3O 
 
 
 iff 
 
 s
 
 ^njizini i
 
 THE 
 
 DOCTRINE 
 
 AND 
 
 FLUXIONS. 
 
 CONTAINING 
 
 (Befides what is common on the Subjeft) 
 
 A Number of NEW IMPROVEMENTS 
 in the THEORY. 
 
 AND 
 
 The SOLUTION of a Variety of New, and Very 
 Interefting, Problems in different Branches of 
 the MATHEMATICKS. 
 
 PART I. , 
 By THOMAS SIMPSON, F. R. S. 
 
 THE SECOND EDITION. 
 
 Revifed and carefully corre&ed. 
 
 LONDON: 
 
 Printed for JOHN NOURSE, in the Strand, 
 BOOKSELLER TO His MAJESTY. 
 
 MDCCLXXVI.
 
 LENDINGJ JBRARi 
 
 , :
 
 StacK 
 Annex 
 
 :T iii 
 
 105" 
 
 111 
 
 TO THE 
 
 RIGHT HONOURABLE 
 
 George Earl of MaccksfaU. 
 
 MY LORD, 
 
 AS I efleem it a very great Honour to 
 be permitted to place the following 
 Sheets under your Lordfhip's Pro- 
 tection, who are not only an Encourager of, 
 but an Ornament to, Mathematical Learn- 
 ing; I have taken more than ordinary Pains, 
 that, What is here ufhered into the World* 
 with fuch Advantage, may not be found al- 
 together unworthy of ib distinguished a 
 Patron. 
 
 I am not vain enough to imagine, that, to 
 One fo deeply read in thefe abftrufe and cu- 
 rious Speculations, as your Lordmip is uni- 
 A 2 verfally
 
 ir DEDIC^flON. 
 
 verfally allowed to be, this Work will appear 
 without Faults : But then, I have the Satif- 
 fa&ion to think, on the other hand, that, 
 whatever is Here to be met with capable of 
 bearing the Teft of an exact and folid Judg- 
 ment, will a/ft have its due Weight, and not 
 fail of receiving your Lordmip's Approba- 
 tion: And if, upon the Whole, there is Merit 
 enough found to entitle me to a favourable 
 Reception, it will gratify the higheft Am- 
 bition of, 
 
 MY LORD, 
 Tour LORDSHIP'S 
 Moft Obedient Humble Servant, 
 
 Tho. Simpfon,
 
 PREFACE. 
 
 HAVING, in the Year 1737, publiflicd 
 a Piece, on this fame Subject, under the 
 Title of A <Treatife of Fluxions (whereof 
 the whole Imprefiion hath been long fmce fold) 
 it may be proper here, firft of all, to affign the 
 Reafons why this Work is fent abroad into the 
 World as a New Book, rather than a Second 
 Edition of the faid Treatife. Which, in fhort, 
 are thefe two : Firft, becaufe the prelent Work 
 is vaftly more full and comprehenfive ; and, fe- 
 condly, becaufe the principal Matters in it which 
 are alfo to be met with in that Treatife, are 
 handled in a different Manner. 
 
 BESIDES the Prefs-Errors with which the 
 faid Treatife abounds, there are feveral Obfcu- 
 rities and Defects (which the Author's Want of 
 Experience, and the many Difadvantages he then 
 laboured under, in his firft Sally, may, it is 
 hoped, in fome meafurc excufe.) But what is 
 A 3 now
 
 PREFACE. 
 
 now offered to the Publick, being a Performance 
 of more mature Confidera.tion and Judgment, 
 it will, I flatter myfelf, be found much more 
 correct, and claim a favourable Reception , ef- 
 pecially, as particular Care and Pains have been 
 taken to put every Thing in a clear L,ight, and 
 to oblige the lower, as well as the more expe- 
 rienced, Clafs of Readers. 
 
 THE Notion and Explication Here given of 
 the firft Principles of Fluxions, are not effen- 
 tially different from what they are in the above- 
 mentioned Treatife, tho' expreffed in other Terms. 
 The Confideracion of Time, which I have in- 
 troduced into the General Definition, will, per- 
 haps, be difliked by Thofe who would have Flux- 
 ions to be nicer Velocities : But the Advantage of 
 confidering them otberwife (not as the Velocities 
 Themfelves, but the Magnitudes They would, 
 uniformly, generate in a given finite Time) ap- 
 pear to me fufficient to obviate any Objection on 
 that Head. 
 
 B y taking Fluxions as meer Velocities, the 
 Imagination is confined, as it were, to a Point, 
 and, without proper Care, infenfibly involved in 
 metaphyfical Difficulties : But according to our 
 Method of conceiving and explaining the Mat- 
 ter, lefs Caution in the Learner is neceflary, and 
 the higher Orders of Fluxions are rendered much 
 
 more eafy and intelligible Befides, tho' Sir 
 
 6 Ifaac
 
 PREFACE. ^ vii 
 
 Jfanc Newton defines Fluxions to he tie Velocities 
 of Motions, yet He hath Recourfe to the Incre- 
 ments, or Moments, generatca in equal Particles 
 of Time, in order to determine thofe Velocities ; 
 which he afterwards teaches us to expound by 
 finite Magnitudes of other Kinds : Without 
 which (as is already hinted above) we could have 
 but very obfcure Ideas of the higher Orders of 
 Fluxions : For if Motion in (or at) a Point be 
 fo difficult to conceive, that, Some have, even, 
 gone fo far as to difpute the very Exiilence cf 
 Motion, how much more perplexing muft it be 
 to form a Conception, not only, of the Velocity 
 of a Motion, but allb in infinite Changes and Af- 
 fections of //, in one and the fame Point, where 
 all the Orders of Fluxions are to be confidsred ? 
 
 SEEING the Notion of a Fluxion, according 
 to our Manner of defining It, fuppofes an uni- 
 form Motion, it may, perhaps, feem a Matter 
 of Difficulty, at firft View, how the Fluxions 
 of Quantities, generated by Means of accelerated 
 and retarded Motions, can be rightly afilgned ; 
 fmce not any, the leaft, Time can be taken during 
 which the generating Celerity continues the fame : 
 Kere, indeed, we cannot exprefs the Fluxion by 
 any Increment or Space, actually ^ generated in a 
 given Time (as in uniform Motions.) Bur, 
 then, we can eafily determine, what the contem- 
 porary Increment, or generated Space i-could be^ 
 if the Acceleration, or Retardation, was to ceafe 
 
 at
 
 viii PREFACE. 
 
 at the propofed Pofition in which the Fluxion is to 
 be found : Whence the true Fluxion, itfelf, will be 
 obtained, without the Affiftance of infinitely fmall 
 Quantities, or any metaphyfical Confiderations. 
 
 THUS, for Example, the Motion of a Ball, de- 
 fcending by the Force of its own Gravity, is con- 
 tinually accelerated , but to have the Fluxion of the 
 Diftance fail'n thro* at any given Pofition of the 
 Ball, we muft find how far the Ball would^ uniform- 
 ly, defcend, from that Point, in a given Time, if the 
 Gravity, or the Earth's Attraction, from thence, 
 was toceafe afting. By which Means we fhall have as 
 clear an Idea of the Fluxion and the trueMeafure of 
 the Velocity of the Ball, at any Point afilgned, as in 
 thofe Cafes where the Motion is, a finally ^ uniform. 
 
 AGAIN, if a Right-line be fuppofed to move 
 parallel to itfelf with an equable Motion, and to 
 increafe in Length, at the fame Time , the Area 
 generated thereby, will increafe with an accele- 
 rated Velocity : But the Fluxion thereof, at any- 
 given Pofition of the Line, will be had by taking 
 that Part of the Increment which would, uni- 
 formly; arife, was the Length (as well as the 
 Velocity) of the Line to continue invariable from 
 the propofed Pofition. For, if the Length be 
 fuppofed to increafe, from the faid Pofition, the 
 Area generated, from thence, will be, evidently, 
 greater than That which would uniformly arife 
 in the fame Time ; fmce the new Parts, produced 
 10 each
 
 PREFACE. xx 
 
 each fucceeding Moment, are greater and greater- 
 Therefore the Fluxion muft be lefs than any Space 
 that can be defcribed, in the given Time, when 
 the Line increafes. And, in the lame Manner, 
 the Fluxion will appear to be greater than any 
 Space that can be defcribed, in the fame Time, 
 when the Line decreafes. It muft, therefore, 
 be equal to that Space, which will arife, when 
 the Length of the generating Line, from the given 
 Pofition, is fuppofed neither to increafe nor de- 
 creafe : Agreeable to Art. 4. 
 
 THUS much it feem'd proper to offer Here 
 with regard to the Firft Principles I fhall now 
 proceed to fay fomething concerning the Order 
 obferv'd in treating, and putting together, the 
 feveral Parts of the Work ; wherein the Eafe 
 and Benefit of the younger Beginner have been par- 
 ticularly confulted : To load fuch an One with a 
 Multitude of Rules and Precepts, before giving 
 him any Talte of their Uie and Application, 
 would, certainly, be very difcouraging -, and like 
 obliging a Traveller to afcend an high Mountain, 
 without allowing him to flop by the Way, to take 
 Breath, and refrefh his Spirits with a Profpect of 
 the agreeable and extenfive View he has to expect 
 when he arrives at the Summit : I have there- 
 fore, after demonftrating the Firft Principles, 
 proceeded immediately to exemplify their Utility 
 in fevernl - tertaining Enquiries, before touching 
 he Inverfe Method, or the more dif- 
 ficult
 
 PREFACE. 
 
 ficult Parts of the Direct. And, fmce that Branch 
 of-the Inverfe Method which treats of the Com- 
 parifon of Fluents is, naturally, Ibmewhat difficult, 
 it is referred to the Second Part of the Work, to- 
 gether with fuch other Matters in the Theory as 
 might appear, either, too tedious or hard to a 
 Learner at firft fetting out. The like Care has 
 been taken in the Difpofal of the reft of the 
 
 Work As to the feveral Particulars whereof 
 
 // is compofed, I muft refer to the Book itfelf, 
 They being too many to be here enumerated : 
 One Thing, however, I muft not omit to take 
 notice of, relating to that Part which treats of 
 the aforefaid. Bufinefs of Fluents: To which it 
 may, perhaps, be objected, That, notwithftand- 
 ing my having infifted fo largely on the Subject, 
 there are a Number of Forms of Fluxions 
 and Fluents to be met with in Authors, that I 
 have not fo much as touch'd upon. This is 
 granted ; but then they are moft of them fuch 
 as, I dare pronounce, can never arife in any In- 
 quiry into Nature : And it would, doubtlefs, be 
 Time and Labour miiapply'd, to fwell the Work, 
 and embarrafs the Learner with a Number of un- 
 necefiary Difficulties, and empty Speculations 
 when what is, really, proper and ufeful, in the 
 Subjefl, is fufficient (it is well known) to exer- 
 cife his utmoft Attention and Jvefolution. 
 
 I CANNOT put an End to this Preface without 
 acknowledging my Obligations to a fmall Tract, 
 
 in-
 
 PREFACE. 
 
 intitled, An Explanation of Fluxions in a Short Effay 
 en the Theory, printed for W. Innys : Wrote by a 
 worthy Friend of mine (who was too modeft to 
 put his Name to that, his firft, Attempt) whofe 
 Manner of determining the Fluxion of a Re&angle, 
 and illuftrating the higher Order of Fluxions, I 
 have, in particular, follow'd, with little or no 
 Variation. 
 
 XI
 
 The following BOOKS are all written by Mr. 
 ^Tbcmas- Simpfo;: y F. R. S. and printed for 
 J. Ncurfe. 
 
 I. rir^HE Elements of Geometry ; with their Ap- 
 plication to the Men-fura-ticm of Superficies 
 and Solids, to tbe Determination of the Maxima and 
 Minima of Geometrical Quantities, and to the Con- 
 ftruction of a great Variety of Geometrical Prob- 
 lems, 8vo. tbe thiid Edition, 5*. 
 
 2. Trigonometry Plain and Spherical, with the Con- 
 
 ftruction and Application to Logarithms, the Second 
 Edition, 8vo, is. bd. 
 
 3. A Treatife of Algebra j wherein the fundamental 
 
 Principles are fully and clearly demonftrated and 
 applied to the Solution of a great Variety of Prob- 
 lems, and to a Number of other ufeful Enquiries, 
 the Fourth Edition, 8vo. 6s. 
 
 4. Select Exercifes for young Proficients in the Mathe- 
 
 matics, 8vo, 51. 6,d. 
 
 5. The Doctrine of Annuities and Reverfions, deduced 
 
 from General and Evident Principles : with ufeful 
 Tables, fhewing the Values of fingle and joint 
 Lives, &c. at different Rates of Intereft, the Second 
 Edition with the Appendix, 8vo, 3*. 
 
 6. Effays on feveral curious and ufeful Subjects in fpe- 
 
 culative and mixed Mathematics ; in which the moft 
 difficult Problems of the Firft and Second Books 
 of Newton's Principia are explained, 410, 6s. 
 
 7. Mathematical Diflertations on a Variety of Phyfical 
 
 and Analytical Subjects, 410. js. 
 
 8. Mifcellaneous Tracts on fome curious and very in- 
 
 terefting Subjects in Mechanics, Phyfical Aftro- 
 nomy, and Speculative Mathematics, 410. js. 
 
 THE
 
 THE 
 
 DOCTRINE and APPLICATION 
 o F 
 
 FLUXIONS. 
 
 PART the Firft. 
 
 SECTION I. 
 
 Of the Nature, and Invejiigatioriy of 
 Fluxions. 
 
 i.'^'N order to form a proper Idea of the Nature of 
 
 Fluxions, all Kinds of Magnitudes are to be 
 
 - confidered as generated by the continual Motion 
 
 u of fome of their Bounds or Extremes ; as a 
 
 -^- Line by the Motion of a Point ; a Surface by 
 
 the Motion of a Line j and a Solid by the Motion of a 
 
 Surface. 
 
 2. Every Quantity fo generated is called a variable, or 
 flowing Quantity : And the Magnitude by which any 
 flowing Quantity WOULD BE uniformly hicreajed in a 
 given Portion of 'Time, with the generating Celerity at any 
 propofed Pofetion^ or Inftant (was it from thence to con- 
 tinue invariable) is the Fluxion of the faid Quantity at 
 that Portion, or Injiant. 
 
 B Thus,
 
 71 
 
 777 
 
 The Nature and Invef.igation 
 
 Thus, let the Points be conceived to move from A, 
 
 and generate the 
 
 m m r variable Right- 
 
 A l line Am, by a 
 
 ** Motion any how 
 
 regulated j and 
 
 let the Celerity thereof, when it arrives at anv propofed 
 Pofition R, be fuch as would, was it to continue uni- 
 form from that Point, be fufficicnt to defcribe the Dif- 
 tance, or Line Rr, in the given Time allotted for the 
 Fluxion : Then will Rr be the Fluxion of the variable 
 'Line Am, in that Pofition. 
 
 3. The P'luxion of a plane Surface is conceived in 
 
 like Manner, 
 S Q by fuppofing a 
 
 given Right- 
 line mn to 
 move parallel 
 
 J * to itfelf, in 
 
 A R r the Plane of 
 
 the parallel, 
 
 and immoveable Lines AF and EG : For, if (as above) 
 Rr be taken to cxprefs the Fluxion of the Line Am, 
 and the Rectangle RrsS be completed ; then that Rect- 
 angle, being the Space which wculd be uniformly de- 
 fcribed by the generating Line mn, in the Time that 
 Am would be uniformly increafed by m\, is therefore 
 the Fluxion of the generated Rectangle Bw, in that 
 Pofition, according to the true Meaning of the Defi- 
 nition. 
 
 4. If the Length of the generating Line mn con- 
 tinually varies, the Fluxion of the Area will Jl':ll be 
 expounded by a Rectangle under that Line and the 
 Fluxion of the Abfciilu, or Bafe : For let the cur- 
 vilineal Space Arnn be generated by the continual, and 
 parallel, Motion of the (now) variable Line mn, and 
 let Rr be the Fluxion of the Bafe, or Abfcifla, Am (as 
 before) ; then the Rectangle RrsS will, here allb, be the 
 Fluxion of the generated ISpace \mn : Becaufe, if the 
 Length and Velocity of the generating Line mn were 
 
 to
 
 A 
 
 R r 
 
 of FLUXIONS. 
 
 to continue invari- 
 able from the Pofi- 
 tion RS, the Red- 
 angle RnS would 
 then be uniformly 
 generated, with the 
 very Celerity where- 
 with it begins to be 
 generated, or with 
 which the Space 
 Amn is increafed in 
 that Pofition. 
 
 5. From what has been hitherto faid it will appear, 
 that the Fluxions cf Quantities art, a/ways, as the 
 Celerities by which the quantities themfelves increase in 
 Magnitude: Whence it will not be difficult to form a 
 Notion of the Fluxions of Quantities otherwife generated ; 
 as well fuch as arife from the Revolution of Right-lines 
 and Planes, as thofe by parallel Motion : But of this here- 
 after. I come now to fliew the Manner of determin- 
 ing the Fluxions of algebraic Quantities ; by which all 
 others, of what Kind foever, are explicable. But firft 
 of all it will be requifite to premife the following Ob- 
 fervations. 
 
 I. 'That the final Letters u, w, x, y, z of the Alpha- 
 bet are commonly put for variable Quantities ; and the ini- 
 tial Letters a, b, c, d, &c. for invariable ones: Thus 
 The Diameter of a given Circle may be denoted by a, 
 and the Sine of any Arch thereof (confidered as varia- 
 ble) by x. 
 
 II. Tnat the Fluxion of a Quantity reprefented by a 
 /ingle Letter, is ujually exprejjed by the fame Letter with 
 
 a Dot or Full-point over it : Thus the Fluxion of x is 
 reprefented by x, and that of y by y. 
 
 III. That the Fluxion of a Quantity which decreafes, 
 injttad of incrcafing, is to be confidered as negative. 
 
 B 2 
 
 PRO-
 
 Ihc Nature and Invejligation 
 
 PROPOSITION I. 
 
 $. 'the Fluxion of a Quantity being given, it is prepofeJ 
 to find the Fluxion of any Power of that Quantity. 
 
 As a clear undemanding of this Problem will be of 
 great Importance throughout the whole Work, it may 
 not be improper to confider it firft in one or two of its 
 moft fimple Cafes. 
 
 Cafe i. Let x exprefs the Fluxion of x, (according 
 to the foregoing Notation) and let the Fluxion of A" 
 be required. 
 
 Conceive two Points m and n to proceed, at the lame 
 time, from two other Points A and C, along the 
 Right-lines AB and CD, in fuch fort, that the Mea- 
 fure of the Diftance CS (y) t defcribed by the latter, 
 may be, always, equal to the Square of that AR (*) 3 
 defcribed by the former moving uniformly. 
 
 A 
 
 
 '' R 
 
 
 B 
 
 
 
 mym f 
 
 S 
 
 
 - - 
 
 x 
 
 1 H 
 
 T\, 
 
 , 7 
 
 a/ 
 
 i 
 
 
 I) 
 
 Furthermore, let r, i, and R, S, be any contem- 
 porary Pofitions of the generating Points, and let the 
 Lines x and y reprefent the refptclive Diftances that 
 would be uniformly defcribed, in trie fame time, with 
 the Celerities of thofe Points at R and S, then thole 
 Lines will exprefs the Fluxions of Am and Cn in this 
 Pofition, (by the Definition^ Art. 2 and 5). 
 
 Moreover, fince C s =. A r* and C S r= A R ? (by 
 Hypothecs), if Rr be denoted by */, we (hall have CS 
 (y) = *% and C s (rr x v] z ) x 1 2xv -f v* y and 
 confequertly S; ( = CS C*) = 2xv -y* ; from 
 whence we gather, that, while the Point m moves over 
 the Diftance v, the Point moves over the Diftance 
 
 2XV
 
 ^FLUXIONS. 
 
 v*. But this laft Diftance (fince the Square of 
 any Quantity is known to incieafe fafter, in Propor- 
 tion, tlian the Root) is not defcribed with an uniform 
 Motion (like the former), but an accelerated one j and 
 therefore is equal to, and may be taken to exprefs, the 
 uniform Space that might be defcribed with the mean 
 Celerity at fome intermediate Point e, in the fame time. 
 Therefore, feeing the Diftances that might be defcribed, 
 in equal times, with the uniform Celerity of m, and 
 the mean Celerity at <, are to each other as v to 2xv 
 v 1 , or as I to ^x v, or, laftly, as x to 2xx vx 9 
 (all which are in the fame Proportion) it is evident, 
 that, in the time the Point m would move uniformly 
 over the Diftance AT, the other Point /;, with its Cele- 
 rity at <>, would move uniformly over the Diftance 2xx 
 vx. This being the Cafe, let r, R, and j, S, be 
 now fuppofed to coincide, by the Arrival of the gene- 
 rating Points at R and S, then e (being always between 
 s and S, will likewife coincide with S; and the Diftance, 
 2xx XX, which might be uniformly defcribed in the 
 aforefaid time, with the Velocity at e, (now at S), will 
 become barely equal to 2xx j which (by the Defin.) is 
 equal to (j), the true FJuxion of Cn or ** a . 
 >. 
 
 * It may, perhaps, feem inaccurate, that the Fluxion) of* 
 and AT* are compared together, and exprejjed both by Lines, 
 auben tbt flowing Quantities themjelves, considered as a Right 
 Line and a Square, admit of no Comparifon, 7 bis Objection 
 would, indeed, be offeree, 'were the Expre^/ions retrained t a 
 geometrical Signification ; but here our Notions are more ab- 
 jlratted and univerfal, not obliging us to regard what Kind of 
 Extenfton, may be defined by this or that Exprejfion, but only 
 the Values of the algebraic Quantities thereby Jtgnified ; to 
 which the Meafures of all other Quantities whatever art ulti- 
 mately referred And, though Quantities of different Kinds 
 
 cannot be compared with each other, their Meafures, in Num- 
 bers, may. Thus, for Injiance, though it would be wrong 
 
 to affirm, that a Square whofe Area is 9 Inches is equal to a 
 Line off) Inches long, yet it is no Impropriety at all to fay the 
 Numbers exprefling their Meafures, in Inches, are equal. 
 
 B 3 7-
 
 Nature and Invcftigation 
 
 7. Cafe 2'. Let the Fluxion of .v 3 be required. 
 Suppofe every Thin^ to remain as in the preceding 
 
 Cafe ; only let Cn be here equal to the Cube of Am 
 (inftead of the Square). 
 
 Then, in the very fame manner, we have S* ( CS 
 
 Cj~ x 3 x -v\ l } = 3x*v $xv i -\- r J* : From whence 
 it appears, that the Diftances which might be defcribed, 
 in the fame time, with the uniform Celerity of w, and 
 the mean Celerity at e y will, in this Cafe, be to each 
 other as v to 3^1; yxv* + v 3 , or as x to 3* a * 
 %xvx-\-v 1 'x : Which laft Expreflion, when j, ^, and S 
 coincide (as before) will become y z x y the true Fluxion 
 of x 3 required. 
 
 8. Univfr/ally. Let Cn be, always* equal to Am K > 
 alfo let x t\" (or x v raifed to the Power whofe Ex- 
 ponent is n] be reprefented by x" ax" v + bx if 
 
 ex' '""v 3 , &. and let every Thing elfe be fuppofed 
 as above. 
 
 T"U r c f n i u } " l L n ~' L i 
 
 rhcn,imcebj \x x il / is ax v bx v 
 
 + ex" v 3 , &c. it is plain that the Spaces which might 
 be defcribed, in the fame time, wiih the uniform Ce- 
 lerity of OT, and the mean Celerity at <?, will, here, be 
 
 v i , i a , 3 , , . 
 to each other as v to ax v bx v-\-cx v 3 , ?r. 
 
 ^ " J 7 n ~ Z . *~3 ! t^J 
 
 or as x to ax x bx vx-\-cx vV, <5c. 
 
 Therefore, all the Terms, wherein v is found, vanifh- 
 
 ing, when j, <?, and S coincide, we have ax x for 
 
 the required Fluxion of C, or x ; which Fluxion, 
 becaufe the numeral Co-efficient of the fecond Term of 
 a BinpmiaJ involved is known to be, univerfally, equal 
 to the Exponent of the Power, will alfo be truly ex- 
 
 prcfied by nx" * x. Q. E. I. 
 
 9. If the Quantity Am (or #*) be generated with an 
 accelerated, or a retarded Motion, inittad of an uni- 
 form
 
 of FLUXIONS. 
 
 71 
 
 form one, the Fluxion of x (or C) will come out 
 exactly the fame : 
 
 For the Spaces rR and jS, actually defcribed in the 
 fame time, being always, to each other, in the Ratio 
 
 of x to ax x bx vx> &c. the mean Celerities, 
 at certain intermediate Points between r, R and s> S 
 muft, alfo, be in that Ratio : Which, when v vanifhes 
 
 (as above) will become that of x to ax x,(ornx x) 
 the very fame as before. 
 
 PROPOSITION II. 
 
 10. To find the Fluxion of the Produft or Reftangle of 
 two variable Quantities. 
 
 Conceive two Right-lines DE and FG, perpendi- 
 cular to each 
 other, to move, 
 
 from two other , 
 
 Right - lines, 
 BA and BC, 
 continually pa- 
 rallel to them- 
 felves , and 
 
 thereby gene- 
 rate the Rect- 
 angle DF. Let 
 the Path of their 
 
 InterfecYion, or the Loci of the Angle H, be the Line 
 BHR; alfo let Dd (x) and Y f (y) be the Fluxions 
 of the Sides BD (x) and BF (y)^ and let dm and fn, 
 parallel to DH and FH, be drawn. Therefore, be- 
 caufe the Fluxion of the Space or Area BDH is truly 
 expreffed by the Rectangle Dm ( = yx * ) and that * Art. 4. 
 of the Area, or Space BFH, by the Rectangle F, and 
 equal Quantities have equal Fluxions, it follows that the 
 Fluxion of theRedhngle xy ^=DF (= BDH-f BFH) is 
 truly exprefled by yx + xy. Q^ E. I. 
 
 B 4 Tfy
 
 The Nature and Invffligation 
 
 The fame ctherwife. 
 
 II. Let xy be the given Re&angle (as before) ; and 
 put z =: * + j, then 2.* being n x* + 2xy + y 1 , we have 
 ^ i** J/ rr *y. But the Fluxion of iz* k** 
 
 1^% (and confequently that of its Equal xy) is zz 
 
 xx yj (by Art. 6) : Which, becaufe x x-j- y and 
 
 K=x+j y is alfo equal to A- + yXx+j *.v yyyx 
 Q.E.I. 
 
 COROLLARY i. 
 
 12. Hence the Fluxion of the Producl: of three va- 
 riable Quantities (yzu) may be derived : For, if x be 
 put rr zu ; then yzu will become zzyx, and its Fluxion 
 z: yx + xj (c.; alove :) But x being rz zw, and, there- 
 fore, x =. zu + ux., if thefe Values be fubftituted in jx 
 
 -f sty, it will become^ x zu -f- uz-\-zuy'^.yzu+ yuz + 
 zuj the Fluxion of ^zw required. Jn like Manner the 
 Fluxion of xyzu will appear to be xyzu 4- Aryzw 4- 
 xyzu + ATJZW, and that of xyzuw zr A^ZW -f xyzuw -J- 
 xyz-uw + xjzuw 4- xyzuw. 
 
 COROLLARY 2. 
 
 13. Hence, alfo, the Fluxion of a Fraction may 
 
 2* 
 # 
 
 be determined. For, putting * = , we have A-Z=, 
 
 M 
 
 and therefore xz 4- zx r= fjj above) ; whence, by 
 
 Tranfpofition and Divifion, i (by 
 
 z z z z ^ ' 
 
 u zu uz 
 
 writing for *) = - ; j which is the true Fluxj- 
 
 " Z Z 
 
 on of *, or its Equal , the Fradion propofed. 
 
 2 
 
 14. Now, from the foregoing Propofitions, and their 
 fubfequent Corollaiies, the following pra&icai Rules, 
 
 for
 
 ofFLUXIONS. 9 
 
 for determining the Fluxions of algebraic Quantities, 
 are obtained. 
 
 RULE I. 
 
 To find the Fluxion of any given Power of a. vari- 
 able Quantity. 
 
 Multiply the Fluxion of the Root by the Exponent of th( 
 Power > and the Produfl by that Power of toe fame Root 
 whofe Exponent it lefs by Unity than the given Exponent. 
 
 This Rule is inveftigated in Prop, i, and is nothing- 
 more than nx" * x (the Fluxion of x") exprefled in 
 Words. 
 
 fience the Fluxion of x 3 is 3*** ; that of x 5 is $x*x j 
 
 and that of a -f y] 7 is -jy X a + >)% (becaufe, a being 
 eonftant, j is the true Fluxion of the Root a+ y, in this 
 Cafe). 
 
 Moreover the Fluxion of a 1 + z 7 } 1 , will be \X.2zz 
 X tf* -f Z^T, or 322 v/a* + z*: For here, x being put 
 ^= a 1 + z z , we have i 222, and therefore | A- **, the 
 
 Fluxion of j; 1 (or a 1 -f z 1 ]*) is =r 32^ v/a* + 2% as 
 above. 
 
 RULE II. 
 
 15. To find the Fluxion of the Produ& of feveral 
 variable Quantities multiplied together. 
 
 A^ultiply the Fluxion of each ^ by the ProduSl of the reft 
 of the Quantities, and the Sum of the Produfts thus arl- 
 Jing will be the Fluxion fought *. ' %A 
 
 Thus the Fluxion of *y, is xy -f- yx ; that of xyz, is 
 xyz + xzy +yzx j and that of xyzu, is xyzu + xyuz + xzuy 
 + yzux. 
 
 RULE
 
 jo The Nature and Li'cefllgation 
 
 RULE III. 
 
 16. To find the Fluxion of a Fraction. 
 
 From the Fluxion of the Numerator drawn into the De- 
 mminator, -fubtraft the Fluxim cf the Denominator 
 drawn into the Numerator, and divide the Remainder by 
 *Art.i3./^ Square of the Denominator *. 
 
 v <* v* >*y J 
 
 Thus, the Fluxion of is - 5} that of JT": is 
 ix ,-*+)** Jl^j , and that of 1+1+?, 
 
 - -^ + I V I M 
 
 *+' 
 
 2 ;<; X*+v x + yXz ,, , , 
 
 or i + JTT IS - - __ ^ ' - ; and fo of others. 
 
 17. In the Examples hitherto given, each is refolved 
 by its own particular Rule ; but in thofe that follow, the 
 Ufe of tv.o, and fometimes cf all the three, Rules is 
 requifite. 
 
 Thus (by Rule i. and 2-} the Fluxion of x*y' i is 
 
 . 
 
 x > that f ~ is 
 
 * 
 
 i i r 
 
 and that of ^- is 
 
 z z~ 
 
 where all the three Rules are neceflary. 
 
 When the propofed Quantity is affected by a Co-effi- 
 cient, or conitant Alultiplicator, the Fluxion found as 
 above, mufl be multiplied^}' that Co-efficient or Mul- 
 tiplicator. 
 
 Thus, the Fluxion of f* 3 is 15*^-. For, the Flu- 
 xion of x 3 being 3* 1 .*, that of 5-v 5 , which is 5 times 
 as great, muft confequently be 5X j^ 2 ^, or i^x. 
 
 And, in the very fame Manner the Fluxion of ax will 
 
 r. I 
 
 appear to be nax' x. Moreover, the Fluxion of 
 
 a I 
 
 -, - -i n or a x x* +_y z i % will be exprefled by 
 * 4-jrJ*
 
 ^FLUXIONS. it 
 
 a X xx 4- yv 
 
 X 'T[X9xte + 9yjXx*+j*] , or 
 
 _ _ f -^^ - 
 
 that of v / -v-f-Vv > or A'-f-^'irj by v^-fl^iJ7 ^* 
 
 _^_ ^ j 
 
 *+7I) f^^ i.) or ll_l_-: 1, or 
 
 j i . * c x + a] 
 and that of == , or 
 
 x Jf a Q XXY.X* a I X 
 
 by Reduction, is 
 
 y^x 1 a~ xxX. x + a _1x x x a x x + a xx v 
 
 x a x Vx- 
 
 x a X y ' x~ a 1 
 
 Having explained the Manner of confidering and de- 
 termining the iirft Fluxions of variable or flowing Quan- 
 tities, it will be proper to fay fomcthing, now, con- 
 cerning the hi-hcr Orders, as Second, Third, Fourch, 
 cffi. Fluxions. 
 
 18. The Secsnd Fluxion of a Quantity is the Fluxion 
 of the variabk or algebraic Quantity expr effing the Fir/I 
 
 Fluxion already defined 
 
 y the Third Fhixian /iArt.s. 
 
 meant the Fluxion if the variable Quantity cxpnjjing tha 
 Second : And by the Fourth^ &e Fluxion of tb; variable 
 Quantity exp^'cjji':^ tk: Third Fluxion: /fr;dj 
 
 Thus, {";)! Kxampie, let the Line AB reprefcnt a va- 
 riable Quantity, generated by the Motion of the Point 
 B, and let the (hrft) Fluxion thereof (or the Space 
 that might be uniformly defcribed in a given Time, with 
 the Celerity of Bj be always exprefieu by the Diitance 
 6 of
 
 1 2 V/je Nature and Invejligation 
 
 of the Point D from a given, or fixed Point C : Then, 
 
 if the Celerity of B 
 g be not every where 
 
 A' ~^ ' the fame; the Dif- 
 
 D tance CD, expref- 
 
 C "" * fing the Meafure of 
 
 F that Celerity, muft 
 
 E ' alfo vary, by the 
 
 jj Motion of D, from, 
 
 G . or towards C, ac- 
 
 cording as the Cele- 
 rity of B is an increafing or a decreafing one : And the 
 FJuxion of the Line CD, fo varying (or the Space 
 (EF) that might be uniformly defcribed in the aforefaid 
 given Time, with the Celerity of D) is the fecond 
 Fluxion of AB. Again, if the Motion of B he fuch 
 that neither it, nor that of D, (which depends upon it) 
 be equable, then EF, expreffing the Celeii'y of D, will 
 alfo have its Fluxion GH; which is the third Fluxion 
 of AB, and the fecond Fluxion of CD. 
 
 And thus are the Fluxions of every other Order to be 
 confidered, being the Mcafures of the Velocities by winch 
 iheir refpeftive fcwing Quantities, the Fluxiom of the 
 'Art z. P rec ding Ordcr^ are generated *. 
 
 19. Hence it appears, that a fecond Fluxion always 
 {hews the rate of the Increafe, or Decreafe, of the firft 
 Fluxion ; and that Third, Fourth, &c. Fluxions, dif- 
 fer in Nothing (except their OrJer and Notation) from 
 Firft Fluxions, being actually fuch to the Quantities 
 from whence they are immediately derived ; and there- 
 fore are alfo determinable, in the very fame Manner, by 
 the general Rules already delivered. 
 
 Thus, by Rule 3. the (firft) Fluxion of x 3 is 3#V : 
 And, if x be fuppofed conftant, that is, if the Root .v 
 be generated with an equable Celerity, the Fluxion of 
 3* 1 .*- (or 3*X# a ) again taken, by the fame Rule, will 
 be 2* * 2JC.X-, or 6xx* ; which therefore is the fecond 
 Fluxion of* 3 : Whofe Fluxion, found in like Sort, 
 will be 6^ 3 , the third Fluxion of * 3 . Farther than 
 
 which
 
 cf FLUXIONS. 
 
 which we cannot go in this Cafe, becaufe the laft 
 Fluxion 6.* J is here a conftant Quantity. 
 
 20. In the preceding Example the Root x is fuppofed 
 to be generated with an equable Celerity : But, if the 
 Celerity be an increafing or a decreafing one, then *, 
 expreffing the Meafure thereof, being variable, will alfo 
 have its Fluxion ; which is ufually denoted by x : 
 Whofe Fluxion, according to the fame Method of No- 
 tation, is again defigned by x ; and fo on, with refpe<St 
 to the higher Orders. 
 
 21. HITC folio w a few Examples, wherein the Root 
 jr, (or y) is fuppofed to be generated with a variable 
 Celerity. 
 
 Thus, the firft Fluxion of x* is 3*** (or 3* 2 Xjr). 
 And, if the Fluxion of 3* 1 x.v (confidered as a Redl- 
 angle) be, again, found (by Rule 2.) we fhall have 
 * l x*=6#x* + * a *> for the fecond Fluxion 
 
 Moreover, from the Fluxion laft found we {hall in 
 like manner get 6xXx l + 6xX2xx -f bxxxx -f- 3**X 
 (or bxi+iSxxx + ^x) for the third Fluxion of x 3 . 
 
 Thus alfo, if y ~nx x, then will y'=.n^n ^ 
 A: x l + nxx ; and if i*~xy, then will 
 
 yx: And fo of others. But, in the Solution of 
 Problems, it will be convenient to make the firft 
 Fluxion of fome one of the fimple Quantities (x or y) 
 invariable, not only to avoid Trouble, but that it may 
 ierve as a Standard to which the variable Fluxions of the 
 other Quantities, depending thereon, may be always 
 referred. The Reader is alfo defired here (once for all) 
 to take particular Notice, that the Fluxions of all Kindt 
 end Orders, whatever, are contemporaneous, or fucb as 
 may be generated together, ivith their rtfpeftive Celtri* 
 tiss, in one and the fame^Tinu.
 
 Solution of Problems 
 
 SECTION II. 
 
 On the Application of Fluxions to the Solu- 
 tion of Problems DE MAXIM is ET Mi- 
 
 NIMIS. 
 
 22. TF a Quantity, conceived to be generated by Mo- 
 [ tion, increafes, or decreafes, till it arrives at a 
 certain Magnitude or Pofition, and then, on the con- 
 trary, grows lefler or greater, and it be required to de- 
 termine the faid Magnitude or Pofition. the Queftion is 
 called a Problem tie Maximh & Minimi;. 
 
 GENERAL ILLUSTRATION. 
 
 Let a Point m move uniformly in a Right Line, from 
 A towards B, and let another Point n move after it, 
 with a Velocity either increafmg, or decreafmg, but fo 
 that it may, at a certain Pofition, D, become equal to 
 that of the former Point ;#, moving uniformly. 
 
 This being premifed, let the Motion of n be firft 
 
 confidered as an in- 
 
 A T\ rt -D creating one ; in 
 
 1 t which Cafe the Di- 
 
 ?l 711 ftance of behind 
 
 m will continually 
 
 increafe, till the two Points arrive at the cotemporary 
 Pofitions C and D ; but afterwards it will, again, de- 
 creafe; for the Motion of , till then, being flower than 
 at D, it is alfo flower than that of the preceding Point 
 m (by Hypothefis) but becoming quicker, afterwards, 
 than that cf TT?, the Diftance mn (as has been already 
 faid) will again decreafe : And therefore is a Maximum^ 
 or the greateft of all. when the Celerities of the two 
 Paints are equal to each other. 
 
