GIFT OF Miss e.t.wf University of California • Berkeley Iti TREATISE OF ALGEBRA. WUEREIN THE PRINCIPLES ARE DEMONSTRATED, AND APPLIED IN MANY USEFUL AND INTERESTING ENGIUI- RIES, AND IN THE RESOLUTION OF A GREAT VARIETY OF PROBLEMS OF DIFFERENT KINDS. TO WHICH IS ADDED, THE GEOMETRICAL CONSTRUCTION OF A GREAT NUMBER OF LINEAR AND PLANE PROBLEMS; WITH THE MKTHOD OF RESOLVING THE SAME NUMERICALLY. By THOMAS SIMPSON, F. R. S. THE TENTH EDITION, CAREFULLY REVISED. LONDON : PRINTED FOR J COLLINGWOOD, IN THE STRAND. 1826- i ^ n y^ 10 cu'.'irniT« PRlNTtD BY W. GLENDINNING, 25, IIAITON GAllDBN. TO THE RIGHT HONOURABLE JAME^ ]EA!R1L OF MOIRTOM^ LORD JBERDOUR, Knight of the most. ancient Order of the Thistle, One of the Sixteen Peers of Scotland^ VICE ADMIRAL OF ORKNEY A.ND ZETLAND, President of the Philosophical Society at Edinburgh, AND Fellow of the Royal Society of London. My Lord, ''1L70UR Character will be a sufficient apo- "*- logy for my desiring the honour to in- scribe the following Sheets to your Lord- ship, and your Good^iess will pardon the liberty I take, as it affords me an opportunity of testifying the high respect and esteem with which I am, My Lord, Your Lordship's most devoted, most obedient, and most humble servant, THOMAS SIMPSON. 243266 THE AUTHOR'S PREFACE TO THE SECX>ND EDITION. THE motives that first gave birth to the ensuing Work, were not so much any extravagant hopes the author could form for himself of greatly extending the subject by the addition of a large variety of new improve- ments (though the Reader will find many things here that are no where else to be met with) as an earnest desire to see a subject of such gene- ral importance established on a clear and rati- onal foundation, and treated as a science, capable of demonstration, and not a mysterious art, as some authors, themselves, have thought proper to term it. ^ j t How well the design has been executed, must be left to others to determine. It is ])ossible that the pains here taken, to reduce the funda- mental principles, as well as the more difficult parts of the subject to a demonstration, may be looked upon, by some, as rather tending to throw new difficulties in the way of a Learner, than to the facilitating of his progress. In order to gratify, as far as might be, the inclination ot this class of Readers, the demonstrations are now siven by themselves, in the manner ot Notes (so as to be taken or omitted at plea- sure) : though the Author cannot by any means be induced to think, that time lost to a Learner HREFAtE. which U taken np in comprehending the grounds whereon he is to raise his superstructure ; his progress may indeed, at first be a httle retard- ed ; but the re«Z knowledge he thence acquires will abundantly compensate his trouble and enable him to proceed, afterwards, with certamty and success, in matters of greater difficulty, where authors, and their rules can yield tliem no assistance, and he has nothing to depend upon but his own observation and judgement. This, second, Edition has many advantages over the former, as well with respect to a num- ber of new subjects and improvements, inter- spersed throughout the whole, as in the order and disposition of the elementary parts: in which particular regard has been had to the ca- pacities of young beginners. Ihe Work, as it now stands, will, the Author flatters himself, be found equally plain and comprehensive, so as to answer, alike, the purpose ot^ die lower, and of the more experienced class of Readers. PS The great reputation of Mr. S i m p s o n "s Treatise 0/ Algebra, and the favorable re~ ception it has universally met with since the jirst publication, and which testifies it to be Jhekst elementary work upon the subject, has induced the froprietor to have this Tenth edition carefuUi/ revised and corrected by an eminent matkemafician: he therefore trusts it will be found as ^^orthy tM approbation ofthepublic, as f revised by the Author himself. THE C O N T E N T S. SECTION I. J^OTATION Page l SECTION II. ADDITION 8 SECTION III. SUBTRACTION U SECTION IV. MULTIPLICATION 13 SECTION V. DIVISION 29 SECTION VI. INVOLUTION 37 SECTION VII. EVOLUTION 43 SECTION VIII. TUB REDUCTION OF FRACTIONAL AND RADICAL QUANTITIES ,.. 46 SECTION IX. OF EQUATIONS 57 ) . The reductum of single Equations ibid, 2. The Extermination of unknown Quantities, or the reduction oftico or more equations to a single one 63 SECTION X. OF ARITHMETICAL AND GEOMETRI- CAL PROPORTIONS 70 SECTION XI. THE SOLUTION OF ARITHMETICAL PRO^ BLEMS 7A CONTKNTS. SECTION XII. THE RESOLUTION OF EQUATIONS OF SEVERAL DIMENSIONS Page 131 1. Of the origin and composition of Equations . .ibid. 2. How to know whether some y or all the roots of an Equation be rational, and, if so, what they are » • 134 3. Another way of discovering the same thing, by means of Sir Isaac Newton*s method of divi- sors ; with the grounds and explanation of that method 135 4. Of the solution of cubic Equations according to Cardan , ... 143 5. The same method extended to other higher Equations , , . . 145 6. Of the solution of biquadratic Equations ac- cording to Des Cartes 147 7. The solution of biquadratics by a new method, without the trouble of exterminating the se^ cond term 130 8. Cases of biquadratic Equations that may be re- duced to quadratic ones 1 53 9. The resolution of literal Equations, ivherein the given, and the unknown quantity are alike affected , ... 156 10. The resolution of Equations by the common me- thod of converging series 158 1 1. Another ivay, more exact 162 12. A third method 1 70 13. The method of converging series extended to surd Equations 1 74 14. A method of solving high Equations, when two, or more unknown quantities are concerned in each 177 SECTION XIIL OF INDETERMINATE PROBLEMS 180 SECTION XIV. THE INVESTIGATION OF THE SUMS OF POWERS 201 CONTENTS. SECTION XV. OF FIGURATE NUMBERS ... Page 2J3f 1. The Sums of Series, consisting of the reciprocalsof Jigurate numbers, icit/i others of the like nature 215 2. The sums of compound Progressions, arising from a series of poicers drawn into the terms of a geometrical progression 219 3. The combinations of Quantities 224 4. A demonstration o/Sir Isaac Newton's Binomial theorem 227 SECTION xvr. OF INTEREST AND ylNNUITIBS 229 1. Annuities and Pensions in Ar rear y computed at simple interest . 231 2. The investigation of Theorems for the solution of the various cases in compound interest and annuities #.•..>• •••••» 234 SECTION XVII. OF PLANE TRIGONOMETRY 241 SECTION XVIIL THE yIP PLICATION OF ALGEBRA TO THE SOLUTION OF GEOMETRICAL PROBLEMS 254 1 . An easy icay of constructing^ orfnding the roots of a quadratic equation^ geometrically 267 2. A demonstration ichy a problem is imppssible when the square root of a negative quantity is concerned 272 3. A method for discovering ichether the root of a radical quantity can be extracted 284 4. The manner of taking atcay radical quantities from the denominator of a fraction, and transferring them to the numerator 288 5. A method of determining the roots of certain high Equations^ by means of the section of an angle 301 y AS APPENDIX, Containing the geometrical construction of a large variety of linear^ and plant Problems ': with the manner of resolving thr same numerically .... 313 TREATISE OF ALGEBRA. SECTION I. NOTATION, ALGEBRA is that Science which teaches, in a general manner, the relation and comparison of abstract quantities: by means whereof such Questions are resolved whose solutions would be sought in vain from common Arithmetic. In Algebra, otherwise called Specious Arithmetic^ Numbers are not expressed as in the common Notation, but every Quantity, whether given or required, is com- monly represented by some letter of the alphabet ; the given ones, for distinction sake, being, usually, denoted by the initial letters a, 6, c, d, &c. ; and the unknown, or required ones, by the fmal letters u^ ?c, x, y. &c. There are, moreover, in Algebra, certain Signs or Notes made use of, to shew the relation and dependence of quantities one upon another, whose signification the Learner ought, first of all, to be made acquainted with. The Sign +- , signifies that the quantity, which it is pre* fixed to, is to he added, Thus a -f 6 shews that the number represented by b is to be added to that repre- sented by fl, and expresses the sum of those numbers ; »o that if a was 5, and b 3, then would a -f 6 be 5 -f 3, 2 NO T A T I O iV. or 8. In like manner a -^ b + c denotes the number arising by adding all the three numbers a, b, and c, together. Note, A quantity which has no prefixed sign (as the leading quantity a in the above examples) is always un- derstood to have the sign + before it ; so that a signifies the same as 4- a; and a + b, the same as 4- a f 6. The Sign — , signifies that the quantity which it precedes Is to be subtracted. Thus a — b shews that the quan- tity represented by b is to be subtracted from that repre- sented by a, and expresseth the difference of a and b ; so that, if a was 5 and b 3, then would a — ^ be 5 — 3, or 2. In like manner a + b — c — d represents the quantity which arises by taking the numbers c and d from the sum of the other two numbers a and b ; as if a was 7, b 6j c 5, and c?3, then would a -f b — c — d be 7 f 6 — 5 — 3, or 5. The Notes + and — are usually expressed by the words plus (or more) and minus (or less). Thus, we read, a + 6, a plus b ; and a — b, a minus b. Moreover, those quantities to which the sign -f is prefixed are called po^/^/re (or affirmative) ; and those to which the sign — is prefixed, ?iegative. The Sign x , signifies that the quantities between which it stands are to be multiplied together. Thus axb denotes that the quantity a is to be multiplied by the quantity by and expresses the product of the quantities so multi- plied; and a y: b X c expresses the product arising by multiplying the quant ities ^ , b, and c, continually to- gether: thus, likewise, a -{- b y. c, denotes the product of the compound quant ity a -\- b b y the si mple quantity c ; and a f 6 4- c x a — b -{- cxa -\- c represents the product which arises by multiplying the three com- pound quantities a + b + c,a — 6 + c, and a + c con- tinually together; so that, if a was 5, h 4 , and c 3, then would a -h Z> f c X a — b -{- c x a -{- c be 1 2 X 4 X 8, which is 384. But when quantities denoted by single letters are to be multiplied together, the Sign x is generally omit- ted, or only understood ; and so ab is made to signify (he same as a x ^; and aba, the uarae as a x i x c. N T A T I O N. 3 It is likewise to be observed, that when a quantity is to be multiplied by itself, or raised to any power, the usual method of Notation is to draw a line over the given quantity, and at the end thereof place the Expo- nent of the Power. Thus a -i- bf denotes the same as a + h X a -f b, viz, the second power (or s quare) o f a + b considered as one quan ti ty : thus , al so, ah -t- 6cl denotes the same as ab -\- be x ab -t be y: ab ^ be, viz* the third power, (or cubej of the quantity ab + ic. But in expressing the powers of quantities repre- sented by single letters, the line over the top is com- monly omitted ; and so a^ comes to signify the same as aa OY a x a, and h^ the same as bbb or b x b x 6 : whence also it appears that a^b^ will signify the same as aabbb ; and a^c"^ the same as aaaaaec ; and so of others. The Note . (or a full point) and the word into, are likewise used instead of x , or as Marks of Multipli- cation. Thus a 4- 6 . a + c and a ■{- b in to a -{- c both signify the same thing as a + /; x a 4- c, namely, the product of a + 6 by a + c. The Sign -f- is used to signify that the quantity pre* ceding it is to be divided by the quantity which comes after it : Thus c -f- 6 signifies that c is to be divided by b ; and a -\~ b -^ a — c, that a + 6 is to be divided by a — c. Also the mark ) is sometimes used as a note of Divi- sion ; thus, a -{- b) ah, denotes that the quantity ab is to be divided by the quantity a -\- b; and so of others. But the division of algebraic quantities is most com- monly expressed by writing down the divisor under the dividend with a line between them (in the manner of c a vulgar fraction). Thus — represents the quantity arising by dividing c hy b; and , denotes the quantity arisingby dividing a-{-bhy a — c. Quantities thus expressed are called algebraic fractions ; whereof B 3 4 NOTATION. the upper part is called the numerator, and the lower the denominator, as in vulgar fractions. The sio-n v/~ is used to express the squarej;oot of anv quantity to which it is prefixed: thus n/23 sig- nifies the square-root of 25 (which is 5, because 5 x 5 is 25) : thus also >/ab denotes the square root of ah ; and ""^^ "" ^'"^ denotes the square root of "" ' ^ ' ■ or of d the quantity which arises by dividing ah-^hchy d: but YJ^-JI^ (because the line which separates the d numerator from the denominator is drawn below / ) signifies that the square root of ah + he is to he first taken, and afterwards divided by ^: so that, ifa^was 2, ,,v/rt/>-f-6c, v/36 ,,, ^ 16, c 4, and d 9, then would j— be — oi -^ . but ^^^' is a/ f , or v/4, which is 2. The same mark /, with a figure over it, is also used to express the cube^ or biquadratic root, &c. of any quantity : thus 1/64 represents the cube root^ 64, (Which is 4, because 4 / 4 X 4 is 64Uind x/ «^ + cd the cube root of ah + cd ; also V i6 denotes the biquadratic root of 16 (which is 2, because 2 x 2 x 2 x 2 is \6)"dnd\/ab f cd denotes the biquadratic root of ab V re/ ; and so of others. Quantities thus ex- pressed are called radical quantities, or suids; whej;e- of those consisting of one term only, as /a and n/ ab, are called simple surds ; and those consistingjDf^^^eral terms, or members, as \/a^ — b' and Va^ — ^' + ^c. compound surds. Besides! this way of expressing radical quantities, (which is chiefly followed) there are other methods made use of by different Authors ; but the most com- modioHS of all. and best suited to practice, is that where the root is designed by a vulgat fraction, placed at the ^nd of a line drawn over the quantity given. Accord- NOTATION. 5 in^ to this Notation the square root is designed by the fraction i, the cube root by ^, and the biquadratic root by ir, &c. Thus c^i^ expresses the same thing with \/a^ viz. the square root of a ; and a^ 4- ab ^^ the same as \/a'' 4- cib^ that is, the cube root of o* 4- at • also T'^ denotes the square of the cube root of a ; and a~+n^ the seventh power of the biquadratic root of a -^ z; and so of others. But it is to be observed, that, when the root of a quantity represented by a single letter is to be expressed, the line over it may be ne- glected ; and so a^ will signify the same as a* S and b'^ the same as P 3- or -v/Zj. The number, or fraction, by which the power, or root of any quantity, is thus de- signed, is called its Index, or Exponent. The Mark — f called the Sign of equality J is used to signify that the quantities standing on each side of it are equal. Thus 2 + 3 — 5, shews that 2 more 3 is equal to 5 ; and x —a — 6, shews that x is equal to the dif- ference of a and b. The Note : : signifies that the quantities between which it stands are proportional : As a : b :: c : d, denotes that a is in the same proportion to 6, as c is to c?, or that if a be twice, thrice, or four times, &c. as great as /;, then accordingly is c twice^ thrice, or four times, &c. as great as d. To what has been thus far laid down on the signifi- cation of the signs and characters used in the Alge- braic Notation, we may add what follows ; which is' equally necessary to be understood. When any quantity is to be taken more than once, the number is to be prefixed, which shews how many times it is to be taken : thus 5a denotes that the quan- tity a is to be taken five times ; and 3bc stands for three times bcy or the quantity which arises by multiplying be by 3 : also 7 v/a^ + b^ signifies that \/ a' + P is to be taken 7 times ; and so of others. B 3 « NOTATION. The numbers thus prefixed are called coefficients ; and that quantity which stands without a coefficient is always understood to have an unit prefixed, or to be taken once, and no more. Those quantities are said to be like that are expressed by the same letters under the same powers, or which differ only in their coefficients : thus 36c, bhc, and bbc are like quantities ; and the same is to be understood of the Radicals 21/ ^-^^ii and 7 V^Ali". -Qxxt unlike quantities are tiiose which are expressed by different letters, or by the same letters under different powers : thus 2a6, 2a6c, 5ab^, and ^hd^, are all unlike. When a quantity is expressed by a single letter, or by- several single letters joined together in Multiplication (without any Sign between them), as a, or 2ah, it is called a simple quantity. But that quantity which consists of two or more such simple quantities, connected by the signs 4- or — , is called a compound quantity ; thus a — <2ah + bahc is a compound quantity; w^iereof the simple quantities o, Qah and bahc are called the Terms or Members. The Letters by which any simple quantity is expressed may be ranged according to any order at pleasure, and yet the signification continue the same ; thus ah may be wrote ba ; for ab denotes the product of a by b, and lia the product of 6 by a ; but it is well known, that, when two numbers are to be multiplied together, it matters not which of them is made the multiplicand, nor which the multiplier, the product, either way, coming out the same. In like manner it will appear that flic, ach, bac, bca, cab, and cba, all express the same thing, and may be used indifferently for each other (as will be demonstrated further on' ; but it will be some- times found convenient, in long operations, to place the several Letters according to the order which they obtain in the alphabet. Likewise the several members, or terms of which any quantity is composed, may be disposed according to any order at pleasure, and yet the Signification be no ways'affected thereby. Thus a — Qab -f baV) may be wrote a 4- ba^b — <2ab, or — 2ab + a + ba^b, kr. for all these represent the same thing, viz. the quantity NOTATION. 7 which remains, when, from the sum of a and 5a*5, the quantity ^ab is deducted. Here follow some examples wherein the several Forms of Notation hitherto explained are promiscuously concerned, and where the signification of each is expressed in numbers. Suppose a — Q,h zz 5, and c r= 4 ; then will a« + 3ah — c* zr 36 -1- 90 — 16 = 110, 2a' — 3a'^h ■{- c^ — 432 — 540 -f 64 =: — 44, a^ X a + b '—'iahc = 36 X 11 — 240 zi 156, a^ , 216 ^ ^-^-^+c' =—+,6 = 12 +16 ^28. \/<2ac + c^ (or 2ac-|-c1^) - v/64 = 8 (for 8 x 8=: 64), 2/;c . 40 = = 2 + — = 7, a'—s/ h"" - ac _ 36 -~l __ £5 _ oa^^yip- + ac "" 12—7 "" 5 "" * v/6^ — cfc -h v/2a!C + c» := 1 + 8 =: 9, v/ ja — ac 4- \/ 2ac + c^ = v/25 — 24 4- 8 = 3. This method of explaining the signification of quan- tities [ have found to be of good use to Young Begin- ners: And would recommend it to Such, who are desirous of making a Proficiency in the Subject, to get a clear idea of what has been thus far delivered, before They proceed farther. B 4 [ 8 ] SECTION ir. ADDITION. ADDITION, in Algebra, is performed by connect- ing the quantities by their proper signs, and join- ing into one sum such as can be united : For the more ready effecting of which, observe the following Rules: 1°. 7/, in the quantities to he added, there are terms that are like and have all the same sign, add the coefficients of those terms together, and to their sum adjoin the letters common to each term, prefixing the common sign. Thus 5a And 5a f 7b Also 5a — 7h added to 3a added to 7a f- 3h added to 7a — 3^ makes 8a. makes isa + loi. makes I2a — lob. Hence C 2 \/ ab -^ 7 \/ he ^^^ the^-f^— ^^ likewise < 3 \/'ah -\- 2 \/l^ ^^^^^ ^^ \ a ~7~ the sum of ( 6 \/ ab -\- g \/~Tc will be \\ \/~ah ^isv^Tcl c willbeli-l^ 2°. IVhen in the quantities to he added, there are like terms, tvhereof sorne are affirmative and others negative, add together the affirmative terms f if there he more than one) and do tlie same by the negative ones , then take the difference of the two sums (not regarding the sig7isj by subtracting the coefficient of the lesser from that of the greater y and adjoining the lexers common to each; to which difference prefix the sign of the greater. The Reasons on which the precedino- Operations are grounded, will readily appear by reflecting a little on the nature and signification of the quantities to be added ; For, with rej?:ard to the first example (where 3a is to be added to 5a) it is plain, that three times any quantity whatever, added to five times the same quantity, must make eight limes that quantity : 'J'herefore 3a, or three times the quantity denoted by a, being added to 5a, or ADDITION. Examples of this Rule may be as follow : .1. 12« — hh 2. — Zah -^^ bhc — 3a f 26 4- lah — ghc Sum 9« — 3^. Sum 4a/) — 46c. 3. Gah \- \9hc — Scd 4. 5 \/«^' — T \/hc -i-8c? — 7^/> — 95c4- 3cc? 3v/a6 + 8v//^ — 12^ — 2a5 — bhc + I2c(i 7 v/a/> + 3 \/^ c + 9c? Sum— 3rt/> — 26C -+ 7cc?. Su ml5\/a^ f 4 \//;c 4-5) we are to consider, that, if the two quantities to be added together had been exactly 5a and 7a ? which are the two leading terms) the sum wou'd, then, have been just I2a : but, since the for- mer quantity wants 76 of 5a, and the latter 3b of 7a, their sum must, it is evident, want both 7b and 3h of 1 2a ; and therefore be equal to 12a — 105, that is, equal to what remains, when the sum of the defects is deduct- ed. A nd by the very same way of arguing, it is easy to conceive that the sum, which arises by adding any Bumber of quantities together, will be equal to the sun4 10 ADDITION. In the last example, and all others, where fractional and radical quantities are concerned, every such quan- tity, exclusive of its coefficient, is to be treated in ^11 respects like a sioiple quantity expressed by a single letter. 3°. When in the quantities to be added, there are Terms without others like to them, write them down ivith their proper signs. Thus a + 2b And aa -f bb added to 3cj\-d added to a 4- b makes a + 26 + 3c +- (/. makes aa + bb-\-a-{-b. Here folJov^r a few examples for the Learner*s exer- cise, wherein all the three foregoihg rules take place promiscuously. 1. 2aa + Sab + See 4- d^ baa — lab -f bcc — d^ — 2aa + Aah + Sec 4- 30 Sum baa * + 1 6cc ^- c?' — c?^ -4- 30. b \/ ax — B\/ ax -^ 6\/ ax — 2. S\/ aa — XX + 12\/ aa + 4xx \b \/ aa — XX — 8 v/ aa + 4xx T s/ aa — XX -V 10 \/ aa 4- ^xx Sum 19 \/ dx * 4- 14v/ aa-f 4xx, 2tt* — Sab 4- qP — 3a* 4c^ ^<£b^ j^bab 4- 100 ^Oab 4- 160' — be — 80 Sum 13a* 4- 22aZ> 4- Sb^ 4- a" — c^ -+- 20 — be. of all the affirmative Terms diminished by the sum of all the negative ones {considered independent of their signs) ; from whence the reason of the second general Rule is apparent. As to the case where the quantities are unlike, it is plain that such quantities cannot be united into one, or otherwise added, than by their signs; thus, for example, let a be supposed to represent a Crown, and b a shillmg ; then the sum of a and b can be neither 2a nor 26, that is, neither two crowns nor two shillings, but one crown plas one shilling, or a -\- If. r 11 ] SECTION in. SUBTRACTION. CiUBTRACTION, in Algebra, is performed by ^ cha?igvig all the Sis^ns of the Subtrahend for con- ceiving them to be changed j and then connecting the quantities, as in addition. Ex. 1. Froiii Sa + 5h Ex. 2. From 8a + 5h take ba -\- 3h take ba — 3 6 Rem. 3a \- 26. Rem. 3a + 86. Ex. 3. From 8a — bh Ex. 4. From 8a — bb take ba 4- Sb take ba — 3b Rem. 3a — 8A. Rem. 3a — 26. In the second example, conceiving the signs of the subtrahend to be changed to their contrary, that of 3b becomes 4- ; and so the signs of 3b and bb being alike, the coefficients 3 and 5 are to be added together, by case 1 of addition. The same thing happens in the third example ; since the sign of 36, when changed, is — , and therefore the same with that of bb. But in the fourth example, the signs of 3b and bb, after that of 3b is changed, being unlike, the difference of the coefficients must be taken, conformable to case 2 in addition. Other examples in Subtraction, may be as follow ; From sOQcT \ bbc — 7«« t;rom 7 \/ax -h 9 v/^ take 12aT — 3bc — 5aa take — 5 s/ax + 12 s/by Rem. Sax + she — 2aa. Rem. 12 \/ ax — 3^ by. From 6 \/ aa — xx -f 10 s/ a^ — a?^ — 7 \/ ^^V ^ c take 9 v/ aa — a-x — . 15 \/ a^ — 0?^ — 9 y aa c Rem. — 3v/aa — xx + ^b \/~ a^ — x^ -\- 2 y aa — « c 12 SUBTRACTION. From 7a' - ~ -h 6 \/ -^' + d take c* + ■ Y ' 1- ^ Rem. 6a^ — -l^+7v/-^+c/- In this last example the quantity a* in the sub- trahend, being without a coefficient, an unit is to be understood; for la* and a^ mean the same thing. The like is to be observed in all other similiar cases. The Grounds of the general rule for the subtraction of alfifebraic quantities may be explained thus : Let it be heje required to subtract ba — 3h fron) 8a f 5h {as in ex. 2.)« It is plain, in the first place, that if the affirmative part 5a were alone to be subtracted, the remainder would then be 8a + bb — 5a ; but, as the quantity actually proposed to be subtracted is less than 5a by 36, too much has been taken away by 36 ; and therefore the true remainder will be greater than 8a -+• 56 — 5a by 36; and so will be truly expressed by 8a + 56 — 5a + 36 : wherein the signs of the two last terms are both contrary to what they were given in the subtrahend; and where the whole, by unitmg the like terxns^ is reduced to 3a 4- 86, as in the example. [ 13 3 SECTION IV. MULTIPLICATION, BEFORE I proceed to lay down the necessary rules for multiplying quantities one by another, it may be proper to premise the following particulars, in order to give the Learner a clear idea of the reason and certainty of such rules. First, then, it is to be observed, that tvhen several quantities are to be multiplied contintially together, the re- sult, or product, will come out exactly the same, multiply them accordino; to what order you ivilL Thus a x 6 x c, a >c c y. b, b X c X a, ^c. have all the same value, and may be used indilferently : To illustrate which we may suppose a — 2, b — 3, and c th 4; then will ax^xczi 2X3X4 = 24 ;aXcx6:r2X4X3zi 24; and 6xcxa=:3X4X2 = 24. Secondly. If any member of quantities be multiplied continually together, and any other number of quantities be also multiplied continually together, and then the two products one into the other, the quantity thence arising will be equal to the quantity that arises by multiplying all the proposed quantities continually together. Thus will abc y.de — a x 6 x c x c? x e; so that, if a was — ^,bzz 3, c — A, d — 5, e — 6, then would abc X (/e = 24 x 30 = 720, and axb Acxdy^e ~ 2x3x4X5x6 z= 720. The general Demonstrations of these observations is siven below in the notes. The following demonstrations depend on this Prin- ciple, i/i^^ if two quantities, whereof the one is n times as great as the other (n being any number at pleasure), be multiplied by one and the same quantity, the 'product, in the one case, icill also be n times as great as in the other. The greater quantity may be conceived to be divided into n parts, equal, each, to the lesser quantity ; and the product of each part (by the given multiplier) will 14 MULTIPLICATIOX. The multiplication of algebraic quantities may be considered in the seven following cases. be equal to that of the said lesser quantity ; therefore the sum of the products of all the parts, Avhich make lip the whole greater product, must necessarily be n times as great as the lesser product, or the product of one single part, alone. This being premised, it will readily appear, in the first place, that 6 x « and a a h are equal to each other : For, b y. a being b times as great as 1 x a (because the multiplicand is b times as great J it must therefore be equal to 1 x a (or a], repeated b times, that is, equal to a x b, by the defimtioii of multiplz-. cation. In the same manner, the equality of all the variations, or products, ate, hac, acb, cab, bca,cba (where the num- ber of factors is 3) may be inferred : for those that have the last factors the same fichich I call of the same class J are manifestly equal, being produced of equal quantities multiplied by the same quantity : And to be satisfied that those oi different classes, as abc and acb, are like- wise equal, we need only consider, that, since ac x b is c times as great as a x i (because the multiplicand is c times as great) it must therefore be equal to a x b taken c times, that is, equal to a x b x Cy by the definition of midtipUcation, Universally, If all the Products, when the number of factors is w, be equal, all the products, when the number of factors is « + 1, will likewise be equal : for those of the same class are equal, being produced of equal quantities multiplied by the same quantity : and to shew that those of different classes are equal also, we need only take two Products which dither in their two last factors, and have all the preceding ones according to the same order, and prove them to be equal. These two factors we will suppose to be repre- sented by r and s, and the Product of ali the preceding ones by/); then the two Products themselves wiii be represented by prs and jo^r, which are equal.^y cusc i» MULTIPLICATION. 15 1°. Simple quantities are multiplied together by multi* plying the coefficients one into the other y and to the product annexiiig the quantity which, according to the method of notation, expresses the product of the species; prefixing the sign + or — , according as the signs of the given quantities are like or unlike. Thus 2a Also Qab And lladf mult, by 3h mult, by be mult, by lab makes 6ab, makes 30ahc. makes llaabdf Thus, by way of illustration, abcde will appear to be = abced, &c. For, the former of these being equal to every other product of the class, or termination e {by hypothesis and equal multiplication), and the latter equal to every other Product of the class, or termination d; it is evident, therefore, that all the Products of different classes, as well as of the same class, are mutually equal to each other. So far relates to the first general observation : It remains to prove that abed x pqrst is=ax6xcx d X p X q A r X s X t. In order to which, let abed be denoted by x, then will abed x pqrst be denoted by a; X pqrst or pqrst x a? (by case 1), that is, by p X q X r X s A t A X; which is equal to x x p x q xrxsxt, or axbxcxdxpxqxrxsxt, by the preceding Demonstration, The Reason of Rule 1 depends on these two Gfeneral Observations : for it is evident from hence, that 2a x 36 {in the first example) is zz 2 xaxsxbzz^x 3Xaxb=:6xaXb — Gab : And, in the same manner, lladf x Tab (in the third example) appears tobezrlj XaxdxfXT xaxb— II X IX. aXaxbxdxf— TTX aabdf ~ 11 aahdf. But the grounds of the method of proceeding may be other- %vise explained, thus : It has been observed that ab (according to the method of notation) defines the pro- duct of the Species a, b (in the first example), therefore the product of a by 3b, which must be three times as great (because the multiplier is here three times as 16 M U LT I PL I C A T I O N. In the preceding examples all the products are affirmatirej the quantities gi\Qw to be iTiiiltif)lied being so ; but, in those that follow, some are affi'rmatice, and others negative, according to the different cases speci- fied in the latter part of the rule; whereof the reasons will be explained hereafter. Mult. + 5a Mult. — 5a Mult. — 5a ^y — ^^^ l^y + 6 b by — 6f) Prod. — 30ab, Prod. — 30^/;. Prod. 3()«6. Mult. + 7 \/ £za? Mult. — 7a \/ aa \- ocx by — 5 \/ cy by — Qh %/ aa — yy Pr0._35 X\/ 6ir/v/c^.Pro. -f42a6 as/ gg-^xx /\/ au-^yy. In the two last examples, and all others, where radical quantities are concerned, every such quantity may be considered, and treated in all respects as a simple quantity, expressed by a single letter ; since it is not the Form of the expression, but the value of the quantity that is here regarded. 2°. A Fraction is multiplied, by multiplying the nume* Tatar thereof by the given multiplier^ and making the pro* duct a numerator to the given denominator, rr,, a . ac . Sac , , Qancd Thus — X c makes -— /-; also—, x 2aa makes—-; — ; great), will be truly defined by 3ah, or ab taken three times-: but since the product of « by 3b appears to be 3fl6, it is plain that the product of 2a by 36 must be twice as great as that of a by 3i, and therefore will be truly expressed by Qab. Thus also, the product of the Species ab and c (in the second example) being abc (by bare notation) it is evident that the product of 6a6 by c will be truly defmed by Qahc, or ahc six times taken, and consequently the product of Qab and 5c, by 30abc, or 6uhc taktn five times, the multiplier here being five times as great. 1 helleasonof Rules'* may be thus demonstrated: Let the numerator of any proposed fraction be denoted by A, MULTIPLICATION. 17 likewise^--- x 7\/ax make3 7—; lastly ^ X Qab makes — \/ua 4-a\r 3**. Fraction}! are multiplied into one another by muUiplyins[ the numerators inp^ether for a new mimerator^ and the denominators together for a new denominator. Thnc ^ c _ ac <2ah bad lOa^bd lOa'v/7 , SflV'Ty" 5h\/aa + 070? — — ; and — 7-==^ X : = 3bbc y/ah a + z a -f « X \/ab the denominator by B, and the given multiplicator by C: AC A AC then, I say, that —^ is equal to ~ x C. For since -^ denotes the quantity which arises by dividing AC by B, and — the quantity which arises by dividing A by B, it is evident that the former of these two quantities must be C times as great as the latter (because the dividual is C times as great in the one case as in the other) and there- fore must be equal to the latter C times taken, that is, AC A -^ must be equal to z^ x C, as was to be shewn. The Reason of Rule 3^ will appear evident from the preceding demonstration of Rule 2°. For it being A AC there proved that g" x C, is equal to -g-, it is ob# A C AC tious that g- X g- can be only the D part of -g-; be- 18 multiplic;ation. 4 . Surd quantities under the same radical sign are multiplied like rational quantities, only the product must stand under the same radical sign. Thus, \/J X \/'5 = \/35 ; Va x \/S = \/ab ; l/Tbc X K^5ad n K/Zbahcd; 3 /oS X 5v/c= 1 3\/^; Sfl t/icp X 3h y/~bax {— ^ah x v^icy x \/ bax) ?^^ . / T04^ cause, -g, the multiplier here, is but the D part of AC the former multiplier C : But -^ is also equal to the AC D part of the same -^ ; because its divisor is D times AC as great as that of -:^ ; therefore these two quanti- A C AC ties, ^ X Y) ^"^ Tvn ^^^"S *^*^ ^^^® P^^* ^^ ^^® and the same quantity, they must necessarily be equal to each other ; ichich was to be proved. As to Rule 4° for the multiplication of similar radical quantities, it may be explained thus : Suppose VA and v/B to represent the two given quantities to be multiplied together; let the former of them be denoted by a, and the latter by b, that is, let the quantities represented by a and b be such that flajnay be == A, and bb — B ; then the product of V'A by \/B, or of a by b, will be expressed by ab^ and its square by ab x ab: but ab x ab is = a x 6 X a X 6 zi aa X bb {hy the general observations premised at the beginning of this section) : whence the square of the product is likewise truly expressed by aa x bby or its equal A x B ; and consequently MULTIPLICATION. 19 5*^. Powersy or roots of the same quantity are muU tiplied together, by adding their exponents: But the exponents liere understood are those defined in p. 5, where roots are represented as fractional powers. Thus, x^ X oc^ is- x^; a \- z\^ x a + zY = a + z\^ ; a-' X x^ z: x^ ^ zz x"^ ; and a?^ x a:^ =: x* = x ; .2 , t .1 also aa + zzr x aa + zz\^ is — aa -h zz\ = aa + ss; 1 »i 1^ »i -4- 4- 4-* and c -\- y\^ X c -{ y\^ — c ^ y\^ ^ = c -f y,^ , the product itself, by \/A x B, that is by the quan- tity which, being multiplied into itself produces A X B. In the same manner the product of y/X x V^B will appear to be ^AB : for if V^A be denoted by a, and i/B by b; or, which is the same, if aaa — A, and 566 — B ; then will \/a X \/^ — a x h (or ab) and its cube ~ ab A ab X ab zz aaa x bbb ~ AB{by the aforesaid observations) when ce th e product itself wiil evidently be expressed by V^AB. * The Grounds of these Operations may be thus explained. First, when the exponents aie whole num- bers, as in example 1 , the demonstration is obvious, from the general observations premised at the begin- ning of the section : For, by what is there shewn, a?* X x^y or XX x xxx is -xxxxxxxxxz x^ (by No" tationj. But in the last example, where the exponents are fractions, let c 4- yF be represented by x; that i&, let the quantity x be such, that xxxxxxxa CO X X, or x^ may be equal to c 4- y ; so shall c 4- y]^ be expressed by x^ ; because, by what has been already shewn, x^ x x^ is — a?** : and in the same manner, will G -\r y\^ be expressed by x*; because x» x a?' x ** ii C 2 20 MULT I P^LI CATION. 6°. A Compound quantity is multiplied by a simple me, by muliipbjing every term of the multiplicand by the multiplier. Thus; a -I- 25 — 3c Also a'^ ^ ba^yx ■\- yh mult, by 3a mult, by 8c makes 3a^^ Qab — Qflc; makes 8a'c-40acv/x + 566c; And 5fl*— 8a6 + 6ac—- 75c + 126' — 9c» mult, by 3a 5c makes l5a^5c-24fl*5'c+18a*5c'-21o5=c''+36a5^c-27fl5c'. likewise — x^. Therefore c + y]^ x c ir yV is — a:^ X x^ — x^ — the fifth power of c + «/ F ; which i&i c + ^/fj ^^ Notation, To explain the Reason of the two last Rules, let it he^Jirstj proposed to multiply any compound quantity, as a -f 5 — c — ^, by any simple quantity /; and, I say, the product will be a/ + bf — cf — df. For, the product ot the affirmative terms, a + 5, will be af-^- bf, because, to multiply one quantity by ano- ther, is to take the multipHcand as many times as there are units in the multiplier, and to take the whole multiplicand (a + b) any number of times {/), is the same as to take all its parts {a, b) the same num- ber of times, and add them together. Moreover, see- ing a + b — c — c? denotes the excess of the affirmative terms (a and b) above the negative ones (c and c?,) therefore, to multiply a -h 6 — c — c? by y; is only to take the said excess / times; but / times the excess of any quantity above another is, manifestly, equal to /times the former quantity, mi?ius f times the latter; but/ times the former is, here, equal to qf -f hf (by what has been already shewn), and / times the latter (for the same reason) will be equal to cf + df, and therefore the product of a 4- 5 — c — d by/, is equal to af + bf— cf — df; as was to be proved. Hence it appears, that a compound quantity MULTIPLICATION. 21 7*. Compound quantities are multiplied into one mnother, by multiplying every term of the multiplicand by each term of the multiplier, successively, and coum necting the several products thus arising with the signs of the multiplicand, if the multiplying term he affir^ mative, but loith contrary signs, if negative^ Thus the product of 5a + 30? multiplied by 3a + 2x is multiplied by a simple affirmative quantity, by multiplying every term of the former by the latter, and connecting the term thence arising with the signs of the multiplicand. But to prove that the Method also holds when both the quantities are compound ones, let it be^ now, pro- posed to multiply A — B by C — D; then, I say, the product will be truly expressed by AC — BC — AD 4- BD. For, it has been already observed, that to multiply one quantity by another, is to take the multiplicand as many times as there are units in the multiplier ; and therefore, to multiply A — B by C — D is only to take A — B as many times as there are units in C — D : Now (according to the method of multiplying com- pound quantities) I first take A — B, C times (or mul- tiply by C) and the quantity thence arising will be AC — BC (by ivhat is demonstrated above). But, I was to have taken A — B only C — D times ; there- fore, by this first Ope^ration,.I have taken it D times too much ; whence, to have the true product, I ought to deduct D times A — B from AC — BC, the quantity thus found ; but D times A — B fby what is already proved) is equal to AD — BD : which subtracted from AC — BC, or wrote down with its signs changed, gives the true product, AC — BC— AD -f ^T> as was to he demonstrated. And, universally^ if the sign of any proposed term of the multiplier, in any case whatever, be affirmative, it is easy to conceive that the required c 3 22 MULTIPLICATION. Likewise the product of a^ + a^b + ab^ + b^ by a — b , a* + a'b + a*b' -1- «/>' / ^ a3^ _^a^l)'i_ah' — b*s Which, by striking out the terms that destroy one another, becomes a* — b\ product will be grenter than it would be if there were no such term, by the product of that term into the whole multiplicand ; and therefore it is, that this pro- duct is to be added, or wrote down with its proper signs, which are proved above to be those of the mul- tiplicand. But if, on the contrary, the sign of the term, by which you multiply, be negative; then, as the lequired product must be less than it would be, if there were no such term, by the product of that term into the whole multiplicand, this product, it is manifest, ought to be subtracted, or wrote down with contrary signs. Hence is derived the common Rule, that like Signs produce 4 , and unlihe Signs — , For, first, if the signs of both the quantities, or terms, to be multiplied are afrirmative fand therefore like) it is plain that the sign of the product must like- wise be alfirmative. Secondly, also if the signs of both quantities are negative (and therefore still like), that of the product will be affirmative, because contrary to that of the mid" tiplicand, by what has been just now prated. Thirdly, but it the sign of the multiplicand be affir- mative, 9nd thut of the multiplier negative (and there- fore unlike y the sign of the product will be negative, because contrary to that of the multiplicand. Lastly, if the sign of the multiplicand be negative and that of the multiplier affirmative, (and therefore still unlike) the si^n of the product will be negative, because the same with that of the multiplicand. And thehC four are all the Cases that can possibly happen with regard to the variation of signs. MULTIPLICATION. ^ Other examples in Multiplication, for the Learner's exercise, may be as follow; from which he may (if he pleases) proceed directly to Division, by passing over the intervening Scholium. I. Multiply by oc^ -f xy + y* a^ — xy ■\- y* x^ -V- x^y -h xy^ — x^y — x'^y- — xy^ + xhf -I- xy' 4- y' product a:* * + xY * + ^*. 2. Multiply by 2a^ — 3^0? + 43?'^ 5^2 — Qax — 2a?« lOa*— loa^o? 4- 20a V — \<2a^x 4- ISa^'x — 24aa?' — 405 a?* -f Qax^ — 8a?* product 10a*— 27a^^4- 34a*x^ — \Bax^ — Sx^, 3. Multiply by 3a — 2b + 2c 2a — 46 -f 5c 6aa — 4:ab 4- 4ac — 12ab 4- 866— 86c 4- 15ac-io6c4- locc product 6a« — I6ab 4- l9ac4-866— 1 86c4- lOcc. 4. Multiply by a3_3^^5 ^ 3a6^ — 6^ a^ — Qab 4- 6^ a5_3a*6 + Sa'b'—aV)' ■^Qa'^b 4- 6a^6' — 6a*6^4-2a6* + ^36'^ — 3a^6^4- 3ab'—b^ product a? — 5a'»6 + 10a^6 — lOa'b^ \5ab'—b^. scholium; The Manner of proceeding in referring the reasons of the different cases of the signs to the multiplication of compound quantities, may perhaps be looked upon as indirect, and contrary to good method ; according to c 4 24 MULTIPLICATION. which, it may be thought, that these reasons ought to have been given before, along v^ith the rules for simple cuantiiies, as it is the way that almost all Authors on the subject have followed. But. however indirect the method here pursued may seem, it appears to me the most clear and rational ; and I believe it wili be found very difficult, if not impossible, without explaining the rules for compound quantities first, to give a Learner a distinct Idea how the product of two simple quantities, with negative signs, such as — b and — c, ought to be expressed, when they stand alone, independent of all other quantities; And I can- not help thinking farther, that the difficulties about the si^ns, so generally complained of by Beginners, have been much more owing to the manner of ex- plaining them, this way, than to any real intricacy in the subject itself; nor will this opinion, perhaps, appear ill grounded, if it be considered that both — a and — h^ as they stand here independently, are as much impossible, in one sense, as the imaginary surd quan- tities \/ — b and \/ — c; since the sign — , according to the estab ished Rules of notation, shews that the quantity to which it is prefixed, is to be subtracted ; but, to subtract something from nothing is impossible, and the notation, or supposition of a quantity less than nothing, absurd and shocking to the imagination: And, certainly if the matter be viewed in this light, it would be very ridiculous to pretend to prove, by any shew of reasoning, what the product of — b by — c, or of %/ — b by \/ — c, must be, when we can have no Idea of the value of the quantities to be multiplied. If, indf ed, we were to look upon — b and — c as real quantities, ttie same as represented to the mind by b and c f which cannot be done consistently, in pure Alge- bra, where magnitude only is regarded) we might then attempt to explain the matter in the same manner that some others have done; from the consideration, that^ as the sign — is opposite in its nature to the sig^n +, it ought therefore to have in ail operations an oppo* site eii'ect ; and consequently, that as the product when the sign + is prefixed to the multiplier, is to be added^ MULTIPLICATION. %5 80, on the contrary, the product, when the sign — • is prefixed, ought to be subtracted. But this way of arguing, however reasonable it may appear, seems to carry but very little of science in it^ and to f'-ill greatly short of the evidence and conviction of a demonstration : nay, it even clashes with First Principles, and the more established Rules of notation ; according to which the signs +• and — are relative only to the magnitudes of quantities, as composed of diffe- rent terms or members, and not to any future operations to be performed by them : Besides, when we are told that the product arising from a negative multiplier is to be subtracted, we are not told what it is to be sub- tracted from ; nor is there any thing from whence it can ^e subtracted, when negative quantities ^re indepen- dently considered. And farther, to reason about oppo- site effects, and recur to sensible objects and popular considerations, such as debtor and creditor, &c. in order to demonstrate the principles of a science whose object is abstract Number, appears to me, not well suited to the nature of science, and to derogate from the dignity of the subject. It must be allowed, that in the application of Alge- bra to different branches of mixed mathematics, where the consideration of opposite qualities, effects, or positions can have place, the usual methods have a better foundation; and the conception of a quantity absolutely negative becomes less difficult. Thus, for example, a line may be conceived to be produced out, both ways, from any point assigned ; and the part on the one side of that point being taken as positive, the other will be negative. But the case is not the same in abstract Number; whereof the beginning is fixed in the nature of things, from whence we can proceed only one way. There can, therefore, be no such things as negative numbers, or quantities absolutely negative in pure Algebra, whose Object is Number, and where every multiplication, division, &c. is a multiplication, divi- sion, &c. of Numbers, even in the application thereof: For, when we reason upon the quantities themsehes^ and not upon the numbers expressing the measures of ^ MULTIPLICATION. them, the process becomes p7irely geometrical, whatever symbols may be used therein, from the algebraic nota- tion; which can be of no other use here than to abbreviate the work. However, after all, it may be necessary to shew upon what kind of evidence the multiplication of negative, and imaginary quantities is grounded, as these some- times occur, in the resolution of problems : In order to which it will be requisite to observe, that, as all our reasoning regards real, positive quantities, so the alge- braic expressions, whereby such quantities are exhi- bited, must likewise be real and positive. But, when the problem is brought to an equation, the case may indeed be otherwise ; for, in ordering the equation, so much may be taken away from both sides thereof, as to leave the renlaining quantities negative ; and then it is, chiefly, that the multiplication by quantities absolutely negative takes place. Thus if there were given the equation a — "T — ^ (in order to find x) ; then by subtracting the quantity a from each side thereof, we shall have r — c —- a; o which multiplied by — b, according to the general Rule, ,T ^ives X =: — cb -\- ab; that is r by — b will give -}- X; chy ' — 6, — cb; and — a by — b, + ab ; which appear to be true ; because the products being thus ex- pressed^ the same conclusion is derived, as if both sides of the original equation had been first increased by -T c, and then multiplied by b ; where both the mul- tiplier and multiplicand are real, affirmative quantities, and where the whole operation is, therefore, capable of a clear and strict demonstration : but then it is not in consequence of any reasoning I am capable of forming rf> about Y and — 5, or about + c and — J, considered MULTIPLICATION. ST independently, that I can be certain that their product ought to be expressed in that manner. So likewise, if there were given the equation a — ~T~ ~ c\ hy transposing a and taking the square root on both sides we shall have \/ — "t~ — V^ — ^' ^^^ this tnultiplied by \/ — /), will give v/J^ (or x) zz \/ — cb 4- ah: which also appears to be true, because the result, this w^y, conies out exactly the same, as if the operations, for finding t, had been performed alto- gether by rea/ quantities : But notwithstanding this, it is not from any reasoning that I can form, about the multiplication of the imaginary quantities "^ ir and \/ — b, &c. considered independently, that T can prove their product ought to be so expressed ; for it would be very absurd to pretend to demonstrate what the product of tw<» express-ions must be, which are impossible in themselves, and of whose values we can form no idea It indeed seems reasonable, that the known rules for the si^ms, as they are proved to hold in all cases whatevc, where it is possible to form a demonstration, should al«o answer here: But the stronLjest evidence we can have of the truth and certainty of conclusions derived by means of negative and imaginar} quantities, is, the exact, and constant agreement ot such conclusions with those determined from more demonstrable methods whereiu no such quantities have place. In the foregoins: considerations, the negative quan- tities — 6, — c, &c. have been represented, in some cases, as a kind of imaginary, or impossible quan-^ tities; it may not, therefore,* be improper to remark here, that s'jch imaginary quantities serve, many times, ^s much to discover the impossibility of a problem, as nnaginaiy surd quantities: for it is plain that, in all question-; relating to abstract Numbers, or such wherein magnitude onlt^ is regarded, and 28 MULTIPLICATION. where no consideration of position, or contrary va- lues, can have place; I say, in all such cases, it is plain that the solution will be altogether as impos- sible, when the conclusion conies out a negative quan- tity, as if it were actually affected with an imaginary surd ; since, in the one case, it is required that a number should be actually less than nothing; and in the other, that the double rectangle of two numbers should be greater than the sum of their squares; both which are equally impossible: But, as an in- stance of the impossibility of some sort of questions, when the conclusion comes out negative, let there be given, in a right-angled Triangle, the sum of the hypothenuse and perpendicular =: a, and the base — 6, to find the perpendicular; then (by what shall here- after be shewn in its proper place) the answer will come out — "^ — , and is possible, or impossible, according as the quantity — is affirmative or negative, or as a is greater or less than b; which will manifestly appear from a bare contemplation of the problem: and the same thing might be instanced in a variety of other examples. C 29 ] SECTION V* DIVISION. DIVISION in species, as in numbers, is the con- verse of multiplication, and is comprehended in the seven following cases. 1°. When one simple quantity is to be divided by another, and all the factors of the divisor are also found in the dividend, let those factors be all cast off or expunged, then the remaining factors of the divi^ dend, joined together, will express the quotient sought* But it is to be observed that, both here and in the suc- ceeding cases, the same rule is to be regarded in rela- tion to the signs, as in multiplication, viz, that like Signs give +■, and unlike — . It may also be proper to observe, that, when any quantity is to be divided by itself, or an equal quantity, the quotient will be ex- pressed by an unit, or 1. Thus tf -f- a, gives l ; and ^ah -f- 2ah gives 1 ; moreover 3ahcd -f- ac, gives 3hd ; and \()bc -f- Sh, gives 2c : for the dividend here, by resolving its coefficient into two factors, becomes 2X8 xbxc; from whence casting olf 8 and b, those common to the divisor, we have 2 x c, or 2c. In the same man- ner, by resolving or dividing the coefficient of the dividend by that of the divisoi', the quotient will be had in other cases : Thus, QOabc divided by 4c, gives Sab ; and -^ 5labV xy y.\/ XX -^ yy, divided by — 1 7a^xy, gives f 36 \/xx + yy. ' ' ..I.I I ,1.. I II I ■ >■ The first Rule, given above, being exactly the con- verse of Rule 1° in the preceding section, requires no other demonstration than is there given. The second Rule (as well as those that follow hereafter upon Frac- tions) depend on this principle, that, as many times as any one proposed quantity is contained in another, just so many times is the half^ third, fourth, or any other assigned part of the former, contained in the half, third, fourth or other corresponding, part of the latter; and in 20 DIVISION. Q\ But if all the factors of the divisor are not to he joitnd in the dividend, cast off those fif any such there he) that are common to both, and irrite down the remainiigjactors of the divisor, joined together, as a denominator to those of the dividend ; so shall the fraction thus arising express the quotient sought. But if, by proceedina thus, all the factors in the dividend should happen to go off, or vanish, then an unit will be the numerator of the fraction required. Thus, abc divided by bed, gives -r : And iGa^hx^ divided by 8ahcx^, gives : Likewise 27^6 \/^ divided by 9 a' x/x^, gives And 8ab\/ay divided by iGa^l^v^ay, gives 2a just so many times likewise is the double, triple, quad- ruple, or uny other assigned multiple of the former contained in the double, triple, quadruple, or other cor- responding multiple of the latter. The Demonstration of this Principle (though it may be thought too obvious to need one) may be thus: Let A and B represent any two proposed quantities, nnd AC and BC their equimuU tiples (or, let AC and BC be thtrtwo quantities, and A AC A and B their like parts) : I say, then, that -r-p = -^ : ' AC For the multiple of -5-7^ by BC is manifestly zz AC; A A and -r- X BC, the multiple of ~ by the same BC is A/BC . , . ,^. ,. ^. , ACB , ., = — rr — (by rule 2 in viultiplicationj— — -r — {vid, p* 14 and 15) =r AC : Therefore, seeing the equimultiples of the two proposed quantities are the same, the quan- titiea^— -SaV+Sfl V — x^ (a^ + Sa'x + Saaj^-f /j.? + 3a^a7— 6a*a?* 4- a^^^ + 3a'^x' + 3a^a?^ — Qd^x^ 4 Qfl^Jg' — 3 ax^ o D 2 36 DIVISION. But it is to be observed, that it is not always that the work will terminate without leaving a remainder ; and then this method is of little use ; and in all these cases it will be most commodious to express the quo- tient, in the manner of a fraction, by writing the divisor under the dividend, with a line between them, as has been shewn in the method of notation. It would be needless to offer any thing by way of de- monstration to the two last rules, the grounds thereof being already sufficiently clear from what has been delivered in the last section, and the rules themselves nothing more than the converse of those there demon- strated. — I shall here shew the reason why, in division (as well as multiplication) like signs produce -f , and urdike — . In order thereto it must first be observed, that according to the nature of division, every quotient whatever multiplied by the given divisor, ought to pro- duce the given dividend ; whence it is evident, 1 . That ■\- a) -\- ah (■\- h \ because 4- a mult, by + h, gives + ah ; 2. That + «) — ah [ — h ; because -f a mult, by — b^ gives — ah ; 3. That — a) -\- ah ( — h ; because — a mult, by — 6, gives + ah ; i. That — a) — (z6 ( + h\ because — a mult, by -f b, gives — ah : And these four, are all the cases that can possibly happen in respect to the variation of the signs. [ 37 ] SECTION VI. INVOLUTION. INVOLUTION is the raising of powers from any proposed root, and may be performed by the followina;' Rules. •o 1°. If the Quantity, or Root proposed to he involved has no index, ihat is, if it be not itself a power or surd, the power thereof ivill be represented by the same quan- tity under the given index, or exponent. Thus, the fifth power of a i s expr essed by a^ ; and the seventh power of a 4- 2 by a + z\^ , 2°- But if the quantity proposed be itself a power, or surd, it will be involved by multiplying its exponent by the exponent of the proposed power. Thus, the cube, or third power of a* is a^' ; the fifth power of x^ is x^^ ; the fourth power of ax -\- yy\^ is ax 4- 2/2/1*' ; and the third power of a — x] is a — x\ • 3°. A Quantity composed of several factors fnulti- plied together, is involved by raising each factor to the power proposed. The first of the Rules, here given, being mere nota- tion, does not require, nor indeed admit of a demon- stration : The second may be explained thus ; let A^^ be proposed to be raided to the power whose exponent is n : then I say, that the power itself will be truly ex- pressed by A'"" : For since (by notation) A'" is the same thing as A X A X A X A, Sec, continued to ?n factois* This raised to the 7/th power, or m>iltiplied n times, will, (by the general observation at p. 13) be equal to AxAxAxAxAxA, &c. cojntinued to n times m factors, that is, to mn factors ; which, by notation, is A""*. But the same thing may be otherwise demon- strated in a more general manner, by means of rule 3^ in multiplication : For, since powers raised from the D 3 3S INVOLUTION. Thus, the square, or second power of ah is a^h^ ; the cube, or third power of 2ah is raV)^, or 8a^b^ ; the fift h power of 3 x aa — xx x « + 5 4- c is 243 X aa — xx'.