fradion ; then, per queftion, is to be a fquare number; but is equal y X y- X xy therefore the queftion is, To find two numbers fuch, that their produdl and the difference of their fquares may be fquare num- bers. In order then to inveftigate two fuch numbers, if pof- r^ . r* fible, put xy zz r% then will x zz — ; therefore y^ — D y. y" per queft. Now it is evident that — will be in the fame ratio 1 • y to J, as the nyp. of a right-angled triangle is to a fide ; that is, Z5 h '. s :: — '• y, *•' y — V- x r. From whence it y h appears that if a right-angled trianrle can be found, of which the hyp. and either fide are rat'onal fiuare numbers, tlie quell, may be anfwered, otherwlie it is impoffible. But th^t it is abfolutely impoflible for fuch a triangle to be formed, will be very evident from the general expreffions for the fides of a commenfurate right-angled triangle, viz. rti' -f- «*, m^ — «% and imn. For fince ni^ + 7r, and m^ — n (or imn) are both to be rational fquare numbers, it is plain that in tlie former expreftion 711 and n muft tliemfelves repreft-nt the two legs of a triangle; and in the latter ;/; mufl alfo repref nt the hyp* and n a leg of a triangle ; hence it follows, that ci'hcr 771 mnft at the fame time have two different values, or that there muft be a right-angled triangle exifting the hyp. of v.hich is eq-ial to one of its fides ; both which conclufions are abfurd. The abfurdity of this queftion may be a'fo fhevvn rather differently, thus; Since x^ — y^ will be always a rational 9S MISCELLANEOUS / + r^ fquare number when x — , and xy will be always a T" rational fquare when ;«• zr — , (where y and r may be taken y x^—y'" at pleafurc) it appears, that if , be a fquare number, ■i- — ' — muft neceffarily be equal to — . Therefore if we fup- ir y pofe r of the fame value in both expreflions, {y being the fame from the nature of tlie queftion) we fhall have 2r^ =: y^ ^ yr^ ; and this fuppofition we have certainly a right to make, fince the numbers are taken at pleafure, confequently if it can poflibly be a fquare number, it muft evidently hold equally good in this cafe as in any other. Now in this equation nothing can be more obvious than if r cr J' that 2r' will be greater than y -{- yr'^, and if r -a^ that zr^ will be lefs than y'^ + yr^ ; therefore this equation cannot poffibly ob- tain in rational numbers unlefs r zz j, and this gives *• r= y alfo; ffom whence nothing more can be inferred than that the required fraftlon muft be -, or unity; which agam flicws n the abfurdity of the thing. The queftion therefore requires an abfolute im.pofllbility. It may, perhaps, be obje£led by the lefs difcerning reader, that thefe are not fufncient pi-oofs of the abfurdity of this queftion, as there are vulgar fractions which are equivalent to rational fquare numbers, and yet neither numerator x\ox de- nominator independently fuch ; and therefore ~ may be a fquare number, when neither x~ — -.y^^ nor xy are rational fquares. To which it may be anfwered, that it can only fo happen, when the numerator and denominator of a fraftion arr! not prim.c to each ot'ier, or in their lowePt terms. But muft evidently be in Us loweft terms when - is foj it xy X is (QUESTIONS. 99 is therefore imnoflible that the expreflion can be a fq^iare y >: number, unlcfs lx)th x" — y^ and xy arc fucl>, which has been before proved to be abfurd. Another queftion equally as abfurd, whicli has been fbmc time handed about, is, To find a triangle fuch, that- not only the lldes may be whole Clumbers, but a!fo a line drawn from the angle at the bafe, and terminating in the perpendicuhr, may be a whole number ; and moreover, that the fquare of the klfer fegment, taken from the fquare of the whole perpeiuli- lar, may leave a whole fquare number. Now the three fides of a triangle are only commenfurabis when m^-hn^ denotes the hypothenuft; and ?ri'^ — n% and 2mn the two fides ; tn and n being numbers talccn at pleafure, fo as m cr ?:, Let then AB (fig. 2#) be denoted by 2Wi»j then can AD lie exprefled only by ;;r -r- n-, if AB, AD, and BD mud be rational numbers. And fince AC^ — AD^ is t) be a whole fquare number, per queft. AD will be alfo the fide, and AC the hyp. of a right-angled triangle ; but AD is equal //r — ;j% and therefoi-e AC can be reprefented only by /7z~-f «% to be a whole number ; and fince AB is denoted by 2wz«, AC can be equal only to m^ — n'. Hence it is evident, that if AB, AC, BC, BD, and AC" — AD" muft be v.hole numbers, AC muft I ■ neceliarily be equal to both m"4-'A ^^'-^ ^^^~ — ^''■y which is ah- \ furd. But as this may not perhaps appear wholly fatiifac- torv to every reader, fince m and n may have different values, and yet irnyi which reprefents the bafe of the triangle continue the fame, I (hall confider it a little differently, Let AB = imn (as before) =z ipq^ wlicre m zr «, /> cr q^ ?n cr/*, and \' qrr K, then will BC rr m'^-\-n-y AC = tri^ — «% BD — f'-Vq"-, and AD = f — q^ y but by the quefl. AC^ + AD"' G 2 zzU, 100 MISCELLANEOUS zz n, that IS m^ — «^i*+ P' — q''\''= D. Now it is obvious, fince the fum of thefe two fquares is to be a fquare whole num- ber, that their roots muft alfo reprefent the two fides of a com- menfurate right-angled triangle; but if p"" — ^^ be one fide thereof, the other can only be equal to 2pg when all the fides are rational, hencr m'^ — h^ = 2pq = 2ww, and AB = AC, confequently BC (=: m'-rjz^] is irrational. That AD and AC may be both exprefTed by /«" — «% and BD and BC by rn' + rr', when tlie bafe is denoted by 2mn, is evident, becauft, as we have juft obferved, the fadors m and fi may be varioufly aflumed, and yet the bafe of the triangle re- tain the fame numerical value. From whence it appears that there may be various right-angled triangles of rational fides, having one common bafe. To illuftratc this, fuppofe we take the iSth Queflion of the 6th Book of Diophantus's Algebra, where it is propofed, that in the right-angled triangle ABC, AB, BC, CA, BD, AD, DC, (hall be all whole numbers. Let 7nm — any compound number, fo that m may be always greater tban n ; fuppofe 24 ; then will w ~ 4, or 6 j and n r= ^, or 2. From the finl: values of m and «, we have AD rr y, BD =. 25, and AB — 24 ; and from the fecond values of m and w, AC =: 32, BC =: 40, and AB = 24 ; and thence DC zz 25. It may here be obferved, that this is a general me- thod of folving the problem ; but Diophantus is obliged to draw the line BD, fo as to bifedl the angle ABC, in order to gc't an equation, viz. AB x DC =: BC x AD, to make the fidts become rational. ]f v/e take 2w« = 28, and ;n zz i/iS, or \/9S, and thence n =r v'7, or a/ 2, (for 2 X a/ 28 x -v/y =28, and 2 X -1/98 X '/2 = 28) we fluall have AD =: 21, BD = 35, AC r: 96, BC =r 100, AB =z 28, and thence DC = 75 ; \yhich are the numbers found by Diophantus's method, 3 Hencs (QUESTIONS. lOI Hence it appears, that it is equally as eafy to firid a rio-ht- angled triangle, with any number of lines drawn from cither of the acute angles, and terminating in the oppofite fide, fo that not only the fides of the triangle, but alfo the lines fa drawn, and the fegments of the oppofite fide formed thereby, fhall be all whole numbers ; provided we can find a compound number to reprefent half the bafe, which may be refolved into as many different faftors. Queftion XXXIV. ^' In triangulo piano ABD, angulus DAB obtufus eft, angulus DAC reftus. Dantur AD + BD = 51, AC per- pendic. ad AD =. 21, AB = 32. Quaeritur AD. (fig. 3.) Solut. Ex A. dimitte perpendiculum AE. Sit AB ~ a^ AC =: bj AB + BD = f, AD ■=: a-; tunc (ex natura tii- , . BD^ + AD^— AB^ , ^^ /_2<:a--j-2x^-— «- aneulorum) t^yt — dat DE — . 2BU 2C — 2X Et ob fimilia trianguia ACD, EAD; erit DC : DA :: Da : DE, AD^ et [per Euc. 47. I.) DC = VAD^ + AC% ergo ^^.^^q -j4 T^T? 17 • J V '^^^ — 2f.v + 2A'- — a' x"- — DE. Exmde erit rr — ^rry hinc ic—ox Vx^rb- acquatio numerofa — 8072.%'^ -\- 501636^'^ — 98 bQ.'iA-^ 4- 141873228^' = 1096735689, ex hoc X rr i3'67g2. ^.E.D. '^- This queftion, widi the folution, as they are here inferted, were font to the Editor of the Town and Countr'/ Magazine in Jan. 1774, and the ijueftion was accordingly propofed in Feb. and anfwered in the next Maga- zine by Mr. G. B. of Coventry. But as the folution there given (if it may be called one) is very inelegant, and the conclufion falfc, I thought this a proper opportunity of giving the true one. It is evident, from a fompaiifon of the folutions, tliat he has either committed an error in the .reduftion of his final equation, which would arife to vcrv high dimenfions ; ^n■ elfe he has found the mcafure of AD trom a mechanical conl^ruftion, y/iiich is little belter than guclTmg at the anfwer. G 3 Queftion J02 MISCELLANEOUS Qyeftion XXXV. There are three ports ABE, whofe bearings from each other are as follow, viz. B rroin A, N by E ^ E ; E from A, E by S, and E froi-n B, S. S. E. A fhip at the port A being bound to a certain ifland, C, bearing N E., faileth thence E by 5, and after running as many miles (by the log.) as the port E is diftant from A, in a current whi;h fettcth N N Ey arrives at her dcfired haven, which is lOO miles diftant from B. Quere the diibncc of the pi .ces from eich other, and the velocity of the current, without Algebra? (fig. 4.)^- Jnfiuer. The figure being conftrufted as per queftion. In the trapezium ABCE we have EC rr lOO, and the z.'s BAE, = 87°, II', i5"> AEB = 56% 15', ABE = 36°, 33', CAE =: 56°, 15', AEC = 101% 15', and ACE = 22°, 30'; from whence we find BAC rr 30°, 56'!, and BEC — 45". ■Now if the fines of all tlie angles be drawn to any determinate radius, it will readily appear from a fimilarity of triangles, and the compolition of their ratios, that as the fin. BAC x fin. ACE : fill. ABE X i\n. BEC :: BD : DC :: 1967 : 4202 :: i : 2-136. * This queflion appeared in the Town and Country Magazine, for Sept. 1772, and was anivvered by the propoler Mr. C. in the next Magazine, the above foJutlon being (as I was informed) fent too late for the month. What this Gentleman there gives us as a folution, may with juft as much propricly be called i'o, as faying the drawing of a geometrical figure is the dcnv>niliating of its, properties. For it plainly appears from the queftion, that the method ot invcftigating the numerical values of the dillances and the veloc. of the current ivitbout having recourfe to Algebra, was the thing r::quired, and not a bare conftruttion of it, which, by the bye, he has alfo rendered f[u:tc tV.lfe, by taking the ang. NAB much lefs than that given ia fhc nuelVioii. The demonfiration and calculation he has omitted; bccaufe, lie fays, tiicy are too evident to be infilled on ; but in this I cannot help thinking he is much miftaken, for I am pretty certain that no one can fee the rcafon of the analogy, upon which the folution wholly depends, without • I'onje recollection. The {QUESTIONS. 103 The fum of the ang. at the bafe of the triang. BDC is 1 12°, 30'; therefore {per the third axiom of plane trigon.) it will be, as 3*136 : 1*136 :: tang. 56°, 15', : tang. 28', 28', half the difference of thofe angles; whence the ^ CBD is 84°, 43', and BCD 27°, 47'. Thefe being obtained, we eafi'.y find al^ the diftances, that is, B A ~ 90*66, BEz: io8'92, AE r: 64*94, AC ~ 166*11, and EC — 140*82. And fince EC is || to Aa (the direction of the current) per conftruc. therefore as AE : EC :: i : 2*13, which is the ratio of the fhip's velocity to the velocity of the current, which are all obtained without Algebra. W.W,R, Queftion XXXVE **• In a plane triangle, having given the vertical angle, the ilifterence of the bafe and one fide, and the fum of the per- pcnd.cular, from the angle of the bafe contiguous to that fide upon the oppofite fide, and the fegment thereby cut off from that oppofite fide contiguous to the othtr angle at the bafe; to conflru6l the triangle"^." * This queflion was propofcd in tlie Ladies' Diary for 1774, by tl>c P.ev. Mr. Lawfon, to which two different folutions were infertcd in the laft year's Diary by the Rev. Mr. WiUlborc, and the author of this Trcatife. But as neitl^er of thefc fohuions were concife enough for the cor.duftor of the matliematical correfpondence in the Town and Country Magazine, he repropofed it } hoping that fome of his ingenious con- tributors would fend a more elegant folution. How far this has been per- formed I fliall not pretend to determine ; but the method of conftru£tion given in the Town and Country Magazine is certainly more fimple than either of thofe in the Diary. But as the figure itfelf is quite falfe, owing to a want of parallehfm in the lines, the triangle produced does not in any one refpeft correfpond to the data.; the vertical angle being much too large, and tlie allumed diflerencc in the qucilion being double of that in the 2ii.r>igle. Upon this account, and the want of a demcnilration, which perhaps may not appear fo evident .(at lealt to the young reader) as he itenis to think, I have iuferted it here iud corrected the figure. G 4 CONSTRUC- IC4. MISCELLANEOUS Construction. ** Make AB (fig. A ) equal to the fum of the perpendicular and fegmenr, the an^jle ABC, 45", and ABE the fupplemcnt of the given vertical one, BE the given diffeience. Produce CB to mtec FED drawn parallel to AB, in D. Join the points A, D, draw BM — BE; from A draw AC and CG parallel to BH and BE relpedtively, then will AGC be the triangle required." Demonstration. Draw BK |j to DA; and join EH, and GK. Then, fince AB is (C]ual t(; the given (urn of the perpendicular and feg- ni'^nt, and the z. ABC half a right angle, by conftruction, a perpendicular let fall from any point as C upon the oppofite fide will be equal to aB (Eu. 9. 4.) .'. Aa -f «C = AB, the given fum. And as the z. ABE is equal to the fup. of the given z. hy c. nllruc. and CG || to BE, the z. CGB is equal to the z. ABl (Eu, zg. i,) and therefore the z. CGA equal to the given vertical angle. And moreover, becaufe DE, DH, D3, BH, BE, are refpectively 1| to BG, BK, BC, CK, CG, tlie triangles BEH, CGK are hmilar. But B.4 is = BE .'. CK is =r CG. And again, becaufe DA is j] to BK, and KA jj to iU^, KA is = BK = bE rr the given difference of the bafe CA, and one fide CG. ^ E. D. Queflion XXX VIE In the mound of an Elliptical Garden, whofe tranfverfe is to the coi/i. as 3 to 2, a peJefla!, 50 yards high, is fo placed, as that the apparent magnitude of an Herculean Figure, 10 feet high, on the top of it, is the greateft pollible to an eye fituated on the tranfverfe, and 20 yards from the center of the Jillipfe. Required from hence the Area of the Garden. Jnfiuer. In Fig. ^. EF reprefents the pedeftal _L to the plane of the femi-ellipfe ABl; FG the fiatue, and D the giyen point in the tranfverfe. Now it is evident lince DE is in the H D I QUESTIONS. 