 But, if n arrives at D with a decreafmg Celerity ; 
 then its Motion being firft fwifter, and afterwards flower, 
 than that of JTJ, the Diftance mn will firft deireafe and 
 
 then
 
 de Maximis ct Minimis. 15 
 
 then increafe ; and therefore is a Minimum, or the lead 
 of all, in the forementioned Circumftance. 
 
 Since then the Diftance mn is a Maximum or a Mi" 
 nimum, when the Velocities of m and n are equal, or 
 when that Diftance increafes as faft through the Mo- 
 tion of ;, as it decreafes by that of , its Fluxion at 
 that Inftant is evidently equal to Nothing *.Art. 2 
 Therefore, as the Motion of the Points m and n may* ^. 
 be conceived fuch that their Diftance mn may exprefs 
 the Meafure of any variable Quantity whatever, it fol- 
 lows, that the Fluxion of any variable Quantity what- 
 ever, when a Maximum or Minimum, is equal to No- 
 thing. 
 
 EXAMPLE I. 
 
 23. To divide a given Right-line AB into two fach 
 Parts, AC, BC, that their Produtt, or Rettangle, may 
 be the greateji pojfible. 
 
 Put the gi- 
 ven Line AB Q 
 
 = *, and let A' |R 
 
 the Part AC, 
 
 confidered as variable (by the Motion of C from A to- 
 wards B) be denoted by x : Then BC being a *, 
 we have AC X.ftC=ax ** : Whofe Fluxion ax^xx 
 being put o, according to the prefcript, we get ax 
 ixx, andconfequently* =: \a. Therefore AC and 
 BC, in the required Circumftance, are equal to each 
 other : Which we alfo.know from other Principles. 
 
 EXAMPLE II. 
 
 24. To find the Fraflien which Jhall exceed its Cube by 
 the greatejl Quantity pofllble. 
 
 Let x denote a variable Quantity, expreffing Number 
 . in general ; then the Excefs of x above x 1 being uni- 
 verfally reprefented by xx\ if the Fluxion thereof be 
 ^f. we fhall have x 3^-0 ; and therefore 
 the Fraction required* 
 
 E X-
 
 1 6 Solution of Problems 
 
 EXAMPLE TIT. 
 
 25. To determine the great eft ReR angle that can be in* 
 fcribed in a given Triangle. 
 
 Put the Bafe 
 AC of the gi- 
 ven Triangle 
 , and its Alti- 
 tude BD a ; 
 and let the Alti- 
 tude (BS) of the 
 infcribed Red- 
 angle me (confi- 
 dered as variable) 
 be denoted by x : 
 Then, becaufe of the parallel Lines AC, and ac, it 
 
 will be as BD (a) : AC (J) :: DS (</*) : 
 
 ac : Whence the Area of the Relangle, or ac x BS 
 
 box bx~ . box ibxx 
 
 will be := : Whofe fluxion being 
 
 a a 
 
 (as before) put = o, we fhall get x [a. "Whence the 
 greateft infcribed Rectangle is that whofe Altitude is juft 
 half the Altitude of the Triangle. 
 
 26. It will be proper to obferve here, that the Value 
 of a Quantity, when a Maximum or Minimum, may 
 oftentimes be determined with more Facility by taking 
 the Fluxion of fome given Part, Multiple, or Power, 
 thereof, than from the Fluxion of the Quantity itfelf. 
 Thus, in the preceding Example, where the genera] 
 baxbx* b 
 
 Expreffion is 
 
 if the conftant 
 
 Multiplicator be rejected, we fhall have ** 
 
 whofe Fluxion ax 2xx being put = o, we get * 4 
 the vtry fame as before, 
 10 '
 
 de Maximis & Minimis. 
 
 The Reafon of which is obvious ; becaufe when the 
 Quantity itfelf (be it of what Kind it will) is the greateft, 
 or leaft poflible, any given Part, Power, or Multiple of 
 it is alfo the greateft or leaft poflible. 
 
 E X^A M P L E IV. 
 
 27. Of all right-angled plain Triangles having the fami 
 given Hypothennfe^ to find that (ABC) wl)oje Area is 
 the greateft. 
 
 Let AC = a, AB=.v, 
 and BC = y : Then, 
 JT -f- y* being IT a* 9 we 
 fhall have y V^a*x\ 
 
 and confequently 
 
 - V * x 1 = the 
 
 2 
 
 Area of the Triangle ; / 
 
 B 
 
 whofe Square being, alfa, a Maximum *, Art. 2 6. 
 
 the Fluxion thereof 
 be =: o, f : Whence 
 
 x*x mufl therefore 
 
 ence *" is found = a V , and y 
 " 
 
 otherwife. 
 
 Since i^y is a Maximum, and X 1 +)*=:#% let the 
 Fluxions of both be taken, and you will have -~xy + kyx 
 =o, and ixx + iyy =: o ; from the former of which y 
 
 xx 
 
 will be = ; and from the latter, it will be =. : 
 
 XX 
 
 Therefore and are equal to each other, and con- 
 x y 
 
 fequently x sz y, (the fame as before.) 
 
 C EX-
 
 Solution of Problems 
 
 EXAMPLE V. 
 
 l8. Of all right-angled plain Triangles containing tfc 
 fame given Area^ to find that whereof the Sum of the 
 two Legs AB + BC is the leaft pojjible. (See the pre- 
 ceding Figure.) 
 
 Let one Leg, AB, be denoted by *-, and the Area 
 of the Triangle by a j then the other Leg will be de- 
 
 la 
 
 noted by , and the Sum of the two Legs will be x -{- 
 
 - j whereof the Fluxion is x -- - -, which, put o, 
 x x 
 
 gives x 
 
 Whence BC 
 
 f1&\ 
 
 (-) is 
 
 alfo = 
 
 V ia. Therefore the two Legs are equal to each 
 other. 
 
 EXAMPLE VI. 
 
 29. To determine the Dimenftons of the leajl Ifofceles Tri- 
 angle ACD that can circumfcribe a given Circle. 
 
 Let the Diftance 
 (OD) of the Vertex 
 of the Triangle from 
 the Center of the Cir- 
 cle, be denoted by x> 
 and let the remaining 
 Part of the Perpendi- 
 cular, which is the 
 Radius of the Circle, 
 be reprefented by a: 
 Then, if OS, perpen- 
 
 dicular to DC, be drawn, we fhall have DS = V V a* ; 
 
 and therefore, fmce DS : OS : : DB : BC, we like wife 
 
 have BC = "-= which multiplied by ~a (BD) 
 v x a 
 
 7 gives
 
 de Max! mis & MInimis. 19 
 
 ives ==== for the Area of the Triangle : Which 
 Vx*~ a* 
 
 being a Minimum, its Square muft be a Minimum^ and 
 
 .. , x 4- a] . -,- , x-\-a\ ,.. . 
 
 confequently - ! -, or us Equal ! , a Mini- 
 
 X ~ Q X (I 
 
 mum a!fo *. Whofe Fluxion, therefore, which is* 
 
 3 
 
 -, being put rr o, and 
 x a \- 
 
 ' x Xx + a}' 
 
 the Whole divided by p- , we alfo get 3 XA- a 
 
 x + a =. o ; whence xia \ Therefore, OD being 
 rz 2OS, and the Triangles ODS and BDC equiangular, 
 it is evident that DC is likewife = aBC = AC ; and fo 
 the Triangle ACD, when the leafl poflible, is equila- 
 teral. 
 
 EXAMPLE VII. 
 
 30. To determine the great eft Cylinder ', dg^ that can bt 
 infer ibed in a given Cone ADB. 
 
 Let a r=BC, the Altitude of the Cone ; 
 :=AD, the Diameter of its Bafe ; 
 x=fg (dh) the Diameter of the Cylinder, con- 
 fidered as variable ; 
 
 . the Area of , he 
 
 V 4 / 
 
 whofe Diameter is Unity. 
 
 Then, the Areas of Circles being to one another as 
 the Squares of their Diameters, we have, i 1 : A-* : : 
 p : (px*) the Area of the Circle figr: Moreover, from 
 the Similarity of the Triangles ABC and A^f, we have 
 
 ii(AC) : a (EC) : : {b '-* (Ad) : df - 
 
 which multiplied by the Area px* (found above) gives 
 
 C 2 abx*
 
 20 
 
 Solution of Problems 
 
 = 0*, confequently 
 
 pabx 1 pax* 
 
 ~~b~ 
 
 for the folid 
 Content of 
 the Cylin- 
 der: Which 
 being a 
 
 Maximum , 
 its Fluxion 
 
 n 
 
 mult 
 b 
 
 = and df - : From 
 
 whence it appears, that the infcribed Cylinder will be 
 the ereateft poflible, when the Altitude thereof is juft 
 | of the Altitude of the whole Cone. 
 
 EXAMPLE VIII. 
 
 31. To determine the Dimenjions of a cylindric Meafure 
 ABCD, open at the Tcp y which Jhall contain a given 
 Quantity (of Liquor, Grain^ &c.) under the leafl in- 
 ternal Superficies pojjible* 
 
 Let the Diameter 
 ABn#, and the Alti- 
 '< C tude AD y ; moreover 
 let p (3114159, &c.} 
 denote the Periphery of 
 the Circle whofe Dia- 
 meter is Unity, and let 
 c be the given Content 
 of the Cylinder. Then 
 it will be i : p ::x : (px) 
 > the Circumference of the 
 Bafej which, multiplied 
 by
 
 de Maximis & Minimis. 
 
 by the Altitude^, gives pxy for the concave Superficies 
 of the Cylinder. In like Manner, the Area of the Bafe, 
 by multiplying the fame Expreflion into ~ of the Dia- 
 meter *, will be found = j which drawn into the 
 
 21 
 
 = i and therefore x 
 
 Altitude y y gives - for the folid Content of the Cy- 
 linder ; which being made = c, the concave Surface 
 pxy will be found , and confequently the whole 
 
 Surface = -' + : Whereof the Fluxion, which is, 
 
 * 4 
 4cx pxx 
 r + - j being put o, we Ihall get Sc -f px 3 
 
 * 2* 
 
 2 \/ - : Further, becaufe a* J 
 
 / 
 
 =. 8r, and px*y = 4^, it follows, that x == 2y ; whence jf 
 is alfo known, and from which it appears, that the Dia- 
 meter of the Bafe muft be juft the Double of the Alti- 
 tude. 
 
 EXAMPLE IX. 
 
 32. Of all Cones under the fame given Superficies (s) t 
 find that (ABD) wbofe Solidity is the greatejl. 
 
 Let the Semi- 
 diameter of the 
 Bafe, AC =. *, and 
 the Length of the 
 flant Side AB =y t 
 and let p ( as in 
 the preceding Ex- 
 amples) denote the 
 Periphery of the 
 Circle whofe Dia- 
 meter is Unity. 
 
 Then
 
 2 2, Solution of Problems 
 
 Then the Circumference of the Bafe will be ~ 2px > 
 the Area of the Bafe />*% and the convex Superficies 
 of the Cone = pxy, (which laft is found by multiplying 
 half the Periphery of the Bafe by the Length of the 
 flant Side) : Wherefore, fince the whole Superficies is 
 
 px*+~pxy j, we have y~ x; whence the Alti- 
 
 PX __ 
 
 titude CB (/AH* AC 1 ) = \/ -^ - ; which 
 
 P * P . 
 
 multiplied by f * - J | of the Area of the Bafe, gives 
 for the folid Content of the Cone. 
 
 ,j sx 
 Which being a Maximum, its Square - muft 
 
 25^ XX 
 
 alfo be a Maximum ; and therefore 
 
 9 
 
 \vhencej 4px t Q t and confequently x 
 
 4? 
 
 which v ( r= x =. ->-- = ^ =: 3*-) will like- 
 
 J \ px px px 
 
 wife be known ; and from whence it will appear that 
 the greateft Cone under a given Surface, (or a given 
 Cone under the leaft Surface) will be when the Length 
 of the flant Side is to the Semi-diameter of the Bafe in 
 the Ratio of 3 to r, or, (which comes to the fame) 
 when the Square of the Altitude is to the Square of the 
 \vhole Diameter in the Ratio of 2 to i. 
 
 EX-
 
 de Maximis & Minimis. 
 
 2 3 
 
 EXAMPLE X. 
 
 33. To determine the Pofitlon of a Right-line DE, which, 
 pajfing through a given Point P, Jball cut two Right- 
 lines AR and AS, given by Pofition ^ in fuch fort that 
 the Sum of the Segment 's, AD and AE, made thereby* 
 may be the leaft poffible. 
 
 s\ 
 
 E 
 
 Make PB, parallel to AS, = a, and PC, parallel to 
 AR, = I, and let BD x: Then, by reafon of the 
 
 parallel Lines, it will be, x : a : : b : CE r: 
 
 X 
 
 Therefore AD + AE := + .* -j-a+ , and its Fluxion, 
 
 x 
 
 abx 
 x -, which, in the requjred Circumftance, being 
 
 o, we have A- 1 ab alfo rz o, and confequently x zr 
 
 y ab ; whence the Pofition of DE is known. But the 
 fame Thing may be otherwife determined, independent 
 of Fluxions, from the general Solution of the Problem 
 for finding the Pofition of DE, when the Sum of the 
 Segments AD and AE (inftead of being a Minimum) 
 fhall be equal to a given Quantity. Of which Problem, 
 the geometrical Conftru&ion may be as follows. 
 
 C 4 Compleat
 
 24 Solution of Problems 
 
 Compleat the Parallelogram ABPC (as before) and, 
 in RA produced, talce Af = AC, and let ^F be equal 
 to the given Sum of the two Segments: Alfo let two 
 Semi-circles be defcribed upon EC and BF, and let AH, 
 perpendicular to Br, interfe<5l the former in H j like- 
 wife let HK, parallel to F<:, interfeft the latter in I ; 
 draw ID perpendicular to Yc, and, through P and D 
 draw DE ; which will be the Pofition required. For 
 AB xA<: being = AH 1 = DI a = BDxDF, we have BD 
 : AB : : Ac (AC) : DF; alfo, becaufe of the parallel 
 Lines, we have BD : AB : : AC : CE ; whence DF = 
 CE, and confequently AD + AE (AD + AC-f FD) is 
 equal to cF, which Conftru6tion is more neat than that 
 in p, 155. of my Geometry. But to (hew how far this 
 may conduce to the Matter firft propofed ; we are to 
 obferve, that, as the Problem here conftruc~ted appears 
 to be impoflible, when the Right-line HK (inftead of 
 cutting or touching) falls wholly below the Circle BWF, 
 the Jeaft poflible Value of BF (and confequently of AD 
 -j- AE) muft, therefore, be when that Right-line touches 
 
 the Circle; that is, when BD = DI~AH=v / ABxAC ; 
 which Value is the very fame with that found above. 
 
 The fame Conclufion may alfo be deduced from the 
 algebraic Solution of the forefaid Problem : For, put- 
 
 ab 
 
 ting^-f x + a+ (AD + AE) = s y and iolving the 
 
 Equation, x will be found = '- -f ^/ s ~ a ~^\ _^. 
 
 4 
 
 Which Equation being no longer poflible than till s ~ a ~ v i 
 
 4 
 ab is o, we have AT, in that Circumftance, 
 
 $ ~ * - fs ^~^ h ^^^ 
 
 -- -= V ab j fnll as before. In like Manner the 
 
 Maxima and Minima may be determined in other Cafes, 
 by finding the Pofition or Circumftance wherein the 
 general Problem begins to be impoflible, (fuppofing the 
 Quantity fjught to be given). But the Operation by 
 
 Fluxions
 
 de Maximis & Minimis. 25 
 
 Fluxions is, for the general Part, much more fhort and 
 expeditious. 
 
 EXAMPLE XI. 
 
 34. Tfie fame being given as in the preceding Example^ to 
 determine the Position, when the Line DE, itfc/f y is the 
 leafl poffible. 
 
 Upon AF let fall the perpendicular PQj make BQ_ 
 =r, and, the reft, as before: Then DP 1 being ( = 
 DB*+BP 2 2BQ_x DB) =**+* 2r, and 
 DP* :: DA 1 : DE% we have x* : x*-\-a z 2cx :: 
 
 2c 
 
 * ** 
 
 whofe Fluxion, which is 2xXb + x X i -- _i -- 
 
 * 
 
 rZ"?* x -^A bein S P ut and tne whole 
 # 2 jf 3 
 
 Equation divided by 2x X />-f x, there will come out I 
 
 i -- r > whence ^? 3 2cx +a x 
 
 -f b + xxcx a* =. 0; that is, (by Reduction) x 3 ex* 
 -\-bcx {fbo : From the Refolution of which Equa- 
 tion, the Pofition of DE is determined. 
 
 LEMMA. 
 
 35. If a Body or Point (n) be fuppofed to move in a 
 Right-line AB, its abfoluie Celerity, in the Direftion of 
 that Line^ will be to the relative Celerity , whereby it tends 
 tOj or from, a given Point C, any where out of the Lint, 
 as the Dijiance C.. is to the Diftance D, intercepted by 
 n and the Perpendicular CD ; or t as Radius to the Co-fine 
 of the Angle of Inclination DC. 
 
 For, putting CD <?, D #, and Cn rr y, * ^ 
 we have z + ..v a =>'', and confequently 2xx = 2yj>*.-aad 5 '. Z 
 
 Whence
 
 Art. 
 and 5. 
 
 Solution of Problems 
 
 A D 
 
 Whence x : 
 y : : y (Cn) : x 
 (D) :: Radius : 
 Co-fine DC: Bur, 
 the Fluxions of 
 Quantities are as 
 the Celerities of 
 their Increafe *, 
 therefore the Truth 
 of the Proportion 
 is manifeft. 
 
 COROLLARY. 
 
 It follows from hence, that the relative Celerities in 
 any two different Directions wE and #C, are directly as 
 the Co-fines of the correfponding Angles DE and 
 DwC. Therefore, when E is perpendicular to Cn 9 
 (and the Angle DE therefore equal to C) the Celerity 
 in the Direction E, will be to that in the Direction 
 C, as the Sine of DnC is to its Co-fine. From whence 
 it appears, that the Celerities in the Directions D, C, 
 and E/z (perpendicular to wC) are to each other as C, 
 D//, and CD refpectively. 
 
 EXAMPLE XII. 
 
 36. To determine the Pafttlon of a Point, from wbcn:e y 
 if three Right- lines be drawn to fo many given Points 
 A, E, C, their Sum fall be the Uajl poffibU. 
 
 Let HPG be the Periphery of a Circle defcrihed 
 about the Point A, as a Center, at any Diftance AG ; 
 in which let the Point P be conceived to move with an 
 uniform Celerity, from G towards H. 7'hen, becaufe 
 thcj relative Celerity thereof, in the Direction PC, is to 
 that in the Direction BP produced, as the Co-fine of 
 the Angle CPH-to the Co-fine of the Angle BPG, (by 
 the pr(ct<i<n Lemma) j anJ> fincethefe Cekiities, when 
 
 the
 
 de Maximis & Minimis. 
 
 the Sum of CP and BP is a Minimum^ muft be equal *,* Art. 
 
 G 
 
 it follows, therefore, 
 that the faid Angles 
 CPH and BPG, as 
 well as their Co-fines, 
 will in that Circum- 
 ftance become equal 
 to each other ; and 
 confequently A P C 
 alfo equal to A P B. 
 From whence it ap- 
 pears, that (take AG 
 what you will) the 
 .Sum of the three 
 Lines, AP, BP, and 
 CP, cannot be the 
 leaft pofiible when the 
 Angles A P B and 
 A P C are unequal. 
 And, by the fame 
 Argument, it alfo appears that their Sum cannot 
 be the leaft pofiible, when the Angles BPA and 
 BPC are unequal : Therefore, their Sum muft be 
 the leaft pofuble, when all the three Angles about the 
 Point P are equal to one another ; provided the Cafe 
 will admit of firch an Equality, or that no one of the 
 Angles of the Triangle ABC is equal to, or greater than 
 of 4 Right Angles (for otherwife, the Point P will 
 fall in the obtufe Angle) : Hence this 
 
 CONSTRUCTION. 
 
 Defcribe, upon BC, a Segment of a Circle, to con- 
 tain an Angle of 120, and let the whole Circle BCQ_ 
 be compleated ; and from A, to the Middle (QJ of the 
 Arch BQC, draw AQ_ interfering the Circumference 
 of the Circle in P ; which will be the Point required. 
 For, the Angles BPQ_ and CPQ^, ftanding upon the 
 equal Arches BQ^ and CQ_, have their Complements 
 APB and APC equal to each other; and therefore, the 
 Angle BPC being 120"" (by Conftruclion) each of the 
 
 laid 
 
 and 22.
 
 28 Solution of Problems 
 
 kid Angles APB, APC, will, likewife be 120 De- 
 grees. 
 
 After the fame 
 Manner, it will 
 appear that the 
 Sum of all the 
 Lines AP, BP, 
 CP, faff, drawn 
 from any Num- 
 ber of given 
 Points A, B, C, 
 faff, to meet in 
 another Point P, 
 will be the leaft 
 pofiible, when the 
 
 Co-fines of the Angles RPA, RPB, RPC, faff, that 
 the faid Lines make with any other Line RS, pafling 
 through the Point of Concourfe, deftroy each other : 
 Which will be when all the Angles APB, BPC, CPD, 
 faff, are equal, in all Cafes where the Pofition of the 
 given Points will admit of fuch an Equality. But, if the 
 Number of given Points be four, the required Point will 
 be in the Interfe&ion of the two Right-lines joining the 
 oppofite Points : For, fuppofing APC and BPD to be 
 continued Right-lines, the Co-fine of RPA will be equal 
 and contrary to that of RPC, and that of RPB equal 
 and contrary to that of RPD. 
 
 EXAMPLE XIII. 
 
 37. If two Bodies move at the fame Time, from two given 
 Places A and B, and proceed uniformly from thence in 
 given DireflionS) AP and BQ_, with Celerities in a 
 given Ratio j ;'/ is propofed to find their Pofition, and 
 how far each has gone^ when they are the nearejl pojjible 
 to each other. 
 
 Let M and N be any two cotemporary Pofitions of 
 the Bodies, and upon AP let fali the Perpendiculars 
 NE and BI>j alfo let QB be produced to meet AP 
 
 in
 
 de Maximis Sc Minimis. 
 
 29 
 
 AM 
 
 ED 
 
 in C, and let MN be drawn : Moreover, let the given 
 Celerity in BQ_be to that in AP, as n to m t and let 
 AC, BC, and CD, (which are alfo given) be denoted 
 by rt, b, and c refpeclively, and make the variable Dif- 
 tance CN =x: Then, by reafon of the parallel Lines 
 NE and BD, we fhall have b (CB) : x (CN) :: c (CD) 
 
 coc 
 
 : CE . Alfo, becaufe the Diftances, BN and 
 b 
 
 AM, gone over in the fame Time, are as the Cele- 
 rities, we likewife have, n : m :: x b (BN) : AM 
 
 = , and confequently CM (AC AM)rrc-f- 
 
 mb mx mx mb\ 
 
 d , (by writing da+ ). Whence 
 
 n n n n / 
 
 will alfo be found 
 
 
 icdx icmx"- idmx inf 
 
 +x 7 + ; whofe Fluxion -- + - 
 
 b tip n n 
 
 _l o V -> 
 
 icdx A.CTTIXX 
 
 b nb 
 
 to be a Minimum) we get 
 
 + 2mncx o; and confequently x 
 
 ndxmb+nc 
 
 by, m* + n L +2mnc 
 are alfo given, 
 
 being made = o (becaufe MN is 
 
 i i bx -f rfbx rfcd 
 mnbd+ rfcd 
 
 ? and MN
 
 30 Solution of Problems 
 
 The fame oihcrivifc.- 
 
 Bccaufe the relative Celerities of the two Bodies, at- 
 M and N, in the Direction of the Line MN (pro- 
 
 n m Co-fwU Co-f.tf 
 
 ducedj are truly exprelled by ^ ,. X ;/z, and " 
 
 Art.35. x, refpeftively *; and as thefe Celerities, when the 
 
 Diftance MN is a Minimum, do become equal to each 
 Art.zz.other f, it follows that, in this Circumftance, m : n : : 
 
 Co-f. N. : Co-f. M : : Secant of M : Secant of N (by 
 
 plane Trig.) 
 
 Whence this Conftruaion. Take CH to CB in the 
 
 given Ratio of m to n t and draw HB j upon which 
 
 produced (if necefTary) let fall the Perpendicular AR j 
 draw RN parallel to AH, meeting CQ_ in N; laftly, 
 draw NM parallel to AR, and it will give the Pofition 
 required. For, firft, ir is plain, becaufe AM (RN) : 
 BN (: : CH : CB) ::;*:, that M and N are cotem- 
 porary Pofitions : It is Jikewife plain, that RN and BN 
 will be Secants of the Angles KNR (CMN) and KNB 
 (CNM) to the Radius NK j becaufe the Angle NKR 
 (r:ARK) is a Right-one. Which Lines or Secants 
 are in the propofed Ratio of m to , as has been already 
 fhewn. 
 
 But
 
 de Maximis & Minimis. 
 
 But the fame Solution may be, yet, otherwife de- 
 rived, independent of Fluxions, from Principles intirely 
 geometrical. For, let m and be any two cotempora- 
 ry Pofitions at Pleafure, and let CH (as before) be to 
 CB, as the Celerity in AP to that in CQ_; moreover, 
 let r, parallel to AP, be drawn, meeting HB pro- 
 duced in r, and let A, r be joined. Then, ilnce CB : 
 CH : : B* : nr (byfm. Triangles] and CB : CH : : En 
 : Aw, (by Hyp.) it follows, that nr and Am, (which 
 ire parallel) will alfo be equal to each other; and there- 
 fore Ar and mn, likewife equal and parallel. But Ar is 
 the lead poflible when perpendicular to Hr. Whence 
 the Solution is manifeft. 
 
 EXAMPLE XIV. 
 
 38. Let the Body M move, uniformly^ from A towards 
 Q^, with the Celerity /, atid let another Body N pro- 
 cced from. B, at the fame time^ with the Celerity n. 
 Now it is propofed to find the Direction (BD) of the 
 latter, fo that the Df/fance MN of the two Bodies, 
 when the latter arrives in the Way or Direftion A ~ 
 the former ', ?nay be the greaieft pojfible. 
 
 Let BC be perpendicular to AQ_, and make AC =z 
 <?, BCrz/', and BN=jc. Therefore, if the Pofition 
 M befuppofed cotemporary with N, we {hall have n : 
 
 mx .^..mx 
 
 m :: x : AM = j whence CM ~.7,andcon- 
 n n 
 
 fequently
 
 3 2 Solution of Problems 
 
 fequently MN (CN CM) = -S^Hp - 4- a 5 
 
 n 
 
 whereof the Fluxion being taken, and made o, we 
 
 x m mb 
 
 get . = ; therefore x = . . , and CN 
 
 V x * b* n VrrS n* 
 
 Vx~-P}=, _ a : Whence, m : n (:: BN : 
 
 CN :: Radius : Co-fine N. The fame Conclufion is 
 otherwife derived, thus, 
 
 Let the Right-line BD be fuppofed to revolve about 
 the Point B, as a Center, with a Motion fo regulated, 
 that the intercepted Part thereof BN may increafe with 
 the uniform Celerity n : Then, the Celerity with which 
 
 n X Radius * 
 *Art.35.CN is mcreafed being ,. ^ , this Lxpreffion, 
 
 when MN is a Maximum^ muft, confequently, be equal 
 fArt.s2.to (m} the Velocity of the other Body f M ; and there- 
 fore m : n :: Radius : Co-fine N, as before. 
 
 EXAMPLE XV. 
 
 39. Suppofing a Ship to fail from a given Place A, in a 
 given Direction AQ_, at the fame time that a Boat t 
 from another given Place B, fets out in order (if pof- 
 fible) to come up with her, and fuppojing the Rate at 
 which each FeJ/el runs to be given ; it is required to find 
 in what Direction the latter muji proceed^ fo that, if it 
 cannot come up with the former , ;'/ may, however^ ap- 
 proach it as near as pofftile. 
 
 Let the Celerity of the Ship be to that of the Boat 
 in the given Ratio of m to n ; alfo let D and F be the 
 Places of the two Veflels when neareft poffible to each 
 other, and, from the Center B, through F, fuppofe the 
 Circumference of a Circle to be defcribed. Then (the 
 Diftance DF being the leaft poffible), the Point F muft 
 be in the Right-line (DB) joining the Point D and the 
 
 Center
 
 dc Maximis & Minimis. 
 
 Center B; be- 
 caufe no other 
 Point in the 
 whole Periphe- 
 ry, at which 
 the Boat from 
 B might ar- 
 rive in the fame 
 time, is fo near 
 to D as that 
 wherein the 
 Line DB inter- 
 feds the faid 
 Periphery. But now, to get an Expreflion for DF, in 
 algebraic Terms, let BC be perpendicular to AQ_, and 
 make AC =<?, BC = , and CD=* ; and then BD 
 will be = yT + x*-, moreover, be- 
 
 ;7< 
 
 eaufe:::AD(fl-f AT): BF,youwillhaveBF = 
 and confequently, DF zr 
 
 m 
 
 l + * a 
 
 whofe 
 
 =f bei "g made = o, we find 
 
 , i whence the Direaion BD is known 
 
 V m n 
 
 And, if the Value of AT, thus found, be fubftituted in 
 that of DF, (found above) we fhall have DF = 
 
 m ~~ n ~ na ; whence the Pofition of F is known. 
 m 
 
 And from which it is obfervable, that, as DF muft be a 
 real, pofitive Quantity (by the Queftion) this Method 
 of Solution can only obtain when m is greater than n 
 and ^vV--**, alfo greater than tia : For in all other 
 Cales the Boat will be able to come up with the Ship. 
 
 The fame otherwife, 
 
 Let the Radius of the Circle EFH be conceived to 
 
 mcreafe uniformly, with the Celerity n 3 whilft the Point 
 
 * D moves 
 
 33
 
 Solution of Problems 
 
 D moves uniformly along AQ_, with the Celerity m: 
 Then, the Celerity at D, in the Direction of BD pro- 
 
 m X Co-fine D 
 duced, being -- ~R~d' - ' e re ' atlve Celerity with 
 
 which the Point D recedes from the Periphery of the 
 faid variable Circle, will be univerfally expreffed by 
 
 ,. , , . -..., . 
 
 n' t which beina; = o, when DF is a 
 Radius 
 
 Minimum^ we have in this Cafe m X Co-fine D=n X Ra- 
 and confequently m : n :: Radius : Co-fine D. 
 
 Therefore, if, at C, a right-angled Triangle Cbd be 
 conftituted, whofe Bafe Cdn^ and its Hypothenufe 
 db m, and parallel to the latter you draw BD, it will 
 be the Direction required : In which, if there be taken 
 BF, a Fourth-proportional to m, n, and AD, you will 
 alfo have the Pofition required. 
 
 EXAMPLE XVI. 
 
 40. To determine the greatejl Parabola that can be formed 
 by cutting a given Cone A CD. 
 
 Let nv t parallel to CA, be the Axis of the Parabola 
 rvm t and rm the Bafe (or Ordinate) thereof; putting 
 
 DC
 
 de Maximis & Minimis. 
 rr<7, CAn, and Dn x; then, becaufe of the 
 
 parallel Lines, it will be, a\l'.\ x : nv : More- 
 
 a 
 
 over, by the Property of the Circle, we have m* 
 (nm 1 DxCn) = axx\ and confequently rm 
 
 , _, __ T <j l)X 
 
 2 v ax ,v a ; which multiplied by x (becaufe ev- 
 
 3 * 
 ry Parabola is f of a Parallelogram of the fame Bafe and 
 
 _ - - 
 
 Altitude) gives V ax x~ for the Content of the 
 
 3" 
 Parabola : Whofe Fluxion, or that of ax 3 x* * being *Art.a6, 
 
 put equal to Nothing ; we find x : Whence qv 
 
 4 
 and the Area of the greateft, 
 
 -. 
 
 4 
 
 X-AC, m^r 
 or required, Parabolas: AC X CD X 
 
 EXAMPLE XViL 
 
 41. To determine the greatcft Ellipfi BTES that can It 
 formed by cutting a givtn Cone ADD. 
 
 Let BE be the 
 greater, and TS the 
 lefler, Axis of the El- 
 Jipfis BTES, confider- 
 ed as variable by the 
 Motion of (the End 
 of the Tranfverfe) E, 
 along the Line AD ; 
 moreover let Ev be 
 parallel to AC the Axis 
 of the Cone, meeting 
 the Diameter BD in v t 
 and let the Diameters 
 EF and np be parallel 
 to BD j whereof the 
 latter np is fuppofed to 
 
 D 2
 
 3 6 Solution of Problems 
 
 pafs through O the Center of the Ellipfis : Then, put- 
 ting AC~ #, CD=^, and Cvrr.v, we fhall have Bv 
 b + x-, alfo, becaufe of the parallel Lines we have CD 
 
 (b) : CA (a) :: Dv (bx) : *** * =Ez;j whence 
 BE 
 
 Furthermore, fince the Triangles EOw, EBD, and 
 BO/>, BEF are equiangular, and EO (=BO) =-J- BE, 
 xve likewife have Oa=iBD=, and Cty = iEF=Cw 
 x ; and confequently On X Op (r=OT% by the Pro- 
 perty of the Circle) bx ; whence ST = i V bx^ and 
 
 therefore BE x ST = 
 
 Now the Area of any Ellipfis being in a conftant 
 Ratio to the Rectangle of its greater and JefTer Axes 
 (namely as 3,14159, b'c. to 4) the laft general Ex- 
 preflion muft therefore be a Maximum, when the 
 Area is fo; and therefore its Fluxion, or that of b*x x 
 
 Art.2a. 2? a ^ a + a*x 3 ) equal to Nothing*; that is, b*x 
 
 fl*A?V = o : 
 
 .,,, 4/^.v X rt a b 1 I" 1 
 
 Whence x* -- - - - -- , and x = 
 
 3* +3* 3 
 
 * P+b\/a 4 l+a- . 
 
 ~ > from which the EU 
 
 - ( f. p 
 
 Jipfis is known. 
 
 But it is obfervable, that, when a 4 i^a l b' t -^b* is 
 negative, this Solution fails, becaufe the Square Root of 
 a negative Quantity is to be extracted. Therefore, to 
 determine the Litr.it, put a* J4^ 1 + ^ 4 cr o ; then, 
 by ordering the Equation, you will get a 2 = b* x 
 74-^/48, and = ^X2 + \/3; and therefore a : b :: 2 
 4-1/3 : I% Hence, if the Ratio of AC to CD be not 
 
 greater
 
 de Maximis & Minimis. 37 
 
 greater than that of 2 + ^3 to i> or (which comes to 
 the fame thing) if the Angle DAC be not lefs than 15 
 Degrees, the Fluxion of the Ellipfis can never become 
 equal to Nothing j but the Ellipfis itfelf will increafe 
 continually, from the Vertex till it coincides with the 
 Bafe of the Cone ; and therefore is greater at the Bafe 
 than in any other Pofition. 
 
 But it is further to be obferved, that this Problem is 
 confined to, yet, narrower Limits. For, either the 
 Ellipfis will increafe, continually, fiom the Vertex, to 
 the Bafe, of the Cone, (which is (hewn to be the Cafe 
 when the Angle DAC is greater than 15) or clfe it 
 will increafe till the Point E arrives at a certain Pofition 
 H, and afterwards decreafe to another certain Pofition b y 
 and then increafe again till it coincides with the Bafe of 
 the Cone, (for it muft always increafe again before it 
 coincides with the Bafe, becaufe, after the Point E is 
 
 5t below {he Perpendicular BQ_, both the Axes of the 
 llipfis increafe at the fame time). 
 The fame thing alfo appears from the foregoing Equa.- 
 
 * 
 
 tion x = - ;__., '- ~ whofe two 
 
 Roots exprefs the two Values of AC (or Cv) at the Times 
 of the Maximum (at H) and its fuccecding Minimum 
 (at h). Hence it is manifeft, that the Ellipfis may ad- 
 mit of a Maximum between the Vertex of the Cone 
 and the Perpendicular B Q_, and yet, that Maxi- 
 mum be lefs than the Bafe of the Cone, unlefs the 
 forefaid Angle DAC be fo much lefs than 15 (above 
 found) that the Increafe from /; to D, be lefs than 
 the Decreafe from H to /;. Now therefore, to de- 
 termine the exadl Limit, let the forefaid Increment 
 and Decrement be fuppofed equal to each other, or, 
 which is the fame in Effect, let the Ellipfis BTESB 
 = the Circle BqVm, or BExST = BD% that is, let 
 
 ^ 
 
 - 1 =&* : From which 
 
 D 3 Equa-
 
 3 8 Solution of Problems 
 
 V- 4 3 * 
 Equation you will get a~ X 
 
 b x] 
 
 =: x - 7-^ - : Moreover, from the Equation 
 x b x 
 
 b*x + \b 3 xx + $b*x-x -f cffx -^-bxx + 3a a jpV=o t (gi- 
 
 , v 4 3* 
 
 ven above) you will, again, get <z = 
 J 
 
 ; Whence, by comparing thefe 
 xx $x b 
 
 4y + 3^ + ^_y + 4faf + y* 
 Values, theieanfes ^ ~ - __, - 
 
 o 
 
 which, ordered, gives x z -\-2l<y ^*r:o, and therefore 
 x bi/2 b. 
 
 a* 4^4-.-?^ + ^ 
 
 Moreover, -^ being , __ ; , if ^* 2^ be 
 
 fubflituted herein for, its Equal, #% it will become 
 SP + lx 
 
 _ 
 / i " " bx x~~ ~~ 3* b 
 
 4+ V/2X4+ 3V/2 _22+l6v/2 
 - 
 
 = n -f 8 V 2. 
 
 ---- - ---- x - 
 4+3/2X4+3^2 _ 2_ 
 
 Hc-ncc we have, i : 1/11+8/2 : \b (DC) : a (AC) 
 : : Radius to the Tangent of the Angle ADC = 78 3': 
 Whofe Complement DAC =r 11 57', is the leaft Li- 
 mit poffible. Therefore, unlefs the Angle which the 
 flant Side makes with the Axis be lefs than n 57", the 
 grcateit Ellipfis will be lefs rhan the Bafe of the Cone. 
 
 EXAMPLE XVIII. 
 
 42. Of (ill Triangles^ having the faxii given Perimeter ', 
 . Inscribed in the fame given Circle j to determine the 
 g'talejl. 
 