^ X a -+^6 + cV ; and the square, or second power of the radical quantity a' x a -f a:f is ,2 « X a -I- arr- 4°. yi Fraction is involved, by raising both the mimera" tor and the denominator to the power proposed. Thus, the second power of -r is -jr ; the third power r, <2ab . Sa^b^ .i r- ., r ^a'b . I6a'6* 01 —-- IS ^ ; the fourth power ot is 3c 27c' '^ 3c^ 81c 3 » the square of — —, or — is - ; the cube of — is -^ 1 ^, . ^1 p aa \ xx^^ . aa -{- xx\^ and the sixth power of ' is a — x\^ a — x\ When any quantity to be involved has the sign — prefixed, the power itself, if the index is an odd num- ber, must be expressed with the same negative sign, but if an even number, with the contrary sign, or +. same root are multiplied by addition of their indices, it is evident that the square of A"* (or A*"' X A'"] whe- ther the exponent m be a whole number or a fraction, will be truly defined by A^"*: whence it likewise ap- pears, that the cube of A''' (or A^"' x A'") will be de- fined by A"'" ; and the fourth power of A"' (or A^^'" x A'") by a'^ &c. The Reason of the third Rule is also grounded on the same general observations : For, in the first ex- ample, where the square of ab is asserted to be 0*6* we know that square to be ab xab fby the definition of a square), which quantity is there proved to be the same with a xb Aa / b, or aa x bb. So likewise, in the se- cond example, the cube of 2ab, or 2ab a 2ab x 2ab, INVOLUTION. 39 Thus the second power of — «, or — a x — «, is + a^ because — into — produces -f-; • also the cube of — a, or + a* X — a is — a^ (because 4- into — produces — ), so likewise the fourth power of — «, or -— «3 ^ «_ fl is + «+, and the fifth power, or + a"^ X — «, is — a', &c. &c. Hence it appears that all even Powers, whether raised from positive or negative Roots, will be positive. 5°. Quantities compounded of several terms are involv- ed hy an actual multiplication of all their parts. Thus if a 4- 6 was proposed to be involved to the sixth power; by multiplying a ^ h into itself, we shall first have a^ + 2a/; -f ^% which is the second power of «+ 6; and this, again, multiplied by a + ^, gives a^ + 3a"bi-Sab^'\'h^, for the third power of « + h : whence by proceeding on, in this manner, the sixth power of a -^ b will be found to come out a^ + Qa^b -f l5tt*6* + QOaV + \5d'b' + Qab^ -f 6^ See the operation. a + ft, the root or first power. a + b aa + ab + ab + V' a' + »2ab + 6% the square or second power. a ^ b will be=:2x«x6x2x«x6x2xax b — 2 A 2 X QA ax a A a X b /. b y. b — 8 X a^ X b^ - 8a^b^, And the case will be the same when radical quantities are concerned (as in the fourth example) : for the square 1 1 I T I oftt'' X a^oc\\ OY a^ X a + ocf x a^ x 0"+^^^ is =z 1 I I a Xa X a ^r x'\ X a \- XV — a X a X a H- a/ X ~~ T I 1 a + a'F: buta^ x a^ (by rule 5"^ in multiplication) is — g' - a, and a \ x\ x a" I- a\^ ~ a -r ^)^; there- fore our square, or its equal product, is likewise ex- pressed by a X a r a?l^. The 4th rule, or case, for the involution of fractions, is grounded on rule 3" in multiplication, and requires no other demonstration than is there given. D 4 40 INVOLUTION. 4- a-h-V Qah^-A- h^ a^ \ 3a'b^- 3ab' + b^ the cube, or third power, a-r b ^ a* + 4a'6+ 6a^6'4- 4a6^+ "^Vthe fourth power. fl4- ^> Q^ + 4a*64- aa'6 4- 4a'6 + a// 4- fl'^64- 4a^5^4- 6g-7)^ f 4flM-{- fe s «M-5a*6+10a^6 -t-10tf"65+ 3a6^+ 6^ the 5th power. g 4- /; a^\ba^b\\ Qa^b'^WQa^b^ \ ha^b'^^ah^ Ar a^b-^ 5aW-{-lOa^h^ h I0a''b''^,',ab^^-b^ <^*4 6a5Z)i-l5ft*6- + 20«'/>Hl5a*6*4-6o65 + ^^ the 6th or required power of a -ib. So likewise, if it be required to involve or raise a — h to the sixth power, the Process will stand thus : a — b a — b a' — ab — abh h G' — <2ab 4- b\ second power, a — b fl3 — 2d'b^ ah' — ■ a:'b-\-2 a b''— ¥ a^ — 3a^b-^'6ab' — 6', third power. a — b ft*__ 3a'^4 3aV;^ — ab^ — a'b-\-3a'b^— 3ab '^ b* a* — 4G^6 + 6a'6^ — 4ab^ + b*, fourth power. a — b a^^ 4a*b\-6a'b^'- 4a-()^-^ at)-" — a'^b-\^4a'h^— 6(vb^-\-4ab'' — b' a^—5a'b^\0a^b*— 10a b^ -{■ 5ab' -^ b^Cifth power. INVOLUTION. 41 a^^Ga'b 'r I5ab' — QUa'd' i- i5a'b' — 6ah' f 6^ the sixth power of u — b ; and so of any other. But there is a Rule, or Theoreni, given by Sir Isaac Newton, ^demonstrated hereafter) whereby any power of a binomial a -f />, or « — 6, may be expressed in simple terms, without the trouble of those tedious mul- tiplications required in the preceding operations; which is thus: Let n denote any number at pleasure; then the 72th power of a 4- ^ will be a^ + wa""^ b -f -^ — ^^^r n-27 9 n.n^X.n — 2 n-S-i-, . ^.^ — X .n — Q,n — 3. « ^+ 1 .. . F-^ ^-*- 1 r2-T-3-T-i 1.2.3*4.5 And the nth power of a — b will be expressed in the very same manner, only the signs of the second, fourth, sixth, &c. terms where the odd powers of 6 are involved, must be negative. An example or two will shew the use of this general Theorem. First, then, let it be required to raise a \-bio the third power. Here n, the index of the proposed power, be- ing 3, the first term, a^, of the general expression, is equal to a^; the second na'^b — 3a b ; the third -— . a' "^b^ — 3ab'; the fourth 1.2 1.2.3 ^w-J^3 _ J3 .^^^^ |.|-jg ^j^j.|^ I . 2 a"" 6\ &c. — nothing. Therefore the third power of tf 4- 6 is truly expressed by a' -f 3a^6 + 3ab' -\- b\ Again, let it be required to raise a '\- b to the sixth power. In which^case the ir.dex, n, being 6, we shall by proceeding as in the last example, have a" zi a% 42 INVOLUTION. na''h zz 6a% ^^^ \ a'-^b' - I5a'b\ &a and consequently q + b]^ = a^ 4- 5aV; 4- i5a7r + 202V>^ 4- 15a'// + 6a/j^ + 6^; being the very same as was above determined by continual multiplication. Lastly, let it be proposed to involve cc + xy to the fourth power. Here a must stand for cc, b for xy, and w for 4 ; then, by substituting these values, instead of a, b, and », the general expression will become c^ -\-4:C^xy-\-6c X'y^ 4. 4c^x^y^ + x^y\ the true value sought. From the preceding operations it may be observed, that the unciae, or coefficients, increase till the indices of the two letters a and b become equal, or change Talues, and then return, or decrease again, according to the same order: therefore we need only find the co- efficients of the first half of the terms in this manner ; since, from these the rest are given. [ 43 ] SECTION VII. EVOLUTION. THVOLUTION, or the Extraction of Roots, being di- ■^-^rectly tke contrary to Involution, or raising of powers, is performed by converse operations, viz. by the division of indiceSy as Involution was by their multiplication* Thus the square root oF x^, by dividing the exponent by 2, is found to be x^ ; and the cube root of x^, by di- viding the exponent by 3, appears to be oc' ; moreover, the biquaflratic root of a +- a:P will be a A- xY' -, and the cube root of aa + xx^^ will be aa v xx^"^. In the same manner, if the quantity given be a frac- tion or consists of several factors multiplied together, its root will be extracted, by extracting the root of each particular factor. Thus the square root of a'^b'^ will be ah\ that of a^h- .,, , ah „81 X «* /"cyTT^^ -n u — — will be — ; and that oi ,, will be c~ c \Q X a—^^ 9a X aa f xx^" ^.-^ ^, ^ r ^ — tl- ' - : Moreover, the square root of aa — xcd ^ 4 X a — X will be aa — ocx\^ ; its cube root aa — a;ai";andits biquadratic root, aa — xx\^ ; and so of others. All which being nothing more than the converse of the operations in the preceding section, requires no other demonstration than what is there given. Evolution of compound quantities is performed by the following Rule. First, place the several Terms, whereof the given quan- tity is composed, in order, according to the dimensions of some letter therein, as shall be judged most commodious; then let the root of the first term be found, and placed in the quotient; ichich termheing subtracted Jet the first term of the remainder he brow^ht down, and divided by twice the first term of the quotient, or by three times its S(/uare or four times its cube, ^c. according as the root to be ex- tracted is a square, cubic, or biquadratic one, <§c. and Icf 44 E V L U T I O X. the quantity thence arising he also lurote down in the quotient, and the whole be raised to the second, third, or fourth, S^c, poller, according to the aforesaid Cases, re- spectitely, and subtracted from the given quantity : and (if any thing remains) let the operation be repeated, by always dioiding the first term of the remainder by the same divisor, found as above. Suppose, for example, it were required to extract the square root of the compound quantity 2ax + a~ + x~: then havins^ ranged the terms in order according to the dimensions of the letter a, the given quantity will stand thus, a^ 4 2ax -h x^, and the root of its first term will be a; by the double of which I divide Qax, (the first of the remaining terms) and add + x, the quantity thence arising to a (already found) and so have a \-x in the quotient : which being raised to the second power, and subtracted from the given quantity, nothing remains : therefore a + a: is the square root required, bee the operation. «* + Qax + a* (a + X ea) Qax ^ fl' 4- Qax f X*, second power of a -f a% ~ O In like manner, if the quantity a* — Qa-x + 3a V — Qax^ -\- .T* be proposed, to extract the square root thereof; the answer will come out c/^ — a.v j x', as appears by the process. a*-^Qn\v-\-3a^x*—Qax^\-x*{a^ — ax fa* Qa-) — Qa^X G* — Qa^x^ a'^x^, second power of «' — ax, " ^aF) Q^x\ first term of the remainder. a* — 2«'a7-f 3o*a* — Qax^ \ x\ square of a^ — ax-\-x*, " Q O Again, let it be required to extract the cube root of a^ — 6a^x \-\2ax' — 8x^, and the work will stand thus: a 3 __ 6a'x 4- 12ax'— 8i' {a — Qx 3a-) — Ga'^x a^ — aa«x + iSfl.T* — 8.T^ cube of a — 2r. EVGLUTIOiV. 45 Lastly, let it be required to extract the biquadratic root of 16^* — 96x^y + 2l6a;y — 2l6xy^ -f 81z/% and the process will stand as follows: l6x^ — Qda^y + 216j:V — QlGxy^ + 81 y^ (Six — 3y 3 23?^) - ijGx ^y " 16T'' — 96Vv h 216A-V — 216j:/ -f 81y* O O O And, in the same manner the root may be deter- mined in any other case, where it is possible to be ex- tracted ; but if that cannot be done, or, after all, there IS a remainder, then the root is to be expressed in the mauner of a surd, according to what has been already i?hewn. As to the truth of the preceding Rule, it is too obvious to need a formal demonstration, eveiy operation being a proof of itself. I shall only add here, that there are other rules besides that, for extracting the roots of compound quantities ; which, sometimes, bring out the conclusions rather more expeditiously ; but as these are confined to particular cases, and would take up a great deal of room to explain in a manner sufficiently clear and intelligible, it seemed more eligi- ble to lay "down the whole in one easy general method, than to discourage and retard the Learner by a multipli- city of Rules. However, as the extraction of the square root is much more necessary and useful than the rest, I shall here putdown one single example thereof,wrought according to the common method of extracting the square root, in numbers; which [ suppose the reader to be acquainted with, and which he will find more expeditious than the general Rule explained above. Examp. a*+4a^a:+6aV4-4aa?^4-a?^ (a* + Qax + a?* + 4a'a7 + 4a'a7'* -i-2aV-f 4qa^^4-a?* C 46 ] SECTION VIII. OF THE REDUCTION OF FRACTIONAL AND RADICAL QUANTITIES. THE Reduction of fractional and radical quantities is of use in changing an expression to the most simple and commodious form it is capable of; and that, either by bringing it to its least terms, or all the members thereof (if it be compounded) to the same denomination. A Fraction is reduced to its least terms, by dividing hoth the numerator and denominator by the greatest common divisor. Thus, -r-, by dividing by 6, is reduced to ~ ; ^abc . 2c And ^^-jT-* by dividing by ba, is reduced to -r-; ,_ 20abd .,, , , , , Ad Moreover, ^ . will be reduced to — , or 4c?: 5ab 1 And A^.^^ will be reduced to-^-. 72a\x^^xy 6a 1 \'2aa — ^2ab , ,. ... ^ ^ Thus also, 5 , by dividmg every terra of the numerator and denominator by 2a, is reduced to 6a— b .2a And = — 3 , by dividmg every terra 6a X + Aax 1 . 4a* — Gar + 3x* by Qax. IS reduced to — —— : •^ * 3a f Qx ^-t.y. iil±i^y,f^^ by dividing both the numerator and denominator by the compound divi- , . , , ^ aa 4- 2ab + bb sor a + 6, IS reduced to — r • But the compound divisors whereby a Fraction can, sometimes, be reduced to lower terms, are not so easily FRACTIONAL QUANTITIES. 47 discovered as its simple ones ; for which reason it may not be improper to lay down a Rule for finding such divisors. Firsty divide both the numerator and denominator hy their greatest simple divisors, and then the qtwtlents one by the other (as is taught in Case 7, Section 5,) alicays observing to make that the divisor which is of the least dimensions ; and if any thing remains, divide it by its greatest simple divisor, and then dhide the last com^ pound divisor by the quantity thence arising; and if any tiling yet remains ^divide it likewise by its greatest simple divisor, and the last compound divisor by the quantity thence arising; proceed on in thi^ manner till nothing re^ mains; so shall the last divisor exactly divide both the nu- Qnerator and denominator, without leaving any remainder^ Note, If, after you have divided any remainder by its simple divisor, you can discover a compound one which will likewise measure the same, and is prime to the di- visor, from whence that remainder arose, it will be con- venient to divide, also, thereby. And, if in any case it should happen that the first term of the divisor does not exactly measure that of the dividend, the whohe di- vidend may be multiplied by any quantity, as shall be necessary to make the operation succeed. Ex, 1. Let it be required to reduce the Fraction to its lowest terms, or to find a b 4- <2a"b- + 2ab' 4- // ' the greatest common measure of its numerator, and de^ nominator. Here, dividing first by the greatest simple divisors, 5a^ and b, we have a* + 2a5 -F 6*, and a^ -h '2(fb -h ^tz/;^ 4- b^ : and if the latter of these be divided by the former, the work will stand thus : a' 4- 2ab + h") a' + <2a^b + Sfl^^ + b' (a a' + '2anj + ab"- where the remainder is h ab^ -{- V\ which be- ing divided by b\ its greatest simple divisor, gives a + Z> ; by this divide a^ + '2ab + b\ and the quotient will come out a -^ b, exactly; therefore the last divisor, a ■\- b, will exactly measure both quantities, as may be proved thus : 48 REDUCTION OF a-^b) 5a^ + 10a*6 + 5a^b^ (5a' + 5a^b 5a^ -f 5 a*b 5a*b -f 5a^b^ 5a*b f 5a'i* a a 4- ^,) a^b + 2a7/ + 2rt6' + b* (a'^b + «&* + b^ a'b + a'6* fl'^^' -V- ab' O In both which cases nothing remains; therefore the .,,1 J J X 5a* 4- 5a^b fraction given will be reduced to — rrr — t? , ta » Ex, 2, Let it be proposed to reduce the fraction ^^ ar" -^-— — 5 — — , to its lowest terms : then the a^ — (rx — ax^ + ^^ Avork will stand as follows : fl3 ^ a'^x -^ ax" + x') a* -{■ 4- + o — :c* (a + x a* — a^x — aV-f flx' a^x 4- a~x^ — ax — x^ Q?x — a^x' — av^ h -r* 4-2aV-+- — 2x' a* f — x^) o? — ax — ax^ -f x^ (a — x q^ — " — ax^ — rt r -f + ^'' — a' I 4- + «' These operations are founded on this Principle, That whatever quantity measures the whole^ and one part of another, must do the like by the remaining part. For, that quantity (whatever it is) which measures both the divisor and dividend, in the first example, must evidently measure a^ + 2«*Z» + ai* {being a multiple of the farmer) : whence, by the Principle above quoted, the same quantity, ^ it measures the whole dividend. FRACTIONAL QUANTITIES. 49 From whence it appears that a'+o — J?% or a* — a;* will measure both a* — a?* and a^ — a'x — ax'^'h x^ ; and, by dividing thereby, the fraction proposed is re- duced to -• a — X Example 3. In the same manner the fraction ^'-3a.r'-8aV-H 8alr-8a- ^jn be reduced to x~^ax' — aa'x + 6a' ^A- 5fiL-h±l' . See the process. :f — 3a .x^ax-8 a'x-1 6a^) x^'-Sax^ -Sa^x^ + ISalr — 8a*(a-2<* X* — ax^ — Sa'^x" 4- 6a^x "^2ax2-f + 12a^x — 8 a* — <2ax^ + ^a\T" 4- 1 6a\v — 1 2a* remainder — 2a"x" — 4a'^x + 4a*; ifvhich divided by — 2a% gives a;^ + 2aa7 — Sa^ for the next divisor. X' + 2aa,' — 2a") .t^ — ax- — 8a"x + 6a^ (a? — 3a a3-h 2a r- — 2a2a? — 3aa:- — 6a"x + 6a^ — 3aa- — 6a'2.T + 6a^ a?- -f- 2ox-2a-) a?* — aax^— 8a~a:^+ IBa^x-Sa* (a:2-5ax+4a^ a:* + 2ax^— 2a^a^ — - 5ax^ — 6a''X" -\- 1 8a^a- — 5aa^3 — 10a%" + lOa^o? + 4a" X- + Sa^x — 8a* + 4a"a-'- + 8a''x— Sa* Now if, by proceeding in this manner, no compound divisor can be found, that is, if the last remainder be only a simple quantity, we may conclude the case pro- posed does not admit of any, but is already in its lowest must also measure the remaining part of it, ub'^ + 5' : but, the divisor we are in quest of, being a compound one, we may cast olf the simple divisor b^, as not for our purpose ; whence a -f 6 appears to be the only compound divisor the case admits of : which, therefore, must be the common measure required, if the example proposed admits of any such. £ 50 11 E D U C T I O N O F terms. Thus, for instance, if the fraction proposed were to be- ; it is pUiin by inspection, that it is not reducible by any simple divi- sor; but to ki:ow whether it may not, by a compound one, I proceed as above, and find the last remainder to be the simple quantity 7xx: whence [ conclude that the fraction is already in its lowest terms. Another observation may be here made, in relation to fractions that have in them more than two ditferent letters. When one of the letters rises only to a single dimension, either in the numeiator or in the denomi- nator, it will be best to divide the said numerator or de- nominator, (v/hich ever it is) into two parts^, so that tlie said letter may be found in every term of the cne part, and be totally excluded out of the other ; this being done, let the greatest common divisor of I hese two parts be found; which will, evidently, be a divisor to the whole, and by which the division of the other quantity is to be tried ; as in the following example, where the . x^ +01- + hx- — 2a-x 4- hax — ^ba- traction given is — -. —- ~-t . ° XX — • ox -{- '2ax — Qao Here the denominator being the least compounded, and ib rising therein to a single dimension only, I divide the same into the parts x* f eax, and — bx — 2ab ; which by inspection, appear to be equal to a: 4- 2a /. x, and r -ir 2a X — b» Therefore a? f 2a is a divisor to both the parts, a nd likew ise to the whole, expressed by „__^^_ X X — b; so that one ot these two factors, if the fraction given can be reduced to lower terms, must also measure the numerator: but the former will be found to succeed, the quotient coming out x2 .^ ax -{- bx — ab, exactly : whence the fraction it- j-c — ax + bx — ab i • i • 4.^^ self is reduced to ^^-^;^ ; which is not re- ducible farther, by x — b, since the division does not terminate without a remainder, as upon trial will be Having insisted largely on the reduction of fractions to their least terms, we now come to consider their reduction to the bame denominator. FRACTIONAL QUANTITIES. 51 Fractions arc reduced to the same denominator by muU tiplying the numerator of each into all the denominators^ except it9 own, for a new corresponding numerator, and all the denominators continually together, for a common denominator. Thus, -7- and — will be reduced to -^-r and 7-7. ^ * b d bd od* a c y € ^ adf chf , bde "J- > -ry and — ' to 7-~> y-jrr» awd -j-r-y, b d J bdf bdf bdf , Qax , 5hx ^ Ga'^x , Sbxcd , and — r-' and , to - — j, and , » cd 3a 3acd 3acd and so of others. But when the denominators have a common divisor, the operation will be more simple, and the conclusion neater, if, instead of multiplying the terms of each fraction by the denominator of the other, you only multiply by that part which arises by dividing by the common divisor. As, if there were proposed the frac- tions—; and — r; then, the denominators having the ad cd ^ factor d common to both, I multiply by the remaining factors a and c; whence the two fractions will be , , bhc , aah , . . . . . reduced to , and - — ^ where d remams as before, acd acd nothing having been done therewith). By the same method — ^ and — —;- are reduced to r^ and 5abc 4aoa _____ ^Oabcd 35bcx^ , 6a\/ax , 7c\/aa +xx ^ ISa^v/oi ^^ , J and ; — and --r ■* to , — QOabcd 5 he 3ab Idaho J 35c'^\^aa -f .1^ and j-^ — • I5abc But, as has been before hinted, the principal use of this sort of reduction is to transform compound quanti- ties to the most commodious forms of expression ; >vhich, for the general part, are more easily managed, (whether they are to be added, substracted, multiplied, or divided) when all their members are brought to the same denomiriation. Thus the compound quantity r + i- will be trana- E2^ 1 52 REDUCTION OF formed to f^- + ^, or to 'lL±j£. for it is evident, oa oa Lid that the quotient which arises by dividing the whole, is equal to the quotients of all the parts, by the same divisor. T„ .1 .,, a c . ad — he , m tne same manner will -^ r- be zr rr^ » b d hd and ?^ 4-1'^ ^ — 2a^?//'c? 4- 1 5aaxd — 5anJ>c^ ^ baa h IT ~ daabdT' , 2dx ^dx h .,, , 2dx + ah, also — — 4- 6 or + — will be z= ~ ; a * a i a . 9ah c^ah -f <7a — ah ah -f aa and - — - -\- a zz -. =: ■— .— c \/ b; which, by extracting the root of a^, becomes a v^/j ; likewise \/a^//V, or «36V 1^" is reduced to v/a^6' x \/bc\ or ab\/hc^: moreover i/l^IlEj^ is reduced to 4 /i£L x ^ be' ^ c* ^ b c V ij * V 8\b''c^--l6jrb^x IS reduced to \/ -^-- x 1/ — = -7 — ,or -r-r X 1/ 816* y c"" — 2bx * 3b — ; and so of anv other : all which is evident from case 4 of multiplication, and case 3 of in- volution. But it is to be observed, that in resolving any expression in this manner, the factor out of which the root is to be extracted, is always to be taken the greatest the case will admit of It afso may be proper to take notice, that this kind of rrduction is ( hitfly useful in the addition and subtraction of surd quantities, and in uniting the terms of compound expressions that are commensurable to each other, where the irrational part, or factor, after reduction, is the same in each term. fined by -^, being d times as great as that defined by-g-, c . y ac must be tr ly expressed by^^ X a, or its equal -^; .as teas to be shncn. RADICAL QUANTITIES. .^5 : 71ius\/i8 + \/32 is reduced to 3v^Q ! 4v/2ror 7v/2; and v^BtP \- \/ 50a^ — x/jQa'^ is reduced to 2a\/2 + 5t7v/2 — 6av/;^ = «>/?. Moreover, by re- ;" . ^ /Tsa-r . ^ / Tsa^ . ^ /48a^ duction,1/ -h V becomes zi v *^ 20 *^ 20 *^ 20 *^ 20 And 3as/'4carx" -y sx^ 4- Socx/ga"^ + I8a"a?^ becomes 6Q!X\/a" -f 2a;2 + Qa^v/'a^ 4- 2x^ r: loa.Tv/a^ + 2a?2. S?^r£? qnantitiesy under diff'erefH radical signs, are reduced to the same radical sign, by reducing their in- dices to the least common denoininator. Thus a^ and a^, reduced to the same radical sign, \yill become a« and a^ (for the indices are here i- and I, and these are equivalent to I and J, where both have the same denominator). In the same manner^"^ and "s^'^ will become 2^^ and Tl', or 8j and "g^ . And, n p universalh/, A '" and B ^ , will when their expo- nents are reduced to the same denomination, become nrr inp At and bl . That the reduction of a radical quantity to another of a different denomination, by an equal multiplication of the terms of its exponent, makes no alteration in the value of the quantity, may be thus demonstrated. vt Let A" be any quantity of this kind; then, the terms of its exponent being equally multiplied by any VI r number r, I say, the quantity A '"", hence arising, is equal to the given one A " . E 4 5i RADICAL quantities;. The principal use of this sort of reduction is, when quantities under different radical signs are to be mul- tiplied or divided by each other. Thus, x/J multiplied by v/To, orTj- into lo^^' will give 123)5 X 1000 or 1 25001"^: also \/ax into y/a"x, or ^- into 'a^^ will give ahi^^ x ^^^ Or uTx^f: and \/ax divided by K^a'x will give 7' ^^ Z\ . Lastly, 207 multiplied into \/3ax, will give v/JI^ X N/soir, or v/fsox^ For, if X be assumed — A~^'\ or, which is the same, if the value of x be such, that a:"'" — A; then the nth root of a - being x'' (by case 2 0/ section 6) and the «th root of A be'mg A'^{by notation), these two quantities 1 a:'* and A" must, likewise, be equal to each other: and, if they be both raised to the ?wth power, the equa- lity will stili continue ; but the mth power of the former (X'*) is — x"^^ (by case 2 of involution)-, and the mth 1 li* power of the latter { A « ) is A " {by notation) ; there« - i_ fore j:"*^ is r= A". But, x being — A"'\ we have x*"^ ■- A "**, br/ notation ; and consequently A '"" — A "^; which was to he proved. I 57 1 SECTION IX. EaUATIONS. AN EQUATION" is, when two equal quantities, differently expressed, are compared together, by means of the sign = placed between them. Xhus, 8 — 2 zi 6 is an equation, expressing the equality of the quantities 8—2, and 6 : and x — a -\- b is an equation, shewing that the quantity represented by X is equal to the sum of the two quantities repre- sented by a and b. Equations are the means whereby we come at such conclusions as answer the conditions of a problem ; wherein, from the quantities given, the unknown ones are determined ; and this is called the resolution, or reduction of equations. REDUCTION OF SINGLE EQUATIONS. Single equations are such as contain only one un- known quantity ; which, before that quantity can be discovered, must be so ordered and transformed, by the addition, subtraction, multiplication, or division, &c. of equal quantities^ that a just equality between the two parts thereof may be still preserved, and that there may result, at last, an equation, wherein the unknown quan- tity stands alone on one side, and all the known ones on the other. But, though this method of ordering an equation is grounded upon self-evident principles, yet the operations are sometimes a little diificult to ma- nage in the best manner ; for which reason the follow- ing Rules are subjoined. l^ Any Term of an equation, may be transposed to the contrary side, if its sign he changed*. * The reason of this Rule is extremely evident; since transposing of a quantity thus, is nothing more than subtracting or adding it on both sides of the equation, according as the sign thereof is positive or negative. 68 EQUATIONS. Thus, if X -f 6 n 16 ; then will x = 16 — 6, that is, X _ 10: And, if X — 4 = 8 : then will x - s + 4, or a; r= 12 : Also, if 3X — 2X -f 24 : then will 3x — 2x — 24, that is, a: = 24 : Again, if 5t — 8 ::: 3x 4- 20: then will 5X — 3x zz 20 f 8, or 2X — 28 : Lastly, if ax i- Z>.r*— c ^~ d — ex —f — g -{-hx — Jcx ; then, by transposition, ax -f- hx — ex — kx -^ kx — f — g -^ c — d ; where all the terms affected by x (the un- known quantity) stand, now, on the same side of the equation. 2'. If there is any qvantity hy which all the terms of the equation, are multiplied, let them all he divided hi) that quantity ; hut if all of them he divided hy any quantity, let the common divisor he cast aicay. Thus, the equation ax =: ah is reduced to x — hi also, lOr ~ 70 (or 10 /^ x zz 10 x 7; is reduced to X — 1 \ and x^ — ax~ -f hx-, is reduced \,ox — a ^ h-. X h Moreover [hy the latter part of the Rule] — ——is , ax^ ahx~ — acx^ , reduced to x — h ; and — =: , to ax^ =z c c ahx'^ — acx^; which, if the whole be divided by ax*, will be farther reduced to x — b — c. 3°. Tf there are irreducible fractions y let the whole equation be multiplied hy the product of all their deno- minators, or, which is the same, let the numerator of every term in the equation he multiplied by all the deno^ minators, except its own, supposing such terms f if any there he) that stand without a denominator, to have un unit subscribed. Thus, in the equation .r 4- 6 = 16 (which by trans- position becomes x = 16 — 6 - 10) the number 6 is subtracted from both sides ; and in the equation .r — 4 = 8 (which by transposition becomes .r zz 8 -f 4 n 12) the number 4 is added on both sides. EQUATIONS. 6S Thus, the equation a: + — 4- ^ = H, is re- X -\- Q duced to 6x -{■ 3x -\- 2x — 66 ; and x + — - — r= 12 + 5 ^-^^^, to 40:c 4- 8x 4- 16 = 4S0 + 5jp — 15 : so 8 X X -\~ likewise a — , is reduced to a*c — ex nz ax a c ox f"X! 4- abi and -h a — ~. to ahx 4- o^h + ahx zz a -{- X acx f cx"^. 4^. /f, in your equation, there is an irreducible surd ^ wherein the unknown quantity enters, let all the other terms he transposed to the contrary side (by rule 1) ; and then, if both sides he involved to the power denominated by the surd, an equation will arise free from radical quan^ titles ; unless there happens to he more surds than one^ in which case the operation is to be repeated. Thus, \/~x 4- 6 — 10, by transposition, becomes \/x (=z 10 — 6) == 4 ; which, by squaring both sides, gives X — i6. So, likewise, \/aa -\- xx — c —x, becomes \/aa + xx -=2 -[-X ; which, squared, gives aa -\- xx — cc -f- Qcx + XX, or aa — cc = 2cx {by rule i). The Reasons of this, as well as of the two preceding rules, depend on self-evident principles: for, when the equal quantities, on each side of an equation, are multiplied or divided b}?- the same, or by equal quantities, or raised to equal pow- ers, the quantities resulting must necessarily be equaK 5°. Having, by the preceding rules fij there is occasion] cleared your equation of fractional and radical quanti- ties, and so ordered it, by transposition, that all the terms, ivherein the unknown quantity is found, may stand on the same side thereof, let the whole he divided by the coefficient, or the sum of the coefficients, of the highest power of the said unknoton quantity. And then, if your equation be a simple one (that is, it the first power or the quantity itself, be only concerned) the work is at an end : but if it be a quadratic, or cubic one, &c. something (TO EQUATIONS. further remains to be done; and recourse must be had to the particular methods for resolving these kinds of equations, hereafter to be considered in a proper place. I shall here subjoin a few examples for the Learner's exercise, wherein all the aforegoing Rules obtain pro- miscuously. Eoc. 1. Let 5x — \6 — 3x + 12 : then {hy rule l) 5x — 3a: — 12 4- l6, or 2j: — 28 ; whence {by rule 5) 28 ^ = - = 14. Ex, 2. Let 20 — 3x —8 = 60 — 7a' : then — 3.r, + 7x zz 60 — 20 + 8, that is, 4a: =z 48 ; and conse- 48 quently X rr — zz: 12. Ex. 3, Let ax — b ~ ex -^ d', then ax — ex :^ d + b , d '\~ b , or a — c X x z= d -f 6 ; and therefore x = [by rule 5.) Ex. 4. Let 6x* — 20X =z I6x -h 2t' ; then dividing by 2x [according to rule 2) we have 3x — \0 := 8 {■ x : whence 3x — x - 8 + 10, that is, 2x zz 18; and I 8 therefore x zz — -- = 9 '• 2 Ex, 5. LetSax' — abx^ zz ax^ 4- Qacx^ : here dividing the whole by aa% we have 3x —-b zz x + 2c ; there- 7 J 2c -F i tore 2x zz 2c + 0, and a: = — • Ex. 6. Let - + - =: 21 : then {by rules) 4x + 3i \j ^ 050 =: 252 ; and therefore x z= ^^-^— 36. 7 Ex. 7. Let "L + — - — =: 16 -— : then, 2 3 4 12x + 12 -f 8x 4- 16 zz 384 — 6x — 18 ; whence J 33 8 S6j: =: 338, and x zz —^ = 13. 26 EQUATIONS. 01 Ex, 8. Let a — c : theu ax — bb = ex; whence ex — cxziz bb, and x ~ . a — c f! ft' f Ex, 9, Let — + -J- + — — c?: then, bcx + acx -^ abx — abed, or be 4- ae + ab y. x zz abed ; and conse- quently x zz^-^-j-^^--^ fly rule 5,J. Ex, 10. Letaa: + b^ - ^?1±J?^: then, ax + 6* x a-f- d: « 4- a; z= ajj- -j- ae^, that is, a-x + ab^ + aj?' -h i"a:zr ax^ + ac' ; whence a^x + ax* -\- Px — ax^ zz ae^ — ab^, or a^x + b'^x — ac' — a5*; and therefore x — ae" -^ a b^ aa + bb ' Ex, 11. Let ; + - =z I: then ax -{- ab -{- bx a -^ X X I ^ zz ax + XX ; whence — xx -h bx — — ab ; which, by changing all the signs (in order that the highest power of X may be positive) gives xx — bx zz ab. But the same conclusion may he otherwise brought out, by first changing the sides of the equation ax + ab -{- bx zz ax + XX ; which thereby becoming ax -[- xx zz ax -\- ab -{- hx, we thence get xx — bx ^ ab, the same as before, Ex. 12. Let ^J-i£ +12 = 17: then l_if - 5, and 3 3 v/ix =15; whence, (6y rule 4) 5x zz 225, and there*- « 925 fore X zz -— = 45. 5 • Ex, 13. Let v/12 + X = 24- n/J": then {by rule 4) 12 + .T = 4 + 4 \/x + X; whence, by transposition, 8 = 4 \/a?; and by division, 2 = y/x; consequently 4 zz X. 62 EQUATIONS. Ex. 14. Let X + s/a- -+ x- = , "^ ■- > Here {by \/a^ ^- X- ru le 3) X y . v/ a' -h a;* + a' -f a-- - 2a^ ; whence x X Va2 -f x« = a2_ j,2^ and a:^ x a-1- a* = a* ~2a'j?- 4- x* (% rule 4) y that is, a^or^ -t- x* —a* — 2a" x- + a,"* ; therefore 3aV — a\ and x' — — - ~ — . * 3a2 3 i:^. 15. Let v/x + \/a + X - —--^~ — - . Then \/a + X \/ax-^xx -\-a^x — 2a, or \/ axi^ xx=:a—^x; whence ax + XX — a' — 2ax + x*, and x =z — — - • ' 3tt 3 Ex, i6. Let >/a:^ — a^ zz x — c: then, by cubing both sides, x'^ — a^ — x^ — 3c.r* + 3c^x — c^ ; whence 3cx" — 3C-X — a^ — c^ and a* — c;^ n by 3C 3 "^ dividing the whole by 3c. Ex. 17. Lei \/aa + xx — \/b^ + x* : the n, by rais « ing both sides to the fourth power we have aa + xx^'^ =: b* -\- x\ that is, a* + eaV + x* zz b* -j- x* ; and consequently a:* iz ^ — zr — — ' q", Ex. 18. Lel^^jz v/Vjf X ^bb }r XX — a: Here a? + a =: va' -i"J^v/ 56 + j- j? : which,, squared gives x^-\' ^ax + «' = a2+ J:v/Z;F4^a', or a^- 4- 2ax- xx/W^cc] divide by x, so shall x + 2a - v^^^ -h 0:0; ; this squared again, gives x^ + 4aa: -f- 4a2 ^r i6 + a-or ; whence 4a^ = i6 — . 4a% therefore x = a. Aa C 63 ] OF THE EXTERMIXATIOKOF UNKNOWN QUANTITIES, OR THE REDUCTION OF TWO Oil M.ORE EQUATIONS TO A SINGLE ONE. It has been shewn above, how to manage a single equation; but it often happens, that, in the solution of the same problem, two, or three, or more equations are concerned, and as many unknown quantities, mixed promiscuously in each of them ; which equations, before any one of those quantities can be known, must be reduced into one, or so ordered and connected, that, from thence, a new equation may at length arise, af- fected with only one unknown quantity. This, in most cases, may be performed various ways, but th^ following are the most general. 1°. Observe luhich, of all your unknown quantities, is the least involued, and let the value of that guantlti/ be found in each equation f^y the methods already explain^ edj looking upon all the rest as known ; let the values thus found be put equal to ea'Ji other (for they are equal, because they all express the same thing J ; whence new equations tcill arise, out of which that quantity will be totally excluded; luith which new equations the opera- tion may be repeated, and the -unknown quantities ex^ terminated, one by one, till, at last, you come to art, equation containing only one unknwn quantity* 2°. Or, let the value of the imknown quantity, ichick you would first exterminate, be found in that equation wherein it is the least involved, considering all the other quantities as known; and let this value, and its poioers, be substituted for that quantity, and its respective powers in the other equations ; and with the new equations thus arising, repeat the operation, till you have only one un^ known quantity, and one equation, 3°. Or, lastly, let the given equations be multiplied or divided by such numbers or quantities, whether known or unknown, that the term which involves the highest power of the unknown quantity to be exterminated, may he the same in each equation: and then, by adding, or subtracting the equations, as occasion shall require, that 64 EQUATIONS, term will tanish; and a new equation emerge, wherein the number of dimensions (if not the number oj unknown quantities) loill be diminished. But the use of the different methods here laid down will be more clearly understood by help of a few examples. EXAMPLE I, Let there he given the equations x -V y — 12, and 5x -^ 37/ zz 50 ; to find x and y. According to the first Method, by transposing y and 3y, we get X — 12 — ?/, and 5x -zz 50 — 3^ : from the last of which equations, x — — ——^. Now, by equating these two values of cT, we have 12 — y — — -\ o and therefore 6o — 5y n 50 — 3y ; from which, y is given = — zz 5 : and ar(— 12 — y — 12 — 5) — 7, According to the second Method, x being, by the firsfe equation, =z 12 — y, this value must therefore be sub- stituted in the second, that is, 60 — 5y must be wrote in the room of its equal 5x ; whence will be had 60 — 5y + 3y — 50; and from thence y rz -- zz 5,as before. But according to the third Method, having multiplied the first equation by 3, it will stand thus, 3.r + 2// — 60 ; from whence subtracting the 2d equation, 5x4- Syzz 50, there remains 2^ zi 1 ; whence y — 5, still the same as before. The first of these three ways is much used by some Authors, but the last of them is, for the general part, the most easy and expeditious in practice, and is, for that reason, chiefly regarded in the subsequent ex- amples. EQUATIONS. as EXAMPLE II. » . < 5x + Sy = 124 ( 3X — 2t/ = 20. Here the second equation being multiplied by 4 (in order that the coefficients of y in both equations may- be the same) we have I2x — 8y = 80. Let this equation and the first be now added together; whence?/ will be exterminated, there coming out 17 a? = 204; from which a; zz -— = 12: therefore, by the f, ^ ,. , 124— 5a? 124—60 64. _ first equation, y [- g — = — ^ — ) =i 8. EXAMPLE III. Given J 5x^3y =: 90 ^^^^^ i 2.T+ 5y =160. Here multiplying the first equation by 2, and the second by 5, in order that the coefficient of x may be the same in both, there arises lOacr— 6y = 180 10.T? -f 25?/ = 800. , By subtracting the former of which from the latter we have 31?/ zi 620: hence y = —*- =: 20; and so, by the first equation, x [=:?l±±y= Sl+J^) = 30. But the value of x may be otherwise found, inde- pendent of the value ofy; for, by multiplying the first equation by 5, and the second by 3, and then adding them together, y will be exterminated, and you will get 930 esx + 6.r r: 450 f 480; whence x zz — - =: 30, the same as before. 66 E Q U A T I O X S. EXAMPLE IV, Given T + f = '« Heve our equation?, cleared of fractious, will be 3x + 9y — 96 9x — 5^ ~ 90. And, if from the triple of the former the latter be sub- tracted, we shall have 6y -{- 5]/ =z 288 — • 90, that is, II7 = 198; whence^^zi 18; andx(= ?^ZllV) := 20. EXAMPLE V. Given -|--=l + s Here 4a? — 96 — Qy + 64, and 12.r + IQy -\- eox — 480 = 30^ — 15a: + 1620; which, contracted, become 4x — Sy =: 160, and 470? — I8y z= 2100 : from the last of which subtract 9 times the former : so shall llx — 2100 — 1440 =: 660 ; therefore 0? = 60, and y (zz4x— 160 ^ X . ^ ^ r: 2X — 80) = 40, \ EXAMPLE VI. to find Xy y, and z. By subtracting the first equation from the second (in order to exterminate x) we have z^- y zz 1 ; to which the third equation being added, y will likewise be ex- terminated, there commg out 2z zz 16, or 2 =. 8; whence y (=: z — 1} = 7 ; and x{zz 13 — yj = 6. EQUATIONS. 6T EXAMPLE VII. Here the given equations, cleared of fractions* become IQX + 8?/ +- 62 = 1488 QOX f 13// + 12s z= 2820 300? + 24y + 20:3 zz 430O. Now (to exterminate 2) let the second of these equa- tions be subtracted from the double of the first ; and also the triple of the third from the quintuple of the second ; whence is had 40? 4- y zz 156 lOa^ -^ 3]/ zz 420. 48 from which 12a:— 10x~ 468— 420, and a? = — zz S4f 2 Therefore^ (= l56-4r)-:60; and«(==^i?^®?^^=^) = 120. Let To the double of the first, let the second equation be added; so shall the a?\s, on the contrary sides, destroy each other, and you will have 300 -f y == 2^ + 42, or 300 = y 4- 4?. Moreover to the triple of the firsts let the third equation be added, whence will be had z + 400 = 6y + 3z, or 400 = 6i/ + 2z. ^ Now, if from the double of this last equation, the former, 300 ^ y -^ 4z, be subtracted, there will come , , 500 out 500 = iiyi and, consequently, 3^ = — = 45/x* EXAMPLE VIII. X + 100 = y + z y + 100 = 20? 4- 22 z + 100 = 3X + 3y, 68 EQUATIONS. therefore z {- 55^^ = 75 — |- z: 75 — 1 if^) = 63iV; and a: (= ?/ 4- 1 — 100 = 109 ^— lOO) iz 9tt- EXAMPLE IX, Let X — y zz Q, and xy + 5x — 6y — 120; to exterminate i\ By the former equation a: zr t/ -f 2 ; which value be- ing substituted in the latter according to the second ge^ neral method, it becomes y + 2xy + 5X2/ + 2 — 6y zz 1 20, that iSj ?/* f- 2^ 4-5^4- 10 — by =. 120, or ^' + 3/ = lio. EXAMPLE X. Let there be given x + y =z a, and of^ + y^ n b; to exterminate x* Here, by the first equation, x zz a — y; and there- fore T^ zr a — yl^; which value being wrote in the other equation, we have a — yl* -{• y* — b, that is, «' — lay \ r^7j'^ — b; and therefore 3/* — ay =: — "^ — . EXAMPLE XI. ^ . i axy -^ bx -{- ex iz d } . ^ . ^ ^^^'^^ifxy^gx -^hy zzk\ ^^ exterminate y. Multiply the first equation by/, and the second by a^ and subtract the latter product from the former; whence yov will have bfx — agx+cfy — ahy = df — ak ; which, . • JJ-. • • df — akjuasx^-'bfx by tr^nspos. and division, gives y = -=2 'T ^ . — ^^^ Let tliis value of y be now substituted in the first equation, and there will arise ad fx — gVcx \ a^gx'^-^ahjx^+c d f-^cak 4- cagx-^ cbfx — c/— ah ■ ; "*■ hx — d: which, mult iplied by cf-^ah, a nd contracted, gives ag'-^df x x* -J- <(/ — ak-^cg-^ bh y.x =. ck-^hd. EQUATIONS. 69 EXAMPLE XII. Supposing ax' ■\-hxArC—0, and /i' + gx' \ h^O ; to exterminate x. Proceeding here as in the last example, we have/^x ■\-fc-^agx — ah n 0; and, from thence, x — yj—^ . ah—fcX 6 X ah-^fc Whence, by substitution, a x ~ — ^ ^ +~7a — Ti^^ JO— ah\ y ""-^"fe I c zz 0. This, by uniting the two last terms, an4 ,. .,. , , 1 , . ah-^fcT hh--ce dividingthewholebya, gives =^=:^—-z + v>t-^— = 0; /b—ag\ fb'—ag ^ consequently ah — fc\' -{-fb — ag x bh—og — O. After the same manner x may be expunged out of the equations ax^ -f bx' -f c.r -f c? — o, and/x* -\^ gx -h h =: O^ &c. But, to shew the use of the above example, sup- pose ther€ to be given the equations x'^ + yx — t/* =0, and x^ + 3xy — 10 zi o : then, by comparing the terms of these equations with those of the general ones, dx^ -f- bx -\- c~ 0, andy^i;^ + gx 4-^ = 0; we have a = 1 , b = y, c = — 2/% /= 1, g = 3y, and A = — - 10 ; which values being substituted in the equation ah-^fc\^ '^ f^ — ^§ X M — eg =z 0, it thence becomes — 10 + y'yl^ + y — 3y x -^ioy + 3y^ = o, that is, 100 — 20?/* + y* 4- 20/ -^ 6/ = O ; or, 100 = 5?/* ; whence ?/ may be found, and from thence the value of X also. F 3 C 70 ] SECTION X. PROPORTI ON. QUANTITIES, of the same kind, may be compared together, either with regard to their differences, or according to the part or parts, that one is of the other, called their ratio. The comparison of quantities according to their differences, is called arithmetical ; but according to their ratios, geometrical. When, of four quantities, 2, 6, 12, 16, the difference of the first and second is equal to the difference of the third and fourth, those quantities are said to be in arith- metical proportion. But, when the ratio of the first and second is the same with that of the third and fourth (as in 2, 6, 10, 30) then the quantities are said to be in geo- metrical proportion. Moreover, when the difference, or the ratio, of every two adjacent terms (as well of the se- cond and third, as cf the first and second, &c. j is the same, then the proportion is said to be continued: thus S, 4, 6, 8, &c, is a continued arithmetical proportion : and 2,4, 8, 16, &:c. a continued geometrical one. These kinds of proportions are also called Progressions, being carried on according to the same law throughout. ARITHMETICAL PROPORTION. THEOREM I. Of any four qvantities, a, b, c, d, in arithmetical progression*, the sum of the two means is equal to the sum of the two extremes. For since, by supposition, 6 — a\s — d — c, there* fore isb -{- c =: d + a,hy transposition. THEOREM II. In any continued arithmetical progression (3, 7> 9. 11 » 13, 15) the sum of the two extremes, and that of every other tico terms equally distant from them, are equal, * Although, in the comparison of quantities accord- ing to their differences, the term proportion is used ; yet theword/jroore^^zo/jris frequently substituted in its room, and is, indeed, more proper; the former term being, in the common acceptation of it, synonymous with ratio, which is only used in the other kind of comparison. PROPORTION. ^1 For since, by the nature of Progressionals, the second term exceeds the first by just as much as its correspon«:f- iijg term, the last but one, wants of the last, it is manifestthat when these corresponding terms are added tocfcther, the excess of the one will make good the defect of the other, and so their sum be exactly the same with that of the two extremes.: and in the same manner it \y ill appear, that the sum of any two other corresponding terms must be equal to that of the two extremes. When the number of terms is odd, as in the progres- sion, 4, 7, 10,^13, 16, then the sum of the two extremes being double to the middle term, or mean, the sum of any other two terms, equ^illy remote from the extremes, must likewise be double to the mean. THEOREM III. In any continued arithmetical progression, a,a + d,a -|-2fl?, (i-^3d, a \-4d ^c. the last, or greatest term, is equal to the first for least J y more the common difference of the terms drawn into the number of all the terms after the first, or into the whole number of the terms ^ less one. For, smce every term, after the first, exceeds that preceding it, by the common ditTerence, it is plain that the last must exceed the first by as many times the common differev^ce as there are terms after the first % and therefore must be equal to the first, and the com- mon difference repeated that number of ti fries. THEOREM IV, The sum of any rank or series of quantities, in con^ tinned arithmetical progression, {5, 7, 9, 1\, 13, \5) is equal to the sum of the two extremes multiplied into half the number of terms* For, because'(by the second Theorem) the sum of the tw^o extremes,and that of every two other terms equally remote from them, are equal, the whole series consist- ing of half as many such equal sums as there are terms, will therefore be equal to the sum of the two extremes repeated half as many times as there are terms. The same thing also holds, when the number of terms is odd, as in the series 8, 12, 16, 20, 24 ; for theii, the mean, or middle term, being eJqual to haif the sum of any two lerms equally distant from it, on contrary »ides, it is ob- f4 72 PROPORTION. yious that the value of the whole series is the same, as if every tei-m thereof was equal to the mean, and there- fore is equal to the mean (or half the sum of the two extremes) multiplied by the whole number of terms ; or to the whole sum of the extremes multiplied by half the number of terms. GEOMETRICAL PROPORTION. THEOREM I. If four quantities a, b, c, c?, (2, 6, 5, 15) are in geometrical proportion, the product of the two means, be, will be equal to that of the two extremes, ad. For, since the ratio of a to 6 (or the part which a is of 5) is expressed by -r-, and the ratio of c to d, in like manner, by- ; and since, by supposition, these two a ratios are equal, let them both be multiplied by bd, and a c the products j- x 6c? and -^ x bd wiW likewise be equal; that is, -r- = -v-j or ad zz cb (by case 2, sect, 4). THEOREM II. Jf four quantities c, 6, c, «f, are such, that the product of two of them, ad, is equal to the product of the other two, hc^ then are those quantities proportional. For since, by supposition, the products ad and be are equal, let both be divided by bd, and the quotients Tk'd^ / ^^^ h^ ("TT ] ^^^^^ ^^^^ ^^ equal ; and there- fore a: b :: c : d, THEOREM III. Jf four quantities a, b, c, d, (2, 6, 5, \5) are propar^ tional, the rectangle of the means divided by either extreme, will give the other extreme. For, by the second Theorem, cd =i 5c (2 x 15 — 6 X 5), ^yhence dividing both sides of the equation by a (2), we havcc? = ~ (15 =: - — ^~\ Hence, if the two means and one extreme be given, the other ex- treme may be found. PROPORTION. 73 THEOREM IV. The 'products of the corresponding *terms of tioo geo^ metrical proportions are also proportionaL . That is, if a : i : : c : ff, and e : f :i g i h, then will ae I bf : : eg : dh. For ~ =-^,and y =~/ » by supposition; whence -T- X -^ — ;v X -y-, by equal multiplication; and coa- ae c* frequently -r? := ;^'(by P* 18); that is, ae : hf'.\ eg : dh^ Hence it follows, that, \f four quantities are pro* portional, their squares, cubes, &c. will likewise be proportional. THEOREM V. If four quantities a, h, c, d (2, 6, 5, 15) are proportional^ 1. inversely, h i a i: d i c (6:2::15: 5) 2. alternately, a : c :: b i d (2:5:: 6:] 5) 3. compoundedly, fl:a + 6::c :c + f/(2:8:: 5:20) c 4, dividedly, a :h — a::c %d — c(2:4::5:l0) ^"^ 5. mixtly, b^-a^.h — a::d\-c:d — c(8:4::2o:]o) 6. by multiplication, ra : rb i: c i d (2r :6r:: 5:1 5) 7. by division, — .;-*-:: c : cff— :— ;: 5:15) Because the product of the meai>s, in each case, is equal to that of the extremes, and therefore the quan« titles are proportional, by Theorem 2, THEOREM VI, If three numbers a, b, c, (2, 4, 8) be in continuef^ pro- portion, the square of the first icill he to that of the second^ as the fir St number to the third; that is, a"^ : b'' :: a : c. For, since a : b :: h : c, thence will ac — bh, by Theorem 1 ; and therefore aac z=l abb, by equal multi- plication ; consequently a" : ¥ :: a : c, by Theorem 2. In like manner it may be proved, that of four quan- tities continually proportional, the cube of the first is to that of the second^ as the first quantity to the fourth. t4 PROPORTION. THEOREM VII. In any coniinned geometrical proportion ( 1 , 3, 9, 27» SI ♦ t;c.) th^ product of the two extremes, and that of every other two terms, equally distant from them., are equaL For the ratio of the first term to the second, bcino^^ the same' as that of the last' but one to the last, these four terms are in proportion ; and therefore, by Theorem 1, the rectangle of the extremes is equal to that of their two adjacent terms : and, after the very same manner, it will appear, that the rectangle of the third and last but two» is equal to that of their two adjacent terms, the second aijd last but. tue; and so for the rest. Whence the truth of the proposition is manifest, TIIEOKEM TUT. Th€ sum of any nitmher of quantities, in contlnned geometrical proportion, is equal to the difference of the rectangle of the second and last terms, and the square of the first, divided by the difference of the first and second terms. For, lei the first term of the proportion be denoted by tty the common ratio by r, the number of terms by 5»,and the sum of the whole progression by x\ then it is manifest that the second term \yill be expressed by a x r, or ar ; the third by ar y r, or ar*; the fourth by ar^ x r, or ar^ ; and the ?/th, or last term by ar ; and there- fore the proportion will stand thus, a f «r -f ar* -f ar^ 11—2 n— 1 . . . . . + flr 4- cr •=. x\ which equation, mul- liplied by r, gives ar + ar- + ar' -f- ar^ + ar -for'' zr rx; from which the first equation being sub- tracted there will remain — a 4- ar^~ rx — x\ whence ar X ar^"^ — aa (ar^—a _ r x ar" ^'^a \ ^ ar — a as was to be demonstrated. C T5 ] SECTION xr. THE APPLICATION OF ALGEBRA TO THE RESOLU* TION OF NUMERICAL PROBLEMS. WHEN" a Problem is proposed to be solved alge. braically, its true design and signification ought, in the first place, to be perfectly understood, so that {it' needful) it may be abstracted from all ambiguous and unnecessary phrases, aud theconditions thereof exhibit- ed in the clearest light possible. This being done, and the several quantities therein concerned being denoted by proper symbols, let the true sense and meaning ot the question be translated from the verbal, to a symbo- lical form of expression ; and the conditions thus ex- pressed in algebraic terms, will, if it be properly limit* ed, give as many equations as are necessary to its solu- tion. But, if such equations cannot be derived without /jome previous operations (which frequently happens to be the easel, then let the Learner observe this rule, viz. let him consider what method or process he would use to prove, or satisfy himself in, the truth of the solu- tion, were the numbers that answer the conditions of the question to he given, or adirmed to be so and so; and then, by following the very same steps, only using unknown symbols instead of known numbers, the question will be brought to an equation. Thus, if the question were to find a number, which being multiplied by 5, and 8 subtracted from the pro- duct, the square of the remainder shall be 144; then, having put a zz 5, b — 8, and number sought to be then 5, or a times that number ^ will be .... .. . . i from which 8, or /; being sub- > tracted, there remains . . J which, squared, is Therefore o*a?* — '^axb 4- i' is = c (or 144) accordr ing to the conditions of the question, In the same man- c zz ■ 144, suppose the 4 (or) X 20 ax . 12 ax—b 144 a'x"— ^axb + h\ ^ 75 THE APPLICATION OF ALGEBRA ncr may a question be brought to an equation when two or more quantities are required. After the conditions of a problem are noted down m algebraic terms, the next thing to be done is to consider whether it be properly limited, or admits of an indefinite number of answers ; in order to discover whiGh, observe the following rules. , RUIiE I. iVhen the number of quantities sought, exceeds the number of equations given, the question (for the general part J is capable of innumerable answers. Thus, if it be required to find two numbers (a: and y) with this one single condition, that their sum shall be 100; we shall have only one equation, riz. .x + y z: 100, but two unknown quantities, x and y, to be determined ; therefore it may be concluded, that the question will ^dmit of innumerable answers. RULE II. But if the number of equations, given from the eoU" ditions of the questions, is just the same as the number of quantities sought, then is the question truly limited. As, if the question were to find two numbers, whose ?um is 100, and whose difference is 20 ; then, x being put for the greater number, and y for the less, we shall have .r + y — 100, and a: — y = 20 : therefore, there being here two equations and two unknown quantities, the question is truly limited ; 60 and 40 being the only two numbers -that can answer the conditions thereof. RULE III. IVhen the number of equations exceeds the ?iumhcr of quantities sought, either the coiiditions of the problem are inconsistent one with another, or what is proposed, in ge- neral term?, can only be possible in certain particular cases. But it is to be observedj^that the equations understood here, as well as in the preceding rules, are supposed to be no ways dependent upon, or, consequences of TO THE RESOLUTION 01^ PROBLEMS. .77 one another. If this be not the case, the question may be either Unlimited, or absurd, or perhaps both, at the same time that it seems truly limited ; as will appear by the following example. Wherein it is required to find three numbers, under these conditions ; that the sum of once the first, twice the second, and three times the third, may be equal to a given number b ; that the sum of four times the first, five times the second, and six times the third, may be equal to a given number c; and that the sum of seven times the first, eight times the ?econd, and nine times the third, may be equal to a third given number d. Now, tlie three numbers sought being respectively denoted by a?, //, and s, the question, in algebraic terms, will stand thus, X -{- Qy + 3z z= I) 4x + 5y -{■ 6z — c 7x -^ 8y + gz - d. Here, there being three equation? and just the same number of unknown quantities, one might conclude the question to be truly limited • but, by reflecting a little upon the nature and form of these equations, the con- trary will soon appear: because the last of them in- cludes no new condition but what is comprised in and may be derived from tlie other two ; for if from the double of the second the first equation be taken away, the value of Tx + 82/4-92 will from thence be given = 2c— 6. Hence it is manifest, that giving the value of 7^ -+- 8^ -f 92, in the third equation, contributes nothing towards limiting the problem ; and that the problem itself is not only unlimited, but also impos- sible, except when d is given equal to 2c — 6. Having laid down the necessary rules, for bringing problems to equations, and for discovering whea they are truly limited, it remains that we illustrate what is hitherto delivered by proper examples. ARITHMETICAL PKOBLEMS, PKOBLEiyi I. To find that number, to- which 7 5 heing]added, the sum shall b€ the quadruple of the said required number. 