105 the plane of the ellipfe, and GE ± to that plane, that the Z. GED is a right one, and therefore if we call GE rad. ED X GE ED will be the tang, of the z. G, and -p ■' that of the Z. F. Moreover it is evident, that the z. FDG under which the ohje£l is feen, is the difference of the angles DFE, DGE, which mn?L be a max. per queftion. Making therefore FE — a^ EG = b, and ED =: /, we have — = the tang, of the greater a angle, and t — that of the lefs; and the flux, of the difference of thefe angles, exprefled in terms of the rad. and tang, is b^i apt , . , , Tr — ^-T — -7: 77-: •> which beincr made rr o, and reduced, gives / = V ab. Hence it appears that DE the dift. of the pedeftal is a mean proportional between GE and FE. Now, it is obvious, from the nature of the queft. that the line DE muft be the neareft difc to the curve, or a min. therefore, putting ID :=: f, and the fern, tranfv. r:: x^ we have (pet queft. and Emerfotii Flux. />. 126) — = Dli; hence (Euc. 47. I.) ab — =: HE% and per prop, of the Ellipfe, 25 ^ - X A- = HE^ ; •.• - X ;<• — ab ; 9 '^5 9 -S 2 s ^ reduced x zz 245*9385, and thence the area of the Eliipic rz 1266811 Iquarefeet*. Queftion "* This cjucflipn I propofed in the Ladies' Diary for 1775, to which .a folution is given in the Diary for tlie year following by Mr, Rowe ; but, l the Neg. fign fhewing that it would require a corredlioa according to the nature of the problem. Q^ieftion XXXIX. In the expreflion -r^, required the relation of;; and s, fuppofmg their nafcent increments to be cotemporaneous, and the fluent correfponding to any given values thereof a mini- mum; alfo the curve defined by the equation expreffing the relation of y and %. Anjvuer. Let y alone be confidered as variable, then the ... 2 Flux, will be ^ — rr^ ; and when y only is made variable, its Flux, becomes -^, Let now the latter be divided by th© former, and the quotient put =: -, and we have — rr - : t y ^ Hence the Hyp. Log. / = Hyp. Log. \y -\. Hyp. Log. w, {m being any conftant quantity) therefore / — my~i i which being equated with the Flux, of the given expreflion, when y alone is variable, gives :^ — niy^, from whence y^y^ =z |-mz"'s:% the Fluent of which is |J^•^ =: 2^ jfnz, ex-* prefTing the relation of y and z. If for -y^y we put 4 then the equation becomes y^ — d^z^ anfwering to a Parabola of the higher order, I Qyeflion ro8 MISCELLANEOUS Qtieftion XL. Required the lat. of the place, and declination of the fun ^ when the length of the day is to that of the night, in the ratio of 3 to 2, and the fun's meridian altitude to his deprefliort at midnight, as 2 to i. See T. and C. Mag, p, 303. 1773. Aufiver. In the Orthographic fchenie (fig. 6.) P reprefents the North Pole, EQ_the Equator, HO the Horizon, AD the Sun's Semidiurnal Arc, and DR the Seminodturnal. From the given ratio of the lengths of the day and night, the arches AD, DR, are both knov/n; therefore putting a — the nat. Verfed fine of AD, b zz the nat. v. f. of DR, and x zz [lib) the na^. fine of G R the Sun's dep'ellion at midnight, per fmi. a's b : a :: X : -- — A«, the fine of the Sun's merid. alti- tude. But per queft. AH zz 2 OR, that is ^x^/l — *- ax ,.V._4 = -. = ha, .♦. — zz 2xis/i—x^; hence x zz y i — zz •320604 the nat. fine of 18°, 42', the Sun's depreflion at midnight; «•. by the nat. of the fphere the comp. of the req. lat. HE (= I X HA + OR) zz 28% 3', and the Sun's declin. = EA (:= I X HA — OR) = 9°, 21'. IV. IV. R. Queftion XLI. The Fraaion illlil^ is equal to < 1 1 1 76 ^iiiiiH. Re^ / / 4172 417-6 quired tlie inveftigation, Queflion XLIL If any plane triangle ABC (fig. 7.) be circumfcribed by a circle, and a right line be drawn from any one of the angular points, fuppofe P>, bifcifling the faid ang. till it meets the circumference in D: 1 fay that a circle de.^cibed with rad. DA, Q_ U E S T I O N S. 109 DA, will pafs through the center of a circle infcrlbed in that triangle, and aUo through the other angular point C. Re- quired a Geometrical demonftration, Qpeftion XUII. To find the center of a circle P (fig. 8.) to pafs through a given point, and to cut two lines given in pofition, fo that the intercepted arch AB may be of a given magnitude ; or, that its chord may fubtend a given z. APB. Queftion XLIV. From a given point C (fig. 9.) on the diam. AB of a circle produced, let a tang. CD be drawn, and another right line from the point C, cutting the periphery in E and F, fo that the fine of the z. BCE, may be a fourth proportional to CB, (confidered as radius) and the fines of BCD and a given z. P. I fay that the acute z. AIF or BIE, formed by the diagonals AE, BF, is equal to the given Z. P. Required a Geometrical demonftration. Qiieftion XLV. Suppofe a femicircular bowl placed on an horizontal plane, at what height and diftance from the bowl muft a light be placed, fo as to illuminate one half the interior furface thereof ? Queftion XLVI. By repeatcjd obfervations on the Northernmoft ftar in thft right feot of Urfa Major (marked by Bayer/) whofe prefent declination is 49°, 15', I have found that its altitude encreaies more in a given time, in a certain latitude, than any other ftar of different declination. Requiied the latitude of the place of obfervation, and the increafe of the altitude of the faid ftar i from 8 to 1 1 when it fouths at midnight. G 7 , Qucftiofn no M I S C E L L A N E O U S Quc-fcion XLVn. In the latituae 5-?% 27', N. on June 25, i775> [I faw i lainbow bearing ESE. Ret]uired the hour of the day. Qticfiion XLVIIT. Sti-ppofe an inflexible rod of iron, 40 feet long and 2 inches diameter, be fo fixed at one end that it may vibrate freely : in what time will the other end of the ro.J, being let fall from an hori/ontal dircdlion, defcribe an arch of a given length, fuppofc c!5 feet, from the comir.er.ccment of motion ? Qiieftion XLIX. Required general expreflions for the fides of a right-ang'ed trian^;le in whole numbers, fo that any given number of lines drawn from one of the acute angles and terminating in the oppofi'ic lidc, as alfo the fegments of the faid fide formed th'. rcby, may be all whole numbers. And moreover it is re- quired, that tiic given number of lines fo drawn may be the mod which that triangle can poiTibly admit of in whole num-^ hers J and that the periphery of the triangle may be a minimum, Suppofe 5 the number of lines to be drawn, reuuired the fides of the triangle, Sic. as above. • Qiieaion L. A cone btipg cut by a plane parallel to the bafe, the are^ cf the fetSlion (— 20*7736 mches) is found to be a mean pro- ■portional between the iuperncies of the tuo parts of the cone ; and the ratio cf the fide of the lov/cr fruftum to the femi- diamcter cf th'.' bale as 23 : So. Required the cimenfions of the cone. Queilioii 7„ g F A (QUESTIONS. Ill Queftion LI. The expreflion s/ax'' — b^ is a rational number. Required a rational value of .v. Q^.eflion LIT. The internal diameter and diagonal of a cylindrical cafk (which is made of the leaft wood poffible) are refpeilively exprcfTed by a- ^5 and v^\ x a|*. Required the content in Ale Gallons when y ■=. x^. Queftion Llir. Let y — a ■=: a^ x a — a p be an equation to a curve. Quere the Abfcifla when _y is a maxivnum. Queftion LIV. How many elementary founds may be formed out of the 24 letters of the alphabet * ? Queftion LV. It is required to divide the area of a circle geometrically into a given number of parts, which may be equal both in area and circumference f. Some i\fefid Remarks ttpon Equations, As the young Algebraift generally meets with fome difficulty in rightly afcertaining the roots of quadratic equations of the * Tacquet in his Arithmetka Theor. p. 3S1, fays. Millc milliones fcriptorum mille annorum millionibus won fcribent omnes 24 litterarum alphabeti pcrmutationes, licet finguli quotidie abfolverent 40 pagir.as, quarum unaqua:c[ue contincret divciTos ordines litterarum 24. f This Paradoxical S^urfllon, or perhaps rather double entendre, is taken fiom Laivjoni D'^Jfertaiion on the Gco!:ietr'tcat Analyfis of the Antieiits. The demonriraticns of all the tlicorems, with the Geometrical Gon(lru£tions of liie problems contained in this book, will be g;\ea at the latter end of An EJfaj on the ufefulnef of hl^'.bcviatUi^ Learning, which will foon be pul;liflicd. 3'' form, 114 Miscellaneous 3"^ form, as alfo in Cardan's and Colfon's Formulas for i\i€ roots of Cubics and Biquadratics, from the irrational Binortiial dt X dt: *^ — y. I fuppofe the following remarks will not be "unaccei^table to him. 1. In the folution of a problem where the final equation is In a *\ /^* this form x^ — ax ■=. — Z-, and thence x — - dz V- — ^, 24 the conditions of the problem muft determine whether the af- firmative or negative fign gives the true value of jt; for if from the nature of the queftion, x be greater than a, we muft evidently vife the affirmative fign ; and if lefs, the negative. As in this problem. To divide the number loo (a) into two fuch parts that their produSi and the difference of their fquares tnay he equal to each other. If we denote the lelTer part by Xy and therefofe the greater by a — x^ we fhall find x zz — z±: \J'- — . But X being given lefs than «, the upper fign (-J-) gives *• too great; fo that x — ^^- — y— rr 38*19658, &c. muft be the true value required. 2. Hence it appears that though there be two affirmative roots in a quad, equation, yet, in general, only one of them will anfwer one cafe, or the particular quePaon propofed. The fame obfervation holds good in equations of all dimenfions; for fuppofe in the folution of a queftion we have derived this final biquadratic equation, x'^ — ax^ -i- bx^ — ex -\- d =: o, where all the roots are affirmative, w^e muft not conclude that the queftion admits of four different anfvvers, for it will often be found upon trial, that three of the roots will pro- duce an abfurdity, and only one value anfwer the particular* cafe propofed. O F E Q^U A T I O N S. 113 3. In a queftion producing a quad, equation of the third form, if the unknown quantity be affumed indeterminately in regard to greater and lefs, then will the affirmative and negative ligns exhibit thofe values refpedlively. For if it was required to find two numbers whofe fum is a and the fum of their alternate quotients by we (hall find that - ± * / ^ 2 V 4 2 + ^ are the numbers required, 4. In the redu£lIon of cubic equations it will be proper to inform the young reader, that Cardan's rule is only of ufe in cafes where two of the three roots are impoffible ; and there- fore it would be in vain to attempt to folve a cubic equation compofed of real roots by this method. As for example. Let the equation z' — 72: = 6 be propofed, whofe roots are — 2, 3, and — I ; the numeral coefficients being written in the formula, we have s ::=: V3 + v + Vs — v — 7;-; which is only an imaginary expreffion ; the fquare root of a negative quantity being impoffible. So in this equation x^ — ()ix = — 330, the roots of which are 5, 6, and — i r, we get X — V — 165 + V — 685-03 -1- V —165 — V— 685*03, which is alfo imaginary. But when two of the roots are im- poffible as in this equation x^ + dx rz 20; then we get X — 'J 10 -\- VioS + Vio — -s/ioS — 2, where the ex-^ preffion is real and poffible; the other values of x being imaginary. c. I Ihail now endeavour to clear up to the learner fome feeming difficulties in finding the roots of a cubic equation by Culfon's Theorem. And as this method principally depends H upon 114 O F E Q,U A T I O N S. upon finding the cubic root of an impofliblc binomial, I fhall firH: (hew the inveftigation of an eafy rule for obtaining fuch roots. Let a -i- \/ b = vA + Bj then by involution A + B — a' + S"^' ^ ^ + 3^^ + ^^. Put fl^ + 3^/^ -^ , and 33 V^ -\- hi := B; then will a^ + ^ab\' — 3«V^ + ^T = «" — 3^** + ^a'b''--'b' ~'^~r^' - A^ — B% .'. ^ = tf.;— A^— B'K; hence, by fubftitution, a> + 3« x «'— A'— B"\^ r: A, that is 4^^ — 3 -^A" -— B' x a = A ; from whence a is eafily found, and fince Vh — \] a^ \/ A^ — B% we have « + \Ja'- — V1^^~^^^W - VA + B. When I] h^ — B^ is a furd, both members of the root wfll be irrational. In that cafe multiply the given equation by fome number till v A^ — B^ comes out rational, remembering to divide the values of a and Vh zX. the laft by the root of that number. 6. Now if the fecond term of any cubic equation, reduced to Colfon's general form, be exterminated, we fhall have by Cardan's rule this value of the new root, % — '\/r-\- Vr^ — q' ■{■ y r — s/r' — q\ And as the cubic root of a binomial may be always inveftigated in a fimilar form, they alTume m ±: <^n — '^r dz y/r^ — ^^, and thereby obtain zr=»2+ '/n + m — v'« = 2/«; but as this gives but one value of z, they derive two more ex- prefllons from this moft obvious principle, viz. that the cube root of any number, being multiplied by the cube root of unity, muft O F E Q.U A t I O N S. it^ muft ftill remain a root of that number, that is, y^x^ y. 1/ 1 zz X ; but the three roots of unity are i, — ■■ ^, and 2 : fo it appears that the cube root of each binomial 2 may be exprefled in three different forms as follow : Vr + '^r^ J. m + \/ri X I ~ OT -f \/fi. II. ;;n^« X r- + v^-^ = ^m-^n^m^-^Ar^^in^ 2 2 , 7n-YVn X -■ — — = — ~ — L, Vr — Vr^ — q^ :=! I. m — V n X I — 7;z — '/n. o m—\/n X ~'~^~^ r= — »» + v/« — ^V — 3 + V^ — 3>? From whence we (hall evidently have thefe nine different cxpreflions for z. I. +1. =:2W-- -- -^ - - -- =rz« ^v'w + 'w v^ — X" — •y — 3^, — m — ^/ — 3« will become real and poflible, that is — m -\- V 3;?, — m — \/ 3;?, which no one can be fo dull as not to fee the reafon of, if he knows that a negative quantity fubtradled becomes an affirmative one. On the contrary if n be affirmative, V — 3^^ is abfolutely impofTible, there being no fuch thing (according to the common definition of the term) as the fquare root of a negative quan- tity ; thofc two roots are therefore then imaginary, and 2m is the only polhble value of s. 8. It appears from the laft Rem. that Cardan's Theorem may be rendered generally ufeful, by folving fuch equations as have three roots real, as well as thofe that have but one roQt real and two impoffible ones. For fmce the three values of z iTray be exprefied by 2;;;, — m -f v^ — S'?? an/ — 3« = - - -^ 2 + \/ 3, 9. One root of any cubic equation may be had by Cardan's Form though the expreiTion becomes impoifible, without having recourfe to the formula m ±: '/n, by extra6^ing the - roots of the binomials, the impoflible terms vanifliing in the addition by being always afFe a root. Suppofe it is required to exhibit, by Cardan's Rule, the three roots of this equation .v^ -}- z= 13, q zz: 56, r zz 92, and j — 48^ then will the roots of the biquadratic ;.