 Let the Diameter DA bifefl the Bafe BC of the re- 
 quired Triangle BEC in H, draw AE, AB and BD ; 
 draw AF perpendicular to BE, and G, parallel to
 
 de Maximis & Minimis, 
 
 BC, meeting AD in G : 
 Then, putting AD r= , 
 half the given Perimeter 
 of the Triangle = b, and 
 DHiry ; We have BH ~ 
 Vay y*, and 
 
 over DH (y) 
 :: DB : 
 
 therefore 
 . More- 
 AD (a) 
 
 :: EF* 
 
 "~ x 
 
 , 
 
 * <* 
 
 B 
 
 therefore AG 
 
 > 
 
 whence 
 
 the Triangle BEC (BHx HG) = ** 
 
 and HG = 
 
 of 
 
 iba 
 
 4 2by, whofe Fluxion iby 
 
 being put = o, 
 
 
 
 gives yVayyy = ^ ba; whence y, and from thence 
 the Sides of the/Triangle may be determined. 
 
 EXAMPLE XIX. 
 
 43. To determine the greateft Area that can be contained 
 under four given Right-lines. 
 
 Though it is dc -nonftrable from common Geometry 
 that the Area will be a Maximum, when the Trape- 
 zium ABCD, formed by (he given Lines, may be in- 
 fcribed in a Circle b , yet I fliall here give the Solution 
 from the Principles of Fluxions, (whole Ufts I am now 
 
 a By Prop. 13. Page 62. Elem. Trig. 
 k See Page 117. of Elem. Geometry. 
 
 illuftrating). 
 
 39
 
 40 
 
 Solution of Problems 
 
 illuftrating. In order to which, let the Diagonal AC 
 be drawn, and upon CB and AD let fall the Perpendi- 
 culars AE and CF j putting AB=*, EC =l>, CD=c, 
 
 and DF = y : 
 Then AE being 
 =VV A-% and 
 
 CF = vV-/, 
 
 the Area of the 
 Trapezium 
 (|BC x AE + 
 
 *Art.22. and its Fluxion 
 
 = o 
 
 and therefore 
 
 dyy 
 
 bxx 
 
 Moreover, 
 
 (=AC l ) n^-ft* iJy, by taking 
 the Fluxion thereof, we have ibx =. 2</y, or dj= 
 bx } which, fubftituted for dy in the foregoing Equa- 
 
 bxy bxx y 
 
 tion, gives : = ./ . . and 
 
 and confequently, V^ .,* ( CF ) : 
 
 (DF) :: v * l (AE) : * (BE) : From which it 
 appears that the Triangles DCF and ABE are fimilar^ 
 2nd that (D-f ABC being = 2 Right-angles) theTrape- 
 Z-ium may be infcribed in a Circle; but this by the Bye. 
 XVe are now to get an Expreflion for the Area in known 
 Terms, and in order thereto we have 
 
 dd+fidy,y =-, and CF = ^"" 
 
 :BE:: DC : D 9 &c.) : Therefore, by Subftitution, P 
 
 .v=^--r* -, and the Area (jBCxAE 
 
 -HAD
 
 de Maximis & Minimis. 
 
 cd 
 
 -fAD x CF) = yVf x 3 - + V a 1 -r- x- = 
 
 ob -f- cd 
 
 V x"- ; and therefore the Square thereof =; 
 
 ab + cd}" ^ 
 
 4 
 
 But fince 
 
 tf a* 2ab -f 2cd + 
 
 lab + led 
 
 lab -f led 
 
 [ z _ ~7 
 
 _ 
 ' 2 ~~ 
 
 1L } and confequently, the Square of the 
 
 which becaufc 
 
 the Difference of the Squares of any two Quantities is 
 equal to a Rectangle under their Sum and Difference) 
 
 11 ir v _ d+c+b a* d+c b + axb + a + ac 
 will alfo be . 
 
 b->[ a d + 
 
 - a x 
 
 xi^+ it + ib + iac X -W+ ic +& + \4-fl. Whence 
 it appears, that, if from i the Sum of all the four Sides 
 each particular Sjde be fubtracted, the continual Pro- 
 duel: of the Remainders will be the Square, or fccond 
 Power, of the Area. 
 
 From this Theorem, the Rule in common Practice, 
 for rinding the Area of a Triangle, having the three 
 Sides given, is deduced, as a Corollary ; For, making
 
 42 Solution of Problems 
 
 a n o, the Trapezium becomes a Triangle, and the fe- 
 cond Power of it3 Area zrTT+lT+lixp 
 
 lb ^7 x d+ ic + \bd: Which, in Words, 
 is the common Rule. 
 
 EXAMPLE XX. 
 44. To find the great eft Value of y in the Equation <?*#*= 
 
 By putting the whole Equation into Fluxions, &c. 
 
 we have 2a*xx^.2xx-t-2yy X 3 X x z -f y'~1 a which in the 
 
 Art.az, required Circumftance, when y o *, becomes 2,a*xx 
 
 6x* x x*+y*f ; whence x*+y*=z -7-, and x + /I 3 
 
 6 ^ 
 
 ;= 7- : But, by the given Equation x*+ y ' 3 = fl 4 #*; 
 3V 3 
 
 confequently a 4 ^r 2 . and therefore x =. 
 
 3v/3 \ 
 <7 */ ; whence y r ( ~r~ .v z 1 T~ 2nd 
 
 jr = a</~-L. 
 
 The fame other-wife. 
 Since xx + yyV is -given = a*x~^ we have x' + y 1 = 
 
 4- a 4 * 
 
 a 1 X 4* 7 , and therefore jp 1 c= a T X .V T # a ; whofe Fluxion, 
 
 4- i /^"^" V \f ^ 
 
 |ff 3 X.v ^ 2A'x, being put = 0, we alfo get - f 
 
 a * x X I a* 
 
 rr .v ; whofe Cube is =: x 3 , or ^- x* - t 
 
 whence 27* 4 =rfl 4 , and confequently x 
 
 ' 
 the fame as before. 
 
 ,
 
 de Maximis & Minimis. 
 
 45. When, in the general Expreffion, whofe Maxi- 
 mum or Minimum is fought, there are two or more in- 
 determinateQuantities, independent of each other, their 
 refpcdive Values, in the required Circumftance, will 
 be determined, by making them flow, one by one, while 
 the others are fuppofed invariable ; as in the following 
 
 EXAMPLE XXL 
 
 IPherein it is propofed to fini three fucb Values of *, y y 
 and z, as Jhall make the Value of b* x 3 X x^z z 3 
 X X y y z the greatefl pojffible. 
 
 Firft, confidering y as variable, and the reft conftant, 
 we have xj> iyy=Q * ; whence^ = ?x 9 and xy /== 
 ^x~. By making z variable, we have x*z yfx := o j 
 
 
 whence z = , and x^z z 3 = - . Now let thefe 
 Values of xy ^ 2 and x x z z 3 be fubftituted in the given 
 
 X* 2X 3 _ 
 
 Expreffion, and it will become x -7- x t> 3 x 3 r: 
 
 4 3v 3 
 
 3^5 _ x * 
 
 7-7 - j therefore ^b 3 x*xSx 7 ^o: Whence x = 
 6/3 
 
 i* X V 5 J ( = W =i* * v/S, and z (= =!*x 
 
 V/3* 
 
 The Reafon of the foregoing Procefs is obvious : 
 For, if the Fluxion of the given Expreffion, when any 
 one of the indeterminate Quantities is made variable, be 
 not equal to Nothing, that Expreffion may become 
 greater, without altering the Values of the reft, whkh 
 are confidered as conftant f: And therefore cannot be | Art. at, 
 the greateft poffible, unlefs the (aid Fluxion is equal to 
 Nothing.
 
 44 Solution of Problems 
 
 EXAMPLE XXII. 
 
 46. To determine the different Values of .*, when that of 
 3#* 28ax 3 + $4.a*x* 96<3 3 A- + 48* becomes a Maxi- 
 mum or Minimum. 
 
 The Fluxion of the given Expreflion being (as ufual) 
 put equal to Nothing, we have i2x 3 84.v a + i68<7 a # 
 g6c 3 = o, or x 3 jax'*+ \\a i x 8a 3 = o : From 
 whence (by the Method of Divifors) we get x a o, 
 X 20 O, or x 40 o : Therefore, the Roots of the 
 Equation, or the three Values of x, are , 20, and 
 4*. 
 
 SCHOLIUM. 
 
 47. It appears, from the laft Example, that a Quan- 
 tity may admit of as many Maxima and Minima (ac- 
 Artzz. cording to the Meaning of the Definition *) as there 
 are poffible Roots in the Equation, arifing from af- 
 fuming its Fluxion equal to Nothing. Now to know 
 which of thofe Roots point out a Maximum^ and which 
 a Minimum ; find whether the Value of the faid Fluxion, 
 a little before it becomes equaj to Nothing, be pofitive 
 or negative j if pofitive^ the fucceeding Root gives a 
 Maximum ; but if negative^ a Minimum : The Reafon. 
 of which is extremely obvious ; becaufe fo long as any 
 Quantity increafes, its Fluxion is pofitive, but when it 
 decreafes the Fluxion is negative. 
 
 As an Example hereof, let the Quantity 3** 2$ax 3 
 4- 84<z a * a qba 3 x -f 48^*, be again refumed ; whofe 
 
 Fluxion is 2xXy 3 jax~+ \^a r x 8a 3 I ix X x a x 
 
 x 2a X * 3: Whereof the Value, before it becomes 
 equal to Nothing, the firft time (or before x a) being 
 negative (becaufe the Product of three negative Factors 
 is negative) its firft Root (a) therefore indicates a Mi- 
 nimum : Whence we may conclude, without confiderr 
 ing farther, that the fecond Root (20) gives a Maxi- 
 and the third (40) another Minimum. But, if 
 
 you
 
 de Maximis & Minimis. 45 
 
 you wouM know whether the firft or third Root gives ~~~ 
 the lefler Value of the two ; it is but fubftituting in the 
 given Quantity, which will come out 48^* 37 4 and 
 48^* 644* refpe&ively; therefore the latter is the lef- 
 fer, and the very kaft Value the propofed ExprefHon can 
 admit of. 
 
 When all the Roots prove impofllble, the Quantity 
 propofed (as its Fluxion can never become rro) muft 
 either increafe, or decreafe, continually ; and therefore 
 can neither admit of a Maximum nor a Minimum, 
 
 Moreover, it may fo happen, that the Roots are pof- 
 fible, the Fluxion = o, and yet the Quantity itfelf be 
 * neither a Maximum nor a Minimum in that Circum- 
 ftance. 
 
 For let us, again, fuppofe the Point n to move after 
 tn, as in the general Illuftration, (vid. Art. 22.) only 
 let the Velocity of n (in the firft Cafe} increafe no 
 longer than 'till it arrives at D ; after which let it again 
 decreafe : Then, though the Fluxion of the Diftancc 
 mn is Nothing, at the Pofition CD, yet the Diftance 
 itfelf will not be a Maximum ; becaufe n (having after- 
 wards, as well as before, a lefs Velocity than m) will 
 ftill continue to lofe ground. In the fame manner the 
 Matter may be explained with regard to a Minimum. 
 And it is evident, that thefe Cafes will always happen 
 when the Fluxion of the given Quantity is of the fame 
 Denomination (with regard to pofitive and negative) 
 both before and after, it becomes equal to Nothing : 
 Which, by the Rules of common Algebra, is known 
 to be when the Equation admits of an even Number of 
 equal Roots. An Example hereof, however, may not 
 be improper. 
 
 Let then the Quantity propofed be 240** $oa l x z 
 f i6ax 3 3** ; whofe Fluxion is 244** 6oa*xx -f- 
 4&** 1 * I2A 3 * = i2A- x a xXa x X iax: Which 
 being made =o, it appears that the two leaft Roots are 
 equal. Therefore there is neither a Maximum nor Mi- 
 nimum when x=:a (becaufe whether x be taken a little 
 lefs, or a little greater, than a, the Value of the Fluxion 
 6 will
 
 46 Solution of Problems 
 
 will ftill be affirmative.) The greateft Root, however, 
 not being affected with another equal one, indicates a 
 Maximum^ according to the Rule above prefer ibed. 
 
 To render what has been obferved above ftill more 
 confpicuous, let the given Exprefiion, 2$a 3 x ^oa'x 1 
 -f i6ax 3 3X 4 , be reprefented by the variable Ordinatc 
 PCLof the Curve AQMNRj whofe Abfcifla AP is (as 
 ufual) denoted by x. 
 
 Then, whilft (iix x a x x a x x ia x) the 
 Fluxion- of the Ordinate continues pofitive, (or till x 
 becomes = a = AB) the Ordinate itfelf will increafe : 
 But at the Pofuion BM it becomes ftationary (if I may 
 be allowed the Expreffion) the Fluxion being then = o. 
 After which, the Fluxion being again affirmative, the 
 Ordinate will again increafe, till * becomes r: za ( =. 
 AC) j when, the Fluxion "becoming Nothing, a fe- 
 
 fM 
 
 cond time,) and afterwards negative, CN will be a 
 Maximum: Soon after which the Curve defcends be- 
 low its Axis, and continues to recede from it in in- 
 fimtum, 
 
 Anotfier Thing there is that ought to be regarded in 
 the Solution of thefe Kinds of Problems, and that is, 
 whether the Maxima or Minima^ found by affutrung 
 the Fluxion =r o, fall within the Limits prefcribed by 
 the Nature of the Queftion or Figure ; which is often 
 reftrained by Conditions that do not enter into the al- 
 gebraic Computation. 
 
 Thus, for Example; fuppofe it were required to find 
 that Point (F) in a given Ellipfis ABHD which, of all 
 
 others,
 
 de Maximis & Minimis. 
 
 others, is the moft remote from the Extreme B of the 
 conjugate Axis BD. 
 
 Then, drawing 
 FE parallel to the 
 Tranfverfe AH, and 
 putting AH = fl, BD 
 = , and BE=r*, we 
 have, by the Property 
 of the Curve BF Z 
 = BE* + EF 2 ) ** 
 
 47 
 
 whence x is found n 
 I ,i. But, from the Nature of the Figure, the 
 
 greateft Value that x (r=BE) can poflibly admit of is 
 b ( =BD), therefore if the Relation of a and b be fuch, 
 
 that I ,1 is greater than b, this Solution is manifestly 
 
 impoffible. -* To determine the Limit, therefore, 
 
 make -7 rt, = b; then it will be found that 2 a = r. 
 a b 
 
 Whence the foregoing Solution can only obtain when 
 2BD* is equal to, or lefs than AH*. 
 
 Again, it ought to be alfo confidered whether the 
 Value of x, found by the common Method, gives a lefs 
 Quantity for the Maximum, and a greater for the Mini- 
 mum, than will arife from the Extremes themfelves by 
 which x is limited. 
 
 Thus, let it be 
 required to deter- 
 mine the greateft 
 and leaft Ordinates 
 in a Curve, APR, 
 whofe Equation is 
 y 3 =. fafx qax' 1 -}- 
 4# 3 , and whofe 
 greateft Abfcifla 
 AD is given equal 
 za.
 
 Solution of Problems 
 
 Here we fhall, by taking the Fluxion, &c. have *=s 
 4-0, or x~a : The former of which Values gives the cor- 
 
 3 > 
 
 refponding Ordinate BP = a \/ L . an( j t h e latter,CCL 
 
 4 
 zr : But the firft of thefe is not the greateft of all 
 
 others, becaufe the Extreme DR exceeds it, being = 
 20 ; nor is CQ_the leaft poflible, becaufe the Ordinate 
 at the other Extreme A is nothing at all. 
 
 Sometimes one, or more, of the Points Q_, S, &c. 
 determining the Maxima and Minima^ will fall below 
 the Axis AF, (as in the annexed Figure). In which 
 Cafe the correfponding Value of the general Expreflion, 
 reprefented by the Ordinate, will be negative : But at 
 the Points , c t d, &c. where the Curve interfets the 
 
 Axis, it will be equal to nothing : Whence (by the 
 Bye) the Reafon why the Roots of an Equation (x 
 
 ax*~ l + b l x"~ % .... +/ ) are imp*ffibleby Pairs 
 is evident. For, feeing Ab, A<r, Ad, A* 1 , We. are the 
 Roots of that Equation, or the different Values of A, 
 
 x-x " * . 71 " * I ** 
 
 when the Ordinate * ax +b x + q 
 
 (MN) becomes equal to Nothing, it is plain, ifP/i, 
 
 exprefling the given Term q , be increafed to Pa, fo 
 that AF (then coinciding with af) may touch the Curve 
 in S, the adjacent Roots Ad and A* will then become 
 
 equal -,
 
 de Makimis & Minimisi. 49 
 
 CqUal ; and if q be farthrr increafed, fo that the Axis 
 may fall wholly below the Curve, not only thofe two, 
 but alfo the other RootSj Ab and Ac, will become im- 
 poffible. 
 
 Various other Obfervations might be made, relating 
 to the Limits of Equations, determined by thefe Maxi- 
 ma and Minima ; but this being foreign to the Matter 
 in hand, I fhall content myfelf with one Remark more, 
 viz. 
 
 Any ExpreJJion which, being put equal to Nothing, ad" 
 mits of two or more equal Roots, has as many fuc- 
 ceeding Orders of Fluxions equal to Nathing, at the 
 fame time, as are expreffed by the Number of thofe Roots 
 minus one. 
 
 Thus, an Equation, having three equal Roots, has 
 both its firft and fecond Fluxions equal to Nothing^ 
 when the Fluent itfelf is equal to Nothing. 
 
 Hence we have another Way (befides that given -f- 
 above) to know when a Quantity may have its Fluxion 
 equal to Nothing, and yet neither admit of a Maximum 
 nor a Minimum: For, fmce this Circumftance always 
 takes place when the Equation admits of an even Num- 
 ber of equal Roots (as has been already fliewn) the 
 Number of Orders of Fluxions, equal to Nothing, 
 at the fame time (including the Firft) muft alfo be 
 even. 
 
 Hence, alfo, we have an eafy Method for difcovering _^. 
 when fome of the Roots of an Equation are equal j 
 and, if fo, what they are. 
 
 Thus, let x 3 Tflx 1 + 4fl 3 o be propounded ) 
 whereof the Fluxion 3* 1 * daxx being aflumed equal 
 to Nothing, we find x~ 20 ; which will alfo be a Root 
 of the given Equation, if it admits of two equal ones : 
 To try it, therefore, I fubftitute 20 for x, and find it 
 anfwers. 
 
 Again, let 8** 280**+ i8a*** + ija'x ija+^o ; 
 whereof the firft and fecond Fluxions being $2x 3 x 
 $4ax*x + sbafxx -f 2j a 3 x and 96*^ ibZaxx 1 + 
 36flV, if the latter of them be aflumed = o, x will 
 
 E be
 
 Solution, of Problems, &c. 
 
 be found = + ^/EpL 3f, O r. One of which 
 
 64 2 4* 
 
 Quantities, if the Equation propofed admits of three 
 equal Roots, will be the Value of each of them : By 
 
 trying , it will be found to fucceed. Whence, by a 
 well known Rule, the fourth Root (being = 
 
 O 2 
 
 X 3 = a) is alfo given. 
 
 The Reafon of thefe Operations, as well as what is 
 aflerted above, maybe thus demonft rated. 
 
 Let r x x r x &c. x A + B# + CA-* c5V. = O, 
 be any Equation, having two or more equal Roots, re- 
 prefented, each, by r : Put> r; t #, and let the Num- 
 ber of the equal Roots be denoted by n; then, by Sub- 
 
 ftitution, we have /xA-tBxr ^ + Cxr y* &e. 
 = o ; which, by expanding the Powers of r y, and 
 putting a =. A 4- Br+Cr 2 &V. b =. B -f 2Cr -f 3Dr% sV. 
 
 will be further transformed to / X a h + O' 1 *? & ft 
 zr : Whofe Fluxion najy*~~ n+i . bjy -f n + 2 . 
 
 cyy* &c. is evidently equal to Nothing, when y, or 
 its Equal r *, is Nothing (provided be greater than 
 Unity. It is equally plain, that the fecond Fluxion 
 
 a , a - . ., ., __ t 
 
 . n*i . . i"~-i . 
 
 n . I . ay y n -f I . nby y +n+2..n+l .cy y 
 &c. will alfo be equal to Nothing, in the fame Cir- 
 cumftance, if be greater than 2, &c . &c. 
 
 Hence, univerfally, let the Number () of equal 
 Roots be what it will, that of the Orders of Fluxions 
 equal to Nothing, at the lame time, will be exprefied 
 by that Number minus one, as was to be (hewn. 
 
 SECT.
 
 Ufe O/TLUXTIONS in drawing 'Tangents 
 to Curves. 
 
 ILLUSTRATION. 
 
 48. T E T ACG be a Curve of any kind, and C 
 \_j the" given Point from whence the Tangent is 
 to be drawn. 
 
 Conceive a Right-line mg to be carried along uni- 
 formly, parallel to itfelf, from A towards Q_, and let, 
 at the fame time, a "oint p fo move in that Line, 
 as to defcribe, or trace out, the given Curve ACG: 
 Alfolet mnz, or Cn (equal and parallel to mm) express 
 the Fluxion of Am^ or the Celerity wherewith the Line 
 mg is carried ; and let S exprefs the correfponding 
 Fluxion of ;/>, in the Pofition mCg, or the Celerity of 
 the Point />, in the Line mg. Moreover, through the 
 Point C let the Right-line SF Ire drawn, meeting the 
 Axis of the Curve (^AQ.) in F. 
 
 E a Novr,
 
 Ufe ^FLUXIONS 
 
 Now, it is evident, if the Motion of p, along the 
 Line mg, was to become equable at C, the Point p 
 would be at S, when the Line itfelf had acquired the 
 Pofition mSg (becaufe, by Hypothefis, Cn and n$ ex- 
 prefs the Diftances that might be defcribed by the two 
 uniform Motions in the fame time). 
 
 And, ifwsg be aflumed to reprefent any other Pofi- 
 tion of that Line, and s the contemporary Pofition of 
 the Pointy (ftill fuppofing an equable Celerity of />) ; 
 then the Diftances Cv and w, gone over, in the fame 
 
 F 
 
 in ro 
 
 Q 
 
 time, by the two Motions, will, always, be to each 
 other as the Celerities, or as Cn to S : Therefore, 
 fmce Cv : vs :: Cn : S (which is a known Property 
 of fimilar Triangles) the Point s will, always, fall in 
 the Right-line FCS : Whence it appears, that, if the 
 Motion of the Point p along the Line mg was to become 
 uniform at C, that Point would then move in the Right- 
 line CS, inftead of the Curve-line CG. 
 
 Now, feeing the Motion of p, in the Defcription of 
 Curves, muft, either, be an accelerated or a retarded 
 one, let it be, firft, con fide red as an accelerated one : 
 In which Cafe the Arch CG will fall, wholly, above 
 the Right-line CD (as in Fig. i.) becaufe the Diftance 
 
 of
 
 in drawing 'Tangents. 3 
 
 of the Pointy from the Axis AQ_, at the End of any 
 given Time, is greater than it would be if the Accelera- 
 tion was to ceafe at C ; and, if the Acceleration had 
 ceafed at C, the Point p would ( it is proved ) have 
 been always found in the faid Right-line FS. 
 
 But if the Motion of the Point p be a retarded one, 
 it will appear, by reafoning in the fame manner, that 
 the Arch CG will fall wholly below the Right-line CD 
 (as in Fig. 2.) 
 
 This being the Cafe, let the Line mg^ and the Point 
 p t along that Line, be now fuppofed to move back 
 again, towards A and m t in the fame manner they pro- 
 ceeded from thence : Then, fince the Celerity of p 
 (Fig. i.) did before increafe, it muft now, on the con- 
 trary, decreafe ; and, therefore, as p-, at the End of a 
 given Time, after repaying the Point C, is not fo near 
 to AQ_, as it would have been, had the Velocity con- 
 tinued the fame as at C, the Arch Ch (as well as CG) 
 muft fall wholly above the Right-line FCD. And, by 
 the fame Method of arguing, the Arch C/>, in the ft- 
 cond Cafe^ will fall, wDolfyy below FCD : Therefore 
 FCD, in both Cafes, is a Tangent to the Curve at the 
 Point C : Whence, the Triangles FmC and OzS being 
 fimilar, it appears, that the Sub-tangent ;F is always 
 a Fourth-proportional to (nS) the Fluxion of the ordi- 
 nate (C), the Fluxion of the Abfciflfa, and the Ordi- 
 nate (Cm). 
 
 Otherwife. 
 
 49. Let ACG reprefent the propofed Curve, and let 
 the Right-line FCD be a Tangent to it, at any Point 
 C, meeting the Axis AQ. (produced if neceflary) in 
 F : ^uppofe a Point p to move along the Curve, from A 
 towards G, and let the abfolute Celerity thereof at C, 
 in the Direction of the Tangent CD, or the Fluxion of 
 the Line A/> fo generated *, be denoted by CS, any, Art<1 
 Part of the faid Tangent : Then, if AH, mp and roSaod 5. 
 be made perpendicular, and Ipn parallel, to AQ_, the 
 relative Celerities of that Point, in the Directions Cn 
 and mC, wherewith lp (=. Am) and mp increafe in this 
 E 3 Pofition,
 
 of FLUXIONS 
 
 Art.35- Pofition, will be truly exprefled by Cn and S * : But 
 the Celerities by which Quantities increafe are as the 
 Fiiixions of thofe Quantities : Therefore (CS be- 
 ing the Fluxion 
 TT /I of the Curve-line 
 
 i > 
 
 Ap) 
 
 and S 
 
 m- m a 
 
 Cn 
 
 are the corre- 
 fponding Fluxions 
 of the Abfcifla 
 Am and the Or- 
 dinate mp : And 
 we have Sw : nC 
 : : mC : mF, the 
 fame as before. 
 
 Hence, if the 
 Abfcifla Am be 
 put r= #, and the 
 Ordinate mp = y t 
 
 we fhall have mF = : By means of which general Ex- 
 
 preffion, and the Equation expreffing the Relation be- 
 tween x and y, the Ratio of the Fluxions x and y will be 
 found, and from thence the Length of the Sub- tan gent 
 (mf) as in the following Examples. 
 
 EXAMPLE I. 
 
 50. To draw a Right-line CT, to touch a given Cirdt 
 BCA, in a given Point C. 
 
 Let CS be perpendicular (o the Diameter AB, and 
 
 put AB rr /?, 
 BS = x and SC 
 = y : Then, by 
 the Property of 
 the Circle, y* 
 (CS 1 ) = BS x 
 
 AS ( xXa A-) 
 == ax x 1 ; 
 whereof
 
 in drawing Tangents. 55 
 
 whereof the Fluxion being taken, in order to determine 
 the Ratio of x and y t we get iy'y ax 2xx j confe- 
 
 x 2y y 
 
 quently = *" =r ; which, multiplied by y, 
 ' y a 2x ?a x 
 
 vx v*" 
 
 gives *r = J = the Sub-tangent ST *. Whence * Art< 4t 
 
 a and AQ. 
 
 (O being fuppofed the Center) we have OS (~a x) : 
 CS (y) : : CS (y) : ST j which we alfo know from 
 other Principles. 
 
 EXAMPLE II. 
 
 51. To draw a Tangent to any given Point C of tht co- 
 nical Parabola ACG. 
 
 If the Latus Reflum of the Curve be denoted by a t 
 the Ordinate MC by;-, and its correfponding Abfcifla 
 
 AM by x ; then the known Equation, expreffing the 
 Relation of * and y, being ax =y z , we have, in this 
 
 x 2 y y^l" 
 
 Cafe, ax = iyy j whence =: , and confequently ?- f Art.48 
 
 2 y* 2ax 
 
 = ~=:-- = 2x = MF. Therefore the Sub-tangent 
 a a 
 
 is juft the double of its correfponding Abfcifla AM : 
 Which we likewife know from other Principles, 
 
 E < EX-
 
 56 tfbe Ufe of FLUXIONS 
 
 EXAMPLE III, 
 
 52. To draw a Tangent to a Parabola of any kind. 
 The general Equation of thefe fort of Curves being; 
 
 m n i-4-B , "* n i . m-^-n i . 
 
 a x =zy , we have na x x m + n X y j, 
 
 ^ i .. 
 
 ^ , 
 
 x Bi+nX^ , yx 
 
 and therefore -r = _ ; ; whence -r- j^ 
 
 v m B j y 
 
 ^ 72fl A 1 * 
 
 a (becaufe y " =: %") =: 
 
 IB a 
 
 na x na x 
 
 - X x the true Value of the Subtangent : Which, 
 
 therefore, is to the Abfcifla, in the conftant Ratio of 
 m -{- n to n. 
 
 EXAMPLE IV. 
 
 53. Tt draw a Tangent RT, to a given Ptint R, in a 
 given Ellipfts BRA. 
 
 If RS be an 
 Ordinate to the 
 principal Axis 
 AB, and there 
 be put (as ufual) 
 
 ABr:<7, and the 
 
 letter Axis = b ; we fhall, by the Property of the Curve, 
 have a* : b* : : axx r (BS x AS) : / (RS*), and there- 
 fore Z>*X ax x^a^y 1 : Whence Fxax 2xx=: 
 
 and -7- = -. - . ..- j and confequently the Sub-tangent 
 y b x 2x 
 
 Art.49- ST f _
 
 in drawing Tangents* 57 
 
 . Whence the Point T being given, through 
 
 which the Tangent muft pafs, the Tangent itfelf may 
 be drawn. 
 
 But if you would derive an Expreffion for the Sub- 
 tangent, in any other kind of Ellipfes (befides the coni- 
 cal) let the Equation a JP Xx"~ + X y" , exhibit- " 
 
 ing the Nature of all Kinds of Ellipfes, be aflum- 
 ed : Then, by taking the Fluxion thereof, you will 
 
 i . vn I n . n i , 
 
 have mx x a x\ x* + nxx x a x] 
 
 m }- n I VX 
 
 y y; and therefore 
 
 x m -f- n X y 
 
 m X a *) X x + nx 
 
 m + ny. a *) X * 
 
 n i -- . 
 
 xa x 
 
 (becaufe f- x 
 
 OT-fWXfl - 
 
 na 
 
 j which is the Sub-tangent required. 
 
 EXAMPLE V. 
 
 54. To draw a Tangent, to any given Point R, in a given 
 Hyperbola BR. 
 
 If a and r be put to denote the two principal Dia- 
 meters of the Hyperbola, the Equation of the Curve 
 be c*x.v + **=*>* : From whence we have c* x 
 
 ax +
 
 T&e Ufe ^FLUXIONS 
 
 t * a y 
 
 x J<2 $> " ~r = T""",' '^7^1 and confequent- 
 
 V r X /7 ^J v 
 
 ^/ * ^N 2," ^r * 
 
 - ST - 
 
 Whence BT (ST 
 
 BS) rz - 
 
 - - 
 
 is alfo 
 
 T !B S known ; and there- 
 
 fore the Point T being given the Tangent RT may b% 
 drawn. 
 
 The Manner of drawing Tangents to all Sorts of 
 Hyperbolas, univerfally^ will be the fame as in the El- 
 lipfes, the Equations of the two Kinds of Curves dif- 
 fering in Nothing but their Signs. 
 
 EXAMPLE VI. 
 
 55. Let the prspofed Curve le that whofe Equation is 
 
 Then we fhall have 2axx -f- >** + ixyy -f 
 O j therefore 2axx +y"x 
 
 ., and confidently*? = 
 " ] y 
 
 E X-
 
 in drawing Tangents* 
 
 59 
 
 EXAMPLE VIL 
 
 56. Let the given Curve be the Ciffoid of Diodes, whofe 
 Equation is / = ^ = ^. 
 
 ?*?xXa x + xx 3 '^ax i x^x*x 
 Here we have lyy = ^ . . = .= ==. - : 
 
 Whence 4-= ^^" ^3, and confequently the Sub- 
 tangent f J = 2 '- x j~ 5 =: 
 
 IX X fl AT 
 
 . EXAMPLE VIII. 
 
 57. irf /^ Conchoid 0/*Nicomedes be propofed; where- 
 of the Nature is fuch, that, if from a Point B, called 
 
 R 
 
 the Pole, any Number of Right-lines, BA, BR, 
 BR, fcf<r. be drawn, the Parts of thofe Lines CA, 
 vR, UR, Wt. intercepted by the Curve and its Axis 
 CT } (hall be, all, equal to each other. 
 
 la
 
 Ihe Ufe of FLUXIONS 
 
 In this Cafe (fuppofmg AB and RS perpendicular, 
 and RH parallel, to CT ; and putting BC == a, Ri; 
 (AC) b, CSrr^r, and RS y) we have, per ftm, 
 
 . a+y (BH) : x (RH) :: y (RS) : _ = Sv .- 
 But Sv (/RV- RS 1 ) is alfo = VP-f- j therefore 
 
 - V/fi^f, or x\y*=a+y\' t X "^/ is the 
 fl*f"_y 
 
 general Equation of the Curve; which, in Fluxions, 
 gives ix^yy + 2y z xx = 2j xa+yx b* y 1 2yj X 
 
 X 
 
 i>* ay 2v x : and therefore -r- =s 
 
 y 
 
 confequently = 
 
 y X. 
 
 = 7 -- : Which being a negative Quantity, the 
 yV bbyy 
 
 Tangent will therefore fall on the contrary Side of the 
 Ordinate, from the Vertex j and fo, by changing the 
 
 Signs we {hall have / for the Sub-tanaent 
 
 y v bbyy 
 
 ST in this Cafe. 
 
 After the Manner of thefe Examples the Sub-tangent, 
 in Curves whofe AbfcifTas are Right-lines, may be de- 
 termined : But if the AbfcifTa, or Line terminating the 
 Ordinate, on the lower Part, be another Curve, then 
 the Tangent may be drawn as in the following 
 
 EX-
 
 in drawing tangents. 
 
 EXAMPLE IX. 
 
 58. Let the Curve BRF be a Cycloid; whofe 
 Abfcifla is here fuppofed to be the Semicircle BPA, to 
 which let the Tangent PT be drawn (as above). More- 
 over let rRH be a Tangent to the Cycloid, at the cor- 
 
 refponding Point R, and let GR* be parallel to TPv ; 
 putting the Arch (or Abfciffa) BP=z, its Ordinate 
 PR=^, AF=, andBPAn;: Then, by the Proper- 
 ty of the Curve, we (hall have c (BPA) : b (AF) :: x 
 
 (BP) :y (PR): Therefore.? = -, and y = ~ = re : 
 But, by fimilar Triangles, re (j) : R^ (= Pv = ) : : 
 PR (y) : PH = ?= z(becaufe^=~). There- 
 
 fore, if in the Right-line PT, there be taken PH, equal 
 to the Arch PB, you will have a Point H, through 
 which the Tangent of the Cycloid muft pafs. 
 
 EXAMPLE X. 
 
 59. Let BPA be a Curve of any Kind, to which the 
 Method of drawing the Tangent cPg is known j Jet 
 
 BRA
 
 be life of FLUXIONS 
 
 ERb be another Curve of fuch a Nature, that the Or- 
 dinatc PR (y) (hall always be a Mean-proportional be- 
 
 tween BS (x) and AS (a x) fuppofing RPS perpendi- 
 *Art48cular to AB : Put Po = x y SP =v, oc = <i> *, and er 
 aad 49. = j : Then, f *w^ er (y) : Re ( = PC =, 
 
 Z!: But, by 
 
 :: RP (y) : PH = 
 
 the Equation of the Curve ^*r=^A xx; whence 2yj z: 
 
 Jf 2AT - 2* 2 
 
 * 2*-^, and n : - -. and therefore rrl =r 
 jr r ' 
 
 e exprefled i 
 ax ixx 
 
 pendent of Fluxions, when the Property of the Curve 
 BP/j, or the Relation of x and v is given : Thus, let 
 BPA b th common Paiabola, and AB its Lotus Rec- 
 
 tum -,
 
 in drawing Tangents* 
 
 turn ; then v being =: V ax, <v w m b _ ~- 
 
 ax 
 
 ax 
 
 _ ? + ff _ ** x 4* + * . an d therefore PH 
 4* 4* 
 
 a x X v 4* 1 -\-ax 
 
 ^xx X 
 
 a ix 
 
 ax 2** 
 
 Thus far relates to Curves whofe Ordinates are pa- 
 rallel to eich other : We come now to Curves of the 
 fpiral Kind, whofe Ordinates all iffue from a Point : 
 Such as the Spiral BAG, whofe Ordinates CB, CA, 
 CG, are all referred to the Point C, called the Center 
 f the Spiral. 
 
 ILLUSTRATION. 
 
 60. Let SAN be 
 a Tangent to the 
 Spiral at any Point A, 
 alfo let CT be per- 
 pendicular thereto, 
 and let the ArchCBA 
 (confidered a variable 
 by the Motion of A 
 towards G) be de- 
 noted by z, and the 
 OrdinateCA byj. 
 
 Then z :y : : AC 
 
 Hence, if upon CA, as a Diameter, a Semi-circle be 
 defcribed, and in it, from A, a Right-line AT equal 
 
 yy 
 to y be infcribed, that Right-line will be a Tangent 
 
 to the Spiral at the Point A. 
 
 EXAMPLE I. 
 
 61. Let the Nature of the Curve CBA be fuch, 
 that the Arch CBA may be, always, to its cor- 
 
 refponding 
 
 7
 
 64. The Ufe of FLUXIONS 
 
 refponding Ordinate CA in a conftant Ratio ; namely 
 as a, to b : Then, becaufe z: y i: a: b 9 we have si =. 
 
 4, i = % , and confequendy AT (4-) = ^ = A x 
 ' V 2 ' a 
 
 AC : Therefore, AC and AT being in a conftant Ra- 
 tio, the Angle CAT muft alfo be invariable. Which ia 
 a known Property of the logarithmic Spiral. 
 
 EXAMPLE II. 
 
 62. Let BAA be the Spiral of Archimedes ; whofe 
 Nature is fuch that the Part EA of the generating Or- 
 dinate, intercepted by the Spiral and a Circle BED de- 
 (cribed about the fame Center C, is always in a conftant 
 Ratio to the correfponding Arch BE of that Circle. 
 
 Suppofe An perpendicular to AC, &c. 
 Put BC Cj CA=^, and let the given Ratio of AE 
 to BE, be that of b to c : Then b : c : : yt ( AE) : 
 
 T = BE : whofe Fluxion therefore is = r Now 
 b b 
 
 if
 
 in Curves of contrary Flexure. 65 
 
 if the Right-line CEA# be fuppofed to revolve about 
 the Center C, the angular Celerity of the generating 
 Point A, in the perpendicular Direction A, will be to 
 that of E as AC to EC ; therefore as the latter of thefe 
 
 Celerities is exprefied by - , the former will be ex- * Art - 5 
 prefled by x y, or'~: Which is to (y) the Celerity 
 
 of A, in the Dire&ion A<7, as - to Unity, or as y to 
 
 b. Therefore CT and AT are in the fame Ratio, (by 
 Art. 35) and confequently AC : CT : : V yy -+ bb : 
 y i and AC : AT : : / vy + bb : b ; whence CT and 
 
 y 1 h 
 
 AT are given equal to , and rr r-n^r re- 
 
 Vyy + bb V yy 4- lb 
 
 fpeflively. From either of which (the Tangent AT) 
 may be drawn by Art, 60. And, in the fame manner 
 may the Pofuion of the Tangent of any other Spiral 
 be determined. 
 