78 THE APPLICATION OF ALGEBRA Let the number sought be represented by x; then will its quadruple be denoted by 4a:; whence, by the conditions of the question, a: + 75 zz 4x; this equation, by transposing a:, becomes . . . . 75 z: 3.i ; from whence, dividing by 3, we have a: = — = 25, which is the number that was to be found (for it is plain that 25 + 25 X 3 z:: 23 X 4 r= lOO). PROBLEM II, What number is that, which being added to 4, and also multiplied by 4, the product shall be the triple of the sum ? Let the number sought be denoted by x\ so shall the sum be denoted by a; + 4; and the product by . , 4x ; whenc e, by the conditions of the question, 4a: ' zz 0. -f 4 X 3; that is, 4.r ~ 3x ^ 12 ; from which, by transposition, x — 12. PROBLEM III. To find two numbers such, that their sum shall be 30, and their difference 12, If X be taken to denote the lesser of the two num* b^rs ; then, by adding the diirerence 12, the greater number will be denoted by.T -t- 12 ; and so we shall hftve 2a: 4- 12 = 30, by the question. From which equation, 2a: = 30 — 12= 18;andconse« 1 8 quently x zz -- zz 9 ; whence the greater number (X 4- 13) is also given =21. PROBLEM IV. To divide the number 60 into three such parts, that the fir^t may exceed the second by 8, and the third by 16. Let the first part be denoted by x ; then the second will be xr— 8, and the third rr — 16; the aggregate of all which, or 3a: — 24 is j= 60, by the question. TO THE RESOLUTION OB' PROBLEMS. 79- 84 Hence 3:i? = 60 + 24 = 84, and x~ — - = 28 : so that 28, 20, and 12, are the three parts required. PROBLEM V. ' The sum of GGOL teas raised (for a certain purpose J by four persons A, B, C, andD; lohereof B advanced twice as 7nuch as A ; C as much as A and B; and D as much as B and C : what did each person contribute 9 Let the sum or number of pounds advanced > by A be called S then will the number of B*s pounds be denoted by 2x, that of C's by 3x, and that of D*s by Hx: the sum of all which is given equal to 66o/. that is, 1 la? = 660: from which a: = — — =^0. Tlierefore, ^O, 120, 180, and 300/. are the respective sums that were to be determined. PROBLEM VI. A certain sum oj money was shared amon^ Jive per^ sons, A, B, C, D, and E; ivhereof B received T^/. less than A; C 16/. more than B; D 5/. less than C; E I5t» more than D: moreover it appeared, that the shares of the two last together were equal to the sum of the shares of the other three : What was the whole sum shared, and how much did each receive f Let X denote the share of A : then)i^i ?> will be the share of ^^' C^ + 16) CE; and therefore so? + 17 = 3.r — 4, hy the question : from whence, by transposition. 21 = x; so that 21, 1 1, ^7, 22, and 37/. are the several required shares ; amount- ing, in thjB whole, to lis/. 80 THE APPLICATION OF ALGEBRA PROBLEM VII. To find three numbers, on these conditions, that the sum of the first and second shall he 15; of the first and third 16 ; and of the second and third 17. If the first number be denoted by .r ; then it is plain, by the question, that the second will be represented by 15— j:, and the third by 16— .r. But the sum of these two last is given equal to 17; that is, 31 — 2a7 n 17; whence, by transposition, 14 = 2a:; and consequently 14 T -zi ~ zi T. Hence 15 — x — 8, and 16 — x = 9; which are the other two numbers required. PROBLEM vili. To find that number , which being doubled, and IQsuh" tracted from' the product^ the remainder shall as much exceed \00 as the required number itself is less than 100. The number sought being denoted by x, the double thereof will be represented by 2r; from which subtract- inor 16, the remainder will be 2x — 16; and its ex- cess above lOO, equal to so? — 16 — 100 : therefore 2.r — 16 — 100 = 100 — Xy by the question', whence 2l6 ,3x zi 216; and consequently x zz -— = 72. PROBLEM IX. 7 divide the number 75 into two such parts, that tliree times the greater may exced seven times the lesser by 15, Let the greater part be = .r ; then will the lesser part = 75 — X, and we shall have 3x — liir 75 — x X 7; or, which is the same, 2x — 15 z= 525 — 7x: from whence 10 x = 540, and consequently x z= 54, PROBLEM X. TuiMpcrsoJis, A and B, having received equal sums of money ^ A out of his paid away 25/., and B of his 60/., and then it appeared that A had just twice as much mo» nty as B : lohat money did each receive ? Suppose X to denote the sum received by each per- son; then A, after paying away 25/, had a? »25 ; and B, will ~ — — =: 12, by the conditions of the prob- TO THE RESOLUnON OF PROBLEMS. si after pav ine: away 60/. had x — 60: hence x — 25 ~ 2r — 120, by the question; and therefore 120— -25x2x— a\, that is, 95 ~ X, PROBLEM XI. To find that numler, whose \ pari exceeds its J part hy 12. Let the number sought be represented by sc ; then X X lem ; which equation (by multiplying every numerator into all the denominators, except its own) gives 4x — 3x — 144, that is, X — 144. PROBLEM XII, What sum of money is that ichoae ^ party ^ part^ and -^ part, added together, shall amount Co 94 pounds 9 If X be the number of pounds required, then will X X X -- + ^ + "T = 94 : from whence, by reduction, 343 J i> 20T+ 1 5x 4- I2x = 94 X 60, that is, 47.T =1 94 X 60 ; and and therefore a^ = 2 x 60 ~ 120. PROBLEM Xril. In a mixture of copper, tin, and lead, one half the whole — I6lb. was copper ; one third of the icliole — 12lb, tin; and one fourth of the lohole + 4 lb. lead : what quantity of each icas there in the composition 9 Let X denote the weight of the whole ; £._l6l ^ copper, 2 X then will •< — 12 > be the weight << tin. i of the j .T "^^ J I lead; and, if all these be added together, we shall have SC XX — 4- — + T "*■ 2^ — ^> ^y ^^^ question. Hence J d 4 82 THE APPLICATION OF ALGEBRA by reduction, I2.r + 8j? 4- 6x~576 n 24x; tlierefore 2x = 576, and x n — - = 288. So that there were 128lb. of copper, 84lb. of tin, and 76lb. of lead. PROBLEM xiy. What sum of money is that, from which 5l. being sub- tracted, two-thirds of the remainder shall he 40l- ? Let X represent the required sum; then, 5 being sub- tracted, there will remain x — b \ two-thirds of which will be X — 5 X -, Qf ^^— — ; and so, by the question^ Or— -10 , we have — = 40: whence ex -^ lo = 120- o J 130 ^ and X zz: — n 65, 2 PROBLEM XV. What number is that, ichich being divided by 12, tht fjuotient, dividend, and divisor, added all together, shall amount to 64 ? Let 0? rr the required number; so shall -^ + :c -f 12 :ir 64, by the conditions of the question. Whence ;t + 12>r = 52 X' 12, or \dx zz 624; and con- .1 ^24 sequently a;- ,z^ •--— :;:; 48. PROBLEM XVI. To find two numbers in the proportion of 2 to i, so that if 4 be added to each, the two sums thence arising shall be in proportion as 3 to 2. Let X denote the lessor number ; then the s^reater will be denoted by ^x ; and sr^, hy the (/uestion, we shall have 2a: -4- 4 : X I 4 :: 3 : 2. From whence, as the product of the two extremes, of any four proportional num- bers, is equal to the product of the two means, fsee TO THE KESOLUTIOX OF PROBLEMS. 83 Sectioyi 10, Theorem 1 ) we have the following equation, viz, 2.x. 4- 4 X 2 :^ X + 4 X 3, that is, 4x + 8 — 3a' 4- 12 ; whence x n 4, and 2:r — 8 : which are the two numbers that were to be found. PROBLEM XVII. A prize of 20001. ivas divided heUceen two persons^ whose shares therein were in proportion as 7 to g : lohat was the share of each ? Ifx — the share of the first, then ^/tai of the second v;ill be 2C0O — x ; and we shall have x : 9000 — x : : 7 : 9- Hence, by multiplying the extremes and means^ gx — 14000 — 1x ; from which x is found =: — .— n^ 16 875/. and 2000 — x zz 1125/. PROBLEM XVIil, A bill of 120l. ivas paid in guineas and moldores^ and the number of pieces of both sorts was just 100 ; to find hoiv many there were of each 9 If a? — the number of guineas, then will 100 — a? be the number of mo'dores : therefore the number of shillings in the guineas being 21x, and in the moidores, 27 X 100 — X, we have 2lx + 27 x 100 — x zz 120 X 20 r: the shillings in the whole sum: hence, by multiplication, 2lx + 2700 — 27^ = 2400; and 300 xzz~ =50. PROBLEAr XIX. A labourer engaged to serve 40 days, on these condi* tionsf that for every day he vmrked he was to receive 20 pence, but that for every day he played, or ivas absent, he was to forfeit S pence; now after the 40 days were ex* pired, it was found that he had to receive, upon the whole, 380 pence : the question is, to find how many days of the 40 he worked, and how many he played. Let the number expressing the days he worked be re- presented by X ; then the number of days he played wiH G 2 S4 TUE AlPLICATION OF ALGEBRA be expressed by 40— a:: moreover, since he was to re- ceive 20 pence tbrevery day he worked, the whole num- ber of pence gained by working, will be 20r j and for the like reason, the number of pence forfeited by play- ing, or being absent, will be 8 x 40 — .r, oi^ 320 — 8x ; which deducted from 20x, leaves 28i^ — 320, for the sum total of what he had to receive : whence we have this equation, QSx— 320 =:: 380 ; from wdiich 28x - 380 -f 320 ~ 700, and consequently j; — ~~ — 25, equal to the number of days he worked ; therefore 40 — 25 zz 15, will be the number of days he played. PROBLEM XX. ^farmer would mix two sorts oj grain, viz. wheat, worth 4s. a bushel, with rye, worth 2s, 6d. the bushel, so that the tchole mixture may consist of 100 bushels, and be worth 3s. and 2d. the bushel: now it is required to Jind how many bushels of each sort must be taken to make up such a mixture ? Let the number of bushels of wheat be put r: a',and the number of bushels of rye will be 100 — x\ but the number of bushels multiplied by the number of pencq per bushel, is equal to the number of pence the whole is worth ; therefore 48^^ is the whole value of the wheat, and 30 X 100 — X, or 3000 — SOx, that of the rye; and conseiquently, 48.r -f 3000 — 30t, the sum of these two, the whole value of the mixture: which, by the question, is equal to 100 x 38, or 3800 pence: hence we have ASx + 3000 — 3Qx — 3800; and therefore X — - — — AA^, the number of bushels of wheat ; 1 8 whence the number of bushels of rye will be loO — 44f- - 53f ' PROBLEM XXI# A farmer sold, to one man, 30 bushels of wheat and 40 of barley, and for the ichole received 270 shillings; and to another he sold 50 bushels of wheat and 30 of barley, at TO THE RESOLUTION OF PROBLEMS. 85 the same prices^ and for the ivhole received 340 shil^ lings : now it is required to find what each sort of grain was sold a^ per bushel ? Let X and y be, respectively, the number of shillings which a bushel of each sort was sold for; then, from the conditions of the question, we shall have these two equations, viz. 30t + 40y = 270, box 4- 30^ — 340; from 4 times the second of which subtract 3 times the 550 first, so shall llOi- =: 550; and consequently x — - — = 5 ; moreover, by subtracting 3 times the second, from 5 times the tirst, you will have 1 lOy, z= 330, and there- r 330 ^ lore // z: zz 3 For PROBLEM XXII. A son asldng his father hotv old he icas^ received the following reply : My age, says the father, 7 years ago, was just four times as great as yours at that time; but^ 7 years hence, if you and I live, my age will then he only double to yours: it is required to find from hence, the age of each person ? Let X represent the age of the son seven years be- fore the question ; then the age of the father, at that time, was 4.r, by the conditions of the question ; and, if each of these ages be increased by 14, it is plain that X -{■ J 4 and 4JC + 14 will respectively express the two ages 7 years after the time in question ; whence, again, by the problem, we have4.r 4- 14 =: 2 x x-\- 14 ; from which X zz 7, and 4t zi 28 ; therefore 7 -+- 7 = 143, and 28 + 7 — 35, are the two ages required. 110 V 30 X 5 4- 40 X 3 - 270, ^ 50 X 5 -f- 30 X 3 zz 340. For : S 35— 7 =i 14— 7 X 4, I 35 4- 7 - 14 -f- 7 X 2. g3 88 THE APPLICATION OF ALGEBRA PROBLEM XXIII. A ^evtleman hired a servant for \ 2 monthf:^ and agreed to allow him 90l. and a livery, if he staid till the year was expired ; hut at the end of 8 moJiths the servant went away and received 12l. and the livery, as a proportional part of his wages: the question is, what was the livery valued at? Let X be the value sought ; then 20 4- x will be the whole wages for 12 months, and 12 -V- x the part there- of which he received for 8 months. But the wages being in the same proportion as the times in which they are earned, or become due, we therefore have, as 12 : 8 :: 20 -f x : 12 4- a^ ; whence 12 ^ T2 ^ .T - 8 X 20 t x,(>x 144 + 12>r =: 160 4- Sx (byTheor. 1. p. 72; ; consequently I2a? — 8r ~ 160 — 144, and X = -, z= 4I. PROBLEM XXIV. Four persons A, B, C, D^, spent twenty shillings in company together ; whereof A proposed to pay y ; B i; C^; and Depart; hut, when the money came to be collected^ they found it icas not sufficient to answer the intended purpose : the question then iSy to find how much each person must contribute, to make up the whole reck- oning, supposing their several shares to be to each other in the proportion above specified ? Let X be the share of A ; then it will be, as J : '^, or, as 4 : 3 : : a: : — - the share of B ; and, as 11 3 f rr ; t-, or^ as 6 : 3 : : x I -7- ~ the share of C ; also, as I : r, or, as 2 : I : : jr : — =: the share of D. 3X 3x X Therefore, hy the qitestion, xh— ~> ~r+'^=20. •^A' hence, AOx + 30jc -f £4a: + 20x = 800, that is 114a: TO THE HESOiUTION OF PROBLEMS. 87 ~ 800 ; and consequently x ~ ~*-^ n 7fr» ^^e share of A ; therefore {^) that of B will be = 5\^ : that of C (^j^ = 4if : and that of D (|) ri 3|f. PROBLEM XXV. A market looman bought in a certain number of eggs at 2 penny, and as many at 3 a penny, and sofd them all out again, at the rate of 5 for two pevce^ and lost four pence by so doing : u^hat 7iumber of eggs did she buy and sell ? Let 0? be the number of eggs of each price, or sort ; then - will be the number of pence which all the first sort cost, and - the price of all the second sort ; bu^t the whole price of both sorts together, at the rate of 5 for two pence, at which they were sold, will be — (for as 5 : 2 :: 2.r (the whole number of eggs) : — \ X C€ Ax hence, by the question, - -f - -— ~ — 4 ; whence ■^ o d ]5x -f lOx — e^x — 120, and therefore^ r- 120. Forli3 + i|?-215? X 2 = 60 + 40-96 = 4. 2 3 5 PROBLEM XXVI. ^ composition of copper and tin, containing 100 cuhic inches, being weighed, its ic eight was found to be 505 ounces : how many ounces of each metal did it contain^ supposing the loeight of a cubic inch of copper to be 5^ ounces, and that of a cubic inch of tin 4-^? Let X be the number of ounces of copper; then 503 — X will be the number of ounces of tin, and we shall have 5i ; 1 (cubic inch) : : a? : — ^ inches of copper^ C4 m THE APPLICATION OF ALGEBRA. 4^ •• 1 (cubic inch) : : 505 — oc : — — ~- inches of tin. Therefore -^ -f — r; — ■ = JOO, by the questhn Whence 4i X a: + 5^^ 505— a: — 5|x4iX 100, that is, 17 X .T , 21X505— cT/ 21X17X100\ 21X17X25. h — « I — J ri » 4 i V 4X4 >' 4 which, by r ejecting t he common divisor, becomes nx + 21 X 505 -^ a? — 21 X 17 X 25 — 8925. or 17^' — 21ar = 8925 — 10605 zz — 1680. From " , 1680 , , . , whence a; n n 420 ; and 505 — x — 85 ; which 4 are the two numbers required. The same otheriinse. Suppose X to be the number of solid inches of cop- per ; then the number of inches of tin being 100 — x, we have 5^ x x -{- 4\ x lOO — x — 505, that is, 5^v -\- 425 — 4jX — 505, or T - 505 — 425 - S(»: which; multipHed by 5^, gives 420, tor the ounces of copper. PROBLj:^ XXVII. A shepherd, in time of tear, fell in with a party of sol* diers, who plundered him of half hisjloch, and half a sheep over; qftencards a second part ij mtt him, who took half wha^ he had Icft^ a^td half a sheep over; and, soon after this, a third partu met him, and used him in the same manner and then he had only fve sheep left : it is required tojind what numher of sheep he had at first ? Let x (as usual) be the number souc^ht; then ac- corcjing to the question, the nuniber of sheep left, after bemg plundered tlie first time, will be expressed by JC X — ~" 1 X — "• I I.or^- ■', the half of which is ; from 2^2 4 ^_ ^ j. will be the number of sheep left after being plu!)- ro TflE RF^SOLUTION OF PROBLEMS. 89 X — - 3 dered the second time: in like manner, if from '— - — 8 ^ (the half of '^—-i) you, again, take f, there will re- main'-^^^^-^ i or --^^ the number of sheep remain- iiig at last. Hence we have ■ zz 5 ; therefore X — 7 - 40, and x — 47. PROBLEM XXVIir. The difference of two mimhers being glvefi, equal to 4 and the difference of their squares , equal to 40 ; to find the numbers. Let the lesser number be.r; then, the difference be- ing 4, the greater must consequently be a; -1- 4, and it's square xx ] Sr + 1 6, from which xx, the square of the lesser being taken away, the difterence is So; + 16: therefore 8.r + 16 zz 40; which, reduced, gives a; = 3; whence ^-i- 4 = 7; therefore the two required num- bers are 3 and 7. All the problems hitherto delivered are resolved by a numeral exegesis^ wherein the unknown quantities, only, are represented by letters of the alphabet ; which seemed necessary, in order to strengthen the Beginner's idea, at setting out, and lead him on by proper grada- tions : but it is not only more masteriy and elegant, but also more useful, to represent the known, as well as the unknown quantities, by algebraic symbols; since from thence a general theorem is derived, whereby all other questions of the same kind may be resolved. As an instance hereof, let the last problem be again resumed; then the given difference of the required numbers being denoted by «, the difference of their squares by b, and the lesser number by x ; the greater will be X + a, and its square x^ -h Qax -h a^; from which, x^, the square of the lesser number, being de- ducted, there remains 2«r -f a" — b : whence if aa be subtracted from both sides, there will remain Qaxzz h-^aa; this, divided by 2a, givesa?z: —; and m THK APPLICATION OF ALGEBRA consequently x -^ a :=: ^ -h — . Hence it appears that, if the difference of the squares be divided by twice the difference of the numbers, and half the dif- ference of the numbers be subtracted from the quotient, the remainder will be the lesser number; but if half the difference of the numbers be added to the quotient, the sum will give the greater number. Thus, if the dif» ference {a) be 4, and the difference (b) of the squares •10 fas in the case above) ; then ( — ) the difference of the squares, divided by twice the difference of the mimbers, w^ill be 5; from which subtracting {<2) half the difference of the numbers, there remains 3, for the lesser number sought ; and by adding the said half dif- ference, you will have 7 — the greater number. In the same manner, if the ditlerence of the two numbers had been given 6, and the difference of their squares 60, the numbers themselves would have come out 2 and 8 : and so of any other. PROBLEM XXIX. Havincr ^hen the sum of tiro iiiimhen^, eqnal to 30, nnd the difference of their s(juares, equal to 120 ; to find the numbers. Put o-:30, and 6 = 120, and let a* be the lesser num- ber sought, and then the greater will be a — x ; whose square is aa — 2ax ~V oc'^ ; from which the square of the lesser being subtracted, we havea^ — ^ax zz b ; this re- duced, gives j-, the lesser number, ::= •— ~ 13. Therefore the greater (a — x) will be =: a 4- h a. h -_n — -I — 17. But if the greater number had been first made the object of our inquiry, or been put = a:, the lesser would have been a — x, and it*s square a^ — oax 4- x'^i which subtracted from a* leave.< TO THE RESOLUTION OF PROBLEMS. 91 iifltx — -a^ ^ h ; whence 2ajp =: Z* -h a% and x r= ^+ ^ ' — 1 7, the same as before. PROBLEM XXX. If one agent A, alone, can produce an effect c, in tk^ time «, and another agent B, alone, in the time h ; in how long time will they both together produce the same effect .^ Let the time sought be denoted by x, and it will be, as a : a; : : e : — , the part of the effect produced by A : (TheoT. 3. p. 7-2?) also, as 6 : rr : : e : -j--, the part pro- duced by B ; therefore ^-^ \- ~ zz e, Dii'ide the •^ a whole by e, aad you will have - — h -r- = 1 ; and this, reduced, gives x zz ~. After the same manner, a \~ if therf be three a£:euts. A, B, and C, the time wherein they will altogether produce the given effect, will come abc out — —. y~ • ab -\- ac -\- be Example. Suppose A, alone, can perform a piece of work in 10 days ; 6, alone, in 12 days; and C, alone, in 16 days; then all three together will perform the same piece of work in 4^V days; for in this case, (3! being = 10, i — 12, c — 16, it is plain that abc / 10 >; 12/ 16 \_ , ab ^ ac + be \\o X 12 -f- 10 x 16 i- I'J A 16/ " ^^^* PROBLEM XXXI. Tivo travellers, A and B, set out together from ^hesame place, and travel both the same way ; A goes 29 miles the first day; 26 the second, 24 the third, and so on, decreasing two miles every day ; but E travels uniformly 20 miles every day : now it is required to find how many miles each person must travel before B comes up again with A ? Dl THE APPLICATION OF ALGEBRA. Let X r: the number of days in which B overtake;? A : then the miles travelled by B, in that lime, will be 200;; and those travelled by A, 28 26 H- 24 + 22, &c. continued to ar terms ; where the last term /^/;?/ Section 10, Theorem 3) will be equal to 28 — 2 ax — i , or 30 — Qx; and therefore the sum of the whole pro. gjession equal to §8 f 30 — 2a' x ix, or 29x — .r^ fby Theorem 4.). Hence we have 20a: ii: 29.r — x^ ; whence 20 =: 29 — X, and x — g; therefore 20 x 9 — i so, is the distance which was to be found. PROBLEM XXXII. To find three numbers, so that | the first, X of the se- condy and ^ of the third, shall be equal to 62 ; f of the first, I- of the second, and y of the third, equal to 47 ; and ^ of the first, 4- of the second, and ^ of the third, equal to 38. Put a = 62, b zz 47, and c — 38, and let the num- Tiers sought be denoted by x, y, and z ; then the con- ditions of the problem, expressed in algebraic terms^ will stand thus. b. 6 =" '' Which, cleared of fractions, become 6v + 4y 4- 35 =z 120, 20X + loy -f 12z =: 60b, I5x f 12«/ 4- 102 = 60c. And, by subtracting the second of these equation? from the quadruple of the first, (in order to exterminate z) we have 4x -[- y — 48a — 6ob ; moreover by takinj,^ 3 times the third from 10 times the first, we have 15.r+ 4y — \<20a. — J 80c; this subtracted from 4 times the last, leaves a: ^ 72a — 2406 + 180c z: 24; whence ^ ' , / 12a — 6a?-— 4y\ y (4Sa ^ C0Z> — 4x) =. 6o, and z I ^ J = 120. .r + y + 2 T 3 T a' 4- JL + z ^3" 4 T .T -f y. + s T 5 ~6 TO THE RESOLUTION OF PROBLEMS. r^jL + I + H2 = ,2 + 20 + 30 = 62, For<4- +— + -^ = 8 4- 15 4- 24 = 47,- I 3 4 3 !24 ^60^ ^^ 6+,2 + 20=:38. ^4 5 6 PKOBBEM XXXIII. yif gentleman left a sum of 7noney to he divided amons^ four servant?, so that the share of the first icas j the sum of the shares of the other three ; the share of the second | 4)f the sum of the other three ; and the share of the tliird - of the sum of the other three ; and iticas also fornid that the share of the first exceeded that of the last by 14/,; the cjuestion is, lohat icas pieiohole stim, and ichat teas the share of each person ? Let the shares be represented by a?, ?/, 2, and «, re- spectively, and let a =- 14 ; then, ly the question, we shall have Z = 3 4 u — X -^ a. Which equations, cleared of fractions, become so: zr 2/ + 2 + w, 3y zz X -^ z -^ u, 4z iz x + 2/ + «, u zz X — a, Now, if X be added to the first, y to the second, and ^ to the third, we shall get (:r +> + 2: + w) zr So? = 4y =z 5z ; and from thence z =z-—, and y = — ; which 5^4 values being substituted in the first equation, we have sr r. H. + -^ + ,,, or« z= ^; bJt, by the 4th 94 THE APPLTCATIOX OF ALGEBRA \3x , equation, u zz x — a; therefore .r — a =: —^ , and * ~ -^ = 40 : consequently y f — j r: 30, 2 (~] ~ 24, and u, {x — 14) — 2^\ and the whole sum [x -{- y ■\- z ^ u) zi 120/. PROBLEM XXXIV, To find four numbers^ so that the first together uilh half the second may be 357 f a J, the second with -j of the third equal to 476 fbj, the third with A 0/ the fourth equal to 595 fcj, and the foiirth with 4 of the first equal to ti^(dj. The required numbers being denoted by or, y, z, and a, and the conditions of the question expressed in alge- braic terms, we have the four following equations : * + -f = ^' 2^ + 4- = *' s + n a + ^=.. From the first whereof we get x iz a — — ; and from the 4th, x=:5rf — 5u; whence a • z: 5c? — 5u, and y =:2a — \od + low ; but, by the second, y zzb — ^; theieforeea — IQdi-lOuzzh ;—, and z = 3t 3 3 — 6a + 30d - 30M ; but, by the third, z zz c ; "^ 4 "Whence 3b — 6a+30rf — 30?^ = c , and 12^ — 4 24a -f lS0(f «— 120K s: 4c — u; consequently u zz TO THE UESOLUTION OF PROBLEMS. 95 125— 24a -4- 1 20d — 4c 119 676 ; whence z [c — ) y 426,^ {—b ) -334, nud x (- a ~-)~ 1 3 ' ^ J 90. Othencise, Let the first of the required numbers be denoted by j£ (as above) ; then, the sum of the first and i the second, being 8:iven equal to a, it is manifest that i the second must be equal to a minus the first, that is =: a — x, and therefore the second number — 2« — ^x: moreover, the sum of the second, and | of the third, being given = 5; it is likewise evident, that 5 of the third must be equal to b minus the second, that is — b — 2a -V- 2x, and consequently the third number itself -- 36 — 6a 4- fia: : In the same manner it will appear that ^ of the fourth number zz c — 36 -f 6a — Qx\ and consequently the fourth number itself, — 4c--l26-h24a — ^24x1 w^hence, hy the question^ 4c — 12& + 24a — 24a: + — =^, J ,, . — 5rf -f 20c ■— 60h -f 12Qa and therefore x = - ' - - — — - — — — ~ — igo; as above. PROBLEM XXXV. To divide the number 90 fa) into four such parts, that if the first be increased by 5 (hj^ the second de^ creased by 4 {cj, the third multiplied by 3 fd), and the fourth divided by 2 fej, the result in each case, shall be exactly the same. Let X, y, 2, and u be the parts required ; then, by the question^ we shall have these equations, viz, X + y -^ z -{- u zz Cy and X -{- b zz y — c zz dz zz — . t Whence, by comparing dz with each of the three other equal values, successively, x zz dz — Z», y tz dz + c, andw := dez\ all which, being substituted, for their equals, in the first equation, we thence geidz — ^-fJs + c-fs-^ dez zz a ; whence dez -h 2dz + z 96 TH£ application OF ALGKBRA zr a -f 6 — c, ^ndz zz -^-—--rj^, - 7- Therefore X (— dz — h) zz 16; y {~ dz ^ c) zz 'J 3; and, u {— dez) zz 42» PROBLEM XXXVI. If A and B together, can perform a piece of icork in S fa J days, A a/?c? C together in g (hj days, and B and C in XOfcJdays : hoio many days will it take each per- ■son, alone, tO perform the same work ? Let the three numbers sought be represented by .r, y, and ;:, respectively; then it will be, as x (days) : a (days) : : 1, the whole work, to — , the part thereof performed by A in a days ; and, as ?/ : a : : i : — , the y part performed by B, in the same time ; whence, hy the qtiestion, ~ + — = 1 (the whole work). And, by X y proceeding in the very same manner, we shall have these two other equations, viz, — -f -- = 1» and V — — 1 : let the first of these three equations y z be divided by a, the second by h, and the third by c, you will then have X y a * X ^ ;r ■" 6 ' y ^ c * which added all together, and the sum divided by 2, give -^-i- — + — :=-— -^ -rr + ir I ^^om ^ X . y z Qa Sb Sc I TO THE RESOLUTION OF PROBLEMS.' 97 whence, each of the three last equations being succes* sirely subtracted, we get J__ L4.i4.JL— —^^ + f^c + g^ z " 2a 2b Qc '^ Qabc ' 1 _^ 1 1 I ^ he — ac }- ab y 2a 2b 20 2abc J 1^ _ be ■\- a 2b 2c " 2a 2ahc 1440 1 I . 1 1 6c 4- «c — ah ,, — r: — +— r = — - ■ ^ r • Hence X 2a 2b 2c 2abc = 23f I. --he ^ ac •\' ab — 90 + 80 + 72 — ^^^^ — 1440 __ ,^ ^ " be — ac t ab " QO — 80 ^ 72 "~ ^*' — ga/^c _ 1440 __ ^ "" bc + ac^ab ""90 4-80—72 "* ^^* Let the work performed by A in one day be de- noted by X : then his work in a days will be ax, and in I) days it will be bx; therefore the work of B in o days will be 1 — ax; and that of C, in b days, I — hx, by the conditions of the problem ; whence it follows that the work of B, in one day, will be expressed by 1 •— ax 1 -— bx , and that of C, in one day, by — 7— ; but the sum of these two last is, by the question, equal to — part of the whole work, that is, — -f .5 2x — — ^ a b c ; ^ 1.1 1 ic -f ac — ah , whence a? = ~~ + ~ ^—-^ . — ~, equal 2a 2b 2c 2ahc * ^ to the work done by A in one day ; by which divide 1 (the whole) and the quotient,? r.will give the required number of days iii which he can finish the whole, H 98 iilE APPLICATION OF ALGEBRA PROBLEM XXXVII. To find three numbers, on these conditiovs, that a times the first, h times the second, and c times the third, shall he equal to a given number p : that d times the first, c times the second, and f times the thirds shall he equal to another given number q ; and that g times the first, h times the second, and Jc times the third, shall be equal to a third given lumber r. Let tiie three required numbers be denoted by cc, y, and z, and theli we shall have ax -{- by -\- cz tiz p, dx + cy + fz — q, gx -\- hy ^ kz — r. From 0? times the first of which subtract a times the second, and from g times the first subtract « times the third, and you will have these two new equations, j.^'2 ,^ hdy — aey -f cdz—afz — dp^aq, i 'bgy — ahy + cgz — nkz h: gp — nr ; or, which are the same, . bd — ae X 2/ + cd — af x z ~ dp — aq, and,/?i>-— vvherice x and y may also be foundi TLxaivple. Let the given equations be X A- y ■\- z zz \<2, 9.x + 3y + 42 =: 38. . 3x 4- G/y 4-102-83; Or, which is the sj^me thing, let «- \,h—\,c~\,p-\'2, d-Q, e-3,f-^, q-38,g-3,hzz6,k-\0, and r-83 : then thp.>e vnlues being substituted above in that of z, TO rilE RISOLUTION OF FROBLKMS. 09 ■♦ -n u 3 — 6 '' 24 — 38 — e — 3 X 36 — 83 It Will become ^ 2 — 4rX 3-r.6 — 2 — 3X 3 Ha vino: exhibited a variety of ej^amples of the use •nnd application of Al::eb«a. in the resoiutipn of prob- lems producins simple eq nations, I shall now proceed to give some instances thereof in such as rise to qii^^ dratic equations ; but, fir>t of all, it will be necessary to premise something, in general, with regard to these kinds of equations. It has been already observed, that quadratic equations are such wherein the hi-' best ppwer of the unknown quantity rises to two dimensions ; of whi^jb there are two sorts, viz. simp 9 quadratics, and ^df c ted ones, A simple quadratic equation is that "wherein tht square only of the unkn ivn quantity is concerned, as xx zz ab ; but an adfected one is, whep both the square and its root are found involved in different terms of the same equation, as in the equation a^ \- <2ax z=. bb. The re- solution of the tir?t of these is performed by, barely, extracting the square root, pn both sides thep of: '■ ')us in the equation x^z= ah, the value of x is given := i/oS (for if two quantities be equal, their squaie roots must necessarily be equal > The method ui solut'on -vhen the equation is adfected, is likewise by extracting the square root; but, first of all, so much is to be added to both sides thereof as to make that where the unknown quantity i;-, a perfect square ; this is usual y (ailed com- pletlnji the square, and is a ways done by taking-; half the coefficient ot the single powder of the unknown quantity, in the second term, and squarmg it, and then adding that square to both sides of the equation, 'i'hus, in the equation xo." -h Qax — b^\ the ( oefiicient of X in the second terrn being <2a, its half wd. be o, which, squared and added to both sides, gives x^ -f 2«.r H 2 lOO THE APPLICAIION OF ALGEBRA h a^ zz b- -^ a^; whereof the former part is, now, fi^ perfect square. "The square being tluis completed, its root is next to be extracted ; in order to which, it is to be obsen^ed, that the root, on the left-hand side, where the unknown quantity stands, is composed of two terms, or members; whereof the former is always the square root of the first term of the equation, and the latter the half of the coefficient of the second term: thus, in the equation, .r^ + 9ax 4- a^ = h~ -\- a', before us, the Fquare root of the left-hand side, x'^ ! Sax 4- «"> will be expressed by x f a (for x -f a x .t -f a = x' + 9ax -f «'). Hence it is manifest that x + a zi s/h~ + a% and therefore .r — \/W\~ar — o ; from which X is known. These kinds of equations, it is also to be observed, are commonly divided into three forms, according to the different variations of the signs : thus x^ -i- <2ax zr h^, is called an equation of the first form ; x~ — ^ax — ¥, one of the second form ; and ar^ — ^ax =: — />*, one of the third form; but the method of extracting the root, or finding the value of x, is the same in all three, except that, in the last of them, the root of the known part, on the right-hand side, is to be expressed with the double sign ± before it, a^ having two different affirmative values in this case. The reason of which, as well as of what has been said in general, in relation to these kinds of equations, will plainly ap- pear, by considering, that any square, as a:^ — ^ax + a^^ raised from a binomial root, x — a (or a — x) is composed of three members; whereof the first is the square of the first term of the root ; the second, a rect- angle of the first into twice the second; and the third, the square of the second : from whence it is manifest, that, if the first and second terms of the square be given or expressed, not only the remaining term, but the root itself, will be found by the method above delivered. But now, as to the ambiguity taken notice of in the tljird form, where x"^ — ^ax zz — h", or.r' — ^ax -\- a- zz a- — h^i the square root of the left-hand side may be either x — «, or a — x (for either of these, squared, produce the same quantity) therefore in :Iie former case, x zz a + \/a'^ — 6^, and in the latter, X - a —\/a'^~^b^\ both which values answer the TO THE RESOLUTION OF PROBLEMS. 101 coiiclitions of the equation. The same ambiguity would also take place in the other ibrms, were not the root (a) coufmei] to a positive value. Wlieu the highest power of the unknown quantity happens to be affected by a coefficient, the whole equa- tion must be divided by that coefficient ; and if the sign of that power is negative, all the signs must be changed before you set about to complete the square. All equations, whatever, in which there enter only two diflerent dimensions of the unknown quantity, whereof the index of the one is just double to that of the other, are solved like quadratics, by completing the square : thus, the equation a:* + ^ax^ — b, by com- pleting the square will become x* + ^ax- -f a- n b + a^; whence, by extracting the root o n both sides, X" ^ a :=: \/b -j- a* ; t herefore x^ — \/b + a^ — a and consequentl}^ x — v^v^d + a^ — a. These things being premised, we now proceed on in the resolution of Problems. PIIOBLEM XXXVIII. To find that number, to which 20 being added, and from which 10 being subtracted, the square of the sum added to twice the square of the remainder, shall be 17475, Let the number sought be denoted by x ; then, by the co nditions of the question, we shall have x + 20*^ + 2 X X — idl^ = 17475 ; that is, x^ + 40a? + 400 + ^x"^ — 40x -h 200 = 17475; which, contracted, gives Zx'* ~ 1^^^' Hence x^ n: 5625 ; and consequently, x ~ V^5625 = 75. PROBLEM XXXIX. To divide 100 into tioo such parts, that, if they be muU tiplied together, the product shall be 2100. Let the excess of the greater part above (50) half the number given, be denoted by x; then 50 + a? will be the greater part, and 50 — x^ the lesser ; therefore^ by H 3 102 THE APPLICATiO^ Oi- ALGhbUA the question^ 50 ^r x .50 — v, or 2500 — x- = 2100; whence ' 400, and cohsequentiv x - \/400 = 50 ; therefore 50 \ x — To — the greater part, and 50 — ^' ^ 30 — the less. PROBLEM XL. What two numhers art those which are to one anotiiti in the r tw of 3 fa) to 5 fbj, and whose squares ^ added together, make i666 fcj 9 Let the lesser of the two required nunibers be a- ; hx then, a :b ::x :— — the greater; therefore, by the question, x'^ rl- —^ = c; whence a"x^ + h^x-- a-c, and 3£- - — — r, ; consequently x zz a/ __1^ (^ 1/ - ^ , 7 1= 21 r= lesser numberj and — rz 35 _ '' a f " the greater. PROBLEM XLI, To frjd tico numbers, ichose difference is 8, and pro^ duct 240. If the lesser number be denoted by x, the areater will be ,r + cS ; and so, by the question, we shall have .t* 4- 6x — 24a. Now^ by completing- the square. ^* H- 8x 4- >6 (i:: 240 V 16) =^ 2^6. and by exrra;e difference shall he 12, and the sum of their squares 1424. Let the lesser be x, and t he gre ater will be :r 4- 12 ; therefore, by the problem, x 4- 12]' -f oc^ = 1424, or TO TflE RESOLUTION OF PROBLEMS. 10:> 2a" -I- 24x \- 141 — 1424; this, ordered, gives j^ 4- 12X ~ 640; which, by completing the square, becomes it- -f- 12.r i 36 (zz 640 + 36) =: 676; whence, extracting the root on both sides, we have.r + 6 =: (v/676) 26; therefore x — 20, and x + 12 zr 32, are the two numbers required. For ^ 32 •— 20 = 12, ? 32- V 20* - 1424- PJIOBLEM XLIII. To divide 36 into three such parts, that the second maj/ exceed the first by 4, and that the sum of all their squares may be 4:64. » Let X be the first part, then the second will be:r -f 4 ; and, the sum of these two being taken from (36) the whole, we have 32 — 2a?, for the third, or remaining part; and so, by the question, x* + xT4\ "" -h 32 — Q.x\ ^ — 464, that is, 6^^ — 12O2' + 1040 zz 464; whence 6x" — 120X =: — 576, and x- — 20.i' =: — 96. Now, by completing the square: x- — 20x -f 100(=: loo — - 96) — 4 ; and, by extmcting the root, x — 10 — 4- 2. Therefore a? — 10 4. 2, that is, x ~ 8, or a? =: 12; so that 8, 12, and 16 are the three numbers re- quired. PROBLEM XHV. To divide the number 100 fa) into two such parts, that their product and the difference of their squares may be equal to each other,. Let the lesser part be denoted by Xi then the greater will be a — x, and we shall have « — x a x — a — j"]* — x^, that is, ax — x- — d^ — 2ax; whence x'^ — 3ax = — ^2. and, by completing the square, x^ — Zax + — = ( — ci- -f ~~r~^ — ; 01 which the root being extracted, there comes out x =: ± i/ ^^^ and H4 104 THE APPLICATION OF ALGEBRA therefore x = ~ ± \/ ^. But cr, by the nature; of the problem, being less than a, the upper sign (-f-) 3a /baa gives a? too great; so that a^z: —- — y —- = 38,19658, &c. must be the true value required. PROBLEM XLV. The sum, and the sum of the s guares, of two numbers hevig given ; to find the numbers. Let half the sum of the tuo numbers be denoted bye, half the sum of their squares by 6, and half the difiierence of the numbers by x ; thtn will the numbers themselves be represented by a — x, and a + a:, and their squares by a^ — 9,ax -4- x^, and ci* + 2ax + a:^ ; and so we have a^ - ^ax + a^ + a^ + <2ax + a:^ ~ s';, by the ques" tion. Which equation, contracted and divided by 2 gives a- -fx^ ir 6; whenbe x^ — b — a-, and conse- quently X z = \/ b — a*. Therefor e the numbers sought are a — \/b — «% and a 4- Vb — a*. PROBLEM XLTI. The sum, and the sum of the cubes of two numbers being given; to find the numbers^ Let the two numbers be expressed as in the preceding problem, and let the sum o f their cubes be denoted by c. Therefore v^^ill a — xV \ a -H a?P = c, that is, by involution and reductioni 2a^ + ^ax- = c ; whence 6aa^ =: c — 2a', x = -— -; :=.--, -r, and x =: 6a ba 3 a* 6a 3 PROBLEM XLVII. The sum, and the sum of the biguadrates forA^th pow- ers) of two numbers being given; to find the numbers. TO THE RESOLUTION OF PROBLEMS. 105 The numbers being denoted as above, we shall here have a— x(* +• a + xY - d, that is, 2a* + 12a'a* -f Sa;*r=rf; from which, by transposition and division^ a:* -t- 6a%* :^ ici — a*; and, by completing the square, X* f 6a a:^ -I- 9a* = ici + 8a^; whence a,-* + 3a^ = \/ld V 8a^; andjConsequently^.rziv/— 3a' + v/|7i+Ba*. PROBLEM XLYIII. T/itf sutYit and the siim of the 5th powers of two nujn* hers being given, to find the numbers. The notation in the preceding problems being still retained, we shall have 2a^ + SOa^x* -f lOax* ~ e; and therefore x* + 2a^x'^ n ; and a;* + a' zr 10a 5 4/_L. 4-i^; whencex = a/ ^JL. . ^a^ PROBLJEM XLIX. fTAflt fico numbers are those, ivhose product is 120 fa) 9 and if the greater be increased by 8 ChJ, and the lesser by 5 (cj, the product of the two numbers thence arising shall be 300 (dj ? If the greater number be denoted by x, and the lesser by y, we shall have xy — a, and X -^ b y y ■{- c ■= d, by the conditions of the question. Subtract the fi rat of t hese equat ions from the second, and you will have x ■\- b x y -t- c — xy ~ d-^a, that is, ex + by + he = d — a\ where both sides being multi- plied by X (in order to exterminate ?/j, we thence have cx^ 4- bxy + bcx — dx— ax ; but xy being zz a, there- fore is bxy — abf and consequently, by substituting this value in the last equation, ex* \. ah ^ hex ~ dx — ax', whence cx^ +• bcx -{-ax — dx — — ah^ and therefore „ , ax dv ah ... . i . , x^ + bx ^ — zz ; which, by making/ ~ 106 THE APPLICATION OF ALGEBRA — /' (= 2S), will become x" — fx rr — -^ hence x^ -fx + if- = -^ + if. T-if=± or z= 12; and consequently y ( -) =z 10, or = 7, 5. 1, W2 / 10 = 120 1 or 12 4- 8 X 10 -f 3 = 300, AlsoS^^x 7,3 = I20_ ^ 16 -h S X 7,3 + 3 =: 300. PROBLEM L. To find tii'o numhers, so that their sum, their product, and the difference of their squares, may he all equal to one another. The greater being denoted by .r, and the lesser by z/, we have x t- y = ^ */, and x f y — x'- — ?/^: the last of these equations, divided by :r -f y, gives l — x — y, Avhence x ~ \ f yy ; this value, substituted for .fin the first equation, gives 1 -^^y — y -^ y' ; therefore y"^ — y = 1, and y zz -- -{■ \/-^\ consequently .t {1 4- y) ~ -^ kWZ PROBLK'M LI. To divide the number 100 fa) into two such parts, that the sum oj their square roots may be 14 fbj^ Let the greater part be x, and the lesser will bee — x; therefore, by the problem, \/ x + v/a — x — b\ and, by squaring both sides, x {• 2\/ ax — xx \- a — x — bb; whence, by transposition and division^ \/ ax — xx rz: — -^^^ : therefore, by squaring again, ax — xx — hb — aV 5 , bb . , or x^ bb^a~\ b* ax (= -^J zr — -_ -f. 10 THE RESOLUriON OF PROBLEMS. 107 a -, and X - ~ ■\- Y '■2 4." 2 '^ 4 -l^<2a — t)^ — 64 — the greater part; whence a — x 1= 36 = the lesser part PROBLEM LII. A grazier bought in as many sheep as cost him 60L •ut of which he reserved 15, and sold the remainder for 54 1. and gained two shillings a head by them : the ques- tioH is, how many sheep did he buy, and what did they cost him a head ? Let the number of sheep be x\ tlien if 1200, the number of sliillings which they all cost, be divided by.r, the quotient, -^^ , will, it is evident, be the number of shillings which they cost him a piece ; and so the number of shillings they were sold at per head will be- -f 2, by the question -y and therefore this, mul- tipliedby.T — 13^ the numberof sheep so sold, will give 1206 V 2^ ' — 30> equal to the whole numbei of shillings which they v/ere ail sold for; that is, 11 70 ^ 2^-^ ii229 =: 1080: hence we have ll70r + 2.?^ X — 18000 = 1080.Y, 2J^- H- 9OX — 18000, JP^ + 43x =r 9000, ando? = v/2306.23 — 22,5 — 75, the number of sheep ; an^ price of each. of sheep; and consequently — — - — 16 shillings, the PROBLEM Liir. Two cowitry -women, A and B, betwixt them brought 100 fcj eggs to market ; they both received the same sum for their eggs, but A. (who had the largest and best J says to B, had I brought as many eggs as you I should have US THE APPLICATION OF ALGEGR.l feceired IS fa J pence for them: but, replies E, had t brought no more than you, / should have received only 8 (bj pence Jor mine: the question is, to find how mami eggs each person had ? If the number of enfgs which A had be =z x, tlie number of B's eggs will be z: c — a: ; therefore, hij the problem, it will be, c — :r : c : : a? : -^^ - the num- c — X ber of pence which A received ; and as a^ : 6 :: c x -. - — ^ — - ~ the number of pence which B received • whence, again, hij the problem, -^^ — — ^i-^_J^; and therefore ax^ — 6 x c — a j^ =. bc'^ — ^hcx + bx^ ; which equation, ordered, gives x* + '^^ a — b a — b from whence x comes out ( ~ V he X c\/ab^hc -r. ' , ^;^l)— ZD — — ^^' But the value of 0? may be othervvisCj more readily, derived from the equation ax"^ := 6 X c — xf , without the trouble of completing- the square; for the square root beinsr extracted on both sides thereof, we have a \/a zz c — .i- x \/b; whence Xy/a^-x\/b — c\/'by and consequently x — - .^ f lOOi/S lOO-v/4 , - n ^ = — 40, as before. PROBLEM LIV. One bought 120 pounds of pepper, and as mamj of ginger, and had one pound of ginger more for a crown than of pepper ; and the whole price of the pepper ex^ ceeded that of the ginger by six croivns: hoto many pounds of pepper had he for a crown, and hoic many of ginger ? Let the number of pounds of pepper which he had for a crown be x, and the number of pounds of ginger TO THE llESOrXTION OF PROBLEMS. 109 will he X + 1 : moreover, the whole price of the pep- per will be — crowns, and that of the ginger 120 120 therefore, by the question, -—— •=. 6; whence X 1.1' "T* 1 120X +120 — 120.r = 6.r* + 6.r, and therefore .t^ + x — 20; which, solved, gives x ~ A — the pounds of pepper, and x -{- \ zz 5 — those of ginger. PROBLEM LI. To find three numbers in arithmetical progression, whereof the sum of the squares shall be 1232 ( aj, and the square of the mean greater than the product of the tivo extremes by 16 fbj. Let the mean be denoted by x, and the common dif- ference by tj ; then the numbers themselves will be x— y, .r, and a; -h y; and so, by the problem, vfe shall have these two equations, X — yV + 07* 4- 0? 4- 7/1' z: a, and X- — X — 2/ X a: i- y -\- b'. these, contracted, become 3.r* + 2,v* - a, and a;' - x^—?/' 4- ft; from the latter, whereof we get t/^ iz 6 — l 6 ; and consequently y ~ \/b — 4 ; which, substituted for y in the former, gives :t^ + 26 zz ci; whence eT^ — ~, and therefore - . / « -- <2b _ 20; so that the three required numbers are 16, 20, and 24. For ^ ^^' ^ ^^' + ^'^' - ^^^^' i 20' — 16 X 24 =: 16. PROBLEM LVI. To fi,>)d two numbers whose difference shall be 10 faj, arid if 600 fbJ he divided by each of them, the difference if the quotients shall also be equal to 10 fa J, 1 he lesser number being represented by r, the greater will be represented bya; + a; and therefore, by the prob^ J b h lem — — -— — ~rt; which, freed from fractions. X X ~\- a ' no THE APPLICATION OF ALGEBRA gives bx + ha — hsr - aoc^ - a^r, that is, ha = ax^ -\- a^a; whence, dividing by t/, and compltting the square. Vie have a: f ax -r ^a'^ _ Z> -hl-a ; theieture x t [a zz \/ b \ -J-o', and consequently x — v b ^a- — la — SO, the leaser number, whence x + a zz ZQ, the greater number. PKOBLEM LVII, To fi'td tivo nnmhers ichjse sfim is SO fa J, and if they he divided a: emattlq by each otktr, the sum c/ the (quo- tients shall be 3^ (hj. If one of the numbers be t, the other will be a — Xj X a ~~~ X and we shall therefore hcive -\ — b: which a — X X equation, brought out of fractions, becomes x 4-fl^ — 2ax -\- x'- - ax — 6a*; and this, by transposition, p:ives 2.x* -f bx' — -^ax — obx — — a\ thjt is, "2 -^ h y x' — 2 h /: ax ~ — a^ ; whereof both sides being divided by 2 -h /*, w'e have x — ax — — 2 + 6 ; whence, by completing the square, a* — ax -i a* a"" a^ , r . . /a- a - _ . , hence x — {a zz ± l/ ., ^ 4: <2 V b y ^ a \- b' andr = — ± a/ - — ^ '^\ , - ^O? or - 20; which 2 ^ 4 ' 2 f 6 two, are the numbers that were to be found. PROBLEM LVII I. To divide the numher 134 fa) into three such parts, that once thefrH, ticice the second, and three times the third, added together, may be - ^7SfbJ, and that the smn of the squares of all the three parts may Lc zz 6036 fcj. Lot the three parts be denoted by x, //, and 7, respec- tively ; then, from the conditions of ttie problem, wf shall have these three equations. .r + 7/ + z -zz a, X ^r 2?/ 1- 32 - b. W THE KESOLUTION OF PROBLEMS. IH Let the first of these equations be subtracted iVoin the second, whence ?/ -f 22 — b — a^ ov y zz b — a — 2z; also, if the double of the first be subtracted from the second, there will come out z — x — b — 9a, orx z= z \- 2a — b : wherefore, if/ be put = b ~ a {— 144), g — h — 2(2 (— 10), and for y and Xy their equals/— 2z and z — g, be substituted, bur third equation, x''^ -f y~ \- 2* = c, will become zz — '2gz -f gg -¥■ (f — 4 A 4- zz =z c; which, ordered, gives 2^ — — L^ — ^- ; whence, bv putting; h -in •—-), and completeing- the square, &c. ;:^ i-^ 3 =: 50 : therefore y { — f — 2c) = 44, and .t ( — z — g) z: 40. PROBLEM LIX. A traveller sets out from one city B, to go to another C, at the same time as another traveller sets out from Cfor B; they both travel uniformly, and in such- proportion^ that the former, four hours after their meeting, arrives at C, and the latter at B, in nine hours after : noic the question is, to find in how many hours each person per- formed the journey ? D 3 j C Let D be the place of meeting, and put a — 4^h — 9, and X — the number of hours they travel before they r.YieX : tlien, the distances gone over, with >he sume uni- form motion, being always to each other as the times m which they are described, we therefore have, BD : DC : : X ftlie time in which the first traveller goes the distance BD) : a (the time in which he goes the distance DC) : and for the same reason, BD : DC ::'h (the time in which the second goes the di'^tance BD) ; x (the time in which he goes the distance DC) : wherefore, since it appears that x is to a in the ratio of B D to DC, and h to X in the same' ratio, it follows that x : an b : x; whence x'^-ab, and ^ ~\/ ab [zz 6); therefore a -^s/ab - 10, <^nd b -f \/ah zz 15, ae the two numbers required. 112 THE APPLICATION OF ALGEBRA PROBLEM LX. There are four numbers in arithmetical progression^ whereof the prodw^t of the extremes is 3250 faj^ and that of the means 3300 (bj ; what are the numbers ? Let the lesser extreme be represented by y, find the common difference by x ; then the four required num- bers will be expressed byy, y -\- x^y -{■ 2r, and y + 3x : therefore, by the question, we have these two equations, viz. y y y -\- 3x^ or y - + 3xy — cr, and y + X y + 2x, or ?/* + ^xy 4- 2a:' = b ; whereof the former being taken from the latter, we get so?- — h — a : and from thence x — \/ ^ = 5. But, to find y from hence, we have given y'^ + 3.rr/ rz a (by the first step) ; therefore, by completing the square, &c. X Q t ^ SX y zz 4/ G 4- = 50 : and so the four ^ V 4 2 numbers are 50, 55, 60, and Q5. PROBLEM LXI. The sum fso), and the sum of the squares (308) of three numbers in arithmetical progression being given; to find the numbers. Let the sum of the numbers be represented by 3//, the sum of their squares by c, and the common diffe- rence by x: then, since the middle term, or number, from the nature of the progression, is — 6, or ^ of the whole sum, the least term, it is evident, will be ex- pressed by b — a:, and the greatest by 6 + x\ and therefore, by the question, we have this equation, /7 — xl' -h b^ "+- b + xV — c ; which, contracted, dves 'z¥ -f 2^2 — c; whence 2a;* - c — z¥, and oc zz y ^ — - 2. Therefore 8, 10, and 12, are the three numbers sought. TO THE RESOLUTIOy OF PROBLEMS. 113 PROBLEM LXII. Having given the sum (h), and the sum of the squares fcj of any given number of terms in arithme- tical progression ; to find the progression. Let the common difference be e, the first term x -{- e, and the number of terms n : then, by the question^ we shall have X f e + .T + 2e + T f 3e x ^r ne — b, and X 4- e|' -1- X + 2eV- + x + 3e\" x -^ ne]" z= c. But {by Sect. 10, Theo. 4.) tlie sum of the first of these progressions is nx 4- -^ — — '— : And the sum of the second (as will be shewn further on) is =: nx^ -h n . n -{- I . 2n + \ . e n , n + I . xe ^ '- ' : therefore our 6 two equations will become nx -i zz 0, and n , n ^ \ . 2n ■\- ] » e- vx^ -[- n , n ^ I . xe H — „ — — = c. o Let the former whereof be squared, and the latter multiplied by n, and we shall thence have n'^x' 4- n^ . ^Tl . ^e + ^-^^+^^'''^' _ h\ and w*a?2 + w"- . w + 1 . xe -h g — = nc let the first of these be subtracted from the second, so ;hall t^IL±^^l±ll^ « ^ ^n^-xV, ^ ^ ^^__^, C 4 W^ . W 4- 1 . 2/2 +- I »- . W -i- 1 > But - * ^ \ ' ' is -«- .w -I- 1 X 6 4 2/2 1-1 w 4- 1 , 8/2 + 4 — 6/2—6 w . /2 t- I X 24 ^ <**3 «i 1 1 HE APPL 27J — 2 24 71- . 7*2 12 ICATION OF ALGEBRA 4M|r Therefore 12 1.62 zz nc — i*, / 6 w + 1. lence x { 12 and e — ^12.c- -12^2 :: : wn - 1 is known. Example, Let the given number of terms be 6, their sum 33, and the sum of their squares 199 ; then, by writing these numbers, respectively, for n, 6, and c, we shall have 6 = 1; whence a; 1= 2, and the required numbers 3, 4, 5, 6, 7, and 8. PROBLEM LXIII. Two post-boys A and B set out, at the same time, from two cities 500 miles asunder, in order to meet each other : A rides 60 miles the first day, 55 the second, 50 the third, and so on, decreasing 5 miles every day : but B goes 40 "^ 7niles the first day, 45 the second, 50 the third, ^c. iU" creasing 5 miles every day ; now it is required to find in what number of days they will meet ? In order to have a general solution to this problem, let the first day's distance of the post A be put = m, and the distance vi'hich he falls short each day of the preceding — d; also the first day's distance of the post B — p, and the distance which he gains each day = e; and let x be the required number of days in which they meet: then the whole distance travelled by A will be 'expressed by ihe fallowing arithmetical progression, 7» H- m — d •+■ m — 2d 4- m — 3d, &c. and that of B by p ^ p -4- e + /> -f 2e + p -r 3e, &c. where each pro- gression is to be continued to x terms. But the sum of the first of these progressions (61/ Sect, lo, Theor,4:.yis =z mx _£2L£ZZJJ1^, and that of the second = px -h cTX x — 1 X e therefore these two last expressions, add- TO THE RESOLUTION OF PROBLEMS. 115 €(\ together, must, by the conditions of the question, be equal to 500 miles, the whole given distance ; which we will call b, and then we shall have p 4- m x ar -h X X X — 1 xe — d , ^.g-a?x X — 1 _ . = b,ovfx + ^- = h, by writing / = /> -f m, and g = e — d\ which equation is reduced to gx"^ -^ gx ■\- ^fx ir 26, or a^* — x -\~ 2 fx qI) ■^ ~ F' ^^^"^^» ^y completing the square, &c. x o o _______________ .L+L But ill g 2 . 1/26 7 1 comes out — V — -{- ^ g g ^ the particular case proposed, the answer is more simple, and may be more easily derived from the first equation X X X — 1 X e — d p-\-mAx\ z= b; for, e being r: J, ^ — will here entirely vanish out of the equation; and therefore x will be barely — ^ p -\- m — — — 5. The same conclusion is also readily derived, without algebra, by the help of common arithmetic only : for seeing the sum of the two dis- tances travelled in the first day is 100 miles, and that the post B increases his distance, every day, by just as much as the post A decreases his, it i^ evident, that between them both, they must travel 100 miles every day ; therefore, if 500 be divided by ipO, the quotient 5 will be the number of days, in which they travel the whole 500 miles, %^ PROBLEM LXIV. Tico persons, A and B, ^e^ o?.it together J rom the same place, and travel both the same n-ay : A goes 8 mileF. the first day, 12 the second, 16 the third, and so on, increasing 4 miles every day : hut B goes 1 mile the first day, 4 the second, 9 the third, a-tid so on; according to the square of the number of days : the question is, to find how many days each must travel before B comes up again with A. I 2 116 THE APPLICATION OF ALGEBRA Let (4) the common difference of the progression 8, 12, 16, &c. be put — e, and the first term thereof minus the said common difference iz w?, and let the number of terms, or the days each person travels, be expressed by X : then the sum of that progression, or the number of ^ X }' -\' 1 X c miles which A travels will he x y. m ^- - — ^ (by Sect, 10, Theor.4.) And [by whatfoUoics hereafter) the sum of the progression l + 4+9 x", or the dis- tance travelled by B, will appear to be^£i!LJlL?li±J : .IP 7 x7 X. 1 -^ X .T-1- 1 X 2.1- 4- I therefore, by the question, we have — — _ X X X 1 \ yc € -=, mx + ^ — : which, divided by x and con- , , . 2.T- -1- 307 4- 1 . ex 4 e tracted, gives ~ = m -f ; whence O 2i „ . 3a? Zex ^ ; 3e 1 , , x" ^r ~ rp = 3m + -; and, by complet- f ,, . , 3x sex ,9 18e ing the square, a- + -- + ---._—- + 9^' „ . 3e 1,9 18e , 96- _ (^ 3m -1- — - - + ^ - -^ + ^^ z. A 48m + I -f 3er , ] = -^ -' ; whence 48m + 1 + 6e 4 9^^^ _ 48m + I -f Te]' -.3 3e v/48m + 1 +^3^]* , a? + -^ — — = ~ , and x =: v/48m -h 1 + 3ei* + 3e — 3 ^ . , „ _. — _ -. 7^ the number of days required. >^ PROBLEM LXV. The sum of the squares faj, and the continual pro- duct fbjy of four numbers in arithmetical progression being given; to find the numbers. TO THE RESOLUTION OF PROBLEMS. 117 Let the common difference be denoted by <2x\ and the lesser extreme hy tj — 3x; then, it is plain, the other three terms of the progression will be expressed hy y — X, y + x, and y ^ 3x respectively ; and so, hy ■^ ^Ae question, we have y — 3x1* -h y— a^r + y + a^P + y +3x1^ rz a, and y — 3x X y — X X y + X x y -{^ 3x zz h, that is^ by reduction, 4y~ 4- 20*'' — a, and from the former of which ?/* = ^a — ■ 5x" : and there- fore y' — Te-tt* — ^ax'' 4- esx"^: these values being substituted in the latter, we have -^-^a- — ^ax- + 25x'* — ^ax^ '\- 50x* -t- 9a?