-* 4- 13^^ + ^6x^ + gix 4- 48, be found to be — i, — 2, — 4, and — 6, and confequently thof' of the propofed equation will be — i, — 2, — 4, — 6, and — 13. Example 2. Suppofi' the fecond term exterminated, and let the refu'tlng equation be z^ -\- Ba^ ± Cz^ =!:: D::: + E — o, and let D n BR' ^, U being the root of this equation a^ — B^ B — vc — C ; then will the five values of z be exprefTed by the roots of thefe two equations z^ -f ez^ -]- a''z 4- - — o, d '£" — a% ■\- d zz o. For thcfe equations being multi-plied produce O F E Q.U A T I O N S. 121 produce an equation fimilar to that propofed, and the value of a is known from the above cubic equation; and by equating the coefficients vi^e have alfo the values of b and c. When this formula is applied to any numerical furfolid equation; as s' + 162;^ — 8202^ + Dz + 320 = o, (the coefficients being affumed at pleafure) w^q firft find the value of a =: 10, and thence we get D zr 1440, d — 16, and c - — 20. Therefore the roots of this equation d X* + i6z^ — 820%" + 1440Z + 320, will be had byrefolving the cubic, z^ + 102^ + 1002 + 20 = o, and the quadratic »* — loz + 16 ^ o, the roats of which are, + 9'93> + 8, and + 2, the other two roots being imaginary. II. Various other literal expreffions may be found, by taking away fome intermediate term, and then finding two equations, the product of which fhali give a refult fimilar to the propofed equation ; but thefe Vv^ill be found to be of ufe only in particular cafes, which feldom occur in pradlice. For though there may be always as many independent equations as unknown quantities, yet it will be found, that by their difFereni combinations, we {hall always recur to an equation of the fame dimenfions with that of which we are endeavouring to inveftigate the roots. And if we ftrike out any one term in either of the affumed generating equations, we fliall then have more equations than unknown quantities, from whence neceffarily arifes a particular relation between fome of the coefficients, 12. But fuppofing a compleat formula for the roots of a furfolid, or an equation of higher dimenfions, were by any algebraic artifice obtained, it would be of little value, for there would be far more trouble in getting the roots this way, 4 than ,2t O F E Q^U A T I O N S. than by the method of converging feries, which is well known to be generally far more expeditious, even in biquadratics and cubics, than the finite theorems. The following general formula for furfolid equations I have deduced from thence, wherein the method of operation is rendered fo fimple, that I think none more eafy need be wifhed for. If x^ + aar* + bx^ + cx"^ -\- dx -\- m z: O, then will x be found by this theorem. i I I ', ^ '~ ~~ ■*> In which if the value of ni^ being fubftituted, there is no remainder, the quotient gives one value of x ; olherwife, this quotient muft be fubftituted for m^y and thus repeated till it either terminates or be as near the true root as neceflary. The rcafon of thus aflutnlng m'^ will be obvious, if we confider, that m is always tlie produft of all the roots (having their fign.s changed) and therefore «t will either be a root, or generally near one. Example i. Required the five roots of this equation, >■' — 23^^ + iS9"25-^'' — 459*^ + 564*75^ — 243 = o* Here ni"^ — 3, which is affirmative becaufc the figns change alternately; and 47/2"? {=4-m) — +972, ^arn^ zz — 5589, 2i;n"» = 4-8599-5, ctn' = —4131,— w = 4-243; d- + 56475, 5mt- = -f 405, ^am> — — 2484, ibrn^ — -f 429975, and ictrT^ — — 2754. Therefore 972 — 55^9 4- 8599*5 — 413^ 4- 243 _ 4-94»5 - ^ ^^^ 564754-405 — 24H 4- 4299'75 — ^754 -r3i'5 a root. O F E Q^U A T I O N S. 123 a root. Now dividing the given equation by x — 3, we re- duce it to this biquadratic, .V* ■ — 20.v^ + 99*25.v' — i6i'25;«' + 81—0; the roots of which will be found to be 1, 4, 13-5 and 1*5; hence the five roots of the propofed equation are, i, i'^, 3, 4, and I3'5, Example 2. Given .v' + 15A:* + 79^^ + 189;^^ + 2o8x' + 84 r= O, required the values of x. Here ni^ zz 2*424, &c. which as there are no changes in the figns mufl: be negative, that is m^ zz — 2'424. But in order to fhorten the operation, either reje(ffc the decimals, or encieafe them to unity; fuppofe the latter, then ni^ =; — 3, and by proceeding as before we , — 972 + 3645 — 4 266+ 1701—8 4 _ +_24. _ 405 — 1620+2133 — 1134+208 ~~ — 8 ~ which, as it terminat-e?, is one root; and therefore by di- viding the given equation by x + 3, we get the biquadratic x'^ -{■ i2x^ 4 43*^ -}- 60.V + 28 =: o, whofe roots are — 1, — 2, — 2, — 7, and thence — i, — 2, — 2, — 3, and — 7, are the roots of the propofed equation. Note. If in fubftituling the value of m'^ in the formula, the whole expreffion fhould vanilh, or become equal d, there will be two or more roots of the fame value in the equation. As in this lafl: Example by rejecting the decimals we have nis zz — 2, from vvhence the formula becomes — 128 4- 720 — 1264 + TS^ — 84 o , . , * J ___I_J — l2 1 — - J hence there arc 208 + 80 — 480 4- 948 — 7.S^ o two values of ,v, viz. — 2, equal to each othei*. The reafoa of which will be evident to thofe v/ho are acquainted with the confirudiion of the theorem, J Exarrple 124 O F E Q_U A T I O N S. Example 3. Given x^ — jx^+zox^ — 155'^"'' + lOOOO — o. Required ,v. Here a = — 7, Z* = + 20, c= — 155, d — Oj — m — — loooo, and, by rejecEling the decimalsj zn'^ — 6, which may be af- fumed either affirmative or negative, if we take the latter, then a few operations, by fubftituting the quotient for m'^y give X zz 4'544i9572, which is true to the 7th or 8th place. Example 4. Let the radius of a circle be i, what is the length of the chord of 12 degrees? The final equation will be x^ — 5.v^ -\- S^' — ^ =0. Here a = o, b — — ^^ c — o, d — -\- ^^ m — -{- i, and m^ — 1, and the rormula sjives =: = + i> 5+5—15 —5 which is one root of the equation, but manifeftly cannot be that anfwering the cafe propofed ; agreeable to what was ob- ferved in Rem. 2. Therefore alTuming m'^ zz '2, the firft operation gives vi^ zz '209, and the fecond w? — '209056926, for the required value of x ; or the length of the chord of 12 degrees, when the radius is unity, which is true to the laft figure. 13. As it may be of f me ufe to the young Algcbraifl-, I fliall now endeavour to illullrate, by a few Examples, the Newtonian method of obtaining the roots of literal equations. Example i. Given, y^-\-axy — a''y — x^ z= o. Requiied the value of _y, in a feries c^mpofed of the powers of n and x, with their coefficients. The O F E Q,U A T I O N S. 125 The firfl: thing to be done is to tabulate the equation, which is eafily effefted as follows : Make a right angle DAB (fig. 10.), and from A towards B write down all the powers of ^, and from A towards D all the powers of x, as high as in the given equation, beginning from unity. Divide the whole into fmall fquares or parallelograms, as in the fcheme, and infert the terms of the equation in their correfponding fquares or parallelograms, that is, in thofe which have the fame powers of X and y ; then circumfcribe the fignificant parallelograms with the polygon FBCDE, and the equation is tabulated or prepared for extraiSlion. Now, in order to determine the firft term of the feries, make any two terms which are placed in two adjacent angles of the polygon equal to Oj that is, y" — ^> = o, — a^y — x^ — O) y — *' = o, The two firft equations give the fiift terms of a feries for y when X is fuppofed very little in refpeft to «; and the laft, when a is little in refpeft to x. We fliall take the firft, viz* from whence y — dt. a', which is the iirft term of the feries. To obtain the next term, put y ~ a -\- p, and fubftitutc this value in the given equation, and we fhall have, J^ f - a' + Sa^p + 3«/ + />%• + axy z:- a^x -f- axpy — O — ay zz — a^ — ay, Selea :26 O F E Q,U A T I O N S. Sele£l two terms of the greateft value; or, which is the fame thing, take two terms where p and x are feparately of the loweft dimenfions, and make them equal '^xq + OA^ = — 2axq =: + 2axr,\ 4 * 3 _ 9JC" _ 9;<'" J Tx f x^ 0a''\ The terms to be compared are — -^ ( r: -^ 2— J and + 2«V; that is, 2flV from whence r -=1 A r~~. the fourth term of the feries. lOfl Laftly, let r = + -y-i + fj this, fubftituted in the laft •quation, ^ves — %axr 128 O F E Q.U A T I O N S* 7^ V — %axr — 5—, occ. — 7^' 8 = — 'Jx^ 8 ' - 3** 64^ zz — 3-' 64a' Here all the terms vanifh but aa^s. and — ^t^-* 64a ( — -^ — ^1 — J; which, being therefore conipared as before, viz, we have j = + -^^7—:, the fifth term of the feriesj I2o<2'' X X" *! X^ CQ^ toy = a _^_L-+ _^, &c. 2 oa iba i28fl It will eafily be perceived, by a proper attention to the method of operation, that every new term in the feries is had by dividing the lowefl of the terms affected with the indefinitely fmall quantity a-, or its powers, without the aflumed ones, p, q, r, &c. by the quantity with which the affumed one of one dimenfion only is multiplied ; thus, the fecond term is had by dividing a^x by 20^; the .third by dividing — by 2«% ^c. .4 From which it Is evident, that when the work is continued fo far, as that the aflumed quantity (p, y, r, &c.) is only of one dimenfion, the remaining terms of the root may be had by x" one divifionj fo, when — r- + j is fubftituted for ^, it oa appears O F E Q^U A T I O N S. 129 appears that r is only of one dimenfion, therefore the remain- ing two terms will be had thus. 2t?' 8 64 "8^ + 59^ 64^ . 04<2 Given Example 2. — -T y -A- a — .V — o. Re- 5 4 quired y in terms of « and a*, which arc here fuppofcd to be nearly equal. In order then to make the feries properly converge, we mufl fubrtitute for their difFerence, that is, put .v — « -}- %, and the given equation will become * 54 3 2 ■\- y — % — o. This being tabulated (as in fig. 11.) we have y — iz, for the firft term of the feries. Put y zz % -\- p, and write this value in the above equation, and there arifes, + |/=r ^ 1 — 1/ r: - l^\ &c. + 1/ r= + J2^ + 2> - 1/ = - Iz^-z;.- if + y = + z+p =: O. The 130 O F E Q.U A T I O N S. The terms to be equated are p, and — ^%% from which the fecond term of the feries is -{- |z^. Put p zz 4z'' + ^, and fubftitute this in the laft equation for /», and it becomes, - i/ = \ &c. ^ — zp — — |z^ — -? I + p =r + |2^ + J J> - o. + 1^ 1^4- _ I~+ 1^3 _J_ 1^3 + i%^ 2' 1^^ Here the terms to be compared are q and — -r-A — 1 — ) '■ b \ 23' but as q is only of one dimenfion, the remaining terms may be found by divifion, thus ; %' %■" + T 6" z* + 24 + ^' &c. Therefore the root is ;» n % + |z* + |s" -|- tV-S 5cc.; or, by reftoring the value of x, X — a]^ X — a\ y ~ X — a -\ -_ 1- -, &c. 2 3.2 3.4.2 where the law is manifell, and may be continued at pleafure, f/z. + ^' X 4" ■^' 3.4.5.2 3-4-S-^-2 , 6cc. It O F E Q^U A T I O N S. 131 It appears from the table that other fcries may be found from the equations, if - %, and 4/ -^ ly 4- 1/ _ i^ - _ I. Remark. Sir Ifaac, or fome of his tranfcribers, feem to have committed an overlight in fuppofing we might put either « -f* z, or a — z for a-, for it is indifpenfably neceflary that x fliould bs greater than :Y . y —a — X -\- + + , &C. 2 3.2 3.4.2 Example 3. Given y^ -^ y^ ■\- y — x^ ■=. O. Required )' in terms of X, Here it is evident that x muft be very great, and there- fore the common method cannot exhibit a true value of y. But ^ I to make the feries converge, put - z: z, and thence will A- z= -. Subftitute this for x in the given equation, and there arifes y^ -\- y'^ •\- y — — r: o. Tabulate this equation ffig. 12.) an'J tlie terms to be compared will be >' and — , I 2 from 132 O F E Q^U A T I O N S. from whence y =r -, the firft tern^ of the feries, and pro- z ceeding as before, the root is found = - — ^ ~ - 2: + ^ 2^ + ^ z% &c. ■339 81 81 ' 12 7 Rcftoring: x it becomes y zz x h -x-—:-, Sec. The other equations are y — —, and ^* -f j' rr — i. The former gives y zr — « + -r, &c. which, as it afccnds in the powers of % in the denominators, muft alfo afcend in the powers of x in the numerators ; and therefore will evi- dently diverge j and the latter gives y zz y — , im- polTible. Example 4. Given V + I j^ -^ -^ / + — / + -^ / + -Pr/'> ^'C- 6 40 112 1152 2bi6 — X :z: o. Required *• in terms of ^. Here y, and confequently ,v, muft be fuppofed lefs than unity, in order that the feries may duly converge. Having tabulated the equation (fig. 13 ) we have y zz x, the firit term of the feries. Put y zz x + pj and fubftitute this value of ^ in as many terms of the given equation 2s the feries is propofed to be carried to; fuppofe to four terms, and the operaiion will ftand as follows, 2816 -^ I.,Z^ 4- + -^ / - 4. _2. ^7 2cc. Hi •' I li + ^ v' = + -- x' + ^ x% cS:c. 40 -^ 40 h ^ -^r "-y' = +J A^ + '-xy -h i.v/-, 5cc. -h y 1:= H-" -v + p A- Hence '^Va,. 10 9' ^^ "^-^ ajcy "^~^-^ 2 + / Y y / c 6 ^B 0%'^.. // y + /' ^ ^/2 ^/■^ /' ^ -f i i/^ 4_ J_ 7y<^ r '/ • / f y^ 8Co. z - J ~^^--~.^ z* \ "~^--^^ z \ ^^--^^^ / \ + '/ + '/' + f 1 y. r — ,r 1 + '/ + 7/ +i»}^ / V r /•' /* r ^ O F E Q.U A T I O N S. ^33 Hence / = — i at'. Put p = -^ ~ x' + q, and fub- ftitute this in the laft equation, and it becomes. 2 ^ 12 2 ' + — A-' 112 40 + i v' *% &c. 252 — T^^' + I^V 120 • i 5^ + ? 