 SECTION IV. 
 
 Of the Ufe of Fluxions in determining the 
 Points of RetrogreJ/ion> or contrary Flexure 
 in Curves. 
 
 : 
 
 63. T T THEN a Curve ARS is, in one Part AR 
 V V concave, and in the other Part RS con- 
 vex, towards its Axis AC, the Point R limiting the 
 two Parts is called a Point of RetrogreiTion, or con- 
 trary Flexure. The manner of determining which will 
 appear from the following 
 
 F ILLUSTRA-
 
 66 
 
 Lye of FLUXIONS 
 
 ILLUSTRA 
 
 TION. 
 
 D 
 
 D S 
 
 n, 
 n 
 n 
 
 Suppofe a Right-line BD to be carried along uni- 
 formly, parallel to itfelf, from A towards C ; and let 
 
 the Point r fo 
 move in that 
 Line, at the fame 
 time, as to trace 
 out, or defcribe, 
 the given Curve- 
 line ARS. 
 
 Then (by Art. 
 48.) while the 
 Celerity of the 
 Point r, in the 
 Line BD, de- 
 creafes, the Curve 
 will be concave 
 to its Axis AC ; 
 
 A. B B B C but when ic in - 
 
 creafes, convex to 
 
 the fame : Therefore, as any Quantity is a Minimum at 
 the End of its Decreale and the Beginning of its In- 
 
 *Art.22. creafe *, it follows that the faid Celerity, at the Point 
 of Inflexion R, muft be a Minimum : Whence, if the 
 
 fArt. 5. Fluxion of the Ordinate Br, exprefling that Celerity f, 
 be (as ufual) denoted by y j then willj' (the Fluxion 
 
 JArt.22. of y) be equal to Nothing in that Circumftance J. 
 
 So far relates to Curves which are, in the former 
 Part concave, and in the latter convex, to their Axes : 
 But if (on the contrary) the Celerity of r fait increafes, 
 and then decreafes, that Celerity, at the required Point, 
 between the Increafe and Decreafe, will be a Maxi- 
 mum \ and therefore its Fluxion (or y) is likewije equal to 
 
 Art.22. Nothing in this Cafe . 
 
 Furthermore, if CS (perpendicular to AC) be now 
 confidered as an Axis, and the AbfcilFa S (or its 
 Complement Br y] be fuppofed to flow uniformly, 
 (as AB was fuppofed before) ; then, by the fame Argu- 
 ment, the fecond Fluxion ( x) of the Ordinate nr 
 
 (or
 
 in CUrves of contrary Flextirt. 67 
 
 (or its Complement AB =. x) will be equal to Nothing. 
 Hence it is evident that, at the Point of contrary Flex- 
 ure, the fecond Fluxion of the Ordinate will become 
 equal to Nothing, if the Abfcifla be made to flow uni- 
 formly 3 and vice vtrfa. 
 
 EXAMPLE I. 
 
 64. Let the Nature of the Curve ARS (fee the pre- 
 
 teding Figure) be defined by the Equation ay a*x~* -f- 
 xx (the Abfcifla AB and the Ordinate Br being, as 
 ufual, reprefented by x and y refpe&ively). Then j, 
 expreffing the Celerity of the Point r, in the Line BD, 
 
 3 _JL 
 
 will be equal to - : Whofe Fluxion, or 
 
 3 l 
 
 that of la 1 *- T -f- 2* (be"caufe a and x are conflant) 
 
 muft be equal to Nothing * ; that is, ^a"-x~^x -f- 2* *Art.6j 
 
 = : Whence <?*/"* = 8, a* = 8*% 64*' = a 1 , and 
 
 - - \ 
 * = |a = AB; therefore BR ( = ****+.** j -, 9 
 
 a s 
 From which the Pofition of the Point R is given. 
 
 EXAMPLE II. 
 
 65. Let the Nature of the propofed Curve be defined 
 by the Equation ayy aax x 9 rz o. 
 
 Then, by taking the firft and fecond Fluxions thereof 
 (fuppofing x conftant) we (hall alfo have zayy aax 
 2**x = o, and lay* + iayy bxxx = o ; whereof the 
 latter, when y is zr o, becomes Zay 1 6xx* =: o, and 
 
 therefore/ = 3-fL; But, by the former y=. * ^* * . 
 
 a *. oav > 
 
 a^x -4- "ixx \ 
 whence PJ- =: = x 9 and confequently 
 
 F 2
 
 68 he life 0/~ FLUXIONS 
 
 zr a* + -x'} * ; but, by the given Equation, izaxy l =r 
 
 12*V -f 12**, therefore I20V* + 12^* =: a 1 -j- 3*'{ 
 or 3-v 4 4- 6^A- a fl* = o: Wherice * will be found = 
 
 _^__ 
 Since <7y* = <?V + ,v s , we have y = ' anc * 
 
 : Whbfe 
 
 therefore y 
 
 Fluxion^ or that of- a i + ^x r X a*x + x 3 )~~ i ( becaufe 
 x is conftant) being put o, we get 6x x aV -f A- 3 ! 
 + ^+3* i x "^ |y a x g^ -}- ^ 3 7 * = o, or 6x x 
 Fx a 3 ' Y : Whence 3*? 4 -f6a 2 Ar* 
 
 a* =. o, and ^ rr ^ V : V ^ ij the fame as be- 
 fore. 
 
 EXAMPLE III. 
 
 66. Let the propofed Curve be the Conchoid of A r ;'- 
 (omedes 9 whereof the Equation is x*y* =: a + y^ x 
 
 Art-57. b* i a *. or x* =. r -^-. 
 
 * * *- 
 
 Here
 
 in Curves of contrary Flexure. 69 
 
 . 
 
 Here we have awn: 
 
 Jg* X +" X J*/ 
 
 a o nb 
 
 5 1 a y x y : Whence, making y inva- 
 
 'ifl^b 3 " "iab 1 
 nable, we alfo have x +xx ' -f- i x j 1 : 
 
 Which, becaufe A? is = o*, will be x 1 = ^ h 1 Art.6j. 
 
 r V* 
 
 X j*. But fmce, by the 
 
 Y 
 
 r IT a V a v 
 
 former hquation, xx =. -- ^ - - ^L x^, we like- 
 
 r 
 
 "^^ ^~"^^~~ ^ 
 wife get ^* = f > X g +^ x ^ and con f e q uent i v 
 
 %a i b i +2aK i 'y _y*x AT*/ a-f->l XS^t*/j : But, by 
 the Equation of the Curve x^* is &-\-y] X ^ 1 v x ; 
 therefore 3g^ 1 -j-2^ a > _y 4 X c -f ^1* x ^ l y* =r 0+^* 
 X atf" + > 3 1% and 3<? ^* + 2fl/' a y ^ 4 xi l ^r a atf'-^ y*)*' 
 whence/ + 4<7/ -f $a~y*2ab 1 y 2 a*b 1 - = oj which 
 divided by 7+0, gives / -f- 3^y z 2^ = o ; from 
 whence jr may be determined. But if ba y the Equa- 
 tion will become more fimple by dividing again by 
 y + a-, in which Cafe we get >*-f 2ay ^a L =: o, and 
 confequently y =: a^/ 3 a. 
 
 EXAMPLE IV, 
 
 67 . Let a+y =. i Sca 3 x* 1 1 oa i x 3 + 300** 3**. 
 Then will a*j = ^6oa 3 xx 330^**;* 4- \2Oax 3 x 
 
 F 3 And
 
 7 The Ufe of FLUXIONS, 
 
 And a*J =r 36c* 3 ** 66co s xi* -f 
 Art.6j. Therefore, 6a 3 1 !**.* + 6**'- * 3 = o : 
 
 Which being divifible by any one of the three Quan- 
 tities a *, 2a *, or %a *, the Root * muft there- 
 fore have three Values, a, 20, and 3^, and confe- 
 quently the Curve, denned by the given Equation, as 
 many Points of contrary Flexure. 
 
 But, if you would know whether the Part of the 
 Curve lying between any two adjacent Points, thus 
 found, be convex or concave towards the Axis ; fee 
 whether the Value of the Exprcflion for the fecond 
 Fluxion of the Ordinate, between the two corr fpond- 
 ing Roots, b r>iitive or negative : For, in t.ie former 
 Art. e Cafe, the Curve is convex, and in the 'atler concave f, 
 and 48. (provided the whole Curve lies on the fame Side the 
 Axis). Thus, in ihe Example before us ; becaufe the 
 fecond Fluxion of the Ordinate is a ways as 6a 3 liaax 
 + 6axx x 3 ( =. a^x x 'ia * x 3* x) and it appears 
 that the Value of this Expreflion, while * is lefs than 
 the firft.lJ.oot fl, will be pofitive ; the Curve, there- 
 fore, at the Beginning, will be convex to its Axis : 
 Hut when x becomes greater than a, the faid ExprelTion 
 being negative, the Curve will then be concave, and fo 
 continue 'till x is equal to the fecond Root la ; after 
 which the Fluxion again becoming affirmative, the 
 Curve will accordingly be convex till * $a ; beyond 
 which Limit the Curvature continually tends the fame 
 Way. 
 
 But it will be proper to obferve, that there are Cafes 
 where the fecond Fluxion of the Ordinate may become 
 equal to Nothing, without either changing its Value 
 from pofitive to negative, or the contrary, (fimilar to 
 thofe already taken Notice of in Sefi. II. p. 45 and 46.) 
 which Cafes always happen when the Equation admits of 
 an even Number of equal Roots : And then the Point 
 found as above is not a Point of Inflexion, becaufe the 
 Curvature on either Side of it tends the fame Way. 
 
 SECT.
 
 SECTION V. 
 
 The Ufe of Fluxions in determining the Radii 
 of Curvature, and the Evcfates of Curves. 
 
 68. A Curve /.OH is faid to be the Evolute of ano- 
 jtlL ther Curve ARE, when it is of fuch a Na- 
 ture, that a Thread ROH, coinciding therewith (or 
 wrapped upon the fame) being unwound or difengagod 
 from it, by a Power acting at the End R, ihall, by 
 that End (the Thread continuing tight) defcribe the 
 given Curve ARB. 
 
 ILLUSTRATION. 
 
 From the Point O, where the Right-line RO (called 
 the Radius of Curvature) touches the Evolute />OH, 
 
 B 
 
 let the Semi-circle SRD be defcribed ; which Semi- 
 circle, having the fame Rar.ius with the given Curve, 
 at R, will confequently have the fame Degree of Cur- 
 vature. But the Curvature in two Curves is the 
 
 fame, when, the Fluxions of their Abfciflas being the 
 
 fame, both the Firft, and Second Fluxions of their 
 
 F 4. cor-
 
 72 Of the Radii of Curvature, 
 
 correfporHing Ordinates R and RAH are reflectively 
 equal : j^ch ether : For, the Firft Fluxions being 
 eq"ui, -he two Curves will have, at the common Point 
 
 *Art.4S. R, r-.j and the kii.. Tnn^ent tRh *: And, if the Se- 
 cond Fluxions be likewife equal, the Curvature, or 
 Deflection from ihat Tangent, will alfo be the fame in 
 both ; l-o-'.'ife thefe laft exprefs the Increafe or Decreafe 
 
 fArt.ig. of Motion in the Direction of the Ordinate f, upon 
 
 JArt.48. whic ; . ; Curvature intirely depends J. 
 
 Thi; t:eing premifed, let the Abfcifia Sm of the Semi- 
 circle (confidered as variable) be put = w, its Ordinate 
 Rmv t Rrrrw, rhw^ and R^rza;: Then, R/; be- 
 
 |jArt.48.$ng a Tangent to the Circle at R ||, the Triangles R/;r 
 and ROw will be equiangular, and therefore ay (Rr) : 
 
 1}% 
 
 x (RA) :: v (Rtt) : RO = -- j which, becaufethe 
 
 C W 
 
 Radius of every Circle is a conftant Quantity, muft be 
 
 . .. vx-t-vZ 
 
 invariable, and conlequently its Fluxion : : 
 
 Whence v is found rr ^- =; (becaufe, w being 
 
 z v 
 
 conftant, and ob* -f ^ = % we have, in Fluxions 
 
 , . t ii%. z 1 \ 
 
 ivv 2 -^ and to ~ = r: ). Therefore hnce^ is = 
 
 ~v/ 
 
 * , r iC/ -v or>f' v *\ ^ v- + <w*]? 
 -,wealfogetSO=:RO I - r } r-rrrr ... : 
 
 i) \ *w s ay 11 iv if 
 
 Which laft is a general Expreflion for the Radius of any 
 Circle, \vhatever, in Terms of the Fluxions of its Ab- 
 fcifla (w) and Ordinate (v). But, by what is premifed 
 above, thefe Fluxions are refye&ively equal to thofe of 
 the Abfcifla An (x) and Ordinate R (y) of the pro- 
 pofed Curve ARB. Therefore, by writing x, j, and j>, 
 
 2. |^ , - X. S ,"3 \ 
 
 inftead of ov, y, and i- , we have y +-< l ( _ ' z j 
 
 *y > -H*f^ 
 for the general Value of the Radius of Curvature, RO. 
 
 6 37*
 
 and the Evolute of Curves. 
 
 73 
 
 The fame ctherwife. 
 
 If the Radius of the Circle be put = R, and every 
 Thing e!fe be fuppofed as above; then (by the Property 
 of the Circle) we (hall have v 1 (Rra*) = 2 Rw vf" 
 (S^xD-Tz) : Whence, in Fluxions (making w conftant) 
 we get 2v-i> 2Raw iww, and 2v*+2v'v = 2iw*: 
 
 T. I i 
 
 From the hift of which Equations v is found V 
 
 1> 
 
 *? r i T> ^ (VX- \ Z- 3 % 3 
 
 r. ; and confequently RO I ) ~ .... -. 
 
 v V ov / w v x'j 
 
 the fame as before. 
 
 Otberwtfe without the Circle. 
 
 Let RO and rO be two Rays perpendicular to the 
 Curve, indefinitely near to each other ; and from their 
 Interfeftion O, let OF be drawn parallel to A, cut- 
 ting R and AF (parallel to Rw) in E and F. 
 
 Therefore, fuppofing RErrv, An=x, Rn=y, & c . 
 (as before) we fliall have, by fimilar Triangles, as RP 
 
 F K O 
 
 (x) :Pq (y) :: RE (v) : EO =: ?; and confequently 
 
 FO (A + EO) =x+ -j: Which Value (as well as 
 
 that
 
 74 Of the Radii of Curvature, 
 
 that of AF) continuing the fame whether we regard the 
 Radius RO, or the Radius rO, its Fluxion muff there- 
 
 fore be equal to Nothing ; that is, x+ 
 
 X 
 
 =: p ; whence v = -rr r;. , and confequently RQ 
 
 yx xy 
 
 ' 3 _ , . , 
 
 : Which, if* 
 
 Xy y X X y y X ; 
 
 z 3 
 is fuppofed conftant, or x = o, will become n. as 
 
 alovt. 
 
 But if j be fuppofed conftant, it will be -~~. And, 
 
 xy 
 
 if x be conftant, it will'then be ~: For, fmce ? -f jr* 
 **, by taking the Fluxion thereof, we have 2^'-f 
 
 XX 
 
 2j>'no; whence j' - ; and therefore RO ( = 
 
 . . y 
 
 K, 3 j J )'g 
 
 rrr. = t=r- ^-7 9 as before. 
 
 mjfyj tf v *, -. *.. -U ' -* 
 
 Now from the feveral Values of the Radius of Cur- 
 vature RO, found above, the cortefponding Values of 
 Ae and *O will likewife be given. 
 
 Thus, if x be made conftant; then, RO being =; 
 
 & 3 v ' 
 , we (hall have Ae (A*+OrasA*-f 4 X RO) = 
 
 *y . ' z 
 
 Ar-f-^'and^O (Rw R=4 xRO R) = 
 xy 2 ' y 
 
 y- 
 
 l 
 
 But, if^ be made conftant, then, RO being = -fr, 
 
 2 ** 
 
 we fhall have AE rz x -f , and ^O = -rr- y. 
 
 AT J* * 
 
 Laftly,
 
 ana the Evolute of Curves. 
 
 Laftly, if z be fuppofed conftant ; then RO being 
 vx y xy 
 
 ' -, we fhall have A* =: x 4- , and eO = y. 
 
 3 ' x x J 
 
 Which feveral Expreffions will ferve as fo many ge- 
 neral Theorems for determining the Quantity of Cur- 
 vature, and the Evolutes of given Curves : But, before 
 we proceed to Examples, it will be proper to obferve, 
 that the Right-line A/>, denoting the Radius of Curva- 
 ture at the Vertex A (to be found by making *, or j, 
 == o) muft always be iubtra&ed from RO and A*, to 
 have the true Length of the Arch ^O, and its corre- 
 fponding AbfciiTa/**. 
 
 EXAMPLE I. 
 
 69. Let the given Curve ARBbethe common Parabola, 
 
 ii i_ i 
 
 whofe Equation is y = a*x* : Then will y =a*xx * 
 
 ax 
 , and (making;*- conftant) y ^ 
 
 2** 
 
 
 Whence
 
 Of the Radii of Curvature, 
 and the Radius of Curvature RO 
 
 * *}' 
 
 Which at the Vertex A, where x=o, will be a = 
 Ap. Moreover A* (*+ ~ J {a + y, and there- 
 fore /<r (Ae Ap} 2#, the Abfcifla of the Evolute : 
 
 Likewife Oe ( v) -- the Ordinate of the 
 
 V y ya 
 
 Evolute. Therefore, (5^)* x a being in a conftant Ratio 
 to^Tj 3 , namely as 16 to 27, the Curve is, in this 
 Cafe, the Semi-cubical Parabola : Whofe Arch pO 
 
 (RO Ap} is alfo given = 
 
 E X A M P L E II. 
 
 70. Let the Curve ARB denote a Parabola of any 
 oiher Kind : Then, becaufe y =. ax" is an Equation to 
 all Kinds of Parabolas, we have y nax" x and y = 
 
 n x i x ax x- : Therefore z (v/^+j 1 ) =: 
 
 MV*"*, RO 
 
 A^ ( X + i 
 
 
 1 X nax 
 
 Which, if =4, will become - ^ ; but, if be 
 greater than 1, it will be = o j and, if be lefs than 1,
 
 and the Evohtte of Curves. 
 
 it will be infinite : Whence it appears, that the Radius 
 of Curvature at the Vertex will be a finite Quantity in 
 Curves whofe fifft (or leaft) Ordinates are in the Sub- 
 chiplicate Ratio of their Abfciflas, and in all other Cafes, 
 either Nothing, or Infinite. 
 
 EXAMPLE III. 
 
 71. Suppofe the given Curve to be an Ellipfis ; whofe 
 Equation (putting a and c for the two principal Dia- 
 meters) is d i y i c 1 X ax x*. 
 
 Here, by taking the Firft and Second Fluxions of the 
 
 given Equation, we have zafyy fx x a 2*, and 
 2fl*^ 2 + ia l yy c^x x 2* 2c** x j whence y 
 
 -, and y = ay t VA;a : Which, by fub- 
 
 ftituting the Values of y and J-, will became y z= 
 
 x ax xx x ac\/ax x 1 
 
 -, and the Radius of 
 ax x~ 
 
 Curvature (- jj = ^ > + g* t'x^-^l*. Which 
 
 2<7 <: 
 
 when the Diameters a and r are equal, or the Ellipfis 
 degenerates to a Circle, will be every where equal to 
 
 , or la i agreeable to the Definition of a Circle. 
 
 EX-
 
 Of the Radii of Curvature, 
 
 EXAMPLE IV. 
 
 72. To find the Radius of Curvature^ and the Evoluie fff 
 the common Cycloid. 
 
 Let ARB be the given Curve, and AOH its Evolute; 
 alfo let R and OS be parallel to AC, and eO and R* 
 
 H 
 
 perpendicular to AC; and put ARB (irzBC) =d, 
 AR = z, An A-, and Rn =y : Then BR r=<z z, B 
 = -s y>__ and, by the Property of the Curve, a* 
 (AB a ) : ^^zf (BR 1 ) : : fr ( BC) : a y ( B^ ) 
 
 2^z z 1 ax zz. 
 
 whence y =. ; therefore y =: . z > 
 
 la a J 
 
 (-*) = 
 
 , and x =. 
 
 .. Whence 
 
 (making z conftant) A: = : ,-i from which 
 
 a
 
 and the E volute of Curves. 79 
 
 we get RO, or AO ( - ) = y^az z% and rO, 
 
 V X J 
 
 or AS ( := y) = ; which, when z =. a , 
 
 or ROH coincides with BH, become AOH (BH) = a, 
 and CH (AG) =i#. Hence, becaufe it appears, that, 
 
 AH) 1 (a*) : AO* (2*2 z a ) : : AG (*) : AS 
 
 (2rfZ Z 1 \ 
 ) it follows that the Evolute AOH is alfo a 
 2a ' 
 
 Cycloid equal, and fimilar, to the Involute ARB. 
 
 If the Evolute had been given, or fuppofed, a Cy- 
 cloid, and the Involute required, the Procefs would have 
 been, more fimple, as follows, 
 
 Let AH (aAG) = a, AO (=RO) = z, AS = *, 
 SO = _y, BR = v , B = w, Rr = i, R/ = w, fcf tf . Then 
 it will be f, 
 
 j : i ( : : Ow : OR) : : R/ (,; : Rr = ^5. 
 
 z^ 
 s :_y .' : z (RO) t O/w -r-> 
 
 :* :: z (RO) : R = ^?, 
 
 2 
 
 Whence we have <y = ~, R (Rw AS) = ^ A-, 
 
 2V 
 
 and An (OS O/n) = y f j which Expreffions an- 
 
 fwer to any Curve whatever. 
 
 But, in the Cafe above propofcd, AH 1 (a 1 ) : AO* 
 
 X 
 
 {z*) : : AG (1<7) : AS (*) } therefore * = , x = ~ f 
 
 i 
 
 j and confequently R 
 
 
 c 
 
 ' Whence
 
 Of the Radii of Curvature, 
 
 ,.,, , r a* 
 
 Whence alfo w 
 
 Artj. 
 
 ^a 
 
 and <u { J 
 \ y 
 
 Therefore it will be -y : w (: : a : 
 
 V 2aw 
 
 V\a : -v/ w ; that is, as Rr : Rf : : V/BC" 
 Which is a known Property of the Cycloid. 
 
 Hitherto regard has been had to Curves where the 
 Ordinates are parallel to each other : But when the Or- 
 dinates are all referred to a given Point, as in Spirals, 
 fs'c. other Theorems will become neceflary ; and may 
 be thus derived. 
 
 73. Let ARB be the propofed Curve, P the Point, 
 or Center, to which its Ordinates are referred, NOL 
 
 the Evolute , 
 and R O the 
 Ray of Cur- 
 vature at R : 
 Moreover, let 
 PH be perpen- 
 dicular to RO; 
 and, fuppofing 
 the Ordinate 
 PR (y) to be- 
 come variable 
 by the Motion 
 of the Point 
 
 N 
 
 noted by z and p refpe&ively. 
 
 R along the 
 Curve, let the 
 Fluxions ofAR 
 and PH (p) t 
 expreffing the, 
 Celerities of 
 the Points R 
 and H in Di- 
 rections per- 
 pejidicular to 
 ROT*, be de- 
 
 Therefore,
 
 ana the Evolute of Curves. 8 1, 
 
 Therefore, the Celerities, of any two Points, in a 
 Right-line revolving about a Center, being as the Dif- 
 tances from that Center, it follows that p : z :: OH : 
 OR ; whence by Divifion (putting RHna) we have 
 
 y~ vpx D . 
 
 i-p : i st , (RH) : RO = ^ = : B-V 
 
 ^yvv 
 
 = yy (by Art. 60.) and therefore RO == - - r ; 
 
 yypp 
 
 which, becaufe^* /* is = v* (and therefore^ pp z; 
 
 _, -M ir u v yy yy 
 
 w] will alto be r z: -r-. 
 
 'the fame cthenvife. 
 
 Let SRD be a Circle defcribed about the Point O, 
 as a Center, and fuppofe the Diftance PR to be variable 
 by the Motion 
 
 of the Point R ^ ..*- T 
 
 along the Arch 
 of the Circle 
 (inftead of the 
 Curve) : Then, 
 drawing O P, 
 and putting OR 
 =r, PR =y, bV. 
 
 as before, we c^ ^lk '"""A 
 
 ihall get OP 1 
 
 (OR'-fPR* 2 OR xRH) = r*+/ 2rv; which 
 (as well as r) being a conftant Quantity, its Fluxion 
 2yj ir<v muft be equal to nothing ; and therefore r := 
 
 *y 
 
 -r, the very fame as above. Nor is it of any Con- 
 
 fequence whether y and v be here looked upon as refpet- 
 ing the Circle, or the Curve ; fince, at R, they muft be 
 the fame in both Cafes ; otherwife the Curvature could 
 not be the fame *. Now from the Value of RO thus *Art.6S. 
 found, which (corrected, when neceflary) will alfo ex- 
 prefs the Length of the Arch NO of the Evolute J, jArt.63- 
 the Ordinate PO and the Tangent OH of the Evolute 
 
 G may
 
 8 2 Of tie Radii (f Curvature, 
 
 if 
 
 may be eafily deduced. For OH (RO RH) = 4? 
 
 ;., andPO (= 
 
 V >U 
 
 whence the Nature of the Evolute is known. 
 
 EXAMPLE I. 
 
 74. Let the given Curve AR be the logarithmic 
 Spiral, whofe Nature i: luch, that the Angle PRQ. (or 
 RPH) which the Ordinate makes wiih the Curve is 
 every where the fame. 
 
 Then (denoting the Sine of that Angle by b, and 
 
 by 
 the Radius of the Tables by a) we have RH (v) ~p 
 
 f yy \ &yy &y 
 
 and therefore RO ( ) = -77 = r; which bein<* 
 v i/ / by b 
 
 to PR (y) in the conftant Ratio of a to , or of PR to 
 RH, the Triangles ROP and RPH muft therefore be 
 fimilar, and fo the Angle POH, which the Ordinate 
 PO quakes with the Evolute, being every where equal 
 to PRQ_, will likewife be invariable. Whence it ap- 
 pears that the Evolute is alfo a logarithmic Spiral, 
 fimilar to the Involute j and that a Right-line drawn 
 from the Center, perpendicular to the Ordinate, of any 
 logarithmic Spiral, will pafs thro' the Center of Cur- 
 vature. 
 
 EXAMPLE II. 
 
 75. Let the Curve propofed be the Spiral of Archimedes; 
 
 by y* 
 
 where we have p -7 ~, and v , 
 
 Yyt + b* v y~ 
 
 (fee Art. 62.) Therefore v r=
 
 find the Evolute of Curves. 83 
 
 y X lyy X f+JF\ 
 
 lyy X y* -f- ^* T 3 V _ 
 
 
 whence the Radius or 
 
 Curvature f is here =jjjg> W hich being =-*, Mlt , j; 
 
 whence, the Arch of the Evolutef, reckoned 'from 
 
 yv 4- bbv- b 
 the V ertex, is therefore =r ~ 
 
 2 
 After the very fame Manner you may proceed in other 
 
 Cafes : But if the Value of -v (dr =-r J changes, in any 
 
 Cafe, from Pofitive to Negative, the Radius of Cur- 
 vature (RO) after becoming infinite, will fill on the 
 other Side of the Tangent, and the correfponding Point 
 of the Curve, when v o, will be a Point of Contrary- 
 Flexure. Whence it may be obferved that the Point 
 of Infie&ion, in a Curve whofe Ordinates are referred 
 to a Center, may be found by making the Fluxion of 
 the Perpendicular, drawn from the Center to the Tan- 
 gent, equal to Nothing, which Cafe is not taken Notice 
 of in the preceding Section. 
 
 G 2 S E G-
 
 Manner of fading FLUENTS- 
 
 \ 
 
 SECTION VI. 
 
 Of the Inverfe Method, or the Manner of de- 
 termining the Fluents of given Fluxions. 
 
 76. TN the Jnverfe Method, which teaches the Man- 
 [ ner of finding the refpe&ive flowing Quanti- 
 ties of given Fluxi ns, there will be no great Difficulty 
 in conceiving the Reafons, if what is already delivered 
 in Sefl. i. on the direft Method, has been duly con- 
 fidered : Though the Difficulties that occur in this Part, 
 upon another Account, are indeed vaftly fuperior. 
 
 It is an eafy Matter, or not impoffible at moft, to 
 find the Fluxion of any flowing Quantity whatever; 
 but in the Invcrfe Method the Cafe is quite different : 
 JFor, as there is no Method for deducing the Fluent 
 from the Fluxion a priori , by a direct Investigation, fo 
 it is impofiible to lay down Rules for any other Forms 
 of Fluxions, than thofe particular ones which we know, 
 from the direct Method, belong to fuch and fuch kinds of 
 flowing Quantities. Thus, for Example, the Fluentof 2xx 
 is known to be #", becaufe it is found in An. 6. and 14. 
 that 2xx is the Fluxion of x~ : But the Fluent of yx is 
 unknown, fince no Exprefllon has been difcovered that 
 produces yx for its Fluxion. 
 
 77. Now, as the principal Rule in the direfl Method 
 is that for the Fluxions of Powers, derived in Art. 8. 
 
 (where it is proved that the Fluxion of x is, univer- 
 
 fatly, exprefied by nx x) ; fo the moft general 
 Rule, that can be given in the Inverfe Method, muft 
 be that arifing from the converge thereof > which Jhtws 
 hnu to ajjign the Fluent of any Power of a variable 
 Quantity drawn into the Fluxion of the Root j and which, 
 exprefled in Words, will be as follows. 
 
 Divide by the Fluxion of the Root, add Unity to the 
 Exponent of the Power, and divide by the Exponent fa 
 incrcafcd. 
 
 For,
 
 Manner of finding FLUENTS. 85 
 
 For, dividing the Fluxion nx n "~*x by * (the Fluxion 
 of the Root x] it becomes nx*~*\ and, adding I to 
 the Exponent (n i) we have **} which, divided by 
 
 tf, gives *", the true Fluent of nx , by Art. 8. 
 Hence (by the fame Rule) the 
 
 Fluent of 3; 
 
 fx will be 
 
 That of SA 
 
 v 
 
 
 3 * 
 
 
 x 6 
 
 That of 2x 
 
 
 
 * ~ T 
 
 That of =|; 
 
 8 
 
 That of aylj = 22! j 
 o 
 
 JW-f-'T 
 
 W -f-I - 
 
 T^U c ~- y - n y 
 
 That Of y y= -^', 
 
 That of f, or 
 
 That oftf-f 2] x zz 
 
 . 
 
 Xz * = 
 
 For /;r^ the Root, or the Quantity under the general 
 
 Index n, being a -f z , and its Fluxion = /wz 
 C//r/. 14.) we (hall, by dividing by tho Uft of thefc 
 
 T/* "* i 
 
 Quantities, have ; whence, increafing the 
 
 m 
 
 G 3 Index
 
 S6 The Manner of finding 
 
 Index by Unity, and dividing by (n-f i) the Index Co 
 
 uicreafed, there comes out 
 
 After the very fame Manner the Fluents of other 
 Expreflions may be deduced, when the Quantity, 
 or Multiplicator, without the Vinculum is either 
 equal, or in a conftant, Ratio, to the Fluxion of the 
 Quantity, urvder the Vinculum : As in the Expreflion 
 
 - m 
 
 a + cz" 1 ' X dz~~*x \ where the Number of Dimenfions 
 of z under the Vinculum (or general Index) being equal 
 to thofe of z without the Vinculum 4- i, the Fluent 
 may therefore be had, as in the preceding Examples} 
 
 ,m4-i 
 
 a + cz> x d 
 and will come out- - . : And, that this (or 
 
 flf X / 4- i 
 
 any other Expreflion derived in like Manner) is the true 
 Fluent will evidently appear, by fuppofing x equal to 
 
 a + cz" the Quantity under the Vinculum ; for then 
 (equal Quantities having equal Fluxions) x will be 
 
 Art. 8. = *cz~*x * ; 'and confequently a + c* x dz " * 
 
 (=* w x I = - ; whofe Fluent is therefore 
 nc / nc 
 
 4- 
 
 d " .__d 
 }Art. 77 . - -I - v - , as before. 
 
 78. In aligning the Fluents of given Fluxions there 
 is another Particular th^t ought to be attended to, not 
 yet taken notice of ; and that is, whether the flowing 
 Quantity, found by the common Rule, above deli- 
 vered, does not require the Addition or Subtraction of 
 fome conftant Quantity to render it complete. This 
 
 9 indeed
 
 Manner of finding FLUENTS. 87 
 
 indeed can, only, be known from the Nature of the 
 Problem under Confideration ; but that fuch an Addi- 
 tion or Subtraction may, in fome Cafes, become ne- 
 ceflary is evident from the Subject itfelf ; fmce a flow- 
 ing Quantity increafed, or decreafed, by a conftant 
 Quantity, ha* ftill the fame Fluxion ; and therefore the 
 Fluent of that Fluxion is as properly exprefled by the 
 whole compound ExprefEon, as by the variable Part of 
 
 it, alone : Thus, for Inftance, the Fluent of nx"~ l x may 
 be either reprefentcd by *" or by #"-[^<7,becaufe(rf being 
 conftant) the Fluxion of x* ^<7, as well as of x n , is 
 
 n i 
 
 nx 
 
 70,. Hence it appears that it is the variable Part of 
 a Fluent only which is aflignable by the common Me- 
 thod ; the con -ant Part (when fuch becomes neceflary) 
 being to be afcertained from the particular Nature of 
 the Problem. Now to do this, the beft Way is to con- 
 fider how much the variable Part of the Fluent, firft 
 found, differs from the Truth, in that particular Cir- 
 cumfrance when the required Quantity which the whole 
 Fluent ought to exprefj, is equal to Nothing } then 
 that Difference, added o, or fubtraded from, the faid 
 variable Part, as occafion requires, will give the Fluent 
 truly corrected : For, f;:ice the Difference of two Quan- 
 tities flowing with the fame Celerity (or having equal 
 Fluxions) is either, Nothing at all, or con/iantly the 
 fame, the Difference in that Circumftance will like- 
 wife be the Difference in all other Circumftances : And 
 therefore being added to the lefler Quantity, or fub- 
 tracted from the greater, both become equal. 
 
 80. To render what is above delivered as familiar a.s 
 may be, I {hall put down a few Example? ; in which 
 the variable Quantities rcprefented by * and v are fup- 
 pofed to begin their Exiflence together, or to be gene- 
 Mte J, at the fame time. 
 
 G 4 I. Let
 
 tfhe Manner of finding FLUENTS.^ 
 
 1. Let y cfxx j then the Fluent, found as ufual, 
 
 *V* *V , 
 will be y -: j where taking y =: o, alfo va- 
 
 -- 2 
 
 nifties, (becaufe then xo by Hypothefis) : Therefore 
 the FJuent requires no Correction in this Cafe. 
 
 2. Let y =. ~a + x\ X x : Here we fir ft have y = 
 
 i but when v o, then becomes = 
 
 4 44 
 
 (fince x by Hypothefis is then o :) Therefore 
 
 " 1* a* 
 
 a .. always exceeds ^by ; and fo the Fluent pro- 
 4 4 
 
 .4- 
 
 perly corrected will be y f Zlf_ fi 1 jf 4- 3 x 
 
 4 2 
 
 #* 
 
 But the very fame Fluent may be otherwife found, 
 without needing any Correction : For the given Equa- 
 tion -- a + *l x A-/, by expanding a + x] , is tranf- 
 formed to j=. a 3 x -f $a~xx + r $ax t x + x 3 x ; whence ^ =; 
 
 ?V A-* 
 
 ^3^ i ^2 }. ^7^3 ^_ j the fame as above. 
 
 2 4 
 
 Hence it appears that the Fluent of an Expreffion, 
 found according to one Form, may require a very 
 different Correction from the Fluent of the fame Fluxion 
 found according to another Form. 
 
 T 
 
 3. Let y a 1 *- z l l X xx j then, firft, y =
 
 The Manner of finding FLUENTS. 
 
 = j therefore * ~~ * is too little by ^p 
 
 a 3 
 
 and fo the Fluent corrected will be y - . 
 
 3 
 
 ^ _ ^ 
 
 4. Let y-=.<T + .v"l X x m ~~* x : Here we 
 
 - iJL_= 5 and making _y = o, the latter Part of the 
 
 m 
 
 
 
 ^1 nn+m 
 
 Equation becomes = a ; whence tho 
 
 Equation, or Fluent, truly co/rected is y r: 
 
 - ,n+I 
 
 " ** 
 
 W X + I 
 
 5. Laftly, let j = a -f 
 
 mbx m I x + w.v" ^ ; then, in the firft Place, we have^y 
 
 m " 
 
 g Y +CJ>r j which corrected, as above, becomes 
 p+1 
 
 , m *4-l 
 
 g+kc -{-ex I a r 
 
 y - 
 p + I 
 
 8 1 . Hitherto x and y are both fuppofed equal toNothing 
 at th" fame time; but that will not always be the Cafe 
 in the Solution of Problems. Thus, for Inftance, 
 though the Sine and Tangent of an Arch are both equal 
 toNothing when the Arch itfelf is equal to Nothing, yet 
 
 the
 
 go f/je Manner of jinduig FLUENTS. 
 
 the Secant is then equal to the Radius : It will be proper 
 therefore to add an Example or two wnerein the Value 
 of y is equal to Nothing, when that of x is equal to any 
 given Quantity a. 
 
 Let, then, the Equation y x r x be hrfl propofed ; 
 
 x 3 
 
 whereof the Fluent (firft taken) is y ; but when 
 
 x 3 a* 
 
 y = o, then = , by Hypothefis j therefore the 
 ' j 
 
 x* a 3 
 
 Fluent, corre&ed. is y rr . 
 
 3 
 
 Again, let the propofed Equation be j =r x x ; 
 +i 
 
 then will v = ; which corrected becomes y 
 
 +i 
 
 n-f-I 4-I 
 
 a x 
 
 Laftly, let y c z -f- bx~Y X xx\ then, firft, y =3 
 3 
 
 ^ j and, when j = and x = *, f 3 
 
 comes = C "^ ' : therefore the Fluent corrected is 
 
 3* 
 
 82. All the Examples hitherto given relate to fuch 
 Fluxions as involve one variable Quantity only in each 
 Term, whofe Fluents are afiignable from the Converfe 
 of the firft General Rule, in Seftiea I. But, befides thefe, 
 various other Forms of Fluxions may be propofed, in- 
 volving two or more variable Quantities, whofe Fluents 
 may alfo be fcund by Help of the other two General 
 Rule*- delivered m the fame Section. 
 