^ — b, and therefore a* zz — — -^ — — ; whence, by completing the square 84 lo X 84 x* ^ + ( = — + 1 = 84 4 X 84 X 84 V 84 84 X 84; ?i^; therefore a- - -^ = ±^^^*±Z 84 X 84 2 X 84 84 and X = y/3« ± 2V/846 + «^. ^j,^„^^ ^ lo8 \/^a — 3x* ) is also known. PROBLEM LXVI. The difference of the means fa), and the difference of the extremes (hj, of four numbers in continued geome- trical proportion being given; to find the numbers. Let the sum of the means be denoted by x\ then the X -\- a greater of them will be denoted by , and the les- ser by : whence, by the nature of proportionals, it I 3 WWi # '- m 118 THE APPLICATION OF ALGEBRA will be.i^J^ : iZZ^ .... -^ , ^EUL, the lesser extreme, and ^Tl? : ^±-^ .. ^±f . ^ "^ «'' thp 2 2 '• 2 • 2x — 2a' "^ greater extreme : therefore, by the problem, we have i^="2"^"~iITiS == ^' ^"d consequently FT^ - a?— a^' = 26 X a?— a x a: + a, that is, 6x'a + 20^ 1= Qb X ct"" — «*; whence x^ =: -7 , and consequently X z= ay b + a b — 3a PROBLEM LXVII. The sum, and the sum of the squares of three numbers in geometrical proportion being given; to find the num^ bers. Let the sum of the three numbers be denoted by a, and the sum of their squares by b, and let the numbers themselves be denoted by x, y, and z : then we shall have a? + y -h 2 = a, cT* f y^ + z^ zz b, and xz = y\ ^ Transpose y in the first equation, and square both sides, so shall x^ + 2az + z^ = a^ — Qay -f- y^ ; from whence subtracting the second equation, we have 2xz ^—y^ — a- — 9oy + y* -- b: but, by the third, Qxz =: 2y'; therefore y' — a- — 2ay +5^* — b; and conse- quently V == =^ • -T—- Now, to find X and 2, y may be looked upon as known ; and so, by the -^1^ second equation, we have given x* + z^ zz b — y' ; ^^ from which subtracting Qxz = Qy^, there arises x' — 2xz i- z^ zz b — 3y^ ; where, the square root being extracted, we have x — z zz \/b — 3y* ; but, by the first equation, we have x -^ z — a — y ; whence, by adding and subtracting these last equations, there results Qxzza-^y \r\/ b—Syy^ and <2z - a—y—s/b—Syy. ^ TO THE RESOLUTlOxN OF PROBLEMS. 119 PROBLEM LXVIII. The sum fsj, and the product fp) of any two num.'* hers being given ; to find the snm of the squares ^ cubes ^ biguadrates, ^c, of those numbers. If the two numbers be denoted by x and ?/ ; then will, ^^ + y = ^ } btj the problem. and a ?/ = p S •; ^ ^ The former of -which, squared, gives xx + Qxy + yy ; fc il' from whence subtracting the double of the latter, we have a;' + y- — s"^ -- 2p, the sum of the squares. Let this equation be multiplied hy x -{• y zz s ; so shall x^ -h xy X xT~y + y^ — s^ — Qsp, that is, x^ + X -^ -H y^ — s^ — Qsp {because xy •=. p, and x -V y —s), nd therefore x^ -i- y^ zn s'^ — 35/), the sum of the cubes. Multiply, again, hy x + y = s, then will x* + xy y ¥M^~^ -f ?/" =. 5* — 35'p, or x' '{- p X s^ — 2p -f y* z= 5* — Ss'^p (because x'^ + y- — s^ — 2/?). Conse- quently cT* -f ?/* z= 5* — 45*jt> -t- 2p^, i/ie 5?, '^122 THE APPLICATION OF ALGEBRA Jrom whence, and yy ^- a — u x y — uu, the value of jr (=• uu 4- ) IS likewise sriven. 4 2 / "^ PROBLEM LXXII. The sum (a), the sum of the squares [h], and the sum of the cubes [c), of any four numbers in geometrical proportion being given; to find the numbers. Let half the sum of the two means be x, and half their difference y ; also let half the sum of the two ex- tremes be z, and half their difference v^ and then the numbers themselves will be expressed thus, z — i?, X — yt X -\- y^ z -^ v\ whence, by the conditions of the problem, we have z- — V + + + X - X- X — -y + X + y X + yi* x+yf + + 2 -f U — a. Z' -tV z + rP h z- ^vV c. % z—v xz + v —x—y X a: -f 1/ (Theor.Lp,72); which, contracted, are, 22* + 2«* + 2a' -f 2?/' =: ^, 2z3 4- ^zv"' 4- 2j?3 4- Qxy" - c, Z«— 2,2 - a;C _ yC, .F Let x^ — z" 4- u*, the value of y', in the last of these equations, be substituted instead of t/^, in the two pre- ceding ones, and we shall have 2z2 4- 2»* 4- 2X- -i- 2x2 — 222 4- 2r2 = b, and 22^ + 6zv^ -^ 2a:'' -4- 6x^ — - 6.12- 4- 6j:'15« - c ; which, abbreviated, become 4x* 4- 4t" =z ^, and 22^ 4- 8x3 — 6^2- + 6j; + 6z X »* IT c. Let :^6 — 0:% the value of p% in the former of these equations, be substituted, for its equal, i n the latt er, and we s hall next have 22^ 4- Sx^ — 6x2- 4- 6.r 4- 62 x ^b — X* — c; moreover, if for 2, in the last equation, its equ al \a — a; be subs tituted, there will come o ut 2 x \a-^xy 4- 8^^ — 6x X {a — x^"" 4- 3a X |6 — xx — c ; m ^Bm * TO THE RESOLUTION OF PROBLEM?. 1^3 that is, 6aa?'' — 3a"x + ~ + — — c; therefore x"" — 4 4 ax c b a^ ^ , a --= ;5 ; and consequently oc — ■- ^ ^'^'T'^ TS» whence, 2, v, and 2/, are likewise ufl 8 4o known. 7^Ae same otherwise. Let the sum of the two means = s, and their rect- angle 1= r; so shall the sum of the two extremes = n — 5, and their rectangle also —r [hy the question; from whence, and Proh. 68, it is evident, that the sum of the squares of the means will be — s^'' — ' 2r\ an d the sum of the squares of the extremes ~ a — s^ — 2r; also, that the sum of the cubes of the means will be — s^ — 3r5, and that of the extremes =: a — s\^ — 3r X a — s\ by means whereof, and the condi- tions of the problem, we have given the two follow- ing equations, viz, s 4- a—s^'- — 4r ~ 5, or 2/^ — 2a.9 — Ar — b — ««; and^^ -}- 10^^^ ^—3ra — c,OY3as' — 3a-s — 3ar — c — a^z divide the former by 2, and the latter by 3a, and then subtract the one from the other, so shall r = —x 6 2 C CL H ; whence the value of .9 ( ~ 3a ^2 V -^^ h 2r -1 , by the first equation) is also given, being {when substitution is made) zr 2* 4 / aa h 2C 12 T ^' PROBLEM LXXIII. Having given the sum, faj, and the sum of the squares (b)^ of any number of quantities in geometrical pro* gression; to determine the progression. i # -^ 124 THE APPLICATION OF ALGEBRA Let the first term be deRote(J by x, the common latio by z, and the given ijumber of terms by n: then, by the conditions of the problem, we shall have a; + xz -{- xz"" + xz^ -f xz* ... f ^2«-^ =: a, x^+ :rV 4- x"z' + x^~z^ + a;V ... + cT's^""^ := h. Multiply the first equation by l — 2, and the second by 1 — 2^ ; so shall X — xz^ ~ a x 1 — z , and Divide the latter of these by the former ; whence will be had x + xz^ — — x 1 + z: let this equation and the first be now multiplied cross-wise, into each other, in order to exterminate x ; so shall a x 1 + s** =: — X 14-2; X I + z-^ z^ + z^ .,, z^'—K If n he an even number, put ^m — n; then our last equation, when multiplication by i -f 2 is actually made, will stand thus, ^ x 1 + z"" :r 1 + 22 f^ 22* .:.. + 2/'"-"2 _j. ^^2m^\ _j_ ^2;.. ^^,|^i^h, divided by s"», becomes -^ + — •+ z"" zz ~ + r + 2.2.2 . -;^:::2....+ -^ + -7 + 2 + 22 + 23= .... + 22'«-2 X- z ■ \r Qz"'—'^ + 2'". Let s be now put { ~ — + z) zz z the sum of the halves of the two terrhs of the series adjacent to (2) the middle one ; then, the rectangle of these quantities being 1, the sum of their squares (or half the sum of the tuo terms of the series next to those) will be — s^ — 2 {by Problem 68) ; and iIjc sum /J- 4- 2^) of half the two next terms to these last = \z^ s^ — 3s, &c. &c. TO THE RESOLUTION OF PROBLEMS. 1^5 Hence, by making d — -~ — —and putting the va- lues of -;;; + z^ (as expressed in the said problem 68) = Q, and then subtracting above, &c. our equa- tion becomes dQ, — l +^-h^* — 2+*^ — 3^ + s^ — 4^-2 -t- 2, &c. continued to m terms ; v^^hence the value of .9 may be determined. Thus, let n, the number of terms given, be four; then m being =z 2, Q (= -^ + %") will be s'^ — 2; and our equation will, here, he d x s^ — 2 1= 1 -\- s. If w be - 6, O (=r -3 + z^) will be = s^ — 3S', and we shall have d :k s^ — 35 = J +5 + 5^ — 2 =: ^^ -f ^ — 1 ; and so in other cases, where n is an even number. If n be an odd number, put 2m =: n — 1 ; and let; both sides of the equation « X 1 + 2" = — X 1 + s X 1 4- 2 + 2' ... 2 be divided by 1 + s ; so shall w~l a X 1— z 4-s'— a'...— 3'*— ^ + s"— ^= - X H-2 4-s^.. + 3«--^ (because 1 -f 2 x l — . z 1- 2^— z^-h s* ...— s'^^^+s"— ^ _ U — 2 + 2- — 2' 4- 2* ... — 2"-2 + 2"-^ * > __ C+3 Z -fx^^z*... 4-2; 2 -f2> 1 — 2") : whence, by transposition, and substituting m. b a — -- X 1 -f 2* 4- 2* ... + 2^^"* = « + "TT ^ z + z' -h 2^ . . . 2'^'"—^ ; put ^^ =: c, and let the ^ aa — b ' whole equation be divided by a x 2"* ; then will a m IZ6 THE APPLICATIOxV OF ALGEBRA f.- 1 + 1 ^.-4 c X ^ I z' 1 w— 3 * * ^.2»^-3 ^ ^^_i^ Now, if m be an even number, the powers of z in the former part of the equation will be the even ones, and those in the latter the odd ones : but if m be an odd number, then, vice versa. In the first case our equation may be wrote thus, 1 1 , „ 1 . Where, since \-z = s,—r -^ z- = s-— Q, -^ -{■ z' z z- z -. ^3 __ 3^ _L ^ 2* - ^' _ 4^* + 2, &c. we shall, z* by substituting these values in each series ( proceedi ng from the middle both ways) h ave 1 + 5- — 2 + 5* — 4s- + 2 + &c. — c into s -i- s^ — 3^^ -f- &c. But, in the second case, where m is an odd num- ber, and the even powers of z come into the second series, we shall, by the very same method, have s~^ s^ _ 35 -4- s' — 55^ + 5s + kc. — c into 1 -f 5'^ — 2s 4- ?~— 45^ f 2 +, &c. In both which cases the terms are to be so fcir conti- nued, that the exponent of s, in the highest of them, «iav be = ^^. Thus, if w, the given number of terms, " 2 be 3, t\\eQm(^-^) being - l, the equation be- longs to case 2, and will be .v - c, barely. U n zz 5, then m = 2: and therefore 1 + s^' - 2 = cs, or ^2 — 1 = C5, ^'T/ case 1, if « be 7, m will be 3; and ^m- TO THE RESOLUTION OF PROBLEMS. 147 so 5 + ^f — 3s zz c X 1 + /^ — 2, or ^^ — Qs zz c X. s^ — I, by case 2, Lastly, if n ~ g^ then m z= 4, and therefore 1 + *" — 2 + s* — 4^^ + 2 = c x 5 + s^-^3s, or i' — 3s" + i zz c X s^ — 25, ^»3/ ca^e 1. PROBLEM LXXIV. - -:r* X ^rui^^'^TL^ ^ :.« X ^!l±-^!_Li be. « 2 '— 1 X 2'* — 1 Z" + 2 + 1 cause _ - ;s' 4- 2 + 1, and — • zz z ^ 2 1 Zn 1 z'* + 1). Let this equation, and the square of the first a- zz x" X ^^'^ ^^ t-i, be now multiplied, cross- 2'' — 2z + 1 wise, irf order to exterminate x ; whence wifl be had b z^"-—" —• 29, here brought out, it is evident that no other dimensions of b, c, d, e, &c. besides the squares, can remain there- in, as no dimensions of «, besides its square, has place in this equation. In order to find the sum of all the cubes, put A(zz ?) — a -{' p — sum of the roots, and B (- P^ — qQ) = a^ -f p^ — 29 zi sum of the squares ; then, by multiplying the two equations toge- ther, we have PB =: a^ -f pa^ f p^a — 2qa -f p^ — Qpq. From whence (to exterminate pa- the next inferior power of a after the highest, a') let QA — pa^ -h p'^a \- qa -f pq (the product of the equations Q arid A) be subducted ; and there will remain PB — QA = a? — 3qa + p^ — Zpq, To this last equation (in order to take away the next inferior power of a) add three times the equation R zi 9a -t- r, so shall PB — QA -f 3R IT a^ + p3 — 3pq -f- sr. From whence it is evident that PB — QA + sR must be the re- quired sum of all the cubes a^ + 6^ + c^ + d^ ^q, K 130 THE APPLICATION OF ALGEBRA, ETC. for reasons already specified with respect to the pre- ceding case. To determine the sum of the biquadrates, put C zz a^ + p^ — 3pq -4- 3r = the sum of all the cubes; then multiplying- by the equation F — a -t- p (as be- fore), we get PC — a^ + pa"" ~\- p^a — 3pc/a + 3ra 4- p* — 3p"q 4- 3pr, From which (to exterminate pa^) subtract QB •= pa? + p^a — ^pqa -\- qa' -f p'^q — 29* (the product of the equations Q and B;) so shall PC — QB = a* — qa} — pqa + 3ra + p* — ^p'q + 3pr 4- 29- ; to this add RA — 9a* -f pqa -\~ ra + rp; then will PC — QB + RA =z a* + 4ra -f p*^4p^q -f 4pr -I- 29*; lastly, subtract 4S = 4?'flc 4- 4^, so shall PC — ■ QB 4- K A — 4S = a* 4- p* — 4p"q 4- 4pr 4- 2g- — 4:S — D, the sum of all the biquadrates. In like manner (the last equation being, again, mul- tiplied by P = a 4- p, the preceding one by Q — joa 4- q, &c. &c.) the sum of the fifth powers will be found =: PD — QC + RB — SA 4- 5T: from whence, and the preceding cases, the law of continua- tion is manifest; the sum (F) of the sixth powers being PE — QD 4- RC — SB + TA •— 6U; and the sum (G) of the seventh powers = PF — - QE 4- KD — SC 4- TB — UA 4- 7 W, &c. &c. But, if you would have the several values of B, C, D, E, &c. independent of one another, in terms of the given quantities P, Q, R, S, T, &c. then will B = P^ — 2Q, C = P' — 3PQ -I- 3R, D r= P^ — 4P^Q 4- 4PR 4- 2Q2 — 4S. E - ps __ 5p3Q ^ 5p:R 4. 5pQr _ 5PS _ 5QR f 5T, &c. &c. which values may be continued on, at pleasure, by multiplying the last by P, the last but one by — Q, the last but two by R, the last but three by — S, &c. and then adding all the products toge- ther ; as is evident from the equations above derived. These conclusions are of use in finding the limits of equations, and contain a demonstration of a rule, given for that purpose, by Sir Isaac Newton, in his Universal Arithmetic, L 131 ] SECTION XII. OF THE RESOLUTION OF EQUATIONS OF 6EVERAL DIMENSIONS. BEFORE we proceed to explain the methods of re- solving cubic, biquadratic, and other higher equa- tions, it will be requisite, in order to render that subject more clear and intelligible, to premise something con- cerning the origin and composition of equations. Mr. Harriot has shewn how equations are derived by the continued multiplication of binomial factors into each other : according to which method, supposing x — a, .T — b, X — c, X — d, &c. to denote any number of such factors, the value of a:, is to be so taken that some one of those factors may be equal to nothing : then, if they be multiplied continually together, their product must also be equa l to nothing, that is, x — a X a? — 6 X x~^c x .T — d, &c. z: 0: in which equation x may, it is plain, be equal to any one of the quantities a, 6, c,c?,&c. since any one of these being substituted insteadof ar, the whole ex- pression vanishes. Hence it appears, that an equation may have as many roots as it has dimensions, or as are expressedby the number of the factors, whereof it is sup- posed to be produced. Thus the quadratic equation X — ax X — b = or a?^ , I x -\- ab — 0, has two roots , a and b ; the cubic equation x — a x x — b X X — c — 0, or — a i ab ^ x^ -^ — 6 C a?' + «c N X — abc — o, has three roots, — c\ be) «, bf andc; and the biquadratic equation, x -r- a x abed = 0. 132 THE RESOLUTION OF EQUATlOxYS has four roots, a, 5, c, and d. From these equations it is observable, that the coefficient of the second term i* always equal to the sum of all the roots, with contrary signs; that the coefficient of the third term is always equal to the sum of their rectangles, or of all the pro- ducts that can possibly arise by combining them, two and two; that the coefficient of the fourth is equal to the sum of all their solids, or of all the products which can possibly arise, by combining them three and three; and that the last term of all, is produced by multiply- ing all the roots continually together. And all this, it is evident, must hold equally, when some of the roots are positive and the rest negative, due regard being had to the signs. Thus, in the cubic equation -«> X — a X X — h X cr f c — 0, or .r5 H h Vx" -\- ■{■ ab^ — ac> X + ahc zz o (where two of the roots, c, h, are — bc^ positive, and the other — c, is negative) the coefficient of the second term appears to be — o-— Z> + c, ^xiAthat of the third, ab — ac — be, or ab^^-a x — c-f /; x — c, conform- able to the preceding observations. Hence it follows, that, if one of the roots of an equation be given, the sum of all the rest will likewise be given ; and that, in every equation where the second term is wanting, the sum of all the negative roots is exactly equal to that of all the positive ones ; because, in this case, they mutually de- stroy each other. But when the coefficient of the second term is positive, then the negative roots, taken together, exceed the positive ones. But the negative roots, in any equation, may be changed to positive ones, and the po- sitive to negative, by changing the signs of the second, fourth, and sixth terms, and so on alternately. Thus, the foregoing equation, (a; — a x x—b X X -{- c —)x^ -\ 6>-:r- — <7cV a? + i- c) ^bcy ahc == 0, by changing the signs of the second and fourth + a} + ab'} terms, becomes a:' 4- + b>x^ -^ ac}- x -^ abc = o, or OF SEVERAL DIMENSIONS. 133 X -^ a X X -\- b X oc — c=:0; where the roots, from 4- a, + &, and — c, are now become — a, — /;, and + c. Moreover the negative roots may be changed to positive ones, or the positive to negative, by increasing or di- minishing each, by some known quantity. Thus in the quadratic equation x" + Sx f 15 = 0, where the two roots are — 3 and — 5 (and therefore both negative) if z — 7 be substituted for x, or which is the same, if each of the roots be increased by 7, the equation will become z — 7^ + 8 x g — 7 + 15 =: 0; that is, z^ — 63 4-8 = 0, or 2 — 2 X 2 — 4 = 0; where the roots are 2 and 4, and therefore both positive. This method of augmenting, or diminishing the roots of an equation is sometimes of use in preparing it for a solution, by taking away its second term ; which is always perfoum- ed by addins*, or subtracting y, -j, or -^ part, &c. of the coefficient of the said term, according as the proposed equation rises to two, three, or four, &c. dimensions. Thus, in the quadratic equation x- — Sa? -f 15 =0, let the roots be diminished by 4, that is, let a? — 4 be put = 2;, or a; = 4 + 2 ; then, this value being substituted foric, the equation will become z 4 4r — 8 x z + 4 + 15 = 0, or 2* — 1 = ; in which the second term is wanting. Likewise, the cubic equation 2^ — az" -i- bz — c = 0, by writing x — h 2, and proceeding as above, 7 + iab ^ will become x^ * , ^Ix -^ c >- = 0; and 01 others. so Hence it appears, how any affected quadratic may be reduced to a simple quadratic, and so resolved with- out completing the square; but this, by the bye. I now proceed to the matter proposed, viz, the Resolution of cubic, biquadratic, and other higher equations ; and shall begin with shewing K 3 134 THE RESOLUTION OF EQUATIONS HOW TO DETERMINE "WHETHER SOME, OR ALL THE ROOTS OF AN EQUATION BE RATIONAL, AND, IF SO, WHAT THEY ARE. Find all the divisors of the last term, and let them be substituted, one by one, for x in the given equation ; and then, if the positive and negative terms destroy each other, the divisor so substituted is manifestly a root of the equation ; but if none of the divisors succeed, then the roots, for the general part, are either irrational or impossible : for the last term, as is shewn above, being always a multiple of all the roots, those roots, when ra- tional, must, necessarily,be in the number of its divisors. Examp, 1. Let the equation x^ — 4a* «— 7x -\- 10 = 0, be pi'oposed; then, the divisors of (lo) the last term being + 1,-1, + 2, -— 2, -+- 5, -— 3, + 10, — 10, let these quantities be, successively substituted instead of x, and we shall have, 1 — 4 — 7 f 10 — 0, therefore 1 is a root; — 1 — 4 -f 7+10—12, therefore — 1 is no root; 8 — l6 — 14 4 10=: — 19, therefore 2 is no root; — 8 — 16 -h 14 + 10 = O, therefore — 2 is another root; 125 — 100 — 35 + 10 ~ 0, therefore 5 is the third root. It sometimes happens that the divisors of the last term are very numerous; in which case, to avoid trou- ble, it will be convenient to transform the equation to another, wherein the divisors are fewer; and this is best effected by increasing or diminishing the roots by an unit, or some other known quantity. Examp. 2. Let the equation propounded be y*--^y^ — 81/ + 32 =0; and, in order to change it to another whose last term admits of fewer divisors, let a? -h 1 be substituted therein for v, and it will become a?* — ^x" — \Qx + 21 =: ; where the divisors of the last term are, 1, — I, 3, — 3, 7, — 7, 21, and — 21; which being, successively substituted for x, as before, we have, 1 — 6 — !6 + 21 — 0, therefore 1 is one of the roots; 1 — 6 -fl()-l-21:=32, therefore — 1 is not a root ; 81 — 54— 48+ 21 = o, therefore 3 is another root. OF SEVERAL DIMENSIONS. 135 But the other two roots, without proceeding further, will appear to be impossible; for, their sum being equal to — 4, the sum of the two positive roots (already found), with a contrary sign (as the second term of the equation is here wanting), their product, therefore, cannot be equal to (7) the last term divided by the product of the other roots, as it would, if all the roots were possible. However, to get an expression for these imaginary roots, let either of them be denoted by v, and the other will be denoted by — 4 — v; which, multiplied toge- ther, give — 4y — v^ — 7; whence 2? — — 2 f \/ — 3, and consequently — 4 — v ~ — 2 — \/ — 3. Now let each of the four roots found above, be increased by unity, and you will have all the roots of the equation proposed. When the equation given is a literal one, you may still proceed in the same manner, neglecting the known quantity and its powers, till 3'Ou find what divisors suc^ ceed ; for each of these, multiplied by the said quantity will be a root of the equation Thus, in the literal equation x^ V- 3ax^ — 4a^i? — I2a^ =: 0, the numeral divisors of the last term being 1, — 1,2, — 2, 3, — 3, &c. I write these quantities, one by one, instead of cT, not regarding a ; and so have I + 3 — 4 — 12 = — 12, therefore a is not a root; — If 34- 4 — 12=-- 6, therefore — « is no root; 8 -f 12 — 8 — 12=0,therefore2aisoneoftheroots; -^8-fl2-f 8 — 12 ± 0, therefore— 2a is another root; 27 f 27 — 12 — 12 = 30, therefore 3a is not a root ; — 27+ 27 + 12-^12 =0, therefore— 3a isthe 3d root; The reason of these operations is too obvious to need a further explanation. I shall here subjoin a different way, whereby the same conclusions may be derived, from Sir Isaac Newtoris Method of Divisors ; which is thus : Instead of the unknoicn quantity substitute, successively three, or more adjacent terms of the arithmetical progres- sion 2, 1, 0, — 1, -—2; and, having collected all the terms of the equation into one sum, let the quantities thus resulting, together with all their divisors, he placed in a line, right against the corresponding terms of the progres- sion ^^ 1,0, — 1, — 2 ; then seek among the divisors an K4 136 THE RESOLUTION OF EQUATIONS arithmetical progression, ivhose terms correspond with^ or stand according to the order of the terms 2,1,0, — 1, — 2, of the first progression, and tchose common difference is either an unit, or some divisor of the coefficient of the highest power of the unknown quantity [x) in the given equation. If any such progression can he discovered, let that term of it ichich stands against the term 0, in the first progression!, he divided by the common difference, and let the quotient, with the sign + or — prefixed, according as the progression is increasing or decreasing, be tried (as above J by substituting it for x in the proposed equation. Thus, let the proposed equation be x^ + 6zz X* 1007 0; then, by substituting successively the terms of the progression, 2, ], 0, — 1, instead ofo, there will arise — lo, — 4, 6, und 14, respectively; which, toge- ther with their divisors, being placed right against the corresponding terms of the progression 2, 1,0, — 1, the work will stand thus : 2 — 10 1 — 4 + 6 — 1 + 14 10 5 4 '3* 2 1.2.5 1.2.4. 1.2.3. 6 1 . 2 . 7 . 14 Now, since the coefficient of the highest power (x^) is, here, only divisible by an unit, I seek, among the di- visors, a collateral progression whose common difference is an unit; and find the only one of this kind to be 5, 4, 3, 2 ; whose third term standing against the term in the first progress ion, I therefore take and divide by unity, and then substitute the quotient, with a negative sign, instead of .r, and there results — 27 — g f 30+ 6 =i O; therefore — 3 is, manifestly, a root of the equation. Again, if the proposed equation were to be 2x^ — 5x' 4- 4.T — 10 — 0, we shall, by proceeding in the same manner, have 2—6 1 — 9 O — 10 — 1 —-21 — 2 1 — 54 6 10 21 6 1 3 5* 7 9 . 9 &c. In which case, I discover, among the divisors, the increasing arithmetical progression, 1, 3, 5, 7, 9; whose OF SEVERAL DIMENSIONS. 137 third term, 5, standing against the term o in the first progression, being divided by 2, the common difference, and the quotient (4) substituted for Xj the business succeeds, tlie positive and negative terms destroying each other. Moreover, if the equation x"^ + x^ — sgx' — 9x 4- 1 80 ~ were proposed, the work will stand as follows : 2 70 1 144 180 — I 160 — 2 90 10 6 5 8 6 14 8 6 10 9 35 9 9 16 10 70 12 &c. 10 &c. 20 &c. 15 &C. 2,5 34 4|3 5'2j4 6 113 7 6 5* Here are discovered no less than four progressions,* whose terms differ by unity: whereof the terms cor- responding to the term 0, in the first progression, are 3, 4, 3, and 3 : therefore the two former progressions be- ing ascending ones, and the two latter descending, I try the quantities + 3,4-4, — 3,-5, one by one, and find that they all succeed. And after the same manner we may proceed in other cases; but, in order to try whether any quantity thus found is a true root, we may, instead of substituting for X, divide the whole equation by that quantity con- nected to 07, with a contrary sign; for, if the division terminates without a remainder, the said quantity is manifestly a root of the equation. Thus, in the last example, where the equation is a?* -{- x^ — 29.r2 — Qiv 4- 180 =: 0, the numbers to be tried being 4-3, +4, — 3, and — 5,1 first take — 3 and join it to x, and then divide the whole equation, a:* -h x^ — QQv^ — Qx + 180 ( = 0) by cT — 3, the quan- tity thence-arising, and find the quotient to come out jj3 _j_ 4J.2 — j^^ — qq^ exactly. Therefore 4- 3 is one of the roots. Again, in order to try 4- 4, the second number, I divide the quotient, thus found, by .t — 4, and there comes out .t^ + 8jc 4- 15; therefore 4- 4 is another root: lastly, I try — 3, by dividing the last quotient by X + 3, and find it also to succeed, the quotient being ^4-5. See the operation at large. 138 THE RESOLUTION OF EQUATIONS ap— 3):r* + J?' — 29^*— 9Y + 180(a?^ + 4x*— IJjr— 60 + 4:i^— S>9i- + 4x^—l2x^ —17^'-— 9^ — 17^"f5la? -GOx f 1 80 -60J? 1 180 « — 4)ar» + 4a?2 0^ — 4a:* — 17^' — 60(a* -h 8 -\-QX' + 8^2. -17J? — SSar -f 15a — 60 -f 15x — 60 .P + 3) a? 2 -J. 8a:-f-15(x ~ 4- 3.r + 5 + 5X + 15 + 5a: 4- 1 5 As another instance hereof, let there be proposed the equation 2a^^ — Sa:^ + i6x — 24 = O; then expound- ing a: by 2, 1,0, and — i, successively, and proceeding as in the foregoing examples, we have 2 + 12 1 — 9 — 24 1 — 45 1.2.3.4. 6 . 12 1.3.9 1.2.3.4. 6 . 8 &C. 1 . 3 . 5 . 9 . 15 . 45 + o + 3 4- 4 + 5 — 1 + 1 + 3* + 5 Therefore, the quantities to be tried being 4 and -f , I first attempt the division by x — 4 ; which does not answer: but trying a: — |, or (its double) 2r — 3. 1 find it to succeed, the quotient being a:' -f 8, exactly. The reason why the divisors, thus found, do not al- ways succeed, is, because the first progression 2, l, o. OF SEVERAL DIMENSIONS. 13U — 1 is not continued far enough, to know whether the corresponding progression may not break off, after a certain number of terms ; which it never can do when the business succeeds. Thus, in the last example, where we had two different progressions resulting, had the operation, or series, 2, I, 0, — 1, been continued only- two terms farther, you would have found the first of those progressions to fail ; whereas, on the contrary, the last (by which the business succeeds) will hold, carry on the progression, 2, 1, 0, — 1 as far as you will. The grounds of which, as well as of the whole method, upon which the foregoing observations are founded, may be explained in the following manner. Let there be assumed any equation, as a:c* -f bx^ -f ex' ^ dx-^ c =0, wherein a, h^ c, d, and e, represent any whole numbers, positive or negative, and \^tpx-\-q denote any binomial divisor by which the said expressipn ax"^ -h bx^ -4- ex" -h dr 4- e is divisible, and let the quo- tient thence arising; be represented hyrx^ 4 sx' 4- tx + r, or, which is the same in effect, let ax^ -4- bx^ + ex* -f- dx-J^e — px -h q X rx^ \- sx'' H- tx V v. This being premised, suppose x to be now, successively expounded by the terms of the arithmetical progression 2,1,0, — 1 , — 2 {as above); and then the corresponding values of our divisor pa? -4- q, will, it is manifest, be expounded by 2/) -f q,p -f <7, q, — p f q, and — 2p ^ q respec- tively ; which also constitute an arithmetical progres- sion, whose common difference is p ; which common difference { p ) must be some divisor of the coefficient ( « ) of the first term, otherwise the division could not succeed, that is, p could not be had in a, without a remainder. Hence it appears that the binomial divisor, by which an expression of several dimensions is divisible, must always vary as x varies, so as to be, successively ex- pressed by the terms of an arithmetical progression, whose common difference is some divisor of the first, or highest term of that expression. It also appears, that the said common difference is always the coefficient of the first term of the general divisor; and that the term {q) of the progression, which arises by taking x — 0, is the second term. Therefore, 140 THE RESOLUTION OF EQUATIONS whenever, by proceed in<>' according to the method above prescribed, a progression is found, answering to the conditions here specified, the terms of that progression are to be considered only as so many successive values of some general divisor, as px + q. Whence the rea- son of the whole process is manifest. After the same manner we may proceed to the in- vention of trinomial divisors, or divisors of two dimen- sions: for, let mx' + px ■\- 9, be any quantity of this kind, wherein m, p, and q represent whole numbers, positive or negative, and let the terms of the progres- sion 3, 2, 1, 0, — 1, — 2, — 3, be wrote therein, one by one, instead of a;; whence it will become gw 4- 3/> + 9, 4m 4 2p -^ qy7n ■\- p + g, q, m — p + q, 4m — ^p + 9, and Qm — 3p + q, respectively; where m must be some divisor of the coefficient of the first term of the given expression; otherw^ise, the division could not succeed. Hence it appears, 1°, That the coefficient (m) of the first term of the divisor must always be some numeral divisor of the coefficient of the first term of the proposed expression. 2<=, That the product of that coefficient by the square of each of the terms of the assumed progression, 3, 2, 1, 0^ — 1^ — 2, — 3, being subtracted from the corres- ponding value of the general divisor, the remainders (3/) + q, 2/) -h 9, p + ^, q, — p + q, — 2p + 9* — 3/> 4- q) will be a series of quantities in arithmetical pro- gression, whose common difference is the coefficient of the second term of the divisor. 3°, And that the term (9) of this progression, which arises by taking x — O, will always be the third, or last term of the said divisor. From whence we have the following rule, histead of x in the quantity proposed, substitute^ successively, four or more adjacent terms of the progression 3, 2, 1,0, — 1, — 2, — 3 ; and from all the several divisors of each of the numbers thus resulting^ subtract the squares of the corresponding terms of that progression multiplied by some inimeral divisor of the highest term of the quantity proposed, and set down the remainders right against the corresponding terms of the progressions 3, 2, 1, 0, — 1, — 2, — 3 ; and then seek out a collateral progression ir/uch runs through these re* mainders; which being found, let a trinomial be assumed. OF SEVERAL DIMENSIONS. 141 ivhereofthe coefficient of thejirst term is the foresaid nu* meral divisor ; that of the second term, the common diffe- rence of this collateral progression; and ivhereofthe third term is equal to that term of the said progression which arises by taking x — O; and the expression so assurned will he the divisor to he tried. But it is to be observed that the second term must have a negative or positive sign, according as the progression^ found among the di- visors, is an increasing or a decreasing one^ Thus, let the quantity proposed be x* — x^ — 5a?'*+ 12.r — 6; and then, by substituting 3, 2, 1, 0, — i, — 2, successively, instead of a:, the numbers resulting will be 39, 6, 1, —6, — 21, and — 26 respectively; which, together vi^ith all their divisors, both positive and negative, I place right against the corresponding terms of the progression 3, 2, 1, 0, — 1,— 2, in the followino^ manner : 13 . 3 . 1 . — I . — 3 . — 13 . — 39 3.2.1.— 1.-2. — 3. — 6 •I 3.2.1. — 1. — 2. — 3.— 6 7.3.1.— 1. — 3.— 7.— -21 13 . 2 . 1 . — 1 . — 2 . — 13 ; — 26 3 39 5 6 1 1 6 1 21 2 26 Then, from each of these divisors I subtract the square of the corresponding term of the first progresion multiplied by unity (as being the only numeral divisor of the first term), and the work stands thus : 3 30. 1 — 6.— 8.— 10,-12.— 22.— 48 2 I 2. — 1.— 2.— 3.— 5.— 6.— 7. — 10 1 o. — 2. 6. 3. 2. 1.— 1.— 2.— 3.— 6 — 1 20. 6. 2. 0.— 2.— 4.— 8.— 22 —2:22. 9.-2.-3.— 5.— 6.— 17.— 30 f4 —6 +2 —3 -fo + — 2 + 3* —4 + 6 —6 + 9 Here I discover, among the remainders, two col- lateral progressions, viz. 4, 2, 0, — 2,-4, — 6, and — 6, — 3, 0, -1- 3, 4-6, +9; therefore the quantity to be tried is either x^ -\- 2j? — 2, or x^ — 3:r + 3 ; by both of which the business succeeds. 142 THE RESOLUTION OF EQUATIOiNS This invention of trinomial divisors is sometimes of wse in finding out the roots of an equation when they are irrational, or imaginary. Thus, let the equation given be a;* — 4tx' + dx"^ — 4x + 1 = ; and let x be successively expounded by the terms of the progression 3, 2, 1, O, and the numbers resulting will be 7, — 3, -^ 1 and 1; which, together with their divisors, being ordered according to the preceding directions, the operation will stand as follows : 3 7. 1.1. -7 — 2 . — 8 . — 10—16 — 2 —8 2 3 . 1,1. — 3 — 1 . — 3 . — 5—7 — 1 —5 1 1 .- -1 * * . — 2 * * —2 0[1 .- -1 * * 1 .— 1 * « 1^- 1 + 1* Here we have two progressions, — 2, — 1,0, 1 ; and — S, — 5, — 2, 1 : therefore the quantity to be tried is either a?^ — x -\- l, or x^ — 3x -h 1 ; but I take the first, and having divided x^ — 4a:^ f 3x^ — 4x + 1, thereby, find it to succeed, the quotient coming out x^ — 3x + 1, exactly* Therefore x* — 40?^ + sV — 4y -4- 1 bein g universally equal to x' — x ■\- l X X' — 3a: -i- 1 , let a?* — x -[■ i be taken — 0, and also a" — 3x + \ — 0; from the former of which equations we have x — ^ -jr \/ — f; and from the latter x -^ ±. \/^, Therefore the fou r roo ts of the given equation arei + >/^^ f- v/^^, i + a/ f and-f — n/^; whereof the two last are irrational and the two first imaginary. And in the same manner, the roots of a literal equation^ as 2* — 4(32^ -f ba'^z'^ — 4a^z + a* = 0, where the terms are homogeneous, may be derived : for, let the roots be divided by a, that is, let x be put — -^, OT ax — z; and then, this value being substituted a for 2, the equation will become x* — 4x^ f 5x- — 4x 4- 1 —0; from which x will be found, as abpve; whence 2 (= ax) is also known. Having treated largely of the manner of managing such equations as can be resolved into rational factors, whether binomials, or trinomials, I come now to ex- OF SEVERAL DIMENSIONS. 143 j)Iain the more general methods, by which the roots of equations, of several dimensions, are determined; and shall begin with THE RESOLUTION OF CUBIC EQUATIONS, ACCORD- ING TO CARDAN, If the given equation has all its terms, the second term must be taken away, as has been taught at the be- ginning of this section ; and then the equation will be reduced to this form; viz. x^ + ax — b ; where a and b represent given quantities. Put a: zz y -{- z; and then, this value being substituted for x , our eq uation becomes y^ + 3y'' z f 3^ 2- + 2 ^ + « X y -^ z zz b, or i/^+z' -f 3yz xy-{-z^axy\-zzzb. Ass ume, now, 3yz — — a ; so shall the terms 3yz x y \- z and a x y ■\- z destroy each other, and our equation will be reduced to y^ ^ z^ —b. From the square of which, let four times th« cube of the equation yz - —^a be subtracted, and we shall have y^-^ 9.y'^z^ -j- ^^ _. j2 ^ 1^ . ^nd there- fore, by extracting the square roots, on both sides, y^ — = s/ 4a 6^ 4- _; which added to, and subtracted 27 from t/3 -f ^3 _ j^ gj^gg 2^3 =zb ^Y b^ + — , and ^z^ = h ^\/bz -f 1^: hencey zz ^ + \/ ^-^f -^' ^ 27 ^ 2 "^ 4 27 and z zz~~ — y -— -f.— ^; and consequentIy;ir(— y ' 2 ^ '^ 4 271 2 »^ 4 271 Which is Carc?a/z*s Theorem: but the same thingmay be exhibited in a manner rather more commodious for 144 THE RESOLUTION OF EQUATIOxNS practice, by substituting for the second term its equal V^4:^«^ =rrC y — 2, because yz zz — 2 ' "" 4 • 27 •J-a). And this being done, our Theorem stands thus, ■ — ^ , ■=^w *T, -a Example l. Let the equation v^ 4- 3y* + gy — ishe propounded ; and, in order to destroy the second term there of, let x — 1 be put — y, so shall jT^i' -h 3 X 07 — l)^ 4-9 XX — 1 i:: 13, or.r^ + Gx = 20; therefore, in this case, a being = (5, and b — 20, we have V ! + •-,- i| / 2 4 "^ 27^ 10-f\/lG0 + S^^ 10 4- n/ioo + sI^ 20,39231 ; =: 2,732 — ,732 — 2 ; and consequently 20,3923|T y(-x-^l) := 1. Examp. 2. If the equation given be y^ — 3y' — 2?/* 8=0; then, by writin.^- x + 1 fo r y'^, i t will be- comeir+TI' — 3 X FTTl* — 2 X x + i — 8 = o, oro;^ — 5j = 12: therefore, a being = — 5, and h zi 12, X will here be equal to 6 -}- \/36 — '^V^^ y — 6 ^ 5,600 1^ + ;,^ 1,6666 &c. <5 + 5,6009!^ 6+V/36-VV S.26376 -f ,736-24 = 3; and consequently y"- {= x -^ 1) := 4; which is the only possible value of y^ OF SEVERAL DIMENSIONS.. 145 in the given equation. And it will be proper to take notice here, that this method is only of use in cases where two, of the three roots, are impossible (except when they are equal); for — + -— being, in all other cases, a negative quantity, its square root is manifestly impossible. I shall now give the investigation of the same ge- neral theorem, for the solution of cubics, by a different method; which is also applicable to other higher equations. Supposing, then, the sum of two numbers, z and y to be denoted by .v, and their product {zy) by p, it will appear ^/rom Prob. 68, p. II9) that the sum of their cubes (2^ 4- ^^) will be truly expressed by s^ — 3ps, If, therefore, 2' + y'^ be assumed — 5, we shall also have s^ — 3ps — b; but, zy being — p, or y = — , our z first equation, z' 4-2/^ = 5, will become x^ + -j- —b; z from which, by completing the square, &c. z is found = 16 + y/^bb—p']^: whence 3^ i— —) is given = z P - ' ■■ ■ , ; and consequently s {zz z-t y) zi ib-^>/ibb-^~p^^ W+y/W—p'^ + -^z::^ r ; which is, evidently, the true root of the equation s'-—3ps = h. From whence the root of the equation x^ + ax - b, wherein the second term is positive, will be given, by wntmg X for s, and ^a for — ;>; whence x is found 146 THE RESOLUTION OF EQUATIONS =i6+v/"" + ^r_ i« the same as before. In like manner, if things be supposed as above, and there be, now, given z^ -\- 2/ :=: b; then, by the prohlem there referred to, v^e likewise have s^ — bps^ + 5p^s— h. But the first equation, by substituting — for its equal y, becomes 2' + -^j- zzibi whence 2*° --b^ z=z — p'. z^ ±\b v\/ i^f^ — P^ and z 1= 16 + v/ ^bb — p^l ^ ^ and consequently 5(=zz4- i/ — 2; + ^\ ~ ib + \/i^6— p'l ' + _ = the true ib -i- \/ ibb -^ f]^ root of the equation s^ — 5p^' + 5p*^ ~ b. Which by substituting x for j, and — -— for p, gives a? n: true root of the equation x^ -{- ax"^ -\- ^a'x — bi Generally, supposing z"" -{■ y"^ = b, or z"" + -^ = b (because y zz ^\ y^e have s-'* — bz"" = — p"; whence 2« = |6 -h s/:J:66— />«, and z =• P |6 -i- V ^66— p'i" : therefore i(z + y = z+--)z: OF SEVERAL DIMENSIONS. 147 \b + >/ibb — p'^" 4- i^ ; which is ib ^• ^/ ^bb — H" the true root of the equation s" — nps'^'^^ -f n . /2 — 3 . n—^ n--4 n^ 5 3 n— 6 w — 5 n — 6 n — 7 . n—S ^ ( » 1 "\ = 6, This equation, by writing x for s, and ~ for — p, becomes x + ax + —^ — • «^ + ""i^^ 3« 2/2 3/2 4/i zz b ; and its root a?=z~r-+V — +~ — ' 2 4: nn{ a X Wherein the two preceding h ^ ./ b' a^ Theorems are included, with innumerable others of the same kind ; but as every one of them, except the first, requires a particular relation of the coefficients, seldom occurring in the Resolution of problems, I shall take no further notice of them here, but proceed to THE RESOLUTION OF BIQUADRa^TIC EQUATIONS^ ACCORDING TO DES CA-RTES. Here the second term is to be destroyed as in the so- lution of cubics ; which being done, the given equation will be reduced to this form, x* + ax"^ -f fix 4- c iz: . wherein a,6,andcmay represent any quantities whatever* L 2 148 THE RESOLUTION OF EQUATIONS positive, or negative. Assume x^ -\- px-\- q x x"-k-rx+s =: a:* + ax* -^ bx -^ c; or, which is the same, let the biquadratic be considered, as produced by the mul- tiplication of the two quadratics x^-^-px-^- 9=0, and x^ '\- rx + s zzO: then, these last being actually mul- tiplied into each other, we shall have x* -f ax- + bx + c z: a?' i Ma:^ 4- Q>x" -^ P^l X ^ qs; whence, by equating the homologous terms (in order to deter- mine the value of the assumed coefficients, p, q, r, and s) we have p + r zz 0, s -\- q -{- pr — a, ps -\- qr — b, and qs zz c; from the first of which r zz —p; from the second s + q (— a — pr) — a '\- p^ ; and iwm the third s — q — — . Now, by subtracting the square of the last of these from that of the precedent, we have 4qs zz a' + Qap^ + />* -, that is, 4c zz a" -{- Qap^ 4- p* (because gs zz c); and therefore p^ + "^ PP tap* i 4 \ P* = ^' 5 from which p will be determined, as in example the second, of the solution of cubits. Whence ^(= la + ip' + — )» andg'(=ia4- I/ — —\ are also known. And, by extracting the roots of the two assumed quadratics x^ -{- px ^- q zz 0, and p «'. + r;c + X zi 0, we have x, in the one, = ^ ± y Pt^q-^ and, in the other, = ^± y j — -^ — i-f. '1/ 2Z — $. because r = — p. Therefore the OF SEVERAL DIMENSIONS. 149 four roots of the biquadratic, x* 4- «a?' + bx -^ c EXAMPLE. Let the equation propounded be y^ — 4y'^ — Sy f 32 r: 0; then, to take away the second term thereof, let X ^ \ zz y ; whence, by substitution, x* * — 6r^ — l6x + 21 - 0; which being compared with the ge- neral equation, a:* * + ax"-\- hx + c — o, we here have a = — 6, b = ^^ l6, and c =z 21 ; and conse- quently p'* — 12p* — 48p' (= p** + 2ap* ^ ^^ J p2) - 256 (= b-). Now, to destroy the second term of this last equation also, make z ^- 4 zz p' ; and then, thij, value being' substituted, you will have z^ — 96^ z= 576; whence, by the method above explained, z will be found (= 288 f \/28sl^ — 32)'] ^ + -^ ) = 12. Therefore p ( = 32 288 4- V^288j* -^ 32! v/TTI) is = 4, ^ (- -|^+ -|- + - ) = 3, and V^f- jc* ♦ ♦ — BV — sBCx— C^S + ex + d; from whence^ by equating the homologous terms, will be given, I.2A4- ia' — B^ = b,or,2A +^a'^b=: B'; 2. aA — 2BC z= c, or, aA — c =2BC ; 3. A2— C2 = d, or. A' — d - CK Let now the first and last of these equations be multi- plied together, and the product will, evidently, be equal to ^ of the square of the second, that is 2A^ 4- laaT ZTb X A' — 2r/A — rf x \aa - 6 { " B^C*) = ^ X a^A* — 2ac'A + c'^ (- B C). Whence, denotin g the given quantities ^^ac — d, and ic' 4- <£ x ^aa — b OF SEVERAL DIMENSIONS. 151 by k and /, respectively, there arises this cubic equation, A' — {hk^ + /t'A — \l zz 0: by means whereof the vahie of A may be determined (as hath been already taught) ; from which, and the preceding equations, both B and C will be known, B being given from thence — \/2A + iaa- b, and C - ^^7"^ 2d The several values of A, B, and C, being thus found, that of X will be readily obtained : for x'' \- \ax ^ AY — Bx ^- Cr being universally, in all circumstances of .T, equal to a?* 4- ax^ + bx"^ 4- ex + c?, it is evident that when the value of x is taken such, that the latter of these expressions becomes equal to nothing, the for- mer must likewis e be — ; and consequently a- -h \ax -^ A]"^ —Ex^ Cp; whence, by extracting the square root on both sides, x^ 4- \ax + A =: ± Bo? + C ; which, solved, gives a?=+iB—|a±V^^aq-iBl^±C — A = ± tB — ia ± VrV «* =f It aB + \W ± C — A ; exhibiting all the four different roots of the given equa- tion, according to the variation of the signs. This method will be found to have some advantages over that explained above. In the first place, there is no necessity Aere,of being at the trouble of exterminat- ing the second term of the equation, in order to prepare it for a solution: secondly, the equation A^ — \bA- 4- kA — \l — 0, here brought out, is of a more simple kind than that derived by the former method : and, thirdly (v/hich advantage is the most considerable) the value of A, in this equation, will be commensurate and rational (and therefore the easier to be discovered), not only when all the roots of the given equation are com-' mensurate, but when they are irrational and even impos» sible; as will appear from the examples subjoined,* Examp. 1. Let there be given the equation x*^ l'2x — 17 = 0. * It is now well-known thst the author's concluding observation, in the above paragraph, is incorrect, as the instances in which the method holds, are very few indeed, compared with those in which it fails. L 4 152 THE RESOLUTION OF EQUATIONS Which, being compared with the general equation a?* 4- ax^ + hx'^ + ex -\- d — 0, we have a =z O, 6 =z 0, c zz 12, and d = — 17 ; therefore k l\ac — d) = 17, ^(tc' + c? X ^(/a — ^) = 36; and consequently A'— |iA* + ^'A— j/=z A^ + 17A— iszro; where it is evident, by bare inspection, that A zz I. Hence B (= v/sA + iaa-^b) -\/¥, C {- ^^^^) =- = ±fv/2 TVt 3\/2 - 4-- Therefore the four roots of the equation are | \/2 + V — 3\/2 — — » .| v/i"- V — 3\/2 - 4", - 1/2+ V 3v/2 — 4' and — I v/s — V 3v/2 ; whereof the first and second are impossible. Examp, 2. Let the equation given be x^ —^x^—dSx^ — 114a— 11 = 0. Here a — — 6, & = — 58, c = — 1 14, and J = — 1 1 ; whence A: (^ac — d) =182,/ (-^cc 4- rf x ^aa — b) := 25! 2 ; and therefore A^ + 29 A"" -f 182 A — l'^56 = O. Where, trying the divisors 1, 2, 4, 157, &c, of the last term (according to the method delivered on p. 134) the third is found to succeed; the value of A being, therefore, zz 4. Whence there is given, B zz v/75 = 5\/3, go y— C z= Y^--r= zz 3v/3, and X (= ± iB -^ ^a ± v/tV^' -T- 4-«B + iB* ± C — A) = ± i\/3 + -J ± OF SEVERAL DlME.VSTONS. 153 Examp. 3. Let there he now proposed the literal equa- tion z' -f ^az^ -^ SI a"z —■ ?>Sa^z -^ a* = 0. This equation, by dividing the whole by a*, and writing a? zi — , is reduced to the following numeral one, x'^ -jr 2a?^ — 31 x^- — 38.r + l n o. If, therefore, cz, 6, c, and d, be now expounded by 2, — 37, —38, and 1, respectively, we shall here have /^ [\ac — d) =: — 20, /(Jc* + c? X ^aa — b) - 399; and therefore by substituting these values, A^ 4 3_7 A^ — 20A - iL^ z: 0. or, 2A^ 4 37A' — 40A — 399 =0.- Which equation, by the preceding methods, will be found to have three commensurable roots, f, — 3, and — 19 : and any one of these may be used, the re- sult, take which you will, coming out exactly the same. Thus, by taking — 3, for A, we shall have x" -^ x — 3 r= ± \/^ X 4jc 4- 2 : but, if A be taken =: i, then will ^* + :r 4- i = ± v/5~ X 3a: 4- f : lastly, if A be taken =: — 19, then /— c?. But, (fB he supposed — ; then will 2A -f \a^ — 6 rz O, and also aA — c ;z o ; whence A = ^6 — ^a" = 1/ = ~; and therefore C (= x/A'—d) -\/^ff^di so that in this case (where c— ?-^ ) the general equation becomes x' + \aX'\-\fzz -f- ^/^ff—d-, which, solved, gives a: = — f a ± \^ ^A"- — \f± V^i#^=^. Lastly, ifC he supposed -=. 0, then will aA — c = 0, and A* — d = ; consequently A =: — = v/"fl?^ and B ( - v/2A 4- |a' — 6 ) =:V — — /: therefore, in re \ this case (where c? =r - — j we shall have x ^ a ^ ' + i«x + -^ From the whole of which it appears, that, if c be fiT cc cc zz -^ ; or d, either, equal to — ., or to — f/ being = 2 ^ 4/ aa^^ b — \aa) ; then the roots of the given equation. OF SEVERAL DIMENSIONS. 5 a X* -^ ax^ 4- 5x^ -\- ex ^ d — 0, may be obtained^ by the resolution of a quadratic, only. Examp. 1. Let there he given x* * — 25x^ + 6ox — 36 iz 0. Here a = 0, b zz — 25, c z= 60, and d zz — 36 ; cc therefore, /(= — 25) being — --^ (— — 25], we have, hy case l, x"^ 4- {ax zz. ± xV — / + \/ — dx that is, a^' =: ± 5r q: 6 : which, solved, gives x zz. ±i ± v/ V =t 6» that is, X zz ^ ± i, or, x zz — 4, ± -1^: so that 3, 2, 1, and — 6, are the four roots of the equation propounded. Examp, 2. Let there be now given x- -r 2*2*'^ + Sq^x'^ + 29^0; — r* - 0. Then, a being — 29', 5 = 3<7*, c =z 2q^, and (^ rr. J? ~r*, thence will/ (=5 — ^aa) — 29% and -^ zz (2?/^) ZZ c; and so, the examp le belonging to case 2, w e have a^ ( = - |a + V^:^^ — i / ± v/T/ ~^^0 = — 19 ± v/ — i7? ± v/'TT^. Examp. 3, Lastly^ suppose there to he given the equa^ tion a?* — 9.T^ 4- 15a;' — 27a7 +9 = 0. Here, a being zz — 9, 6 zr 15, c = — 27, and ^ =: 9,* cc it is evident that — ( — 9) zz d [zzg)i therefore by c / Qc case 3, we have x^ -f \ax + — zz ± a? V — . -^iaa — 5, a ^ a that is, x"'— 4f^ -{- 3 [ — ± X \/ Q ^ %' — lb ) zz ± f x\/ 5 : which, solved, gives 9 ± 3v/ 5 ± V^TS ± 54\/^ X zz : • 156 THE RESOLUTION OF EQUATIONS ^HE RESOLUTION OF LITERAL EQUATIONS, WHERE- IN THE GIVEN, AND THE UNKNOWN QUANTITY, ARE ALIKE AFFECTED, Equations of this kind, in which the given and the unknown quantities can be substituted, alternately, for each other, without producing a new equation, are always capable of being reduced to others of lower dimensions. In order to such a reduction let the equation, if it be of an even dimension, he first divided by the equal powers oj its two quantities in the middle term; then aS" sume a new equation, by putting some quantity for letter) equal to the sum of the two quotients that arise by divide ing those quantities one by the other, alternately ; by means of ivhich equation, let the said quantities be eX' terminated; ichence a numeral equation will emerge , of half the dimensions with the given literal one. But, if the equation proposed be of an odd dimension, let it be, first, divided by the sum of its two quantities, so will it become of an even dimension, and its resolu- tion will therefore depend upon the preceding rule. Examp. 1. Let there be given the equation x^'—4ax'^-{' Here, dividing by a'^x^, we have -^ >— 4-5 — ° "^ aa a 4a , aa ^ , XX , aa ^ x , a ^ ^ — -I =0, (or -— 4X 1 \- 5 X XX ^ aa XX ax zz 0, byjoining the corresponding terms) ; and by making X a z — 1 -, and squaring both sides we have also a X XX , aa o \ x^''^ o"- z"' zz — +2+ — , or z^ — Q — ' — +.— . aa XX aa xx Therefore, by substituting these values, our equa- OF SEVERAL DIMENSIONS. 157 tioii becomes »* — 2 — 43 + 5 — o, or 2* — 42 = — 3 ; whence z = 3. But -— + — beinff =r z^ we ax have x"- — zax = — a- ; and consequently x =: \zii ih \/ ^a^z- — aa =z |(2 X 2± \/ 22; — 4= = ^a X 3± \/ 5, 2?2 ^Ae present case, Examp, 2. Le^ there he given x"^ + 4a;c* — I2a-a?^ — 12a^a/2 + 4a*a7 + a^ zi 0. In this case we must first divide hy x -V a, and the quotient will come outx* -h 3ax^ — 15a*x* 4- 3a^x + fi* — : whence, by proceeding as in the former ex- am ipJe, we have + C(7 ax + 3 X -a + a X 15 = 0, or 2* — 2 4- 32 — 15 ~ 0, and from thence T v/77 — 3 Examp. 3. Suppose there to he given 7x^ — Q6ax^ — Qda^x + 7a^ — 0. Which divided by a V, becomes 7 x -y -I 7- *— A X ^ a?^ , a* TVT , . , „ X a 25 X - + -^ ziO. Now, making, as before, z^ - -|- — , a- a?'' ^' ' a a?* ^* «* we have 2* — 2.=: -^ + - ; and multiplying again a X X a a?' by 2 iz 1 , we likewise have 2' — 22 zz — ? 4- "^ a X ' a^ — -I -1 -— -5-4-2+ — r; and therefore, 2^ — 32 zi -1- + -1-: which values being substituted above, our equation becomes 7 x z^ — 32 — 26 x if* — 2 = 0, or 72=^ — 262' — 212 + 52 = 0. Where, t/);ing the divisors of the Jast term, which are 1, 2, 158 THE RESOLUTION OF EQUATIONS 4, 13, &c. the third is found to answer; z, conse- quently, being = 4. Examp. 4. Wherein let there he given ^x' — 13a'a?' — I3a5A* + 2a^ ~ O. Here dividing, first, by x + a, the quotient will be ^r 2a^ = O; which, divided again by a^a'% gives .T^ a^ xa^- X , a a^ a^ tf- a;* ax H = 0, that is, 2 X 2^ — 33 — 2 X 2^ — 2 — \\z + 11 zi O, or 22^ — 22-- — 172 4-13 = {vid, p. 119): whence z — 3. A literal equation may be made to correspond with a numeral one, by substituting an unit in the room of the given quantity (or letter) : and equations that do not seem, at first, to belong to the preceding class, may sometimes be reduced to such, by a proper substi- tution; that is, by putting the quotient of the first term divided by the last, equal to some new unknown quantity (or letter) raised to the power expressing the dimensir>n of the equation. Thus, if the equation given be 2,r' + 24x' — Sldx' + 2l6x + 162 zz 0; by put- ting -__ ZI !/♦, we have x — Sy, whence, after substi- tution, the given equation becomes l62i/* + 648y^ — 28331/ 4- 648?/ f 162 zz 0: which now answers to the rule, and may be reduced down to 23/*+ ^y^ — 33^*-i- 62/ -f 2 = 0. OF THE RESOLUTION OF EQUATIONS BY APPROXI- MATION AJS'D CONVERGING SEIUES's. The methods hitherto given, for finding the roots of equations, are either very troublesome and laborious, or else confined to particular cases; but that by con- verging series's, which we are here going to explain, is universal, extending to all kinds of equations; and thouf;h not accurately true, gives the value sought, with BY APPROXIMATION. U^ little trouble, to a very great degree of exactness. When an equation is proposed to be solved by this method, the root thereof must, first of all, be nearly estimated (vt^hich, from the nature of the problem and a few trials, may, in most cases, be very easily done) ; and some let- ter, or unknown quantity (as z) must be assumed, to ex- press the difference between that value, which we call r, and the true value {x) ; then, instead of x, in the given equation, you are to substitute its equal r ± z, and there will emerge a new equation, affected only with 2 and known quantities; wherein all the terms having two, or more dimensions of z, may be rejected, as inconsiderable in respect of the rest; which being done, the value of z will be found, by the resolution of a simple equation; from whence thatof a:(— r± z) will also be known. But, if this value should not be thought sufficiently near the truth, the operation may be repeated, by substituting the said value instead of r, in the equation exhibiting the value of z; which will give a second correction for the value of x. As an example hereof, let the equation x^ 4- lOx* 4- ^Ox — 2600, be proposed : then, since it appear* that X must, in this case, be somewhat greater than 10, let r be put _- 10, and r -h z zz x; which value being substituted for x, in the given equation, we have r^ + 3r-z 4- 3rz^ + z^ + lOr^ -f 20rz + lOz* 4- 50r 4- 502 =z 2600: this, by rejecting all the terms where- in two or more dimensions of z are concerned, is re-* duced to r^ 4- 3r~z + lOr^ + 20rz 4- 50r + 5O2 — 2600; whence . comes out = gSOO-r^-lOr^-^Or 3r2 4 20r 4 50 = 0,18, nearly: -which, added to \0 { zz r), gives 10,18 for the value of x. But, in order to repeat 'the operation, let this value be substituted for r,in the last equation, and you will have z — — ,0005347 ; which, added to 10,18, gives 10,1794653, for the value of x, a second time corrected. And, if this last value be again, substituted for r,you will have a third correction of x; from whence a fourth may, in like manner, be found; and so on, until you arrive to what degree of exactness you please. im Tin: resolution of equations But, in order to get the general equation from whence these successive corrections are derived, with as little trouble as possible, j-ou may neglect all those terms, which, in substituting forcf and its powers, would rise to two or more dimensions of the converging quantity : for, they being, by the rule, to be omitted, it is better entirely to exclude them, than to take them in, and afterwards reject them. Thus, in the equation a?' + x^ + a:' — 90, let r -|- 2 he put z= .r, and then, by omitting all the powers of 2 above the first, w^e shall have r^ + 2rz - rc^, and ■i^ \- 3fz — .t\ nearly; which, substituted above, give ■r^ -h 3r's ^r r^ '\- '2,rz ■\- r -^ z:=z 90; whence z is found = — -. Therefore, if r be now taken equal to 4 {which, it is easy to perceive, is nearly the true value , . I 1, 1 , 90 — 64 — 16—4 6 X o{ x) we shall have z(— ■ :=—-) — ^ ^ 48 i- 8 + 1 57/ 0.10 &c. which, added to 4, gives 4.1, for the value of jc unce corrected; andjif thisvalue of o" be now substituted 90 — T^ — r^ — T\ for r, we shall have z { - — ^-— __-_ — ) -^ 00283 ; which, added to 4.1, gives 4.10283, for the value of j-, a second time corrected. In the same manner, a general theorem may be de- rived, for equations of any number of dimensions. Let ax" + /a:"-^^ -h cx"-2 -f dx""-^ 4- ea""^^ &c. = Q, be such an equation, where ??, a, h, c, d, &c. represent jmy given quantities, positive, or negative ; then, put- ting r \^ z — X, Vie have, by the Theorem in p. 41. x" 1= r" 4- nv^^'^z &c. x"-^ - r""-^ + 71—1 X r^'-^z &c. ^v^2 ^ ^n-2 ^ 72—2 X f«-^2 &C. &C. Which values being substituted in the p ropos ed equa- .tion,27 becomes ar" -f nmn^^z + br^'^^ -h n — l X br^'^^z BY APPROXIMATION. 161 &c. =1 Q. From which z is found = As an instance of the use of this Theorem, let the equation — x^ + 3000? z: 1000 be propounded. Here n being = 3, a=z — i, 6irO, czz 300, and Q = 1000, we shall, by substituting these values above, have lOob 4- r^ — 300r . .