5. = o. Here y is of one dimenilon only, therefore. ^ 2>'^ 120 ^252 V^ 120 C040 ' ^^* + — ^^ 120 + 240 5040 5040 Hence the root is y = at — -7 x^ -\ x^ — x"^ &c 6 120 codo ' or, y-x -■-- + ■ + a:9 5040 6 4.5.6 4.5.6.7.6 4.5.6.7.S.9.6 4.5.6.7.8.9.10.11.6* &c. the law of continuation being evident. u This 134 O F E CL^' A T I O N S. This lafl example is ufually called the reverfion of a feries. There are fcveral other methods of performing it; but this, I think, is as eafy to be unclcrflood as any of them. The youncT fludent may here fee with what facility thefe in- tricate affairs are managed when freed from all that unneceffary prolixity we find them embarrafied w'ith in moll authors. What before feemed almofl: jnfuperable to him may be now only a pleafant and agreeable exercife, as he cannot fail of undcrftanding the method of operation, if he but attentively e:camine thefe examples. Indeed, there may be a great many literal equations propofed which may n ;t coincide with any of the examples we have given. But when the equation is tabu- lated, and two or three of the firft terms of the feries obtained, it will immediately appear whether it properly converges or no. If all the equations diverge, or have impoflible root?, a little artifice mud be ufed, fuch as augmenting or diminifliing the roos by fome known quantity; or by taking the recipi'ocal of an indefinitely large quantity, and fo on. So that in almoft all cafes the feries may be made to converge, and the root ob- tained by the foregoing method. Some iifeful Remarks en the nature and method of Fluxions, As I have always found the following remarks of fervice to the learner, I lliall make no apology for infertin^^ them in this tre.itile, which, 1 have before obfervcd, is purely defigned as an help to tlie young reader, in removing fome of thofe little ohilaLles which he mnft unavoidably meet with at his firft cnirdHce on thefe fludies. Tiie doflrine of ptirne and ultimate ratios^ by which the iliiNions of quamiiips are gi-nerally invefligated, or demon- i:j..ittdj cbnta'.ns in it fomcthing fo very obfcure and unin- /; telligible OF FLUXIONS. 135 iclliglble to the learner, that it is rather more apt to confufe than give a proper arrangement to his ideas on the fubjefl *. The * Tlie fiift Lemma of Sir Ifa.ic Newton's Piincipia appears to many to be very exceptional)le ; liis wonls are, — ^antitates, lit et qitantitatUT/i ratione!, qua ad aqualitatem tempore quovis Jinito conjianter tendunt, et ante finem temporis iliuis propius ad iniicem accedunt qtcam pro data quauis I'dfferentia, Jiiint ultimo aqnaleu And then adds. Si negus; fiant ultimo inaquales, et fit earum ultima differentia D. Ergo nequeunt propitis aft aquaVttcitem accedere quam pro data differentia D : contra hypothejin. Now, though this method of dcmonilnition is far from being faiisfaftory, beinj^ a kind of rcafoning tnutatis mutandis ; vet, it is plain that the pohtion may be rcndily admitted if tlie decrements only, or parts dejlrcyed, are to be proportional. For inftance, fuppofc two lines, the one 20 inches, the other 12, to be diminiflied by fome cotemporary decrements, of which the ratio is rtlpeftivcly as 3 to i. Here then it is evident, that the quantities 20 and 12, would foon become equal by fuch a diminution. Thus, in the firft portion of time, lei tlie former line be reduced to 14; then the latter will l)ecome 10. In the lecond portion of time, the former becomes 8, and alio the latter 8. They have therefore converged to equality by a cotemporary diminution with proportional decrements. But when the c^uantitics thcmiclves are fuppolcd to be proportionably diminiflied, and thereby to obtain the ratio of equalltv, 1 think it will apjicar that there is nothing more ablurd. For fuppofe the quantities 20 and 12 to be anv-how proportionaljly diniinilhcd, iuppolc by a continual bifcftion, and it is evident, that they will converge to equality in rcfpecl of their difierence, and yet retain their oiiginal proportion. Thus, the firlt difference of the propofed quantities 20 and 12, is 8 inches, the firll bifc£tion reduces that difference to 4 inches, and the quantities thcmfelves to 10 and 6 ; but 10 is to 6, as 20 is to 12 ; therefore the proportion is not altered. The lecond bik£lion makes the diirereuce only 2 inches j but ftill the quantities are in the fome ratio, for 5 is to 3, as ao to 12. The third biieclion reduces the difFcrence to 1 inch, and ttie quantities themfclve; to 2 I and I \, which are flill in tlie fame proportion. Hence then it plainly a[)pears, that two quantities may converge to equality in relpect ot their difference, and that that difference may become lefs than any aflignable quantitv, yet the quantities themfcl\ es can never become cqv.al ; tor )ri whatever ratio tliey were ori-ginally, in the fri'-ie r:.'io v/iil thev remain, it I 4 .limi'.idied -136 OF FLUXIONS. The moft natural and eafy way of acquiring a right notion of fluxions, is by the introducing of time into the account. For by this means we do not confider them as mere velocities, which diminiihed Jine fins, according to the mathcniatical doSrinc of the infinity divifibility of matter. How abfurJ then muft it ?.ppear, to attempt to find the laft ratio of the cotemporary increments of two flowing quantities by a continual dimi- nution of thofe increments, fince it is obvious they will always remain in the fame ratio, however fnisU they are taken. And yet after this manner have Colfon, Ditton, Hayes, and feveral other writers on fluxions, pre- tended to find the laft ratio of the vanifhing increments. Thus if the increment of x be denoted by x, tlie cotemporary increment of x" will be cxprelled by nx^ ^v -j x'^ ^x^ -\ f — : x"~ixi, &c. 2 2.3 Here then, fay they, by continually diminifliing thefe increments, we ap- proach nearer to the ratio with which they firft arife or begin to be gene- rated ; and therefore, when x is become infinitely fmall, the higher powers of it muft vanifh fidl, and leave the infinitely fmall increments in the ratio of X to nx"~"'x; which is therefore their prime or ultimate ratio. But does this appear in the leaft fatisfaftory ? Is it not rather a mere quibble ? For if the increment of x have a real value, though ever fo fmall, it is obvious, that the increment of x" cannot be accurately nx''~^x. And / / ■ ■• ' -' ■ if .v', ,v^ &:c. be abfolu;:ely nothing, certainly the root itfclf muft be fiOt/ji>tg alfo ; and confequently the whok exprefiion muft yanifli together. Befidt-c, this is plainly contradictory to the lemma; for, if that be admitted, the lafl ratio muft be that of equality and not of i to /;x"~*, fince the quantities arc proporticnably diminished in tlie fame time. Some other writer:, wlio, I imagine, faw the fallacy of this method of rcai'oning, have endeavoured to obviate thefe difficulties, by reprcfenting the affiur in a di.ferent light, as follov/s : If we confider that the increment of x" ic much different from the fluxion of it, the foriper being defcribcd by an accelerated motion, and the latter by an uniform one, ic will not be fo hard to conceive, that by continually diminifhing the increment of the fj;r!ple quantity .r, the increment of the compound quantity *•" will come 4 nearer OF FLUXIONS. 137 which naturally involve the mind in metaphyfical difficulties; but as the magnitudes they uniformly generate in a given finite time, fuppoling the fiuent or fpace to be defcribed by an uniform motion. And if the motion by which any magnitude is generated be not uniform, but accelerated or retarded, the idea of a fluxion will flill be the fame : For though we cannot exprefs the fluxion by any fpace usually generated in a given time, as in uniform motion ; yet we can readily aflign the magnitude, or (as it is commonly called) the cotemporary increment, that would be uniformly generated, if the accele- ration or retardation were to flop at any point in which the fluxion is required to be inveftigated. Now, as our ideas of magnitude arife from a comparifon of the propofed obje£l with fome other of determinate dimeniions : fo, in the method of fluxions, we fix on a given magnitude, which is fuppofed to have been uniformly generated in a given time by the motion of a point, line, or plane, as a fiandard, nearer to an uniform velocity ; and therefore, juft in the inftant of vanifii- ing, or when it becomes nothing, the velocity muft be uniform, and truly exprefs the fluxion at that point. Hence then it appears, that it is by con- founding the increment and fluxion, that thefe feeming abfurdities arife. For if we imagine the fluxion of the uniformly generated quantity to be an infinitely fmall, yet certain magnitude, it will readily appear, by keeping our ideas of the fluxion and increment diftinfi:, how the ratio of the fluxions may be had, when the increments themfelves vanifh, or become nothing. Thus, let x denote the fluxion of the fimple quantity x, or that infinitely fmall quantity which would be uniformly defcribed by the gene- rating point in one inftant of time ; then, fince the ratio of the increments, / / fi^ fj / _ «* — n __ ' X ■■ nx" » X A jf"— ^ .v^, &c. is the fame as i : nx" ' -{ x" ^x, a a &c. or, by multiplying by a-, as x ■ nx^ '-v -}- x^^^xx, &:c. it will be evident, that if we fuppofe the increment of x to vanifh, the ratio will tiien become accurately, x : nx^" x. But this is too metaphyfical for mofi readers. wherewith I38 O F F L U X I O N S. wherewith to compare any other magnitude, which is fuppofecl to have been generated in the fame time, by an accelerated or retarded motion. Tlius (for the (ake of illuftration) fuppofe a ball to roll on an horizontal plane, in a ftraiglit dire6lion, at the uniform rate of 20 feet in a minute ; and alfo another ball to move uniformly in the fame direflion, at the rate of 40 feet in the fame timej here then it will be plain, that the magnitudes generated in any given time mufl be in the ratio of 2 to I ; and therefore the fluxion of the latter will be doubfe that of the former. And from hence it appears, that if the fluxion of x be .v, that of 2x will be 2a-, 3^ will be 3a-, &c. and generally, that of fix will be 77a-. But if, while one ball moves along with an uniform velocity, the other is fuppofed to move with an accelerated motion, and that the law of acceleration is fucli, that the fpace dcfcribed by the latter, from the commencement of motion, is always fomc power of that defcribed by the former, fuppofe the fquaie of it; then, if the magnitude by which the fpace that is uni- formly defcribed is increafcd in a given time be denoted by x, that magnitude which the accelerated motion would 'uniformly generate in the fame time, and commencing from the fame jnllant, will be exprefled by 2xx, Thus, in the cafe propofcd, if the firft bail has uniformly defcribed a fpace of 10 poles, the other muft have run 100 poles ; but the former ball moves uniformly at the rate of 20 feet in a minute, therefore the magnitude or fpace, which the accelerated ball would uniformly defcribe from the fame inflant in one minute, will be 400 feet. The fluxions will be therefore at that point in the ratio of 400 to 20; or of 20 to I. This is eafily demonftrated as fallows : Let x — /, and x, rcprcfent the fnaces defcribed by the uniform motion at any two given time;, then (by Hy- pothchs) will the cotemporary ipaces, palled over by the ac- celerated motion, he (x — /'^ =r} ;<•' — ixt -{- /", and a'; and '7 ilicrefove OF FLUXIONS. '39 tlierefore .v" — x^ — 2xt + f — ixt — t^ will be the differ- ence of thole fpaces. From whence it follows, that, while the firft ball runs uniformly over the fpace /, the other runs over the fpace 2xt — /\ Now, as this fpace is not generated by an uniform but accelerated motion, it cannot reprefent the fluxion of either of the fpaces exprcfl'ed by x-^^^ or x^ ; but it may evidently reprefent the magnitude which might be uniformly defcribed with the mean velocity at fome point between X — /|^ and x^. But this magnitude is to that generated in the fame time by the uniform motion as o.xt — t^ is to /j or, by ip.ultiplying by - as 2Xx — tx to x', therefore, when x — /by the uniform motion of the ball becomes x, t vaniflies, or is equal to nothing, and confequently the point of mean velocity then coincides with x ; and hence the above ratio becomes barely, as q.xx : x. Therefore, according to the commoi^ phrafe, the fluxi-.n of x~ is 2xx. This may be generally reprefented thus : Let the law of ac- celeration (or retardation, prefixing the negative fign) be univerfally expreffed by .v", and by the fame method of rea- foning, by help of the binomial theorem, we {hall find the cotemporary fluxions to be in the ratio of nx"~^x to x. For if A- — / and x reprefent the fpaces uniformly defcribed as before, the cotemporary fpaces defcribed by the accelerated motion will be expreffed by *• — 1\" and x" -y but the difference of thefe fpaces x" — Z — /]" is equal nx"'^t — < . *""V, &c. therefore the ratio of the magnitudes, generated in the fame time by the uniform motion and the mean velocity, wilt Z be expreffed by nx"~'t — .*■""''/", &:c. to t; or, by muhiplying by -, as wx" " '^ . x^'^i.if &c. to x i r 2 and H^ OF FLUXIONS. and when x—t is equal x, t will evidently be zz o, hence all the terms wherein / is found muft vanifh, and the point of mean velocity coincide with x; confequently the ratio of the fluxions will be as nx^'^x to x. Hence then it appears, that we have the mofl rational notion of fluxions from the confideration of time in the generation of the increment or decrement, and that the fluxion of any variable quantity may be truly defined, Th magnitude by which any flowing quantity would be increafed in a given time with the gene- rating Telocity at a given irjlant, fippofing it from thence to proceed uniformly sr invariably. And with regard to the higher orders of iluxions, how much more obfcure are our notions without the idea of time in the operation of the fluent generating the increment; lince by having recourfe to the firft ratio of the nafcent increment, or the laft ratio of the evanefcent incre- ment, even to obtain only the firft fluxion of a variable quan- tity, we unavoidably fall into this abfurdity. That a velocity which csntinucs for no time at all a5lually defcribes a fpace. How then can we form any conception not only of fuch a fpace or increment, but alfo of an infinite variety of magnitudes of it, generated in one and the fame point and inftant of time, in which it h well known ail the orders of fluxions are con- fidercd, when nothing, I think, can be more evident than jfhat che magnitude or increment imagined to be generated rauft in fuch a cafe be purwn putum nihil^ or fl:ri£lly and ab-^ folutely nothing. If a doubt of the exiftence of an increment under fuch circumftances be deemed incredulity and a fpecies of infidelity*, I am afraid I fhall be fligmatizcd with thofe appellations; for I confefs it is paft my comprehenfion bow a mere point can contain in itfelf gn infinite variety of magnitudes, and which are all at tlie fame time * Sec Colfon's Newton's Flux. p. i8, Preface. equal O F F L U X I O N S. ,4, equal to one another. Thefc unnecefTary quibbles, and meta- phyfical niceties, by which feme have attempted to explain the principles of fluxions, have not only rendered them quite obfcure to the learner, but alfo expofed them to the ridi- cule and feverc criticifms of feveral writers of great abilities in the mathematics. But thefe criticifms, it is probable, were not intended to invalidate the method of fluxions (which it is evident may be ftriftly mathematically demonftrated) but to fhew the futility of the method they had taken to elucidate the principles; in which light It is well known the incomparable inventor never intended they Ihould be viewed. From what has been faid we may draw thefc pra£lical ob- fervations. I, That the common rule for finding the fluxion of a flow- ing quantity, viz. Multiply the jiuxion of the root by the exponent of the power and the affixed coefficient, and the produ5i by that power of the fame root of which the exponent is lefs by unity than the given exponent^ is general, and without exceptions, being applicable to any expreflion whatever confifting of one variable quantity with a conllant exponent. 