 Thus
 
 yhe Manner of finding FLUENTS. 91 
 
 Thus the Fluent of yx + xj is exprefled by xy* j thatArt.io. 
 Z- f j that of ax + xj+yx by ax + xy % ;.j. Art . I3 . 
 
 and that of wyy + y x n*x xX y"x-ax * ^7 
 __H- W 
 
 n "\~~m~ 
 
 mxyx-ax\ . f diyidin / in the Jaft 
 
 the Fluxion of the Root y x ax , which (by Art. 
 
 rti n n1 
 
 14 and 15) is nxy j+y x nax x t we nrft have 
 
 " 1 
 
 yx ax"\ m ; whence, adding Unity to the Exponent 
 
 ~, and dividing by the Exponent fo increafcd, we get 
 
 p + m 
 
 m 
 
 ent of the Quantity proppfed. But it feldom happens 
 that thefe Kinds of Fluxions which involve two dif- 
 ferent variable Quantities in one Term, and yet admit 
 pf known, or perfe#, Fluents, are to be met with in 
 Practice : I fliall therefore take no further Notice of 
 them in this Place (but refer the Reader to the fecond 
 Part of the Work) my Defign here being to infift only 
 upon what is moft general and ufeful in the Subject ; 
 which b-ings me to further confider thofe Forms of 
 Fluxions, involving one variable Quantity only, that 
 frequently occur in the Solution of Problems, whofe 
 Fluents may (after proper Transformation) be found, 
 by the Rule already delivered in Art. 77. 
 
 83. It
 
 *Tbe Manner of finding FLUENTS. 
 
 83. It has been already hinted, that if a Fluxion of 
 
 m 
 
 the Binomial Kind, as a + cz ' x dz i, has the In- 
 dex (n i) of the variable Quantity (z) without the 
 Vinculum -f I, equal to () the Index of the fame Quan- 
 tity under the Vinculum^ the Fluent thereof may be then 
 truly found by the forementioned Rule. But the fame 
 Observation may be farther extended to thoft Cafis. 
 where the Index without the Vinculum increajed by Unity 
 is eyial to any Multiple of that under the Finculum ; as 
 
 in the Expreffions, a+cz* X dz" z, a -f cz'\ X 
 
 |W 
 
 dz V ~* z t a + <' X ^z 4 *"" 1 ^,^. Whofe Fluents arc 
 thus determined. 
 
 * XQ - K I 
 
 Put/7+cz ^:jr, then will 2; -, and?;z x. 
 
 * an( ^ therefore z = 
 
 * Art.S. 
 
 i whence by Subftitution we get a -f- 
 
 _ , 
 d x 
 
 x xax x 
 
 nc TIC 
 
 Whofe Fluent (by Art. 77.) is therefore = = X 
 
 nc 
 
 W-j-2 W-pI 
 
 --- ; which, by reftoring the Value of x. 
 m+2 m + I 
 
 becomes , X
 
 The Manner of finding FLUENTS. 93 
 
 a + cz 
 X 
 
 m + 2 m+i 
 
 cz 
 
 W + 2XW+I ; thetrue Fluent of 
 
 ^nl 
 
 az %. 
 
 m 
 
 Again; for the Fluent of a + cz"i x dz^'~ l z t be 
 caufe z"* = -I, and 2"- ^Zf, we have z 3 *" 1 * 
 
 Wf C 
 
 nc 
 
 i w I 
 
 Whence, + fz"' being r: .* , we get-f rz ' x 
 
 . w ' 
 
 efz z= dx x 
 
 luxx 
 
 f-f-I tr. . 
 
 2fl.v .V + O'AT A- ; whofe Fluent is there- 
 
 , <* x 
 
 fore x 
 
 m+i 
 
 dxa + 
 
 m -f 3 m + 
 
 z lacz 
 
 J **' * I j 
 
 
 + 3X
 
 94 *rhe Manner of finding FLUENTS. 
 
 Univerfat/y, let r denote any whole pofitive Number 
 
 l 
 whatever, and let the Fluent of a + czl xflfc'*"' 1 * be 
 
 required; then, by putting a + cz =x y and proceed- 
 ing as above, our propofed Fluxion is transformed to 
 
 J n ~ r I 
 
 ax x > r r i 
 
 ' x xa\ ; which, expanding x a\ 
 
 nc 
 
 ( by the Binomial Theorem ) becomes x 
 
 M 
 
 i x ta * + r i x 
 
 r. tfhofe Fluent is therefore = x 
 
 nc 
 
 r r 
 
 ri* ax _j_ r ix r7. x a * x 
 
 -r 2 
 
 2 
 
 Where, r being a whole pofitive Number, the Mul- 
 
 tiplicators i,r l,r i X r 2,r I X r 2 X r 3,^r. 
 will therefore become equal to Nothing, after the r firft 
 terms ; and fo, the Series terminating, the Fluent itfelf 
 will be truly exhibited in that Number of Terms : Ex- 
 cept when IB 4- r is likewife a whole pofitive Number, 
 lefs than r ; in which Circumftance the Divifors m + r 9 
 m + r i, m + r 2, &V. becoming equal to Nothing, 
 before the Multiplicators, the correfponding Terms of 
 the Series will be infinite. And in that Cafe the Fluent 
 is faid to fail, fince Nothing can then be determined 
 from it. 
 
 84. Be-
 
 Manner of finding FLUENTS. 9 1 
 
 84. Befides the foregoing, there is another Way of 
 
 
 deriving the Fluent of a -f cz x dz z, in Term? 
 of the original flowing Quantity z ; which will afford a 
 Theorem more commodious for Practice than that above 
 given : The Method of Inveftigation is thus. 
 
 Let dxa+cz n 
 &V. (where />, i/, A, B, C, &V. denote unknown, but 
 determinate, Quantities) be aflumed for the Fluent 
 fought : Then by taking the Fluxion of the Quantity fo 
 afTutned we (hall have 
 
 py 
 
 j fff A P~*U p~"2i*& 
 
 z X-f cz"\ xAz 4-Bz -f Cz 4- 
 
 l+I "" /.-I 
 
 Dz fcff. -f dXa + cz ' *MZ z + p v X 
 
 z+p2vxCz f l z &c. which being put * Art - 8 - IO 
 
 " \ m 
 
 71 I *"" - * 
 
 equal to the given Fluxion, a + cz X dz c, and 
 
 - )" 
 
 r l x 
 
 the whole Equation divided by a + cz ' X dz , there 
 
 comes out 
 
 p-v p-l-v 
 
 2 p -fCz +Dz 
 
 !*<' ? 
 ^. 2W > _ z 
 
 ! feV. J 
 
 x/Az 4-/> i 
 
 Whence, by colleding the Coefficients of the like 
 Powers of z, we have 
 ? 
 
 - i=0 
 
 z + paAz 
 
 Where, comparing p + n and rw, the two greateft Ex- 
 ponents of <x, we find p~rn nr I x ; and by com- 
 paring the two next inferior Exponents/* 4- n v, and/,we 
 
 like wife
 
 96 'The Manner of finding FLUENTS: 
 
 likewife get v n ; which Values being fubftituted 
 our Equation is reduced to 
 
 " 
 
 Where, putting m + r=s, and comparing the Coeffi- 
 
 cients of the homologous Terms *, we have A r= 
 
 i _ r i XaA r IX 
 
 * 
 
 r I x r 2 X r -2 x a 3 
 - == - - -- ' 4 > &c. Vc. 
 
 s x s ix s a x s 3 x nc 
 
 which Values, with thofe of p and v, being fubftituted 
 
 in the aflumed Fluent, it becomes d~> 
 
 rn n rn in ' , rn in 
 
 z r I X az r i x r lY.az 
 
 SX S--IXS 2Xf* 
 rn in 
 
 " of 
 
 n ~~ l zi which was to be determined: 
 Which Fluent therefore, when r is a whole pofitive 
 Number, will always terminate in as many Terms as 
 are expreflcd by that Number; except in that particular 
 Cafe, fpccificd'in the laft Article^ Thus, if r2, or 
 
 the 
 Vid. p. 1 8 1 of my Treat i/e of Algebra.
 
 Manner of finding FLUENTS. 
 
 the given Fluxion be tf-f-<rz"' X dz"" * ; then, i 
 (w-fr) being w-f 2, the Fluent itfelf will become 
 
 which is exactly the fame with 
 
 the firft of thofe found in Art. 83. by a different Method. 
 The like Agreement will likewife be found, when r 
 {5 = 3: But when r, either denotes a broken, or a 
 negative, Number, the Series for the Fluent will theft 
 run on to Infinity \ becaufe no one of the Multiplicators 
 r -i, r 2, r 3, r 4, &c. can in that Cafe be 
 equal to Nothing. 
 
 85. The foregoing Fluent, ift may be obferved, was 
 
 found by afluming ^Xfl-f-fz"! xAz -f Bz -f-Cz 
 &c. and comparing the two greateft Exponents, of 
 the Equation thence refulting : But if, inftead of 
 
 p ft -z; 6 ~2<z/ P 
 
 Az +Bz -fCz &c. an afcending Series, as Az -J- 
 
 Bz -fCz &c. (where the Exponents of z con- 
 tinually increafe) be taken, and the two leaft Indices 
 of z in the Equation (in like Manner refulting) be 
 compared together, the fame Fluent will be had ac- 
 cording to a different Form, which will be of good Ufe 
 in many Cafes, when the foregoing fails, or runs out into 
 an Infinite Series. 
 
 Thus, ifp + v, p-+-2v, &c. be wrote in the Room 
 of p v, p 2v, &V. refpe&ively, in the firft Equation 
 of the laft Article, it will appear that 
 
 H
 
 The Manner of finding FLUENTS. 
 
 z -fp 
 Which Equation may be reduced to 
 
 T 
 
 I ___ 
 
 J - 
 
 +p 3 " +P+v i 
 
 Where, by comparing the two leaft Exponents,^, p 
 
 will be found =. rti, v a ; A = ; B =: 
 
 pa rna 
 
 p-\-nxm-4-tKcA . r + w;-fiX<:A 
 
 vxa 
 
 r+ i x na 
 
 Therefore, denoting r + w by s (as above) the Fluent of 
 
 , 
 
 rf-f-rz ' X dz" x, will f^j be truly represented by 
 
 rn ~~~ 
 
 .WTI js J 4" I 
 
 rn+ 
 
 d X a 4- ^2 
 
 5+iXi-r-2Xf J s 
 
 rXr+i 
 
 n?^ r X r-H X 
 
 Jffr. or its Equal- 
 
 rna 
 
 x i 
 
 Which Series will terminate when s (or r + m] is a 
 whole negative Number j and therefore in all fuch Cafes 
 
 the
 
 Marnier of 'finding FLUENTS: 99 
 
 the Fluent is exadlly determined j provided r be not 
 alfo a negative Integer lefs than s ; for in this parti- 
 cular Circumftance the Fluent fails, the Divifor firft be- 
 coming equal to Nothing. Vid. Art. 83. 
 
 The Uiis of the two foregoing general ExpreflJons, 
 
 rn I 
 
 for the Fluent of '+,*" x dz , will appear 
 from the following Examples. 
 
 EXAMPLE I. 
 
 bxx 
 
 86. Ltt it be required ta find the fluent of i, OF 
 
 a + *i* 
 
 By comparing the Fluxion here propofed with 
 
 % 9 we have aa^ f = i, z^x, n r, 
 
 m o d b t rn~i for r i) = i ; whence rz=2, 
 and s (r + m) = | ; whereof the former being a whole 
 pofitive Number, let thefe Values be therefore fubftitutcd in 
 
 ( 
 
 ,. , rs n 
 
 dXa + cz ' z 
 
 I ^_ X -. _ + 
 
 \ snc i 
 
 -- &-,. the firft of the two ge- 
 
 j ixs ixc* y 
 
 neral ExprefKons for the Fluent, and it will become 
 
 x x - ~ ~ 
 
 - 
 
 Quantity fought in this Cafe. 
 
 H 2 EX-
 
 or 
 
 ioo The Manner of fnding FLUENTS. 
 
 EXAMPLE II. 
 
 87. Let tbt Fluxion propofed be 
 
 a+fx^X.bx 3 "~ l x. 
 
 Here, by proceeding as above, we have a a^ c~f* 
 zsrx, = rt, m - t d-=b,r=.T^ and s (r + m] 
 ^ : Whence, by fubftituting thefe feveral Values in 
 
 the fame general Exprtffion, we get bxa+fx x 
 
 \f ^*i/ x 
 
 b x a +fx I 
 
 ? 3 ~ X 
 
 6 /'V* 1 ' 
 
 IS 
 
 EXAMPLE III. 
 
 y V ~P* + v* 
 
 88. tyherein the Quantity propofed is - 6 > or 
 
 Here we have ^ = ^% c I, z =j, = 2, w z: f, 
 
 6 
 
 d i, rn I (or2r 1)= 6jwhencer(= - 
 
 = , and s (r + m) = 2 ; whereof the lat- 
 
 ter being a whole Negative Number, let the feveral 
 Value* here exhibited be therefore fubftituted in 
 
 * +
 
 The Manner of fading FLUENTS. 
 
 loi 
 
 / 
 
 I a + cz 
 
 .m+X 
 
 X I 
 
 rna r-fi x a r-f 
 
 .) the latter of the two general Expreflions above 
 
 3 
 >>' _c 
 
 " -^ "I Xjy 
 
 derived, and it will become 
 
 ~~ 
 
 5S 
 i the true Fluent 
 
 required. 
 
 EXAMPLE IV. 
 
 1 
 . La/My, let the given Fluxion be a fo*l x 
 
 x. 
 
 Then, a being = a, c /, m=i, d i,r=r-, 
 
 and the reft as in the general Fluxion a -f rz"' x 
 
 Jz*~~ l x j we (hall, by fubftituting in the fecond 
 Form (becaufe s is here equal to ( 3) a whole ne- 
 
 . n |i "5 _V /X." 
 
 fl _ /z I Xz 
 gative Number) have J _ L 
 
 X i - 
 
 
 90. Having infifted largely on the Manner of finding 
 
 fuch Fluents as can be truly exhibited in Algebraic 
 
 Terms i it remains now to fay fomething with regard 
 
 H 3 to
 
 1 03 T^he Manner of finding FLUENTS. 
 
 to thole other Forms of Expreffions, involving one va 
 riable Quantity only, which, yet, are fo affe&ed by 
 compound Divifors and radical Quantities, that their 
 Fluents cannot be accurately determined by any Method 
 whatsoever ; of which there are innumerable Kinds : 
 But there is one general Method whereby the Fluents 
 of fuch Exprefiions are approximated, to any afligned 
 Degree of Exaclnefs ; namely, the Meehod of Infinite 
 Series; which it will, therefore, be necefiary to ex- 
 plain ; fo far as relates to the Manner of expounding 
 the Value of any compound Fraction, or furd Quan- 
 tity, by Help of fuch a Series. 
 
 EXAMPLE I. 
 
 91. Let t then, the fraftion be, firji, given j to be 
 
 Q3f 
 
 converted into an Infinite Series. 
 
 Divide the Numerator ax by the Denominator a .v, 
 as is taught in Compound Divifion of common Algebra j 
 then the Operation will ftand as follows i 
 
 Where
 
 The Manner of fading FLUENTS. 
 
 ** X* X* 
 
 Where the Quotient, or Series *-r-~ + ^ + ^ + 
 ~Z + ~i^ infinitely continued, is taken to expound 
 
 the Value of the propofed Fraction " ; 
 
 a x 
 
 92. But, though the Series thus arifing ought to be 
 carried on to an Infinity of Terms, to have the true 
 Value of the Quantity firft propofed ; or, though the 
 Quotient, continued to ever fo great a Number of 
 Terms, will be^/?/// fomething defective of the Truth ; 
 yet, if the Value of the Quantity (#) in the Numerator 
 be but fmall in Comparifon of the Quantity (a] in the 
 Denominator, the Remainder, after a few Terms in 
 the Quotient, will become fo exceeding fmall, as to be 
 neglected without any confiderabJe Error ; and then 
 the Value of the Whole, or of the Quantity firft pro- 
 pofed, will be, very nearly, exhibited, by taking 3 
 fmall Number of the leading Terms only. 
 
 Thus, for Inftance, let the Value of a be expounded 
 by 10, and that of x by Unity; then the Remainder 
 
 ( ) after the two firft Terms of the Quotient, being 
 r: , this Value, divided by the given Divifor 
 
 (a *:= ) 9, will therefore give = 0,01 1 1 (I i i,&V. 
 
 for the Defect, by taking the two firft Terms only : 
 But, if the three firft Terms be taken, the Defect will 
 be jftill lefs confiderable ; amounting to no more than 
 i 
 
 -" , or 0,001 1 1 1 1 1, cfc. 
 900 
 
 This may likewife be made to appear, without any 
 regard to the Remainder, by collecting into one Sum, 
 
 the Values of all the Terms to be taken : For, if only 
 *\ 
 
 the firft two (x + - J be propofed, their Sum will be 
 
 HA s
 
 IP4 'Tie Manner of finding FLUENTS. 
 
 = !)}; which, deducted from the true Value of the 
 
 _. ' . ox f io\ 
 
 given Fraction - ( zz I = i,niiiii &c. the 
 a x \ 9 ' 
 
 Difference will come out e.oim, the -very fame as be- 
 fore. 
 
 Thus, alfo, by collecting the Sum of the three, four 
 and five, C3V. firft Terms of the Series, you will have 
 1,1 1 j i>i ii and 1,11 1 j fcfV. uhich, being fuc- 
 ceflively deducted from uiiiiiim &V. (as above) 
 there will remain c,ooim &c- o,coouii &V. 
 0,00001 in C5V. for the Errors or Defects in thofe 
 Cafes refpe&ively. 
 
 93. From what has been faid in the preceding Ar- 
 ticle jt appears, that Infinite Seriefes, in Algebra (ac- 
 cording to a common Obfervation) are fimijar to, or 
 correfpond with, Decimal Fractions in common Arith- 
 metick : For, as a Decimal Fraction may be carry 'd on 
 to any propofcd Number of Places, however great, 
 and yet never amount to a Quantity, which but a very 
 little exceeds the Value of the three or four firft Places ; 
 fo a Series may be infinite with regard to the Number 
 of its Terms, and yet a few of the leading Terms only, 
 may be fufficient to exprefs the Value of the Whole^ 
 very nearly : Provided, always, that the Series has a 
 fufficient p.ate of Convergency, or that its Terms de- 
 creafe in a pretty large Proportion ; For, otherwife, 
 tven y a great Number of Terms may be ufed to little 
 
 x* 
 Purpofe : Thus, in the foregoing Scries, # -f -f. 
 
 
 # 
 
 *r &c. if x be taken rr <z, no Number of Terms will 
 a 
 
 be fufficient to exhibit the Value of the correfponding 
 
 fix 
 
 Fraction it being infinite in that Circumftance. 
 a x 
 
 94.. Having endeavoured to (hew, that the true Va- 
 lue of an infinite Series may be nearly obtained by ad- 
 ding together a few of the firft Terms only, I fhall 
 how proceed to give other Examples of the Manner of 
 
 con-
 
 Manner of finding FLUENTS,. 
 
 Converting fractional, and furd, Quantities into fuch 
 Kinds of Meriefes, in order to the Approximation of ibc 
 Fluents of" Expreflions affe&ed by them. 
 
 EXAMPLE II. 
 
 Let the Quantity propofed be the Fraflion ^ - - t ; 
 
 then, by proceeding as in the firft Example, you will 
 have 
 
 Where, from a few of the firft Terms of the Quo- 
 tient, the Law of Continuation is manifeft ; the Nu- 
 merators being in Arithmetical Progreflion i and the 
 Signs, -{- and , alternately. 
 
 EXAMPLE HI. 
 
 I -f- X* 2x* 
 95. Let the Quantity given be - - 
 
 Then the Quotient will be i 4- *-f-3* l + 4* J -f 5**-f 
 9* s + 1 4* 6 &c . where the Law of Continuation is ma- 
 nifeft ; being fuch that the Coefficient of each fuc- 
 ceeding Term is equal to the Sum of thofe of the two 
 Terms immediately preceding it. 
 
 EX-
 
 jc6 The Manner of folding FLUENTS. 
 
 EXAMPLE IV. 
 96. Let tae Radical Quantity v'cf + x* be proofed. 
 
 Here, according to the common Method of ex- 
 trading the Square Root, the Proccfs v/ill {land as 
 follows : 
 
 AT* X 4 \ / ** X 4 
 
 ; ) aa + xx [ a+ 5- &V. 
 
 a ^cPJ \ ^a 8 3 
 
 aa 
 
 97. The Law of Continu.atieh in Sericfes, thus arifing ? 
 from radical Quantities, is not eafily difcovered : But, 
 if you would carry on the Series to any propofed Num- 
 ber of Tems, the Work wiH be a good deal fhortned, 
 by dividing the Remainder by the Divifor, when half 
 that Number of Terms is found (as in, consmon Di- 
 vifion) and observing, a,t the &jne time* to neglect all 
 fuch Tcrfliswbofe Indices would exceed the greateff, or 
 the greateil Plus the common Difference, in the faid 
 Remainder, according as the whole Number of Terms 
 propofed to be found is odd", or even. 
 
 Thus, if it were propofed to continue the foregoing 
 
 x 
 
 Series a H --- TJ-J to 6 Terms, then the Divifor 
 
 (o,
 
 Manner of fading FLUENTS. 107 
 
 ** ** 
 (or double Quotient) being ja-f "J""^ 1 ' 
 
 Remainder -^-r ^ (as appears from the laft Ar- 
 ticle) the reft of the Operation will ftand thus : 
 
 5*' ^ 
 
 # 10 
 
 640' 
 
 5* g jc^ 
 ~~ 64**+ (>W* 
 S*'___5* t0 - 
 
 ~ t^a 6 I28a* 
 
 Which three Terms thus found being added to thofe 
 
 AT* AT* *' 
 
 found above, we have a H -- ^r - j^$ 
 
 7A- 
 7 -f , -, for the 6 firft Terms of an infinite 
 
 7 9 ' 
 
 Series exhibiting the Value of r7+* 
 
 98. Another Way of refolving any radical Quantity, 
 is to affume a Series (with unknown Coefficients) for 
 the Value thereof; and then the Scries fo aflumed being 
 raifed to the fecond, third, or fourth Power, &c. ac- 
 cording as the Root to be extracted is a fquare, cubic, 
 or biquadratic one, &c. an Equation will be obtained 
 (free from Surds) from whence, by comparing the ho- 
 mologous Terms, the aflumed Coefficients, and con^ 
 fequently the Series fought, will be determined j as in 
 
 EX-
 
 io8 *Ihe Manner of finding FLUENTS. 
 
 EXAMPLE V. 
 
 . Where it is propofed to extratt the Square Rost ef 
 cf + x 1 " in an Infinite Series. 
 
 In which Cafe, afluming A + B* M -f C* 4 " 
 
 -f E* 8 " &c. for the required Series, and taking the 
 Square thereof, we have 
 
 + 2AC* 4 "+ zAD 
 4 " + iBC* 6 " 
 
 and confequently 
 
 4" . A TN 6" 
 
 ^ 
 
 .. 
 
 A* +2AB* 
 
 *" 
 
 , a 
 
 i i 
 
 x -J- 
 
 Therefore A 1 a a "=ro, aAB 1=0, 2 AC + B x =o, 
 aAD-t-2BC=o, 2AE + 2BD4-C 2 =o, * &c. From 
 
 which we get A=/ ; B ( = -^-)= -', C ( = 
 
 2A / 20 
 
 B'\ I BC\ I 
 
 - ; D --- 
 
 2 BD-fC*\ 5 
 
 ( 7 I -* ^f whence we have 
 
 2*"V ^ Q 7 
 
 A+B"+C,*+D <P 6fc (= ' / /V : )= 
 
 * rV./. 18 1 o/'wy Treati/e of Algebra.
 