-. , -^ K,r m which (as it appears, by z z: 3r^ + 300 inspection, that one of the values of at must be greater than 3, but less than 4) let r be taken z: 3 ; and z 127 v/ill become — — o.5, and consequently x (z=. t 4-2) = 3'5, nearly. Therefore, to repeat the opera- tion, let 3.5 be now wrote instead of r, and z will come out zz -~-i — = — 0.027 ; which added to 3.5, gives 3.473, for the value of x, twice corrected* And, by repeating the operation once more, x will be found — 3,47296351 ; which is true to the last figure. If the root of a pure power be to be extracted, or, which is the same, if the proposed equation be x'^ = Q- then, a being — 1, and h, c, rf, &c. each := ; z, in this case will be barely — ^ "^^ ; which may serve .w— I as a general Theorem for extracting the roots of pure powers. Thus, if it were required to extract the cube root of 10; then, n being — 3, and Q :=, 10, z will be =: —r~i — ; in which, let r be taken =r 2, and it will become 'z — ^ n0.l6: therefore xi= 2.16; from whence, by repeating; the operation, the next Value of X will be found zz 2.1544. M 162 THE RESOLUTION OF EQUATIONS The manner of approximating hitherto explained, as all the powers of the converging quantity after the first are rejected, only douhles the number of figures at every operation. But I shall now give the investi- gation of other rules, or form uia\ whereby the number of places may be tripled, quadrupled, or even quin- tupled, at every operation. Let there be assumed the general equation az + bz^ + cz^ + dz* &;c. = p; z, as above, being the converging quantity, and a, 6, c, d, &c. such known numbers as arise by substituting in the original equation, after the value of the required root is nearly estimated. Then, by transposition and division, we shall have z — ^ — -^^ — - — &c. from whence, by a a a a rejecting all the terms after the firsthand writing g =~~ there will be given z — q: which value, taking in only one term of the given series, 1 call an approximation of the first degree, or order. To obtain an approximation oi the second degree, or such a one as shall include two terms of the series, let the value of z found above, be now substituted in the second term — , rejecting all the following ones ; so shall z = -^ ^ =z g —, which triples a a ^ a ^ the number of figures at every operation. For an approximation of the third degree, let this last value of z be now substituted in the second and third terms, neglecting every where all such quantities as have more than three dimensions of g : whence z will heh^d(=q-!!l' +^^-^)=q- \ ^ a aa a J ^ b . Qbb — ac^ — 9« + g\ a ^ aa ^ The manner of continuing these approximations is BY APPROXIMATION. 163 sufficiently evident : but there are others, of the same degrees, dilVeriug in form, which are rather more com- modious; and whereof the investigation is also some- what different. It is evident from the given equation, that s = — —J — — ^- — i-T-o— . If, therefore, the first va- lue of 2, found above, be substituted in the denomi- nator, and all the terms afier the second be rejected, we shall have z — — ^^-p- zz — ^^-- • w^hich is aa a + bg aa -\' bp approximation of the second degree. bq'^ But, if, for z you write its second value, q —* you will then have z (= ^i^t^ ) = a -^ bq ^ + c^' a , , ; being an approximation a + bq c . 0* a ^ of the third degree. Again, by writms^ q q" -f . q^ in the ° ^ a ^ aa room of 2;, and neglecting every where all such terms as have more than 3 dimensions of q, you will have (= ^ a ^ aa ' ^ a ^ -■■ — ? — : which .7 ^^^ o , 26' 3bc , . a + bq .— c.o'2+ -\-d.q^ * a ^ aa a is an approximation of the fourth degree. It is observable, that the powers of the converging quantity 9, in the former approximations, stand, all of them in the numerator; but here, in the denominator : but there is an artifice for bringing ihem, alike, into M 2 164 THE RESOLUTION OF EQUATIONS both, and thereby lessening the number of dimensions, without taking away from the rate of convergency. To begin with the approximation z = = — , which is of the third degree. a + bn c , n ' a '■ b c put s =: 7- — the co-efficient'of the last term of the '^ a b denominator divided by that of the last but one; so shall z zz , ■ — -; whereof the numerator and the a -f bq — hsq^ denominator being, equally, multiplied by l 4- sq, it becomes 2 — j -^ Zt-^ — — — . a -{- oq — bsq* 4- asq + bsq^ — bs-q^ but, the approximation being only of the third degree, bs-q^ may be rejected, and so we have z zz P + Pq s a + sp .p a + b + as . q ~ aa -^ b -^ as , p In the same manner, in order to exterminate the third dimension of q out of the equation. P z — ===== — ^ a ^ aa a ^ , 26 ad — be , ^ . « , , put ?o n: — \- ~j~, — the co-efficient of the last term ■^ a bb — ac of the denominator divided by that of the last but one ; then will z — ^ :=,== a + bq c . ^ + c . ivq^ '■ a ^ a ^ •— * — — _^_____ I hppftiisp s ""^ . ^^ \ • "~ a -{■ bq — bsq^ -f bsicq^ \ a b ) * whereof the terms being equally multiplied by 1 + tcq, &c. we thence have z zz r — ^-7— a — a •\- bq — bsq^ -f awq 4- bwq"^ BY APPROXIMATION. 165 JD X 1 -^ wq b -\- aw , q + 10 — s * hq'^ __,«P_x,lill^ : which is an ap- a y, aa -\- h ->f aw . p •]- 10 — s , pp proximation of the fourth degree, and quintuples the number of figures at every operation. By pursuing the same method, other equations might be determined to include 3 or more terms of the given series ; but, then, they would be found more tedious, and perplexed in proportion ; so that no real advan- tage, in practice, could be reaped therefrom. I shall, therefore, proceed now to illustrate what is laid down above by a few examples. Examp, 1. Let the equation gioen be x'^ -f 20x zz 100. Here, x appearing, by inspection, to be something greater than 4, make 4+2 =: a:; then the given equa- tion, by substitution, becomes 282 -f z^ —4:. There- fore, in this case, a = 28, b zz 1, c = 0, &c. and p zr 4; and consequently ^ ( ~ - =1 1 = ^ ^ J aa\-bq ^ 788 197/ 0.14213; which is one approximation of the value of z. b c But, if greater exactness be required, then s ( -r- ) I . , 1 1 / 26 . ad — bc^ 1 being here — - , and 10 ( — ^ + n \ = — > we ^ 28* \ a bb — ac) 14 shall, according to our two \nstformulce, have / a -^ sp , p \ __ _28 -h_T_x 4_ _ V aa ^ b -{■ as , p) ~ 28 X ~28^"2 X 4 " 28 + i 197 ^ , , ^8- ^7+2 = "13^ - 0.14213564, nearly; and ______ ap X a 4- lop 28 X 4 X 28 + f Z [ — — T—- ___ ___ . — '.^.=-.^ ^ a /. aa-\-b \- aw.p-^w — s,pp 28 x 784-r 12 + -f — -S X 28T^ _ 28 X 198 _ 5544 " 7 X 796 +- f "" 49 X 796 + 1 ~" "39005" ~ 0.1421356236, more nearly; which value is true to the last figure. M 3 M w 166 THE RESOLUTION OF EQUATIONS Examp. 2. Suppose the given equatiori, when prepared for a solution, to he 7682 f 482^ + z^ zz — 96. In this case a — 768, h — AS, c ~ \, d — 0, p — ^\ a ) 8 V a b ) \Q 48 1 , f ^h , ad — hc\ 1 48 =: — , and w ( — — + T. ) = ^4 "< a bb — acJ 8 48x48-768 — ~^ — , -^ == — • I hcrefore z — ■, — - 8 48—16 32 a+ 6+ 05 . 7 - — 96 — 96x— i>c^V _ — 96^f J _ -191 768 + 48 + 32 X —i 768-6—4"" 1516 "^ —0.1259894, nearly; or^ - 'P ± 131^ a + 6 f aicq + z^ — s ,hqq __ — 96 — 96 X — i X ^^ —96 -h f 768 + 48 172 X—I + ^VXtI 768-6 — 9 + t4t» — 96x 128 + 9X 16 12144 = — ^ ^-^ — z: — O.I 259894802 753 X 128 + 5 96389 .i-^^yoj^ou^ more nearly. In the same manner the roots of other equations may be approached: but, to avoid trouble in preparing the equation for a solution, you may every where neglect all such powers of the converging quantity z as would rise higher than the deoTee or order of the approxima- tion you intend to work by. And further to facilitate the labour of such a transformation, the following general equations for the values of p, a, h, c, d, &c. may be used. p zz k — ar — /3r* — yr^ — cr* &c. a = a + 2/3r + 3yr* + 4'>' &c. ft = ^ + 3yr + 6icr* + lOar &c. c = y -\- 4?r 4- 10ir^+ Sec, d — B + 5ir + &;c. The original equation being ax + ^.t* + yx^ + ^x* -f £X^ Sec. — k: from whence, by making r + z :=. .r, the above values are deduced. The better to illustrate the use of what is here laid down, I shall subjoin another example; wherein let n [.^ I BY APPROXIMATION. 167 there be given x^ -f 22* + 3.r^ 4- 4x- f 'ix (or 5x + 4a?2 + 3x3 ^. 2x' 4- ^') = 51321; to find x by an approximation of the second degree. In this case, k being = 51321, a z: 5, /S =: 4, y =z 3, ^ 1= 2, and e =: 1 , we have p — 54321 — 5r — Ar" — 3r^ — 2r' — r', a - b -^ Sr ^- 9/- t- 8r 4. 5r*, and 6 = 4 + 9'' + 12r^ -I- lOr^ Which values, by assuming r ir 8, will become p zz 11529, a zz 25221, and h — 5964; whence q (zz p\ , , P \ 11529 r) = °''*'' ""•* ^ '= rrr,) = 25221 ^ 2683 = 0.41 ; and therefore x {zz r + z) zz SAl, nearly. To repeat the operation, let 8.41 be now substituted for r; so shall p — 135.92, a zz 30479, b zz 6876, q(=^)= 0.00445, and z (= -^-j-) = -1|M£_ ^v «/ \ a \- bq) 30479 f 30 =: 004455: which, added to 8.41, gives 8.414455, for the next value of x. The formulcB, or approximations determined in the preceding pages, are general, answering to equations of all deo^rees howsoever affected; but in the extraction of the roots of pwre powers the proces-* will be more simple, and the theorems themselves very much abbreviated. For let x" ~ k be the equation whereof the root x is to be extracted ; then, by assuming r nearly equal to x and making r x I -^ z zz x, our equation will become r" X 1 4- zf zz k, or l \- zf zz -7, that is, 1 + nz n — l„ n — In — 2 , n — 1 + n, z- -\r n , , . 2^ -f 72 . — - — * ■ . - " . 2* &c. = — : from whence, by trans- 3 4 ^n position and division, z + — „ — • 2 H .z^ M 4 168 THE RESOLUTIOxN OF EQUATIONS a — 1 W---2 n — 3 ,„ k — r» + — »_ , — . — — , z*(x.c,z=. . 2 3 4 wr» Here, by a comparison with the general equation, az + bz^ -f cz^ + dz' &c. = p, we have a = 1, , __ n — 1 __ w — 1 7i — 2 . __ w — 1 w — 2 2 2 3 2 3 w — 3p J A; — r** , .P. — - — &c. and p zz : whence o ( ~ ) = » ; s (5 c\ n — 1 w — 2 W4-1 , /26.. — ^ : J = — — -— = — ^ — ; and w I h a bJ 2 3 6 \a -^ — c\ n — 1 ^V- w — 2 . ??— 3 — i . 7?— 1 . w— 2 i __ n — 1 w — 2 . w — 3 — -_2« — '2 , n — 2 __ ^1 2 . « + 1 ~" 71 — 1 7j — 2 n — 1 . : J ==4- X «— 3 - 2/J + 2 = : + 1 2 , w f 1 1 n — 2 n — 1 n — 2 n —, X — 72 + 1 = — : -— = -- . There- 2 . ?^ + 1 1 2 2 fore, for an approximation of the third degree, we have 1 _ a -\- sp,p __ 1 •+- ip . n, 4- 1 .p "" aa+bT~as.p "" , , « -^ 1 , w + 1 », "T" — 1 — *P P±n+ ^ •Ip'' . r . . — — ~ : and lor an approximation of the 1 + 2/2 — 1.-5/? fourth degree z z: p X 1 + ?r7 p \- \np^ 2 2 i( 6 2 P # • BY APPROXIMATION. 16'J ._ P + i"P' -. Hence it is evi- 2/i — 1 2/2—1 n — 1 a dent that the rootx (r x 1 + z) of the given equation ^"^ = k, will be equal to r + ^^T ^ ^i '^ "» wear/^; and equal to r -| ^ ^ - . 1 , . 2«— 1 . 2/2— l.w— l^p" 1 + -2— p+ [^ — more nearly. But both tliese theorems will be rendered a little more commodious, by putting v z= ^, and substituting^ tC — — r — , in the place of its equal p, whence, after proper reduction, x will be had n r -f — , V X 6o -t- 4/2 — 2 nearly; and equal to r -| r x 2!? f tz _^ 2;X2i?-|-2/2 — l + J.w — 1.2/1—1 TMore nearly, I shall now put down an example, or two, to shew the use and great exactness of these last expressions. 1. Let the equation ,:^lven be x"^ — 2, or, which is the same, let tlie square root of 2 be required. Then, assuming r zz 1.4, we have n — 2, k — 2, / nr"" \ 2 X 1.96 , , n ' [V^"nJ^ 2~::r7.g6 = ^^^ ^'^^ therefore r + ^-JL^SEEJ = 1.4 + hL^i-^ = 1.4 + V X 6v -{- 4/2 — 2 98 X 594 — l-'^ -^ • ;,^ 1= 1.41421356; which is 70X198 ' 13860 the value of j; according to the former approximation 170 THE RESOLUTION OF EQUATIONS. but, according to the latter, the answ er will come out 1.4 + ---- - 1.41421356236; which is true to the last figure: and, if with this number the operation be repeated, you will have the answer true to nearly 60 places of decimals. 2. Let it be required to extract the cube root of 1728. Here, taking r — 1 1, we shall have v ( ^ n -5— — 10.03793 ; and therefore r 4- 397 r X 2y -h « -===== ; = --— = 11.99998; 2n \~ Qn — 1 xtJ-h-g-xw — IX 2w — i which differs from truth by oniy part of an 50000 unit. 3. Let it be proposed to extract the cube root of 500. Here, the required root appearing to be less than 8, but nearer to 8 than 7, let r be taken - 8, and / 3 ^ 512\ we shall have i^ [— . - ) = — 128; and there- r X 2r -f n fore r -h 2t7 + 2/2 — 1 XV + T>^« — 1 X2« — 1 ^8 — — i_ — 7.937003259936 ; which number is 96389 true to the last place. 4. Lastly, let it be proposed to extract the first sur- solid root of 123000. In which case k being = 125000, n -- 5, r =z 10, and v — 20, the required root will be found - 10.456389. Besides the different approximations hitherto deli- vered, there are various other ways whereby the roots of equations may be approached; but, of these, none more general, and easy in practice, than the follow- ing. BY APPROXIMATION. 171 Let the general equation, az +• ftz" + cz^ -f dz* -f ez^ &c. — p, be here resumed; which, by division, be- comes z — ; i = _^+l, V^ z-^+^z^-^ ^ z^ &c. p p p p p if, therefore, we make A = — ; and neglect all the ^ " 1 . terms after the first, we shall have —- ; being an ap- proximation of the first degree. And if this value of z be now substituted in the se- cond term, and all the following ones be rejected, we 1 A A shall then have z — -, = j — -rr p p A p p by making B = -^- ; which is an approximation of the second degree. In order now to get an approximation of the third degree, let this last value be substituted in the second term, neglecting all the terms after the third; so shall • : but here, in the room of a , h A c p p i> p x% either of the squnres of the two preceding values of z, or their rectangle may be substituted, that is, either A^ 'a' W '^'~^*^^~A ^ R ' ^"^ ^^^ ^^^^ ^^ ^^^^^^ (=: -—-) is the most commodious; whence we have z zz B B . ^ aB^-/;Afc . supposmsf C — . P P P Again, for an approximation of the fourth degree, we . h 5Bc„cBAcA have — 2 — — X —; — Jj- =; — X — '< -^ = — X 7^- p p K> p pCiipC' , d ^ d B A 3 d 1 ... and — z^ zz — X -p,- x — - X -r- = — x -7^; which p p C Jd A p t 172 THE RESOLUTION OF EQUATIONS I lies being substituted in the general equation and all the terms after the four first rejected, there now comes out 1 C 2=- ■ =: ~ p pL pC pC p p p p C , , . _ aC + 5B + cA + df = g-; by makmg D = ^ In like manner, for an approximation of the fifth de- unu f> b C c ^ cCB gree.we shall have — szr — x-rr, — 2== -X^Xts p pup pDC _cB d ^ _ d C B A_dA . e ^ _ -^pD'^^ - p ^D^C ^B"-^'^""* p^ - C B A 1 € , D~ C" ^ B ^ "a ~ oD ' consequently 2 D D = ^ T 2 e - F ' supposing p p p p p tinuationismanifest; whereby it appears, «/iai if there be . A « u flA + 6 ^ aB + 6A 4- c taken A =: — , B = , C zz — ^ , p p P ^ aC^bB^-cA+d ^ aD + ^C+cB+^A + e JL> zz '— i sL, ~ — — > p P ^ rtE+ftD+rC+r/B+eA+f^ aF+bE+cD+nC-\-eB+f\+g ^ C D E / &c. then will-i.A -C-.4' E'T' -f' ^''- ^^ '" many successive approximations to the value of z, as- cending gradually from the lowest to the superior orders. An example will help to explain the use of what is above delivered ; wherein we will suppose the equation given to be 12s f 6z* + 2^ m 2. Here a == 1 2, i = 6, c = 1 , rfiz 0, e = 0, &c. and ;> = 2 ; whence A ( = --) z: 6, B (=: — ^ — J= ^ aB + AA-f-c 12 X 39 4-6 /6fl\ = 39, c( = — = ^ ; BY APPROXIMATION. 173 505 p. aC-\-bB-[-cA-hd _ 6 X'5Q5 + 6x 39"f-6 A 2 Therefore, -^ — — zz. z, nearly, a 13 zz 1635, &C B 78 , -— zz — — z, more nearly* C 305 ' ^ _ — — - = z, 5^7/ nearer, S^ From the same equations the general values of B, C, D, &c. may be easily found, iu known terms, inde- pendent of each other. Thus nizz 1 zi-^ A (because A zi— 1; ^ P V J V V pJ also C/iz — + h— j =-!-+ — ;^- + ~: \ p p p ) p p" p' and D ( z: — -f — + — + - — ) =z -^ -f . + \ p p p p J p^ p- ^ r — - 4- — &c. Therefore P P A ap B" ^ a'-t bp ' B _ p X a^ ■\- bp "C "" a^ + 2ahp + c/>' * C p X q^ -t- 2a6p + cp^ 15 ~" a* + 3a'6p + 2ac~T~bb . p" + c?p^ D p X a' i- Sa-bp -V- 2rtc + bb . p^ 4- c?/)' £ a^ -h 4a^bp + 3«c + 37>/) . ap^ + be {-ad, 2p^ -f ep^ &c. which are so many different approximations to the value of /. Thus far regard has been had to equations which consist of the simple powers of one unknown quantity, and are no ways afiected, either by surds or fractions. If either of these kinds of quantities be concerned in an equation, the usual way is to exterminate them by multiplication, or involution (as has been taught in 174 THE RESOLUTION OF EQUATIONS Sect. IX.) But as this method is, in many cases, very laborious, and in others altogether impracticable, es- pecially., where several surds are concerned in the same equation, it may not be amiss to shew how the method ot" converging series*s may be also extended to these cases, without any such previous reduction. In order to which it will be necessar}'^ to premise, that if A + B represents a compound quantity, consisting of two terms, and the latter ( B) be but small in comparison of the former ; then will. A-r-B ^ A B 1 A A^B 2Ai 2A 1 1 B 1 B 3°. 7 =77 J or — A^" 2Ai A^ ^.nearly. A^ A^ 3 A X A^ 6°. X+Bl* r: A^ +— , or A* 4- -?A 4A^ ^^ ^ ! B 1 — or — — B 4A k 4Ax A All which WMJI appear evident from the general theo- rem at p. 41 ; from whence these particular equations, or theorems, may be continued at pleasure ; the values here exhibited l)eing nothing more than the two first terms of the series there given. But now, to apply them to the purpose above mentioned, let there be given v/i \- x^ + v/2 1 .<" f n/3 +■ 0^ - 10, as an exam- ple, where, x being about 3, let 3 + e be therefore sub- stituted for X, rejecting all the powers of e above the iirst, as inconsiderable, and then the given equation em BY APPROXIMATION. 175 will stand thus, ^ lOf 6e + n/i 1 + 6e + \/l2 + 6« — 10: but, by Theorem 2, n/io + 6e will be =: Vu) ^- 3v/l0 X e^ nearly; for, in this case, A = lo, 10 I I A^B and B zz 6e, and therefore A^ -f — r- — \/l0 4- 2A 3\/l0 X e 10 in like manner is n/ii -\- 6e — \/ n ^ 3v/ 1 1 / e „ , ^, /— - . 3 >/To X e , — , &c. and consequently v 10 4 r^r + y/ll 4- Ti +v 12 f — = 10; which contracted, gives 9.944 + 2.7l8e= 10; whence 2.718C zz .056 and e zz .0205; consequently x zz 3.0205, nearly. Wherefore, to repeat the operation, let 3.0205 + e be now substituted for x; then will v/l0.12342 + 6.041e + v^U. 12342 -f 6.041e + \/ 12. 1^-^342 4- 6.04 le = 10; whence, by Theorem ^3, , 6.04 le / ^^^-^'^^^^ + 2v/iai2342 + V/1 1.12342 -f 6.04 le / 6.04 le 4-\/l2.12342 + — 7 ~ = 10, or I 2\/ 11.12342 2v^l2. 12342 9.9987814 + 272246 z: 10: from which e comes out zz .000447, and therefore x zz: 3.020947; which is true to the last place. Again, let it be proposed to find the root of the equa- 20.T x/'ir 4- T^ *'- ^7TF^^T^ + ^^^ = ''' P"' ^° + e — x: then, by proceeding as before, we shall have 400 + 20£_ ^ 20 4- g X v/405 + 40e ^^ . ^^^ v/516 + 45e 25 (6v Theorem 3.) ^ .. is nearly =: ■ , — ;=- — ^ ^ \/5l6-i- 45e V516 176 THE RESOLUTION OF EQUATIONS 45e • iosF^Tv/Md' ^"^ ^^^ Theorem 2.) >/403 + 40e = 20e which values being substituted \/ 405 above, our equation becomes — 1 45 + 2gt, is trans- ■ equations will stand thus, • 20/+/rl + 3 xW+7? "" fi-- gt /^ + ^' + ^^ x^Tjf^. -7^:=^, ^yrz:^ = 13 : which equations, if / be assumed = 5, and ^ = 4, will be reduced to 5.6462 4- .01043 x 36^ -t- mt >t- 6.3245 -f .6324 ^ =r 12, and 6.4031 4- 78U + .625 1 + 20 + 5« + 4^ X .3333 — .1852^ -\- .1482^ = 13 ; whence 1.008>9 4- .418« =.0293, and 1.59^ — 5.255^ = ,0698, therefore, in this case, A = 1.008, B rr 0.418, Q = .0293, a = 1.59, Z> = — 5255 and q = .0698 : consequently 5 ( n: ^-— |) == 0.305, and * ' ^ K b — -ff-B ^ ~ '~ .0040; therefore x ~ 5.0305 and y — 3.9960. N 2 [ ISO ] SECTION XIII. OF INDETERMINATE, OR UNLIMITED PROBLEMS. A Problem is said to be indeterminate, or unlimited, when the equations, expressing the conditions thereof, are fewer in number than the unknown quan- tities to be determined ; such kinds of Problems, strictly speaking, being capable of innumerable answers : but the answers in whole numbers, to which the question is commonly restrained, are, for the general part, limited to a determinate number ; for the more ready discover- ing of which, I shall premise the following" LEMMA, Supposing — to be an algebraic fraction, in its c lowest terms, x being indeterminate, and a, h, and c, given whole numbers ; then, I say, that the least inte- ger, for the value of x that will also give the value of ?^ —■■ an integer, will be found by the following c method of calculation. Divide the denominator (c) hy the co-efficient fa) of the indeterminate quantity : also divide the divisor by the rejnainder, and the last divisor, again, hy the last re- mainder; and so on, till an unit only remains. Write doivn all the quotients in a line, as they follow; under the first of which write an unit, and under the se- co7id write the first ; then multiply these two together, and having added the first term of the loiver line for an unit J to the product, place the sum under the third term of the upper line : multiply. In like manner ^ the next two corresponding terms of the tivo lines together, and, havifig added the second term of the lower to the product, put down the result under the fourth term of the upper one : proceed on, in this way, till you have multiplied hy every number in the upper line. INDETERMINATE PROBLEMS. 181 Then multiply the last number thus found hy the abso- lute quantity fbj in the numerator of the given fraction, and divide the product by the denominator ; so shall the remainder be the true value of x, required ; provided the number of terms in the upper line be even, and the sign of h negative, or, if that number be odd and the sign of i) affirmative ; but, if the number of terms be even, and the sign of b affirmative, or vice versa, then the differ- ence between the said remainder and the denominator of the fraction will be the true answer. In the general method here laid down a is supposed less than c, and that these two numbers are prime to each other : for, were they to admit of a common mea- sure, whereby 6 is not divisible, the thiog would be im- possible, that is, no integer could be assigned for x, so as to give the value of ~ an integer : the rea- c son of which, as well as of the lemma itself, will be explained a little farther on : here it will be proper to put down an example or two, to illustrate the use of what has been delivered. Examp, 1, Let the given quantity be — ^ — ~ — , Then the operation will stand as follows : 87)256(2 82)_87(I 2, 1, 16, 2 5) 82^(16 1, 2, ,3, 50, 103 2)^(2 50 1 256)5150(20 30 = X, Examp. 2. Given — ~ _ , 89 71 ) 89( 1 18^) n^(3 1, 3, ,^ 17)j_8(l 1, 1, 4, 5 f . JO 50 =: X. N 3 182 THE RESOLUTION OF Examp.3, Given ^llf=l^. 450 377)_4£0(1 1^ 5, 6, 73) 377(5 I, 1, 6, 37 12 ) 73_(6 250 1 1850 74 450 ) 9250 ( 20 250 450 200 :z X Examp, 4. Given 9STX + 651 1235 987) 1235 ( 1 1,3,1,48, 1, 1 248)987(3 1,1,4, 5,244,249,493 243 ) 248 ( 1 651 5 ) 243 (48 493 3)£_(1 2465 2)J_(1 2958 1 1235) 323094 3(259 7394 12193 ij|l^ 1078 1235 157 := X. These four examples comprehend all the different cases that can happen with regard to the restrictions specified in the latter part of the rule. I shall now shew the use thereof in the resolution of problems. PROBLEM I. To find the least whole number, which divided by n, shall have a remainder of 7 ; but being divided by 26, the remainder shall be 13. Let a- be the quotient, by 17, when 7 remains, or which is the same, let 17a + 7 express the number INDETERMINATE PROBLEMS. 183 sought? tlien,since this number, when 13 is subtracted from it. is divisible by 26, it is manifest that 17^7+7 — 13, I7.r — 6 ^ V 11 I il — Z_! 1 or - — -^ — must be a whole number: 26 26 whence, by proceeding according to the lemma, x will be found = 8; and consequently nx ■\- 'J =i 143, the number required. See the operation 17)26(1 17 1, 1, 1 9)17(1 9 "8)9_(1 1 1, 1, 2 ► 3 6 18 26 8 zz X. PROBLEM II. Supposing 9x + 1 3y = 2000, it is required to find all the possible values of x and y in whole positive numbers. By transposing I3y, and dividing the whole equation , 2000—131/ ^^^ , 2—4?/ by 9, we have x = =222—2/ + ; which, as ^c is a whole positive number, by the question, must also be a whole positive number, and so likewise — ■; from which the least value of ?/, in whole numbers, will come out — 5 ; and consequently the. corresponding value of a? = 215. From whence the rest of the answers, which are 16 in number, will be found, by adding 9, continually to the last value of y, and subtracting 13 from that of .r, as in the annexed table, which exhibits all the possible answers in whole numbers. ar =2l5|2()2ll89ji76ll6S[l50]l37l124|lll]9Ri85l 72' 59| 4G| SSl 201 7 y = 51 14| 231 32l 4l| 5o| o9| 68| 77|8c|95ll04l f 13|l22ll3l|l40|t49 In the same manner, the least value of ?/, and the greatest of a; being found, in any other case, the rest of the answers will be obtained, by only adding the co^i- efficient of cT, in the given equation, to the last value of z/, co;iiim<^////,and subtracting the co-eilicient of 7/ from the N 4 184 THK RESOLUriOx\ OF corresponding value of x, Heiice it follows, that, if the greatest value of a: be divided by the co-efficient of y, the remainder will be the least value of x, and that the quotient + 1 will give the number of all the answers. But it is to be observed, that the equations here spoken of, are such, wherein the said co-eflicients are prime to each other; if this should not be the case, let the equation given be, first of all, reduced to one of this form, by dividing by the greatest common measure. PROBLEM III. , To find hoiv many different loays it is possible to pay lOOl. in guineas and pistoles, only ; reckoning guineas at 21 shillings each, atid pistoles at 17, Let X represent the number of guineas, and y that of the pistoles; then the number of shillings in the guineas being 9.\x, and in the pistoles, 17^> we shall tlierefore have 2la7 f I7y= 2000, and consequently a — which being a whole 17?/ — 5 2000 — ^iy^ q.-i f 5^ -17.V 21 21 21 number, by the question, it is manifest that must also be an integer : now the least value of y, in whole numbers, to answer this condition, will be found = 4, and the expression itself = 3 ; the corresponding, or greatest value of x being — 92 ; which being divided by 17, the co-efficient oi y (according to the preceding note) the quotient comes out 5, and the remainder 7 : therefore the least value of a? is 7, and the number of answers (= 3 + Ij = 6 : and these are as follow. re — 92 Z/ - 4 75 j 58 I 41 25 46 1 67 PROBLEM IV. 24 7. 88 109. To determine whether it he possible to pay lOOl. in guineas ami moidores only; the former being reckoned at 21 shiflings each, and the latter at 27. Here, by proceeding as in the last question, we have INDETERMINATE PROBLEMS. 185 2 la? 4- 27y n 2000; and consequently x — — ^ — 05 «-. ?/ LZJ : where, the fraction being in ^21 its least terms, and the numbers 6 and 21, at the same time, admitting of a common measure, a solution in whole numbers (by the note to the preceding lemma) is impossible. The reason of which depends on these two considerations ; that, whatsoever number is divi- sible by a given number, must be divisible also by all the divisors of it ; and that any quantity which exactly measures the whole and one part of another, must do the like by the remaining part. Thus, in the present case, the quantity (yy — 3, to have the result a whole number, ought to be divisible by 21, and therefore di- visible by 3, likewise (which is, here, a common mea- sure of a and c) : but 6//, the former part of 6?/ — 5, is divisible by 3, therefore the latter part — 5 ought also to be divisible by 3 ; which is not the case, and shews the thing proposed to be impossible, PROBLEM V. A butcher bought a certain number of sheep and oxen, for ichich he paid lOOl. ; for the sheep he paid 17 shil^ lings apiece, and for the oxen, one with another^ he paid 7 pounds apiece, it is required to find how many he had of each sort ? Let X be the number of sheep, and y that of the oxen; then, the conditions of the question being ex- pressed in algebraic terms, we shall have this equation viz. \Tx + 1401/ — 2000; and consequently x — 2000—140// ^^^ ^ 4v — 11 ,., ,. ___ - 117 — 8?/-^ \l' ' ^^^^^^ ^^^^^' a whole number, ^ "~ — must therefore be a whole number likewise : whence, by proceeding as above, we find y — 7, and x — 60 ; and this is the only answer the question will admit of; for the greatest value of x cannot in this case be divided by the co-elhcient of y, that is 140 cannot be had in 60; and therefore, ac- 186 THE RESOLUTION OF cording to the preceding note, the question can have only one answer, in whole numbers. PROBLEM VI. A certain number of men and women being merry- making together^ the reckoning came to 33 shillings, to- icards the discharging of which, each man paid 3s. 6d. and each iuo?nan Is, ^d,: the question is, to find how many persons of both sexes the company consisted of? Let a? r^resent the number of men, and y that of the women; so shall 42a: + \Qy — 396, or 21a: -f 8y , ., 198 — 21jf = 198 ; and consequently y — z: 24 — 2x — 1- : whence, y being a whole number, -^—- — must likewise be a whole number; and the value of a:, answering this condition, will be found = 6 ; and consequently that of ?/ (zz 24 — 12 — 3) = 9; Avhich two will appear to be the only numbers that can an- swer the conditions of the question; because 21, the co-efficient of x, is here greater than 9, the greatest value of y, PROBLEM VII. One bought 12 loaves for 12 pence, whereof some were two-penny ones, others penny ones, and the rest farthing ones : what number were there of each sort ? Put X zz the number of the first sort, y — that of the second, and z that of the third ; and then, by the conditions of the question, we have these two equa- tions, viz, X -V r/ 4- 2 r: 12, and 8a: + 4.y 4- z = 48 Whereof the former being subtracted from the latter, in order to exterminate 2, we thence get lx-\-Sy — 30, and therefore y - — - — - 12 — 2r — — ; whence 3 o it is evident that the value of .r - 3, and consequently that y = 5, and s — 4 ; which are the numbers that were to be found. INDETERMINATIi PROBLEMS. 187 PROBLEM VIII. To find the least integer^ possible, which being divided hy 28, shall leave a remainder of IQ; but, being divided by 19, the remainder shall be 15; and, being divided by 15, the remainder shall be 1 1, First, to find the least whole number that can an- swer the two first conditions, let the quotient by 28, the first of the given divisors, be denoted by x, or which is the same, let the said number be expressed by 26a: 4- 19; then this number, when 15 is subtracted from it, being divisible, by 19, it is manifest that , or its gx -4- 4 equal x -\ — - must be an integer; from whence the least value of x will be found — 8 ; and conse- quently QSx + 19 zz 243; which is the least whole number that can possibly satisfy the two first condi- tions. This being found, let the least number that is exactly divisible by both the said divisors 28 and 19, be now assumed ; which, because 28 and 19, are prime to each other, will be equal to 28 x I9, or 532 : then, since the number required, by the nature of the pro- blem, must be some multiple of 532, increased by 243, it is plain that the said number may be represented by 532X -V- 243 : from which, if 1 1 be subtracted, and the (53Qx ' 1 23'^ 15 =- 35x + 15 -h -^~ — j will be a whole number ^by 7 T 4- 7 the question, and consequently - — : — - a whole num- ber also; from whence the least value of x will be found =: 14 ; and consequently that of 532.r + 243 zz 7691 ; which is the number that was to be found. In the same manner the least number, possible, may be found, which being successively divided by four or more given divisors, shall leave given reniuinderb. 188 THE RESOLUTiOx^ OF PROBLEM IX. Supposing Six 4- 256^ =: 15410; to determine the hast value of x, and the greatest of y^ in whole positive mtmhcrs. By transposition and division, we have 13410— 87a: ^ 87* — 50 1 ^1 r t/ = -r-x — —^ 60 — : where the frac- •^ 256 256 tion .being the same with that in Exanip. I. to the pre- mised lemma, the required value of a? will be given from thence — 30 ; from thence that of y will likewise be known. But I shall in this place shew the manner of deducing these values, independent of all previous con- siderations, by a method on which tire demonstration of the lemma itself depends. In order to this, it is evident, as the quantity 87a: — h (supposing h — 50) is divisible by 236, that its double 3 74j: — 26 must be likewise divisible by 256. But 236^ is plainly divisible by 256; and if from this the quantity in the preceding line be subtracted, the re- mainder, 820? 4- 2 A, will be likewise divisible by the same number ; since whatsoever number measures the ichole, and one part of another, must do the like by the remaining part : for which reason, if the quantity last found be subtracted from the first, the remainder,5a; — 3b, will also be divisible by 256: and, if this new remain- der multiplied by 16, be subtracted from the preceding one (in order to farther diminish the co-efFicient of .r), the difference 2x 4- 50^ must be still divisible by the same number. In like manner, the double of the last line, or remainder, being subtracted from the preceding one, we have a; — 103b, a quantity, still, divisible by 256: but — - = 20 + — -; therefore a: — 30 must 256 ^56 be divisible by 256 ; and consequently x be either equal to 30, or to 30 increased by some multiple of 236; but 30, being the least value, is that required. It may not be amiss to add here another Example, to illustrate the way of proceeding by this last method; . . . , 987.r4-65I wheremletussupposetnequantitygiveutobe — -~- — INDETERMINATE PROBLEMS. 189 Then making h — 651, the whole process will stand as follows : From 1235.17 sub 987^1^ 'X- h 1, rem 248a: — 6 1 . rem. X 3 744.Y —zh 2. rem 2433? 4- 4 6 3. rem. bx — 5h 3. rem. X 48 240.r — 240^ 4. rem ,,.. 3a? + 244/; 5. rem ^x — 249^^ 6. rem oc -\- 493^: where, x being without a co-elTicient, let 493^; or its equal 320943 be now divided by 1235, the common measure to all those quantities, and the remainder will be found 1078; therefore a: -+- 1078 is likewise divi- sible by 1235 ; and consequently the least value of x (zz 1235 — 1078) = 157. The manner of workinir, according to this method, may be a little varied ; it being to the same effect, whether the last remainder, or a multiple of it^ be subtracted from the preceding one, or the preceding one, from some greater multiple of the last. Thus, in the example before us, the quantity 248X — 6, in the third line, might have been multi-. plied. by 4, and the preceding one subtracted from the product; which would have given 5x — bh (as in the sixth line) by one step less. If the manner of proceed- ing in these two examples be com|)ared with the pro- cess for finding the same values, according to ihclemma, the grounds of this will appear obvious. PROBLEM X. Supposing e,/, and g to denote given integers to deter- • '»-> __ g ^ -f mine the value of x. such that the quantities ^ , ~* ^ ' i>8 19 X —— fT and • , may all of them he integers. 190 THE RESOLUTION OF By making zz y.'we have x =z <2Sy + e ; which ' value being substituted in our second expression, it becomes— ^^^^ ^ ; which, as well as y. is to He a 19 whole number: but —"i ^ by making b zz c — /, will be zr 2/ + ~ ; and therefore igy and 18?/ + ^b being both divisible by 19, their difference y — 2/> must be also divisible by the same number; whence it is evident, that one -value of 'r/ is 2i ; and. that 26 f IQz (supposing z a whole number) will be a general value of y ; and consequently that x { — <2Sy 4- e) = 5323 -v 56b 4- e is a general value of x, an- swering the two first conditions. Let this, therefore, jf or be substituted in the r€maining expression——^; whtch, i o 14-1,. u 5322-}- 566 4- e —^ by that means, becomes ^ — 35z 1 o + 3/> -h '^ (^supposing /3 = lift f e — g — I2-. Now to have all the terms affirmative, and their co- efficients the least possible, let w be taken ::= — c + 2/ -f- g; whence there results 4b45e + 4200/ + 6916^, for a new value of x : from which, by expounding e, /, and^, by their given values, and dividing the wiiolc by 7980, the least value of .r, which is the remainder of the division, will be known. INDETERMINATE PROBLEMS. 191 PROBLEM XI. If 5x 'i- ly + llz zz 224; it is required to find ail the possible values of r, y, and z, in tvhole numbers, * • In.this, and other questions of the same kind, where you have three or more indeterminate quantities and only one equation, it will be proper, first of all, to find the limits of those quantities. Thus, in the pre- sent case, because a? is = ~ -, and because the least values of y and z cannot (by the question) be less than unity, it is plain that x cannot be greater than —^ ^^ — ,* or 41 : and, in the same manner it will 5 appear that y cannot be greater than 29, nor z greater than 19 ; which therefore are the required limits in this ,* . . 224 — 7?/ - llz onse. Moreover, since .r is — — 43 5 I + Qy h z , , , — y — 22 ■ zz a whole number, it is ma- nifest that~4 ^ must also be a whole number: let 2 + 1 be therefore considered as a known 'quaji- tity, and let the same be represented by b, and then the last expression will become ; from which by proceeding as above, we shall get y — Q,h — ^z 4- 2 ; whence the corresponding value of x comes out ^ 42 — 5z. Let z be now taken = 1, then will a; = 37 and y — 4; from the former of which values, let the co-elli- cient of y be, continually, subtracted, and to the latter, let that of a?be continuallyadded, and we shull thence have 37, 30, 23, 16, 9, and ^, for the successive values ofx; and 4, 9, 14, 19i 24, and 29, for the correspond- ing values of y. which are all the possible answers when 2 =1. 192 THK RESOLUTION OF Let z be, now, taken =: 2, then x :z 32, and 3/ = 6; let the former of these values be increased or decreased by the multiples of 7, and the latter by those of 5, as far as possible, till they become negative; so shall we have 39> 32, 25, 18,11, and 4, for the successive values of a?, in this case, and 1, 6, 11, 16, 21, and 26, for the respective values ofy; which are all the answers when z zzQ. Again, let z be taken = 3 ; then, by proceeding as above, the corresponding values of x, and y will be found equal to 34, 27, 20, 13, 6; and 3, 8, 13, 18, 23, respectively. And so of the rest: whence we have the following answers, being 59 in number. 2 y X 1 4. 9« 14. 19.24. 29 . 37 .30. 23 .16. 9.2 '2 1 . 6. 11. 16.21. 26. 39. 32 .25 . 18 . 11 .4 3 3. 8« 13. 18.23. 34 .27 . 20. 13 . 6. 4 5. 10. 15 . 20. 23. 29^ 22 . 15 ., 8 • 1 . 5 2. 7 ' 12. 17.22. 31 .24 ,17 . 10 . 3 . 6'4. 9' 14. 19. 26. 19* 12. 3 . 7 1 ' 6. 11 . 16. 28 . 21 . 14 • 7 . 8 3 " 8. 13. 18. 23. 16. 9. 2 . 9 5 " 10. 15. 18.11 . 4. 10 2 7. 12. 20. 13 . 6. 11 4 > 9- 14. 15 . 8 . 1 . 12 1 6. 11 . 17 . 10. 3 . 13 3 . 8 . 12. 5 . 14 5 7- 15 2 7. 9- 2, 16 4 4. 17 1 6. 18 3 1 . Ii>J DETERMINATE PROBLEMS. 193 >«OBLEM XII. //17^ rf 19y + 21z = 400; it is proposed to find all the possible values of x, y, and z, in ivhole positive numbers. When the co-efficients of the indeterminate quaiiti- ties Xy tj, and z, are nearly equal, as in this equation, it will be convenient to substitute for the* sum of those quantities. Thus, let x + y + z be put — m ; then by subtracting 17 times this last equation from the pre- ceding one, we shall have Qy A- 4z — 400 — l7??i ; and by subtracting the given equation from 21 times the assumed one x -h y + z — m, there will remain AX + 2y zz 21m — 400. Therefore, since y and z can have no values less than unjty, it is plain, from the first of these two equations, that 400 — 1 7m cannot be less ♦ 400 6 than 6, and therefore m not greater than — , or 23 : also, because by the second of the two last equa- tions, 21m — 400 cannot be less than 6, it is obvious that m cannot be less than , or 19 : therefore 21 19 and 23 are the limits of m, in this case. These being determined^ let 4x be transposed in the last equa- tion, and the whole be divided by 2, and we shall hare yzz 10m — 200 — 2a' -1- — : which being a whole 711 number, by the question, — must likewise be a whole number, and consequently 7n an even number ; which, as the limits of m are 19 and 23, can only be 20, or 22 : let, therefore, m be first taken = 20, then y will be- come =: 10 — 2x, and z (m — x — y] = 10 + 0?; wherein a; being taken equal to l, 2, 3, and 4, suc- cessively, we shall have y equal to 8, 6, 4, 2, and z equal to 11, 12, 13, 14, respectively, which are four ' of the answers required. Again, let m be taken =: 22 ; then will 1/ = 31 — 2.v, and 2 zz a^ — 9 : wherein, let .The interpreted by 10, li, 12. 13, 14, and 15, succes- sively, whence y will come out 11, 9, 7, 3, 3, and 1 ; o iw 194 THE RESOLUTION OF ind X equal to 1,2, 3. 4. 5, and 6, respectively. There- fore we have the ten following answers ; which are all the question admits, of. y = 8 X = 11 2 3 4 10 11 12 13 14 6 4 S 11 9 . 7 5 3 12 13. 14 1 2 3 4 5 15 1 6 .* PROBLEM XIII, Supposing 7x •{- 9y -h 232 = 9999 ; i< t? required to determim the number of all the answers,, in positive integers, - ' <* In cases like this, where the answers are very many and the number of them only is required, the following method may be used. In the general equation, ax -\- by -}- cz zz ^ (where a and b are supposed prime to each other) let z be assumed = ; and find the greatest value of or, and the least of y, in the equation «x + 6y == /r, thence arising; "de- noting them by J" and /: find, moreover, the least posi- tive value of n (in whole numbers) from the equation am 4- ^« =c, together with the corresponding value'of m, whether positive or negative; then, supposing 9 to represent an integer, the general value of ct may be ex- pressed by g ■— bq — mz, and that of y by I ■{- aq ~-nz; as will appear by substituting in the general expression ' ax -{- by ^ cz, which thereby becomes ag — abg — amz + 6/ + abq — bnz + cz zz k (as it ought to be), because ag -\- hi ■=. k, and all the rest of the terms ' destroy oae another. And it may be observed farther, by the bye, and is evident from hence, that any two corresponding values of m and «, determined from the equation am + In =:c, will equally fulfil the conditions of the general equation ; but the least are to be used as being the most commodious. As to the limits of z and 9, these are easily determined ; the former from the original equation, and the latter from the general va- lue of x; by which it appears that g cannot exceed^ S ~~!? i ^ ; wherein the greatest, or the least value of 2 is to be used, according as the second term, after substit^• ; INDETERMINATE PKOBLEMS. 195 tion for m, is positive or negative. But, besides this, there is anotherUmit,or particular value of 9 to be determined, which is of great use in finding the number of answers. It is evident from the given equations, that the values of x will begin to be negative, when z is so increased as to exceed -; and that those of m y will, in like manner, become negative, when z is taken greater than — ^ — - : therefore, as long as S — ^Q M ^ -\- aQ , . 1 ' contmues greater than ? (supposing the value of g to 'be varied) so long will x admit of a greater assumption for z than y will admit of, without producing negative values ; and vice versa. By making, therefore, these two expressions equal to each other, the value of a will be given ( = -^^ — j-—) - ^ ^ ; ; ^ ° ^ am -i- nb ' c expressing the circumtsance wherein both the values of X and y, by increasing z, become negative together. But this holds only when w is a positive quantity; for, rn the other case, the last term ( — mz) in the ge- neral value of X being positive, the particular values do not become negative by increasing, but by diminishing the value of z; it being evident, that no such can re- sult from any assumption for «, but when 9 is greater than -f* To apply these observations to the equation, 7x 4- 9^ -k- 232 - 9999, proposed, we shall, in the first place, by taking 3=0, have x =z 1428 — y ?^ZlL: whence the least value of y is given = 5 ; and the greatest of a' r= 1422. Again, from the equation am -f bn zr c, or 7??2 -{- gn — 23, we have ?n zz 3 — n — — - — ; in which the least positive value of n is given = 1 : and the corresponding value of m = 2 ; and so the general values of x and y do here become 1422 o 2 196 THE RESOLUTION OF 9g — 2r, and 5 -f- 7q — 2, respectively. From the former of which the greater limit of q is given n 1422 — 2 , or 157^; and from flQ on I , expressing the ^ y.~ N.Ans. 5—2 4 1 12 — 2 11 2 19 — 2 18 3 ^6 —z 25 4 33—2 32 &C. &C. &C. 9 ' ' c lesser limit, we have 61, for the value'of 7, when the least value of x becomes equal to that of y. These limits being assigned, let q be now interpreted by 0, 1, 2, 3, 4, 5, &c. successively, up to 61, inclusive: whence the number of answers, or variations of y cor- responding to every interpretation, will be found as in the margin. From whence it appears that the arith- metical progression 4 4- 11 + 18 + 25 + 32, &;c. con- tinued to 62 terms, will trul^ ex- press the number of all the an- swers when q is less than 62 : which number is therefore given = 4T6Tir7^4 X 31 z= 13485. In all which answers it is evi- dent, that x\ as well as y, will he positive (as it ought to be) : because it has been proved that the least value of x, till q be- comes ( zn ^ ' — J = 6ly, will be greater than that of y ; which is positive, so far. But now, to find the answers when 9 is upwards of 6I, we must have re-' course to the general value of x ; which, in these cases, by thedifferent interpretations of 2, becomes negative before that ot'y. Here, by beginning with the greatest limit, and writing 157, 156, 155, 154, 8tc. successively, in the room of 9, it will ap- pear, that the number of answers will be truly ex- pressed by the series 4 + 8 4 1 3 + 1 7 + 22, &c. con- tinued to 157 — 61 terms : which terms being united in pairs (because in every two terms, the same fraction in the limit of z occurs) the series 12 f 30 -f 48 f- &c, thence arising, will be a 9 157 "■r ~ 2-13 N.Ans. 9—22 4} 4 156 18—2;: 9' 8 155 27— 2« 131 13 154 30—22 18 Iz 153 45>-^2: 22| 22 &c. &C. &c. &c. INDETERMINATE PROBLEMS. 197 true arithmetical progression ; whereof the common dif- ference being 1 8, and the number of terms — =: 48, the sum will therefore be given — 20880 : to which adding 13483, the number of ayiswers when q was less than 62, the aggregate 34365 will be the whole number of all the answers required. PROBLEM Xir. To determine how many different ways it is possible to pay 1000 1. icithout using any other coin than crowns, guineas f and moidores. By the conditions of the problem we have 5x -f Qly ■j- Q7z zz 20000; where taking z — o, x \s found — 4000—4?/ v-> ^"cl from thence the least value of 2/ = (0 being to be included, Jiere, by the question) : whence the greatest value of .t is given = 4000. More- over, from the equation 5m f Qin — 27, we have 71 — — 2 m — 5 — 4« — ; from which « =: f , and m = 3 — 3 : so that the general values of x and y, given in the preceding problem, will here become 4000 — 21^7 + 32, and 5q — 22. Moreover, from the given equa- tion, the greatest limit of z appears to be ~ — 740; whence we also have iJ=^" = J"°0+3X740 6 21 = 296 zz the greatest limit of q ; and -y zr ^522 ^ 190, expressing the lesser limit of 9, when the value of X, answering to some interpretations of z, will become negative, while those of y will continue affirmative. To find the number of all these affirmative values, up to the greatest limit of q, let 0, 1,2, 3, 4, 5, &:c. be now wrote in the room of 9 (as in the margm). Whence it is evident that the said number is composed of the o 3 l&S THE RESOLUnON OF series 1 +3 + 64-8 + li + 13, &c. continued to 297 terms; which terms' (setting aside the first) be- ing united in pairs, we shall have the arithmetical progression 9 + 19 + 29 &c. where the number of terms to be taken being 148. and common diffe- rence loathe last term will therefore be J 479, and the sum of the whole progres- sion 110112 : to which adding (i) the term omitted, Ave have 110113, for the number of all the answers, in- cluding those wherein the value of x is negative ; which last must therefore be found and deducted. ' 9 y - Quot. N.Ans. 0—22: 1 1 5—22 2[ 3 2 10—22 5 6 3 15—22 n 8 4 20—22 10 11 5 95—23 12| 13 &c. &C. &c. &c. In order to this we have already found, that these ne- gative values do not begin to have place till q is greater than 190 : let, therefore, 191,192, I93,&c.besubstituted successively, for q\ from whence it will appear that the number of all the said negative values is truly exhibited by the arithme- tical progression 4+ 11 + 18 + 25, &c. continued 10296 — 190 terms; where- of thesum is 39379; which subtracted from 110113, found above, leaves 70734^ for the number of answers required. 9 191 X, Quot. N.Ans. 3z— 11 4 192 32—32 m 11 193 32—53 I7i 18 194 3z— 74 24 1 25 &c. &c. &c. &c. After the manner of these two examples (which il- lustrate the two different cases of the general solution, given in the preceding problem) the number oFansvvers may be found in other equations, wherein there are three indeterminate quantities. But, in summing up the numbers arising from the different interpretations ofq, due regard must be had to the fractions exhibited in the third column expressing the limits of z ; because, to have a regular progression, the terms of the series in the fourth column, exhibiting the number of answers. . INDETERMINATE PROBLEMS. 1S9 must be united by twos, threes, or fours, &c, according as one and the same fraction occurs every second, third, or fourth, &c. term (the odd terms, when there happen any over, being always to be set aside, at the begin- ning of the series). * And it may be observed farther, that, to determine the sum of the progression thus arising, it will be sufficient to find the first term only, by an actual addition ; since, not only the number of terms, but the common difference also, will be known; being always equal to the common difference of the limits of z {or of the quotients in the said third column) multiplied by the square of the number of terms united into one ; whereof the reason is evident. But all this relates to the cases wherein the coefficients of tho inde- terminate quantities, in the given equations, are {two of them at least) prime to each other : I shall add one example more, to shew the way of proceeding when those coefficients admit of a common measure. PROBIEM XV. Supposing I2x + 15y + 20?: r: 100001 ; it is required to find the number of all the answers in positive integers. It is evident, by transposing 202 : and dividing by (3) the greatest common measure ofx and y, that 4x -f 5y, and consequently its equal 33333 — 6z — ff-ZILr^ must be an integer, and therefore ez — 2 divisible by 3 : but 32 is divisible by 3, and so the difference of these two, which is z + 2, must be likewise divisible by the same number, and consequently z — 1 + some multiple of 3. Make, therefore, l + 3m — z {u be- ing an integer) : then the given equation, by substituting this value, will become 12j7 -f- 15?/ 4- 60u + 20 =: 10000 J ; which, by division, &c. is reduced to 4x + 5^ -h 20M =: 33327 : wherein the coefficients of x and y are now prime to each other, and we are to find the number of all the variations, answering to the different interpretations of u, from to the greatest limit, in- clusive. o 4 200 LNDETERMINATE PROBLEMS. By proceeding, therefore, as in the aforegoing cases, we have x = 8331 — y — - — ~ ; whence the least value of y is given zz 3, and the greatest of a? = 83^8. Moreover, from the equation 4 w 4- 5 /z — 20, we have m — 5 — n : whence n = 0, and m z: 5. 4 Therefore the general values of x and y [given in Problem 13) do here become 8328 — 5q — du, and 3 + 49 ; from the former of which the greatest limit of 9 is given 8328 = — — - =: 1663. Now, since the value of v will here 5 continue positive, in all substitutions for 7 and u (as r.o negative quantity enters therein) ; the whole num- ber of answers will be determined by the values of x alone. In order to this, let q be successively expounded by 1663, 1664, 1663, &c. and it will thence appear thatthe said numberwill be truly defined by 1666 terms of the arithmetical progression 1 + 2 f 3 + 4+3 &c. . whereof the sum is found to be 1388611. When there are four indeterminate quantities in the given equation, the number of all the answers may be determined by the same methods : for, any one of those quantities may be interpreted by all the integers, suc- cessively, up to its greatest limit (which is easily de- termined); and thenunlberof answers, corresponding to each of these interpretations may be found, as above; the aggregate of all which will consequently be the whole number of answers required : which sum, or aggregate may, in many cases, be derived by the me- thods given in Section 14, for summing of series's by means of a known relation of their terms. But this being a matter of moro speculation than real use, I shall now ])ass on to other subjects. 7 X Quot. N.Ans. 1665 1664 1663 &c. ■ 3— 3W 8—5u \3 — 5U &c. 01 H &c. 1 2 3 &C. [ 201 ] SECTION XIV. . the investigation of the sums of powers of 'numbers in arithmetical progression. BESIDES the two sorts of progressions treated of in Section 10, there are infinite varieties of other kinds; but the most useful, and the best known, are those consisting of the powers of numbers in arithmeti- cal progression ; such as V + 2^ -h 3* + ^- . , , . n-, and 1^ + 2' + 3^ + 4' . . . . w^, &c. where n denotes, the number of terms to which each progression is to be continued. In order to investigate the sum of any such progression, which is the design of this Section, it will be requisite, first of all, to premise the following LEMMA. If any expression, or series, as An + Bw« + C«^ + D;^* &c.; . , . ,, ^an^ hn^ - en' - dn^ &c.^ mvolvmg the powers of an indeterminate quantity w, be universally equal to nothing, whatsoever be the value of n ; then, I say, the sum of the co-efficients A — a, B — Z>, C — c, &c. of each rank of homologous terms, or of the same powers of n, will also be equal to nothing. For, in the first place, let the whole equation An -V Bn" ^Cn'hcf , ...... ^an-^ bn" — cn^ &c. S = ^» "^ divided by n, and we shall have I _t l^l^ ""J^H = 0; ^ni this being universally so, be the value of n what it will, let, therefore, n be taken =z o, and it will become |__^ | =0; which being rejected, as such, out of the last equation, we shall next have 4- B/i + Cw^ 4- D/z^ &c. > ^ ,: ., — hn — c?i' — dri^ 8icA - ^ » whence, dividing again by n, andproceeding in the very same manner. » 202 THE INVESTIGATION OF B-'b is also- proved to be iz o ; and from thence, C — c, D — (/, &c. &c. , Q.E.D. Now, to apply what is here demonstrated to the pur- pose above specified, it will be proper to observe, first, that, as the value of any progression (l^-h 2^ -f. 3« "t 4' w') varies according, as [n) the number of its terms varies, it must (if it can be expressed in a ge- neral manner) be explicable by n and its powers with determinate CO- efficients; secondly, it is obvious that those powers, in the cases above proposed, must be ra- tional, or such whose indices are whole positive num- bers ; because the progression, being an aggregate of whole numbers, cannot admit of surd quantities; lastly, itwill appear that the greatest of the said indices can- notexceed the common index of the progression by more, than unity ; for, otherwise, when n is taken indefinitely great, the highest power of n would be indefinitely greater than all the rest of the terms put together. Thus, the highest power of w, in an expression univer- sally exhibiting the value of P 4- 2' + 3^ . . . . . n*, cannot be greater than w' ; for 1 * + 2^ -j- 3* ..... n* is manifestly less than n^ (or n^ -]- n^ + n'^ -[- &c. con- tinued to n terms) ; but «*, when n is indefinitely great is indefinitely greater than n^, or any other inferior power of n, and therefore cannot enter mto the equation. This being premised, the metliod of investigation may be as follows. ^ Case i". To find the sum of the progression 1 + ^4-3 -+- 4 . . . . ;?. Let An' v B?2 be assumed, according to the foregoing observations, as an universal expression for the value of 1+2 + 3 + 4 n\ where A aud B represent unknown, but determinate quantities. Therefore, since the equation is supposed to hold universally, whatsoever is the number of terms, it is evident, that, if the num- ber of terms be increased by unity, or, which is the same thing,'if 71-J- 1 be wrote therein, instead of w, the equa- Jity will still subsis^r, and we shall have A X « -f-T-f THE SUMS OF PROGRESSIONS. 503 B X n + 1 = 1 f 2 + 3 4- 4 n V n ^- \, From which thejjrst equation being subtracted, there re- mains A X n + 1 r — A.n' 4- B X n \- i — B;2z=w -h 1 : this contracted will be 2 A.« -I- A -f B ~ n -{- i ; whence we have 2A — i xn-f-A4-B — izzo: wherefore, by taking 2A — 1 =: 0, and A -f B — i — o (acccording to the lemma) we have A zz f , and B z: ^ ; and con* sequently 1 + 2 4- 3 4-4 . n (— .\n^ 4- B«) z= n* n n X n 4- 1* — + ^. or -^-^ Case 2**. To find the sum of the progression 1^4-2*^ 3' »% or 1 4-44-9+1^ • • • • ^*- Let A«' 4- Bn* 4- C«, according to the aforesaid ob- servations, be assumed — l"" + 2^* 4- 3" 4- 4- . . . . «2; the^,_by^easoningasintheprecedingc^se^eshall have A X « 4- iV 4- B X n 4- ll" + C x /<4-l - 124-2*4- 3* 4- 4' ....;?* 4- n ^ l\^ ; that is, by involving n -\- l to its several powers. Aw' 4- sAri' 4- 3A« 4- A 4- Btj* 4- 2Bw 4 - B 4- C« 4- C zz 12 + 2* 4- 32 4- 4' n^ 4- w 4- iT I from which, subtracting the former equa- tion, we ge t 3A^i- + sAw -h A 4- 28/2 4- B 4- C ( = n 4- if) z= «- 4- ^/2 4- I ;. and consequently * In this investigation it is taken for granted, that the sum of the progression is capable of being exhibited by means of the powers of n, with proper co-efficients : which assumption is Verified by the process itself: for it is evident from thence, that the quantities An^ -r B^« and 1 4- 2 4- 3 4- 4 ... «, under the values of A and B there determined, are always increased equally, by taking the value of n greater by an unit: if, therefore^ they are equal to each other, when ?? is :=: (as they actually are) they must also be equal when « is 1 ; and so likewise, when n is 2, &c. &c. And the same reason-, ing holds in all the follovving cases. m 204 THE INVESTIGATION OF 3A-— 1 X w» -f'sA + 2B — 2 xw-fA + B + C-l = O; whence {hy the lemma) sA — 1 — 0, 3 A + 2B — 2 = 0, and A + B I- C — 1 zz ; therefore A=i-,B=2ZI£A^2_.C=1_A-B=4; 3 3 2 6 3 ^ and consequently 1 4- 4 + 9 + 16 ....