2. If the expreflion be a compound one, that is a binomial, trinomial, or any multi- nomial, the fluxion of each term muft be found feparately, and connedted with their refpe£Vive refuliing ligns; the fum arifing by fuch addition is the fluxion of th;; compound expreflion. 3. If the expreflion confifts of the produft of tv/o or more variable quantities, each quantity muft flow leparately, while the others are fuppofed to be conilant, or as coefficients to that variable quantity ; the fum of thefe fluxions will be that of the given expreflion. This follows from the general ex- preflion n/'-'x. Thus, let the fluxion oi yz be propofed to beiinVefligated. Put y-^z — Vy then will j^ ■^iyz\--z^ — v'; hence 142 O F F L U X I O N S. hence yz rr — iv^— \y' — |z*. And, from what has been before fhewn, the fluxion of this will be 'ui — y) — zi ; but V IT j» + z, and i zn y-\-z^ •.* by fubftitution, the fluxion of ^z is yz -\- "zj. And in the fame manner will the fluxion of xyz be found to be xyi. -f xzj -{■ yzi; and that of uxyz zz uxyz + ux%y + uyzx + xyzii ; &c. We (hall now give a few examples in order to fhew the propriety of thefe remarks. Let the fluxion of ;<• be a- ; that is, let the magnitude or fpace which is pafTed over in a given time by the uniform motion of a point in generating the fpace x be denoted hy x; then will the fluxions, or relative magnitudes, which would be uniformly generated in the fame time, of the cotemporary fluents, or fpaces already pafled over when the fluxions are compared, be obtained by the general rule as follows. Varlahle OF FLUXIONS. 143 1 1 *, 1^1 « + c! H\H ^ ii •>^ 11^ ■«,l (Si X SI « X • • •>< 10 11 ?= CN » CO 1 1 n 11 11 II 1 ■^'i •K CO •K ■H •», 1 ;. 1 II »-t T X J.J it 1 1 'h • ** Ik. c« « ;H »0 •H SI « 51 a X X J X % X X « -- CO r^ 1 -»* 1 is- II c It? H I H il ^ ^ .x .^-^ J« H *y J? d « ""it w^ , . , , ►-1 « CO '^ ■«.l »!- ■^ rv =s " w 1 1^ <• 1 k . 144 OF FLUXIONS. X "^ K^ ^ 3 •?^ n M I I I X + + «3 I H *i II II - H H4^ ^ V «3 > »% + ^ ^ ^ •'^ s «IS .s 3 "1=; 11 -5 ^ .-I '5 ^ S *» 3 *■* ft I "^ «IS = S '3 o E .5 •6 ^ Sic 13 O* OF FLUXIONS. H5 K ^'^ /•^ 1 ? N s V V 8 s + -{ S; •N «=''€ 1 s ^s I- fe K i + » I ■»« I + •H 1 S H K n S i » + 1 R R is + •t: 4 1! -^ il s + >> ?0 ^ v» • SO » 9 r^- » K 146 OF FLUXIONS. {^ ,J3 a c c o a. fcO »: ^ ctt CO o .2 S o c ». o c ^5 •^ n + X •^ + + ' I ^ + X B + CIO o o a o ">< 3 si o ^ ^ 6 bO O o a o bO O O ^ — ' u CO ^ 3 < o c .2 3 3 •J^ 0. < < • U & U- J J < • ^H (U y_, > n J3 a (U ja > !> < -a •"? •i -3 £3 h4 OF FLUXIONS, H7 04 o to 3 • g|S a. ^ ex, -M II 3 to 13 C rt • « I 1 V ^!^ >« H f" c *i rS E c 1 H K. ^ 4) 4-« N O B rt i-i u _L. =5 + .2 j3 O 1 ^ -4^ ^ « is h 'ii ^M rs • •J .2 V 3 N •'^ •XJ 1 1 U J 1 ^ > t> . •H *e ? c4 N X a CO + t> u N c .'5 ?s o C 1 + ,3 (4 3 H 1 *H cr • 4> u J 1 •£ 4-> •^J 0\ M J JU ^ 1 •P 1 Ji N N *C u 5> • Rt J2 1^ » "£ ^ ^« o T3 I-I h4 + '3 cr u II 1 CO 5^ 1 JC ^ 1 J ;> • •H *§ « J 1 C4 r £3 H • *. Vm a ^ ct O 3 C! II 'ii- o 5^ 1 ■4.i 3 1) 1 SM « rr >e <-i OJ ^ ♦* Kz In r48 O F F L U X I O N S. In the inverfe method of fluxions^ nothing can be more eafy than to proceed in a retrograde order from the fluxion to the fluent (in expveffi jns that will admit of it) by the converfe of the general rule, namely, Strike out the fiuxionary letter^ add one to the index, and divide by the index fo increajed. Example. Required the fluent of nx^'^ x. The procefs will ftand nx^ thus; nx'^'^x, nx''^ nx"''*', nx", — , *•", the fluent. In expreflions afFe£led with a radical, if the fluxion without the radical, or vinculum, be the fluxion of the quantity under it, the fluent may be obtained by the general rule, obferving to ftrike out, not the fluxionary letter only, but all that part which appears to be the fluxion of the quantity under the radical. Example. Required the fluent of v/a'" + x" X mx^'^x. The pro- cefs will be as follows ; « -f 1 the fluent. If the fluxion without the radical be not that of the quantity unJer it, but in a given ratio to it, the fluent may ftill be had by the general rule ; obferving to increafe or decreafe it in that given ratio. Example. Required the fluent of a + hx""]" x cx"'''x. Th« O F F L U X I O N S. 149 The fluxion of hx^ is bmx"~'i; therefore the ratio of the fluxion of the quantity under the radical is to that without it, as Im : c. Hence the procefs will be thus ; « + I then, as bm : c :: : ^ the fluent. " + ^ bm X n + I If the fluxion without the radical be neither the fluxion of the quantity contained under it, nor in any given ratio to it, yet, in many forms of expreflions, the fluents may ftill be had in finite terms, by help of the binomial theorem, which will always terminate, when a number equal to the exponent of the flowing quantity without the radical comes to be fubtrafted from it. Example. llequired the fluent of x^x ^a + .v. Here the ratio of the fluxion of the quantity under the radical is to that without it, as i : x", and confequenrly not given; therefore the eafieft way is to proceed by fubftiturion, as follows; put a -{■ x z=.Vf then will x rr v — a, x" z=.v — rf]", and X — -if, .♦. x"x^a + ■*■ — 'i^'"'i> X- 'u — cY' But by the binomial theorem, v—a]" ~ v" — nv^ ^a -\ » v" ^a^^ Sic, 2 So that when « is a whole number, it is evident the ferles will terminate, and the fluent be had in finite terms. Suppofe « = 2, and fn = 3, then the fluxion will become t 1 I y^-3- 6^z -yl of Vi becomes — . a + x\ 4- — ,a + x] . c + a ' lO ' ' 4 7 ' thp required fluent. And when the Piwing part of the quantity under the radical is a power, it always follows by iubftitution, that if the ex- ponent of the flowing quantity without the radical be an even number, the fluent may be had in finite terms ; but if an odd number, it will run into an infinite feries, becaufe the index of the binomial then becomes a fradtion. Example. Required the fluent of x^x y/cC^x\ Put a^AT* - -y, then will x = ^v—a^, x" = v — a^^, and;^=— — — Z - ; .'. x^x ^a^ -\-x'^ — ~ ^ ^ ^^ r _ 2x 1NV — a' IfJv- — a^ - . From whence it appears, that if n be an odd number, the index will be an integer, and the feries will terminate; but if it be an even a mber, the index will be a fradlion, and the feries will run Qx\ fine fine. Suppofe « rz 5, and m = 2, then the expreflion becomes -v'^v x v — a^l ' ; 2 which being expanded, the fluent of each term taken by thq common rule, and the value of v rellored, gives - . a- + x'V + - . a"- + x^Y a* . (2- -f A■^^ for the re* 7 3 5 quired fluent. O F F L U X I O N S. ,51 If the flowing part of the quantity under the radical be raifed to fome power, and the fluxion without it be only that of the root, the fluent muft be obtained by infinite feric-s. This, indeed, may be often avoided ; for if the given fluxion can be reduced to a form fimilar to that of the arch of a circle, then the length of that arch, which is always exprefl'ed in terms of either the fine, verfed fine, tangent, or fecant, and the radius, will be the required fluent. Example. Required the fluent of >s/4<:'' — 4/' _, „ C'v . CC'y c'y The expreflTion . ' ~ is — ■■■ ,r--^ ^ = j^cx—p===' W^c^ — 4_y" 2va- — yy \ cc — yy Cy but -7===. appears to be the fluxion of the arch of a circle, vrc — yy of which the radius is c, and fine y; and therefore when the degrees in the arch are known, the length of the arch is known rtlfp, which being multiplied hy \c gives the fluent qf *>J^c'-—^y^* And when it happens that we cannot reduce the fluxion to any of thofe forms, yet, if we can by any reduction difcover, that the numerator of the exprefTion is the fluxion cf the de- nominator (which frequently occurs in the folutions of proi- blems) then will the hyperbolic logarithm of the denominator be the fluent required. Example. Required the fluent of -^ 1 • K 4 Tlie i$i O r F L U X I O N S, rt-i /»» L X , . , 1CX . '2.CX The expreflion ^ ; \%-\c x -^ ;; and is tz • ' '• c —X 2CX lex X — — ; but this expreflion is the c-\-x X c — X c — xy c-\^x _ . 1CX c-\-x icic X c— ^\"^ lame as ^ -. , or — — , where the nu- c — ^ c — X c-^xxc — *j * merator Is evidently the fluxion of the denonynator j therefore C ~i~ X the hyp. log. of being multiplied by |c* will be the required fluent. But as it would be rather foreign to our prefent purpofe, and fwell the book beyond its intended fize, to endeavour to explain all the various methods of finding fluents by a proper number of examples, we fhall therefore only give one or two more of reducing furd fluxionary expreflions by infinite feries, and Emerfon's tables; which will ferve as a fpecimen of the method of operation with all other fimilar forms. Suppofe we feleft one from Emerfon's Trigonometry p, 27, fecond Edit* ry where he gives for the fluxion of the arch of % \rr — yy circle. Firft Method* ry _ V;-^ — y'^ X Ty *J rr — yy r^ — y^ ftand as follows. , therefore the redu£^ion will r*— / O F F L U X I O N S* ,53 r 8rV 2r 4r 2r r ^ ; 8r* And the root multiplied by ry gives ^. __ /y __ // _ _/y_ •^ 2 8r' i6r* " this divided by r* — ;''' will be in this form, viz. •^ 2 8r^ i6r* laSr*"' ^-f* '54 QF FLUXIONS. 30 (S c5 + CO o i^'M ;;^ (3 o ^ ^ 1 1 * OF FLUXIONS, «55 and the fluent is ^ 3.2r^ ^ S.2.4r* ^ 7.2.4.2r^ ^ 9.2.4.2 8 r*** * pr, V + -ZL ^. M- + 3-5/ . _3:5i7/__ - Second Method, ^ ^i X -^ \ ; t^at is, Multiply each term of the firft feries into all the terms of the latter, reje£ling thofe which exceed the power you intend .carrying it to, and it will ftand thus, . , /i , y^'y , fy , fy . J + -^ + -;i + -^ + ^,&c. ^ fy __ y^ __ fy __ /l ^^ ' 2r^ 2r-^ 2r^ 2^** * 4. 6' 8> JVJ' yjc >^ o /i _ jv'i - I28r^' - 2r" ^ 2.4r" 2.4.2r^ 2.4.2.8/'' from whence the fluent is had as before. Third J56 OF rLlJXiONS. Third Method. ry _ . =r r X r'-—fy^ X i, By Form i6, in Emtrfon's excellent Treatife of Fluxions, we have c r: r% ^ = — i, 2 =: j', *7 =r 2, - =r — |; (or f* = — I, y =: 2,) and w rr oj then by fubflitution TT-}- I + I 1 - r' A=r-'y, J=-iI, v y ^ r r _ / ff + 1 4-1 0+ i + 2 3 2^" , B = + ^,, .-. 2r f*— F —1 — 2 / / 2* 4 2r^ ^' _ .^v' -rv+l + l" 0+1+4 ~ 5-2.4r* C _ + - — li . . 2.4r" -2v —1—4 3/ ^ / 3;. ^ _ 2.4r^ r^ _ 3?/ ___ &c ^+i + 3» 0+1 + 6 7.2.4.br'' Therefore the formula gives ^ + jL + ^ + ^li^, &c. r 3.2r' 5-2. 4r* 7.2.4.5r' y • • • for the fluent of — — ; which being multiplied by r^ s/r'—y' •p 2y' we have y 4- + — —^^ , &c. as before, 3.2/-^ 5.2 4? + It O F F L U X I O N S. 157 It alfo appears, from the conftruftion of the fluxion, that the fluent will be exprefled by the arch of a circle, of which the rad. is unity, and fine -; which may be very conveniently had from the Tables of Nat. Sines, &c. thus ; Find the degrees, &c. correfponding to the given value of -, which call d'y then it will be, as 180 : 3*14159 :: d : Z, 180 the required fluent. This one example indeed might be fuflicient to fliew the method of procedure with all fuch expreflions; but in order to make it as plain as pofTible, we fhall take another example, which is more complex, from p. 30 of the fame book, viz^ ; being the fluxion of the arch of a circle, ex- <^irv — vv preflTcd in t^rms of the radius and verfed fine. ^"^ \/ irv — vv i^2rv — %v zrv—^vv ,58 GF FLUXIONS. CI ^ ^ 1> I Ei N Ei ^ ti I u ^ "+ « 1 ^ r^ ^ )^ 1 d "j. i"' ^ w ci ■a CM I I s OF FLUXIONS. ta^ to 'V n6 •» T C4 i 5" I ^5 ' Of !^ 1^ £3 + + 'i^-L The i6p .':QiF F L U X I O N S, The quotient multiplied by rv gives " ' =t1 ^' — '' — \, ' — 7 — >.~> ^^' 2rv 2.2rv] 4..2.2rvy 6.4.2. Zr'yp J 3 or, ^^2r,v ^t; + I X ^^^-^ + -^ X -^-r- + /^ -"^ 2 r 4.2 2'.r 6.4.2 2^.r And the fluent is, ^ 2 .3r 2\4.5r^ 2^.4.6. yr^ or, v/2r^ ^ : i + TT— + T^— "z + -tH — J» ^c. 2 ..3r 2 .4.5^ 2*.4.6.7r^ The fluent may alfo be had by Form 16, Emerfon's FluxioQs, as, follows : T'V or -- — zz r X 2r — 'V\ - X v ^-y, therefore, a = ar, ^2rv — vif. j3 = — I, z = V, r, = I, ff rr — f , and i^ n -^ 1 j ^ n — I, and » =: 2. And by fubftitution, ^vz'^+^ 2n ~^'* 2y'''y I s/ _L \y ^ A. — iX-V=X — = Aj -^^71; 2r 2/- -' 2V^V ^^l + „ — i+I + J "" 5.2V/^* ^ 13 OF FLUXIONS. /* " -r, I 2 VV^ V 2» ^ __ 4 '"aVv/ar'' 2r n.'i^v v^ 2 V^2r.4.2^5r* C = 3^ -^^ 4.2Vy^2r 3" ^ _ ^ 4.2VV2r 2r ^ a.g.^ ^^^l &c. Hence we have Sv-y iv^v i.Tpyv^ 2.3.51' 'z/^r -7=3 + -j=. -4 pr H ~7=j ^C. V2r '3^.%''r\2r 5.4.2 Wsr 7.4.6. 2 War for the fluent of — 1_ : ■ ^ : . Multiply by r, and it becomes ^/irv — w , s/arv.v s/T.rv.'i.v'^ , 4/2^1;. 2. c.-y' . * ^ 2 .3r "^ '\-V 2*.4.6.7r^ ' for the fluent of — — , as before. N 2,rv — vv Or thus, Ihis correlponds to rorm 10, >y 2rv — vv s/ ir — v Vvhere z = ■y, ^ = 2r, — ^ =: — l, (or /3 = i,) and », = i ; and the fluent is, 2Nr x Degrees of the arch of a circle, of L which i6a O F F L U X I O N S. which the radius is i, and natural line V — • Or, puttlns: d for the degrees, it will be, as i8o : 3*14159 "• 2^r ; \ , the fluent. •^ 180 As thefe examples will be fufficient to explain the method of obtaining fluents by infinite feries, I fhall therefore only farther obferve to the learner, that, in the folutions of queftions, the fluent firjft found generally wants corredting, by the addition or fubtraclion of fome conftant quantity ; which is always to be determined by the nature of the quefl^ion^ and r^iay be efi'eded by this eafy rule : Subftitute for the variable quantity in the fluent firll found, that particular value which it is known to havd when the whole fluent is fuppofed to be equal nothing; then, if the refulting quantity be affirmative, fubtraft it from the fluent before found, but if negative add it ; and it will be truly corre£led. If the whole exprefliion fhould vanifli, the fluent needs no correilion. An example or two will make it plain. Thus, in the fluxionary expreffion j X a + y\ % the fluent fird found is ■■■ ^ ■ . Now fuppofe when the whole fluent which this ought to exprefs is equal to nothings that y is alfo equal to nothing, then will — be- come -{- — 5 hence — exceeds the whole fluent by — 5 5 ' ^^ fl-f;!' — . which beino; therefore fubtrafted from it, leaves 5 for the correct fluent. For if we put % for the whole fluent, then ought this equation % =: — to hold good in all the cotcmporary values of % and y. But it appears that when X — o, if wc fuppofe y zz Q alfo, the cxpreiJion for the fluent, iuilcad OF FLUXIONS. 