 T6e Manner of fading FLUENTS. 109 
 
 5* w,. which Se- 
 
 41 
 
 6" 
 
 ~~~7, """* , n -f- 
 la oa Iva 
 
 I28* 7 ' 
 
 ries, if n be expounded by Unity, will become a -f- 
 
 ** ^4 
 
 - & c . the very fame with that in the pre- 
 20 8<2 3 
 
 ceding Article found by the common Method. 
 EXAMPLE VI. 
 
 VlT . 
 
 99. Let it be required to refolve a + bx" ' into an 
 Infinite Series. 
 
 Here, by afluming A + B 
 cubing the fame, &e. we have 
 
 + D 
 
 -a bx" + 3AB 1 * + 6ABCx* 
 
 c. and 
 
 =: o 
 
 Therefore A = a 7 j B (= 
 
 = 7; C (= 
 
 and confequently, a-f i 
 
 And, in the fame Manner, may the Root of any 
 other Quantity be extracted : But as the celebrated Bi- 
 nomial Theorem, difcovered by the illuftrious Sir Ijaac 
 Newton, is vaftly more eafy and expeditious, in raifmg 
 Powers and extracting Roots than that, or any other, 
 Method, I {hall now explain the Ufes thereof; but, 
 
 firft
 
 Manner of fading FLUENTS. 
 
 firft of all, it may not be amifs to fhew how the Theo- 
 rem itfelf, from the Principles of Fluxions, may be de- 
 rived. 
 
 Let, then, I +_> be a Binomial whofe firft Term is 
 Unity, and its fecond Term any propofed Quantity y j 
 and let the Quantity to be expanded or thrown into a 
 
 *p 
 
 Scries be i+^i ; where the Exponent v is fuppofed to 
 denote any Number whatever, whole or broken, po- 
 fitive or negative. 
 
 Now it is evident that the firft Term of the required 
 Series muft be Unity ; becaufe when y is = o, the other 
 
 Terms all vanifb. ; and, in that Cafe, \-\-y\ is equal to 
 
 Unity. Let, therefore, I + Ay" + B/ -f C/-|- D/ 
 &c. be affumed to exprefs the true Value of the faid 
 Series, or, which is the fame, let 
 
 I -f y\ = i -f Ay + B^ + C^ -f Dy (3c. where 
 A, B, C, D, yr. m, n, p t q, &c. denote unknown, but 
 determinate Quantities : 
 
 Then, by taking the Fluxion of the whole Equation, 
 
 (fuppofing y variable) we fhall have vj> x T-t- y\ = 
 
 m i l f i f I 
 
 my Ay + nyoy -^fj^y ~*~9J>Dy &c. 
 Whence, multiplying the Sides of the two Equations, 
 
 crofs-wife, and dividing by j x T+y\ , there comes 
 
 /> f 
 
 ^ +vD^ &c. which, by Re- 
 duttion, is 
 
 m l X _ P"~ l <t * 
 
 mAy +nEy +pCy 
 
 * 
 
 Now,
 
 Manner of fading FLUENTS. in 
 
 Now, fmce we are at Liberty to take the Exponents 
 of y what we will, fo as to anfwer the Conditions of 
 the Equation, or fo that all the Terms here put down 
 may mutually deftroy each other j Jet them, there- 
 fore, be fo taken that the Terms themfelves may be 
 homologous, that is, let m irro, I *, 1=, 
 q i />, &c. Then, m being zri,=:2, /> 3, J 4> 
 t3c. ir" thefe feveral Values be fubftituted above, the 
 Equation itfelf will become 
 
 Where, taking A w=o, 26 -f A vh~Q, 
 vB=o, 40 + 30 vCrno, &c. fo that every Column 
 of homologous Terms (and, confequently, the whole 
 Exprcffion) may vanifh, we alfo get A=v ; B (= 
 
 vA. A A x v 1\ i;x v i p , _ vB aB 
 
 Bxt ^) = ^x^- 1 x ^=- 2 ; D ( = 
 
 3 / 2 3 
 
 - 3x v ! v 2 ^ 
 
 3 4 
 
 Whence, by writing thefe Values, with thofe of m, , 
 />, y, ^. in the Series i -f A/*-f B>-"-f Cy &. firft 
 alTumed, we, at length, find i-f-j) = i -f- vj + x 
 
 I" X ~~~* x ~ x y T ~* X 
 
 2 123 12 
 
 f 2 W 3 
 
 r~ X x >* -f &V. which wis to be invefti- 
 
 6 4 
 
 gated. 
 
 From the Series here brought out, any Power or 
 Root, of any other compound Quantity, whether Bi- 
 nomial, Trinomial, sV. is eafily deduced : For, if p 
 be put to reprefent the firft Term of any fuch Quan- 
 tity, and Q_ the Quotient of the reft of the Terms di- 
 10 vided
 
 *be Manner of finding FLUENTS* 
 
 vided by the firft ; then the Quantity itfelf will be ex-* 
 prefled by P-J-PQ_or Pxi+Q_, and the v Power 
 thereof by P v x i + C, which therefore is equal to 
 
 V V~~ 1 1) 7J I V - 2, 
 
 7 x xQ! + T x x x 
 
 V Vl 112. V'i 
 
 Qf -f- X X X - xQ^ + esV., by what is 
 
 juft now determined. 
 
 But when v is a Fraction, as in the Notation of 
 Roots, the Theorem here given will be render 'd fome- 
 \vhat more commodious for Practice, if, inftead of -u t 
 
 a Fra&ion as be fubftituted j by which means it will 
 
 become P x i + Q? " =P " x i + Q -f x 
 
 n ^- n 
 
 tn m m n m a_. m m 
 
 Q - + ;r*-ir*r^- 1 -i* x 
 
 m 2 tn 3 
 
 X ' Q! + &c. whofe Ufe, in converting 
 
 radical Quantities into Infinite Seriefes will appear from 
 the following Examples. 
 
 EXAMPLE VII, 
 
 ICO. JPherein it is propofed to extra fJ the Square Root 6? 
 a 1 4- **, in an Infinite Series. 
 
 Here the Quantity to be expanded being ~a r 
 
 ^f x , . I i by comparing it with the general Form, 
 
 aa 
 
 ~ > ~ x* 
 
 P " X i -i-Ql , we have P=a% Qjr m ~ i f 
 
 it 
 
 and
 
 Manner of finding FLUENTS. 
 
 and nz : Whence, by fubftituting thefe Values in the 
 laft general Equation, we get 
 
 *-f* r f = axi-Kx^ + ^x--;-X^. + X 'i 
 
 # 6 v * 
 
 x -| x ? + \ x -i x - x -| x ^- + (ft. = + 
 
 ** * 6 5** 
 
 - - + r 7 TT^ & Which Series agrees 
 3 ' 7 
 
 exactly with thofe found in Art. 97. and 98. by different 
 Methods. 
 
 EXAMPLE VIII. 
 
 10 1. Let it be required to extraft the Cube- Root of 
 I 3 y*, in an Infinite Series. 
 
 Here by 
 
 ~~y*Y ( - T\ 
 
 comparing r l r X I p- V = p .y* \ 
 
 with P X oT , it will be P=b*, Q_= ' 
 
 m~i andn = 3: Therefore, by Subftitution, we get 
 
 1\ 
 
 .T y 3 ] } V 3 
 
 P (^Xl-Ji)/ =*XI **#- + yX 
 
 - i i v a v > v ___ ^-~ m v""* v ~< \f 
 
 jbij^^A 5'^ 19 T7A T ^"~ 
 
 / / sy 9 io/* 
 F " =* 3 4*""^ "" sT^ 8 ^J 77
 
 1 14 T&f Manner of finding FLUENTS. 
 
 EXAMPLE IX. 
 102. Let the Quantity to be converted into an Infinite 
 
 Series be 
 
 V ax xx 
 In this Cafe the given Quantity being firft transformed 
 
 " x ' ~~ a an d I - afterwards com- 
 
 _ "* 
 
 pared with 7+Q\ , we have Q_= * , m = l 
 
 a 
 
 t 
 
 and n~2 j and therefore i fTTTr^ - 
 
 a v * '^.i IT 
 
 the Quantity pro- 
 ^ ^^ 
 
 pofed. 
 
 103. It may not be improper to obferve here, that, 
 when both the Terms of the propofed Quantity are af- 
 firmative, and its Exponent alfo affirmative and lefs 
 than Unity, the two firft Terms of the equal Series 
 will be pofitive, and the reft negative and pofitive, al- 
 ternately ; but if only the firft Term of the Binomial 
 be affirmative, all the Terms of the Series, after the 
 firft, will be negative : Moreover, if the Exponent of 
 10 the
 
 Manntr of fading FLUENTS. 
 
 the given Quantity be negative, and both the Terms 
 affirmative, the Signs will change alternately*; but if 
 only the firft be affirmative, all the Terms of the equal 
 Series will be pofitive. 
 
 EXAMPLE X. 
 104. Let the Quantity propofed be the Trinomial 
 
 Here, by dividing the reft of the Terms by the 
 ?<:. our given Quantity is reduced toTr^F x 
 
 Therefore, in this Cafe ?=*% Q. 
 2*4~ 3*% w =I j and n = 3: Whence (by Subftitu- 
 
 ______________ t 
 
 tion) * 3 + 2* 4 + 3* 5 ) 3 *X 1 + 4 X 2A-+3**! + T X 
 
 + ; x ~| x -| x 
 
 5x2* 
 
 1x* 
 
 Which, reduced to fimple Terms, is =: x + - ^_ 
 
 68** , 
 
 9 -TT^' 
 
 105. When the propofed Expreffion confifts of a ra- 
 tional, multiply'ti by an irrational, Quantity, the Series 
 anfwering to the irrational one nmft be firft found, and 
 afterwards multiply'd by the rational Quantity : But, if 
 two, or more, compound irrational Quantities are to b* 
 drawn into each other, then take the Series anfwering 
 to each Quantity, feparately, and multiply them toge- 
 ther ; obferving, always, to neglect all fuch Terms 
 >ynofe Indices would exceed that of the laft, or higheft, 
 I 2 Term,
 
 u6 The Manner of finding FLUENTS. 
 
 Term, which the Series fought is propofed to be con- 
 tinued to. 
 
 EXAMPLE XL 
 
 jo6. Lst the Quantity propofed be i -f x X I x\ 
 
 Firft we have ,^" = T - ^ - _ Q x 2Q 
 gx igj? 3 QX 19 
 
 _ 
 
 > X 
 sV. Which, mul- 
 
 10X20X30 10X20X30X40 
 
 ______ I 
 
 tiply'd by I+A?, produces i+#x i x?* = I + 
 gx_ 2 9 .v a _ 9-4Q* 3 . _ 9.19.69** ^ =14. 
 
 10 10.20 10.20.30 I0.20.30.4O 
 
 10 200 2000 80000 
 
 EXAMPLE XII. 
 107. Where the Quantity to be exprejjed in an Infiniie 
 
 r, or a 1 *n X c 1 # 
 
 Here we have, ^?1 ( x i - 
 
 ~ 
 
 An<J
 
 The Manner of fading FLUENTS. 1*7 
 
 And 7" 
 
 H- --J x r + * * 
 
 ** ?** 5* 6 
 
 j -f ^-y + -~ 7 &c. Whence, multiplying thefe 
 
 two Values, one by the other, we get 
 
 ., _fL l * 3 a * l 
 
 "I* _ ? *" " X A" ~T~ 77~""c ^ ^ ~" ^ ^"^ r\ ^^ X A* T 
 
 3 - s 3 
 
 S a 3 i i 
 
 T~T x A- 6 + ^. for 
 
 i6t 7 i6flc s 
 
 the four firft Terms of the Series fought. 
 
 D 
 
 EXAMPLE XIII. 
 JC8. Lettbt Quantity to be expanded be the Multinomial^ 
 
 or infinite Series, x + ax + bx + cx + &c. |; 
 whofe Exponent v denotes any Number whatever^ whale 
 or broken^ pcjitive or negative. 
 
 Here, dividing by the firft Term, the given Quantitv is 
 
 fv n -in ~.n 4*1 \ 
 
 transformed to x xi + ax+bx +cx +dx +&.$ 
 which, if ax -f bx -{-ex &c. be put =y, will become 
 
 t*y .- . -\ 
 
 x X i-f ^j ; which laft Expreflion (by Art. 99.) is 
 
 pv V V I V 1> 1 V 2 
 
 x f X i +vy 4- yx -^- X >*+ 7 x -Y~ x 
 
 X/ +~ffr. Whence (for Brevity fake) putting A=v, 
 y <y i -y v I v 2 ^ V 
 
 \ " i 3 ' " i 
 
 I 3 v i
 
 Ji8 *The Manner of finding FLUENTS* 
 
 V 1 V 2 V ? 
 
 - x X - & and fubftituting for _y, there 
 
 2 3 4 
 
 comes out x + 
 
 + Aa + B X 
 
 EXAMPLE XIV. 
 
 109. To extra ff the Square Root of a~ AT% and from 
 thence to determine the Fluent of x V f a i x i ^ in an 
 Infinite Series. 
 
 By proceeding as in the foregoing Examples, the Value 
 of V a 1 x~ in an Infinite Series will be iound to be a 
 
 "."' . &c. Which 
 ia 
 
 ! i i . . / , . *.* 
 
 plied by x gives x y a x" ~ ax TT- : 
 
 la 8a 3 
 
 v v Cjt* X 
 
 '- * <3c. Whofe Fluent therefore (by Art. 
 
 * " x r 
 
 ~ ta 4oa 3 " ~ 5 - 
 
 Which was to be determined. 
 
 EX-
 
 "The Manner of finding FLUENTS. 119 
 
 E X A M P L E XV. 
 IIO. Let it be required to approximate the Fluent of 
 
 , in an Infinite Series. 
 
 It appears, from Example 12, that the Value of 
 
 a> " **' , exprefled in a Series, is _ , a ' 
 
 i c 2c 3 iac 
 
 j a 
 
 i_ _ L-. x x 6 -f tf<r. Which Value being 
 
 therefore multiplied by x" *, and the Fluent taken (by 
 the common Method) we get . -^ + r~j - 
 
 j^ \x / " 2 fl^ 
 
 , 35 i JL 
 
 "" ~" X 
 
 3 
 
 i6<: 5 
 
 I 4 EX-
 
 <Tbe Marnier of fading FLUENTS. 
 
 EXAMPLE XVI. 
 
 III. IFhereinit is propofed to approximate the Fluent of 
 
 in a Series. 
 
 Here, if A be put = v, B r^rvX , C = v ., 
 
 2 2 
 
 , D=v x x - - x -, &c. the Quantity 
 
 <> />+ /'i-i^ P-K^" l v 
 
 x +ax +bx* +cx &c.\ expanded, will 
 
 fv 
 
 ' 
 
 _ 
 
 + A^ + Bo 
 x x^ + " 
 
 * x * + ^. as appears from Art. 108. There- 
 
 ir.l 
 
 fore this Expreflion being multiplied by x x, and the 
 
 pw\-m 
 
 Fluent taken fas ufual) we {hall have - - 4- 
 
 pv + m 
 
 fv+m+n " A . . r> x ^ pv+m+zn 
 
 Aax At> + Ba*Xx 
 
 -- 
 
 Ac 
 
 v + ^z + 
 
 a l b -f Da 
 
 + 4 -F &V, for 
 
 the Quantity propoied to be found. 
 
 S E -
 
 121 
 
 SECTION VII. 
 
 Of the Vje of Fluxions in finding the Areas 
 of Curves. 
 
 CASE I. 
 
 E 7* ARC be a Curve of any Kind whofe Or- 
 dinatei are perpendicular to an Axis AB. 
 
 112. 
 
 Imagine a Right-line ^R^ (perpendicular to AB) to 
 move parallel to itfelf from A towards B ; and let the 
 Celerity thereof, or the Fluxion of the Abfcifla A, in 
 any propofed Pofition of that Line, be denoted by Id; 
 
 Then it will ap- 
 pear, from Art, 4. 
 that the Re&angle 
 (bn) under bd and 
 the Ordinate R, 
 will exprefs the 
 correfpondirig Flu- 
 xion of the gene- 
 rated Area abR. : 
 Which Fluxion, if 
 A x, and bR=y, 
 will therefore be 
 ~vx : From whence, by fubftituting for y or x (ac- 
 cording to the Equation of the Curve) and taking the 
 Fluent, the Area itfelf will become known. 
 
 CASE II. 
 
 113. Let ARM be any Curve whofe Ordinates CR, CR 
 are all referred to a Point or Center. 
 
 Conceive a Right-line CRH to revolve about the 
 given Center C, and let a Point R move along th 
 
 faid
 
 H 
 
 A 
 
 The UJe of FLUXIONS 
 
 faid Line, fo as to trace out, or defcribe the propofed 
 
 Curve Line ARM. 
 
 Now it is evident, that, if the Point R was to move 
 
 from any Pofition Q., without changing its Direction and 
 
 Velocity, it would 
 proceed along the 
 Tangent QS (in- 
 ftead of the Curve) 
 and defcribe Areas 
 QjC, QSC about 
 the Center C, pro- 
 portional to the 
 Times of their De- 
 fcription ; becaufe 
 thofe Areas, or Tri- 
 angles, having the 
 fame Altitude (CP), 
 are as the Bafes Qj 
 and QS, and thefe 
 are as the Times, 
 becaufe the Mo- 
 tion in the Tangent 
 
 (upon that Suppofition) would be uniform. 
 
 Hence, if RS be taken to denote the Value of (z) 
 
 the Fluxion of the Curve Line AR, the correfponding 
 
 Fluxion of the Area ARC, will be truly reprefented by 
 Art. 2 the, uniformly generated, Triangle QCS * : Which, 
 ? nd 5- putting the Perpendicu!ar (CP) drawn from the Center 
 
 QSxCP 
 
 to the Tangent, = j, will therefore be ( = =. 
 
 Ctg 
 
 - ; from whence the Area itfelf may be determined. 
 
 2 
 
 But, fince in many Cafes, the Value of z cannot be 
 computed (from the Property of the Curve) without fome 
 Trouble, the two following Expreflions, for the Fluxion 
 of the Area, will commonly be found more commo- 
 
 svy y 1 ^ 
 
 dious, viz. and ; where / ~ RP and x =: the 
 it 20 
 
 Arch BN of a Circle, defcribcd about the Center C, at 
 
 any
 
 in finding Areas, 1 23 
 
 any Diftance <t (CB). Thefe Expreflu-ns are de- 
 rived from that a'oove, in the following Manner 5 viz. 
 
 z :j : : y (CR) : f (RP) * j therefore = ^ j andMit. 3 s, 
 
 confequently ; which is the firft Expreffion. 
 
 Again, becaufe the Celerity of R in the Direction of 
 the Tangent is denoted by z, that in a Direction per- 
 pendicular to CQ_ (whereby the Point R revolves about 
 
 CP 
 the Center C) will therefore ^ e ( QR x *) *' = Art. 35. 
 
 ifUS 
 
 ; which bejng to (x) the Celerity of the Point N 
 
 (about the fame Center) as the Diftance (or Radius) 
 CR (y) to the Radius CN (a) we fhal), by multiplying 
 
 QSZ 
 
 Extremes and Means, have =yx -, and confequently 
 
 2 
 
 : which is the other Expreffion. 
 2 la 
 
 The Method of applying this, together with the pre- 
 ceding Forms, will appear at large from the following 
 Examples : Wherein *, y t z, and u are all along put to 
 denote the AbfciiTa, Ordinate, Curve-line, and the Area 
 refpe&ively, tinlefs where the contrary is exprefsly 
 fpecilied. 
 
 EXAMPLE I. 
 
 114. Let it be propofcd to deter mine tfe Area of a right- 
 angled Triangle AHM. 
 
 Put the Bafe AH=a, the Perpendicular HM= j and 
 let AB (x) be any Portion of the Bafe, confidered as a 
 flowing Quantity, and let BR (y) be the Ordinate, or 
 Perpendicular, correfponding : 
 
 Then,
 
 J24 tte-Ufe 0/" FLUXIONS 
 
 Then, becaufe of the fimilar Triangles AHM and 
 ABR, it will be, a : b :: x : y =r -- . When ce y 
 
 A 
 
 B 
 
 H 
 
 *Art.ii*.(the Fluxion of the Area ABR*) is, in this Cafe, ;=: 
 
 bxx 
 
 j and confequently the Fluent thereof, or the Area 
 
 fArt ' 77 ' itfelf = f : Which therefore, when *=, and BR 
 2a 
 
 ab AHxHM 
 
 coincides with HM, will become - 
 
 2 2 
 
 the Area of the whole Triangle AHM; which we alfq 
 know from other Principles. 
 
 EXAMPLE II. 
 
 
 
 115. Let the Curve ARMH, wbofs drea you would find, U 
 the common Parabola. 
 
 In which Cafe the Relation of AB (x) and BR (y) 
 
 being exprefled by y*ax (where a is the Parameter) 
 t_ i . 
 
 JArt,iia. we thence get^ = a*x* ; and therefore a ( = yx % ) 
 
 = tfV^ : Whence u = | X 
 
 = a 
 
 x x\yx 
 
 (becaufe
 
 in finding Areas. 125 
 
 (becaufe aV =y) = | X AB x BR : Hence a Para- 
 bola is $ of a Retlangle of the fame Bafe and Altitude. 
 
 R 
 
 A B 
 
 The Area is here found in Terms of * ; but it will, 
 many times, be more eafily brought out in Terms of y 
 (without radical Quantities) as in the very Cafe laft 
 
 y* 
 
 propofed : Where A- being = , we therefore have *== 
 
 2yv 2y v 
 
 ; and confequently u (yx) = - : Whence u - 
 
 ^- = 3 x - = ^x* = jxABxBRs tbtfam 
 3* 3 3 
 
 as before. j 
 
 EXAMPLE III. 
 
 1 1 6. Let ARM (fee the preceding Figure) be a Para- 
 bola of any Kind', whereof the general Equation is 
 
 n-f-n m * 
 
 y a x . 
 
 Therefore, by extracting the Root, or dividing each 
 
 n 
 
 Exponent by m + n, we have y =: a X x j whence
 
 126 7&> Vfe of Fiirxrt>Ns 
 
 IM I 77 WT 1 J '# 
 
 z/ C^ = a x xx j and confequently (the trut 
 
 Fluent, or Area) = a*~ * x 
 
 m -f n 
 
 AB x BR. 
 
 No Notice lias been yet taken of any conftant Quan- 
 tity to be added to, or fubtra&ed from, the variable 
 One, firft found, in order to render it complete, agree- 
 able to the Obfervation in Art. 78. 
 
 But that no fuch Correction is required in any of tha 
 preceding Examples, is evident from the Nature of the 
 Figure j becaufe, when x and y are nothing, the Area 
 (u) ought alfo to be nothing, which it actually is ac- 
 cording to the Equations above exhibited. 
 
 The Fluent found in the fucceeding Example, will, 
 however, ftand in need of a Correction. 
 
 EXAMPLE IV. 
 
 117. ffbere ft is propofed to find tbt Arta. of the 
 ARH, whofe Equation is x* aV-M 1 ^* = o 
 
 Here, the given Equation is reduced to y 
 
 . 
 
 + whence *( = ;*; = ** ""* '' x ** : 
 
 Art.77. Whereof the Fluent (by the common Rule *) i$
 
 in finding Areas. 
 \\ 
 
 127 
 
 "* *' : Which, when x0 and arro, becomes 
 
 3 " ,* 
 
 -- ; this therefore fubtracted from a * ' . , leaves 
 3 - 3* 
 
 i 
 
 for the Fluent corre&ed, or the true 
 
 g * 
 
 Value of the Area ABR *. 
 When the Ordinate BR 
 
 *Art. 7 8. 
 
 becomes 
 
 equal to Nothing, and B coincides with H, then * will 
 
 become =rtf=AHj and therefore the Area of the whole 
 
 ji 
 
 Curve ARH will be barely = = f AH* 
 
 3 
 
 EXAMPLE V. 
 
 118. Let it be required to detumine the Area of tbt 
 hyperbolical Curve whofe Equation is x* y" = 
 
 *f 
 
 ft ~ n 
 
 In this Cafe we have y = - = a X x 
 
 and
 
 iz8 The Ufe of FLUXIONS 
 
 m\n -m 
 
 and therefore u (=y*) = a " x x * x: Whofe Fluent 
 
 m\n m m\n r m 
 
 a* X x " _ na " X x " 
 
 is jjj JM * wnicn wnen * * 
 
 J 
 
 = o, willalfobe =:o, if n be greater than m: There- 
 fore, the Fluent requires no Correction in this Cafe j 
 the Area AMRB, included between the Afymptote 
 AM and the Ordinate BR, being truly defined by 
 
 m\n n m\ 
 
 I the Quantity above determined. 
 
 ( 
 
 n m 
 
 But, if n be lefs than m, then the Fluent, when #=0, 
 
 will be infinite (becaufe the Index 
 
 - being nega-. 
 
 \ 
 
 :) Whence the Area 
 
 tive, o becomes a Divifor to na 
 AMRB will alfo be infinite. 
 
 But, here, the Area BftH comprehended between the 
 Ordinate, the Curve, and the Part BH of the other Afymp- 
 
 na 
 
 m n 
 
 tote, is finite, and will be truly expounded by 
 
 the fame Quantity with its Signs changed. For the 
 
 Fluxion
 
 in fndirtg A 
 
 1.29 
 
 Fluxion of tV Part A V !RB being- a x * ' that 
 of its Supplement BRH muft confequentiy !>e 
 
 " xx * x : Whereof the Fluent is 
 
 tn ' | w fi^**fff 
 
 lL a _*-2!5_L_ = the Area BRH : Which wants no 
 m n 
 
 Correction ; bccaufe, when x is infinite, and the Area 
 BRH = o, the faid Fluent will alfo intirely v uifh, 
 
 feeing the Value of x n (which is a Divifor to 
 is then infinite. 
 
 EXAMPLE VI. 
 
 -t-" \ 
 
 a " ' 
 
 119. Where let it be required to determine the Area of 
 the circular Seftor AOR. 
 
 Then, putting the Radius AO (or OR) = a, the 
 
 K 
 
 Arch AR (confidered as variable by the Motion of R) 
 ^ z, and Rr = x, the Fluxion of the Area will here 
 
 K be
 
 The life of F L tr x j ot? s 
 
 Att,n 3 . be exprefled by ( = the Triangle ORr * :) Whence 
 
 az 
 the Area itfelf is = = AO X | AR : From which it 
 
 appears that the Area of any Circle is exprefled by a 
 Re&angle under half the Circumference and half the 
 Diameter. 
 
 EXAMPLE VII. 
 
 J2.0. Wherein it is propofed to determine the Area CBAC 
 of the logarithmic Spiral. 
 
 Let the Right-line AT touch the Curve at A ; upon 
 which, from the Center C, let fall the Perpendicular 
 CT ; Then, fince by the Nature of the Curve the 
 
 Angle TAC is every where the fame, the Ratio of AT 
 (t) to CT (s) will here be conftant : And therefore the 
 
 Art-iis-Fluent of x ^* = x - = the Area which 
 
 t 2 '4 
 was to be found. 
 
 E X-
 
 in finding Areas. 
 
 E X A M P L E VIII. 
 
 121. Let the Curve ARM be the Involute of a given 
 Circle 
 
 In which Cafe the intercepted Part of the Tangent 
 RP (t) being every where equal to the Radius CO (a) 
 
 of the generating Circle, we therefore have CP (t ) = 
 
 Whence u (=^ 
 
 2t 
 
 ** 
 
 vy* **jy j ., _ 
 
 - - & ; and confequently = <. _ 
 
 2a 6a 
 
 CP 
 
 the required Area ACR : 
 
 Which will alfo exprefs the Area ARO generated by 
 
 the Radius of Evolution RO; becaufe, RO being = 
 
 K 2 the 

 
 132 
 
 The Ufe of FLUXIONS 
 
 *Art.n 9 .the Arch AO, the Seder ACO (i AO X OC *) is 
 equal to the Triangle CRO (i; RO x OC) which equal 
 Quantities being fucceffively fubtra&ed from CARO, 
 there remains AORzrACR. 
 
 EXAMPLE IX. 
 
 122. Let the Curve CRR, whofe dreaCRgC you would 
 ) be the Spiral of Archimedes. 
 
 Let AC be a Tangent to the Curve at the Center 
 
 C, about which Center, with any Radius AC (a] 
 fuppofe a Circle A'gg to be defcribed ; then the Arch 
 (or Abfciffa) fig correfponding to any propofed Ordi- 
 nate CR, being to that Ordinate in a given, or con- 
 ftant, Ratio (fuppofe as m to n] we have* (A^) =. 
 
 mv ' v * my y f i 
 
 ; therefore u - = - J , and confequently u 
 ArMis. n 2 2tf 
 
 = ^- = the Area CRR^C. 
 
 tan 
 
 EX-
 
 in finding Areas. 
 
 '33 
 
 E X A M P L E X. 
 
 123. Let the Equation of the Spiral CRR (fee the lajl 
 Figure) be ^ 
 
 Then, x being = l>j + 2<yj + ^dy 1 } + ^ey 3 j -f &c. 
 
 .. y*x\ by*} lcy*y 
 
 we fhal! have u ( ~ ; = + 
 20 / 2a 20 
 
 20 
 
 20 
 
 3^ $ 4^y 6 
 , + _ 
 
 loa na 
 this Cafe. 
 
 r. and therefore = r 
 ta 
 
 true Value of the Area i 
 
 n 
 
 EXAMPLE XL 
 
 124. Let it be pr ope fed to find the Area of a Semi- 
 circle AREH. 
 
 . Here, putting the Diameter AHzr*, AB~ *, and 
 
 BR =y bV. (as ufual) we have /- (BR a ) = ax y* 
 
 ABO H 
 
 (AB x BH), and confequently u (yx) x ^ax xx =5 
 
 i_ 
 
 a~x x x i I : Which Expreflion not being of the 
 
 Kind defg-ibed mdrt. 83 and 85. that admit of Fluents in 
 t finite
 
 134 * e - 
 
 finite Terms, let it therefore be refolved into an In- 
 
 i i 
 
 Art. 90 finite Series * and you will have rr *** x x 
 
 and 99. ; : - . 
 
 X X 1 X 1 5** _ \ T . 
 
 I rj-i ri t- 4 t$f' a X x x 
 
 2a 8a ioa 3 i2Qa 4 
 
 4. . 
 
 XX XX XX 
 
 -TT 7-3 &c. From whence, the Fluent o f 
 
 2a oo iva - j 
 
 every Term being taken, according to the common 
 
 .j. 5^ 
 
 Method, there will come out u =r a * x 1- ^__ 
 
 3 5" 
 
 t 
 
 X X 
 
 720* 704^ 
 
 > - 
 
 c . rr x V ax X 
 
 a x * * 5* 4 , . , A 
 
 ----- - - -1- vc. = the Area 
 <7 2oa 2a* 
 
 ABR. Now, when, AT = * <?, the Ordinate BR will 
 
 coincide with the Radius OE ; in which Cafe the Area 
 
 - - : - : - : - 
 becomes = ~ aV \aa~A \ T ' -^- T TT T ^ 
 
 0,6666 0,1 0,0089 
 
 0,0017 0,0004 &V. = 0,1964^*; which, multiply'd 
 by 2, gives 0,39283* for the Area of the Semi-circle 
 AEH, nearly. 
 
 As the foregoing Series, in finding the Area of the 
 whole Quadrant AOE, converges but flowly, a con- 
 fiderable Number of Terms ought therefore to be taken 
 to have the Conclufion hut tolerably ex2<5t, the five 
 firft Terms above collected being fufficient to bring 
 cut no more than three Places of Figures that can be 
 depended on. For which Reafon it may be of Ufe to 
 confider, whether, by computing the Area of fome par- 
 ticular Portion (ABR) of the faid Qiiaclpnt, that of the 
 whole may not be deduced j where x being fmall in 
 
 com-
 
 m finding Areas* 
 
 comparifon of , the Series may have fuch a Rate of 
 Convergency, that a fmaller Number of Terms will 
 be fuffccient *. 
 
 Now, in order to this, it is well known that, if the 
 Arch AR be taken = iAE (or 30 Degrees) the Sine 
 BR will be i AO ; and confequently AB (x) AO 
 OB:=AO /OR 1 BR a ; which, if the Radius AO 
 be expounded by Unity, (to facilitate the Operation) 
 will be =. 0,1339746 very nearly : This therefore, with 
 the Value of a, being fubftituted in the forementioned 
 
 Series, \/ax 3 x ~ -~ &c. we have 
 
 0,0694505 x 0,6666666 0,0133975 0,0001603 
 
 0,0000042 C3V. = 0,0693505 x 0,6531046 = 
 0,0452931 =. the Area ABR : Which added to the 
 Area OBR ( = OB x ^8R = /Jx i = 0,2165063) 
 gives 0,2617994, for the Area of the Sector AOR j 
 the treble whereof, or 0,7853982 (becaufe AR = 7 AE) 
 will therefore be the Content of the whole Quadrant 
 AOE : W T hich Number, found by talcing four Terms 
 of the Series only, is true to the laft Decimal Place. 
 
 This Conelufton may be otherwife brought out, 
 by finding a Series for the other Part of the Area, in- 
 cluded between the Radius OE and the Ordirute BR ; 
 wherein the Co-fine OB (inftead of the verfed Sine AB) 
 will be the converging (or variable) Quantity. 
 
 For, putting OB =r x, and OR (OA) = b y we 
 
 have y (BR = V OR* OB a = 
 
 confequently (yx) the Fluxion of the Area OBRE * = Art.iu. 
 
 _ _ _ 
 
 ""~ 
 
 7V IC * X* 
 
 ! , fcff. Whence the Area itfelf is = bx ~r , 
 256^^ 6ff 
 
 4ob 3 
 
 Now,
 
 Now, if A- (OB) be aflumed = AO (fo that the 
 Arch ER may be = ; AE) and the Value of b (AO) 
 bs expounded by Unity, we (hall have 
 
 (=A-XAr*=,5X|- y =,125 
 
 4 
 
 ,03125 
 
 X 7 \ 
 
 ^:-- j = , 
 4' 
 
 AT' (-A-'X^^:-- = ,0019531 
 
 #" (=* 9 X**=^-) = ,0004883 
 4 
 
 Which Values of the Powers of * being refpe&ively di- 
 vided by 6, 40, 112, 1152, 2816, &c. there will refult 
 0,5000000 0,0208333 0,0007812 0,0000698 
 0,0000085 0,0000012 0,0000002 &c. = 
 0,4783057, for the Area OBRE in the forementioned 
 Circurnftance, when OB r= \ OA^_ From which, de- 
 dueling the Triangle OBR (= V \ X \ =0,2165063) 
 the Remainder ,2617994 will confequently be the Area 
 of the Sector EOR ; the treble whereof (becaufe ER 
 is, bere^ = j AE) will give the Area of the whole 
 Quadrant, 0,7853982; as before. 
 
 EXAMPLE XII. 
 
 125. Let the Curve, wbofe Area you would find y be the 
 
 x 3 
 
 Ci/old of Diodes ; whereof the Equation is / = a _ x ~ 
 
 x* x x*"x 
 
 *Art.ii2, Here we have it (jx*) = ^-^ ~ "~T 
 
 X 
 
 X I 
 
 a
 
 in finding Areas. 
 
 T . 71""* 
 
 __ * * x i - : Wh 'c h being none of the Kind 
 
 ~ -i a I 
 
 a 
 
 that admit o' Fluents in finite Terms *, let it therefore Art.8j, 
 be refolved inr> .<n T nfinite o'-ries, and you will have u 
 
 5* 3 
 
 1 Crf TIT. / t. 
 
 H 
 
 2 a 
 
 I 2JC x . 
 
 Area itfelf ) will come out = - X -- h HP 
 
 rv* Vv- o x x' 
 
 5 i ffrV, ***; '-* ~ x * 
 
 ' a 5 
 
 5* 3 
 
 * \ ftff 
 
 CtO 1 I * 
 
 EXAMPLE XIII. 
 
 126; Let the propofed Curve CSDR be of fuch a Nature^ 
 that ( fuppofing AB Unity) the Sum of the Areas 
 CSTBC and CDGBC answering to any two propofed 
 Abfcl/jas AT and AG, jhall be equal io the Area 
 CRNBC whofe forref ponding Abfciffa AN drawn into 
 AB is equal to, AT x AG, the Produtt of the Meafures 
 of tint two former Abfcijjas, 
 
 Firft, in order to determine the Equation of the 
 Curve, (which mud be known before the Area can be 
 found) let the Ordinates GD and NR move parallel to 
 themfelves towards HF ; and, then, having put GD=y, 
 
 NR=z,
 
 '38 
 
 NR = z, AT = *, AG = j, and AN = *, the Fluxion 
 of the Area CDGB will be reprefented by ys 9 and that 
 
 . of the Area CRNB by ay * : Which two Expreffions 
 muft, by the Nature of the Problem, be equal to each 
 other ; becaufe the latter Area CRNB exceeds the for- 
 mer CDGB by the Area CSTB, which is here con- 
 fidered as a conftant Quantity ; and it is evident that 
 two Expreflions, that differ only- by a conftant Quan- 
 tity, muft always have equal Fluxions. 
 
 Since, therefore ys is = zu, and- u = aj, by Hypothecs, 
 it follows that u as^ and that the firft Equation (by 
 fubftituting for u), will become ys =: azs t or y =: az t or 
 laftly ys=zs, that is, GD x AGrzNR x AN: There- 
 fore- GD : NR : : AN : AG ; whence it appears that 
 every rc ii n:it e of the Curve is reciprocally as its cor- 
 refpond4ng Abfcifla. 
 
 Now, to find the Area of the Curve fo determined, 
 put BC = b, and BG = x : Then, fmce AG ( I + x) 
 
 : AB (i) : : BC (b) : GD (y) we have y = and 
 
 confequently u ( =. yx-) 
 
 ^ 
 
 b X x xx + X*x 
 
 10 
 
 Whence, BGDC, the Area it- 
 
 fejf
 
 in fading Areas. 139 
 
 be =*x* ^ + ; - + ^ &c. Which 
 
 %} 
 
 was to be found. 
 
 It may here be obfer,yed that the Areas of the Spaces 
 above mentioned, are analogous to, and have the very 
 fame Properties as Logarithms ; and that thofe Spaces, or 
 Logarithms, may be of different Forma or Values, ac- 
 cording as you take the Value of the firft Ordinate BC, 
 which may be aflumed at Pieafure : Thus, if BC be 
 taken =. AB Unity, the Curve will become an equi- 
 lateral Hyperbola whofe Center is A (becaufe then AG 
 X GD =. AB 1 ) and in that Cafe they are called hyper- 
 bolical Logarithms : But, if BC be taken 0,434294.48 
 (fo that the Logarithm, or the Area of the Space 
 CDGB, anfwering to the Abfcifla AG, when exprefled 
 by the Number 10, may be expounded by Unity, or 
 AB 1 ) we fhall then have the common, or Erlgean Form 
 of Logarithms. 
 
 From thefe Logarithms (given by the Tables) the 
 Bufinefs of finding Fluents, is in many Cafes, very 
 much facilitated : For, if the Fluxion <;iven appears to 
 agree with the Fluxion of any known Logarithmic Ex- 
 prcflion, its Fluent may, it is evident, be had by the 
 Tables, ready calculated, without the Trouble of ai\ 
 Infinite Series. 
 
 But, now to know what Kinds of Fluents are ex- 
 plicable by Means of Logarithms, it will be necefTary 
 to obferve that, the Fluxion of any hyperbolic Logarithm 
 is always exprejjed by the Fluxion of the corresponding 
 Number divided by that Number : This appears from 
 above, where (\x) the Fluxion of the Area (or Lo- 
 garithm) BGDC, when BC AB = I, is truly repre- 
 
 X 
 
 fented by ; where i -f x ( AG) may ft and for 
 
 1 f * 
 
 any Number whatever ; and x for its Fluxion. 
 
 Hence
 
 140 The Ufe of FLUXIONS 
 
 Hence the Fluent of will be exprefled by 
 
 ^* : 
 
 the hyperbolical Logarithm of x + v'x* + a 1 : For the 
 
 Fluxicn of (x -}- V ' x~ -jr a 1 ) the Number itfelf, bf ing x 
 
 ** _ * / x* a + ** __ * 
 
 t/V o 1 /*~ z ~ V x" a- 
 
 X /a? 1 ^ <f + x, this laft Quantity, divided by that 
 
 Number, gives . - -- , the very Fluxion firft 
 V x 2 - a 
 
 propofed. 
 
 It alfo appears that the Fluent of - : will be 
 
 y tax + x*- 
 
 truly exnounctd hy the hyperbolical Logarithm of a + 
 x + V iax -f x* Becaufe the Fluxion of the Number 
 
 ax -f xx 
 
 =^- " : X ^/2ax + xx + a + x j which divided 
 
 V2GX + XX 
 
 bv that Number produces -= 
 
 V2a# + xv 
 
 Likewiie the Fluent of -5 5 will be reprefented by 
 
 ti m ""' ^f 
 
 _ I . 
 
 the hyperbolical Logarithm of - - : Beeaufe, the 
 
 T-I f a+x x xa- x + x X tf-f x _ 
 
 Fluxion of - , being -..,, 
 a x> ^ 
 
 if the fame be therefore divided by - -, we (hall have 
 
 1 a x 
 
 a x lax lax 
 
 xf a+ x~ a x X a -f x ~~ a? ** 
 
 Laftly,
 
 in finding Areas. 14* 
 
 IGX 
 
 Laftly, the Fluent of == wi ^ be denoted 
 
 by the hyperbolical Logarithm of a a ^ . for 
 
 + /***' 
 
 ^\-1Cx 
 
 here the Fluxion of the Number is -rr =r. x 
 
 V? j-* 1 x 
 
 xx 
 
 4- 2axx 
 
 - ._>* ; which divided by 
 2 
 
 ? X ^ J # ~t^ 
 the Fluxion pro- 
 
 a -^x i^-^-x xr u ~rx 
 
 pofed. 
 
 Thefe four are the principal Forms of Fluxions ; 
 whofe Fluents may be found from a Table of Loga- 
 rithms of the hyperbolic Kind: Which Table, upon 
 Occafion, maybe eafily fupply'd by a Table of the com- 
 mon Form : For, fince the hyperbolical Logarithm of 
 any Number is to the common Logarithm of the fame 
 Number, in the confta^t Ratio of Unity to 0,43429448 
 (as appears from ah-ive) it follows that if any common 
 Logarithm be, either, divided by 0,43429448, or mul- 
 tiply'd by its Reciprocal 2,30258509, you will thence 
 obtain the hyperbolical Logarithm corresponding. 
 
 EX-
 
 142 Vbe Vfe of FLUXIONS 
 
 EXAMPLE XIV. 
 
 127. Lit it be required to determine the Area of the Curve ; 
 whofe Equation is efyx^y <4 3 0. 
 
 Art.m- In which' Cafe y being = - ^ we have it (yx] * 
 
 & ~x 
 
 x*x X 6 x 
 
 
 H, 
 
 B 
 
 M 
 
 X' 
 
 f * f 
 
 Whence = ax -I- + r + T -j- - 4- 6ff. 
 3 5<a 3 7<j 5 ga 7 
 
 =r the Area fought. 
 
 But the fame Area (or Fluent) may be found with- 
 out an Infinite Series, by Means of a Table of Lo- 
 garithms, agreeable to the Obfervations in the laft Ar- 
 ticle: For, fmce it there appears that the Fluent of 
 
 - t .. '- is truly exprcfled by the hyperbolic Logarithm 
 
 a * "X 
 
 a + x , . a 3 x / 2ax \ 
 
 of - , it follows that that of-^ -; ( = -r ; x -la* ) 
 
 ax a ~x \ o ~~x / 
 
 will be exprefled by the fame Logarithm multiply'd by 
 if. Thus, for Example fake, let a (=AC) be 
 
 taken
 
 in finding Areas. 
 
 taken = 10, and*(:=AB) =5; then will =3; 
 
 whofe Logarithm taken from the common Tables 
 is 0,4771213; which muhiply'd by the Modulus 
 2,30258509 (Tee the laft Article) gives 1,09861228 
 
 for the hyperbolical Logarithm of : j and this again 
 
 multipiy'd by 50 (**) produces 54,930614 for the 
 true Value of the Area ABRC, in the aforefaid Circum- 
 ftance, when AC = 10, and AB=r5. 
 
 EXAMPLE XV. 
 128* JVhere the propofcd Curve is that wbofe Equation is 
 
 '43 
 
 Here, by reducing the given Equation, we get y = 
 
 : Therefore^ = 
 
 * 
 
 * Art. 1 1. 
 
 Whence, the Fluent of ^ ^ being =: hyperb. 
 
 14 | & 
 
 c 
 
 A B H 
 
 Log. of x + vV-f#* (by ^r/. 126. that of 
 
 will confequenriy be =: the fame Logarithm multiply'd 
 by a*. 
 
 But to find whether the Fluent thus determined does 