«« = — + — ' "J M n n . n ■\- I • ^n -\r I + -g-, or g . Case 3°. To determine the sum of the progression P+ 2^+3^+4' w% or 1 + 8 4- 27 + 64 «^ By putting A?^* 4- B;2^ + C«* + D« = 1 + 8 + 27 + 64 n^, and proceeding as above, we shall have 4A7J^ +- 6A«" + 4A;z + A + 3B«' + 3B» + B + 2C« + C + D( = % + l)' ) - n^ + Sn"" + 3« + 1 ; and therefore 4 A — l x w^ + 6A + 3B — 3 x »' + 4A+3B + 2C— 3X w + A + B + C + D— 1 =: o; , A 1 T>, 3— 6A. I ^, 3— 4A~3B^ henceA -— ,B(= --^— ) =— . C(=: ^ ) iz-L, D (zr 1 — A — B — C) = O; and therefore 4 1^ + 2^ + 3^ + 4^ w^ = •^+ "I" + "^^ ^^ In the very same manner it will be found that 1- 4 2« + 3- ....»*=- + -+---. * ' ft « f, riJ ^ n^ ^ n^ n^ ^ n l»+2^ + 3-....,.^:=-+~ +_-.- + ^-^.. / &c. &c. In order to exemplify what has been thus far deliver- ed, let it, in the hrst place, be required to find the sum of the series of squares 1+4 + 9 + 16, &c. continued to 10 terms : then by substituting 10 for n, in the ge- THE SUiMS OF PROGRESSIONS/ 205 neral expresssion- — (^^"i""*" T ~6'^* found by case 2°, there will come out 385, for the re- quired sum of the progression ; which, the number of terms being here small, may be easily confirmed, by ac- tually adding tihe 10 terms together. Secondly, let it be»required to find the number of cannon shot in a square pile whose side is 50 ; then , by w riting 50 for u in the same expression, —^ -^ , we shall have , 50 X 51 X lOr • ^1 I r { X } 42925, expressing the number ol shot in such a pile. Lastly, suppose a pyramid com- posed of 100 stones ©f a cubical figure; whereof the length of the side of the highest is one inch ; of the • second two inches ; of the third three inches, &c. Here, by writing 100 instead of «, in the third general expression, we have 25502500, for the number of solid inches in sUch a pyramid. Hitherto regard has been had to such progressionsas ' have unity for the first term, and likewise for the common difference ; but the same equations, or^ theo- rems, with very little trou^^le, may be also extended to those cases where the first term, and the common difference, are any given numbers, provided the for- . mer of them be any multiple of the latter. Thus, sup- * pose it were required to find the sum of the progression 62 ^ go _j_ j,q2 ^^^ ^Qj. 36 4. 64 + 100. &c.) conti- nued to eight terms : then, by making (4), the square of the common difference, ageneralmultiplicator,thegiven expression will be reduced to 4 x 3M~4M^~5^ , .10": but the sum of the progression I'** 4- 2f -h 3^ 4-4^ . .10* is found, by the second Theorem, to be 385 ; from which, if (3), the sum of the two first terms (which the series 3- 4- 4^ 4- 5^ ........ 10^ wants) be taken away, the remainder will be 380; and this, multiplied by 4, gives 1520, for the true sum of the proposed progres- sion: and so of others. But if the first term is not divisible by the common difference, as in the progression, 5* 4- 7* 4- 9' &c. the speculation is a little more difficult; nevertheless, 206 • THE INVESTIGATION OF f the sum of the series, in any such case, may be still found, from the same Theorems. Let th e series m + e]'^ + m -\r 2e i* + m + 3e|* m + neY be proposed, where m and e denote any q'uan- - tities whatever, and where n represents the number of * terms. Then, by actually raising each root to its se- cond power, and placing the terms in order, the given expression will stand thus that the sum of the first rauk, or series , is w X m' t also the sum of the second, or 2me x 1 + 2 4- 3 -f 4 . . . .n 4- 7W* . . . . 7W* ^ f 6me .... Qnme >. Now, it is evident 4- ge^ ;i'e« 3 appears (5y case 1) to be Qme x,^ ^ " ^J ; and that * ofthe third, ore' X I + 4«+ 9 4- 16\. . .n^bycsise^) — e' X : therefore the sum of the o , whole progression, 7/2 + ep + w + 2el' 4> wTiel* . . . . m + we!^ is - « . 7?j« 4- ;? . « + 1 . me 4- w . n 4- I . g;z 4- ) . e^ In like manner, if the series proposed be m^-e]^ 4- nTTlel^ 4- m 4- 3el' m4-we|^ then may it be resolved r i -^ 1 4- 1 1 X w^ 1 into <' f±-14-4^^^l^^ ^ 3^^ !. : whose sum, [r+ 8 4- 27' 72' X e« J ' ' by the afor ement ioned Theo rems, w ill appear to be *>i5 I ^* ^ + I « 3m"e « . « 4- I . 2« 4- 1 . twe« , "*• ' 4-, ^ 4- ^ + — '- *' — i And, by following the same method, . the sums of other series's may be determined, not only of powers, but likewise of rectangles, and solids, &c. • • THE SUMS OF PROGRESSIONS/ 207 » provided that their sides, or factors, are in arithmetical progression. Thus, for exanjple, let there be proposed the series of rectangles m+ e.p-\-€+m-{-2e ,p +2e -t- m + 3e . p ■\- 3e .... + m + ne , p -j- ne> Then, the factors being actually multiplied together, and the terms placed in order, the given series will be resolved into the three following ones : * ^^ "^ ^P "*" .^^ + ^jg .. .. + nip m'^p.e+m-\-p*Qe+m-{-p,3e-\-m-i-pAe .,,. -{-m^p.ne Whereof the, respective, 'sums (by case 1 and 2) are mp X w, m+p . e X ^ '^ , and e* x and the aggregate of all these, or nxmp-{ ^—, m+p, e 4- ^ '— . e'.js con- 3 o sequently the true sum of the series of rectangles pro- posed. » • ^ From this last general expression, the number of can- non-shot in an oblong pile,* whether whole or broken, will be known. For supposing e zz i,'our series of rectangles becomes m ^ i . jo -f 1 + m 4- 2 . p 4- 2 + m4-3.p + 3 '\- 771 -{■ n k p -^ n; and the sum ^u £■ ' . w -h 1 — , n + i , 2«Tl thereof = n x mp -{ . m + p + -— ~ 2 O :„ the number sought : where m 4- 1 and p 4- 1 repre- sent the length and breadth of the uppermost rank, or tire ; n being ^he number of ranks one above another. But the expression herfe brought out may 436 reduced to . --- X 2m4-«4-l . Qp + 7i-\-i 4- ^ ^ ^ '^ ~ \ ; which ^ ' « 3 . is better adapted to practice, and which, expressed in words, gives the following rule. To twice the length, and to twice the breadth* of .the uppermost rank, add the number of ranks less one, and multiply the two sums together; also multiply the number of ranks less one, by that number more one, and add | of this product to the former; then i of the 208 THE INV^ESTIGATIOX OF sum multiplied by the number of ranks will be the . answer. As a rule of this sort is of frequent use to persons concerned in artillery, it may not be imprdper to add an example or two, by way of illustration. » 1. Suppose a complete pile, consisting of 15 tires, or ranks, and suppose the nuniber of shot in the upper- most (which in this case is a single row) to be 32. - Then th e fi rst prod uct mentioned in the rule will be 64 + 14 X 2^+ 14 — 78 X 16 zi 1248; and the se- cond = 14 X 16 =: 224; ^ whereof is.74f, and this, added to 1248, gives 1322|; whereof :J: part is 330j-: which, multiplied by 13, gives 4960, for the whole number of shot in such a pile. • .^ 2. Let the pile be a broken one, such that the length and breadth of the uppermost tire may be 25 and 16, and the number of tires \\, •• - Here, we have 50 -h lO X 32 + 10.r;60x42zz2520 for the first product; and 12 x 10 n 120,' for the se- 2560 . ^ cond; therefore x -11 = 64o x *11 t= 7040, is 4 the true answers Having exemplified the use of the Theorem, for find- ing the sum of a series of rectangles, I shall here subjoin one instance of that preceding it^ for determining the sum of a series of cubes; wherein the value of 'the first 10 terms of th e progressi on 2 + y^2]^ + 3 + 2 V 2 ^^ + 4 + 3 -v/ 2lM- 5 4- 4\/?l' &c. is required. Here, e being 1 + -/S, m will be :r i ; therefore, by writmg 10, 1, and 1 4- \/2 for «, ?», and e, respectively , in the •^ . .. .„!. * 10.11.3. I + a/S general expression, it will become 10 4 10 . 11 . 21 .1 -+- \/2l^ -loo. 121 . i + V^ ' _ + ; ^ + I - 24815 -f- 17600v/2, the value sought If any one is desirous tp see this speculation carried further, so as to extend to series's of powers, whose in-" dices are fractions; such as square roots, cube roots, &c. I must beg leave to refer to my Essays, where it is THE SUMS OF PROGRESSIONS. 209 treated in a general manner. Here I must desire the reader to observe, once for all, that the Theorems above found will hold e qual ly, in c ase of a descend ing series, such as m — e]* + m — 2e]* &c. or m — el^ 4- m — '2eV- &c. provided the signs of the second, fourth, &c. terms be changed ; as is evident from the investi- gation. Although the subject of this section has, already been pretty largefy insisted on, yet it may not be im- proper to add a different method, whereby the same conclusions will, in many cases, be more easily de- rived; in order to which it is necessary to premise the subsequent LEMMA.. If a 4-^-fc-fc?-f-e + &c, be a series, whereof the terms, a, b, c, d. Sec. are so related to each other, . that the sum, or value thereof, can be universally ex- pounded by an expression of this form, viz. An 4- B X ;^ X n-rl -tO xn x w—l xn^^ + D x w x n — 1 X 71 — 2 X 71 — 3 &c. n being the number of terms to which the series is to be continued, and A, B, C, D, &c. determinate co-efticients ; then, Isay,the values of those co-efficients will be as hereunder specified, viz, A — a, — a 4- b B = 2 p _ a — 26 4- c D = E — a 4-36 — 3c -]- d 2.3.4 ' a — 4b + 6c — 4d 4- e 2.3.4.5 ' &c. &c. For, since the equation Ax»4-Bx7iX72— -1 + C X n X n — i X « — 2 4- D x-n X'^— 1 x n — 2 X '« — 3 &c. zza-\-b+c + d + e, &c. is supposed to hold universally, let the number oiterms be what it p 2ro THE INVESTIGATION OF will, let n be expounded by 1, 2, 3, 4, &c. successively, and the general equation will become 3\ 3A-f 6B f 6C=ra4-54-<:, 4^ 4A hlsBf 24C+ 24Dz=a4-64-c4-'/, 5°. 3A-f 20B4-6oC f l20D-f l20Ei=a4- i + c + d + e, &c. &c. Now, the double of the first of these equations being subtracted from the second, its triple from the third, and its quadruple from the fourth, &c. we shall have ♦2Bzi^»— -a, 6Bf 6C= — 2a-i-6 + c, 12B + 24Cf 24D-— -3a^-6fc-ff/, 20B f 60C+120D+120E-— 4(J ^\■h^c^d^-e, &c. &c. Again, if the triple of the first of these be subtracted from the second, and its sextuple from the third, &c. we shall, next, have ♦6C z= a— 2^ 4- c, 24 C -f 24Di=3a— 56 -fc+rf, 60C ^- 120D-f 120E=r 6a — 9& + c+^4-e. Moreover, by taking the quadruple of the first of these from the second, &c. we get *24D — — a + 36 — 3c 4- ^, and 120D 4- 120E - —4a 4- 116 — 9c 4- (/ + 6", from the latter of which subtract the quintuple of the former, and there will remain *120E - « — 46 + 6c — 4(/ 4- c. Now divide each of the equations marked thus, *, by the co-efiicient of its first term, and there will come out the very values of A, B, C, D, &c. above exhibited, Q. E. D. COROLLARY. If every term of the proposed series a, i, c, c/, &c. be subtracted from the next following, the first of the lemainders, — a 4- ^, — h 4- c, — c 4- ' 4th \^ I Figurate numbers of the 4th order f ^ | i 5th f g i Figurate numbers of the 5th order I | ^ j 6th ] Lpigurate numbers of the 6th order J ,^ ^"^ L7th J Therefore the figurative numbers fist order "^ f 1 . l *. i I 2d order | 11.2.3 of the<< 3d order > are << l . 3 . 6 I 4th order | | 1 . 4 . 10 i_5th order J Ll . 5 . 15 Hence it is manifest, that, to find a general expression for a figurate number of any order, is the same thing as to find the sum of all the figurate number of the preceding older, so far. Let n be put to denote the distance of any such number from the beginning of its respective order, or the number of terms in the pre- ceding order whereof it is composed : then it is evident, by inspection, that the sum of the first order, or the nth term of the second, will be truly expressed by n, the number of terms from the beginning. It is also evident, from Sect. 14, p. 203, that the sum of the second order, 1 +24-3 4-4....w,willbe^+— f = — 2 ^ 2 ^ 1 .which, according to the preceding observation, is also the value of the wth term of the third order. Hence, P 3 1 . 1 • &C. 4 . 5 . &c. 10 . 15 • &c. 20 . 35 . &c. 35 . 70 . &c. 214 THE SUMS OF FIGURATE NUMBEllS. if th^ numbers, 1 , 2, 3, 4, 5, &c. be successively wrote instead of «, in the general expression -^ + — > we shall thence have } + i, f + |. i + 5, V 4- f . ¥ + f, &c. for the values of the first, second, third, fourth, fifth, &c. terms of this order, respectively ; whence it appears, that the series l-4-3-f6-flo + l5-f21, &c. may be resolved into these two others, viz, 1 + ^ -f 1 + ¥ 4- ¥ -h ¥ &c. and i + f+f-ff-ff-Vt &c. The former of which being a series of squares, its sum will therefore be r: -— • 4- -7^+ y^ (% ^^^^ ^' ?• 203) and that of the latter series [by case l. p. 203) appears to be -f ---: and the aggregate of both, which is — + -+ Y(or - X -^ X -—-Jwillbethe true value of the proposed series 14-3 + 6 4- 10 4-15 &c. continued to n terms, and therefore equal, like- wise, to the nth term of the next superior order, 1 4- 4 4- 10 4- 20 -4- 35, &c. Let, therefore, 1, 2, 3, 4, 5, &c. (as above) be successively wrote for w in this general expression, — - ^ -i , and it will be- come j4-i 4- i, f 4- 1 4- f, 4- V 4-1 + i, V 4- V 4-4, &c. for the values of the first, second, third, fourth, &c, terms of the fourth order respectively ; whence it ap- pears that the series 1 4- 4 + 10 4- 20 4- 35, 5^c may be resolved into these three others, ziz, 1 4- 8 -h 27 -f 64 + 125 + 216 ii^ 6 » 14-4 4- 9 4- 16 4- 25 4- 36.... ^2 2 » 1424- 34 44- 5 -\t 6....n AND THEIR RECIPROCALS. 215 , r^\ ^i* ^'' «' '^^ . w^ n whereof the sums are J \- — . ^ — ~ { 24^ 12 24' 6^ 4 ^12 and "g" + "g" (^1/ P' 202, and 203) the aggregate of which, or. - + -- + —+-( =:-x -^ X X ) will consequently be the true value of tlie whole series. After the same manner the sum of the filth order will appear to be — x x — - — I — j— *^' 12 3 4 X ; from whence the law of continuation is o manifest. And it may not be amiss to' observe here, that though the conclusions thus brought out, are deri« ved by means of the sums of powers determined in the preceding section, yet the same values may be other- wise obtained, by a direct investigation, from either of the two general methods there laid down. In order now to find the sums of the reciprocals of any series of figurate numbers, suppose 1 -\- b -{- he ^ bed -f bcde 4- bcdef 4- &c to be a series whose terms con- tinually decrease, from the first to the last, so that the Jast may vanish, or become indefinitely small : then, by taking the excess of every term above the next follow- ing on e, we shall have i — 6, 6 x I — c, he x \ — d, bed X 1 — e, bcde x 1 — /, &c. The sum of all which is, evidently, equal to the excess of the first term above the last, or equal to the first term, barely ; because the last is supposed to vanish, or to be indefinitel y smal l in resp ect of the fi rst. Hence it appears that i — 6 4- b X 1 — c+ be X 1 —d 4- bed X 1 — e -{-bcde x 1 •— / &c. = 1. Let b be now taktm zz — , c z: , d = : — a a 4-p « -f 9 € - — 111-., /~ —11-, &c. Then, 1 - b bemg = P 4 216 THE SUMS OF FIGURATE NUMBERS, Q'-ni a— 771 a—m a-^m «' a-^p. a+9 a+r' &c. we shall, by substituting these several values in the above equation, have ^ -f — ^ X — \ x ^ a a a + p a m^-p a — m, 7n m + p tti ■{- q a — 7n - — -^ X — ; — 4- — X — --^ X — --^ X — — + &c. «+/> a + 7 a a + p a ^ g a -i- r , ^, m 7)1 771 -\- p m 1 ; and consequently 1 -i \ x a-\- p a + p a -i- fj m m At p 771 •\- q . a + X T^ X --7--^ + &c. = ; by a -\- p a -\-q a -\- r a — m -^ dividing the whole by ■ - . Hence, if q be taken = Qp,r — 3p, s — 4jd_, &c, 771 and /3 be put — a + p, we shall have l + "t "^ m»77i-{-p 771 , m \-p»7n 4- 2p , m, 7w 4- p . i7i -\- Qp , 771 + 3p WTJTp'^ (S.(3-\-p.(S-^Qp /3.6 -ip.^+Sp.^+Sp ^ p -f &c. ad infinitUTTit — r-;-- — -— - ; which, when . m , 771 .771 ^ \ , 772 . 77Z + 1 . Wi 4- 2 + &C. — — ^— ^ : this by taking m — \ and 1.2.3 w. « 4- 1 .^ 4- 2. w + 3 ^^— 2 ting the general value of a series of the reciprocals of figurate numbers, infinitely continued ; whereof the or- der is represented by 71: from whence as many parti- cular values as you please may be determined. Thus, by expounding n by 3, 4, 5, &c. successively, it appears that AND THEIR UECiPROCALS. 2l7 1 -I- JL ^ 1 -1- i -f -L &c. = 2. 3 ^ 6 10 15 4 10 20 35 2 And so on, for any higher order; but the sums of the two first, or lowest orders, cannot be determined, these being infinite. By interpreting /3 and m by diflferent values, the sums of various other series's may be deduced from the same general equation. Thus, in the first place, let /3 = 7w + 2 ; so shall the said equation become 1 + m + 2 m,m -\- I ^m,m -h I . 7n .m ■\- l „ &c. m + 2.m + 3 m -\- 3 . m -\- 4: m4-4.m-f5 ::= w 4- 1 ; which, divided by m ,m + 1 , gives m.m-^l 771+1, m^2 w4-2.m + 3 7?z4-3.m+4 &c. 1= i. 711 Again, by taking ^ = m + 3, and dividing the whole equation by m . m + 1 . 7/2 + 2, we have 1 1 + — — -- — - , -^- 1;^:= + m.m + l.m+2 ?/z-fl.//i-f2.w-h3 - &c. z: 7/2 + 2 . m -f 3 . m + 4 m.m -\- l ,2 In like manner we shall have — _ -f 772,772 h 1.7724-2.772 + 3 ' -.^ &c. = ^ . 772+1 .772 + 2 . 772 + 3 . 772 + 4 772 .772+1 . 772 + 2.3 From Avhence the law for continuing the sums of these last kinds of series's is manifest; by which it appears «18 THE SUMS OF that, if instead of the last factor in th<2 denominator of the first term, the excess thereof above the first factor be substituted, the fraction thence arising will truly express the value of the v^^hole infinite series. A few other particular cases will further shew the wse of the general equations above exhibited. 2 2 4 Let the sum of the series l 4 — -- -f -— x -;=- + 2 4 6 2 4 6 8 o ad infinitum, be required. Here, by comparing the proposed series with l -f + -7; — w + &c. ( nz — -^- ^- — ) we have rw n 2, ^ = 5, and/> zz 2 ; and consequently — ^ — -- — zz 3 n the true value of the series. Let the sum of an infinite series of this form, viz* * + ^ ^ I o — + ^ . ^ o + &c. be m 1.2.3 &c. 2.3.4 &C. 3 . 4 . 5 &c demanded. Here, (according to the preceding rule) we have 1.2 2.3 3.4 1.1 1.2.3 2.3.4 3.4.3 1.2.2 4 &c. — 1.2.3.4 2.3.4.5 3.4.3.6 "" 1 .2 . 3.3 ~ 18* &C. &c. If, instead of the whole infinite series, you want the sum of a given number of the leading terms only ; then let the value of the remaining part be found, as above, and subtracted from the whole, and you will have your desire. COMPOUND rilOGRESSIONS. 219 Thus, for instance, let it be required to find the sum of the ten first terms of the series \ 4- — i- 1.2 2.33,4 &c. Then the remaining part. rt^/e a5oue) and the whole series =. i , the value here sought will therefore be l i- =H. The like of uU u 11 others. The sums of series's arising from the multiplication of the terms of a rank of figurate numbers into those of a decreasing geometrical progression, are deduced in the following manner. By the theorem for involving a binomial {given at p. 40, and demonstrated hereafter) it is known that =J^ (orT=3-«j is = 1 + ^x + m . !1±1 . ^e ^ 3 2 3 4 &c. In which equation let m be expounded by i, 2, 3, 4, 5, &c. successively, so shall ^''- rZTi = ^ + ^-^ 3!'' + 3c' + X* + x' -\- he. 2' 3^ ^^==j3 = 1 + 3a: + 6a?' -f lOl'^ + 15t* + Six* &c. . 1 4 • f==74 = 1 4- 4a; + 10x*4- 20x3 + 35a:* + 56x'kc;. ^ 5 =1+50;+ 15.T2-f 35a:'+70x*+ 126a:' Sec. 1—0^ ^ *f^^6 = 1 + 6j7+2la;H56x^'l-126.i**+252.i^&c. 220 THE SUMS OF All which series's (whereof the sums are thus given) are ranks of the different orders of figurate numbers, multiplied by the terms of the geometrical progression From these equations the sums of series's composed of the terms of a rank of powers, drawn into those of a geometrical progression, such as 1 + 4a7 + 90:'^+ l6x^ &c. and 1 4- ScT + 27a?- 4- 64^^ &c. may atso be de- rived ; there being, as appears from the former part of this section, a certain relation between the terms of a series of powers and those of figurate numbers; the lat- ter being there determined by means of the former. To find here the converse relation, or to determine the former from the latter, it will be expedient to multiply the several equations above brought out, by a certain number of terms of an assumed series 1 + A^r + Bx^ + CcT^ &c. in order that the co-efficients of the powers of 0? may, by regulating the values A, B, C, D, &c. become the same as in the series given. Thus, if the series given be 1 -\- 4x -\- 9x" + l6x^ -{- 25^* &c. ; then, by multiplying our third equation, J 4- Ax by 1 + A 07, we shall have ,3 =; i + 3-1-A x x 1 — x\ + 6 + 3A X ct"" + 10 + 6A X x^ -{- &c. which scries, it is evident by inspection, will be exactly the same, in every term, with the proposed one, if the quantity A be taken zn l. The sum of the said series, intinitely 1 ■{• X continued, is therefore truly represented by - . 1 — xf In like manner, if the fourth equation ^ :7 ~ 1 — - x\ 1 V- 4t + 10.r* f 20x^ + 35.r*&c. be multiplied by 1 + At 4- B^" 1 f Ax 4- B.r^ there will arise .. — 1 1- 1 — xf 4 4- A X .T 4 10 f 4 A 4- B X a- 1- 20 4 lOA 4- 4B x .r' &c. where, the several terms of the series beinn- coni- purod with those of the series l \- Sx 4- 27^c* 4- 64.v^ &;c. COMPOUND PROGRESSIONS. 221 we have '4 + A = 8, and 10 + 4A + B := 27 ; whence A = 4, and B =: 1 ; and consequently, by sub- 1 4- 4x + x^ stituting these values — . -.4 ■■ - - =1-1- Sx -f 27 x- 1 — — 0P\ 4- 64a?' + 125a:* &c. Again, by multiplying the fifth equation, ■■:■ >s ^ I + 5x + I5x^ -{- 35x^ &c. by 1 + Ax + Bx" f Cx\ 1 + A* + Ba?" + Cx it becomes ,5 z= 1 4-5fAxa:-f 1 — x\ 15 + 5A + B X .t' + 35 4- 15A + 5B h C xa?^&c. And, by comparing the several terms of the series with those of 1 + I6:r + 8Lt' -f 256x^ &c. we get 5 + A z= 16, 15 + 5A 4- B zz 81, and 35 + 15A 4- 5B 4- C — 256: whence A = n, B (rz 81 — 15 — 55) = 11 andC (= 256 — 35— 220) =1 1 ; and consequently 1 -I- 1 1 T 4- 1 1 1* 4~ x^ -^ ^ .3^ = 1 4- I6a? -f 81a;'» + 256x^ &c. 1 —x\ By proceeding the same way it will- be found, that 1 4- 26.r 4- 66x'' + 26x^ 4- a?* ' = 14- 2'':r 4- 3^x2 4- 4^x^ 4- &c. &c. * , ' . „ . 7 w 4- 1 And, universally^ puttmg a — m, :=: m , — czz7n. ^ •+• 1 /» 4- 2 ^ ^^^ ^^^ multiplying the gene- ral equation, ' - zzl i- ax+ hx^ 4- cx^ 4- dx"^ &c. by 1 4- Aa: 4- B;c2 ^ Cx^ + Da* &c. there arises » + ^- + ^^' ^"- = , + «-r-A X . + 1 — x\ i V «A 4- B X 0?'^ 4- c 4- 6A + aB 4- C X a:" &c. The terms of which series being compared with ihose 222 THE SUMS OF of #ie series i 4- 2"jc + 3V f 4"a:' -H 5"a;% &c. \ye have A z= 2" - a, B = 3" — rzA — 5, C = 4^ •— aB — /yA — c, D - 5** — aC — 5B — cA — J, &c. where the law of continuation is manifest, and where, from the law observed in all the preceding cases, it ap- pears, that the value of m must exceed the index w, of the given series of powers, by an unit ; and that the series 1 -f Aa? + Bx' -f Cx^, &c. will always con- sist of 72 terms ; whereof the co-efficients of the first and last, the second and last butone,&c.willbe respectively equal to each other : so that having found from the preceding equations as many of the quantities A, B,C, &c, as are expressed by {n — 1, the others will be given 1 + Ax-\- BxM-CV&c. from thence, and consequently, „ . i 1 — x) the true value of the proposed series l -|- 2"ar -f S^^ -f 4"x' &c. Thus, for example, let n zz 6 : then m = 7 = a, i = 28, A = 64 — 7 = 57, B = 729 — 399 — 28 = 302 ; and therefore 1 4- 57x -f 302X* 4- 302^3 -f- 57x* -f a;' , . ^n . .7 — . — — iz 1 4- 2" a; 4- 1 — a I S^X' 4- 4V &c. and so of others. These equations, or theorems, give the sum of the whole series, infinitely continued ; but from thence the sum of any assigned number of terms may be deter- mined, not only when the co-efficients are a series of powers, but likewise when they are produced by factors that are unetiual : the method of which I shall in- stance in find in "f the sum of t terms of the series / — p^g — q'^'-^f—^P^g — '^q. 2''+" 4- / — 3p. j^ — 37 . 2''+^*' 4- &c. Which series, by actually mul- tiplying the factors together, is resolved into the three following ones, /gz"" X 1 -i- 2^ -f- z^"" f 2^*^ 4- z^'^ &c. — /7/ X 1 + 42« 4 92^'' 4- 16^'^" &c.> ,<,ve shall here have &c. for the sum of the whole infinite series; and if we make a' = a — tp, b' = h — tp,r' — r -\- tv, &c. it is evident that the sum of the remaining terms, afte^j the t tirst, will be truly expressed by z'' X p .- ^'^ Ry .TT^ _^ s y^i. 4-^ -f a^ 1.-^ 1-0? 1^=^^ Tzm^ &c. where a? = 2^ and P', Q^ R', S^ &c. are the same in relation to a\ I/, c\ d\ &c. as P, Q, R, S, &c. in respect to a, b, c, d,8ic. A multitude of other cases and examples might be given, there not being, in the whole scope of the mathematical sciences, a subject of greater variety and intricacy than this business of series's : but to pursue 27 farther Aere would be inconsistent with the general plan of this work. Such, therefore, who are desirous of a greater insight into the matter, may, if they please, turn to my Miscellanies, where it is carried to a greater length. From the series's for figurate numbers, derived in the former part of this section, the investigation of a general theorem for determining how many dif- ferent combinations any number of things will admit of, when taken two by two, three by three, &c. may be very easily deduced. Let the number of things in each combination be, first, supposed two, only ; and let n be, unii er sully i^ put to represent the ivhole number of OF COMBINATIONS 225 things, or letters, a, b, c, d, h)C, to he combined. When the number of things is only two, as a and b, it is evi- dent that there can be only one combination {ah) ; but, it" 7J be increased by 1, or the letters to be combined be three, as a, 6, c, then it is plain that the number of combinations will be increased by 2, the number of the preceding letters a and h ; since, with each of those^ the new letter c may be joined; and therefore the whole number of combinations, in this case, will be truly expressed by 1 f 9. Again, if « be increased by one more, or the whole number of letters be four, as a, h, c, ci, then it will appear that the number of com- binations must b« increased by 3, since 3 is the num- ber of the preceding letters, with which the new letter d can be combined, and therefore will, here, be truly expounded, by i 4-2+3. And, by reasoning in the same manner, it will appear, that the whole number of combinations of two, in five things, will be 1 4- 2 -f 3 + 4 ; in six things, 14-2+34-4 + 5; and in sevv^n, 14-2+34-44-34-6, &c. Whence, universally, the number of combinations of n things, taken two by two, isz: 1-4-2 + 3 + 4 + n — \: which being a series of figurate numbers of the second order, where the number of terms is w — i, the sum thereof, by case l,p. 203, will therefore be truly de- ^ J , n — 1 n n — i fined bv — y^— , oxn x • •'12' 2 Let now the number of quantities in each combination be supposed to he three. It is plain, that, in three things, a, h,c, there can be only one combination; but, \in be increased by l, or the number of things be 4, as a, b, c, d, then w^ill the number of combinations be increased by (3) the num- ber of all the combinations of two, in the preceding letters a, b^ c; since with each two of those the new letter d may be combined ; therefore the number of combinations, in this case, is 1 + 3. Again, if n be supposed to be increased by 1 more, or the number of letters to become five, as a, 6, c, r/, e ; then the number Q 2^6 OP COMBINATIONS. of combinations will be increased by six more (=: l + 9 + 3), that is, by all the combinations of two, in the four preceding letters, «, A, c, d; since (as before) with each two of those, the new letter e may he combined. Hence the number of combinations of 7^ things, taken three by three, appears to be i -f- 3 + 6 + 10, &c. continuedto?? — Sterms; which being a series of ligurate - numbers of the third order, the value thereof, by what is before determined (p. 214) will be truly expressed by n — 2 » — 1 M . .71 n — 1 w— 2 -y-X --^ X- . or. us equal,- X -j- x -^. And, universally, since it appears, that increasing the number of letters by l, always increases the number of combinations by all the combinations.of the next inferior order with the preceding letters (for this obvious rea- son, that to each of these last combinations the new letter may bejoined), it is manifest, that the combina- tions, of any order, observe the same law, and are ge- nerated in the very same manner as ligurate numbers ; and therefore may be exhibited by the same general expressions; 'only, as there are 2, 3,4, 5, &c. things necessary to form the first, or one single combination, according to the different cases, it is plain, that the number of terms must be less by l, 2, 3, Sec, respec- tively, than («) the number of things; and therefore, instead of //, in the aforesaid general expressions, we must substitute n — l,w — 2, or n — 3,&c. respective- ly, to have the true value here. Hence, the number of combinations of two things, in ?i things, will be 71 — 1 n n n — 1 r.., «- 2 n—\ n -^ X J, or - X - -- ; of three. -_ x -^ x - , n n — 1 n — 2 ^^ -n — 3 n — 2 n~i ^'^'^ -^x-^;offour,-^^X-^ x-^ - (vid. p. 215) : whence, universally, the number of combinations in the number, n, of things, taken two by two, three by three, n n — 1 n — 2 n — 3 &c. will be expressed by ~ x — ~ X — y- X — j— , OF THE BINOMIAL THEOREM, 227 &c. continued to as many factors as there are things in each combination. From this last general expression, shewing the coni- binations which any number of quantities will admit of, the known theorem for raising a binomial, to any given power, is very easily, and naturally derived. For, it is plain that a^^^^a-f6c = a+-6x a -i- c ; -f 6 ^ he ^ which, multiplied by a + rf, gives a' + c [-a* f- hd^a {- + d} cd} hcd zz cTVb X a^ c x a f rf ; and this, again, multiplied by a + ^5 gives + he ^ hcd hce -hb") + bd) 4- ^de^ a'^h X a-fc X a^d x a-^e. J J ^ a -\- bcde = hde cde Whence it appears, that the co-efTicient of a, in the second term, is always the sum of .111 the other quanti- ties 6, c, (/, &c. added together ; and that the coefficient of the third term is the sum of all the products of those quantities, or of all their possible combinations, taken two by two ; since, from the nature of multiplication, they must be all concerned alike, in every term : whence it is also manifest, that the coefficient of the fourth term must be the sum of all the solids of the same quantities, or of all their possible combinations, taken three by three, &c. &c. Hence, if the number of tht quantities b, c, d, e, &c. or the number of the factors, a + 6, a -\- c, a ^ d, to be multiplied continually together, be denoted by n ; it follows, that the number of letters, or qnanti- ties in the coefficient of the second term of the pro- duct will likewise be denoted by n ; that the num- ber of all their products, or of all the combinations of (22 ei8 OF THE BINOMIAL THEOREM. two, in the coefficient of the third term, will be w X -~— (it having been shewn above, that the number of com- binations of n things, taken two by two, is n x — — J ; and that the number of all the solids of those quantities, or all the combinations of three, in thecoefficient of the fourth term, will be w X -^^— X —~, &c. Therefore, 3 o if all the quantities 6, c, d, e, &c. be nov y take n equal to each other, it is evident that a^-b x a-^-c x a-Vd xa + e, &c. will become a-^b x ahb X tf-i-6 X a+b &c. or u + b\^: and that the coefficient of the power of a, in the second term of the product, will be ?ib ; in the fi I third;tx- b- (since all the rectangles, as well^as all the solids, 8cc. do here become equal) ; and in the fourth n x ^^-^ X '^-^— ^% &c. But it is eviden t,from the nature of multiplication, that the powers of a, in the second, third, fourth, &c. terms of a H- 5 raised to ' the power fty are d!"^ , a"" , a"*" , &c. Therefore ^"^IT)", o\ a \ b raised to the power w, is truly ex- pressed by a^ + «^a"~^+ « X ^^ *' ^'"''^ +«x ^^ X !Lri63A"-^&c.o^ a« + «fl"-'6 + nyX^a^^^b'^ 3 '^ _|. /I X ^^ X ^^^^a""'^^^ &c. as was to be shewn. 2 "^ OF INTEREST AND ANNUITIES. 220 SECTION XVI. Of Interest and Am^uities, INTEREST may be either simple or compound : simple interest is that which is paid for the loan of any principal, or sum of money, lent out for some li- mitted time, at a certain rate per cew^ agreed upon be- tween the borrower and the lender, and is always pro- portional to the time. Thus, if the rate agreed upon, be 4 per cent, per annurn^ or, which is the same, if the interest of 100/. for one year be 4/. then the simple in- terest of the same sum for two years, will be 8/. for three years, 12/. and for 4 years 16/. and so on for any other time, in proportion. Compound interest is that which arises by leaving the simple interest of any sum of money, after it be- comes due, together with the principal, in the hands of the borrower, and thereby converting the whole into anew principal. Thus, he who lets out 100/. for one year, at the rate of 4 joer cent, has a right to receive 104/. at the year's end ; which sum he may leave in the bor- rower's hands, a second year, as anew principal, in order to receive interest for the whole; and this interest (which will be found 4/. 3s, 2jc?.) together with 4/. the interest of the first principal, for the first year, will be the compound interest of loo/. for two years : and so on, for any greater number of years. But I shall first give the investigation of the theorems for simple interest. Let the rate per cent, or the interest of I0o7. for one year = r ; the months, weeks, or days in one year z=. t ; the months, weeks, or days which any sum, a, is lent out for — n ; and the amount of that sum, in the said time, viz. principal and interest, = h. Then it will be as i oo is to r (the interest of 1 00/.) >so is the proposed sum [a) to -^, the interest of that sum, 100 for the same time. Again, as t, (he time in which the Q3 230 OF JNTEKl^ST AND ANNUITIE's. said interest is produced, is tow (the time proposed) so is -— ", the interest in the former of these times, to . , that in the latter ; which added to, a, the principal, gives « + ——=:/; (the whole amount): from whence, we also have a - -■--—■ , r - — r , and n = 1 00^ -t- nr an loot y. h — a . r X ' X. ■ the use oi which equations, or theorems, will appear by the following examples. Examp, 1. What is the amount of 550/. at 4 /)er cent, in seven months? In this case we have a z=. 550, r z: 4, i zz I2,n zz 7 ; and consequently a + -^^ =:550-f '^'^^^ ^ ^^=5624/> ^ -^ 100^ ICO X 12 ^ or 562/. 16^. 8(/. the true value sought. Examp, 2. What is the interest of i/. for one day, at the rate of 5 per cent. ? Here r being — 5, t m 365, a zz l, and « = 1, we have zz — - zz zz 0.0001369863, 100« 100X365 100 X 73 ^ ' &c. zz: the decimal parts of a pound required. Examp, 3. What sum, in ready money, is equiva- lent to 600/. due nine months hence, allowing 5 per cent, discount? Here r being zz 5, t zz 12, n zz 9y and b zz 600, we h^wea{by Theorem 2)- —^ "^f^^l ^! = 578,213/. ^ ^ ' 100 X 124-9X5 ' or 578/. 6s. 3[d, which is the value required, Examp, 4. At what rate of interest will 300/. in iifteen months, amount to, or raise a stock of 33o/. ? In this case we have given « =z 12, n = 15, « =: 300, and b zz 330 ; whence (by Theorem 3) r will come out ^' " — 8 ; therefore 8 per cent, is tlie rate 300X15 required. OF INTEREST AND ANNUH1ES. 231 Exmnp. 5. In how many daysUvill 365/. at the rate of 4 joer cent, amount to, or raise a stock of 400/.? 100 X 365 X 35 Here {by Theorem 4) we have n — — -j^^ ^ ^ = 875 = the number of days required. Of Annuities or Pensions in arrear, computed at Simple Interest. Annuities or Pensions in arrear are such, which, being payable, or becoming due, yearly, remain unpaid any number of years : and we are to compute what all those payments will amount to, allowing simple inte- rest for their forbearance, from the time each particu- lar payment becomes due ; in order to which, rA— the annuity, pension, or yearly rent. J An zz the time, or number of years, it is forborn. ^ \r — the interest of l/. for one year. ^m = the amount of the annuity and it*s interest. Then, as l : r : : A : rA, the interest of the proposed sum or pension A, for one yen r; which, as the last year's rent but one, is forborn only one year, will express the whole interest of that rent, or payment : moreover, since the last year's rent but two is forborn two years, it's in- terest will be 2rA : and, in the same manner, that of the last year's rent but three, will appear to be 3rA, &c. &c. whence it is manifest that the sum total of all these, or the whole interest, to be received at the expiration of ;2 years, for the forbearance of the proposed annuity or pension, will be truly defined by the arithmetical pro- gression r A f 2rA-f 3rA-\- 4rA-f 5rA, &c. continued ton — 1 terms, that is, to as many terms as there are years, excepting the last. But the sum of this pro- gression is equal to « x — — x rA {by Theor. 4. Sect.lO.) Therefore, if to this, the aggregate of all the rents, or wA, be added, we shall have nA ~\ x rA — m ; Q4 232 OF INTEREST AND ANNUITIES. whence we, also, have A zz n— 1 7? -f w X X r 2 — ^^ — 2nA "" w X n — 1 X -^, and w zi V ?£ + |,2_p . sup- posing/) - -i- - — . Examp. 1. If 600/. yearly rent, or pension, be for- born five years, what will it amount to, allowing 4 per cent, interest for each payment, from the time it be- comes due ? Here we have given A = 600, w — 5, and r ~ .04 (for as 100 : 4 : : 1 : .04) which values substituted in fi J TheoTem 1. give m — {iiK -J- ^ X Ar — 3000 4- 240) zr 3240/. for the value that was to be found. Examp. 2. What annuity, or yearly pension, being forborn five years, will, in that time, amount to, or raise a stock of 3240/. at 4 per cent, interest ? In this case we have given n zz d^r — .04, and m m 3240, and therefore, by Theorem 2, A ( =: ; = — zz > \ = 600 ; which is the annuity required. Examp, 3. At what rate of interest will an annuity of 560/. in seven years, raise a stock of 4508/. ? In this case we have given A ~ 560, w = 7, and m — Qjfl - 2wA 4508 ; whence (hy Theor. 3) we have r I — 7JX72— ixA = -t;. r? — ) ^ '05 =: the interest of l/. for one 42 X 56o / year ; therefore it will be as 1 : .05 : : 100 : 5 [per cent,) the rate required. Examp, 4. How long must an annuity of 560/. be forborn, to raise a stock of 4508/. supposing interest to be 5 per cent. } OF INTEREST AND ANNUITIES. 233 Here, we have given A — 560, r — .05, rn z=. 4508 ; whence, by Theorem 4, we also have p (= ) ~ 19.5 ; and consequently n {=z y -t-^ p* — p)— 7 ; which is the number of years required. Note. If the rent or pension, is payable half-yearly, or quarterly, the method of proceeding will be still the same, provided n be always taken to express the num- ber of payments, and r the interest of l/. for the time in which the first payment becomes due. Thus, if it were required to find what 300 /. half-yearly pension would amount to in five years at 4 per cent, interest : then, the simple interest of l/. for half a year being — .02, and the number of payments — 10, we, in this case, have A =: 300, r z: .02, and « — 10 ; and con- sequently m [by Theorem i) = rA -f wx X rA zz 3270/. which is the value sought. And the like is to be observed in what follows hereafter. Of the present values of Annuities^ or Pensions, com- puted at Simple Interest, Let C A — the annuity, pension, or yearly rent. J r = the interest of iL for one year. 1 « — the number of years. \^v =: the present value of the annuity. Then, because the amount of the annuity, in n years, is found above to be nA -f in . n — i . rA, and since 1/. present money, is equivalent to l -f ??r to be re- ceived at the end of the time w, we therefore have 1 + wr : I : : nA f in , n — 1 .rA (the said amount) nA 4- {71 . tT^ 1 . rA . , . , , . : L __^_ ^ Its required value, m present money. But it may be observed, that this method, given by authors for determining the values of annui- ties, according to simple interest, is, in reality, a parti- cular sort, or species of compound interest : since the allowing of interest upon the annuity, as it becomes 234 OF INTEREST AND ANNUITIES. due, is nothing less than allowing interest upon inte- rest; the annuity itself being, properly, the simple in- terest, and the capital, from whence it arises, the prin- cipal. It is true, the sum, l 4- nr, expressing the amount of iL is given, strictly speaking, according to shnple interest: but the conclusion (as a late author* very justly observes) would be more congruous, and answer better, were the same allowances to be made therein, as are made in finding the amount of the annu- ity ; thht is, were interest upon interest to be taken once and no more. Agreeable to this assumption r, the interest of l/. being considered as an annuity, it's amount in w years (by writing r for A, in the ge neral formula above) will be given =: wr -f f w . w — 1 . r- : to which the pr incipa l i/. being added, the aggregate 1 + wr -f \n . n — 1 . r" will therefore be the whole amount of 1 /. in the time n ; and so we shall have 1 -h nr -f In . n — 1 . ?-' : 1 : : «A + fw . n — 1 . rA ; QnX-\-n.7i — 1 .rA ■■■ c= V, the true value of the annu- Q^^nr-^n .n — 1 . r^ itv, according to the said hypothesis. From which equation others may be derived, by means whereof the different values of A, n, and r, may be, successively, determined. But, as this method of allowing interest upon interest, once and no more, is arbitrary, and the valuation of annuities, according to simple interest, a matter of more speculation than real use, it being, not only custonmrv, but also most equitable to allow com- pound interest in these cases, I shall not stay to exem- plify itt but pro( eed to The resolution of the various cases of Compound Intc^ rest, and Annuities as depending thereon, r D — / l^he amount of l/. in one year, viz. prin- Let \ ~" \ cipal and interest. I P zz any sum put out at interest. * 31i\ liardij in his Annuities. OF INTEREST AND ANNUITIES. 235 n n the number of years it is lent for. a — it's amount in that time. Let / A. = any annuity forborn n years, m = it*i> amount. __ ^the present value of the annuity for the ^' — i same time. Therefore, since one pound, put out at interest, in the first year is increased to R, it will be as i to R, so is R, the sum forborn the second year, to R^, the amount of one pound in two years ; and therefore as i to R, so is R\ the sum forborn the third year, to K^ the amount in three years : whence it appears that R", or R raised to the power whose exponent is the number of years, will be the amount of one pound in those years. But as l/. is to it's amount R , so is P to {a) it's amount, in the same time ; whence we have P x R — a , More- over, because the amount of one pound, in n years, is R", it's increase in that time will be R'' — 1 ; hut it*s interest for one single year, or the annuity answering to that increase, is R — 1 ; therefore as R — i to R — i, A X K" I so is A to m. Hence we get — 5 =:: m, Fur- K — 1 thermore, since it appears that one pound, ready-money, is equivalent to R", to be received at the exp iration of n A X R" 1 n years, we have, as R to 1, so is — r^ — (the sum in arrear) to r, it's worth in ready money ; and there- Ax I— -i f R" fore-j^— ^— From Which three original equations, others may be derived, by help whereof the various questions relating to compound interest, annuities in arrear, and the pre- sent values of annuities, may be resolved. 236 OF INTEREST AND ANNUITIES. Thus, because PR" is n a, there will come out P = •^1 j,„» and R = --- K &c. or, by exhibiting tbe same equations in logarithms (which is the moBt easy for practice} we shall have 1^ Log. a zz log. P + w X log. R. 9^ Log.? zz log. a — n X log. R. 30. Log.Rz:^^g'^7^^--^. ^ • ^ - log. R • Which four theorems, or equations, serve for the four cases in compound interest. Again, since 772 is rr — p -, we shall have 1°. Log. m =log. A + log. R^' — 1 — log. K — l. 2^. Log. A =log. w— log. R« — 1 4- log-R — L 00 ^ _ log. mR — 7» -f A — log. A 3.n^ i^^ns • A A To which the various questions relating to annuities in arrear are referred. Moreover, seeing A x is — », we thence have, R—l r. Log.T? = log. A + log. 1 -. J. — log. R — 1 2°. Log. A - log. V \- log. R — 1 _ log. 1 — j77r 3". w - ^Qg A —log. A 4- v~vU lo-?. R o .1^ ir-*-^ --- -f 1 X R« -t- ~ OF INTEREST AND ANNUITIES. 237 The use of which theorems, respecting the present values of annuities, as well as of the preceding ones, for compound interest and annuities in arrear, will fully appear from the following examples. Examp. 1. To find the amount of 575/. in seven years at four per cent, per annum, compound interest. In this case we have given P = 575, R = 1,04, and n — 7 'y therefore, by Theorem 1, log. a = log. 575 -f 7 log. 1,04 = 2,8789011 ; and consequently a = 756,66, or 756/. 13^. ^^d., the value required. Examp. 2. What principal, put to interest, will raise a stock of looo/. in fifteen years, at 5 per <:ent. ? Here, we have given R=:l,03, w=zl5, and aiziooo; therefore, hy Theorem 2, log. P zz log. looo — 15 log. 1,05 zz 2,6821605; and consequently P zn 481,02, or 481/. OS. 4|df., the value sought. Examp. 3. In how long time will 575/. raise a stock of 756/. 13^. 2|:6?., at 4 per cent, f In this case we have R - l,04, P n 575, and a — 756.66; whence, &y Theor. 4, n^""^' 756.66-log. 575 log. 1,04 n 7, the number of years required. Examp. 4. To find at what rate of interest 481/. in fifteen years, will raise a stock of looo/. Here we have given P = 481, a =i iooo, and «=15; ., r. T ^, I T> log. 1000 — lo^.48i theretore, hy Theorein 3, log. R — ^^^ = .0211903, whence R =: 1,05 ; consequently 5 per cent, is the rate required. The four last examples relate to the cases in com- pound interest ; the four next are upon the forbearance ofannuities. Examp. 1. If 50/. yearly rent, or annuity, be for- born seven years, what will it amount to, at 4 percent, per annum , compound interest ? Here, we have R zz i,04,, A zz 50, and n — 7 ; and 238 OF INTEREST AND ANiNUlTIEx therefo re, hi/ T heor. l , log. m ( = l og. A -^ log. R" — l — log. R — 1) — log. 50 f log. i^^— 1,— log. ,04 = 2,396597: and consequently m zz 395/. the value that was to be found. Examp. 2. What annuity, forborn seven years, will amount to, or raise a stock of 395 /. at 4 per cent, com- pound interest? In this case we have given R = 1,04, n zz 7, and m — 395 ; whence, by llieorem 2, log. A ( = log. m — log. R « — 1 + log. R— 1) - log. 395 — log. l,04t' — 1 4- log. .04 = 1,6989700; and consequently A zi 50/. which is the annuity required. Examp, 3. In how long time will 50/. atinuity raise a stock of 395/. at 4 per cent, per annumy compound interest ? Here, we have R zz 1,04, A rz 50, vi — 395 ; and therefore, by Theor. 3,„(^ log^^R-^^A--log^^ -. » — 7 the number of years required. ,0170333 J i Examp, 4. If 120/. annuity, forborn eight years, amounts to, or raises a stock of 1200/. what is the rate of interest ? In this case we have given n — 8, A — 120, and m •=1 1200, to find R ; therefore, by Theorem 4, we have R*^ — lOR 4- 9 = ; from which, by any of the me- thods in Section 13, the required value of R will be found — 1,06287 ; therefore the rate is 6,287 or 6/. bs. gd, per cent, per annum: The solution of the last case, where the rate is re- quired, being a little troublesome, . I shall here put down an approximation (derived from the third gene- ral /or7«w/a, at jt>. 165) which will be found to answ^er very near the truth, provided the number of years is OF INTEREST AND ANNUITIES. 239 «.«—!. A ^ Let Q =: -■ . ; then will 2 . ?» — w A 3000 Q +- 272— 1 .400 ^ , be the rate 6Q . 5Q + 3/2 — 4 4- i.^*— 2. 11«- 13 per cent, required. Thus, for example, let « = 8, A = 120, and wzzzs 5Q 120 1200; then will Q =: -— rr = 14, and the rate it- 2 . 240 .r, 42000 4 6000 ^ ^„ 7 self = --— - 6 . 287, as above, 84 X 90 + 75 The preceding examples explain the different cases of annuities in arrear ; in the lol lowing ones the rules for the valuation of annuities are illustrated. Examp. 1. To find the present value of 100 /. annu- ity, to continue seveh years, allowing 4: per cent, per afinum^ compound interest. Here, we have given R = l,04, A r; lOo, and ?^ — 7 ; and therefore, by Theorem l, log, » ( = log. A + log. 1 — _ — log. K — 1 ) - log. 100 + log, 1 — c=-— log. ,04 = 2,778296; and consequently l,04r -I J 1) — 000,2 = Goo/. 4^. which is the value that was to be found. Examp. 2. What annuity, or yearly income, to con- tinue 20 years, may be purchased for looo/. at3f per cent, ? In this case, R =: 1,035, ?2 = 20, v = lOOO ; whence, by Theore m 2, we have log. A ( ~ log. v + log. R - 1 — log. 1 — -ji; J =: 1,847336; and con- sequently A zz 70,36, or 70/. 7^. '2d, ^40 OF LXTEUESr AND ANxNUmES. Examp, 3. For how loni^ time may one, with 6oo/. purchase an annuity of 100?. at ^per cent. ? In this example we have R zr i,04, A =: 100, and V — 600 ; and therefore , by Theorem 3, w ( = log. A^— log. A + V — vR \ _ J ~ 7. years required. , ^ I _- « , the number of Examp. 4. To determine at what rate of interest, an annuity of 50/. to continue 10 years, may be purchased, for 400 /. Here, A = 50, n zz lo, and y ■=! 400 ; whence, hy Theorem A^K^^^ -f- 1 X R« + ~ being = o, we have R"-— 1,125 R'° -1- ,125 = 0; which equation resolved, gives the required value of R :=: 1,042775 ; and consequently the rate of interest, 4,2775/. per cent, per annum. The solution of this last case being somewhat tedi- ous, the following approximation (which will be found to answer very near the truth when the number of years is not very large) may be of use. ^ « . w + 1 . A , „ Assume Q = — ^-r- — - — ; so shall ^ 2/2A — 2u 3000Q — 2w + 1 '< 400 _^ express 6Q . 5Q — an-— 4 4- ^ . w 1- 2 . 11« + 13 the rate per cent, very nearly. Thus, for example, let A (as above) be r: 50, A r. cu ^ u • 10 >C 11 X50 72 z: 10, and v = 400 ; then, Q bemg zz 82500— 8400 . «^„. = ^^'' • '"' ''«^^ 103X103.3 +240, ' ^"•- *•'"'• for the rate, per cent, the same as before. [ 241 ] SECTION XVIL OP PLANE TRIGONOMETRY* DEFINITIONS. 1. "pLANE Trigonometry is the art whereby, hav- JL ing given any three parts of a plane triangle (except the three angles) the rest are determined. In order to which, it is not only requisite that the peri- pheries of circles, but also that certain right lines, in and about the circle, be supposed divided into some assigned number of equal parts. 2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees ; and each degree into 60 equal parts, called minutes, and each minute, into 60 equal parts, called seconds, or second minutes, &c. Any part of the periphery is called an arch, and is measured by the number of degrees and minutes, &c. it contains. 3. The difference of any arch from 90 degrees, or a quadrant, is called its complement, and its difference from 180 degrees, or a semi-circle its aupplement. 4. A chord, or sub- tense, is a right line drawn from one ex- tremity of an arch to the other ; thus BE is the chord or sub- tense of the arch BAE, or BDE. 5. The sine (or yight sine) of an arch is a right line drawn from one extremity of the arch perpen- dicular to the diame- ter passing through the other extremity : thus BF is the sine of the arch AB, or BD. 242 OF PLANE TRIGONOMETRY. 6. The ver$ed sine of an arch is the part of the dia- meter intercepted between the arch and its sine : so AF is the versed sine of AB, and DF of DB. 7. The co-sine of an arch is the part of the diameter intercepted between the centre and the sine; and is equal to the sine of the complement of that arch. Thus CF is the co-sine of the arch AB, and is equal to BI, the sine of its complement HB. 8. The tangent of an arch, is a right line touching the circle in one extremity of that arch, continued from thence to meet a line drawn irom thecentre through the other extremity; which line is called the secant of the same arch : thus AG is the tangent, and CG the secant of the arch AB. 9. The co-tangent and co-secant of an arch are the tanc^ent and secant of the complement of that arch ; thus BK and CK are the co-tangent and co-secant of the arch AB. 10. A trigonometrical canon is a table exhibiting the length of the sine, tnngent, &c. to every degree and minute of the quadrant, with respect to the radius which is supposed unity, and cenceived to be divided into 10000000 or more decimal parts. Upon this table the numerical solution of the several cases in trigono- metry depend; it will therefore be proper to begin with its construction. PROPOSITION I. The number of degrees and minutes, S^c, in an arch being given ; to find both its sine and co-sine. This problem is resolved, by having the ratio of the circumference to the diameter, and by means of the known series for the sine and co-sine (hereafter de- monstrated). For, the semi-circumference of the circle, whose radius is unity, being 3,141592633589793 &c. it will therefore be, as the number of degrees or mi- nutes in the whole semi-circle is to the degrees or minutes in the arch proposed, so is 3,141592(55358 &c. to the length of the said arch ; which let be denoted by a ; then, by the series above quot id, its sine will be ex- OF PLANE TRIGONOiVlETRY. 243 pressed by g — - — - + 3 2.3.4.3 2.3.4.5.6.7 &c. and its co-sine by l -f ~ 2 2.3 &c. 2.3.4.5.6 2.3.4.5.6.7.8 Thus, for example, let it be required to find the sine of one minute: then, as 10800 (the minutes in 180 de- grees) : 1 :: 3,14159265358 &C. : .000290888208665 — the length of an arch of one minute : therefore, in this case, a ^ .000290888208665, and -^ (—---) — .000000000004102, &c. And consequently .000290888204563 =: the required sine of one minute. Again, let it be required to find the sine and co-sine cf five decrees, each true to seven places of decimals. Here .0002908882, the length of an arch of I minute (found above) being multiplied by 300, the number of minutes in 5 degrees, the product .08726646 will be the length of an arch of 5 degrees : therefore, in this case, we have a - ,08726646 -5- := — ,00011076, o a} 4- -^ = + ,00000004, &c. and consequently ,08715574 iz the sine of 5 de- grees. Also -^ - .00380771, a zz .00000241 ; 24 and consequently ,9961947 = the co-sine of 5 degrees. After the same manner, the sine and co-sine of any other arch may be derived ; but the greater the arch the slower the series will converge, and therefore a greater number of terms must be taken to bring out the conclusion to the same degree ot exactness. R 2 244 OF PLANE TRIGONOMETRY But there is another method of constructing the trig- onometrical cannon ; which, though less direct, is more geometrical ; and that is by determining the sines and tangents of different arches, one from another, as in the ensuing propositions. PROPOSITION II. The sins of an arch being given ; to find its cO'Sine, tangent, co^tangent, secant^ and co-secant. Let AE be the proposed arch, EF its sine, CF its co-sine, AT its tangent, DH its co-tangent, CT its secant, and CH its co-secant : then, (hy Euc. 27. 3 .) we shall have CF = \/CE' — £F^; from whence the CO sine will be known ; and then bv reason of the similar triangles, CFE, CAT, and CDH, it will be, 1. CF : FK :: CA : AT whence tlie tangent is known. 2. CF : CE : : CA : CT whence the secant is known. 3. EF : CF : : CD : DH whence the co-tangent is known. 4. EF : CE :: CD : CH ; whence the co-secant is also known. Hence it appears, 1 . That the tangent is a fourth proportional to the co.sine, the sine, and the radius. 2. That the secant is a third proportional to the co-sine, and the radius. 3. That the co-tangent is a fourth proportional to the sine, the co-sine and the radius. 4. That the co-secant is a third proportional to the sine, and the radius. 5. And that the rectangle of the tangent and co- tangent is equal to the square of the radius. OF PLANE TRIGONOMKTRY. 245 PROPOSITION III, The co^sine CF of an arch AE, being given ; to find the sine and cosine of half that arch. From the two extremities of the diameter AB draw the subtenses AE and BE ; and let CQ bisect the arch AE in Q and its chord (perpendicularly) in D ; then since the angle BEA is a right one {hy Euc, 3i, 3.) the triangles ABE and ^^ ^^^K ADC are similar; and therefore AC being =z i AB, AD must be — | AE, and CD =: I- BE: bu t AE is =\/AB X AF ; and j^ BE = \/AB X BF; therefore oUAE, AD nfy/ AB x AT -v /^AC x AF -the sine > CD -f v/AB X BFn-s/TAC x BF=:the co-sine S Hence it is evident, that the sine of the half of any arch, is a mean proportional between half the radius, and the versed sine of the whole arch ; and its co-sine, a mean proportional between half the radius and the versed-sine of the supplement of the same arch. PROPOSITION IV, The sine AD, and co-sine CD, of an arch AQ bei?ig given ; to find EF the sine of the double of that arch (see the preceding figure, J Since AE = sAD and BE - sCD, and the triangles ABE and AEF are alike {by Euc. 8. 