363 inflcad of vanlfhing, remains equal to - ; this quantity therefore muft evidently be either fubtraaed from " '^l or added to » to preferre the equality. From whence the rule is obvious. If when the whole fluent is fuppofed equal to nothino- y fhould be known, from the nature of the problem, to have a certain value, Hill the correaion will be performed in the fame manner. Thus, when the zvhole fluent is equal nothin g, let y = b ; then the above fluent corrected will be ! — . As this IS a very material point in the ufe of fluxions, it may not be amifs to illuftrate it by the folutions of a few queftions, where the fluents will require a correftion. I. Required the area of a curve, of which the equation is By reduaion we have y — t — *^ " ~^ . niultiply both fides by the fluxion of the abcifs, and we get a — yx — the fluxion of the area ; the fluent of which found by the common rule, is . Now when the area, or whole fluent, is fuppofed to be rr o, it is evident that the abcifs X mull be alfo zz o ; hence the above expreilion be- 2 X ~z ">!■ comes , and therefore will be the corrcdt 3 3 S'* fluent, or true value of the area. 2. Let the equation of a curve be x^y^ + d^y^ — «*, required the area. hT, By ,64 O F F L U X I O N S. ■ By reducing the equation we get y — — ; hence is the fluxion of the area. And becaufe =rrr=r X \/ a' -\- x' + *• IS — ^ ==- , where the numerator is ^a'^^ + X evidently the Huxion of the denominator, therefore the fluent is a^ X hyp. log. of AT + y/^" -j- x' . Now put a- = o, then the fluent will become a^ x hyp. log. of a^ .*. a^ x hyp. log. of AT + ^a"- + x"- — a^ X hyp, log. o^ a = a'' X hyp. log, of IS the correct value or the area, a 3. To find the fuperficies of an hyperbolic Conoid. Put the femitranfverfe of the generating hyperbola — /, the fcmiconjugate rr c, and the diflance of any ordinate from the center — x; then by the property of the curve we have y ~ - 4/;^^ — ^% and therefore j — — -= — ; hence the fluxion ^ to^x'' — r of the area will be (putting. 3' 141 592 = p) l^cx / ■ - / '* \ -~- X *v t"- -^ c"- X a'- — ^*; or (by putting a' = -j— ; — rj Q_pCX - k/ x^ — a"" . Now fince the fluxion without the radical a ^ is not in any'given ratio to that of the quantity under it, but is the fluxion of the root only, therefore, in order to avoid an infinite feries for the fluent, let the variable part of the fluxion , X y. x'^ — r/ ,. , . . A-^v — \a^xx pe changed to — , which is equal to — V^" — a' Va'-* — aV •I n x'; . Here then the numerator of the firft term Y^A-* — •a''x'^ .... O F F L U X I O N S. 1% is in a given ratio to the fluxion of the quantity under the radical, whence we get for the fluent of that term } from which fubtra£ling the fluent of the other term ~ — — ■ ' '■ , there anies pea x hvp. loo-, y/v — a " of X 4- v/at"^ — a'' . But from the nature qf the problem it is evident, that when the area is fuppofed to be = o, that x will then be — <7 ; therefore the correiSl fluent will be ^-s/ x' — a" — pe ^ pcd X hyp. log. of ~--I--^l == the true area of the fuperficies. ac t + - t Before we clofe this fubjeil, it may not be amifs to give a ihort Explanation of Trigonometrical Fluxion.-. Thefe fhould by all means be well underftccd by the young Mathematician, On account of their great ufefulnefs in Aftronomy, Navigation, Dialling, &c. For fincc the places of the heavenly bodies, the times of their riling, fetting, he. their right afcenflon, declination, latitude, longitude, 2mplltude, azimuth, &c. are all calculated by fpherical triangles, of which the fides and angles are varioufly affected by parallax, refraction, pre- ceflion of the equinoxes, obliquity of tiae ecliptic, &c. it is obvious, that where accuracy is required, it is neceifary to make a co'rredtion of the variable parts, by fuppoling a fmall Continued motion in one or more of the great circles by v/hich the triangle is formed, and thence by the fiuxionary increafe Or decreafe of the fides and angles to determine the quantities fought. r . This fubje£t naturally divides itfelf into four cafes as follow : L cj 1. Whea i66 O F F L U X I O N S. I. When an angle and a fide adjacent are conftant. 2. When an angle and the fide oppofite are conftant. 3. When two of the fides are conftant. 4. When two of the angles arc conftant. CASE I. Fig. 14. In any fpherical triangle ABC, if the angle B and a fide adjacent to this angle, fuppofe BA, be conftant or invariable, then ihall we have the following analogies, 1. BC : AC :: R : cof.C. 2. BC : A :: fin.AC:fin.C. 3. AC : A :: fin.AC x cof.C : R x fin.C :: fin.AC : tang. A. 4. BC: C :: tang.AC : fin.C. That is, 1. As the fluxion of the variable fide adjacent to the conftant angle is to the fluxion of the fide oppolite, fo is radius to the cofine of the an^le oppolite to the conftant fide. 2. As the fluxion of the variable fide adjacent to the con- ftant angle is to the fluxion of the angle oppofite to this fide, fo is the line of the fide oppofite to the conftant angle to the fine of the angle oppofite to the conftant fide. 3. As the fluxion of the fide oppofite to the conftant angle is to the fluxion of the variable angle adjacent to the conftant fide, fo is the rectangle under the fine of this fide and the cofine of the third angle to the re£langle under the fine of the faiT.e angle and the radius, fo is the fine of the fide oppofite to the conftant angle to the tangent of the angle oppofite to the conftant fide. 4. As O F F L U X I O N S. 167 4. As the fluxion of the fide adjacent to the conftant angle is to the fluxion of the angle adjacent to this fide, fo is the tangent of the fide oppofite to the conftant angle to the fine of the angle oppofite to the conftant fide. In order to give a clear demonftration of thefe analo^^les, ic Will be neceflary to premife the following L E M M A* If from the three angles of any fpherical tdangle ABC (%• ^5*) tal<-en as poles, there be formed another fpherical triangle DEF, we fhall have, DK = A, UF = C, EE r= A ; and D == AC, E = AB, F iz: BC. For it is dcmonftrated by moft writers on fpherics, that each fide of the new trianf»le fo defcribed will be the fupplement of the angle which is at its pole, and each of its angles the fupplement of that fide of the triangle ABC, to which it is oppofite ; and it is well known that an arc and its fupplement have the fame fine, co- fine, tangent, &c. therefore the fluxions of the fides DE, EF, and ED, of the triangle DEF will be refped1:ively equal to the fluxions of the oppofite angles in the triangle ABC ; and the fluxions of the angles D, E, F, the fame as thofe of the oppofite fides AC, AB, BC. Suppofe now in the triangle ABC that the great circle of which AC is a part, by a motion round A as a pole, comes into the pofition AC, and the fides BC, AC will become B^, Cf, refpeftively, whicli produce till they be each 90 degrees, and from the pole A dcfcribe the little circular arcs CC, EE; then will the indefinitely fmall quantities Cr, Cc, be the re- fpeftive fluxions of the fides BC, AC, and the arc EE the meafure of the variation of the angle Aj and becaufe the L 4 triangle i68 OF FLUXIONS. triangle C Cc^ which is right-angled at C, may be eftecmed right-lined on account of its fmallnefs, the fides will be pro- portional to the fines of their oppofite angles. But as the angle at c is eflentially the fame as the angle at C, and con- fequently the angle cCC the complement of that angle, we fhall evidently have, C . -^ fhT.C fii^AB ^ . ^ -^ ^ ^_ C : AB :: — r-r : — — ,— :, that is C : AB :: tang.C : tang. AB, col.C col. Ao o ' the firfl: analogy. After the fame manner it may be proved that, A : BC :: tang. A : tang BC, the fecond analogy. And if D/f be fuppofed equal to DA, DC equal to DC, and the angle at D very fmall ; the angles at J and C may evi- dently be efteemed right angles, JC equal to ac, and therefore j^a equal to Cc. Hence in the right-angled triangle AJa, we have Aa : Ja :: R : cof. A; and in the right-angled triangle CCf, Cc : Cc :: cof. C : R; therefore Aa X Cc : Cc X Ja :: R X cof. C : R x cof. A, that is A^ -.Cc :: cof.C : cof. A j or, AB : BC :: cof.C : cof. A, the third analogy. And by the fame manner of reafoning in the triangle DEF ffig. 15.) u'e have Dii. : Db' :: cof. F : cof. E, therefore per lem. it will be A : C :: cof. BC : cof.AB, the fourth analogy. ^ E. D. CASE. III. Fig. 17. If any two of the fides AB, BC, are conflant, thefe analogies will obtain, 1. B : A :: R X fin. AC : fin.BC x cof.C. 2. B : C :: R X fin. AG : fin. AB x cof. A. 3. B : AC :: R" : fin. C X fin. BC :: R" : fin. A x fin. AB* 4. Cr AC :: cot.A : fin.AC. 5. A : AC :: cot.C : fin. AC* That OF FLUXIONS. jyi That is, I, 2. The fluxion of the angle formed by the two conftant fides is to the fluxion of either of the other two angles, as the rectangle under the fine total and fine of the variable fide is to the re6tang!e under the fine of the fide oppofite to the latter angle, and the cofine of the third angle adjacent to this fide, 3. As the fluxion of the angle formed by the conftant fides is to the fluxion of its oppofite fide, fo is the fquare of the radius to the redangle under the fide of either of the other angles and the fine of its adjacent fide. 4, 5. As the fluxion of either of the angles adjacent to the variable fide is to the fluxion of this fide, fo is the cotangent of the other adjacent angle to the fine of the faid fide. Demonstration. Let the angle ABC become ABC, and the fides BC," BC, AC, A'."", be produced to quadrants; then will Ke evidently mealurc the variation of the angle at B, and G^ that of the angle at A. From A defcrlbe C^, and there will be formed the triangle CcC, which on account of the fmallnefs of the angle CAC may be efteemed right-angled at c, as alfo the an^le cCC tlie complement of the angile C. From hence, and the fimilar fedlors, we have thefe proportions, Ee : CC :; R. : fin. BC ; Cc : G^ :: fin. AC : R; CC : Cf :: R : cof. C; the corrcfponding terms of which being multiplied, and the values of the arcs Ee, Gg, fubfti- tuted in the produft, we fhall have B : A :: R X fin. AC : fin.BC x cof.C, the iirft analogy. By a fimilar procefs we get, B : C :: R x fin. AC : fin. AB x cof. A, the fecond analogy. And by multiplying the correfponding terms 172 OF FLUXIONS. terms of tbefe two proportions, E^ : CC :: R : fin. BC ; CC: cC::R: fin. C, we find Ee '. cC :: R^ : fin. BC x fin. C, or, B : AC :: R* : fin. BC x fin. C ; and if we fuppofe the angle B to flow towards A, we (hall in like manner get B : AC :: R^ : fin. AB x fm.A, which are the third anafogies. By Cafe II. we have B : C :: R x fin. AC : fin. AB x cof. A, and AC : B :: fin. C x fin. BC : R^ ; therefore, by multiply* ing thefe two proportions together and reducing the produiV, we have, AC : C :: fin. AC : cot, A ; or by inverfion, C : AC :: cot. A : fin. AC, the fourth analogy. And by the fame method we find A : AC :: cot. C : fin. AC, the fifth analogy. ^E. D. CASE IV. Fig. 16. When any two of the angles B, C, are conftant, the fol- lowing analogies are derived, 1. AC : AB :: fin. B x R : fin. C x cof. BC. 2. AC : BC :: fin. B x R. : fin. A X cof. AB. 3. AC : B :: cofec. AB : fin. A :: cofec. BC : fin. C, 4. AB : B :: cot. BC : fin.B. 5. BC : B :: cot. AB : fin.B. That isj I, 2. As the fiuxion of tlie fide oppofite to the variable angle is to the fluxion of either of the other fides, fo is the reftangle under the radius and fine of the variable angle to the rectangle under the fine of the angle oppofite to this other fide and the cofine of the third fide. 3. As GF FLUXIONS. 173 3. As the fluxion of the fide oppofite to the variable angle is to the fluxion of this angle, fo is the cofecant of either of the other fides to the fine of the conftant angle adjacent to this fide. 4, 5. As the fluxion of either of the fides oppofite to the conftant angles is to the fluxion of the variable angle, fo is the cotangent of the other fide to the fine of the faid angle. Demonstration. By applying this cafe to fig, 15. the two fides DF, DE, will be variable, therefore by the laft cafe we fhall have, D : E :: R X fin. FE : fin.DE x cof. F ; or, by fubftituting the correfponding values in the triangle ABC, i\C : AB :: R x fin. B : fin. C x cof. EC, the firft analogy. In like manner may the fecond analogy be obtained. And becaufe the fquare of the radius divided by the fine of an arc is equal to the cofecant of this arc, the third analogies will become AC : B :: R" : fin.Axfin.AB :: R'" : fin. 6' x fin.BC, and thefe proportions are eafily invcftigated by the triangle DEF, and the third analogy of Cafe III. The fourth and fifth analogies are immediately deduced by applying thofe of Cafe III. to Fig. 15. ^.E.D, I {hall now fhew the application of this theory by an example pr two. Example i. Required the hour of the day or ni,2,ht, by the obferved al- titude of a flar; and alfo the error of time, fuppofing the error 'm the obferved altitude to be known. ^ ' Solution* 174 O F F L U X I O N S. Solution. In the fpherical triangle PSZ, (fig. i8.) wherein P reprefents the pole, Z the zenith of the place of obfervatioii, and S the place of the ftar, ZS will be the complenjent of the ftar's altitude, P> the complement of its declination ; PZ the complement of the latitude, and the angle ZPS a variable hour-angle contained betwixt the conftant fides PS, PZ. Now by Cafe III. we get, P : ZS :: R^ : fin. Z x fin.PZ; but P is evidently the mcafure of the error in time, therefore R^xZT ^ r. ^ we have P - ■ .. r P7 = ^-^.A lin.Zx fin.PZ ^• R* Cor. I. From this equation it appears that, fince r? — =■ is lin. y^ a conftant quantity, if the fame error be fuppofed to be made in different obfervations in the fame latitude, the error of time will not be altered, whatever the altitude of the ftar be. 2. If the latitude be ftill fuppofed the fame, the error of time will be the leaft when the ftar is obferved on the prime vertical ; for then the angle at Z is a right angle, and there- R^ fore -: — — becomes barely R. And the error will be the leaft linZ pofTible if the obfervation be made at the equator when the ftar . , r , , n- R^xZS is on the prime vertical, for then the expreiTion :: — ^ ■. -= ^ * ^ iin.Zxtin.P^ will evidently be a minimum, fince fin. Z X fin. PZ will be the greateft pofliblc. Example II, Required the corre£lion ncceffary to be made in determining the inftant the fun comes to the meridian from two equal altitudes ; fuppofing the fun's declination to undergo a little change during the interval of the obfervations. SelutfOH^ OF FLUXIONS. 