 not need a Correaion k let x be taken = oj then the j A 
 
 Fluent
 
 144 
 
 life of FLUXIONS 
 
 Fluent will become hyp. t ng. a : x c~ : Which, there- 
 
 fore, mufi h" fubtracled, to have the tru^ Value of the 
 
 Art.yS.Area ACRB*i and then rhere refults a 1 X hyp. Log. 
 
 x + l/fNh** <? * hyp. Log. a = a* x hyp. Log. 
 
 _ 
 
 EXAMPLE XVI. 
 
 
 
 129. Let it be propofed to fnd the Area of the Hyperbola 
 ABD, and afro the Area of the hyperbolical Seftor 
 CAD j fuppoftng C to be the Center^ and A the prin- 
 cipal Vertex of the Curve. 
 
 Here, putting the Semi-tranfverfe Axis CA.=a t the 
 Semi^conjugate =: c y and CB =: x j we have, by the 
 
 Property of the Curve, y (=BD) = <\f ** aa; 
 
 ex 
 
 and therefore a = yx = ^/ AT* a 1 = the Fluxion 
 
 JArt.jii.of the Area ABD t 
 
 But to find the Fluxion of the Se&or CAD, it is 
 to be obferved, that as the faid Sedor is = CBD 
 
 ABD : ; a, its Fluxion will therefore be = 
 2
 
 in finding Areas. 145 
 
 *y y x *v y* 
 
 7 + T * -"t~T (becaufe *=J**) which, * Art . na . 
 
 by fubftituting for y and j, their Equals \/x* a* 
 
 r*.r 
 
 and /-r - => is at length reduced to 
 "V x a 
 
 ./ , -i : Whereof the Fluent (by >/rf. 126.) is 
 
 V.* 1 a z 
 
 X hyp. Log. * -f VV a ; vfrhich correaed (by 
 making * = a) will become x hyp. Log. x -f 
 
 ^ a a* X hyp. Log. a = x hyp. Log. 
 
 * 4- V x *~a* 
 
 - - - = th Sedor ADC : Which, fubtraded 
 
 2 
 ~ 
 
 2 
 
 from -;-" (= . = the Triangle ABD) 
 
 leaves * "" ^ x hyp. Log. *"*****' 
 
 for the required Area of the Hyperbola ABD. 
 
 EXAMPLE XVIL 
 136. Let the Curve propofed be the EHlpfu AEBf. 
 
 Then, putting the tranfverfe Axis ABrirrf, and the 
 Conjugate ( 2 CE) c ) we {hall, by the Property of 
 
 the Curve, have ^ (DR) = ^ Vox xx> and there- 
 
 fore (yx) s= - X x \/ ax xx a t he Fluxion of 
 a 
 
 the Area ARD. 
 
 I- But
 
 146 The Ufe of FLUXIONS 
 
 But* Vax xx is known toexprefs the Fluxion of 
 the cbrrefponding Segment AD of the circumfcribing 
 
 E 
 
 L 
 
 C D B 
 
 Semi-circle; whofe Fluent is, therefore, given, by Art. 
 
 f T I 
 
 124 j which being denoted by A, that of x * V ax x* 
 
 will, confequently, be = x A. Hence, the Area 
 
 of the Segment of an Ellipfis, is to the Area of the 
 correfponding Segment of its circumfcribing Circle, as 
 the lefler Axis of the Ellipfis is to the greater; whence, 
 it follows that the whole Ellipfis muft be to the whole 
 Circle in the fame Ratio. 
 
 EXAMPLE XVIII. 
 
 131 . Let the Curve AR &c. whofe Area CARS you would 
 ) be the Conchoid of Nicomedes. 
 
 Whereof the Equation (putting BC = a, and RV 
 
 (= AC) = b) is *y =7T7V~x I =7 1 (Vid. Art.tf.) 
 
 a */ * ,,* 
 Which, by Rcdudion, becomes x = - -- Z. j.
 
 '47 
 
 VV /: But, to bring it down to a, Jlill, more 
 fimplc Form, make v/^ / (= SV) = Z j then.? = 
 
 . #2 
 
 VJ? z 1 i whence, by Subftitution, x = "y 
 4- z ; and confequently x 
 
 z 
 
 But now, to exhibit the Fluent hereof; upon C, as a 
 Center, with the Radius AC (b) let a Quadrant of a 
 Circle AED be defcribed, and let RH, produced, meet 
 the Periphery thereof in E, alfo let EF be parallel to 
 AC, and let CE be drawn : It is evident (becaufe CE 
 (CA) = VR and EF = RS) that CF is alfo = VS 
 
 = z ; and therefore, EF being ( = /CE 1 CPJ = 
 
 VV z% it appears that VV z a (the fecond 
 
 L 2 Term
 
 148 'The Ufe of FLUXIONS 
 
 Term of our given Quantity) exprefles the Fluxion of 
 the Area AEFC : Whence, if to this Area (found by 
 the Table of Segments) the Fluent of the firft Term 
 
 ab*% "b -f- z 
 
 Art.ii6. 71 1> or ^ e kyP* k2' f L '__ x * a ^ *> be added, 
 
 the Sum will be the whole Area ARCS, that was to be 
 determined. 
 
 EXAMPLE XIX. 
 
 132. Let it be required to determine the Area ASRA 
 included by the common Cycloid ASM and its generating 
 Semi-circle ARH. 
 
 Put the Radius AO (or RO) =*, the Sine BR = j, 
 the Co- fine OB=#, and the Arch AR (= RS, by the 
 Property of the Cycloid) = z : Then AB being = a 
 
 M 
 
 ABO 
 
 H 
 
 A-, its Fluxion will be *; whence () that of the 
 Art, iii. Area ARS is = zx *. Now to find the Fluent there- 
 of, make w =: zx ( = the Fluent, if z was con- 
 
 ftant)
 
 in fndtng Areas. 
 
 149 
 
 ftant) then w being = zx xz* 9 we (hall have* Art ra 
 u ( zx) = <iv + xz. But C y Art. 35. ) * 
 (AR Fluxion) : y (BR Fluxioh) : : Radius : Co-line of 
 the Angle ARB, or its Equal ROB : : OR (a) : OB (x): 
 Therefore, by multiply ing Extremes and Means, we get 
 xk ay: Whence, by Subftitution u ( = <u>4-*) r= iw 
 + <7j; and confequently, by taking the Fluent, 14 = 
 w + ay = z* -f ay ~ AO x BR BO x AR 
 the Area ARS. 
 
 Hence it follows that the Area (AEFA) Ahen RB 
 coincides with the Radius FO, is barely = AO x FO 
 = AO" : And that the whole Area AMHFA is truly 
 defined by ARH x OH, or by ARH x OH; that is 
 by four times the Area of the generating Semi-circle. 
 
 EXAMPLE XX. 
 133. Let the Curve fropofed be the Catenaria DAB. 
 
 Then, drawing BS and b$ parallel to the Axis AC, 
 and AS and cbn perpendicular to the fame ; and making 
 (as ufual) Ac ^ fb=y and A^= z, we (hall have, by 
 
 tfcc Property of ihc Curve, tax H AT* = zz : Whence *= 
 
 J , and x = -7= . -- ~: From which the 
 * 
 
 Value
 
 150 The Ufe of FLUXIONS 
 
 Value of y (which in all Curves is = v'i' x* *) 
 
 will here be found = \S & -- ^ 
 
 '- 1 ~ 
 
 ^_ 
 
 ~ ' anc * 
 
 _ 
 multiplied by Va* + z* a 
 
 
 (-bs) gives ax -7~=f ( =t he Rectangle S*} 
 ^ a z -f- z 4 
 
 tt =r the Fluxion of the Area A: f. From whence, by 
 taking the Fluent, the Area itfelf is found = az, a 1 
 
 Z -f- Y/ a , 
 
 JArt. 12 6. X hyp. Log. - ^_ j . Which therefore de- 
 
 duded fromjhej^eaangle se ( = yx=y^a 
 
 leaves y /a a -f z * ay az, -f a* X ^ Z^. 
 
 - for the required Area Abe. But, fince j 
 
 1 . T * + VV -f- 2 a 
 
 we have y=.a X hyp. Log. ~ _ j 
 
 whence, by Subftitution, the Area, at laft comes out 
 rr y y* 'a 1 + z a az, or z=: a Va~ -f- 2 X ^y/:, <^. 
 
 SCHOLIUM. 
 
 134. At the Beginning of this, and in the preceding 
 Sections, we have feen how the Fluxions of Quantities are 
 determined, by conceiving the generating Motion to be- 
 come uniform at the propofed Pofition ; according to the 
 
 4 Art. a. true Definition of a Fluxion : But hitherto no parti- 
 cular Notice has been taken of tbt Method of Incre- 
 ments^ or indefinitely little Parts, ufed (and miftaken) 
 by many for that of Fluxions : In which the Operations 
 are, for the general Part, exaaiy the fame ; and which, 
 (tho' lefs accurate) may be applied to good Purpofe in 
 finding the Fluxions themfelves, in many Cafes. For 
 which Reafons it may not be improper to add here a 
 
 1 9
 
 in finding Areas. 151 
 
 a few Lines on that Head, to (hew the. Beginner how 
 the two Methods differ from each other ; especially as 
 we (hall be enabled, from thence, to draw out fome 
 Conclufions that will be of Ufe in the enfuing Part of 
 the Work. 
 
 It hath been frequently inculcated in the foregoing 
 Pages, that the Fluxions of Quantities are always mea- 
 fured by how much the Quantities thetnfelves would be 
 uniformly augmented in a given Time. Therefore, if two 
 
 B 
 
 r 1 
 
 M 
 
 
 D 
 
 A 
 
 c N 
 
 Quantities or Lines, AB and CD be generated together, 
 by the uniform (or equable) Motion of two Points B 
 and D, it follows, that any two. Spaces B and D</ 
 aftually gone over (whereby AB and CD are aug- 
 mented) in the fame time, will truly exprefs the Fluxions 
 of the generated Lines AB and CD : Whence it appears 
 that the Increments (or Spaces actually gone over) and 
 the Fluxions are the fame in this Cafe, -where the gene- 
 rating Velocities are equable. 
 
 But if, on the contrary, the Velocities of the two 
 Points, in generating the Increments M and NV, be 
 fuppofed either to increafe, or to decreafe, the Lines or 
 Increments fo generated will, it is plain, no longer ex- 
 prefs the Fluxions of AB and CD ; being greater, or 
 lefs than the Spaces that might be uniformly defcribed, in 
 the fame Time, with the Velocities at M and N. 
 
 If, indeed, thofe Increments, and the Time of their 
 Defcription, be taken fo exceeding fmall that the Mo- 
 tion of the Points during that Time may be confidered 
 as equable, the Ratio of the faid Increments, will then 
 exprefs that of the Fluxions, or be as the Velocity at 
 Ivl to that at N, indefinitely near j but cannot be con- 
 L 4 ceived
 
 152 %he UJe 
 
 ceived '. r be JlriRly fo j unlefs, perhaps, in certain par* 
 ticular Cafes. 
 
 Hen^e we fee that the Differential Method, which 
 proceeds upon thefe indefinitely littlelncrements (actually 
 generated) as we do upon Fluxions (or the Spaces that 
 might be unifymly generated) differs little, or nothing, 
 from the Method of Fluxions, except in the Manner 
 of Conception, and in Point of Accuracy, wherein 
 it appears defective : And yet it is very certain the 
 Conclufions this Way derived are mathematically true ; 
 which has afforded Matter of Wonder to fome : But the 
 Reafon why they are fo is very eafily explained. For, 
 although the whole complete Increment is actually un- 
 derftood by the Notation and firft Definition (of this 
 Method) yet in the Solution of Problems the exa6l 
 Meafure thereof is not taken, but only that Part of it 
 which would arife from an uniform Increafe, agreeable 
 to the Notion of a Fluxion j which admits of a ftricl 
 Demonftration : But, after all, the Differential Method 
 has one Advantage above that of Fluxions, which is, 
 we are not there obliged to introduce the Properties of 
 Motion. Since we reafon upon the Increments them- 
 felves, and not upon the Manner in which they may be 
 generated. 
 
 It has been hinted above, that, though the Increments 
 of Quantities are not, Jlriflly, as the Fluxions, yet 
 from them the Ratio of the Fluxions may be deduced j 
 and it appears that the fmaller thole Increments are 
 tjken, the nearer their Ratio will approach to that of 
 the Fluxions. Therefore, if we can, by any Means, 
 find the Ratio to which the faid Increments, by con- 
 ceiving them lefs and lefs, do perpetually converge, and 
 which they may approach, before they vanifh, nearer 
 than any affignable Difference, that Ratio (called here- 
 after, for Diftin6tion Sake, the Ratio limiting that of 
 the Increments) will be, J?rifl/y, that of the Fluxions. 
 
 This will more particularly appear fr<~>m the follow- 
 ing Inflancts , wherein the Manner of deriving the 
 Ratio of the Fluxions, from that of the Increments, 
 is fhewn.
 
 in fading Area** 
 
 I". Let it be propofed to determine the Ratio of tljt 
 Fluxions of x and **. 
 
 Now, if * be fuppofed to be Augmented by any 
 
 (fmall) Quantity x, fo as to become x -f- *, its Square 
 
 "7]* / / / 
 
 (**) will be augmented tax + x = x* -f ixx + xx ; 
 
 whence the Increment of x* will be ixx + x x- t which 
 
 / t 
 
 therefore is to (x} the Increment of r, as 2#-f x to r 
 
 Hence, becaufe the lefler x is taken, the nearer this 
 Ratio approaches to that of ix to i, which is its Limit, 
 the Ratio of the Fluxions will therefore be exprefled by 
 that of 2* to i, or, which is the fame, by that of 
 to x (as in Art. 6,} 
 
 2. Lst the Ratio of the Fluxions of x and x* be 
 required. 
 
 Then, if x be augmented to x + x, x will be aug- 
 
 mented to x -f x =r x -f nx x -j , x 
 
 n n i n 2 5 / 
 
 99. Whence the Increments of x and x* will be to 
 
 n i n ni n z' , n 
 each other as j to nx -f- x - x x-\ -- - 
 
 12 X 
 
 fl ^ I ft __ O n --_ -5 ' ' 
 
 X - X - x 3 # x &c. Where the fmaller 
 
 2 3 
 
 / 
 
 x is taken, thq nearer the Ratio will approach to that 
 
 of
 
 154 ^ e W e f FLUXIONS 
 
 of I to nx i which appears to be its Limit : There- 
 fore this laftRatio, or that of x to nx *, is the Ratio 
 of the Fluxions required. (Fid. Art.%.) 
 
 3. Let it be propofed to determine the Proportion of the 
 Fluxions of the Sides AC and BC, of a right-angled^ 
 plane "Triangle ABC ; fuppoftng the Perpendicular AB 
 to remain invariable. 
 
 B 
 
 If Cd be afTumed to reprefent any Increment of BC 
 and LWj the correfponding Increment of AC (rzAD) 
 the Ratio of thofe Increments will be, univerfally, ex- 
 prefled by that of the Sine of the Angle CDd to the 
 Sine of the Angle DCd (by plane Trigc.xotnetry) and the 
 lefs the Increments are fuppofed to be, the nearer will 
 the Angle CD^ approach to a right one, or to an Equa- 
 lity with B ; which is its Limit : And the nearer will 
 DCd approach, at the fame time, to an Equality with 
 BAG. Therefore the Ratio here limiting that of the 
 Increments is that of the Sine of B (or Radius) to the 
 Sine of BAG : Which alfo exprefles that of the re- 
 quired Fluxions. (Vid. Art. 35.,) 
 
 In the fame way the Proportion of the Fluxions of 
 other Kinds of algebraical and geometrical Quantities 
 
 may
 
 in finding Areas. 
 
 may be inveftigated ; but it will be unnecefTary to dwell 
 longer upon this Head : I (hall therefore only add one 
 other Obfervation from hence (which will be of ufe 
 hereafter) relating to the Value of an algebraic Fraction, 
 in that particular Circumftance when both its Numerator 
 and Denominator become equal to Nothing, or vanifh, 
 at the fame time. Which VJue (it follows from above) 
 will be found by dividing the Fluxion of the Numerator by 
 that of the Denominator. 
 
 For, fmce the Value of any Fraction, in that Cir- 
 cumftance, is to be looked on as the limiting Ratio to- 
 wards which its two Terms converge, before they va- 
 nifh, and feeing the Fluxions are, always, exprefled by 
 that Ratio, the Truth of the Rule, or Pofttion, is 
 manifeft. 
 
 An Example, however, may not be improper : 
 
 x * a * 
 
 Let therefore the Fraction be propounded, to 
 
 x a 
 
 find the Value thereof when x~ a. In which Cafe, 
 the true Value fought, or the Fluxion of the Nume- 
 
 2 xx 
 
 rator divided by that of the Denominator, is = r- 
 
 x 
 
 = 2x=2<7. And that this is the true Value, may he 
 confirmed by common Divifion, whereby the Fraction 
 propofed is reduced to x+ a ; whofe Value when x a t 
 js therefpre 2<?, the very fame as before. 
 
 SEC-
 
 The life a/" FLUXIONS 
 
 SECTION VIII. 
 
 'The U/e of Fluxions in the Rectification, or 
 fnding the Lengths, of Curves* 
 
 CASE I. 
 
 135. T ET ACG le a Curve of any Kind whofe Or- 
 \ j a'inates are parallel to themfehes and per- 
 pendicular to the Axis AQ; 
 
 If the Fluxion of the Abfcifla AM be denoted by 
 or by Cn (equal and parallel to Mm) and S, 
 
 M in 
 
 equal and parallel to Cr, be taken to reprefent the cor-. 
 
 refponding Fluxion of the Ordinate MC ; then will the 
 
 Art.48 Diagonal CS (touching the Curve in C *) be the Line 
 
 Id 49 which the generating point (p) would defcribe, was its 
 
 Motion to become uniform at C (Vid. Art. 48 and 49.) 
 
 which Line, therefore, truly exprefles the Fluxion of 
 
 7 Art. tithe Space AC gone over, according to the Definition f. 
 
 Hence, putting AM =#, CMrry, and ACrrz, we 
 
 have ~ ( = CS = Vc^TS^) = v/F+7 ; from 
 which, and the Equation of the Curve } the Value of a 
 may be determined. 
 
 CASE
 
 in finding the Lengths of Curves. 157 
 
 CASE II. 
 
 136. Let all the Ordinates of tie propoftd Curve 
 ARM bt referred to a Center C. 
 
 Then, putting the Tangent RP (intercepted by the 
 Perpendicular CP) = f, the Arch BN, of a Circle de- 
 fcribed about the Center C=x; the Radius CN (or 
 CB) =*, &c. (Vid. Art. 113.; we have z : y ::> (CR) 
 
 : t (RP*) and confequently =: ^-: From whence* Art. 35, 
 
 the Value of z will be found, if the Relation of y and 
 * is given. 
 
 But in other Cafes it will be better to work from the 
 
 following Equation, viz. % = 
 
 . Which 
 
 is thus derived. 
 
 Let the Right Line, CR, be conceived to revolve 
 about the Center C i then fince the Celerity of the ge- 
 
 nerating
 
 Derating Point R in a Direction perpendicular to CR is 
 to (x) the Celerity of the Point H, as CR (y) to CN 
 
 V^ 
 
 (a) It will therefore be truly reprefented by : Which 
 
 being to (y) the Celerity in the Direction of CR, pro- 
 
 Art. 35 .duced, as CP (s) : RP (/) * it follows that ^4" : / :: 
 
 y* jc a 
 j* : t* : Whence, by Compofition, + y 1 : j> 2 :: j* 
 
 + l* (;'):' 5 therefore } ~ + / = , and 
 
 confequently x-f- + / (:=-- = * ; as was to 
 
 be (hewn. 
 
 But the fame Conclufion may be more eafily deduced 
 from the Increments of the flowing Quantities, accord- 
 
 ing to the preceding Scholium. 
 
 i 
 
 For, if Rw, rm and N be aflumed to reprefent (z, 
 / 
 y and x) any very fmall correfponding Increments of 
 
 AR, CR and BN, it will be as CN (a) : CR (y) :: 
 
 / 
 
 x (the Arch N) : the fimilar Arch Rr = > -- And, 
 
 if the Triangle Rrm (which, while the Point m is re- 
 turning back to R, approaches continually nearer and 
 nearer to a Similitude with CRP) be confidered as 
 
 t, we fhall alfo obtain z 2 ( = Rw a = Rr a + r^) 
 / 
 
 y 1 x 1 ' 
 -^ \-y z : Whence, by writing z, x and y for 
 
 a 
 
 i i i 
 
 z, *and^ (according to the Scholium) there comes 
 
 v" x 1 
 out x* = T + j'% as before. 
 
 EX- 

 
 infolding the Lengths of Curves, 159 
 
 EXAMPLE I. 
 
 137. Let the Curve ARM whofe Length is fougbty be the 
 
 Semi-cubical Parabola. 
 
 i 
 
 Whereof the Equation being ax*=y* y or x , 
 
 % 
 a 
 
 i 
 
 Oy y w^^^^,* 
 
 we thence have* r= =~ : Whence z. ( Vj,*+x 2 *') *Art.i 3 5 
 
 (found by the common Rule) is2L9ZL. which, 
 
 L 
 
 corrected ( by making y z: o ) becomes 
 
 EX-
 
 i6o 
 
 EXAMPLE II. 
 
 138. Let the Curve propofed le a Parabola cf any 
 (other) Kind. 
 
 n 
 y 
 
 Then x r: n _ I being a general Equation to all 
 a 
 
 T 4 
 
 Kinds of Parabolas, we here have * n _ t > & ^d 
 
 2 2tf 2 .2 
 
 therefore s (=v j a . x 
 
 a 
 
 ? X I -f- -^ : Whofe Fluent, univerfally ex- 
 
 2.B 2 
 
 a 
 
 2 2 I 
 
 n y ____ 
 
 prefled in an Infinite Series, is y + r _ 27^z 
 
 ,. 2 I X 2 
 
 4 4 3 6 6 5 
 
 w y y e., 
 
 =r + ^ A* ^- = z 
 
 "" AB 4 /- OB^O ' 
 
 4^ 3 x8<r 6 5 x ioa 
 
 But, when 2 2, the Index of y, in the given 
 Fluxion, is either equal to Unity, or to any aliquot Part 
 of it, the Fluent may be accurately had in finite Terms, 
 
 by Article 84. 
 
 * 
 
 For, by putting == v, and 2 = c > our 
 
 Fluxion 
 
 (2 2I 2 
 T 4 ny 
 2-2 
 ^ 
 
 X j) is, in the firft 
 
 reduced to i +cy v \ X j; Which being compared 
 7 with
 
 in finding the Lengths of Curves* 
 
 with a + cz \ x dz x, the general Expreflion in 
 the forefaid Article, we have a =r i, %=ry, w=r > 
 
 = o, or i = o ; 
 
 whence r=v> s (r -f m] = v -f . an d confequently 
 
 f. * * Art. 84. 
 
 3 
 
 \ _ ^ = the Fluent 
 
 -s , ^ ; which was to be determined, arid 
 which will (it is plain) always terminate in v Terms, 
 
 when Vy or its Equal ^r~ > is a whole pofitive 
 
 Number. 
 
 ii} -j- i I \ 
 
 If (derived from v = I be fubfti- 
 
 2V in 2/ 
 
 tuted for its Equal , the Equation of the Curve, wiM 
 
 be changed to ax y"" ; which, if v be expounded 
 by 1,2, 3, 4, &c. fucceflively, will become ax z = y 3 9 
 ax*y*i ax 6 =.y 7 y ax*-=zy 9 &e. refpedively : In all 
 which Cafes the Length of the Curve may therefore be 
 accurately had from the Fluent above exhibited. 
 
 M More-
 
 162 The Ufe of FLUXIONS 
 
 Moreover, if n be affumed :=r 2 (ori;=^) the ge- 
 
 n 
 
 y 
 
 neral Equation, x ~ , will then become x rr 
 a 
 
 v* 
 
 j anfwering to the common (or conical) Parabola. 
 
 2 2 2 
 
 n y 
 
 And therefore in that Cafe x ( i + - 
 
 2 
 
 
 
 (by putting b = J = 
 
 4- ~~T= " ;= into 
 
 * 
 
 Where, the Fluent of the firft Term (of the Fluxion 
 fo transformed) being == \ Vb* y * -\- f (or 
 
 by the common Rule ; and that of the fecond Term 
 
 = k V" X hyp. Log. - - - L , * it follows 
 
 b 
 
 that the Length of the Curve will, in this Cafe, be 
 
 W 
 
 T 
 + i* X hyp. Log. 
 
 E X-
 
 in fading the Lengths of Curves* 
 
 EXAMPLE III. 
 
 139. Let the Curve propofed be the Involute of a Circle^ 
 whole Nature is fuch, that the Part PR of the Tangent 
 intercepted by the Point of Contact and the Perpendicular 
 CP, is every where equal to the Radius CO of the ge- 
 
 '63 
 
 yy & i 
 nerating Circle : Therefore z ( = ~- J being here= Art, 
 
 jy y 1 
 
 , we firft get z = j which corrected, by makino- 
 a ia 
 
 / a* f CP 1 \ 
 y a ( = AC) becomes ("QT/ tne true 
 
 Meafure of the required Arch AR. 
 
 M 2 
 
 EX-
 
 164 The Ufe ^FLUXIONS 
 
 EXAMPLE IV. 
 
 140. In which the Spiral of Archimedes is propofed. 
 Where, the Value of / (AT) being denoted by 
 y. a (Vid* Art. 62.) we get z ( = - J 
 
 "*" ^ : Which Fluxion being exactly the 
 
 fame as that expreffing the Arch of the common Para- 
 bola, found in Article 138. its Fluent will therefore be 
 truly reprefented by the Meafure of the faid Arch, or by 
 
 Value there exhibited. 
 
 E X-
 
 in finding the Lengths of Curves. 1 65 
 
 EXAMPLE V. 
 
 141. Let the Curve be a Spiral whofe Equation is 
 <T~ l x^y* (Vid. Art. 136.) 
 
 . m I 
 
 In which Cafe x being = -^- , it is evident 
 
 m i 
 
 that z ( 
 
 Which Value 
 
 f <* | A r-\ uit \jut ~f i x"\ & wi* 
 
 may be otherwise had, without an Infinite Series, when 
 - is a whole pofitive Number, Vid t Art. 138. 
 
 EXAMPLE VI. 
 
 142. Where, the Right-fine, Verfed-fine, Tangent, or 
 Secant of an Arch of a Circle, being given, it is re- 
 quired to find the Length cf the Arch it/elf in Terms 
 thereof. 
 
 Put the Verfcd-fme A =r x, the Right-fine 
 the Tangent AT 
 
 t> the Secant OT 
 =5, the Arch AR 
 
 z, and the Radius 
 AO, or RO, - a ; 
 alfo let R;z .v, nr 
 y and Rr =. z : 
 Since the Angle 
 r*R ( = Right- 
 
 angle) rr O^-R, and 
 
 r R ( = Right- b 
 
 an^le RO) = OR, the Triangles rR and OR 
 
 M 3 are
 
 166 ?be Ufe of FLUXIONS 
 
 are therefore equi-angular ; and it will be, "Rb (y) : OR 
 
 caufe, by the Property of the Circle i/zax xx = y. } 
 Alfo, Ob (vV /) : OR (a) : : nr (y) Rr (ij = 
 
 ay 
 -7=-. ?. Thefe two Values exhibit the Fluxion of 
 
 Vrf* jT 
 
 the Area in Terms of the Verfed-fine and Right- 
 fine refpeclively : But, to get the fame, in Terms of 
 the Tangent and Secant, we have (by fim. Triangles) 
 
 QT (=s = V'TT?) : OA (a) : : OR (a) : Qb = 
 
 ' 
 
 : Hence AZ> = a "=. a 
 
 whofe Fluxion is therefore = ~r ~ 5-: Whence 
 
 * ^T?] 1 
 
 (again by fimilar Triangles) AT (= V s 1 a* = <^ : 
 
 OT (=s= 
 at 
 
 Now, from any one of the four Forms of Fluxions 
 
 here found, the Value of the Arch ifcfelf (by taking the 
 Fluent, in an Infinite Series) will like wife become 
 known. 
 
 But the third Form, exprefled in Terms of the 
 Tangent, being intirely free from radical Quantities, 
 will be the moft ready in Practice, especially where the 
 required Arch is but fmall ; though the Series arifing 
 from the firft Form, always, converges the fafteft. 
 
 if.
 
 in fading the Lengths of Curves. 1-67 
 If, therefore, x be now converted to an In- 
 finite Series, we fhall have =r / ?+T ~ 
 
 t$c. and confequently 'z =r / -f - g + 
 
 5 fcfr. = AR. Where, if (for Example Sake) AR 
 
 be fuppofed an Arch of 30 Degrees, and AO (to ren- 
 der the Operation more eafy) be put = Unity, we 
 
 iliall have / */~~\ .5773502 (becaufe O 
 *R(i)::OA(i): AT (/) = ":) 
 
 Whence 
 / 3 ( =/ x / Z rr/x4) = .1924500 
 
 i* ( sr<?X/=^J = .0641500 
 
 r: - -.0213833 
 
 =r 7 x/"= = .0071277 
 
 = =r .0007919 
 
 r= = -0002639 
 
 And therefore AR r= .5773502 
 
 3 
 
 .0641500 .0213833 .0071277 .0023759 
 5 7 9 n 
 
 M 4
 
 168 The Ufe of FLUXIONS 
 
 '.0007919 .0002639 .0000879 
 
 .0000097 
 
 15 
 
 .0000032 
 
 -f 
 
 -r- 2I 23 ~ -5235987 = Which mul- 
 
 tiplied by 6 gives 3.1411,92 + for the Length of the 
 Semi-periphery of the Circle whofe Radius is Unity. 
 
 At Article 1 26. certain Forms of Fluxions were pointed 
 out, whofe Fluents are explicable by means of hyper- 
 bolical Spaces, or a Table of Logarithms : Which Forms, 
 it is obfervable, a ree in every thing, but the Signs (and 
 ccnftant Quantities) with thofe exhibited above, for the 
 Arch of a Circle. And th fe lafr, like them, may 
 ferve as fo many (r-ther) Theorems for finding Fluents 
 by means of a Table of Smcs, Tanger t- and Secants. 
 But, as fuch a Table is ufually calculated to a Radius 
 of i,OOOOOO &c. (or Unity) the following Equations, 
 derived from thofe above, being adapted to that Radius, 
 will be rather more commodious. 
 
 
 f w 
 
 g 
 
 Verfed-fine 
 
 1 
 
 V/ law a/ 2 
 
 
 
 
 f 
 
 
 
 <u 
 
 ay 
 
 
 
 Ri^ht-fine 
 
 
 3 
 fo ^ 
 
 
 h -J1 
 
 
 . w 
 
 is , and 
 
 "". 
 
 aiv 
 
 o 
 
 Tangent 
 
 Radius Unity. 
 
 a~ + w~ 
 
 r5 
 
 
 15 
 
 
 
 h 
 
 a*iv 
 
 cr 
 u 
 
 Secant 
 
 
 
 i/ * ~* - 
 
 
 
 The way of deducing thefe Exprefiions, from the 
 foregoing ones, is extremely eafy : For, if A be put to 
 denote the Arch whofe Radius is Unity, and whofe 
 
 IV 
 
 Vcrfed-fme, Right-fine, Tangent, or Secant is (ac- 
 cording to the different Cafes here fpecified). Then, 
 becaufe fimilar Arcs, of unequal Circles, are as their 
 
 Radii,
 
 in finding the Lengths of Curves. 169 
 
 Radii, it will be i : a :: A : (a A] the Length of the 
 Arch AR (fee the Figure.) Therefore, the Fluent of 
 
 ax a*iu . 
 
 ... -s (or - . -, putting w = x) being 
 
 v iax xx yiaw w 
 
 *ia 
 
 = aA (AR), that of , . muft neceflarilvr be 
 
 yiaw w 
 
 n: A : And in the very fame Manner the other Forois 
 are made out. 
 
 EXAMPLE VII. 
 143. Let thepropofcd Curve be the common Cycloid. 
 
 Then, if the Radius AO of the generating Semi-circle* * See 
 be denoted by a, we fhal] have BR = Vzax x*; and'*'' ' 
 
 the Fluxion thereof -f* ** ; Which bein? 
 1/2^ x* 
 
 added to ( ax J\ t he Fluxion of AR or its 
 \ V zax xx / 
 
 Equal RS (given by the preceding Article) we 
 thence et 2 ** ~ Xx - 
 
 V / 2ax x 1 _ "T X 
 
 2* x\ i fr the true Fluxion of the Ordinate BS of 
 
 thc'Cycioid. 
 
 Hence z f V/^T? t) = V** + il^JfHf - 
 
 ^ ~ f Art. 
 
 /^L - I " ~^ 
 x \/ ~ 2{i \ x ' x ; 
 
 and confequeatly, by taking 
 
 the Fluent, zr:I7 X JL 2 V 7 2^ = the Arch 
 AS of the Cycloid. 
 
 EX-
 
 *Tbe Ufe ^FLUXIONS 
 
 EXAMPLE VIII. 
 
 1 44. Wterein it is required to determine the Length of the 
 Arch of the common Hyperbola. 
 
 In this Cafe (the Semi-tranfverfe Axis being repre- 
 fented by b, and the Semi-conjugate by t) we have 
 
 = 2bx + *' i and therefore * = b ^ + ** 
 f 1 c 
 
 Lyy ,- ____ _______ r- 
 
 b: Hence * = -7==, and i (= Vj*+i*) 
 
 _ _ <' a +/ _ 
 
 / ^*y l y x y b*y* 
 
 v ? * J^TT? =>' v i + TMTTy ' which > 
 &y 
 
 by converting; - - r-r into an Infinite Serie! y becomes 
 f* + c^ 
 
 ^V ^v* AV ^*v* 
 
 i+^--4-+Tr-!-TT ** But flill 
 
 t L C * 
 
 we have the Square Root to extradl ; In order thereto 
 let it be affumed = I + A/ + B/ + C/ + D/ &fr. 
 Then, by fquaring, and tranfpofing (Vid. Ait. 98.^ 
 th'cre arifes 
 
 -J-AV-f zAB 
 
 =0 
 
 Hence A = ~j B zr - ^ - - - 
 
 i+ i 1 ^ l b* b 6 
 
 . C" - AR _ 4- -- H --- 
 
 """' 8 " " 8 ' C ll > 
 
 c 
 
 fc-V. esfr. Therefore i ( = j v^ i + -+tsc. y X
 
 in finding the Lengths of 'Curves. IJi 
 
 c. And confe- 
 
 By the very fame way of proceeding the Arch of 
 an Ellipfis may be found, the Equations of the two 
 Curves differing in nothing but their Signs. 
 
 145. 
 
 SECTION IX. 
 
 Application of FLUX IONS in invefii gating 
 the Contents of Solids. 
 
 LE T ABC reprefent any Solid ; conceived to 
 be generated (or defcribed ; by a Plane PQ_ 
 pafll \^ over it, with a parallel Motio:i : Let Hh (per- 
 pemiicular to PQJ be taken to exprefs the Fluxion of 
 AH (x) or the Velocity with which the generating 
 Plan is carry 'd ; 
 alfo let the Area of 
 the Part, ErnFn, 
 of the Plane inter- 
 cepted by, or con- 
 tained in, the Solid, 
 be denoted by A: 
 Then it fol'ows, 
 from Art. 2 ?.nd 5. 
 that the Fluxion of 
 the Solid AEF. will 
 be exprefled by Ax. 
 From whence, by 
 expounding A in Terms of #, (according to the Nature 
 of the Figure) and then taking the Fluent, the Content 
 
 of
 
 172 
 
 V/e of FLUXIONS 
 
 of the Solid (which wefhall, always, hereafter reprefent 
 by s) will be given. 
 
 But, when the propofed Solid is that arifmg from the 
 Revolution of any given Curve AEB about AHD, as 
 an Axis, the Fluxion (s) of the Solidity may be ex- 
 hibited in a Manner more convenient for Practice : For, 
 Art. 124. putting the Area (3,141592 &c*) of the Circle, whofe 
 Radius is Unity, = />, and the Ordinate EH y, it 
 will be i 2 :/::/>: (pyr) the Area of the Circle Em Fn, 
 which being wrote above inftead of A^ we have s 
 - rz py*x. The Ufe of which will be fufficiently fhevyn 
 in the following Examples. 
 
 EXAMPLE I. 
 
 146. Let it be propofed to find the Content of a 
 Cone ABC. 
 
 Put the given Altitude (AD) of the Cone ~ a, and 
 the Semi-diameter (BD of its Bafe = b: Then, the 
 Pittance (AF) of the Circle EG, from the Vertex A, 
 being denoted by x, &c. we have, by fimilar Triangles, 
 
 as a : b :: x : EF (y) = . Whence, in this Cafe, i 
 
 and 
 
 i 
 consequently s r 
 
 which, when x~ a 
 
 for the Content of the whole 
 __ Cone ABC. Which appears, 
 
 from hence, to bcjuft -V of a Cylinder of the fame Bafe 
 and Altitude.
 
 in fading the Contents of Solids. 
 
 EXAMPLE II. 
 
 147. Inhere ', let the Solid propofed be a parabolic Ganoid* 
 or that arifing from the Revolution of any Kind of 
 Parabola about tts Axis* 
 
 _ . mn t m 
 
 Then, from the Equation a x - y , of the ge- 
 
 m H 
 
 nerating Curve, we get ya m xx m , and j (=/>/*) 
 
 zm an 2 
 
 = pa x xx m ; and therefore s pa X 
 
 2W- 2 
 
 = /./ X - - = the Content of the Solid j 
 
 which therefore is to (py*x) the Content of thecircum- 
 fcribing Cylinder, as m to 2n + m. Whence the Solid 
 generated by the conical Parabola (where m^=2, and 
 =i) appears to bejuft i of its circumfcribing Cy- 
 linder. 
 
 EXAMPLE III. 
 
 14?. Let the propofed Solid AFBH be a Spheroid. 
 
 In which Cafe, putting the Axis AB, about which 
 the Solid is generated, rr#, and the other Axis FH, 
 of the generating Eilipfis = /, it follows, from the 
 Property of the Eilipfis, that *:*::* x a * 
 
 (AD x BD) : / (DE) 1 = ^ x ax ^ xx . Whence . 
 wehave \ (-pft*} - tf. x ^_^i and . Aftt 
 
 5 = X iaxxlx* = the Segment AIE. Which, 
 
 when
 
 when AD (x) = AB (a), 
 
 - 
 becomes l~: xia 3 i 
 
 J pal* = the Content 
 of the whole Spheroid. 
 Where, if* (FH) betaken 
 = a (AB) we fhall alfo 
 get pa 3 for the true Con- 
 tent of the Sphere whofe 
 Diameter is a. Hence a 
 Sphere, or a Spheroid, is \ 
 of its circumfcribing Cy- 
 linder } for the Area of the 
 Circle FH being expreffed 
 
 by , the Content of the Cylinder whofe Diameter 
 
 is FH, and Altitude AB, will therefore be - ; 
 
 4 
 of which i pab r , is, evidently, two third Parts. 
 
 EXAMPLE IV. 
 
 149. Let tie Solid, whofe Content you would find, be the 
 hyperbolical Conoid. 
 
 Then, from the Equation,/ 1 = xax -f xx, of 
 
 the generating Hyperbola, we have s (fy ) = 
 
 pb 1 
 
 
 
 _ _ 
 
 X axx + x*x, and confequently s -j-x r a** + i f 3 
 
 == the Content of the Conoid j which therefore is to 
 
 f>b z _____ 
 
 ^T x ax-\-x' 1 X *) that of a Cylinder of the fame 
 
 Bafe and Altitude, as a + f # too-f *. This Ratio, 
 if # be extremely fmall, will become as i to 2 very 
 nearly ; Whence it may be inferr'd, that the Content 
 
 of
 
 infolding the Contents of Solids* 
 
 of a very fmall Part of any Solid, generated by a Curve, 
 whofe Kay of Curvature at the Vertex is a finite Quan- 
 tity, is half that of a Cylinder of the fame Bafe and Al- 
 titude, very nearly : Becaufe any fuch Curve, for a fmall 
 Diftance, will differ infenfibly from an Hyperbola, whofe 
 Radius of Curvature, at the Vertex, is the fame. 
 
 This might have been inferred, either, from the 
 common parabolic Conoid, or the Spheroid, in the pre- 
 ceding Examples} but other Obfervations would not 
 allow Room for it there. 
 
 EXAMPLE V. 
 
 150. In which the propofed Solid is that arifmg from tht 
 Rotation of the Ctffoid 0/~ Diodes, about its Axis, 
 
 x 3 
 Here, _y* being =: _ , ' we have s (fy*x) Art. 
 
 px*x 
 
 - . But, in Cafes like this, (where the Denominator 
 
 G^'^X 
 
 is rational arid the variable Quantity in the Numerator 
 of feveral Dimenfions) it will be neceflary to divide the 
 latter by the former, in order to obtain the Fluent, by 
 leflening the Number of Dimenfions : Thus, dividing 
 px 3 x by #+<*, according to the Manner of compound 
 Quantities, the Work will ftand thus : 
 
 px 3 x o ( px'xpaxx pa'x 
 
 px 3 x pax z x 
 
 -\-pax*x o 
 
 xx o 
 
 +pa 3 x 
 
 \Vhere, the Quotient being px z xpaxxp a ' x , and the 
 
 Remainder pa*x, the Value of the given Fradion 
 
 a x* 
 
 will
 
 the Ufe of FLUXIONS 
 
 will therefore be truly exprefled by px*x - paxx -* 
 pa*x -f : Whofe Fluent, properly corre&ed, is 
 
 Q 
 
 3 f ' a~x 
 
 Vid. Art. 126. . 
 
 EXAMPLE VI. 
 
 151. Let the Solid be that arifmg from the Rotation of the 
 Conchoid of Nicomedes about its ' 
 
 The Sub-tangent ~r of this Curve being = - 
 y 
 
 (Vid. Art. 48 and 57.) we have x =7?^==^, an d 
 
 J ' b 1 
 
 if', *\ faV'ypfj 
 Art 145. therefore s (py x*} = / , = 
 
 y b 3 - 
 
 "' ~' But, in order for the more eafy frnd- 
 Y fr 1 y 1 " 
 
 ing the Fluent thereof, put vV/ = j and then, 
 
 uu 
 
 y being = V * * and j =: ' we fhall, 
 
 * 1 
 
 paF'u 
 
 by Subftitution, get s z + p X b*u u*u . 
 
 Whence, the Fluent of -===. being exprefled by 
 vb 1 u 
 
 the Arch (A} of the Circle whofe Radius is Unity and 
 Sine f, the Fluent of the whole Expreffion will be 
 
 * x >/+/> X PU y 3 - Which, when>>_ o, or ub-, 
 
 gives C/^ a X \p + p X | ^ 3 ) />i a X i^ + yi for the 
 Content of the whole Solid, when its Axis becomes in- 
 finite. q 
 
 EX-
 
 in fading the Contents of Solids. 
 
 177 
 
 EXAMPLE VII. 
 
 151. U^nere it is required to find the Content of a parafafic 