6.) we have, as AB (2AC) : AE TsAD) : : B*E I2CD) : EF ; whence it appears, that the sine of double any arch is a fourth proportional to the radius, the sine, and double the co-sine of the same arch. PROPOSITION V. The sine CD and tangent BE, (fa very small arch are, nearly^ in the ratio of equality. For, the triangles ADC and ABE being similar, K 3 2^6 OF PLANE TKIGONOMKTRV. thence will AD : AB : : DC : BE ! but as the point C approaches to B, the ditJerence of AB and AD will become indefinitely small in respect of AB, and there- fore the difference of BE and DC will like- wise become indefinitely small with respect to BE or DC. CoroL Because any arch BC is greater than its sine and less than its tangent ; and since the sine and tangent of a very small arch are proved to be nearly equal, it is manifest that a very small arch and its shne are also nearly in the ratio of equality. PROPOSITION VI. To find the sine of an arch of one minute. The sine of 30 degrees is known, being half the chord of 60 degrees, or the radius ; therefore by Prop. 2 and 3, the sine of 15 degrees will be known : and, the sine of 15 degrees being known, the sine of 7° 30' will be found {by the same Propositions), and from thence the sine of 3° 45' ; and so likewise the sine of htiUthis ; and soon, till 12 bisections being made, we come, at last, to the sine of an arch of 52'', 44' ^^ 03'''', 45"''" ; which sine(6y CoroL to the preceding Prop.) will (as the co-sine is nearly equal to the radius) be nearly equal to the arch itself. Therefore we have, as 52'^ 44"', 03"", 45""', is to l', so is the length of the former of these arches (found as above) to the length of an arch of one minute or that of its sine, very nearly. If it be taken for granted, that 3,1415926535, &r, is the length of half the periphery of the circle whose radius is unity, we shall have, as 10600, the number of minutes in 180^, or the wlwle semi-circle, is to one minute, so is 3,1415926535, &c. the whole semi-circle tot),000290888208, the length of an arch of one minute, or that of its sine, very iicarlv. OF PLANE TRIGONOMETRY. 247 PROPOSITION VII. Jf there he three equidifferent arches AB, AC and AD, it will be, as the radius is to the co-sine oj their common difference BC or CD, so is the sine C¥, of the mean, to half the sum of the sines, BE + DG, of the two extremes: and as the radius is to the sine of the common difference, so is the co'sine FO of the mean, to half the difference of the sines of the two extremes. For, let BD be drawn, cutting the radius OC in m, also draw mn parallel to CF, meeting AO inn, and BH and mv parallel to AO, meeting DG in H and v : then because the arches BC and CD are equal to each other, OC is not only perpendicular to BD, but also bisects it (Euc. 3. 3.) ; whence it is evident that B/w, or Dm, will be the sine of BC or CD; and Om its co- sine ; and that m?^, being an arithmetical mean between the sines, BE and DG, of the two extremes, is equal to half their sum, and Dv equal to half their difference. Moreover, by reason of the similarity of the triangles OCF, Omn, and Dmv, it will be as, OC : Om : : CF mn and as, OC : Dm :: FO : Dv Q. E, D, COROL. I. Since, from the foregoing proportions, mn is = Om X CF .^ , „^ Dm x FO . . — ^^ — , and DiJ { =z i;H) = ^^ — , it is evident Jl 4 248 OF PLANE TRIGONOiVlETRY. that uQ^zzmn-^-uv) will be — —^ o^^ni?/ TTx Om X CF — Dm X FO ^ andBE(=m« — cH) ;= ^^ ^; iVom whence it appears, that the sine (DG) of the sum (AD) of any two arches (AC and CD) is equal to the sum of the rectangles of the sine of the one into the co^sine of the other, alternately, divided by the radius ; and that the sine (BE) of their difference (AB) is equal to the difference of tlie same rectangles, divided also by the radius. coaoi« 2. Moreover, seemg, DG + BE (2w n) is — ^.-^ — and DG - BE (n DH = 2Dv]=z ^^'^^^^ , from the former of these, we have DG — ^^p ^^» ^"^ from the latter, DG z= — ^^ + BE; which, ex- pressed in words, gives the two following Theorems. ' Theor. 1. If the sine 0/ the mean of three equidlfferent arches (supposing the radius unity J be rmdtiplied hy twice the co-sine of the common difference, and the sine of either extreme he subtracted from the product, the rem mainder will be the sine of the other extreme, Theor. 2. Or, if the co-sine of the mean he multiplied by twice the sine of the common difference, and the pro- duct be added to, or subtracted from the sine of one of the extremes, the sum or remainder icill he the sine of the other extreme. These two theorems are of excellent use in the con- struction of the trigonometrical canon : for, supposing the sine snd co-sine of an arch of 1 minute to be found, by Prop. 6 and 1, and to be denoted by p and q, respec- tively/then the sine of 2 minutes being given from Prop. 4, the sine of 3 minutes will from hence be known, being = 27 x sine 2' — sine V [by Theor, l) or r= 2p xco-sineof 2' f sineof l'}hy Theor, 2.) After the same OF PLANE TRIGONOMETRy, 249 manner of the sine 4' will be found, being zi 2q x sine of 3' — sine of 2^ or — 2p x co-sine of 3' -f sine of 2'. And thus the sines of 5, 6, 7, &c. minutes may be suc- cessively derived by either of the 'I'heorems ; but' the former is the most commodious. If the mean arch be 45°, then its co-sine being := V^i, it follows (from Theor. 2.) that the sine of the ex- cess of any arch above 43"*, multiplie. 11 1 <^ dx the greater segment AD will be = -^ + aa -I- dx 4aa But AD + 6" = 2 + DC* = AC*; X' + 2dx -f dd 2a that is, whence TO GEOMETRICAL PROBLEMS. 261 466 — dd , . , I J . ^ / aa ■\- 4bb which, solved, eives x =z a \/ , , ° ^ aa — dd PROBLEM XI, The base AB, the sum of the sides AC + BC, a7id the length of the line CD drawn from the vertex to the middle of the base, being given, to determine the triangle. by AC + BC = c, C Make AD (z= BD) = a, DC and AC zi a?; so shall BC = c — 07. But AC* + BC* is = 2AD' + 2DC« (by El. 12. 2.); that is, X- -f c — a-i* zz 2a* + 26* ; which, l)y reduction, becomes :r* ^~ -- ex zz a" -\- b' '— ^ J^ ^ Jc' ; whence x is found zi |c ± v^aa 4- 66 ^- ice. PROBLEM XII. The tico sides AC, BC, and the line CD bisecting the vertical angle of a plane triangle ABC, being given ; to find the base AB. Call AC, a; BC, 6; CD, c; and AB, x; then a + 6 : ^ : : a : AD zz ax a + h' b: DB and a + b : X But (btj hx a -\' b El, 20. 3.) AC X CB — AD X DB zzDC^ that is, ahx'^ ab (I -V 6] c*; from whence x will be found 7- . / ab — cc S 3 2(52 THE APPLICATION OF ALGEBRA PROBLEM XIII. The perimeter AB +'BC + CA, and the perpendicular, BCf falling from the right angle Byto the hypothenuse AD, being given ; to determine the triangle. Let BD zz a, AB = Xy BC =: ^, AC = z, and AB -\- BC 4- CA IT h : then, by reason of the similar tri- angles ACB and ABD, it will be as 2 : ?/ : : ^ : a ; and therefore xy iz az : more- over, 0?' + y- =z z- {by Euc, 47' 1.) and x -h y + z z= b (by the question). Trans- pose 3 in the last equation, and square both sides, and you will have x^ -f 2xy + 7/2 = b' — Qbz + z"-, from which take x' + y^ — z~, and there will remain Qxy — b- — 26s; but, by the first equation, Qxy is tz 2az; therefore 2az zzh^ — 2bz and z — J-; whence z is known. But to find 2a + 26 X and V from hence, put , n c, and let this value of z be substituted in the two foregoing equations, X + y zz b — z, and xy — az, and they will become X ■\- y — b — c, and xy zz ac : from the square of the former of which subtract the quadruple of the latter, so shall x^ — Qxy 4- j/* = b — t'l* — 4ac; and conse- quently X — y — y/b — cf — 4ac. This equation be- ing added to, and sub tracted from x + y — b — c, gives Q x — b — c -f v "6 — cl" — 4ac, and Qy zz: b — c — \/b — Tl^ — 4ac, PROBLEM XIV. Having the perimeter of a right-angled triafigle ABC, and the radius DF, of its inscribed circle ; to determine all the sides of the triangle. TO GEOMETRICAL PROBLEMS, 263 From the centre D, to the angular points, A, B, C, and the points of contact E, F, G, let lines DA, DB, DC, DE, DF, DG be drawn ; making DE, DF, or DG = a, AB =r :r, BC = t/, AC = z, and X -{- 7j {- z — b. It is evident that -—- + ~ + —1, ^ or its equal — (expressing the sum of the areas ADB, BDC and ADC) will be zz ^ := the aiea of the whole triangle ABC; and consequently Qxy =: Qab: moreover {by Euc. 47. 1.) x* -f y'^ = z^; to which if Qxy — 2ab be added, we shall have x^ + 2xy -f y^, or x~T~yf = 2* + 2ab ; but, by the first step, x -f 2/1' is =: b — z\- — h^ — (ibz 4- z- ; therefore, by making these two values of x + yf equal to each other, we get s" -I- ^ab — b^ — <2bz -f- 2^ : whence Qa — b — Qz, and z — \b — a. But, to find x and y, from hence, we have now given x -\- y (~ b — z) ■=: \b + a, and xy zz ab ; the former of these equations, multiplied by bx Xf gives X- ^- xy — -^ + oj? ; from which the latter xy — ab being subtracted, we havea:- =|ia: -\- ax — ab, <2a ^ b or x^ — A X ~ — ab'. whence, by completing s 4 264 THE APPLICATION OF ALGEBRA the square, &c. x -zz 2a + 5 ± s/^a" — l^ab -f h* so that the three sides of the triangle are, \h — a, 2«+Z>4-\/4a*~12a6+62 , 2(2+ h - \/ 4:a"- - \^ah -\- b* — _ , and — . Otherwise, The right-angled triangles ADE, ADG, having the sides DE, DG equal and AD common, have also AE equal to AG ; and, for the like reason, is CE m CF; and consequently AC (AE + CE) - AG + OF. Whence it appears that the hypothenuse is less than the sum of the two legs, AB -f- BC, by the diameter of the inscribed circle, and therefore less than half the pe- rimeter by the semi-diameter of the same circle. Hence we have AC =: |6 — «, and AB + BC =2 16 + a. Put, therefore, \h — a zz c, \h -^ a — d, and half the dif- ference of AB and BC 1= x; then will AB =z d + .t, and BC = d--x; and consequently 2d " + 2x" ( A B" + BC-) zzc- (AC^), when ce x is fo und =1 \/ic" — d^; therefore AB is =: ic? + v/4c2 — d% and BC z= ic? — n/ic^ — d", PROBLEM XV. ^11 the three sides of a triangle ABC being given ; to find the perpendicular f the segments of the base, the area, and the angles. Put AC ~ a, AB == b, BC =: c, and the segment AD =: x; then BD being — b — x, we have c- — b-'xl" (= CD2) zz a" -^ X-, that is, c- -— b" + Qbx — a' =: a^.. — a- ; whence 2bx = aa f bb — ccj and TO GEOMETlllCAL PHOIibEMS. ^^aa+bb-cc,^ Now CD"- = AC- 2b 265 3« + 6/; — cc A /~^ I A T\ s^ A/^ Ar> ^ 1 V AU + AJJ A AU — AJJ = a i" 2b ^- bb 2b c" — X — aa + bb — cc Qah + aa sb =- . "^•^x Qab — aa — bb-^cc a -{- b\" — c- Qb " <2b 2b » hence CD = \ x l/a -t- ^f - c' and the area (^° "^ ■^^) = X c^~ . a- -il"; i i/ fl + ^1" — c* X c^ — a — Z>|% But, because the difference of the squares of any two lines or numbers, is equal to a rectangle under their sum and difference, the factor a + ^l"" — c" will be zr a -{- b H- c X a 4- b — c; and the remainjng factor c- — 'a — Z>|2 ~ c~+a — b x c~^ a -\- b ; and so the area will be likewise truly expressed by Iv a + b + c X a -{- b — c x c + a — b x c ^ a-\- b J — - — — - — , , . a-Vb'V c — Vs.s — c,s — b,s—-a\ bymakmg5=: — «4-6-Vc a^b — c c -{- a — b c — a+6 — X :; X — — X 2 2 2 2 2 In order to determine the angles, which yet remain to be considered, we may proceed according to Prop. ii. in Trigonometry, by first finding the segments of the base ; but. there is another proportion frequently used in practice ; which is thus derived : let BA be produced to F, so that AF may be =: AC ; and then FC being joined, it is plain that the angle F will be the half of the angle A; and DU (— AC' + AD) will be given 266 I HE APPLICATION OF ALGEBRA {from above) = — -^^ — . — -—^ x 20 - b 2s ~b ZZ -.- X S — c : but DF (■ es X s — c ) is to DC iW s,s — c . s — b '") , so is the radius to the tan- gent of F ; and consequently s x s — c : s — b X s — a : : sq. rad. : sq. tang, of F ; that is, in words, as the rectangle under half the sum of the three sides, and the excess of that half sum above the side opposite the re- quired angle, is to the rectangle under the differences between the other two sides and the said half sum, so is the square of the radius, to the square of the tangent of half the angle sought. PROBLEM XVI. Having given the base AB, the vertical angle ACB, and the right line CD, which bisects the vertical angle, and is terminated by the base; to find the sides and angles of the triangle. Conceive a circle to be described about the triangle, and let EG be a diameter of that circle, cutting the base AB perpendicularly in F ; also from the cen- ter O, suppose OA and OB to be drawn, and let CD be produced to E (for it will meet the pe- riphery in that point, be- cause the angles ACD and BCD, being equal, must stand upon equal arches EA and EB). Now, because the angle AOB at the centre, standing upon the arch AEB. is double to the angle ACB at the periphery, standing upon the same arch (Euc. '20. 3.) TO GEOMETRIC A I. PKOBf.EMS. 267 that angle, as well as ACB, is given ; and, therefore, in the isosceles triangle AOB, there are given all the angles and the hase AB ; whence AO and FO will be both given, by plane trigonometry, and consequently EF (AO — FO) and EG (~ 2AO). Call, therefore^ EF zz rt, EG zzz h, CD =: c, and DE = x ; and sup- pose CG to be drawn ; then, the angle ECG being a right one {Euc. 31.3.) the triangles EDF and EGG will be similar ; whence x -. a : xb i x -\- c \ therefore, by multiplying extremes and means, we have x- + ex — ahy and consequently x zz \/ab 4- ^cc — |c; from which DF (\/~ED2 — EF^), half the difference of the segments of the base, will be found, and from thence all the rest, by plane trigonometry. Before I proceed further in the solution of problems, it may not be improper, in order to render such solu- tions more general, to say something here, with regard to the geometrical construction of the three forms of adfected quadratic-equations. X' 4- ax — be. viz, -s^ X- — ax n be. C ax — x^ — be. CONSTRUCTION OF THE FIRST AND SECOND FORMS. With a radius equal to ^a, let a circle OAF be de- scribed ; in which, from any point A in the periphery, apply A B equal to & — c (b being supposed greater than c)and produce the same till BC becomes — e; andfrom C through the centre O, draw CDE cutting the periphery in D audi E ; then will the value of X be expounded by BC, in the first case, nnd by CE, in the sccunc EC = AC x CB {Euc. 35^ 3.) that is, ax — x-zzi bcyas icas to be shown. The method of construction, when b and c are equal is no-ways different ; except that it will be unnecessary to describe the whole circle; for, AC being, here, per- pendicular to thediameter ED, if a right-angled triangle OCA be formed, whose hypothenuse is ^w, and one of its legs (AC) = b, it is evident that the sum (EC) and the dillcrcnce (DC) of the hypothenuse and th6 other kg will be ihc two values of x required. TO GEOMETRICAL PROBLEMS. 169 Note. If b and c be given so unequal, that b — c, in the two first forms, or ^ •+- c, in the last, exceeds (a) the whole diameter ; then, instead of those quantities, you may make use of any others, as ^5 and 2c, or|6 and 3c, whos6 rectangle or product is the same ; or you may find a mean proportional between them, and then proceed according to the latter method. PROBLEM XVII. The base AB, the vertical angle ACB, and the right line CD, drawn from the vertical angle, to bisect the base, being given ; to find the sides and perpendicular. Suppose a circle to be described about the triangle : and let CQ be perpendicular to AB, and ED equal, and parallel toCQ ; moreover, from the centre F, let FA, FB, and FC be drawn ; also let CE be drawn (parallel to AB.) Put the sine of the given angle ACB, to the radius 1, — m, its co-sine zz w, the semi-base BD z=. c, the bisecting line CD zz b, and the perpendicular CQ (DE) - x; then, since (by Euc. 20. 3.) the angle BFD is equal to ACB, it will (by plane tri- gonometry) be, as m (sine of BFD) : a (DB) : : n (sine of DBF) na m zz DF; and, as m (sine of BFD) : a (DB) :: 1 (sine of BDF) : — n the radius BF, or FC m whence EF (ED — DF) na mx — na — or ' m m But {% Ewe. 12.2.) DF^ 4- FC^ -V 2DF x FE = DC' 270 THE APPLICATION OF A[X;EBRA that IS, in species, — v- -\ f - — x — =0% or — r r- H =: 6 : but, since the sum of the m m m square of the sine and co-sine, of any angle whatever, is equal to the square of the radius, or, in the present case, m* -f «^ =: 1 , therefore is 1 — n^ — ?»-, and conse- Quently — :; r» (or — 77 x 1 — n") = — « x m =z a-; QflCtX whence our equation becomes a" -\ := 6*; which, ^ m J , . m X b^ — a^ m DC* — DB^ ordered, eives a^ — — =: — x r-T> m DC+DBx DC — DB , m = — X -T-n ; where — expresses the tangent of the angle ACB : therefore, in any plane triangle, it will be, as the base is to the sum of the semi-base and the line bisecting the base, so is their difference to a fourtii proportional; and, as the radius is to the tangent of the vertical angle, so is that fourth pro- portional to the perpendicular height of the triangle : whence the sides are easily found. The same otherwise. Let the tangent of the angle ACB, or BFD, be re- presented by p, and the rest as above ; then it will be (hy trigonometry) as p : 1 (the radius) : : a (BD) : — =: DF; therefore FE (DE — ■ DF) zz a^ — -?-, and FC^ P (= FB^ - DB"- + DF») =z a* + %; and consequently ~ +a"^4-~4- - X .T— -(DF2fFC- + 2DFxFE)=: hh [— DC2) that is, «'- f "— — b"; whence .r — p x P b- — a^ , , ^ — , tn(^ same as before. TO GEOMETRICAL PROBLEMS. 271 PROBLEM XVIII. The area, the perimeter, and one of the angles of any plane triangle ABC, being given; to determine the tri. angle. Suppose a circle to be inscribed in the triangle, touch- ing the sides thereof in the points D, E, and F ; also from the centre O, suppose OA, OD, OC, OF, OB, and OE to be drawn : ^ and upon BC let fall the a^ perpendicular AG; put- / l\ ting AB + BC 4- AC n / I \ b, the given area = a~, / I \ the sine of the angle ACB .^"'l^VV (the radius being 1) = m, D/^ I ^J^ the co-tangent of half that 4 ^*^J^^*^\ p angle (or the tangent of /I J^^^^**n\' DOC) rz n, and AC =: x. / 3<^-*^''^X^ \ Therefore, since the area >^L : ^ y^\ of the triangle is equal to ^^ -^==*-^f=^^ ^ 4AB X OE+ iBC X OF ^ ^ ^ 4- 4AC X OD, that is, equal to a rectangle under half the perimeter and the radius of the inscribed circle, we have — x OE = aa; and therefore OE = —r-* But 2 b AD being — AE,and BF = BE; it is manifest that the sum of the sides, C A + CB, exceeds the base AB, by the sum of the two equal segments CD and CF; and so is greater than half the perimeter by one of those equal segments CD ; that is, CA + CB = 4& 4- CD : but (by trigonometry) as 1 (radius) : n (the tangent of DOC) :; ^ (OD) : DC - ^"; whence CA + CB (= \b f CD) - |6-i- -^- ; which, taken from {h) Qna'^ the whole perimeter, leaves ^b r- zz the base AB. Make now 4i I ; — — c; then will BC — c-^x; also o (by trigonometry) it will be, as l (radius) : in (the sine 272 THE APPLICATION OF ALGEBRA of ACG) : : x (AC) : mx zz AG ; half whereof, miil- — = a-, the area tiplied by c — :r (BC j, gives i!!££— ^^^' of the triangle: from whence .t comes out zz |c Qaa PROBLEM XIX. The kypothenuse, AC, of a right-angled triangle ABC, and the side of the inscribed square BEDF, being given ; to determine the other sides of the triangle. Let DE, or DF zz «, AG = h, AB zz x, audBCzzy; Q then it will be, as x : y : : x — a (AF ) : a (FD); whence we have ax zz yx — i/o, and consequently xy =z ax -{• ay. Moreover, xx -h yy zz hb : to which equation let the double of the former be add- ed, \ind there arises .t^ + 2xy ■\- if zz h" 4- qax 4- 2flr/; t hat i s, a- -f yf zz b- -\- 2a y. X -\- y, or'.'i + y\- — 9.a y, x -^ y :zib"\ where, by con- sidering 07 4- 7/ as one quantity, and completing the square, we have x-\- yr — 2a j< x -\- y -f a^ z= h- 4- a-; whence x ^ y — a zn s/^h" + a-, and x -V y ^=- \/a" 4 ^- 4 a , which put zz c : then by substituting, c — x in- stead of its equal [y] in the equation xy zz ax + ay, there will arise ex — x- zz ac ; whence x will be found = ic 4- \/, — \/icc — ac. -^cc — ac, and y z It appears from hence that c, or its equal \/aa 4 bh 4- a, cannot be less than 4a, and therefore b- not less than 8a-; because the quantity ^cc — ac, under the radical sign, would be negative, and its square root im- possible; it being known tliat all squares, whether from positive or negative roots, are positive; so that there cannot arise any such things as negative squares. TO GEOMETRICAL PROBLEMS. 273 unless the conditions of the problem under considera- tion are inconsistent and impossible. And this may be demonstrated, from geometrical principles, by means of the following LEMMA. The sum of the squares of any two quantities is great- er than a double rectangle under those quantitie^t by the square of the difference of the same quantities* For let the greater ofthe two quantities be represented by AB, and tjfie lesser by BC (both taken in the same right line) . Upon AB and BC let the squares AK and CE be constitut- ,^ ed; take AP = H I I^^ BC and complete the rectangles PH and CF. There- fore, because AB = AH, and AP n: BC, it is plain that PH and PD are equal to two rectangles under the proposed quan- tities AB and BC : A E X) B but these two rectangles are less than the two squares AK and CE, which make up the whole figure by the square FK, that h, by the square of PB the difference ofthe two quantities given: as icas to be proved. Now, to apply this to the matter proposed, let there be gi ven the qu adratic equation x" 4- 6' ~ Qax, orx — a + v/ aa — bb : then, I say, this equation (and conse- quently any problem wherein it arises) will bs impossi- ble, when aa — bb is negative, or b greater than a. For, since b is supposed greater than a, 2bx will likewise be greater than 2ax ; but ^ax is given =z xx + bb^ there- fore 2bx will be greater thanao? + bb, that is, the double rectangle of two quantities will be greater than the sum of their squares^ which is proved to be impossible. 274 THE APPLICATION OF ALGEBRA PROBLEM XX, The base AB {^aj and the perpendicular BCfbJ of a right-angled triangle ABC, being given; it is proposed to find a point D in the perpendicular ^ so that, if two right lines be drawn from thence, one to the angular point A, and the other (DE) perpendicular thereto, the triangles DEC, ABD, cut off by those lines, shall be to one another in a given ratio. Let AB be produced to F so that the angle BFD may be equal to the angle BCA ; putting AC iz c, CD =z .t. ^ B A and the given ratio of the triangle DEC to ABD,as m to n. Then, by reason of the similar triangles ABC, DBF, it will be, a (AB) : b (BC) :: b — x (BD) : BF - P^bx a c' — bx ; whence AF — a -\- b"- — bx a -f b' — bx (because a'^ + b^ zz c"). Also, as ADE is a right angle, the angles FAD, EDC will be equal: there- fore, the angles C and F being equal (by con.) the tri- angles AFD, DCE, must be similar ; and consequently AF^( gERVcDV^) ,: AFxBD 6-^. X c.-^-^/.r) a' 2 2a h — T X QX^ the area of the triangle AFD :( ^=^===) the area of the triangle DEC: wherefore, the area of the tri- TO GEOMETRICAL PROBLEMS. 275 angle ABD being , or we shall have. m • ji ^' "" ^ ^ "^' : ^.""^ ^ i{by the question) : and * * 2 X c'^ — 6a; 2 J— mSa? mc' consequently, wx^ =: w X c^— ox, ora; + -^ ~~^' which, reduced, gives a: r: y --- + —^ The geometrical construction of this problem, from 7TluX T)ZC the equation a?* + = , may be as follows. In CB let there be taken, CH :CB::m: n, and let H K be drawn parallel to BA ; then CH being :z: — ,andCK z= ^, our equation will be changed to a:^ 4- a? x CH n * =:AC X CK,ortoCDxCD + CH = AC x CK. Upon CH asadiameter letthe circle CTHQ be describ- ed, in which inscribe CG =z AK; andinCG produced, take CS = CA ; and from S, through the centre O, draw the right line S TOQ, cutting the circumference in T and Q, and make CD rz ST; then will D be the point required. For CG being — AK, and CS = CA; therefore will A CxCKnC S x GSzz ST x SQ { Euc, 37. 3.) =: ST X ST~TTq z= CD x CD + CH, the very same as above. PROBLEM XXI. Having the perimeter of a right-angled triangle ABC, and three perpendiculars DE, DF, and DG, falling from a point within the triangle upon the three sides thereof; to determine the sides, SupposeDA,DB,andDC tobe drawn; and let DE = «, DF - b, DG - r, AB - x, BC = y, AC =z. 276 THE APPLICATION OF ALGEBRA and the perimeter, AB +BC + AC, =:p: then, the (3 area of ADB being ex- pounded by — ; that of BDC by ^^ ; that of ADC, by- ; and thatofthe wholeABC, by -~ we therefore have ox . hi/ , cz XV -^ A- ^ -r — =— , or ax + Inj \ cz = xy : more- over, we have .t' + y" — z", and .t -f ?/ + 2 =: p, by the conditions of the problem. Let z be transposed in thelastequation, and both sides squared, soshaUa^-^^xy \' y'^ — p'*- — 9.pz -+• 2^, from which, if a'* + ?/^ r= 2' be subtracted, there will remain 2a\v zz p" — 2pz = 2f/.T h 2hy + 2C2 {by the first equation) -.whence \pp : from this last equa- ax ^ hy + c -^ p X z tion subtract a times a 4- ?/ + z rrp, and there wi remain by — ay + p -^ c ^ a Y. z — -J-p- — op ; also, if from the same equation, b times x \r y -^ % = p be subtracted, there will remain ax — bx + p \e — 6x2= ^p- — bp ; which two last equations, by putting d zz. b — a^e z:ip \ c — a^ f — \p' — ap^ g =z p + c — b, and h n \p~ — hp, will stand thus, cfy-'\' ez zz f, and — dx + gz = h; whence y -zzz f ', and a': (Tz — h Let these values of x and ?/ d ' d be substituted in a* + y'^ zz z" and we shall have d~ d' X Jt- — '2ef + Qgh y z zz — /- — h~ : ])ut c- + g'- — c?- - /t, (f -\- gh zz /, and/2 ^ ^i _ ^^ ; so shall /. :^ — oJz zz m 2/2 wz , / whence z- r- — — -j- an(| ;: z: ~ TO GEOMETRICAL PROBLEMS. 277 V 7.2 h l±\/ I' km from which x ( m 'z-'h, f ~~~ €Z ) and y( — ^ — — - ) will also be known. d d If a, 6, and c are all equal to each other, the point D will be the centre, and each of the given perpendiculars a radius of the inscribed circle ; and the value of z in this case, will be barely equ al to |p— «; for the equa- tion, by-—arj-\' p Jf c — a x z — ip'^ — ap, above found here becomes pz zz ip* — ap. But, if only a and b (or DE and D F) be equal, then the equation will become p + c — a y. z ::zip- — ap ; — 7)^ — — (IT) and therefore z — ^~ — ^; in which.ifcbetakenzzO, p -j-c — a :;willbez= ^'P^' ~ p — a scribed square. ~: where a is the side of the in- PROBLEM XXII, The perpendicular CD, the difference of the sides AD — BD, and the vertical angle D, of any plane triangle ABD, being given ; to determine the sides. From B, upon AD (produced if need be) let fall the perpendicular BE : let the sine of the angle BDE zr s, its co-sine = c (the radius being unity) ; also let the perpendicular CD zip, the lesserside BD —x, and the greater DA — J? 4 d: then {by Prop.Q.in trigonometry) as 1 : 5 : : 07 : 5T = B E ; and, as l : c \: x : ex — ED. Now AB\ being = AD^+D3--AD X 2DE [Euc. 13. 2.), will be ex- pounded by X f dY -1- X- — X + d X 2ca:, or 2x- V 2dx \- d^ — 2c,r' — 2cdx ; T3 A C B whence, by reason of the 278 THE APPLICATION OF ALGEBRA similar triangles ABE and ADC, it will be, as Qx'^ + 2dx -f tZ* - 2ca?2 -. cicdx (AB*) : sH^^ (BE*) : : .t» 4- 2xd 4- dd (AD") : p^ (DC)% and consequently by multiplying extremes and means, s^x* + Qs'dx^ + s^d*x'^ — ^p'^x^ + Qp-dx-\'p^d^ — 2p^cx'^ — Qp'cdx ; from whence, bv transposition and divisio n, we have j* + if — i-— _ — o. Which equation answering the condi- tions of the second case of biquadratics, explained at p, 154, we shall therefore have x'^ + dx + - — —^ — V ^^ — h ^ ; and consequently a: rz — |rf-|- 4 ^* ^' Otherwise, Supposing ^, c, and jo to be the same as before, put half the given difference of the sides = a, and half their sum zr x; then the greater side AD will be =: a? + «, and the lesser BD =: x — o; wherefore (% trigonometry) 1:5:: X — a : s x x — a zz BE; and, 1 :c :: x — a : c X x'^- DE: but ABMs = AD^ + DB* — sDE X AD — X 4- a^^i4- a; — o*'^ — 2c x x — a" x ^ + flf = 2.r" + 2a' — 2CcT* + 2ca-; whence by reason of the similar triangles ABE, ADC, it will be 2a" + 2a^ — 2ca;2 + 2ca^ (AB-) . ^» ^ ^TT^* (BE*) :: rT~?' (AD") : p" ( DC) ; a n d consequentl y s^ x jTirjs x V -\-a\'^ rr 2r' + 2a' — 2c.i* -*- 2ca" x p", or s"x* — 25'aV* + ^'a* — 2/)-.r< — 2rjo%t2 -4- 2/>'a* + 2f/>'a' ; whence by transposition and division, x"^ — 2aV — -^V- + 2 — = -^o— 4* : "• Substitute c* c* e" oV TO GEOMETlllCAL PROBLEMS. 279 then the equation will stand thus, x * — 2/a;« = g : whence x is found zz v// ± v//^+g. If, instead of the difference, the sum of the sides had been given, in order to find the difference, the method of operation would have been the very same, only, in- stead of finding the value of .r in terms of a, by means of the equation s''x* — ^is'^d^x^ •\- s\t' — 2/>^x*— 2cp'a:* -f 2/)V f ^cp-a", that of a must have been found, in terms of a:, trom the same equation. PROBLEM XXIII. Having one leg AB of a right-angled triangle ABC ; to find the other leg BC, so that the rectangle under their difference (BC — AB) and the hijpothenuse AC, may he equal to the area of the triangle. Put A Biz fl!, and BC=.r; so shall AC ~ \/ aa^ xx\ and -^— x — a. \/ aa\xx^ by the conditions of the problem. By squaring both sides C^ of this equa^n we have jaV = ^•'i -J(2.ax +• o:^ X aa -\- xx : in which the quantities x and a being concerned exactly alike, the solu- tion will therefore be brought out from the general method for ex- tracting the roots of these kinds of equations (delivered at p. 156): according to which, having di- vided the whole by a:'x'\ we get — = — X — 4- — ; which, by makino; z — — -\ — ; will be la ^ a X reduced down to \ =: z — whence z is given zz \ ^\/ i. T 4 2 / z, or 2' But since — + '2z — t- a 280 THE APPLICATION OF ALGEBRA we have x^ — azx — — d^\ and therefore x = -^ + ^22 _4; which by sub- stituting the value of 7, becomes x — — x 1 + V^l 4-n/:v/5 — -T. PROBLEM XXIV. To draw a right-line T>F from one angle D of a given rhombus ABCD, so that the part thereof FG intercepted by one of the sides including the opposite angle and the other side produced, may he of a given length. Let DE be perpendicular to AB ; and let AB ( = AD) n a, AE z= h, FG = c, and AF =: x ; then DF^ ( = AF* G +AD'^-2AE/AF) = XX f a a — ^hx ; and by similar tri- angles, XX h aa — E i\ F 2KDF-^):.Tx(AF^) : : cc (YG^) : x — a^* (BF') ; and consequently XX ■\- aa — 2hx y. xx — 2aa' -\- aa — ccxx, Makemazn by and na — c \ so shall our equation become XX -{-aa — ^max x xx — ^ax -\r aa — n'^a'^x^ ; which, di- t! a X (1 vided by a'a:%gives -+- — 2mx-+- — 2 zi ax ax jp fi n^ : this, by making ~ = — I - j becomes z — 2?w x a X z — 2 =: w*^ : therefore z'^ — 2//2 +2 x z — n""- — 4?n, and 2 - 1 f m +v/«* + 1 — w)* - a 4 7; ^- v^c» + a ~ l>\^ , by restoring tlif values of 7n and w. From whence the value of r xvill be also TO GEOMETUICAL PROBLEMS. 281 known ; for ^ -f - being = z, we have, by reduc- a X tion, x' — azx — — aa\ and therefore x ~ J" ^ S 4- V Z2 — 4. TROELEM XXV. The diagonals AC, BD, and all the angles, DAB, ABC, BCD, 2i\\6.QT>k, of a trapezium ABCD, being given, to determine the sides. Let PQRS be another trapezium similar to ABCD, whose side PQ is unity; andletQPandRS be produced till they meet in T : also let PR and QS be drawn, and make Rv and Sid perpendicular to TQ, Let the (natural) sine of the given angle STP, to the radius i, be put = m; that of TSP'or PSR, = w; that of TRQ zi p ; the co-sine of SPQ =: r ; that of RQv zz s ; AC — a ; BD=:/^; and PT-o:. Then thy plane trigono- mx metry) ni m -. i x i YS — — ; and l mx n (PS) rmx r:Yw — — -: whence, [by Euc.i3.2,) QS^ ( + PS2-2PQX P..)z=l 4-'^^"^'^ '""'"^ QP* nn Again {by trigonometry) p : m QR = m 4- mx ; and 1 : s m 4- mx + 07 (TQ) (QR) : Q.V - ms 4- msx And therefore PR= ( = PQ- -f QR2 282 THE APPLICATION OF ALGEBRA PP P because of the similar figures ABCD, PQRS,it will be, AC=^ : BD*^ : : FR^ : QS'^ that is, a' : b' : : m -{- mxY Qms 4- Q7nsx PP p fin n and consequently a" ^- — 6- + - — ^ "^ nn n pp , Qh"m^x b'^m'^x'^ '^h'^ms Qh'msx , -f- .-{- — : whence writ- PP PP bhre anrm pp n we have Jx^-\- 2ga; - h gives :r =\/j+ f-7= f'«"» whence SQ will also be known : and then, a^^ain, by reason of the similar figures, it will be as QS : QP (unity) : ; ^Yy : AB ; which, therefore is known, like- wise: from whence the rest of the sides BC, CD, and DA will all be found by plane trigonometry. The last problem is indeterminate in that particular case, where the trapezium may be inscribed in a circle, or where the sum of the two opposite angles is equal to two right ones; for, then, there can but one diago- nal be given, in the question, because the value of the other depends entirely upon that. PROBLEM XXVI. Supposing BOD to be a quudrant of a given circle; to find the semi'diamctcr CK, or CL, of the circle CEGL.inscrihed therein; and likewise the scmi-diameter ofthe little circle n¥mV^ touching both the other circles DLB, LmB, and also the right line OB. Let BQ, P;?,and CE, be ptrptndicular to BO ; join C, n and O, n ; and draw OC uietting BQ in Q. TO GEOMETRICAL PROBLEMS. 283 and nr parallel to BO, meeting CE in r : put OB ( =r 6Q) = 1, OQ (= \/2 by Euc. 47. l.) - b, and ?n( — nm)=:x; Then, by reason of the similar tri- angles OBQ, OEC, it will be, OQ : BQ : : OC : CE; whence, by compositio?i, OQ 4- BQ : BQ : : OL (OC 4- CEj : CE ; that is, 6 4- 1 : 1 : : 1 : CE = 1 . b-^i . h ~ b 1 = v/2 b i-l^ 6 + 1 xT=ri^-/r— 1 — 1 ; which let be denoted by a, then we shall have Cn 4- Or — 2a, and Cfi — Cr — 2a? ; and therefore nr (v^C/z + Cr X Cn — Cr) =: 2V^ax, Moreover, O;^ 4- ?n bei nff =: 1 , and O/i — P« = 1 — Qx, thence will OP =l_\/l — 2x; which also being = PE 4- OE (2\/ax 4- a), we therefore have n/F^^^x = 2\/Gi 4- a ; whereof both sides being squared, there arises 1 — 2r =: 4^0? 4- 4:a\/ax -+■ a'^, or 1 — d^ — 2ar — 4ax =z 4a\/a x ; which, because l — aa is 2a, will be a — 1 4 2« X 0? = Qav^ax: this, squared, gives a' — 1 + 2a X 2a£ 4 1 fi^il' X X- zz 4 air ; whence 1 -1- 2al^ X x^ — 1 f 2a X 2(^r — 4a^^ = — • aa; which, by writing 284 THE APPLICATION OF ALGEBRA ^ — 1 instead of its equal a, becomes 26 — i|* x a?' 7<5> — 9 X Qx =: Qb — 3 ; therefore x' — ^2 X 2x — . ; from whence x is found r= 26—11"" 26"^^* 26 — 1)* 2~6"=D^ ,^_,^v/8/.3+ 296-- 112^.4-78^ ^^.^^^^ ^^, ^^.,^ 26— 1^ ing \/ 2 for 6, becomes 7y 2-~9±v^l36— 96 \/2 2v/2~— iV Tv/ 2— 9 ± 6\/2 T 8 ,, . . - ^ 13v/2 - 17 *^ ^» that IS, equal to 2v/2 — ir or to - J!!--^ ' c:, ; which last is the root required, the 2v/2 — ir " other being manifestly too large: but this value will be xetluced to — ^^^^^—, Therefore OP (= v/l — 2.c) 49 ^ . ^ /3l — lo-v/2 5\/2 — 1 IS given = Y — — ^ = 2?n; and consequently BH — IBQ ; from whence we have the following construction. In the tangent BQ, take BH = ^BG; drawIIO; catting the circumference BDL in F, and make the angle OFP = ^OHB, and draw Pn parallel to BQ, meeting OH in n, the centre of the lesser circle re- quired. SCHOLIUM. In the preceding solution it was required, not only to extract the S(]uare root of the radical quantities 136 — TO GEOMETRICAL PROBLEMS. 285 gov's and 51 — 10-/2, but likewise to take away the radical quantity from the denominator of the fractiou \/2— 1 Wr — — 7' ^"^ confine it, wholly, to the numera- tor: all of which being somewhat difficult, (and,forthat reason, omitted in the introduction, as too discouraging to a young beginner) L shall therefore take the oppor- tunity to explain here the manner of proceeding in suck like cases, when they happen to occur. ^First, then, with regard to the extraction of roots out. of radical quantities, let there be proposed A ±n/B, A being the rational, and y^B the irrational part thereof; and let the root required be represented hy\/lc'±\/y^ ; the square of which will be a: + y ± 2 \/^y, or ar + let the two last equations be added together,and the sum be divided by 2 : so shall x - ^-^v^^' + ^- ^^1^^ ^ 2 and by multiplying the same equations together, we getcT-'— y — A' — Bp, and consequently y — x' — A* — B)^, whence y is likewise known. Universally, let the index of the root to be extracted be denoted by w, and let the root it?elf be represented by x±\/ y (as above). Then this expression raised to the «th power, will be x''±nx''''^\/~y + w x ^^^^ a"'^ y n — 1 n — 2 r?-5 /— o r 1 ± n X — -— X — -— X y V y &c. from whence^ still following the same method, we shall here get »j J 7i — 1 n-2 A 1 X -T n A - - - X y &c. •=. A, and TO GEOMETRICAL PROBLEMS. 287 «a"-VF+ n X ~ X ~x''~'^y-/y &c. = ^/B: let, therefore, the root of the sum, and also of the dif- ference of these two last equations, be taken, and you will have, x + \/^ zz A + |/Bl~ , and x c/d \/y iz 1 A — -/ BlV; which two equations being added to- ^j ^_i ,. -n 1 f ^ Ai-/Bl« ± A— v/B\« gether, a? will be lound z: ^^^ 1 1 — Z — !__ -f ; and if the same 2 — __^L 2>cA+v^Bl» equations be multiplied together, you will have ^i 1 x^(/^y=: A'- — B|«; whence 2/ r: a:* + A* — B^.n; The use of which conclusions will appear by the fol. lowing examples. First, let it be proposed to extract the cube root of the radical quantity 26 + 15 \/ 57 or 26 + x/'gtT. Here, A being iz 26, B = 675, and n z= 3, we have ;! 26 + \/673r ^ 1 _3,732±,268 xi 2X26 + v/675)'^ 2 J zz 2*; and 2/ ( = 4 - 676—6751^ ) = 3 ; and conse- quently X ■\-s/y ~ 2 4-V/3 — the value required : for 2 f ^/ 3 X 2-h>s/3 X 2 4- V3 = 26 + v/675. Again, let it be required to extract the biquadratic root of 161 4- v/ 25920! In this case, A being 161, B = 25920, and n = 4. we have x ( = ^ll^^l^ . 1 _ 4, 236 4- ,236 , . 2 X 321,99 &c )* 288 THE APPLICATION OF ALGEBRA y (— 4 + 23921 — 25920V) = 5 ; therefore the root sought is, here, :z 2 + v^^T Lastly, if it were required to find the first sursolid root of 76 f v/ssos; then, by proceeding in the same •11 1. r . , 2,732X ,732, , J manner, :r will be found ( = -^-^ — ) = I, and y (= 1 — 5776 — 5808V) ~ 3: and so of others. But it is to be observed, that the second part of the value of a', to which both the signs + and — are pre- fixed, is to be taken affirmative or negative, according as that or i^z^ shall be found requisite to make the value of 0? come out a whole, or rational number ; and that, if neither of the signs give such a value of a:, then this method is of no use, and we may safely conclude that the quantity proposed does not admit of such a root as we would find. It may also be proper to remark here, that, if the upper sign in the value of x be taken, the upper sign in that oiy must be taken accordingly ; and that the application of logarithms will be of use to fa- cilitate the extraction of the root A + \/~^\ ' ^s be- ing sufficiently exact to determine whether .r be a whole number, and, if so, what it is. Thus much in relation to the extraction of the roots of radical quantities; it remains now to explain the manner of taking away radical quantities out of the denominator of a fraction, and transplanting them into the numerator. In order to which, supposing r to denote a whole uumber, it is evident, in the first place, that since, by an actual multiplication, the product appears tobe^ f :^ ; ^, where all the terms, except the first and last, destroy one another Hence, by inual division, we htive TO GEOMETRICAL PROBLEMS. «89 in the very same manner, it will appear that j: + y "" a?*- ±yr where the sign + or— , in the denominator, takes place, according as the number r is even or odd. Let now x - A"*, and y =: B"; then our equations will become A B * J^rm _ grn a" -i-B" A"" ± B'" From which theorems, or general formuiw, the mat- ter proposed to be done may be effected with great fa- cility : for, supposing -— or to be a J » ff ^ A"~B" A'^+B'* fraction having radical quantities A''*, B" in the deno- minator, it is plain, that its equal value, given by the said equations, will have its denominator entirely free from radical quantities, if r be so assumed that both rm and rn may be integers. To exemplify which, let the fraction yr=z , or V^2 — 1 — be propounded; then, A being z: 2, B z: l, m zz I and n-.l, we shall, by taking r z: 2, have(/ro7» Theorem l) — i— = ^.±1^ v^+ 1. ■^. - 1 2-1 Again, V 290 THE APPLICATION OF ALGEBRA Again, let the given fraction be - _= -y=r- or _j - j^ — — . In which case, A being =: ca:, B = c^ -f a?', m zi i, and n z= |, we shall, by taking T = 4, have -—-^ iz cxl^ + c* + ^^ If the numerator is not an unit, you may proceed in the same manner, and multiply afterwards by the numerator given. T'hus, in the case mentioned at the beginning of this scholium, we had given .^ ^J^lJ: s which may be reduced to -/sT— i x -7 x x 8^ — 1 s"^-! — --— , or to y/T^:r~i X -r-^— X -x-^, : but-; is found (by Theorem 1) to be zi ■ 8i — 1 ^ 8—1 21/5" 4- 1 , . , —- — — ' — : whence our expression becomes V2--1 X X T : which, by multiplication, C3c, is reduced, at length, to -^- — -. PROBLEM XXVII. EavinfT one leg AC, of a right-angled triangle ABC, to find the other leg EC, so that the hypothenuse AB shall he a mean proportional between the perpendicular CD falling thereon^ and the perimeter of the triangle. TO GEOMETRICAL PROBLEMS. 291 Put AC = a, and BC z=z x ; then will AB = \/xx + a«, andCD (= therefore, 5?/ — bb l6aa 4a 4a V 3 ^94 THE APPLICATION OF ALGEBRA From the same equations the problem may be re* solved, when the distances from the three angular points to the circumference of the inscribed circle are given : for, denoting the said distances by/, ^, and /i, you will have AO zz x + f, BO=: x + g, and CO zzx + h ; which values being wrote in the room of a, 6, and c, there will arise an equation of six dimensions : by means whereof x may be found. PROBLEM XXX. To draw a line NM to touch a circle D, given in ma,gt nitude and position, so that the part thereof AC, inter- cepted by two other lines BK, BL, given in position^ shall he of a given length. Suppose CP and DE to be perpendicular to AB, and DF and DG to AC and PC, respectively ; and let DA DC, and DP be drawn ; putting DE = a, DF = h, AC = c, BE zi di PC = X, PA = ^, and the tan- N/ E gent of the given angle BCP, to the radius 1, = /. Then, by trigonometry , 1 : ^ : : a? : «j: = BP ; there- fore DG ( = PE; ~ d — tx\ which, multiplied by i7C, or |., gives ^'^ — ^^\ for the area of the tri- angJe GDP: in like manner the area of the triangle PDA will be found = ?^ he ; and that of ADC = - ; which three, added together, are equal to the wholearea A on 4 1 4- ^^ — tx"^ ay he xy J ACP; that IS, i _^ 4. — — -_:^; and conse- 2 ^22 2 ' TO GEOMETRICAL PROBLEiMS. ^95 iquently he -{- dx — tx" = ocy — ay» Let both sides of this equation be squared, aud you will have be -{•"dx — tx'f — X — a\'^ x y" = x — a]^ X c^-^x" ; that is, b'-c- + 2hcdx — Qbctx" + d-x- — ^dtx^ + fx'^ — — a?* + ^ax^ — a-a:3 + cV — 2ac^x -4- a"c- \ whence TV? x x' — 2a~V~2di x a^H a" — c'^^- d'^ — '2bct X X- -\- Qac^ 4- 2bcd x x — ft-c* + &Vz=0 : from which the value of a: may be found : and then, the value of y (—\/c'' — x-) being known, the position of the points A and C, through which the line must pass, will also become known. If the given angle B be a right one, the point B will coincide with P ; and therefore t in this case being zz 0, the equation will become x* — ^ax^ + a^ — c^+d* X a:2 + 2ac2 .^ ^bcd y. x — a-c- -\- b^c^ — 0. When the circle touches the right line AB, a will then be equal to b : and, i n that case^ the equation will be l~fT X X-' — ioTs^iXor'+a' — c* -f d'^-^Yact X X -h ^ac^ + <2.acd — 0, because the two last terms — aV' 4- b^c'^ destroying each other, the whole may, here, be divided by x. Lastly, if Z> be =: 0, or the line AC, instead of touching a circle, be required to pass through a given point the equation will then become 1 -f t' y x* — 2a + ^dt X a:^ -f d' — c' + rf* X ^' f Sac'a:— aV=:p, ,. PROBLEM XXXI, Supposin 1 • v/iooo X 3e , , , transformed to v/iooo + -^—^ + v 1300 f 130 v/3400 f v/l800 f \/3400 X 5n 340 v/ 1 800 X 3e 180 — 50 — Ge. — ; this, contracted. gives 1,8 + l,37e = ; whence e - — 1,3, and con* sequently a^ z: 28,7, nearly. Let, now, 28,7 be put z= X ; and then, by proceeding as above, we shall have ,0083 + 1.43e = O; hence e = — ,0058, and X = 28,6942 ; which is true to the last figure. PROBLEM XXXH. The perimeter, AB 4- BC + AC, and the perpend i^ cular CP of a triangle ABC whose sides are inharmonic proportion (AB:hQ :: AB — AC : AC — BC) being given; to determine the triangle^ TO GEOMETRICAL PROBLExMS. 297 hetabc be another triangle, similar to the proposed one; and let a6 rz 1, be — x, ac = y, CP =: a, and AB + BD -f AC — ^>: then, half the sum of the three 1 -I- X A- 1/ sides of the triangle abc being — — — — ? , if from the same, each particular side be subtracted, and all the re- B « /" / mainders be multiplied continually together, and that product, again, by the said half sum, we shall have 1 i- x — y 1 + y X y 2 equal to the second power of the area abc {by prob, ]5) : which, as the base is unity, also expresses | of thesquare of the perpendicular. But the squares of the sides, as well as the sides of similar triangles, are propor- tional, &c. and therefore 1 + j? -f ?/p : b* : : 1 +x — yx 1 — X ^- y Xy ^~x — I X 1 — x -^ y a^ wh ence we h ave 4a* x I i- x + y = \ +x — yx 1 — x^y xy^-x—l X 6Z>; but the sides AB, AC, and BC, be- ing given in harmonic proportion, therefore, 1, ?/, and x, must likewise be in the same proportion ; that is, l : x ; : l — y : y — x; whence y — x —x — xy, and there- 2t fore y — — — -— ; which, substituted above, gives 4q" X 1 + 4.r + x"- _ I H-.r'^ l+2x---^' 2.r ■^■x'—X 1 + a: """ 1 -f .r 1 f 07 \ ^ x X 6 % or 4a- X 1 f AX V x^ x 1 + ^' ~ 1 + J^~ X 1 -f- 2j: — a* X 2x -f x'^ — T X 6*; from which x will be found, and 298 THE APPLICATION OF ALGEBRA Qx ; and from thence the required side3 also 2/ (=1-^.^ of the similar figure ABC, will, by proportion, be like- wise known. PROBLEM XXXIII. Let there he three equi-different arches^ AB, AC, afid AD ; and, supposing the sine and co-sine of the mean AC, of the lesser extreme AB, and of the common dif- ference ^Q for CT>) to he given, it is proposed to find the sine and co-sine of the greater extreme AD. Upon the radius AO let fall the perpendiculars BA, Cc, and T>d\ join B, D, and from the centre O, let the radius OC be drawn, cutting BD in n : also draw «R parallel to Cc, meeting AC in R ; then, because of the similar triangles OCc and OnW, it will be, OC : 0«;:Cc: «R ; and, OC : 0/7 : : Oc : OR : whence we have A/ ''Kr/ „ Cc X On wR zz— , , and OR=: Oc X On but. since BC OC '" OC equal to CD (and therefore ^n equal to D«), r^R w'ill, it is plain, be an arithmetical mean between B/j and D xy) — xy^ — 2.V7/ ; sine of 5A ( = sine of 4 A xy — sine of 3 Ana?^'— 2a'2/^ — xy^ -f x) = a?/* — 3jc?/* + x ; sine of 6A {~ sine of 5 A xy — sine of 4A = xy'^ — 3xy^ + xy — xy^ + ^xy) — xy^ — 4:xy^ + 3xy ; sine of 7 A ( 1= sine of 6A xy — sine of 5A—xy^ — 4cxy* Sxy- — xy^-\- Sxy- —x) — xy^— bxy'^ + Qxy^—x : whence, universally, the sine of the multiple arch wA, where n denotes any whole positive number, whatever, will be truly expressed by a x n-i n — 2 „-3 , n — 3 72—4 Z^ n — 4 y j— xy + —j- X ~j~ xy — 7i— 5 72 — 6 n-7 X X X y , &c. Moreover, irom the 2 3*^* second theorem, we have co-sine of 2A ( — co-sine of Ax^— co-sine of 0= i) — ^—-— ; 300 THE APPLICATION OF ALGEBRA Co-sine of 3A (= co-sine of 2A y y — co-sine of A— 2 -^ 2 ' 2 ' Co-sine of 4a ( zz co-sine of 3 A x y — co-sine of 2Az= y'—3y\ y" — g x _ y' — ^y' 4- 2 , 2 2 ^ 2 * whence, universally y the co-sine of the multiple arch ?iA will be truly represented by ■? — -f ~ n n — 4 71 — 5 „_6 n n—5 X X X ?/ -f - X 2 2 3 "^22 X y^~ &c. which series, as well as X 71 — 2 ■3 X y"- -4 X 72 — 6 X n — 4 7 3 .that for the sine, is to be continued till the indices of ^ become nothing or negative. But, if you would have the sine expressed in terms of cT only, then, because the square of the sine f the S(]uare of the co-sine is aiw^ays equal to the square of the radius, and therefore, in this case, X'-\- i y" z= 1, it is manifest that the sines of all the odd multiples of the given arch A, wherein only the even powers of y enter, may be exhibited in terms of x only, without surd quantities: so that 4 — 4 1'* being substituted for its equal ?/-, in the sines of the aforementioned arches,we shall have 1st. Sine of 3 A — 3.r — Ax\ 2d. Sine of 5 A z: bx — 20 r» f l6.r' ; 3d. Sine of 7A =: "Jx— 5Qx^ + 112.i^ — 64r^ 4th. Sine of gA zz ac— 120a;^' f 432^' — 576.r'l 256.c'; h^c. ^^. Aw^, generally, if the multiple arch be denoted by Wx\, then the sine thereof will be truly represented by 2.34.3 I —25 n n'^ — 1 n — j) «"— 2.^ w' — 49 , „ . TO GEOMETRICAL PROBLEMS. 301 From this series the sine of the sub-multiple of any- arch, where the number of parts is odd, may also be found, supposing (.9) the sine of the whole arch to be given: for let x betherequiredsineof thesub-multiple, and n the number of equal parts into which the whole arch is divided ; then, by what has been already shewn, t_ 11 L ^ r{^—\ , . ^ rf" — 1 we shall have nx — — x — — r- x x^ + -7 x - - — X X a-^ <5rc. zr s: from the solution of which equa- 4.5 ^ ^ tion the value of .r will be known. Hence also, "sve have an equation for finding the side of a regular po- lygon inscribed in a circle: for seems: the sine of any arch is equal to half the chord of double that arch, let \xi and \w be wrote above for a* and 5 respectively, and ^, ^. .„ , m n n^ — 1 v^ then our equation will become x ■ — -- x ~ ^ 2 1 2.3 8 n n^ — 1 w^-9 »^ -, ^ n + T^T:T ^ ITT^ 3^^^.^-l^^^or..-. -X 2.3 4 1 2.3 4.3 \Q pressing the relation of chords, whose corresponding arches are in the ratio of 1 to n, But,when the greater of the two arches becomes equal to the whole peri- phery, its chord {w) will be nothing, and then the equa- tion, by dividing the whole by wy, will be reduced to n^—\ tj^ n^ — X n" — Q v' n"— l ^ "" T:^ ^ 4 ^ 2TF ^ TTT ^ Yd 2TT ^ fi^ q ^2 __ 2 ^ ^6 •- — f X — — ~ X — ~ ^c, — ; where 7i is the num- 4.5 5.6 64 ber of sides, and v the side of the polygon. From the foregoing series, that given by Sir Isaac Newton, in PkiLTrans, mentioned in p. 242 of this Trea- tise, may also be easily derived. For, if the arch A and its sine x be taken indefinitely small, they will be to one another in theralioof equality, indefinitely near, by 30*2 THE APPUCJATION OF ALGEBRA what has been proved at p. 246 ; in which case, the ge- neral expression, by writing A instead of a-, will become 7?A — - X -— — X A^ + - X - — -- X - — - X 1 2.3 1 2.3 4.3 Therefore, if n be now supposed indefinitely great, so that the multiple arch fiA may be equal to any given arch z, the squares of the odd numbers, 1, 3, 5, ^*c, in the factors w*— l, 7^*— 9, n^-— 23, ^c may be rejected as nothing, or inconsiderable, in respect oUi^; and then n^ A^ n^A^ the foregoing series will become n A— —-— -f r - w^A^ ^•c. wherein, if for wA, its equal 2.3.4.3.6. 7 2, be substituted, we shall then have z — ■ " - -h _ A-c. which is the sine 2.3.4.3 2.3.4.3.6.7 of the arch z, and the same with that before given. Moreover the aforegoing general expressions may be applied, with advantage, in the solution of cubic, and certain other higher equations, included in this form, ^,^. z --az + 2- "^ ^ ' - "^ "" ^^ "^ ® ^ 2/1 3/^ 4w For, if 2 be put m ?/ \/-, the equation will be trans- formed to — n X v" w „— .o w w — 3 ^c. zz /, and consequently ^ — g'^^ +2^ "T" TO GEOMETRICAL PROBLEiMS. 303 „.4 n n — 4 n — 5 »?~6 „ ^ \f ^^^ 2 ^ -^ ""Tlw. f whence, as it is proved above, that the former part of the equation (and therefore its equal) represents the co-sine of n times the arch whose co-sine is |?/,we have the followincr rule : Find ^ from the tables^ the arch whose natural co-sine is n 2" ^ or its log. co-sine = log, |/ log. - the radius li being unity ; take the ni\\ part of that arch, and find its co-sine, tvhich multiply by 2 k — ,andtheproductwiil be the true value ofz, in the proposed equation z^ — az^"^ , ^—3 , n— 4 n—4: n — 5 „ w— 6 ro + -—-' X a^z X y. aH (^c. Qn 2n 3n Thus, let it be required to find the value of 2;,in the cubic equation 2' — 4322 =z 1728; then, we shall have n — 3, « — 432, and/ zz 1728 ; consequently •^^ ( =: ~) — ,5, and the arch corresponding -a^ l44l1r 71] thereto := 60"; whence the co-sine of (20**) * thereof will be found ,9396926; and this, multiplied by 24 ( -: 2 v — ) gives 22,53262 for one value ofz. But besides this, the equation has two other roots, both of which may be found after the very same manner : for, since 0,5 is not only the co-sine of 60°, but also of 60° + 360°, and 60^+ 2 x 36o", let the co-sine of (1 40°) |. of the former of these arches be now taken, Avhicli is — ,7660444, and must be expressed with a negative sign, because thearch corresponding is greater than one right angle, and, less than three. Then, the value 504 THE APPLICATION OF ALGEBRA thus found being, in like manner, multiplied by 24 { — ^Y ^), we shall thence get — 16,38306 for an- other of the roots: whence the third, or remaining root will also be known; for, seeing the equation wants the second term,the positive and negative roots do here mu- tually destroy each other ; and therefore the remaining root must be— 4,16756, the dili'erence of the two for- mer, with a negative sign. PROBLEM XXXV. From a given circle ABC H it is proposed to cut off a segment ABC, such, that a right line DE draicn from the middle of the chords AC, to make a given angle therewith^ shall divide the arch BC of the semi-segment into two equal parts. Let the chord BC be drawn, and upon the diameter HDB let fall the perpendicular EF ; put the radius OB^ of the circle zn 7*,and the tangent of the given angle C DE (answering to that radius) =: f, and let OF = z; then will EF =r \/rr — zZi and BC ( = 2EF):r: ^s/rr — zz, and consequently BD ( = BC2 _ 4r«— 42"- BH 2 2r T -k- z , r — z from which takino- BF=r--2, we have DF zr r-h2z xr — r But, by trigonometry, EF : DF : : rad. : tang. DEF, f : t. Whence r + 2s . r— 2 that is, s/rr^zz : we have FT^' x 7*^=^' = ^' x »^ — »*; where TO GEOMETRICAL PROBLEMS. 305 the whole being divid ed by r — 2, their results r + Szi^ X T — z — i- X T -\- z', which, ordered, gives Zrr — tt X z zi — ATT — tU 4 3rr — ti Put — a, and — vc rr — tt zz f ; then it 4 4 will be z^ — az zz f. Therefore find, from the tables, If the arch whose cosine is — ^4=- (the radius beine: unity) ; take i- thereof, and find its co-sine ; which, multiplied by 2v/i«> gives the true value of z (see the last problem*) Now, hy logarithms, it will be log. \f — log. ^a — If flog, la =z — 1.9425328 zi log. ' - ^ z= log. co- •sci\/-la sine of 28^ 50' ; whereof the third part is 9° 36F, whose log. co-sine (to the radius 1) is — 1.9938609; which added to the ^ log. of ia (z: — 1.6826316) gives — 1.6764925 = log. of 0.47478^ whose double .94956, is the true value of 2;, or FO : whence the correspond- ing arch BE =: 18^ l6Y, and consequently BC {- 2BE) =: 36"^ 33'. — By means of this problem that portion of a spherical surface representing the apparent figure of the sky is determined. PROBLEM XXXVr. The base AB, and the difference of the angles at the base being givcjiy while the angles themselves vary; to find the locus of the vertex E of the triangle* Let the base A B be bisected in O, and the angle BOD so constituted as to exceed its supplement AOD by the given difference of EAB and EBA; and let ED, APQ, BSF, be perpendicular, and EFF parallel to OD : then, since the angle BCE (BOD) as much exceeds X 306 THE APPLICATION OF ALGEBRA ACE, as CAE exceeds CBE, it is evident that the sum of the two angles BCE, CBE, of the triangle BCE, is equal to the sum of the two angles ACE, CAE of the triangle ACE; and, consequently, that the remaining* angles A EC and BEC, are equal the one to the other : therefore, by reason of the similar triangles EFB, EAP, we have EF : EP : : BF : AP, that is, OD + OQ:OD —OQ A : : QA + DE : QA — DE ; whence, by composition and di- vision, 2OD : 2OQ i: 2QA : 2DE ; wherefore 0.0 x DE is = OQ X QA; which is the known property of an equilateral hyper- bola with respect to its asymptote. PROBLEM XXXVII. To find the solidity of a Cimical ungula BFCB, cut off by a plane BRFSB passing through one extremity of the base-diameter. Let EPF be parallel to the base-diameter BC, cuf- tinp; AD the axis of the cone in P ; also let An be per- pendicular to BF ; join P, ??, and let RS be the conju- gate axis of the elliptical section BRFSB : then the part ABF, above the said section, being an oblique el- liptical-cone, its solidity will be expressed by '7834 x SR X BF X ^^', that is, by the area of its base BRFSB TO GEOMETRICAL PROBLEMS. 307 drawn into ^ of the perpendicular height. But the triangles BCF and hVn will appear to be equi- angular ; for, APF and AwF being both right-angles, the circumference of a circle, described on the diame- ter AF, will pass through P and n ; and so the angles AF« (BFC) and APw, as well as AFP (FCB) and A«P, insisting on the same arch, are respectively, equaL Hence we have BC : BF : : Am AP; and therefore BFxA?2 = BCxAP: this value being substituted above, the content of the part A BF becomes SR x BC x AP x •2618: which, because SR is known to be=:v/BCx EF, is farther reduced to BC x AP Xv/BCx£F x .2618. This subtracted from,BC^ x AD x .2618, the content of the whole cone ABC, leaves BC^x AD — BC x AP x n/BC^TEF x .2618 for the required solidity of the ungula BCF ; which, because ,^ D P X BC ,.^ DP X EF AD = -^^ :i=r^, and AP = ^^^ rr^^ will be re* BC — EF' BC— EF' ducedto?^-^?ii^. ..6,8BP X BC. X 2 308 THE APPLICATION OF ALGEBRA PROBLEM XXXVm. Let A and B be two equal weights^ made fast to the ends of a thread, or perfectly flexible line pVnQq, sup- ported by two pins, or tacksy P, Q, in the same horizon^ lal plane; over which pins the line can freely slide either way; and let C be another weight, fastened to the thread, in the middle, between P and Q ; now the question isy to find the position of the iveight C, or its distance beloio the horizontal line PQ, to retain the other two weights A and B in equilibrio. Let PR (=iPQ) be denoted by a, and R« (the distance sought) by .t ; and then P//, or Qn, will be re- presented hy\/a' + x\ Therefo re, by the resolution of forces, it will be, as \/a- + x"^ (P?z) : x (Rw) : : the whole force of the weight A in the direction P/2, to A Y — it's force in the direction nR, whereby iten- deavours to raise the weight C ; which quantity also expresses theforceof theweightB inthesamedirection : but the sum of these two forces, since the weights are supposed to rest in equilibrio, must be equal to that o A "*< of the weight C ; that is, •>• v =: C ; whence we have 4A"a,^ =: CV -{- C^r^ and consequently x = __ aC v/4A'-^~ C' TO GEOMETRICAL PROBLEMS. 