175 Solution. If the fun's declination was the fame at both ob- fervations, half the interval would afcertain the inftant of noon ; but as this can happen only at the folftices, at all other times of the year this half interval muft want a corrccSlion, by Ibmething being fubtraited when the fun is in the afcending ligns, and added when in the defcending figns ; becaufe in the former cafe the fun muft evidently arrive later at the fame altitude, and in the latter cafe fooner. In order then to de- termine this corre 2ncl in M. De la Lande's Aftronomy. But it may not be improper, however, to obferve by way of caution to the young reader, that folutions obtained by this method cannot be depended on as ftriftly true, unlefs the arcs under confideration be in their evanefcent ftate ; and therefore the nearer they approach thereto, the nearer will the folution ap- proach to precifion. The following problems (among many others) are invefti- gated by the Dlre£i Method of Fluxions, i. The Maxima and Minima. 2. The finding of Tangents. 3. Determining the points of contrary Flexure in a Curve. 4. Finding the Evo- Jutes of Curves. 5. Inveftlgating the Catacauftic and Dia- cauftic Curves, he, he. And the Inverfe Method is applied, I. To the finding of the Lengths of Curve Lines. 2. To the Quadratures, or finding the Areas of Figures. 3. To in- yeftigating the Surfaces of Solids. 4, To determining the Solidities of Bodies ; in which is included all Menfuration, ^. To finding the Centers of Gravity, Percuffion, and Ofcil- lation. 6. To inveftlgating the Law of Centripetal Force in a given Curve. And laftly, to the folving of all Phyfical Problems whatfoever. Since then fluxions are univerfally ap- plicable to all kinds of problems ^^, I would advife the young algebraill ♦ There may be cafes propofed wliich are too fimple for the method of fluxions, fuch as finding the areas of parallelograms, the fuperficies and folidities of parallelopipedons, prifras, cylinders, &c. fince a figure or folid fimilar to tliefe is evidently always aiTumed in the very nature of the fluxion ; and therefore nothing can be inferred from thence. And, on the other hand, there may be caics propofed which fcem t^. be inacceflible by a fiuxionary calculus. Such is the problem refpe£ling the equality of the areas of the hyperbola and triangle of equal bafes, when the altitudes of both are fuppofed to be infinite. Here the method of increments anfwcrs fucl> E /6' OF FLUXIONS. 177 ^Igebralft to endeavour by all means to make himfclf thoroughly acquainted with theni. 1 know of no book on the fubjefl fa . n and a by the equation — X 360°+! So" — 2; — cof.s — r — a=.o; whc:i ■' m I II III the bodies will meet tbr^e times when a is between and a, u and ^, and / // only once wlicn a is between « and a. It may perhaps appear to the reader, that wlicn the characleriilic of a changes, the particular values thereof might be determined from changing the order of thefe equations, that is by a reciprocal fubfdtution of one cafe for the other; but from )ience we fhould find either that fomc values of a were negative, or feme greater than thofc wliich refult from § 5. whicir therefore ought to be re- jected. iJence Vr'c fee the inconvenience attcndinc^ fuch a rrnccdurc. relaiica , O F CURVES. 189 relation between the abcifs and ordinate at the extremity m of the finuofity kmm\ and the equation n X 3to'' + 1 80° — "z — zq{.% — r — a — o determines the relation between the abcifs and ordinate at thi; extremity of the finuofity m m m. Now becaufe (by the hypothefis) the two fmuofities have the fame tangent ? m m, or rather becaufc they coincide at that point, we have at the fame time the two following equations, — z + cof. z — r — a =: o; — X q6o° + 180'' — z — cof. % — r — a =r o; m '^ which, by equating the two values of a derived from hence, give for the condition of the problem n -^60 + 1(3'! — X • z — col. 2; — o : ?n 2 ' . n fin.z or, becaule — ~ , m r - ?oo + ibc - lin. "z X 2 — f CGI. z rr o. 2 An equation depending on the quadrature of the circle, Frotw hence we eafily find o J '/ n 217? I % ~ arch 12, Q2, 50, — =:: — — — ; — , and a ~ 2371. •^ -' //; 1 0000 4*603 ' 17. Hence it appears that if the ratio of the motions of the two bodies T and R be as looco is to 2173, and that when the body T is at t, R is diftant 2371, they will meet five times in the fame perpendicular. In all other circumftances they can meet only three times, 18. We 'go A NEW SPECfES 1 8. We fhall now enquire what the ratio of m to n ougKt to be, when a is fuppofed zz o, that the equation — ;:; -f cof. z — r — a zz o may have five real roots. 7/i If the ratio of m to « be yet farther diflant from that of /// // /// equality, the finuofity m m m is not limited by the ordinate ' ' " . , . V mm, but paffcch beyond it j and there is one particular cafe "where it touches the ordinate correfponding to the abcifs zr O. To determine which we muft obferve that the equation - X ^60"" + 180^ — % — cof. z — r — a zz o, m cxprefleth the relation between the abcifs and ordinate at the // /// // /// extremity 711 of the finuofity 7n m m (agreeable to § 16.) If " then we fuppofe that the point m in the curve hath its abcifs ■=. o, we (hall have ^ z: o in the laft equation, therefore — X 300^ + 180" — % — cof. z — r zz o; m n fin.% , 1 /- 1 fi- . but — = , hence by lubltitutionj m r 260^ + 180^ — % X fin. z — r X r -f cof z zz Oy an equation depending ftill on the quadrature of the circle..- From hence we get o / // o o o y // » ZZ 12, 23, 48 J 360 + 180 — K = 527, 36, 12 > n _ 2147 _ I m ~ loooo ~ 4'^58* 19. From whence it appears that if the ratio of the motion* of T and R be as lOOOO to 2147, and that they both begia to move at the fame time from the point /, they will firft btf ilk OF CURVES. 191 in the fame perpendicular at the commencement of motion, a fecond and third time when T hath defcribed two certain arches, and a fourth time when it hath defcribed an arch of 527, 36, 12, from the point /, and this laft meeting will be double. If the value of a be contained between o and 2341, n determined by the equation — z + cof. z — r — a = 0, 7n there will be five meetings of the two bodies in the fame per- pendicular; but if a be greater than this laft number they can meet only three times. But befide the above limit for ^, the particular value 134920 giveth yet five meetings of the bodies in the fame perpendicular; this being the diftance run by R during one revolution of T in its orbit, agreeable to what has been already obferved in § 5. Let A (fig. 28.) be the origin of the curve, to which the correfpondent abcifs is n o, and the abcifies AP, AP, refpe^iively equal to 2314 and 134920. Here then it is evident that the finuofities of the curve arc bounded by the ordinates hm^ Fm, Vm', and in the fame man- ncr as the finuofity vi m rn is limited by the ordinate hm of /// // ih which the abcifs is 0, fo is the finuofity m m m limited by the II II ^ ordinate V m of which the abfcifs A is 134920. This is then III II III the cafe where the part of the curve m m m beginneth to an- fwer the queftion confidered aftronomically, while the ana- // // logons part of the curve m m m ceafeth to refolve it. 20. If the ratio of w to n continueth to be ftill farther re- moved from the ratio of equality, the finuofity of the curve II II _ m m m will then not be limited by the ordinate A/w corre- fponding to the abcifs ir o, but will extend itfelf over the negative fide cf the abcifi-, as wc haye before obferved in § 15, anit 192 A NEW SPECIES /// // /,/ and in a fimilar manner will the finuofity 7n m tn extend itfelf // // // beyond the ordinate V m correfponding to the lafi: abcifs A P. If in this cafe we denote the firft abcifs by o ; the lafl abcifs derived from § 5, by « ; the two values of a refpedively de- termined by the equations — s + cof. z — r — a ~ o, in — X 2 X 360" + 1 80'' — z — cof. z — ^ r — a zi o. by «, a ; and the leaft angle deduced from the equation % ■= — by z ; m the two bodies will meet five times when a is between and a^ it III a and a ; but will meet only three times when it is between locoo , 1 0000 a and a. If the value a be comprized between and 2173 2147' / // and if a, a, refpeiiively reprefent the two values of a de- termined by the equations ' — X 360" + iSo*^ — % — cof. z — r — a — o, 7n ' — z + cof. z — r — a rr 0, the bodies R and T will meet m . . , / // /// but three times when a is between o and a, a and a\ and / // five times when between a and a. 21. If we would Inveftigate the ratio of m to n, fo that tftc original equation might have feven real roots, v/e fliould fiiKf for the condition of the problem - 2 X 360 + 180° ^ ^ iin.z X % — r cof. % z=i o\ 2 * o / // . from whence we may eafily find 2 — arch 7, 22, 31 j n 1284 J -no - — — ; a — 788. m loooo 7*791 And y ■ ^■y.'J" .;, c^ .>,.- . ^ ^— — 1 -_-^-" r^- p '~Z-^'. - h^- OF C U R. V E S. t93 And in the fame manner we might proceed to determine the "l-ario of m to », fo that the original equation would be fuf- ceptible of any afligned number of real roots. 22. We fhall now exhibit in a general manner, how to de- termine in all cafes the number of real roots of the original equation. Write down in the following order the equations which have been deduced in the preceding Articksj fin. 2 1. -*■ '" ■ % 4* ecu z — r — a =o; r fin. 2; -- - » 2. X loO — z — col. z — r — a = o ; r fin. 3. X 360^ + 1 80" — % — cof. % — r — a ■=. o\ c . 4. — '— X 2 X 360'' + 1 80° — z — cof. % — r — tf r: o } fin.z —— ■^—- . 5. — ^ X 3 >*^ 300 + loo — a — cof; s — r — • X 1 80° — s — cof. % — r — « = o; N 2 'in 196 ANEWSPECIES in which equations wemuft remember that % is determined by fin.z n . . n . . . , the equation =: — , and that -- is eiven in the queftion. r m m 24.. This laft difficulty maybe eafily obviated; for fince the equations (2) and (3) § 23. cannot both obtain at the fame time, if the equation (2) for inftance, be that which we ought to ufe, the equation (3) will give eflentially a value of a greater than a ; but if the equation (3) be that we fhould make ufe of, the equation (2) will give a value of a lefs than o. We (hould then immediately try the equation (2), and if it give a value of a greater than o, this is the value we have denoted by ^, and the equation (i) will give «. If on the contrary the equation (2) giveth a value of a -kfs than o, it is of no ufe, and the equation (i) will determine a^ as the equation (3) will give a. In the firft cafe the propofed equation will have y — 2 real / // ill roots when the value of a is between and a, a and a ; and » real roots when a is between a and a. In the fecond cafe, the equation will have » real roots, when a is between and Three effential real roots, and may have five according to the different values of a% ^Five effential real roots, and may have feven, by the different values of o* Inclination of the ring to the ecliptic - - - 31°, 20, O. Inclination of faturn's orbit to the ecliptic - - 2, 30, 20. Lon^^itude of thofe points of the earth's ~| _ „ '^. . 2, 17, 5, o. orbit which the plane of the ring )• touches when produced, J > 7> 5> • Some new Geometrical PropofJionSy which will be often found ufefiil in the foliUions of Problems. Pfopofition I. Theorem. Fig. 29. If from the middle of the chord of a given fegment of a circle ADB, a perpendicular be drawn to D, and AD pro- duced to E, fo that DE be equal to AD, and through E another fegment be defcribed on AB ; then, if a line be drawn from the end of the chord as AF, the part intercepted be- tween the two peripheries GF will be always equal to GB. For DB is evidently zz DE j and AFB is = AEB, alfo AGB = ADB (Euc. 21. 3.) therefore EDB - FGB, and the triangles BDE, BGF are fimilar, confequently EG ;;; GF. Propofition II. Problem. The fame conftru£tion being made as in the laft propofition, it is vt:quired fo to draw the line AF, that either the fum, difference, ratio, redangle, fum of the fquares, or difference of the fquares of AG and GF, may be of a given magnitude. For PROPOSITIONS. 199 . For the Sum. With diftance AF z= the given fum defcribe aa arch to cut the periphery of the greater feament in F, and join A, F ; and AG + GF will be zr the given fum* This is evident j as alfo the limitation, that the given fum mufl not be greater than 2AD, nor lefs than AB. The Difference. With diftance AL zr the given dif- ference of AG, GF, defcribe an arch ab; and on AB de- fcribe th-- fep;ment of a circle containing an angle equal to the fup. of DAB, (Euc. 33. 3.) and through the interlcition L draw ALFj fo will AG — GF — AL the given difference. For fince AL is equal the given difference per con. GF will be equal GL ; but GF rr GB pL-r con. therefore GL = GB; hence ALB z= LGB + GBL (Euc. 32. i) = ADB + DBA =z fup. of DAB. Th-,- limitation is evident, -the given difference muft not be gneater than the diameter of the leffer circle. The Ratio. Divide AB in the given ratio of AG to GF in O, complete the circle ADB (Euc. 25. 3.), and produce DC to ?; then through O draw PG, and through G draw AF; and AG will be to GF in the given ratio. For fince the arches AP, BP, are equal, the angle AGB is bifefted by G? (Euc. 27. 3.) and therefore as AO : OB :: AG : GB :: AG : GF, per Euc. 3. 6. and conftruclion. The fame may be effefted independent of the property of the circle. — Having divided AB in the given ratio in O, in any angle draw the indefinite line AH, and take AH to HI alfo in the given ratio; join lO, parallel to which draw HK, and from center K and diftance KO defcribe the arch OG ; then draw AGF, and AF will be divided in the given ratio In G. For by conflruaion AH : IH :: AO : OB, and be^ faufc of the parallels lO, HK, AH : HI :: KA : KO, - N4 .-.KA zco GEOMETRICAL .'. KA : KO :: AO : OB, and by alternation and divifioi^ 1i4:KA-AC>::KO:KO-OB, .hat is KA: KO:: r^O: KB, or KA : KG :: KG : KBi hence »he triangles .\KG, GKB are fiTiiiar (iiuc. 6. 6.j und the other fides wili be proportional, or AG : GB :: AK : KG ;KO) ;; All : HI, which are in the given ratio by conllruftion, Th- Re£langle. — Let M reprefent the fide of a fquarc equal to the given rc£la: gle (Euc. 14 2.\ Find the centre of the leffer circle V (Euc. 25. 3.) and on AB produced take /K eqial to the diameter thereof j make the normal KT a third proportional to AK and M, and draw TG |1 to AK j t!ien through G draw AGF, and AG x GF will be equal M^. ■ For thioujh the center V draw GS, join SB, and demit the ± ^Rj then are the triangles AGR, SGR fimiiar (Euc. 21. 31. 30aiidAG:GR::SG:GB, hence AG xGBirGR xSGj but AK = SG per con. and TK - GR, .-. AG X GB = AG X GF r= AK X TK = M"" per conftrudion. Limitation. M^ muft not exceed AD x D£, or the fquare of the femi- diameter of the greater circle. The Sun; of the Squares, — Let 2M* exprefs the fum of the fquares, and take CQ^ =: M^ — AC'i with diftanc« CQ_ and center C defcribe ihe arch QG, and through G draw AF, and AG" + GF" will e4ual 2M*. For AQl = QC^ + AC" (Euc. 47. I.) and AQ^ + Q_B'- = 2AQI = zQC' + 2hC'; afo AG* + GB^ = AC- + CGV+ 2 AC X CR -f AC* + CG' — 2AC x CR (Eac. 12. 13. 2.) = 2 AC* + 2CG*j but CG =z CQ, p^r con. therefore AG^ + GB" = AG* + GF" = 2AQf- 2CQ1+2AC* r= 2M* per conftruftion. — — Limitation. N'P muft not be greater than AC + CD% nor lefs than | AB\ c The PROPOSITIONS. 201 The Difference of the Squares. ■ — Let M* — the givea jdifFerence of the fqua.cs, and take CR rr to half a third pro- portional to AB and M ; then drav/ RG ± to AB and through the interfeaion G dr^w AF ; fa will AG* — OF* = M\ : For AG? — GBMs = AR^ + GR* — BRM- G^ (Euc. 47. I.) = AR' — BR* =z AR -i-'RB x AR~-I^r^B (Euc. 5. 2.) = (becaufe AR evidently exceeds RB by 2CR) AB X 2CR = M* per conftrudlion. -Limitation. M muft 11 ot exceed AB. Propofition III. Theorem. Fig. 30. If the bafe of a triangle ABC be produced both ways to D and E, fo that CE = CB, and AD = AB ; and the center of a circle F be found to pafs through the points D, B, E, (Euc. 5. 4.) and B,F he joined, the angle ABC will be biieded by BF. For join BD, BE, FA, FD and FE, then is ACB = CBE -h BEC - 2BEC (by con. and Euc 32. i.) = BFD (Euc. 20. 3.), and in like manner BAG z: BFE; hence ABC = EDF + DEF = 2FDA; and becauf the triangles BAF, DAF are evidently equal and fimilar, ABF ::= FDA = i ABC, therefore ABC is bifeded by BF. Propofition IV. Problem. Fig. 31. There is given DC the diftance of the indefinite line AB from the center of the given circle IFE; it is required io to draw a line through I, that the part intercepted between AB and the concave periphery of the circle may be of a given length. Ort IE produced (if neceffary) take IP equal to the p,iven line, on which defcribe the femicircle lOP. Make IS a mean proportional between CI and IE, and Xto CE j draw SO || to pE, and from I apply IL = SO, which produce to F, and LF 2C2 G E O M , E T R I C A L I>F V/ill be equal the given line. For demit the normal OT, and Join F, E. Then becaufe ICL, IFE are right angles, and CIL rr FiE (Euc. 15. j.) the triangles ICL, JFE are fimilar ; therefore CI : IF :: LI : IE, hence CI x IE zr LI X IF ; bjJt SP = CI X IE per con. alfo IP r: the given line per con. .*, IT x TP = TO^ (Euc. 13. 6.) = SP =1 JL,I X IF 3 and LI — IT (— SO) per con. confequently IF =z TP, and LF =; IP the given Vine per conftru6lion. Proporition V. Theorem. Fig. 32. If tviTO circles ACD, FDG touch each other, and another circle be any v/here delcrihed cutting both the former circles, as in the points A, C, G, f , the chords AC, FG being pro-» duced will meet the tangent drawn from the point of contafl D in the fame point B. For in the circle ACGF, AB xBC -= FB X BG; and in the circle ACD, AB X BC = BD"; alfo in the circle FDG, FB X BG := BD" (Euc. 36. 3.) hence DB is a common tangent to both circles, and confequently the lines A 8, DB, FB meet in the fame point B. The converfe of this is alfo true. That if two lines be any how drawn from a point B to cut a circle ACGF in C, A, G, F ; atid upon one of the chords as GF a circle be defcribed FDG, and from the point B a tangent BD be drawn to this circle j then, if on ED produced the center of a circle be tarken to pafs through the points A 'and C, it will alfo pafs throu Ed. Lat. EouiTE Aurato.J & Eng» I2S. 2 Vol. 1747 12s, 2 Vol, James Fergufon, 2d. 1770 7s. 6d, F.R.S. James Fergufon, 3d. 1772 5s. F.R.S. O a Subject* 212 A S E L E C T Subjeft. Author. Aflr'^nomy explained James Fergufon, i![)onSirI(aacNcw-| F.R.S. ton's Principjes. A Gyftem of" Aflronf-- my, containing the In veP '.nation and DeiTjonftration of the Elements of that Science, Mr. Emcrfon. Ediiion. London Prices. 5'^'. 1772 i8s.4to< 9s. 8vo. 1769 6st 'Other Writers of Eminence in the feyeral Branches of the Mathematics, &c. are as follow : Arithmetic. Chapman, Cockin, DiUvorth*, Ewing, Eadon, Fifher, Hardy, Hill, Hayes, Kirby, Lowe, Pardon, Perry*, Sadlej", Scotr, Thompfon, Ward, Welch, and Willon. Algeora. Afliby, Hammond*, Kerfey, Martin% Ronayne, Wallis, Weft, and Woliius. ArchlieSlure. Emerfon, Gibbs, Hutton, Salmon, and Langley. The Italian Authors are, Vincenzo, Scamozzi, Pier Cata-? neo, Andrea Paliadlo, and Baftiano Serlio. And in Fiench, Le Clerc, and Davilcr. Jjlronomy. Dunn, Gregory, Heath*, Harris, Halley, Keill, Leadbetter, Long^, and Whifton. The Aftronomical Tables of moft note are, Halley's, 410. 1752. i8s. Fergufon's 8vo. 2S. 6d. Mayer's /[.to. 1 7 70. 13s, fcwed. Clairaut's ^vo. 1765. 3s. fevvcd. The Durham Tables, 4:0 1766. . 3s, 6d. boards. Bock- CATALOGUE. 213 Book-keeping, Cooke, Everard, Gordon, Hutton, Mair, Pcriy, Quin, Roofc*, and Webfter*. Chance^ and Annuities. BufFon, Clark*, De Molvre*, Du Pre, Emerfon, Halley, Price, and Simpfon. Chronology. Blair, Emerfon, FerguTon, Du Frefnoy, Marfliall, Newton, Strauchius, and Whifton. Conies, De rHoipita), De la Hire, De Wit, Emerfon"^*, Ha- milton, Jack*, Muller, Milne, ]\Iydergius, Ozanam, Simp- fon, Vivani, Vincentio, Ward, and Wallis. Decimals. Cunn, H. Clarke, Drape, Martin, Robcrtfon, an4 Wilfon. jyialling. Emerfon*, Fergufon, Leadbetter, Leybourn, Mar- tin-, and Potter. EleSiricity. Franklin, Freke, Fergufon, Hoadly, Lovett, Martin, Prieftly*, Watfon, Wclley*, and Wilfon. J<'luxions. Ditton, Hayes, Maclaurin*, Muller, Sanderfon, and Stone, Fortification. Emerfon, Muller*, Pleydell^, and Vauban. Gauging. Clark*, Emerfon, Leadbetter, Mofs, Ovcriey, Shirtcliffe, Symons"^, and Turner. Geography. Emerfon, Fenning, Guthrie*, Jones*, Salmon, Varenius, and Wel!^. Geometry, Anderfon, Cunn, Donn*, D'Omerique, Fletcher, Gregory, Ghetaldus, Herigon, Keill, Lawfonf, and Rudd, t Tlie Rev. Mr. Lawfon, Rt 1:or of S\vanl'cotnI)e in Kent, who, among Ills other works, has publifhed a moft ufeful peiformance for the exercifc of young Geometricians, intitltil, A Dilfertation on tlie Geometrical i^ualvfis of the Antients. 1774. a. 6(1. fewed. Gunnery^ 214 SELECT Gunnery. Emerfon, Gray, Holliday, Robertfon, Robins, and Simpfon, MenfuraUon. Fletcher Hawney, Hutton*, and Robertfon. Mechanics. Emerfon*, Fergufon, Fletcher, Watts, and Wells. Mufic^ the Mathematical Principles ofy Emerfon*, Helftiam, and Martin. Navigation. Atkinfon, Crofsby, Emerfon, Harris, Martin, Mafkelyne*, Patoun, Robertfon*, Henry Wiifon, and William Wiifon*. Optics, Baker, Emerfon, Jurin, Martin, Newton, and Smith. Terfpe£live. H. Clarke, Emerfon, s'Gravefande, Hamilton, Kirby, Lamy, Malton, Martin, Noble, Prieftly, and Taylor. The Latin Authors are, Marolois, and Andrea Pozzo. The Italian^ Daniello Barbaro, and Jacopo Barozzi da Vignola. The French^ Cerceau, Jan Vredeman Friefe, Lamy, and the Jefuit. Philofophy. Blifs, Clare, Clarke, Cotes, Defagullers, Green, s'Gravefande, Gregory, Hales, Helfham, Hawkfbee, Fran. Jacquier, Keill, Mufchenbrock, Abbe Nollet, Penberton, Prieftly, Rowning, Shaw, Stewart, Worfter, Whifton, and White fide. Surveying. Breaks*, Burn, Emerfon, Fletcher, Gardiner*, Grey, Hammond, Lawrence, Wilde, and Wiifon. Trigonometry. Boad, Keys, Hawney, Payne, and Wiifon. The Tables of moft note are, Gardiner's, and Shcrwin's. That Edition of Sherwin's, revifed by Mr. Sam. Ckrk, is the moft correft, Thefe C A T A L O G U E, &c. 215 Thefe Authors I have felecSed from among feveral others • but, as there are but few who have time and inclination to read fo many, I have diftinguiflied with an aftenfk. thofe which I dare venture to pronounce, from my own reading and occafional ufe of them, the beft written treatifes on the fubjeft, and that will afford the moft information in the leaft time. Thofc Books, which are particularly diftinguiflied by their Date Price, &c. I would ftrenuoufly recommend a very careful and reiterated perufal thereof, as being undoubtedly, not only the beft adapted to the capacity of a learner, but containing as it were the very balls on v/hich the young mathematician is to build his whole fuperftrudure. ERRATA. pAGE 32, /. 10, r. 999999; P' 95> ^* 4> fiom bottom, for + in the num. r, x ; p, 127, /. 2, r. — — , &c. ; oa p. 167, /. 10, r. FE = B; p. ib. I. 5, from bottom, r. Ac-, p. 168, /. 4, r, C. FINIS. AT THE COMMEP^CIAL AND MATHEMATICAL SCHOOL, In SALFORD, MANCHESTER, Youth are Boarded, and Infl:rii£ied in all thofe Branches of Learning, which quaUfy them either for the Army, Navy, Counting-houfe> or any Artificer's Bufinefs. By H. C i: a R K E. A COURSE OF LECTURES O N GEOGRAPHY and ASTRONOMY, Commences Twice every Year, viz. On the Firfl of February^ and ends about Midfummer ; and on the Firft of July, and ends in November. At which Times, any Perfon may enter for the Courfe. The Times of Attend- ance are, 3 o'Clock on Thuri'days, and 2 o'Clock on Saturdays, in the Afternoons, at the Lefture Room under the School. The Terms are, 2s. 6d. Entrance, and One Guinea the Gourfe. The Heads of the Leftures are as follow : A N introdu£lory Difcourfe, (hewing the Neceflity, Ufcfiil- -^"^ nefs, and Excellency of the Knowledge of Geography, a Defcription of the Globes, with their Appendage?, and other neceflary Definitions of Terms relating thereto, with their Derivations, Of Of the feveral Pofitions of the Sphere in refpetft to the Horizon, and of the Inhabitants of the Globe in rcfpcct of their Situations to one another. All the ufefjl Problems performed on the Terreftrial Globe, Of the Ufe of Maps, general and particular, and of the con- ftituent Parts of the Terraqueous Globe, as they are delineated thereon. A general Defcription of the Earth and Seas; and a particular Defcription of Europe, in which is given the Geography of ancient Britain. A Defcription of Afia, with the ancient Lefler Afia, a-^d the ancient Mefopotamia, Aflyria, Babylon, or the Chaldeans, and Armenia. A Defcription of Africa and America, with aComparifon of the Extent of the Four ancient Monarchies, and of the Ff^ur prefent general Religions. The Navigation of Ulyfles, according to Homer ; and of JEne&s. accordino- to Virgil. Romafiufn Imperium ad Acmen eve^nm : or, a Defcription of' the Roman Empire at its utmoft Height. The Geography of the Four ancient Monarchies, of all the Places mentioned in the FourGofpels, and of the Travels and Voyages of St. Paul, and the other Apoftles. Of the Solar Syftem. The Sun and its Properties, with the agronomical Problems relating thereto. The Laws, Nature, Magnitude, Didances, and Motions of the Planets, and the aftronoinical Problems relating to x.\:cm. Of the fixed Stars, their Diftance, Magnitude, Order, Nuni- ber, Names, and Appearances, with Problems how to know, and where to find them in the Firmament. Of the Nature of Comets. Of thefeveral Syftemsof the World, ancient and modern, &c. And of the Origin and Rcafon of the Charaflers, Figures, and Names, given to the Planets, Signs, and Conilcllarions; with the claflicai Stories and poetical Fictions relating thereto. In the Courfe of thefe Lectures, which are particularljf adapted to the Capacities of young Gentleaikn .ftudying t^e Claffics, and to fuch as are defirous of being acquainted with tlie Alirorum Scicritia, abftra^led from mathematical Cd.-is- lations, is fiiewn the Alethod of finding the Latitude, fro.n the ©bfcrved Altitude and Azimuth, by an accurate L;iL--.;:rjcaT. Alio Alfo the Method of making an Obfervation at Sea, by the Hadiey's Q^iadrant, and Smeaton's Top, &c. The Globes, Maps, Ciiarts, ;,nd oiher explanatory Schemes, mnde Ufe of for illuftration, are all new, and of the beft Conftruflion. N. B. The Tickets for Admiffion are not transferable ; but will admit the Purchafer g>atis in any future Courfe. No more than twelve Perfons can enter for the fame Courfe* Jujl PubUJhcd, Upon a new Plan, purely adapted to the Ufe of Schools, and divided into Fifty-two Leflbns, PRACTICAL PERSPECTIVE. lUuflrated with 33 Copper-plates, and moveable Schemes. Vol. I. 8vo. Price 5s. in boards. By H. Clarke, Teacher of the Mathematics, Salford, Manchejier, London: Printed for the Author, and fold by Mr. Murray, N° 32, Fleet Street. 'Speedily ivill be TuhUJljedy by the fame Author, AN Eflay on the Ufefulnefs of Mathematical Learning; wherein is ftiewn the progreffive Growth of the Mathe- matics, from their Infancy to the prefent Time ; and a Com- parifon drawn between the Ancients and Moderns ; proving the high Eftimation they were held in by the former, as com- prehending wa'vla m MaO^'pola, or the whole Circle of Human Learning. With an Alphabetical Account of the mofi: eminent Geometers and M ithematicians, ancient and modern, and the Works they have publiflid.. To which is added, A Treatife on Magic Squares, tranflated from the French of Frenicle, as pubiifhed in Les Ouvrages de Mathcmatique par Mejfieurs de rjcadetnie Royals de Sciences, with feveral Additions and Re- marks : A fubjecl:, though not very interefting in itfelf, yet which affords the Mind a pleafmg Satisfa£lion in obferving th© Wonderful Properties of Numbers. EMENDATIONS. The latter part of Art. 23, Ihould be as follows,— — Add the terminate part to the numerator of the given fraftion ; remove the decimal point, and dafh off the repe- tends as before ; then increafe the right-hand place by the difference of the nenu terminate part and that which was added, and the circulate will be correal. Art. 32, fhould have been thus expreffed, — — If the circulates to be added have the places of their repetends in a geometrical progrcffion, of any ratio, t'lie Sum, &c.