309 PROBLEM XXXIX. To determine the position of an inclined plane AE, along which a heavy body descending by the force of its own gravity from a given point A, shall reach a right line BP, given by position, in the least time possible. Through the given point A, perpendicular to the horizon, let there be drawn the right-line RB, meeting BP in B ; also conceive the semi-circle AER to be de- scribed, touching BP in E; then let AE be drawn, which will be the position required ; because the time of descent a- long the chord AE being equal to that along any other chord Ajz, it will consequently be less than the time of the descent along Ae, whereof An is only a part : therefore, if AQ and OE be now made perpendi- cular to BP, we shall have, (by reason of the similar triangles) AB : AQ : : AB + AO : (OE) AO ; whence, by multiplying extremes and means, AB x AO = AQ x AB f AQ X AO ; therefore AB x AO - AQ X AO z= AQ X AB, and AO (OE) = ^^^^q ; f^om which BE and AE are also given. The geometrical construction of this problem is ex- tremely easy; for, if AQ (as above) be drawn perpen- dicular to BP, and the angle OAQ be bisected by A E, the thing is done : because, OE being drawn parallel to AQ, the angle OEA is = QAE = EAO; and so, AO being =: OE, the semi-circle that touches BP, will pass through A. x3 310 THE APPLICATION OF ALGEBRA PROBLEM XL. A ray of light, from a lucid point P in the axis AP of a concave spherical surface, is reflected at a given point E in that surface ; to find the point D where the refiectec^ tay meets the axis. Draw EQ perpendicular to AP, and from the centre C let CE be drawn ; also make CE = a,CQ,zz b, CP =: c, CD =z x; and {by Euc, 12,2.) P E will be = wherefore, the an- gles of incidence and reflection,CEP and CED being e- qual, we have, as PC (c) : CD (:r) : : PE (\/ a'' + c^ + 26c) : ED zi X\/a^ + c2 -f 26. angles, Ca (c) : aE (x) : : CD (c + J) ; DS = ^i^±^ ; and by the law of refraction, m ; n -. ; DS : S5 zz n ^, xj^jjrj. ^viience D^ [zz DS — S^) = m n X X c -\r d - X , m c and vs {= aE ■— Ds) - 314 THE APPLICATION OF ALGEBRA, &c. re X r — — , by making r — — , and g — i — r. Now, c^ : E?j (BD) : : aE : aQ (BQ) ; that is, X XT-^^ I d :i XI ^^_ , =: EQ ; which is given from hence. Again, in the very same manner, Qc (b) : ce (y) : : OF (6 + 2) : FR = 12LiLl_f ; and m : « : : FR : Rr — ~ X - — -J — : whence Fr zz l x ~ — -. mo mo ay X b -^ z J .-^ . qz , zz— — T , and wr (Fr — ce) — y x -. r ; and W6 X C6 bz therefore cQ ( rr ) — -^ — ^^-: from which sub- wr qz — br tracting the value of BQ, found above, we get this equation, viz, ; : =z t : whence the va^ ^ qz — or cr — (jd cd lue of z, by making the given quantity t -l ^-, = g, / c ' q ci comes out z: ^. But, if you had rather have the qg - 6 same in original terms, it is butsubstituting forg ; whence, rbcd -\- rbt X re — qd after reduction, z = qd X 6-f c — rbc ■\' qt x re — qd' which, by restoring m and w, becomes mnbcd -f nbt x nc — m — n x d m — n X md x b-\-c — mnbc~{- m — n x tx nc — rn — nxd where, if/ be taken equal o, we shall have nbcd z — . r , the very same as was jfi — n xd X b-{-c — nbc found by the preceding method. C 315 ] APPENDIX: CONTAINING THE CONSTRUCTION OP GEOMETRICAL PROBLEMS, WITH THE MANNER OF RESOLVING THE SAME NUMERICALLY, PROBLEM I. The base, the sum of the tico sides, and the angle at the vertex of any plane triangle being given, to describe the triangle. CONSTRUCTION. DRAW the indefinite right-line AE, in take^AB equal to the sum of the sides, and make the angle ABC equal to half the given angle at the vertex, and upon the point A, as a centre, with a radius equal to the given base, let a circle nCm be described, cutting BC in C ; join A, C, and make the angle BCD zi CBD, and let CD cut AB in D; then will ACD be the triangle that was to be constructed. which 316 THE CONSTRUCTION OF DEMONSTRATION. Because the angles BCD and CBD are equal, there, fore is CD = DB (Euc, 6. 1.) and consequently AD + DC :=AB: likewise, for the same reason, the angle ADC ( =:BCD + CBD, Euc. 32. 1.) is equal to 2CBD. Q. E. D. Method of calculation. In the triangle ABC are given the twosides AB, AC, and the angle ABC, whence the angle A is known ; then in the triangle ADC will be given all the angles, and the base AC ; whence the sides AD and DC will also be known. PROBLEM II. JTie angle at the vertex, the base, and the difference of the sides being given, to determine the triangle. CONSTRUCTION. Draw AC at pleasure, in which take AD equal (o the diflbrence of the sides, and tnake the angle CDB equal to the complement of half the given angle to a right angle ; then from the point A draw AB e- qual to the given base, so as to meet DB in B, and make the angle DBC z= CDB, then will ABC be the triangle required. DEMONSTRATION. Since, {by construction,) the angles CDB and DCBare equal, CB is equal to CD, and therefore CA — CB z= AD : moreover, each of those equal angles being equal to the complement of half the given angle, their sum, which is the supplement of the angle C, must therefore be equal to two right angles — the (whole) given angle, and consequently C = the given angle. Q. E. D. Method of calculation. In the triangle ABD are given the sides AB, AD, GEOMETRICAL PROBLEMS. 317 and the angle ADB, whence the angle A will be given, and consequently BC and AC, PROBLEM III, The angle at the vertex, the ratio of the including sides, and either the base, the perpendicular, or difference of the segments of the base being given, to describe the triangle. CONSTRUCTION. Draw CA at pleasure, and make the angle ACB equal to the angle given ; take CB to CA in the giveu ratio of the sides, and jpin A, B ; then, if the base be given, let AM be taken equal thereto, and draw ME parallel to CA meeting CB in E,and make ED parallel to AB; but if the perpendicular be given, let fall CF, perpendicular to AB, in which take CH equal to the given perpendicular, and draw DHE parallel to AB : GMN lastly, if the difference of the segments of the base be given, take FG — AF, and join, C, G, and take GN" equal to the difference of the segments given, drawing: NE parallel to CG, and ED to BA (as before) ; then will CDE be the triangle which was to be constructed. DEMONSTRATION. Because of the parallel lines AB, DE ; ME, AC; and NE, GC ; thence is DE = AM, and EI =: NG ; and also CD : CE : : CA : CB [Euc. 4. 6.) Q.E.D. 318 THE CONSTRUCTION OF Method of calculation. Let AC be assumed at pleasure; then, the ratio of AC to BC being given, BC will become known ; and there- fore in the triangle ACB will be given two sides and the included angle, whence the angles R and A, or E and D will be found ; then in the triangle EDC, EHC, or EIC, according as the base, perpendicular, or the dif- ference of the segments of the base is given, you will have one side and all the angles, whence the other sides will be known.- PROBLEM IV. The angle at the vertex^ and the segments of the base, made by a perpendicular falling Jrom the said angle, being given^ to describe the triangle. CONSTRUCTION. Let the given segments of the base be AD and DB ; bisect AB by the perpendicular EF, and make the angle EBO equal to the difference between the given angle and a right one, and let BO meet EF in O; from O, as a cen- tre, with the radius OB, de- scribe the circle BGAQ, and draw DC perpendicular to AB, meeting the periphery of the circle in C ; join A, C and C. B, then will ACB be r. ^,.,,-— J - — -^ ^i Vn \ 1/ •••'' o\ \ /..•"""''(' ^^'^ Q the triangle that was to be constructed. DEMONSTRATION. The angle ACB, at the periphery, standing upon the arch AQB, is equal to EGB, half the angle at the centre, standing upon the same arch; but EBO is equal to the difference of the given angle and a right one (by construction) therefore ACB (EGB) i3 equal to the angle given. Q. E. D. Method of calculation. Draw CFG parallel to AB; then it will be. as the base AB : to the difference of segments CG (: : EB : CFj : : the sine of the given angle at the vertex (FOB) : GEOMETRICAL PROBLEMS. 319 to the sine of (COFzrCBG) the difference of the angles at the base; whence the angles themselves are given. After the same manner a segment of a circle may he described to contain a given angle, when that angle is greater than a right one, if, instead of BO being drawn above AB, it be taken on the contrary side. PROBLEM V- Having given the base, the perpendicular, and the angle at the vertex of any plane triangle, to covistruct the tri- angle. CONSTRUCTION. Upon AB the given base fsee the preceding figure J let the segment ACGB of a circle be described to contain the given angle, as in the last problem ; take EF equal to the given perpendicular, and draw FC parallel to AB, cutting the periphery of the circle in C; join A, C and B, C, and the thing is done: the demonstra- tion whereof is evident from the last problem. Method of calculation. In the triangle EBO are given all the angles and the side EB, whence EO will be known, and consequently OF ( == DC - EOj ; then it will be as EB : OF : : the sine of EOB (the given angle at the vertex) to, the sine of OCF, the complement of (COF or CBG) the difference of the angles at the base; whence these an- gles themselves are likewise given. — This calculation is adapted to the logarithmic canon ; but by means of a table of natural sines, the same result may be brought out by one proportion, only : for BE being the sine of BOE, and OE and OF co-sines of BOE and COF (answering to the equal radii OB and OC) it will there- fore be, BE : EF :: sine BOE (ACB) : cosine BOE 4- co-sine COF; from which, by subtracting the co-sine of BOE, the co-sine of COF ( = CBG} is found. PROBLEM VI. The angle at the vertex^ the sum of the two including sides, and the dJfference of the segments of the base bei'r^ gicen^ to describe the triavgle. 320 THE CONSTRUCTION OF CONSTRUCTION. Draw the right line AC at pleasure, in which take AB equal to the difference of the segments of the base, and make the angle CBE equal to h'alf the supplement of the given angle ; and from A to BE, apply AE equal to the given sum of the sides ; make the an- gle EBD -BED and let BD meet AE in D, and from the centre D, with the radius DB, describe the circle DBG, cut- ting AC in C, and join D, C ; then will ACD be the triangle required. DEMONSTRATION. The angle EBD being zz BED, therefore is DE = DB =1 DC, and consequently AD + DC i= AE. Moreover the angle CDE, at the centre, is double to the angle CBE, at the periphery, both standing upon the same arch CE; which \2ist f by construction) is equal to half the supplement of the given angle, therefore CDE is equal to the whole supplement, and consequent- ly ADC equal to the given angle itself. Q. E. D. Method of calculation. In the triangle ABE, are given the two sides AB,AE, and the angle ABE, whence the angle A will be given ; then in the triangle ABD will be given all the angles and the side AB, whence AD and DC (DB) will be also given. PROBLEM VII. The angle at the vertex , the sum of the inciudifig sides, and the ratio of the segments of the base being given ; to determine the triangle. , CONSTRUCTION. Let AG be to GB, in the given ratio of the segments of the base, and, upon the right-line AB, let a segment GEOMETRICAL PROBLEMS, 321 of a circle be described, capable of containing the given angle ; draw GC per- pendicular to AB, meeting the periphe- ry in C; join A, C and C, B, and in AC, produced, take CH zz CB; join B, H, and in HA, take HD equal to the given sum of the sides, draw DE parallel to AB, and £F to BC ; then will DEF be the triangle required. DEMONSTRATION. Let ¥n be perpendicular to DE. Whereas (by con-- struction) CH is equal to CB, and FE parallel to CB, therefore is FE ■= FH [Euc, 4. 6.) and consequently FE + FD z= HD ; also, because FE is parallel to CB, therefore is the angle DFE zz ACB : moreover, the triangles ABC, DEF, being equiangular, it will be, as AG : GB : : Dn : wE. Q. E. D. Method of calculation. From the centre O, conceive AO and OC to be drawn; supposing KOI perpendicular, and CI parallel to AB : then it will be, as AK is to CI (KG) so is the sine of AOK [— ACB, see Prob. 4.) to the sine of COI, the difference of the angles ABC and BAG; which are both given from hence, because their sum is given by the question : therefore in the triangle DHE are given all the angles and the side HD, whence the base DE will be known. PROBLEM VIII. Having the angle at the vertex, the difference of the in' chiding sides i and the difference of the segments of the base, to describe the triangle. CONSTRUCTION. Take AB equal to the difference of the segments of the base, and make the angle AB;j equal to halftiie V 322 THE CONSTRUCTION OF given angle ; from A B C, join O, C; then is AOC the triangle sought. apply AE =: the differ- ence of the sides; produce AE, and make the angle EBO =z BEO, and let BO meet AE, produced in O, and from the centre O, at the dis- tance of OB, de- scribe the circumfe* rence of a circle, cut- ting AB produced in DEMONSTRATION. Because the angle EBO is = BEO (bij construction) ; therefore is EO zr BO = CO, and consequently AO — 00 =: AE. Furthermore because the angle *AOC 19 double to ADC, and ADC - ABE {Euc. Corol29. 3,) therefore is AOC also double to ABE. Q. E. D. Method of calculation. The two sides AB, AE, and the angle ABE being given, the angle A will from thence be found; then in the triangle ABO will be given all the angles and the side AB, whence OB (OC) and AO will be known. PROBLEM IX. The angle at the vertex, the difference of the including sides, and the ratio of the segments of the base being given, to determine the triangle, CONSTRUCTION. Let AG be to GB in the given ratio of the seg- ments of the base, and up- on the right-line AB let a segment of a circle ACB be described (by Frob,4,) capable of the given an- gle ; draw GC perpendi- to AB, meeting the periphery in C, and join A,C GEOMETRICAL PROBLEMS. 3«S and B, C ; in AC take AP =: BC, and draw BP ; also, in AC, take CQ equal to the given difFerence of the sides, drawing QE parallel to PB,and ED to BA ; then will CDE be the triangle which was to be described. DEMONSTRATION. The angle DC£ is equal to the given angle by con- struction ; also EQ being parallel to BP, DE to AB, and AP =z BC, therefore must DQ = EC {Euc. 4. 6.) and consequently DC — EC zr CQ. Moreover, if CG be supposed to cut DE in n, thenDn : En :: AG : GB. Q. E. D. Method of calculation. Let Cm be equal to CE, and let Em be drawn. It will be, as AB is to AG — BG, so is the sine of ACB to the sine of the difFerence of CBA and CAB [by Prob, 4.) then in the triangle DEm will be given all the an- gles and the side Dm, whence DE will be given. PROBLEM X. The angle at the vertex, the perpendicular and the dif- ference of the segments of the base being given, to con» struct the triangle. CONSTRUCTION. Draw RS at pleasure, in which take DE equal to half the difference of the segments of the base, and make EC perpendicular to RS and equal to the given per- pendicular, and the angle DEtz equal to the difference between the given angle and a right one ; join D. C, and draw DnO parallel to CE, and in DC take the Y 2 324 TflE CONSTRUCTION OF point p, so that 7?p, (when drawn) may be equal to wE ; draw CO parallel to np, meeting D«0 in O ; and upon O as a centre, with the radius OC, describe the circle BCA, cutting RS in B and A; join A,"C and B, C, and the thing is done. DEMONSTRATION. Join O, B and O, A: since OC is parallel iopn, therefore is OC : DO : : p?z : ?2D, or OB : DO : : n% : nD ; and consequently the triangle OBD similar to the triangle wEt) [by Eucl.Q.) Therefore, seeing, the angle DE« is (hy construction) ex\n^\ to the excess of the given angle above a right one, ACS must be equal to the angle given [hy Prob,4:.) Moreover since, AD is ~ DB, A E — BE will be equal to 2DE, w4iich is the given difference of the segments (by construction). Q. E. D. MetJiod of calculation. In the triangle CDE, right-angled at E, are given both the legs DE and EC, whence the angle EDC will be known, and consequently ODC ; then as the radius is to the sine ofDBO(::OB: DO : : OC : DO) so is the sine of ODC to the sine of OCD ; whence DOC, the difference of the angles ABC, BAC, [see Prob, 4.) is also given, and from thence the angles themselves. PROBL-EM XI. The ani^le at the vertex, the perpendicular, and the ratio of the segments of (he base being given, to con-^ struct the triangle. CONSTRUCTION. Take AF to FB in the given ratio of the seg- ments of the base, and upon the right-line AB describe a segment of a circle ACB capable of the given angle ; make FC perpendicular to AB meeting the circumference of the circle in C, in which GEOMETRICAL PROBLEMS. 325 take CG equal to the given perpendicular ; draw DG£ parallel to AB, meeting AC and CB in D and E; and then DCE will be the triangle required. DEMONSTRATION, Because of the parallel lines DE and AB, it will be as AF : DG (: ; CF : CG) : ; FB : GE, or AF : FB : : DG : GE ; whence it appears, that DG and GE are in the ratio given. Also the angle DCE and the perpendicular CG are respectively equal to the given angle and perpendicular, by construction. Q. E. D. Method of calculation. As AB is to AF — BF (see Prob. 4.) so is the sine of ACB to the sine of the difference of A and B ; whence both A and B will be given, because their sum, or the angle at the vertex, is given : then in the triangles DGC, EGC, will be given all the angles and the per- pendicular CG, whence the sides will also be known. PROBLEM XII. The base, the sum of the sides, and the difference of the Jingles at the base being given, to describe the triangle. CONSTRUCTION. At the extremity of the base AB, erect the perpen- dicular BE, and make the angle EBC equal to half the given dif- ference of the an- gles at the base ; from the point A, toBC, apply AC equal to the sum of the sides ; and make the angle . — — -^j- CBD ~ BCA; A «• then will ABD be the triangle required. DEMONSTRATION. From the centre D, with the radius CD, describe the Y 3 32C THE CONSTRUCTION OF semi-circle CHF, and join F, B. Then, whereas by construction the angle CBD is zz BCD, therefore is DB — DC ; whence it appears that AD 4- DB is = AC, and that the semi-circle must pass through the point B: therefore, the angle CBF, standing in a semi- circle, being a right angle, and therefore — ABE, let FBE, which is common, be taken away, and there will remain ABF = EEC ; but DF being equal to DB, it is manifest that ABF (EBC) is equal to half the dif- ference of the angles ABD and DAB. Q. E. D. Method of calculation. As the sum of the sides (AC) is to the base (AB) so is the sine of ABC, or of the complement of half the given difference, to the sine of (C) half the angle at the vertex; whence the other angles BAD and ABD are also given. PROBLEM XIII. The base, the diff'erence of the sides, and the difference of the angles at the base, hO'ing given, to determine the tri^ angle, CONSTRUCTION. At the extremity B of the given base AB, make the angle ABD equal to half the given difference of the angles at the base; and from A toBD apply AD z: the ditference of the ^ sides; draw ADC, B A and make the an- gle DBC zr BDC, and ABC will be the triangle re- quired. DEMONSTRATION. Because the anlge DBC is = BDC, CD will be =: CB and AC will exceed BC by AD. Moreover, since A + ABD - (CDB) CBD r^?/c. 32. \.J therefore is % + 2 ABD (zz CBD + ABD) zz ABC, and conse. quently ABC — A z: 2 ABD, equal to the dilferencc given. Q.; AH =z AC x AG, or AC : AP : : AH : AG ; therefore the triangles ACP, AHG being equi-angular {Euc, 6. 6.) we have AP : CP : : AG : GH (AK) : : AF (aj : AI (c). Q. E. D. Method of calculation. In the triangles AFG, AIK are given all the angles and the sides AF and Al, whence AG, FG, and AK (GH) will be found ; then in the triangle FGH will l)e given all the sides, to find the angle HFG ; which, ad- ded to AFG, gives AFH (APB) from whence, and the two given sides AF and FII else is readily determined. niclLuiiii' it, evL-ry thiui GEOMETRICAL PROBLEMS. 36^ PROBLEM LIV. To describe a triangle (ABC) similar to a given one AMiNT, such that three lines (AP, BP, CP) may be draim from its angular points to meet the same point (P) so as to be equal to three given lines AD, AF, and AK, re- spectively. CONSTRUCTION. Draw DE and KG, making the angles ADE and AKG, each, equal to the given angle N, and intersecting AN in E and G ; from the centres D and E, with the intervals AFand AG, let two arcs be described, intersecting in H ; draw AH, in which take AP = AD; , . and from P, to AM A K K D B and AN, apply PB and PC equal, respectively, to AF and AK, and let B, C be joined; so bhall ABC be the triangle that was to be determined. DEMONSTRATION. The three lines APjBP, CP, are, respectively, equal to the three given lines AD, AF, AK, by construction ; we therefore have only to prove that the triangle ABC is similar to the given one AMN. Now supposingDH and EH to be drawn, it will be AP : PC (or AD : AK) :: AE : AG (EH); whence the triangles A PC and AHEwiil beequi-angular(£2^c. 6.6.) and consequently AC : AH : : AP (AD) : AE : : AN : AM {Euc, 5. 6.) : but the triangles ABP and ADH (having AP z: AD, PB — DH [by construction) and the angle DAP com- mon) are equal in all respects; therefore, by substituting AB in the room of AH, our last proportion becomes AC : AB : : AN : AM ; whence it is manifest tliat the triangles ABC and AMN are equi-angular. Q. E. D. 16S THE CONSTRUCTION OF Method of calculation. Id the triangles ADE, AKG, are given all the an- gles and the sides AD and AK, from which AE, DE, and AG will be known; then in the triangle DHE will be given all the sides, to find the angle EDH, which added to ADE gives ADH ; from whence, and the two given sides including it, AH {= A3) will be known. ' PROBLEM LY. In the triangle ace, besides the angle c, are given the segments of the sides ah and de, and the angles aeb and dbe stthtended therehi) ; to describe th(t tria ngle. CONSTRUCTION. Upon AB, equal to a&, let a segment of a circle be described to contain an angle equal to aeb ; make the angle ABF = ace, BA7^ - dbe^ and the line BF — eti; from the point n, where kn cuts the periphery of the circle, through F, draw ;iFE, meeting the periphery in E; join A, E, and B, E, and draw EC parallel to BF, meeting AB, produced, in C ; and then the thing is done. DEMONSTRATION LetBD be parallel to FE. Since the lines BD, EF, and ED, FB, are parallel, jtherefore is ED = BF {- cd\ and the ans^le ACE also ZL AUF [ace) Euc, 28. 1. Moreover, the anL;le BE/i GEOMRTRICAL PROfU.KMS. 369 (DBE) is equal to BAn {dbe), both standing, upon the same segment Bn, Q. E. D. Method of calculation* Join B, n; then in the triangle ABn will be given all the angles and the side AB, whence Bn will be known ; then in the triangle nBF will be given B«, BF, and the included angle «BF, whence BFn (CDB) and all the rest of the angles in the figure will be known. PROBLEM LVI. To make a trapezium, ichose diagonals, and two oppo- site sides, shall he all of given lengths, and whereof the angle formed by the given sides, tvhen produced till they meet, shall also he given. CONSTRUCTION. Draw the indefinite right-line AC, and take therein AB equal to one of the two given sides ; make the angle CBG equal to the given an- gle, and let BG be made equal to the other given side ; up- on the centres A and G, with in- tervals equal to the two diago- nals, let two arches be de- scribed, cutting each other in D; make DE equal, and parallel, to GB; join D, B, and E, A ; then ABDE will be the trapezium required. DEMONSTRATION. Draw DG, DA and BE, and let BA and DE be produced to meet each other in F. The lines BG and DE are equal, and parallel, by construction; therefore BE is z= DG, which last (by B B 370 THii CONSTKUCTlOff OF construction) IS equal to one of the given diagonals, as AD is equal to the other: moreover the sides AB and ED (BG) are equal to the given sides, by construction ; and the angle F is equal to the given angle CBG, be- cause DF is parallel to GB. Q. E. D. Method of calculation. Suppose AG to be dravi'n ; then in the triangle ABG will be given the two sides B A and BG,and the included angle ABG, wiience the side AG and the other two angles will be known; then in the triangle ADG will be given all the sides, vi^hence the angle AGD will be known, and from thence the whole angle BGD ; lastly, in the triangle BGD will be given the two sides BG and GD, and the included angle BGD, whence the side BD will likewise be known. PROBLEIVI LVI[. The ^e^ments of the base AD, DB, and the line DC bisecting the vertical angle ACB, of a plane triangle be- ing given, to describe the triangle. ^^ CONSTRUCTION. In AB, produced, lake DO to AD, au DB to AD — DB. and ^ -^ from the cen- tre O, with the radius OD, describe the circle DCQ; also from the centre D, at the given dis- tance DC, describe the circle mCn, and from C, the intersection of the two circles, draw CA and CB, and the thing is done. DEMONSTRATION. Since DO : AD : : DB : AD — DB: therefore (by the lemma in p, 334,) AC : CB : : AD : DB : whence CD bisects the angle ACB (by Em. 3. 6.) Q. E. D. (i KOMF^rRICAl. PltOliLEMS. J7I Method of calculation. Draw CP perpendicular to AQ. ^.T^ - AD X BD Because, by construction, OD is =: ~a n .HrRn* therefore will DQ — - . .^ ^ p ^ ; whence, by reason AD — DLJ of the similar triangles, DCQ, DPC, it will be, as 2AD X BD AD~BD--^C=-^C^^P = whence AC and CB are given. AD — BD X DC^ "sAD X BD TROBLEM LVIII. Having given the base, the angle at the vertex ^ and the line drawn Jrom thence to bisect the base ; to con* struct the triangle. CONSTRUCTIOK. Upon the ofivenbase AB describe (% Prob, 4.) a seg- ment of a circle ADB ca- pable of the given angle ; and, from the point F, in vjrhich the perpendicular DF bisects AB, with a ra- dius FC equal to the bisect- ing line, describe 7?Cm, cutting the periphery ACB in C ; join A, C and B, C, and the thing is done. The demonstration of which is evident from the construction. Method of calculation. From the centre O let OA and OC be drawn ; then in the triangle AOF will be given all the angles and the side AF, whence FO and OC (OA) will be known- and m the triangle CFO will be given all the sides' whence the angle FOC, and its supplement DOC* expressing the difference of the angles at the base, will aUo be known. 372 THE CONSTRUCTION OF PROBLEM LIX. The base, the difference of the angles at the base, and the line drawn ffom the vertical angle to bisect the base of any plane triangle, being given ; to describe the triangle. CONSTllUCTION. Upon AB, equal to the given base, let a segment of acircleAHEB bedescribed to contain anangle equal to the difference of the angles at the base ; bisect AB in C, and take CD to AC in the duplicate ratio of AC to the given bisecting line KL; make CS and Dl perpendicular to AB, cutting the circle in S and 1 ; draw AI^ cutting CS in G ; and, through G, draw the chord EGH paral- lel . Q. E. D. m THE COxNSTRUCTION OF Method of c a Icula Ho n. Since m h n -{- p : m : : AB : AO m ^- n ■\- p : 71 :: AC: AriQOJ, both AQ and QO will be given from thence; then, in the triangle AOQ, will be given two sides and the in- cluded angle, from which everything else will be known. PROBLEM LXIV. To divide a given trapezium A BCD, ichose opposite sides AB, CD are parallel, according to a given ratio, bij a right-line QN, passing through a given point P, and falling upon the two parallel sides, CONSTRUCTION. Bisect AD in G, and draw GH pa- rallel to AB (or DC) meeting BC in H ; then divide GH in M, accord- J\ ^ j^ i\ n j-jitio, and through P* M draw PQN, and the thing is done. DEMONSTRATION. Draw EMF and IHK parallel to AD, nieelini;- DC and AB in E, I. K and F. Because of the parallel lines, we have GD — ME = HI, and AG ~ FM = KH ; whence, as GD is — AG [by construction) ME will be =i FM^ and HI — HK; and the triangle EMN will be = FxMO. and IHC =: BHK {Euc. 4. 1.) whence it appears that the trapezium AQND is also equal to the parallelogram DF, and the trapezium QBCN equal to the parallelo- gram FI ; but these parallelograms are to one another as their bases, or as GM to MH (E?(c. 1. 0.) ; there- fore GM : MH : : AQND : QBCN. Q. £. D. Method of calvulation. Whereas A Band DC are parallel, GH i» an arithme- tical mean between them, and thcrpfore equ'd to half GEOMETRICAL rROULIi^MS. 379 their sum. Therefore, as the whole line GH and the ratio of its parts GM, MH are given, the parts them- selves will also be given. PROBLEM LXV. To cut offjrom a given trapezium ABCD, whose oppO" site sides AB^ CD, are parallel, a part AQND equal to a rectangle given, by a right-line passing through a given point P, and falling upon the two parallel sides. (See the figure to the last problem,) CONSTRUCTION. Bisect AD in G, and draw GH parallel to AB ; upon AD {by Euc, 43. i.) describe the parallelogram A n EF equai to the rectangle given, and through the intersection of GH and Eb' draw PQN, and the thing is done : The demonstration whereof is manifest from the preceding problem. PROBLEM LXVI. To divide a given trapezium ABCD, whose sides AB and DC are parallel, into two equal parts, by a right* line parallel to those sides. CONSTRUCTION. Produce AD and BC till they meet in H, and make AG equal, and perpendcular to HD ; draw HG ^ and bisect the same with the perpendicular PQ= HP; join H, Q, and in HA take HE e- qual to HQ, and G B C F parallel to AB draw EF, and the thing is done. DERtONSTRATION. Since HE^ ( = HQ^ = HP* + PQ^ = 2HP« zz HG« HA» + AG" HA* + HD\ . "^- = o = o ) ^8 an arith- 380 THE CONSTRUCTION OF metical-inean between HA^ and HD^, it is evident that the triangle HEF will also be an arithmetical-mean between the triangles HAB .apd HDC (or ABFE zz EFCD) ; because those triangles, being similar, are to one another as (HE% HA", HD^) the squares of their homologous sides. Q. E. D. Method of calculation. Since all the sides and angles of the trapezium are supposed given, the side CD and all the angles of the triangle HDC will be given ; there fore HP a nd AH _ ./ HD'4-HA^ willbe known ; whence HE, will also be given. But the same thing may be had without the angles; for since DC is parallel to AB, we have AB — DC : AD : : DC : HD ; whence HE will be given, as before. PROBLEM LXVII. To divide a given trapezium ABCD according to a ginen ratio, by a right-line LH cutting the opposite sides AC, BD in given angles. CONSTRUCTION. Produce the said opposite sides till they maet in E ; draw AD, and CF parallel to it, meeting BE in F ; divide BF in G, accord- ing? to the given ratio ; and, having made EAK equal to the given angle which LH is to make with AC, take EH a mean- proportional be- tween EG and EK ; then draw HL [)arallel to AK, and the thing is done. DExMONSTRATION. By construction, EG : EH : : EH : EK : : EL : EA {Euc. 5. 6.) whence it follows that EG a EA =: GEOMETRICAr. PROBLEMS. 3«l EH X EL, and consequently that the triangles EEIL and EAG are also equal to each other {Euc. 15. 6.) from which taking away EDC, common, the remain- ders CDHL and CDGA will be equal likewise, and consequently ALHB — AGB, being the diflerences between those remainders and ACDB, But the tri- angle ADF is =; ACD, standing upon the same base AD and between the same parallels; therefore (by adding AGD, common) AGF is also =z CDGA (== CDHL); but AGF (CDHL) : AGB (ALHB) : : GF :GB{Euc. I. 6). Q. E, D. Method of calculation. In the triangles ABE and ABK are given all the angles and the side AB, whence BE, BK, and EC will be known; then, in the triangle EFC will be given all the angles and the side CE, whence EF, and from thence FG and EG, will be known ; lastly, from the know n values o f EK, EG, and EF, the value of FH {= v/EG X EK — EF) will be found. PROBLEM LXVIII. Two ri^ht4ines AG and AH, meeting in a point A, being given by position : it is required to draw a right- line wP to cut thos& lines in given angles ^ so that the triangle AnV, formed from thence, may be equal to a given square ABCD. CONSTRUCTION. Let the angle ABE be equal to the given angle APfz and let BE meet AG in E; draw EF perpendicu- lar to AH, make BQ equal 2EF, and upon AQ describe the semi-circle AmQ, cutting BC in w; draw mn pa- rallel to AH, meeting AG in n, and nV parallel to EB, and AwP will be the triangle required. D C G, \ 772. ny^ k^ >^ \^ A F J B 5 •Q P H 382 THE CONSTRUCTJON OF DEMONSTRATION. The triangles AEB and AnP, being similar, are to one another as the squares of their perpendicular heights EF and mB («S) : but wB* is =r BQ x AB = 2EF X AB; therefore it will be, as the triangle AEB (EF X iAB) : the triangle An? : : EF« : sEF x AB : : EF : 2AB : : EF x ^AB : AB^ (Euc, 1. 6.) where- fore, the antecedents being the same, the consequents must necessarily be equal, that is, Aji? rz A BCD. Q. E. D. Method of calculation. In the triangle ABE are given all the angles and the side AB, whence EF will be given, and conse- quently Sn (= y'AB X 2EF); whence AP and An are also given. LEMMA. If from any point C, in one side of a plane an^le KAL, a right-line CB he drawn, cutting both sides AK, AL in equal angles (ACB, ABC); and from any other point D in the same side AK another right-line he drawn, to cut off an area ADE equal to the area ABC ; / say, that DE will he greater than CB. DEMONSTRATION. Complete the parallelogram DCBG, and join B, D, and in BG (produced if need be) take BF z= BE, and draw FD. Since the triangles ABC, AED are equal, by suppo- sition, and have one ^ angle. A, common, therefore will AD : AC : : AB (AC): AE [Euc. 15. 6.) and consequently AD -h AE greater than AC H- AB ( Euc, 25. 5. ) whence it is mani- fest that C\^ must be greater ihan EB, or BG than BF. Moreover, because the angle ABC GEOMETRICAL PROBLEMS. 383 (=: ACB = CDG) is = GBC, it will be greater thaa GBD, which is but a part of ORG ; and therefore^ ABD must, evidently, be greater than GBD ; where-; fore, seeing BF and BE are equal, and that DB is com- mon to both the triangles DBE, DBF, it is manifest that DE is greater than DF {Euc. ig. 1.) ; but DF is greater than DG (by the same) because the angle DGF (DCB), being obtuse, is greater than GFD, which must be acute {Euc. 32. 1.) : consequently DE is greater than DG, or its equal CB. O. E. D. PROBLEM LXIX. From a giveti polygon ABCDEF, to cut of a ginen area AFEIK hy the shortest right-line, KI, possible. CONSTRUCTION. Let the given area to be cut off be represented by the rectangle LMNO ; and let the sides AB and DE, by which the dividing line is terminated, be produced till they meet in G ; make upon OL [by Euc. 45. 1.) a rectangle OQ equal to AFEG, and let a square GSTV be constituted (by Euc. 14. 2.) equal to the whole rectangle QN: bisect the angle BGD by the right-line GH, and make GR perpendicular to GH $ and draw 384 THE CONSTRUCTION OF KI, by the last problem, parallel to RG, so as to form the triahffle KGI equal to the square GSTV, and the thing is done DEMONSTRATION. Since, by construction, KGI (z= GSTV) = QN, let AFEG zz OQ be taken away, and there will re- main AFEIK z= LN. Moreover, since the angle HGI is =z HGK, and the angle IHG (HGR) a right one, the angles I and K are equal ; and therefore, by the preceding lemma, IK is the shortest right-line that can possibly be drawn to cut off the same area. Q. E. D. Method of calculation. Let the area of the figure AFEG be found, by di- viding it into triangles AFG, EFG, and let this area be added to the given area to be cut off; and then, the square root of the sum being extracted, you will have GS the side of the square GT; from whence GI will be determined, as in the last problem. Note, In the same manner may a given area be cut off, by a right line making any given angles with the opposite sides. PROBLEM LXX. Through a given point P, to draio aright line PED to cut two right-lines AB, AC given by position, so that the triangle ADE, formed fi om thence^ may be of a given magnitude* CONSTRUCTION. Draw PFH parallel to AB, intersecting AC in F ; DB GEOMETRICAL PROBLEMS. 385 and upon AF let a parallelogram AFHI be constituted equal to the giveu area of the triangle ; make IK perpendicular to A I, and equal to FP ; and, from the point K, to AB, apply KD = PH ; then draw DPE, and the thing is done. , DEMONSTRATION. Supposing M to be the intersection of DE and IH,it is evident, because of the parallel lines that all the three triangles PHM, PFE, and MDI are equiangular ; therefore, all equiangular triangles being in proportion as the squares of their homologous sides, and the sum of the squares of PF (IK) and DI being equal to the square of PH (KDJ, hy construction and Euc, 47- 1. it is evident that the sum of the triangles PFE and DMI is — the triangle PHM ; to which equal quantities in fig, 1, let AFPMI be added, so shall APE be likewise equal to AFHI : but, in fig. 2, let PFE be taken from PHM, and there will remain EFHM = DMI; to which adding AIME, we have AFHI =: ADE, as before, Q. E. D. Method of calculation^ By dividing the given area by the given height of the point P above AB, the base AI of the parallelogram AFHI will be known, and consequently PH (nKD) ; whence DI (= v/K 02 — PF*) will likewise become known. — This problem, it may be observed, becomes impossible when KD (PH)is less than KI (PF) ; which can only happen, in case i, when the given area is less than a parallelogram under AF and FP. PROBLEM LXXI. To cut off from a given polygon BCIFGH, a part EDBHG equal to a given rectangle KL, hy a right-line ED passing through a given point P. CONSTRUCTION. Let the sides of the polygon CB and FG, which the dividing line ED falls upon, be produced till they meet in A; upon ML (^:>z/ (Eiic^b, \.) make the rectangle c c 3£6 THE COxXSTRUCTION OF MN equal to AGHB, and, by the last problem, let ED be so drawn through the given point P, that the triangle AED, formed from thence, may be equal to the whole r rectangle KN ; then will EDBHG be equal to KL : for since AED is z= KN, let the equal quantities AGHB and MN be take^ away, and there will remain EDBHG = KL. Method of calculation. Let the area of the figure AGHB be found, by di- viding it into triangles, and let this area be added to the area given, and the sum will be equal to the area AED, or the rectangle KN; from whence AD will be found, as in the last problem. i PROBLEM LXXII. Having the base^ the vertical angle and the length of the line bisecting that angle and tenninating in the base, to describe the triangle. CONSTRUCTION. Upon the given base AB let a segment of a circle ACB be described {by Problem 4.J to contain the given angle, and, having completed the whole circle, from 0,"the centre thereof, perpendicular to AB, let the ra- dius OE.be drawn ; draw EB, and make BG piipcn- GEOMETRICAL PROBLEMS. 387 dicular thereto, and equal to half the given bisecting line ; and from G, as a centre, with the radius GB, let a circle BHF be described, intersect- ing EG (when drawnj in F and H ; from E to AB draw ED - EF, and let the same be pro- duced to meet the cir- cumference in C ; join A, C, and B, C ; so shall ABC be the triangle re- quired. DEMOKSTRATION. The triangles CBE and BDE are similar, because the angle BEG is common to both, and the angles BCE and DBE stand upon equal arches BE and AE ; there- fore EC : EB : : KB : ED, and consequently ED x EC - EB" : but (67J Rue. 36. 3.) EB^ - EF x EH zz ED X EH [hy construction). Hence ED x EC - ED X EH, and consequently EC z: EH ; from which tak- ing away the equal quantities ED and EF, there remains DC ,c= FH — the given line bisecting the vertical angle {hy construction) : and it is evident that DC bisects the angle ACB, since ACD and BCD stand upon equal arches AE and EB. Q. E. D. Method of calculation. If BE be considered as a radius, BR (fAB) will be the co-mne of the angle EBR, and BG the tangent of BEG ; therefore BR : BG (or AB : DC] : : co-sin. EBR (ACE) : tang. BEG, whose half-complement EHB is likewise given from hence : then, the angle HBb (sup- posing EB produced to b) being the complement .of EHB, we shall have tang. EHB : rad. (: : sin. EHB : co-sin. EHB : : BE : EH : : EB : EC) : : sin. ECB : sin. CBE = sin. EDB =: co-sin. OED, half the dif- ference of the angles (ABC and BAC) at the base. (J c 2 388 THE CONSTRUCTION OF FROBEEM LXXIII. Having given the two opposite sides ah, cd, the two diagonals ac, bd, and also the angle aeh in which they in" tersect each other ; to describe the trapezium, CONSTRUCTION. In the indefinite line, BP, take BD equal to hd, and make the angle DBF equal to the given angle aeb, and BF z= flc ; also from the centres D and F, with the radii dc and ab, let two arches mCn and rCs be de- scribed intersecting each other in C ; join D, C and F, C, and make BA equal and parallel to FC ; then draw AD, AC, and BC, and the thing is done. DEMONSTRATION. Since (by construction) AB is equal and parallel to CF, therefore will AC be equal and parallel to BF {Euc. 53. 1.) and consequently the angle A EB(jEwc. 29. 1.) = DBF = aeb. Q. E. D. Method of calculation. Join D, F ; then in the triangle DBF will be given two sides DB, BF and the angle included, whence the angle BFD and the side DF will be known ; then in the triangle DFC will be given all the three sides, whence the angle DFC will be known, from which BFC (BFD— DFC) =z BAG will also bekno'^'u. GEOMETRICAL PROBLEMS. 389 PROBLEM LXXIV, HaDing given the tioo diagonals and all the angles, to describe the trapezium. CONSTRUCTION. Assume AB at pleasure ; and, having produced the same both ways, make the angles QAC, RBC equal, respectively, to two opposite angles a and e of the tra- pezium; moreover, make ACF equal to ace, one of the remaining angles ; and from F, the intersection of CF and BQ, take FG zz the given diagonal dc, and draw GH parallel to CB, meeting FB in H. Then from A and B {by the lem, p, 334.) let two lines AE and BE be drawn to meet in FC, so as to be in the given ratio of ac to FH : in AE take AN z= ae, and draw NM parallel to FC, meeting AC in M; lastly, draw NP making the angle MNP zz ced, and meeting FB in P ; so shall A MNP be the true figure required. DEMONSTRATION, Let ED be parallel to NP, and let DC and PM be drawn. It is evident, by construction, that the diagonal AN and all the angles of the trapezium, are equal to the respective given ones ; it therefore remains only to prove that PM is equal to the other given diagonal dc. Now, the angle RBC being z= CED (% constr,}, the cir- cumference of a circle may be described through all the four angular points of rhe trapei/um BCED ; and so the triangles FBE and FCD (as both the angles FBE and FCD stand upon the same chord ED) will be similar ; c c 3 3SK) THE CONSTRUCTION OF and consequently BE : DC (: : FB ; FC) : : FH : FG {dc). But [by construction) AE : BE : : a<; : FH ; there- fore, by compounding these two proportions, we have AE : DC \\ ae : dc ', but (because of the similar figures ADEC, APNM) we also have AE : DC :: AN (ae) : PM ; and consequently PM = dc. Q. E. D. Method of .calculation. All the angles of the triangles ABC, FAC, and FBC being given, we shall have sin. ACB x sin. F: sin. ABC X sin. ACF:;AB:AF; and sin. FHG (FBC) : sin. FGH (FCB) : : FG [dc) : FH ; whence AF and FH are known. „. , . T-^ AB X ae , ^^ AK x FH Fnid AK = rrT-f , and KO = -rrn ; i^H I- ae* l^H -- ae which last is equal to (OE) the radius of the circle de- termining the point E [see the aforesaid lemma). There- fore, in the triangle FOE are given two sides FO and OE, besides the angle F, whence the angle FOE will ]be given ; then in the triangle AGE will be given OA, OE aad the included angle; whence the angle 0/\E, which the diagonal i\N makes with the side AP, will be known, and from thence every thing else required. This problem, as the circle described from O cuts FC in two points, admits of two different solutions (except, only, when FC touches the circle). If the circle neither cuts nor touches that line, the problem will be impossible ; the limits of the ratio of AE to BE (and consequently of ae to dc) growing narrower and narrower, as A B becomes less and less, with respect to AC, or according as the sum of the opposite angles a ^ e — QAC + 1-lBC) approaches nearer and nearer (to two right-angles; so that, at last (supposing AC and BC to coincide) A E and BE will be, cver}/-where, in the ratio of equality; therefore cd can here have only one particular ratio to ae; and the diagonal ANE may be drawn at pleasure, the probleui being, in this case, in^ determinate. GEOMETRICAL PROBLEMS. 391 PROBLEM LXXT. Supposing the right-lines m, w, p, to represent the lengths of three staves erected perpendicular to the horizon, in the given points A, B, C; to .find a point P, in the plane of ihe horizon ABC, Cijually remote from the top of each staff, coksthuction. Join A, B and B, C, and make AE and BF per- pendicular toAB; also make BG and CH perpendi- cular to BC, and let AE be taken ~ m, CH =: p, and BF and BG each zz n\ draw EF and GH, which bisect /V yi.,.-./..,.. .\i '—'.^s;* Jl by the perpendiculars LN and IK, cutting AB and CB in N and K; make KP and NP perpendicular to BC and BA, and the intersection P of those perpendiculars Avill be the point required. DEMONSTRATION. Conceive the planes AEFB and BCHG to be turned up, so as to stand perpendicular to the plane of the ho- rizon ABC and intersect it in the right lines AB and BC; then, because BF and BG are equal to each other, and perpendicular to the plane of the horizon, it is ce 4 392 THE CONSTRUCTION OF evident that the points F and G must coincide, and that AE, BG (BF) and CH will represent the true position of the staves -.suppose KG, KH, PG, PH, PE, and PF to be now drawn; then, since (by construction) Gl = HI, and the angle GIK = HIK, therefore is GK =: HK. [Euc, 4. 1): moreover, since KP is (by con^ struction) perpendicular to BC, it will also be perpendi- cular to the plane BCHG, and consequently the angles PKG and PKH both riglit-angles : therefore, seeing the two triangles GKP, HKP have two sides and an included angle equal, the remaining sides PG and PH must likewise be equal [Euc. 4. l). After the very same manner it is proved that PF (or PG) is equal to EP. Q. E. D. Method of calculation. Draw Ir perpendicular, and Hq parallel to BC ; then, by reason of the similar triangles H9G and IrK, it will R f 4- C H be as BC(H9):BG— CH(G9) :: Z 0^"^) ^ BG — CH X BG V CH , . , , , , , : Kr — ; ll____. which subtracted 2BC ' from Br ( z= i BC) gives BK : and, in the same manner will BN be toiyid ; then in the trapezium KBNPwill be given all the angles and the two sides BK and BN; from whence the remaining sides, &c. may be easily determined. PROBLEM LXXVI. The base, the perpendicular and the difference of the sides being given, to determine the triangle, CONSTRUCTION. Bisect the base AB in C, and in it take CD a third proportional to 2 AB and the given difference of the sides MN; erect DE equal to the given perpendicular, and draw EK parallel to AB, and take therein EF - MN; draw EAG, to which, from F, apply FG - AB; draw AH parallel to FG meeting EK in H ; then draw BH, and the thini;- is done. GEOMErRICAL PROBLEMS. 393 DEMONSTRATION. By reason of the parallel lines, FG (AB) : FE (MN) : : AH : EH (DP; ; therefore AB x DP i= AH x MN, or2ABxDP=2AH ,^ ^ ^ ^ X MN ; to which last i^ -^ -H iLi ^ equal quantities adding 2ABxCD =MN2 fbij constructionj we have 2AB X CP = 2AH X MN f MN^- ; but 2AB X CP is := BH^ — AH* [by a known property of triangles) ; therefore BH^ — AH^ = 2AHxMN + MN-, or BH^ = AH^^ + 2 AH X MN -f MN* = AHTmnI* [Euc. 4. 2.); consequently BH z= AH f MN. Q. E. D. Method of calculation. In the right-angled triangle ADE we have DE and MN* \ AD(=iAB - ^^^ 1, whence the angle DAE (PEG) will be found; then in the triangle EFG will be given two sides and one angle, from which the an* gle GFK ( - BAHj will also be known. PROBLEM LXXVII. The base, the perpendicular^ and the sum of the two sides being given, to describe the triangle. CONSTRUCTION, Bisect the base AB in C, and in it produced take CD a third proportional to 2AB and the sum of the sides, MN; erect DE equal to the given perpendicular, and draw HE parallel to j\B, and take therein EF zi MX; draw EAG, to which from F, apply FG ~ AB, draw AH parallel to FG, meeting EF in H, then draw BH, and the thiiig is done. • S94 THE CONSTRUCTION OF DEMONSTRATION. Because of the parallel lines, FG (AB):FE(MN) ::AH : EH (DP); and therefore 2MN x AH - 2AB X DP; wliich equal quantities being subtracted from MN;" = 2AB x CD (% construction) there will M~ .. remain MN* — 2MN x AH ir sAB x CP =: BH^ -^ AH" ; whence, by adding AH"" to each, we have MN^ — 2 MN X AH + AH^ = BH-, that is, MN ~AH\- = BH- ; therefore MN — AH - BH, orMN =; BH + AH. Q. E. D. Method of calculation. In the triangle AED are given (besides the right- angle) both the legs, whence the angle DAE [— FIlG) will be given; then in the triangle FEG one angle and two sides will be known, from which the angle EFG ( zz BAH) will be determined. PROBLEM LXXVIJI. The differcJice of the two sides, the perpendicular, and the vertical angle being given, to determine the triangle, CONSTRUCT ION. Upon the indefinite line FEQ erect the given per- pendieufar DC, making the angle DCE n: half the given angle; let EF, expressing the given difference of the sides, be bisected by the perpendicular GI, meeting- EC in I ; also let liC be biisected in H, and make EK perpendicular to CE, and equal to Eli and having GEOMETRICAL PROBLEMS. 395 drawn HK take HL, in HE produced, equal ttiereto; from L to FQ apply LB - £K, and join C, B ; also draw EM, making the angle CEM equal to DEC, and cutting CB in M; then from C to QEF apply CA :r CM, so shall A.CB be the triangle required. DEMONSTRATION. Upon EM let fall the perpendicular CN, and join L, M and F, L Now LB" = EK^' {by construe* tion) ri HK + HE X HK - HE [Euc, 5. 2,) =z HlTCH X HL — HE(Z)i/ construction) = CL x EL ; whence CL : LB : : LB : EL; therefore the triangles CLB and ELB must be equiangular {Euc, 6. 6.) and consequently LBM - LEB - CED = CEM [hy con- struction). Therefore, since the external angle CEM of the trapezium LEMB, is equal to the opposite inter- nal angle B, the circumference of a circle will pass through all the four angular points; and consequently the angle LMB will be = LEB, both standing upon the jsame chord LB; but it is proved that LBM is = LEB, Ltherefore LMB —LBM zz FET; and so the triangles BLMand EIF, being isosceles, and having LMB =:EFI, and also LB — EA [by construction), they will be equal in all respects, and consequently BM — EF; whence BC — AC ( - BC — CM :^ BMj - EF, the givei) 396 THE CONSTKUCTION OF difference {by construction). Moreover, CEN being — CED [by construction), CN will be = CD; and so, CM being zz CA, ACD will be - MCN, to which add- ing DCM, common, we have ACB — DCN - 2DCE. Method of calculation. Seeing EG and EH are the sine and tangent of EIG and EKH, to the eq»uil radii EI and EK, it ■will therefore be EG : EH (or EF : EC^ :: sin. EIG (ECD):tang. EKH. But, EC : CD :: the radius : co-sin. EC D; whence, by compounding these pro- portions, EF : CD : : rad. x sin. ECD ; co-sin. ECD T7trtj rad. X sin. ECD , ^ i-r>T^N X tang. EKH : : . — ^„^ (= tangent LCD) : * co-sm. ECD ^ ^ ^ tang. EKH; from which EKL, half the complement of EKH will be also given : then it will be as the ra- dius : tang. EKL (: : KE : EL : : LB : EL) : : sin. LEB(CEp) : sin. LBE (3CE); which proportions, expressed in words, give the following Theorem. As the difference of the sides is to the perpendicular, so is the tangent of half the vertical angle to the tangent of an angle; and as the radius is to the tangent of half the complement of this angle, so is the co-sine of half the vertical angle to the sine of half the difference of the an- gles at the base. rnOBLEM LXXIX, The perpendicular, the difference of the sides, and the difference of the angles at the base being given, to deters mine the triangle, CONSTRUCTION. Q Let a triangle ABC be constructed, by the last prohlon, whose per- pendicular and diffe- rence of the sides shall be the same with those given, and whereof the vertical angle ACB is also equal to the given diflbrence of angles : GEOMETRICAL PROBLEMS. 397 then upon C, as a centre, with the radius CB let an arc be described, intersecting AB, produced, in D; join C, D, and ACD will be the triangle required. For CD being - CB, the angle CDB will also be - CBD = A + BCA {Etic, 32. i). The method of calcula. tion is also the same as in the preceding problem. PROBLEM LXXX. The perpendicular, the sum of the ttvo sides^ and the vertical angle, being given ; to describe the triangle, CONSTRUCTION. Upon AB, the given sum of the two sides, erect AC equal to the given perpendicular ; and make the angle ACD equal to the complement of half the given an- gle : upon AB {by Prob, 72.) let a triangle ABF be iF constituted, whose vertical angle AFB shall be equal to the given one, and whereof the bisecting line Fli (terminating in the base) shall be = DC ; then draw CG and CH parallel to FB and FA, so shall GCH be the triangle required. DEMONSTRATION, It is evident that the angle FICG is = AFB = the given one. Moreover, if EM and EN be taken as perpendiculars to AF and BF, they will be equal to each other, and also equal to the given one AC, because all the angles EFN, EFM, and ADC are equal, by construction, and EF is likewise — CD ; whence, as the angles AHC, AGC are i>espectively equal to EAM, 398 THE CGNSmUCTION OF EBN, it is evident that HC = EA, and GC - EB^ and consequentU^, that HC + GC ( = EA + EB) - AB. Q. E. D. Method of calculation. By the problem above referred to, AB : CD (EF) :: co-sin. ADC (AFE) : tang, of an angle; which let be denoted by Q. Now CD : CA : : rad. : sin. ADC ; which propor- tion being compounded with the former, we have AB : CA : : co-sin. ADC x rad. : tang. Qx sin. ADC co-sin. ADC x rad, . ^ ^ at^/^n . r^ • • sin. ADC (cotangent ADC) : tang. Q. Then, by the same problem, it will be as tang, -f Q. : rad. : : sin. ADC : co-sin. of the difference of the an- gles (G and H) at the base. The ahove proportions, given in words at length, exhibit the following Theorem. As the sum of the sides is to the perpendicular, so is the cotangent of half the vertical angle to the tangent of an angle ; and, as the tangent of half this angle is to the radius, so is the sine of half the vertical angle to the co- sineof half the difference of the angles at the base. PROBLEM LXXXI. To constitute a trapezium of a given magnitude under four given lines. CONSTRUCTION. Make a right- angle h with two of the given lines A6, he ; and with the other two complete the tra* pezium AbcD : upon AD let fall the perpendicular cE, in which pro- duced (if neces- sary) take EF, so that the rectangle GEOMETRICAL PROBLEMS. 399 under it, and AD, may be double the given area: moreover, take a fourth-proportional to AD, A 6, and be, with which, from the centre F, let an arch be de- scribed, meeting another arch, described from D with the radius Dc, in C ; join D, C ; and from A and C draw the other two given lines AB, CB so as to meet; and they will thereby form the trapezium ABCD^ as re- quired. DEMONSTRATION. Draw Ac, AC, and FC ; upon AD and AB let fall the perpendiculars CP, CQ; and make FG perpendi- cular to PCG. Because AD'^ + DC- + 2AD x DP {- KC\ Euc. 12. 2.) iz: AB- -f BC- + 2AB X BQ, and AD^ 4- Dc^ -f 2AD X DE ( = Ac-) 1= A6^t+ he'' {Euc. 47. l.j it follows, by taking these last equal quantities from the former, that 2AD x DP — 2AD x DE (2 AD X EP)— 2AB X BQ, and consequentlv^ that BQ : EP (FG) :: AD : AB : : BC : FC (% construction) whence the triangles BCQ, FCG are similar, and so CO : CG : : BC : FC : ; AD : AB [by construction) and therefore CQ x AB zz CG x AD; hence, by adding CP x AD to each, we have CP x AD 4- CQ X AB ( ;z twice the area ABCD) = CP x AD -f- CG X AD rz: EF X AD — twice the given area {by constructiun). Q. E. D. Method of calculation. From DE (=: — ■ ^^ J and EF (— -T-y-) the value of DP, and likewise that of the angle ADF, will be found: then, all the sides of the triangle DCF being known, the angle FDC will like- wise be known •; which, added to ADF, gives (ADC) one of the angles of the trapezium. It may so happen that a trapezium, having one right-angle, cannot be constituted under the four given lines; in which case it will be necessary (instead of forming the trapezium AbcD) to lay down AD fust, and in it (produced if needfiil) lo 400 THE CONSTRUCTION, &c. take DE equal to ill±_m-m-0^, that is, equal to the altitude of a rectangle, formed on the base 2AD, whereof the contained area is equal to the difference of ABI^ + BCl^ andTD,"- 4- 'DCl' (which line DE is to be set ofl'on the other side of D, when the latter of these two quantities is the greater) : this being done, the rest of the solution will remain the same, as is manifest from the first and second steps ot the demonstration ; the process, from thence to the end, being no-ways different. It may be further observed that the problem itself becomes impossible, when the two circles, described from the centres D and F, neither cut nor touch ; the greatest limit of the area, and consequently of EF, being when they touch each other; in which case, the sum of the radii DC, FC becoming = DF, the point C will fall in the line DF, and the angle DCF will become equal to two right-angles : but the sum of the opposite, external angles CDP and CBQ is always equal to DCF ; because CDP (supposing C» parallel to AP) is =: DCw, and CBQ ( - CFG) = FCn : hence it is evident that the limit, or the greatest area, will be when the sum of the opposite angles is equal to two right-angles or when the trapezium may be in- scribed in a circle. FINIS. \V. Glendinn'mg, Prinier, 2j, Hation Garden, London. w ^^^- ER ^TY OF r\LJ^OX?N THE UNIVERSITY OF CALIFORNIA LIBRARY