 Spindle ; generated by the Rotation of a given Parabola 
 ACB about its Ordinate AB. 
 
 Put CM (the AbfcifTi of the given Parabola) = a y 
 and the Semi-ordinate AM (or BM) = b\ and, fup- 
 pofing ENF to be any Sedlion of the Solid parallel to DC, 
 let its Diftance MN (or EP) from DC, be denoted 
 fcy w : Then, by the Property of the Curve, we fhall 
 
 .F, 
 
 have AM* (T-) : EP 1 (w*) :: CM (a) : CP = 
 
 =5-: Therefore EN (= CM - CP) = =- = 
 
 V 
 
 con f eq uently p x EN 1 = x 
 
 * __ 2i*w* -f w * = the Area of the Se&ion EF : 
 Which multiply'd by (w) the Fluxion of MN, gives 
 
 &'. 
 b* 
 
 ov 1^1^ + tti 4 aw for the Fluxion of the 
 
 Solidity, * whofe Fluent, -^ x b+w ^b l w 3 -f }w s , * Art. 145* 
 
 fQ. 2 7 \ 
 
 fopa b \ 
 when w becomes =: b, is ( J half the Content 
 
 N EX- 
 
 of the Solid.
 
 178 be Vfe of FLUXIONS 
 
 EXAMPLE VIII. 
 
 153. Let the Solid ACBD (fee the la/1 Figure) be a 
 Spindle, generated by the Rotation of the Segment of a 
 Circle, ACB, about its Chord, or Ordinate, AB. 
 
 Then, if the Radius OE be put = r, OMr=</, and 
 EP = w &c. ( as before ) we fhall have OP ( = 
 y/OL EP^nyV w l , and EN ( =OP OM ) 
 := VV w 2 " d: Therefore s, in this Cafe, is = 
 
 v'r* w 7 - d\ p w x r* w* + d 1 2^ v /r 1 
 
 =. <w X r l d 1 w~ -*- >au X ^d V ' r~ - w~ - - 
 
 Whence, the Fluent of the Part, pw x ld^/r i w L 2^ 1 
 
 ( = T.dp X nu X V'r 2 w z d :=r 2dp x w X EN ) 
 * Art. nz. being exprefled by idp x Area MNEC * the Fluent 
 of the Whole, or the true Value of s, will be ex- 
 prefled by pw X r z d 7 - w z -idp x Area MNEC, 
 or by its Equal p x MN x AM 1 4MN 1 2/> x OM 
 X Area MNEC: "Which, when MN = MA, gives 
 p x | AM 3 zp X OM X y/ra yfCAf, for the Con- 
 tent of half the Solid : Where the Area ACM may be 
 found by Art. 124. Or more eafily by the common Table 
 of the Areas of the Segments of a Circle j to be met 
 with in mofl Books of Gauging. 
 
 EXAMPLE IX. 
 
 154. Let it be propofed to find the Content of the S-Jld 
 AEGB; whofe four Sides AH, AF, CH, CF are 
 plane Surfaces, and its Ends ADCB, EFGH given 
 Rectangles, parallel to each other. 
 
 Let the Sides AB and AD, of the Bafe, be denoted 
 by a and b ; and thofe of the Top (EH and EF) by c 
 and d refpeiiively ; moreover, let h exprefs the perpen- 
 
 dicular
 
 in finding the Contents of Solids. 
 
 dicular Height of the Solid ; and let x (confider'd as 
 variable) be the Diftance of (IL) any Section thereof 
 (parallel to the Bafe) from the Plane EG. 
 
 F 
 
 It is evident, from the Nature of the Figure, that 
 the Se&ion IL is a Re&angle ; and that 
 b : x :: AB-EH : IM EH :: BC HG : ML HG. 
 
 From thefe Proportions we have IM EH= * 
 
 b 
 
 .d M L-HG= O*2 : 
 
 U TUT 
 
 Hence IM = 
 
 a ~~ C X * 
 
 r, and ML = - - -f </; and confequently the 
 
 AreaoftheReaangle(IL) = 
 ad 2cd+cb 
 
 j 
 
 X x -f eel: Which being mukiply'd by 
 *, and the Fluent taken, there refults ~ C x b ~ d * ** 
 
 ad 2cd+cl>xx' t 
 
 + cdx for the Content of IFGL : 
 
 N 2 Which, 
 
 179
 
 180 
 
 The Ufe 0/" FLUXIONS 
 
 Which, when x = h, becomes 
 
 / icrl+cb X 
 
 AB x AD-f EHxEF + nb + ,hixAD + EFx h 
 the Quantity propofcd to be found. 
 
 If EF (d) be fuppofed to vanifh, and the Lines EH 
 and FG to coincide, the Planes AEHB and DFGC 
 will form an AnjHe or Ridge, at the Top of the Solid 
 (refembling the Roofs of fome Buildings, whofe Ends 
 as well as Sides run up Hoping) and, in this Cafe, the 
 Content, found above, will become morefimple, being; 
 then exprefled by lab + be X {A, or its Equal 2AB + EH 
 
 But, if EF be fuppofed EH, and ADrr AB, the Solid 
 will then be the Fruftrum of a fquare Pyramid ; and its 
 Content =a* + ac + c 1 X j&, AB*+ ABxEH-f-EH 
 X y h: From whence, by taking EHr= o, the Content 
 of the whole Pyramid whofe Bafe is AB% and its Al- 
 titude b, will alfo be given, being c= AB*x | h. 
 
 EXAMPLE X. 
 
 5^. Let the propofed Solid be tin at ^ commonly known by tke 
 Name of a Groin; whofe Sections parallel to the Bafe 
 are, all, Squares, and whereof the two Sections per- 
 pendicular to the Bafe, through the Middle of the 
 oppofite Sides, are Semi-circles. 
 
 Let bcdef be 
 any Section paral- 
 lei to the Bafej 
 andletitsDiftance 
 
 V 7 A^ from the Ver- 
 
 tex of the Solid, 
 be denoted by x ; 
 alfo let areprefent 
 the Radius AB 
 (or BN) of the 
 cir-
 
 infolding the Contents of Solids. 1 8 1 
 
 circular Section ABNA, perpendicular to the Bafe. 
 Then, bn being {by the Property of the Circle) = 
 Viax y.y, the Side of the Square </f, will be = 
 2 V iax xx, and therefore the Area 4 x lax xx ; 
 whence s 4* X lax xx, and confequently s ^(tx* 
 
 ~-}3 
 
 A-X *2,fl\ 
 
 : Which, when x = a, becomes =; the 
 
 Content of the whole Solid. 
 
 If the Solid be a Groin of any other Kind, or fuch, 
 that its two Sections perpendicular to the Bafe, through 
 the Middle of the oppofite Sides, are any other Curves 
 than Semi-circles, the Content may, {till, be found in 
 the fame Manner ; and will be always in proportion to 
 the Solid generated by the Revolution of the faid Curve 
 about its Axis, as a Square, is to its infcribed Circle. 
 But, if the forefaid perpendicular Sections be Curves of 
 different Kinds, the Sections parallel to the Bafe will 
 no longer be Squares, but Rectangles ; whofe Sides are 
 the correfponding (double) Ordinates of the refpective 
 Curves. Thus, for Inftance, let one Section be a Cir- 
 cle and the other a Parabola, whofe Ordinates, to the 
 comrnon Abfcifla, *, are cxprefled by */ dx .vxand y/ax t 
 refpectively; then the Sides of the rectangular Section, 
 
 parallelto the Bafe of the Groin, will be 2 Vdxxx and 
 2 */ax : Whence the Area of that Section is =; 4* 
 VW <?*, and therefore j ^ Xx ^/ a j ax . 
 Where, by taking the Fluent, * s 
 
 3 * 
 
 Content of fuch a Solid. 
 
 = the true 
 
 N 3 EX-
 
 The Ufe of FLUXIONS 
 
 EXAMPLE XI. 
 
 156. Where the Solid BA.CD propofed is a kind of Cone + 
 or Pyramid ; form'd by conceiving Right-lines to be 
 drawn from every Point in the Perimeter of any given 
 Plane BDC, to a given Point, or Vertex, A above 
 that Plane. 
 
 Let EFG be any 
 Se&ion parallel to BDC, 
 whofe perpendicular Di- 
 ftance (AQJ from the 
 Vertex let be denoted by 
 x i moreover, let the 
 whole given Altitude 
 (AP) of the Soli J be put 
 sz 0, and the Area of 
 the Bafe BDC (which is 
 alfofuppofed given) = b. 
 In the firfl place, it is 
 eafy to conceive that the 
 Planes BDC and EFG 
 muft be fimilar : And 
 therefore, fince ftmilar Figures are to each other as the 
 Squares of their like Sides, or Dimenfions, it follows 
 
 that AP* (*) : AQ! (**) :: BDC (b) : EFG = ^ 
 
 la 
 
 confequently s = -~ - 
 
 bx*x 
 
 Whence s = r> 
 
 when x a. Therefore the Solidity of a Cone or Py- 
 ramid, let the Figure of its Bafe be what it will, is 
 always had by multiplying the Area of the Bafe by -J- 
 of the Altitude, 
 
 EX-
 
 infolding the Contents of Solids. 
 EXAMPLE XII. 
 
 157. Where it is propofed to find the Content of the 
 ' Vngula EFGC, cut off from a given Cone, ABC, 
 by a Plane EFG pafling through the Bafe thereof. 
 
 183 
 
 Let AD be the perpendicular Height of the Cone, 
 alfo let AM be perpendicular to HE, the Axis of the 
 Seftion PEG, and let FAG be another Sedion of the 
 Cone, thro' FG and the Vertex A. 
 
 Since the Solids CAFG and EAFG, whofeBafes are 
 FCG, and FEG, come under the Form fpecified in the 
 preceding Example, their Contents will therefore be ex- 
 prefled by FCG x \ AD and FEG x f AM refpeaive- 
 
 FCGxAD FEGxAM 
 ly : Whofe Difference, - , 
 
 is the Solidity of the Ungula CEFG : Where the Bafes 
 FCG and FEG being conic Sections, their Areas will be 
 given by Art. 115. 124 and 129. from whence the whole 
 will be known. Thus, if HE be fuppofed parallel to 
 AB, the Sedion FEG, then being a Parabola, its Area 
 will be = -| x FGx EH * : Whence the Solidity of the Art. 115. 
 N 4 Segment
 
 184 Vb* W e ^FLUXIONS 
 
 Segment EFGA is = y X FG x EH X AM : Which 
 being deduced from that of CFGA (found by Help of 
 the common Table of circular Segments) the Re- 
 mainder will be the Content of the Ungula. But, if the 
 Axis EH produced, cuts AB, the Section FKG will be 
 a Segment of an Ellipfis EFKG ; whofe conjugate 
 Axis (fiippofing EN and KL perpendicular to AD) is 
 Art. 41. r: 2 v'ENxKL*. Now, in order to compute the 
 Content, the eafieft way, in this Cafe, let the Ratio of 
 EH to EK (which is given by Trigonometry) be ex- 
 ported by that of m to Unity, and let the Ratio of CH 
 to CB, be as n to Unity : And from the common Ta- 
 ble of Segments (adapted to the Circle whofe Diameter 
 is Unity) let the Areas anfwcring to the verfed Sines m 
 and , be taken and denoted by M and N refpeclive- 
 ly : Then, the Area of FEG being =: M X EK X 
 t Art. i 4 2 ^EN x KL, and that of FCG = N X BC l t, the 
 and 130. Content of the Ungula, by fubftituting thefe Values, 
 wiUbecome = j N* EC 1 X AD $Mx EK x AM X 
 2 v'ENxKL: But, fmce AM : AE :: KQ. (perpen- 
 dicular to AC) : KE; and AN : AE :: KQ_: Ki, it 
 follows, by Equality, that AM x KE = AN x KI ; 
 whence the Content of the Ungula is alfo exprefied by 
 
 px AD t MX ANxKIx 2VLN x KU 
 Which, if H be fuppofed to coincide with B, and K 
 
 with BC, will become ' fck. X BC 1 x AD 
 
 ? 
 
 ' 7 5 ^ 9 fcfg. x AN x BC x 2 1/hN x BD = 0.26179 
 
 &c. x BC x BC x AD 2 AN x v'ENx BD. 
 
 When the SecYion EFG is an Hyperbola, its Area 
 may be found by means of a Table of Logarithms (in- 
 ftead of a Table of Segments) whence the Content of 
 the Ungula will likewife be had in that Cafe. 
 
 EX-
 
 in finding the Contents of Solids, 
 
 185 
 
 EXAMPLE XIII. 
 
 158. Let AFC, or AGD, be a Curve of any Kind j 
 whofe Area, and the Content of the Solid arifmo; from 
 its Rotation about its Axis, or Ordinate, AB, arc 
 both known ; it is propofed to find, from thence, the 
 Content of the Solid generated by the Revolution of 
 that Curve about any other Line PR parallel to the 
 faid Axis or Ordinate AB f 
 
 Let AP, FQ_, and CR 
 be all perpendicular to AB 
 and to the Axis of Motion 
 PQR ; alfo let AP ( or 
 EQJ = a, AE, confidered 
 as variable, = iv y the Area 
 AFE, or AEG = M t and 
 the Solid, arifing from its 
 Revolution about AB, 
 N. It is plain that the 
 Area of the Circle gene- 
 rated by QF will be p X 
 
 FQ^ * = p X a + EF|* * Art< 
 
 =r pa r + Ipa X EF -f- p X 
 EF* ; from which de- 
 ducting the Area, />*, ge- 
 nerated by QE, the Remainder, ipa X EF -f t 
 will be the Area of the Annulus generated by EF: 
 Whence the Fluxion of the Solid generated by AEF 
 is truly reprefented by ipa X EF x w + pw x EF a f : 
 And, in the fame manner, it will appear that the t Art -MS- 
 Fluxion of the Solid generated by AEG is ipa x EG x *v 
 r p<w x EG 1 . But the Fluent of EF x w (or EG x w) 
 is = the Area (M) of AEF (or AEG) t, and that of 
 p.-wx EF* (or^wx EG*) equal to (N) the given Solid * Art ' 112 - 
 arifing from that Area ; therefore the Fluent of the 
 Whole) or the Solidity required, is ipaM+N, in the ***.**$ 
 former Cafe, and ipaM N'm the latter j where ipa^ 
 
 in
 
 1 86 '-The Ufe of FLUXIONS 
 
 */ / 
 
 in either Cafe, exprefles the Periphery of the Cylinder 
 defcribed by AB, about the Axis of Rotation PR. 
 
 Hence, if ABC and ABD are equal and fimilar to 
 each other, then the Value of M &c. being the fame in 
 both Cafes, it follows that the Content of the Solid ge- 
 nerated by AFG will be expreffed by ipa x iM y or 
 2/>? x Area AFG. 
 
 Now, if (for Example fake) ACD be fuppofed a 
 Circle, whofe Semi-diameter is d t the Area of that 
 Circle being == />^% the Solid generated by its Revolu- 
 tion "(representing the Ring of an Anchor) will therefore 
 be ~ ipa x pd* =. rfad' 1 ' But if you would know 
 the Content of the Part generated by the upper Semi- 
 circle BAG, or the lower one BAD, let the Content 
 
 *Art.i48. / _ ) * f a Sphere whofe Semi-diameter is d t be wrote 
 
 3 
 
 for A T , in each of the two foregoing Exprefiions, and you 
 
 will then get^W* -j- j and p*ad* 
 
 Again, if AFC, and AGO be taken as Right-lines, 
 
 _ _ AB x BC AB x BD\ , ... 
 
 you will have M =. - ..... (or -- J and N 
 
 f Art. 146. -^xBC a x-|-AB (or/>xBD*x |AB) f : Hence the 
 Solid generated by the Triangle ABC is ( = 2pa X 
 
 . AB x BC + t x DC 1 x AB) = p x AB x BC x 
 2 . T 3 
 
 RB + -JBC j and that generated by ABD ( = 2pa X 
 AB_x_BD _ L BDa = x AB x BD x 
 
 2 3 
 
 Laftly, let ABC (or ABD) be confidered as a Pa- 
 
 rabola, whofe Ordinate is AB, and Axis CB (or DB) : 
 
 JArt. 115. Then M being here = f AB x BC (or f AB X BD) } 
 
 *A rt . I52 . and A^= ^ x AB x BC 1 (or ^ x AB x BD') 
 
 t
 
 in fading the Superficies of Solids. 187 
 
 it follows that the Solid generated by ABC will be 
 (- 2pa x t- AB x BC + ^ x AB x BC 1 ) = 4^ x 
 
 AB x BC x 
 
 
 generated by ABD 
 
 = ^ x AB x BD x 
 
 cBR 2BD 
 - - j -- 
 
 SECTION X. 
 
 Ufe of Fluxions in fading the Superficies 
 of Jolid Bodies. 
 
 E T FAF repre- 
 
 fent a Solid ge- 
 
 I59 *T 
 
 nerated by the Revolution of 
 any given Curve AF about 
 its Axis AH ; alfo let a 
 Circle, whofe Diameter is 
 the variable Line (or Ordi- 
 natc) RBR, be conceived 
 to move uniformly from A 
 towards FF, and to dilate 
 itfelf fo, on all Sides, at the 
 fame time, as to generate, 
 by its Periphery, the pro- 
 pofed Superficies RAR: 
 Then the Length of that 
 Periphery, or the generating 
 Line, being exprefled by 
 3, 141592 * fcfc. x RR 
 ( = ipy) and the Celerity 
 with which it moves by z f 
 the Fluxion of the Superficies RAR, or the Space that 
 10 would 
 
 *Art,i 4 a, 
 t Art. 135;
 
 i88 
 
 Art. 135. 
 
 The U/e of FLUXIONS 
 
 would be uniformly generated in the time of defcribing 
 x y will therefore be truly reprefented by 2pyz. 
 
 Hence, if w be taken to reprefent the whole Surface 
 RAR, generated from the beginning (according to the 
 Method obferved in the three laft Sections) we fhall 
 
 have W = : 
 
 may be found. 
 
 2py **-}" j* * ; whence w itfelf 
 
 f An, 159. 
 
 EXAMPLE I. 
 
 160. Let it be propofed to determine the convex Super- 
 fries of a Cone ABC, 
 
 Then, the Semi-diameter of the Bafe (BD, or CD) 
 being put = , the flaming Line, or Hypothenufe, 
 AC = <:, and FH {parallel to DC) = y &c. we fhall, 
 from the Similarity of the Triangles ADC and Hmb 9 
 
 have b ' c:: J ( m k) ' * ( H *) = : Whence w (2pyz f) 
 
 and confequently w = *--, This, when 
 
 y r= b t becomes = pcb rz p 
 X DC x AC = the con- 
 vex Superficies of the whole 
 Cone ABC : Which there- 
 fore is equal to a Rectangle 
 under half the Circumference 
 of the Bafe and the flaming 
 Line. 
 
 E X,
 
 in finding the Superficies of Solids. 
 
 EXAMPLE II. 
 
 161. Let the Solid, whofe Surface you would find, 
 be a Sphere AEBH. 
 
 In which Cafe, putting the Radius OH a, AF = .f, 
 Um *, &c. we fhall (by reafon of the fimilar Tri- 
 angles OHF and Hmh *) have y (FH) : a (OH) :: A rt .6g. 
 
 ax 
 
 * (Urn) : k (Hk) = - : Therefore <* fayz) = 
 
 Zpax ; and confequently 
 the Superficies (w) itfelf 
 = ipax AF x Peri ph. 
 AEBH. Which if the 
 whole Sphere be taken, 
 will become AB x Pe- 
 ripb. AEBH ~ four times 
 the Area BEAHO. 
 
 Hence the Superficies 
 of a Sphere is equal to 
 four times the Area of 
 its greateft Circle : And 
 the convex Superficies of any Segment thereof, is to that 
 of the tVhole, as the Axis (or Thicknefs) of the Seg- 
 ment to the Diameter of the Sphere. 
 
 EXAMPLE III. 
 162. Wherein let the parabolic Conoid be propofed. 
 The Equation of the generating Parabola being 
 
 y 
 
 y* t or x rr , we have x =. 
 
 2yv 
 
 and therefore 
 
 Hence zv (2 P y~) = ^ 
 
 1 i whereof the 
 Fluent
 
 'The Ufe of FLUXIONS 
 
 _ 1 
 
 Fluent is p x * ,~^ ^ y ; which corrected (by fup- 
 
 ta 
 
 Art. 79. pofing^ = o *) gives 
 perficies fought. 
 
 6a 
 
 - -, for theSu- 
 o 
 
 163. ^/ it be required to determine the Superficies 
 of a Spheroid. 
 
 Let ACFHG reprefent one half of the propofed 
 Spheroid, generated by the Rotation of the Semi-elHpfis 
 FAG, about its Axis AH ; put AH =*, FH (or HG) 
 =<:, BH=A-, BC=y, FC=z, and the Superficies ge- 
 nerated by FC (or GD) = w : Then, from the Na- 
 
 D 
 
 ture of the Ellipfis, we h2ve.y = *-** 5 whence 
 t Art. 1 3 5. y = a ^~^ a " d confequently ( = ** + / 1 )
 
 in folding the Superficies of Solids. 191 
 
 x**" X xx 
 
 r W b *x* 
 
 7~==r- :r (by putting (the Excentricity) 
 
 -r.' = "^^- : Therefore > i 
 
 this Cafe, w (ipyz) : - ^/ _ ^ . ^^ofc 
 
 M 
 
 Fluent, in an Infinite Series, is zpcx x 
 
 But the fame 
 
 Fluent may be, otherwife^ very eafily exhibited by means 
 of the Area of a Circle : For, if from the Center H, 
 
 aa 
 with a Radius equal to -7-, a Circle SER be defcribeer, 
 
 and the Ordinute BC be produced to interfed it in E, 
 
 it is evident that BE = V' j^ xx^ and that the 
 Fluxion of the Area ESH3 will be exprefied by * 
 
 /a* 2pbcx /a* 
 
 \/ x~; which being to X V -. JT, 
 
 bb aa bb 
 
 the Fluxion before found, in the conftant Ratio of i to 
 Z , their Fluents muft therefore be in the fame Ra- 
 tio ; and fo the latter, expreffing the Superficies CFGD, 
 
 .i Kf-f 
 
 will confequently be = -^ x BESFH = 2p X 757: 
 aa lib 
 
 x BESFH. 
 
 This Solution, it may be obferved, obtains only in 
 Cafe of an oblong Spheroid, generated by the Rotation 
 of the Ellipfis about its greater Axis j for, in an oblate 
 
 Spheroid,
 
 Ufe of FLUXIONS 
 
 Spheroid, generated about the lefTer Axis, the Value of b 
 \V <? f) will be impoflible; fmce, in this Cafd 
 HF is greater than HA. But, if we, here, put b = 
 
 a 1 , and d-Tt the Value of w (found above) 
 
 2 i>c 
 
 -* X x V a 1 - + x* : Whofe Fluent may be 
 
 brought out by help of a Table of Logarithms : 
 For, let the variable Part x V^ 2 -f * a be tranf- 
 
 formed to 
 
 (*** + 
 
 fj XX ~l~ XX 
 
 meratorof the firft Term *,- * ==-(nowinagiven 
 
 Yd x + x* 
 
 Ratio to the Fluxion of the Quantity under the radical 
 
 * Art. 77. Sign) may be had by the common Rule * j by which 
 
 means we get { V ' d^x i + # 4 , for the true Fluent of the 
 
 {aid Term j to which adding the Fluent of the other 
 
 Term */j*^ , ..* or t /jl . -i ( S iven b 7 Ar ** 
 
 126.) there arifes \x)/d* + x 7 - + ? d* x hyp. Log. 
 * + VV + A-% for the Fluent of x ^d* + ** : And 
 
 t Art. 78. th ' s > corrected f and multiplied by , gives 
 
 Vd l + .v l + ^rf x hyp. Log. J ^ , for the 
 
 Superficies in this Cafe, where the propofed Spheroid is 
 an blate One. 
 
 E X-
 
 in finding the Superficies of Solids. 193 
 
 EXAMPLE V. 
 
 164. Let the Solid, wboft Superficies ;'; ftttght^ be the 
 hyperbolical Conoid. 
 
 Let the femi-tranfverfe Axis, of the generating Hy- 
 perbola, :=<?, the femi-conjugate ~c, and the Diftance 
 of any Ordinate from the Center thereof = x ; then 
 from the Nature of the Curve you will have y =. 
 
 v . . ..- 
 
 ~ V x* a 1 ; whence j 
 
 att 
 
 uXxxa+i which laft Value, if rf* be put = 
 a* 
 ^'z , ; will be more commodioufly exprefled by 
 
 2 f!fx 
 
 -^- v .v* d* : whereof the Fluent, by proceeding 
 
 as in the latter Part of the foregoing Example, will 
 
 pcx y' x v _ jj 
 come out = - 1 - pcd x . Z. 
 
 AT + v/^ ^ 4 : Which correaed (by taking x=a ) 
 
 PCX , - 
 becomes - V xx dd pc\ ptd X hyp. Log. 
 
 x + \/A- a d* 
 
 - - - , the true Meafure of the required Su- 
 a + _ perficies. 
 
 
 
 EXAMPLE VI. 
 
 165. Let it be propofed to find the Superficies of the 
 Solid called a Grain. (V id. Art. 155.) 
 
 Let M/be any Sedion of the Solid parallel to the 
 
 Bafe therepf, and let x denote ics Diftance from the 
 
 Q Vertx
 
 194 T&e Ufe ^/FLUXIONS 
 
 Vertex A, alfo put z equal to the cfcfre"(pondino: Arch 
 An of the femi-circular Section NA &c. whole Radius 
 AB or BN let be denoted by a. 4. 
 
 It appears from Art* 161. that z -, = ; 
 
 V lax xx 
 
 Which Value, multiplied by (2 V lax Ar.vjthatof 
 ds ( 2bn) gives lax * for the Fluxion of one of the four 
 equal convex Superficies by whjch the Solid is bounded. 
 Hence the whole Superficies (excluding theBafe) comes 
 out = 80* : Which therefore is exadly equal to twice 
 the Bafe. 
 
 If the Solid be fuppofed a Groin of any other Kind, 
 fuch that its two equal Se&icns, through the Middle of 
 the oppcfite Sides, are other Curves than Circles, the 
 Superficies may Jlill be had in the lame manner ; and 
 will be always in proportion to the Superficies arifing 
 from the Revolution of either of the faid equal Curves 
 ?.bout rts-Axis, as a Square is toils inscribed Circle. 
 Thus, the Superficies cf a parabolic "Conoid being = 
 
 ba 
 
 _ -- (fy Art. 162. ) the convex 
 
 Superficies of the Groin, fuppofing the generating 
 Curve AffN to be a Parabcla, will therefore be =: 
 
 " 
 
 
 6a 
 
 E X-
 
 m fading the Superficies of Solids. 
 
 EXAMPLE VII. 
 
 1 66. Wherein let It be required to find the convex Super- 
 ficies of a cinical Ungula ECFD ; formed by a Plans 
 DFE pafling through the Bafe of the Cone. 
 
 Let a right-angled Triangle AOM (whofe BafeOM 
 is the Radius of the Circle BDCE) be fuppofed to re- 
 volve about the Axis AO ; whilft a Right-line NP, 
 drawn perpendicular to OM from the interfetion of 
 AM and the Arch EFD, traces out, upon the Bafe of 
 the Cone, the Curve-line EPGD. 
 
 IfMPOANand 
 mpOAn be confi- 
 dered as two Po- 
 fitions of the ge- 
 nerating Triangle 
 indefinitely near to 
 each other, it is 
 evident that the 
 Space MA;w, ge- 
 nerated by AM, 
 will be to the 
 Space MOw, ge- 
 nerated by OM, 
 as AM to OM, 
 or OB. Whence, 
 MNand MP be- 
 ing proportional 
 Parts of AM and 
 OM (becaufe NP is parallel "to AO) it is likewise 
 plain that the Spaces MN/w and MP/>*, generated bv 
 thofe Parts, will be to each other in the fame Ratio of 
 AM to OB. And fmce this ever/ where holds, ic 
 follows that the whole Space (ENM) &c. generated bv 
 MN, will be to that (EPM) generated by PM, as AM 
 to OB : And fo the whole required Superficies (generated 
 
 by AM) is truly reprefented by , Area EPGDCE. 
 
 O 7 
 
 But
 
 tte Ufa ^FLUXIONS 
 
 But now, to find this Area, EPGDCE, it is ob- 
 fervable that the Area of the Plane DFE (being the 
 Segment of a Conic-fediion) is given, by Art. 115. 129 
 or 130. And it is very eafy to apprehend and de j 
 monilrate that the Area fo given will be to that of 
 EGDH, as the Radius to the Co-fine of the Angle of 
 the Inclination of the faid Plane to the Bafe, or as HF 
 
 HC* 
 to HG. Therefore, feeing EGDM is == J^T x EFD, 
 
 we have EPGDCE ( = ECDHE EGDH ) = 
 
 ECDHE TjT. X EFD j and confequently 7^7 x 
 rir (Jo 
 
 FPrrnr - AM FCHHF AM x HG , 
 EPGCDE = x ECDH OBxHF x 
 
 EFD the convex Superficies that was to be found. 
 
 If the Point H be fuppofed to coincide with B, 
 ECDHE will become the whale Circle CB; and EDF 
 will become a whole Ellipfis, whofe greater Axis is BF, 
 
 Art 41. and its lefler Axis = 2V/OBxOG. * Therefore, the 
 f Art. 124. Area of the former Figure will beexprcfledby p x BO 1 -^ 
 
 and that of the latter by p X - BF X \/Uttx OG j 
 and fo the convex Superficies of the Part BFC will be 
 AM AM x BG , 
 
 VOB x OG) = p x AMxOB^xAMx IBGx 
 
 F^T 
 
 : Which being deducted from (p X AM x 
 
 OB 
 
 OB) the Superficies of the whole Cone BAC, there 
 
 refts p x AM x ' BG x \f. -~ p for the Superficies of 
 the oblique Cone BAF ; which from hence is alfo given. 
 
 SCHO-
 
 in finding the Superficies of Solids. 
 
 197 
 
 SCHOLIUM. 
 
 167. In moft of the Examples, delivered in 
 the four laft Sections, the Part of the propofed 
 Figure next the Vertex, 
 whether, a Curve, Solid, 
 or Superficies, is firft 
 found ; from whence, by 
 taking the Altitude (#) 
 of that Part equal to ( a) 
 the Altitude given, the 
 Content of the wholt is 
 deduced : But, if the 
 Content of the lower Seg- 
 ment ( BCED ) of any 
 Figure (ABC) arifins; by 
 taking away a Part(ADE) 
 next the Vertex, be required ; then the Difference be- 
 tween the jybole and the Part taken away (found as 
 before explained) will be the Quantity fought. 
 
 Thus, for Example, lee ABC be the common Pa~ 
 rabola 9 and let it he propofed to find the Content of the 
 Part, BCED, included between any two Ordinates 
 BC (b) and DE (f) at a given Diftance BD (d) from 
 each other : Then, the Equation of the Curve being 
 
 2> 7 > 
 
 
 
 = -,. ArttIJI . 
 
 ax =y\ we have x , and therefore 
 a 
 
 2V 3 
 
 whofe Fluent is a general Expreffion for the Area 
 
 comprehended between the Vertex and the Ordinate^; 
 Whence, expounding;-, by b and c fucceflively, we get 
 
 ; and for the correfponding Values of ABC and 
 
 ADE ; whofe Difference 
 
 
 is the required Area 
 
 BCED : But, to exprefs the fame independent of <7, it 
 bp, by the Property of the Curve, b~ \ c 1 :: AB : AD; 
 O 3 whence,
 
 The Vje of FLUXIONS 
 
 whence, by Divifion, F : i> z ~ r : : AB : BD (d) and 
 
 V-? l"~ 
 confequently - -rr: a > which firft Value being 
 
 wrote inflead of a, there refults BCED = 1 l^J 
 
 _ _ // 4 fo + f* 
 
 * *" " X / , * 
 
 3 b + c 
 
 After the fame Manner, the Segment? of other Fi 
 jnires may be found ; but in many Cafes they will br 
 more readily had from a direct Investigation, without 
 either finding the Whole or the Part taken away. 
 
 Thus, in the Cafe at>ove, if the Exccfs of any Or- 
 iginate RP above DE (c) be denoted by w;, we fhall 
 have, by the Property of the Curve, b z c~ (BC 1 
 
 DE 1 ) :7T S (RP* DE*-) : : DB (d) : DP = 
 
 d X 7 CIV -r iS . , T-,. ' . 2W-f2U 
 
 iL'l-li-i-^ - j whofe Fluxion fJ X ~ ^ __ f 
 
 u ~~ i> 
 
 multiplied by c 4- to ( = PR ) gives d x 
 
 2<rVu 4- - -- ,- i r-1 r u 
 
 - ' j - - - , for the Fluxion of the Area 
 
 u " * C 
 
 DPRE : Whereof the Fluent ( which is idiu x 
 
 fl + f + , 4 ' C; ') will, when ; = *--. (or RP = BC) 
 u c / 
 
 , , , id X A <: X i //" + i ^f -|- f i--" 
 
 be truly expounded by -- : - XZ - ^ - i_2 
 
 /r c z 
 
 id' I* 4 ^f 4 t" . , 
 or its Equal, - x b ^ c ' the f anic as l ' c f ore ' 
 
 O ^ 
 
 Az,ain-, for another Example, let CEDrr be confi- 
 
 dercd a$ the lower Fruftrum of an Hemifphcre, whofe 
 
 Center is the Point C : Then, LP bein^ here, denoted 
 
 by w, we (halt have y- ( - BR^ BP^ - ^ w% 
 
 * An. 145. and confequently />> Vu * :rr /> x i-'-w -n/w ; whofe 
 
 Fluent
 
 in fading the Superficies of Solids. 
 
 199 
 
 Fluent (p x /*; \ w z 
 
 X ^ ;" 
 
 x 2Z> 
 
 = -J x BP X 
 
 + PR 1 ) is the true Content of the PartCED^; 
 which will alfo hold when the Figure is a Spheroid. 
 
 This laft Method, of finding the Content of a Por- 
 tion of a Figure, remote from the Vertex, will be of 
 Service, when the general Value, for the IVliole, can- 
 not be exprefled without an infinite Series ; becaufe 
 fuch a Series, in that Cafe, not converging, becomes 
 ufelefs *. 
 
 By dividing the whole propofed Figure, AHW, into 
 a Number of fuch Portions, HV, GT, FS, &c. the 
 Content thereof may be obtained, v.hen to find if at 
 once-, by a Series, commencing from the Vertex, would 
 be altogether impracticable. 
 
 H 
 
 G 
 
 Art. gj. 
 
 But, to render fuch an Operation as (hort and eafy 
 as may be, it will be proper to find each Part (DQ_, &V.) 
 of the Figure, by means of a Series proceeding both 
 Ways, from the middle Ordjnate (MN) between the 
 two correfponding Extremes (CR and DR). 
 
 Thus, let the Value of MN (found by the Property of 
 the Curve) be denoted by a ; and let the Value of DR, 
 in a Series, be reprefented by a-\- bx -{- ex* -\- dx 3 + ex+ + 
 fx % -f &c. where x MD ; then the Area MDRN will 
 be reprefented by the Fluent of K x a + bx + ex* + dx 3 + 
 O 4 trV.
 
 200 
 
 The Ufe of FLUXIONS 
 
 y */ 
 
 bx , ex* . dx^ 
 &c. or by x x a + + T -j- &c. And 
 
 2 3 4 
 
 by writing x inflead of*, the Ordinate CQ_will be ex- 
 preffed by a bx + cx*dx 3 fafc. and the Area MCQN, 
 
 bx cx z dx 3 ex* 
 by x x a \r -f C5V. whence the 
 
 r t. 'y A C 
 
 _ i CM VJIi , . 
 
 Area CDRQ. is = 2.v x a + -f + 6 -f CV. 
 
 Therefore, if DE, EF, FG, and GH be fuppofed, 
 each, = BC (ZA-J and the Areas DS, ET, tic. (found 
 
 1 / 
 
 as above) be denoted by 2* X a -\ \- - - ts'c. and 
 
 3 5 
 
 a i 
 
 a (x i- ex * 
 ix X a -\ + -~ &c. refpeclively, it follows that 
 
 the Area CR + DS -f ET will be reprefented by 2*X 
 
 * + * + a 
 
 i a 
 e + e -f e fc 
 
 Q. 
 
 I\ n 
 
 A 3 X c 4- c -f c tf^. -f | 
 
 An Example will 
 the Ufe of this laft Expref- 
 fion : Let CHWQ_ be a 
 Portion of a Quadrant 
 HAW of a Circle, whofc 
 Bafe HC (conceived tJ be 
 divided into four equal 
 Parts) is equal half the Ra- 
 dius AH, reprefented by 
 Unity. Then, putting CM" 
 
 A CM. 1)^ H 
 
 H; ( f ) y, w. have, by the Property of the Cir- 
 
 cle, * (MN) = 
 
 -//>, and 
 DR
 
 in finding the Superficies of Solids. 201 
 
 DR (= v'HR 4 HD 1 ) = v/i l^"** = 
 
 /> ? -f 2/>jr * x = vV + 2/ur **; which, 
 
 in a Serics> is (=# + 
 
 *a W 
 
 = a +tl L -j- .?-, x x 1 &c. Therefore, in this 
 
 a 2a l(r 
 
 
 -i, ^' Which Va - 
 
 2 J 
 
 lue of r, by \vrii ing i a 1 for its Equal j>% will be 
 reduced to -- '-:. From whence it is alfo evident 
 
 2<2 
 
 t-rr _'_ (fuppofing fl fwJ = Vi J 1 ) 
 
 i 
 
 a ..3 
 
 ,3 
 Confequently 2^- x a+ a +a crV. -j" j x X <: -f f -f- c. 
 
 "" 
 
 / 
 
 ' 
 
 &f. + |* J x ^ -f <r + t esto ( + rtX2^J-^ff X 
 
 2* 3 \ "" ~ 
 TV = 
 
 X - 
 
 04 04 4 
 
 y^' 2x6 3 /&$ 64x8x3 
 
 _. 
 
 sx 55_^55 3x63 
 
 3 2 ' 3 X 55 X 55 3x63x63 ~ 
 
 0,4?73 0: f the Area > CP1WQ_, that was to be found. 
 
 This Example, chofcn as an iliuftration of the fore- 
 gging Method, may indeed be wrought the common 
 Way ; whence the very fame Conclufun is brought out 
 
 (Vide
 
 202 The Ufe of FLUXIONS 
 
 K r 
 
 (Vide Art. 124.) But that Method is alfo applicable to 
 any other Cafe, whether the Part propofed be near to 
 the Vertex, or remote from it ; and whether the Figure 
 itfelf be a Curve, Solid or Superficies ; fmce the Mea- 
 fure thereof may, always, be exprefled by the Area of 
 a Curve. 
 
 There is another Way, well known to MathematU 
 cian?, whereby the Area of a Curve may be deter- 
 mined, by means of a Number of equidiftant Ordi- 
 nates; which Method, derived from that of Differences, 
 may, alfo, be ufed to good Purpofe, in Cafes like thofe 
 above fpecified : But, it having been treated of by feveral 
 ether 's, and alfo in my DiJ/ertatiem, the Reader will ex- 
 cufe me, if no further Notice is taken of it here. 
 
 SECTION XI. 
 
 Of the Ufe of FLUXIONS in folding the 
 Centers of Gravity, PercuJJion, and Ofcih 
 lation of Bodies, 
 
 168. r~r\ HE Center of Gravity is that Point of a 
 Body, by which, if it were fufpended, it 
 would r.eli in Equilibrio, in any Pofition. 
 
 LEMMA. 
 
 .169. Letp, q, ?', 5, eft. be any Number of given Weights, 
 banging at an inflexible Line (or Rod) AM fufpended 
 in Kquilibrio, in an horizontal Pofition, at the Poirti 
 Oj to determine the Pofition of that Point. 
 
 Since (by Mechanics) the Force of any Weight (p) 
 to raife the oppofite End (M) of the Balance, is as that 
 Weight drawn into its Diftance (BO) from the Ful- 
 crum,
 
 in finding the Centers of Gravity, 6cc. 203 
 
 crum, we dial], from the Equality of thefe Forces, 
 have p x OB + q X OC-f A X OD=j x OE+/xOF, 
 
 M^-l; Eorn c B , A 
 r "7 /' 
 
 isj> X AO AB + fX AO- AC-j-rXAO 
 s x AE AO-f t x AF AO~, andconfequently AOrr 
 p x AB + q x AC + r x AD -f- * x AE +' f x AF 
 
 From which it appears, that, if each Weight be multi- 
 ply d ly its Dijla,,ce from the End (or any given Point) 
 of the Axis, the Sum of all the Products divided by the Sum 
 of all the Weights, will give the Dijlance of the Center of 
 Gravity from that End (or Point.) 
 
 Note. The Produfta here mentioned are, ufually, 
 cali'd the Forces, of their refpecHve Weights; not iu 
 refpe6l to their Action at the Center O (which is ex- 
 p re fled by a different Q^iantity) but v/ith regard to the 
 Effects they have in the Conclufion, or the Value of 
 AO j which appear to be in that Ratio. 
 
 . 
 PROPOSITION I. 
 
 170- To determine the Center of Gravity of a Line, 
 Plane, Superficies, or Solid (admitting the three former 
 capable of" being affected by Gravity.) 
 
 Let AMBC be the propofed Figure, and G the 
 Center of Gravity thereof; thro* which, parallel to 
 the Horizon, let the Line EF be drawn, interjecting 
 AC, at Right-angles, inO; alfo let AK and NM ba 
 perpendicular to AC, and parallel to EF. 
 
 171. Cafe
 
 204 
 
 . 
 
 
 *Tbe Ufe of FLUXIONS 
 
 171. Cafe i. If 
 the Figure AMBC 
 be a Plane ; let it 
 be fuppofed to reft 
 in Equilibria upon 
 the Line EF ; and 
 then, if the Line 
 MN be confider'd 
 as a Weight, its 
 Force (defined a- 
 bsve) will be ex- 
 prefled by MN 
 drawn into its Diftance (AN ) from the End of the Axis 
 AC; that is by yx (fuppofing, as ufual, ANrz* and 
 MNiry.J This, therefore, multiply'd by the Fluxion 
 cf AN, gives yxx for the Fluxion of the Force of the 
 Plane AMN ; whofe Flyent, when A-=AC, exprefles 
 the Force of the whole Plane, or the Sum of all the 
 Produces of the Ordinates (or Weights) by their re- 
 fpective Diftances from AK : Which Fluent being, 
 therefore, divided by the Area ABC, or the Fluent of 
 yx (according to the foregoing Lemma) the Quotient 
 
 { -' *.*} will give (AO) the Diftance of the Center 
 of Gravity from the Line AK. 
 
 172. Cafe 2. // the Figure be a 5c//W; let MN be a 
 Section thereof by a Plane perpendicular to the Ho- 
 rizon ; then, the Area of that Section being denoted by 
 4*, the Force thereof (confidered as above) will be ex- 
 prefied by Ax, and the Fluxion of the Force of the Solid 
 AMN by Axx \ whofe Fluent, divided by the Content 
 of the Body, or the Fluent of 4*', gives AO, in this 
 Cafe. But, if the Solid be the half (or the whole) of 
 that arifmg from the Rotation of a Curve AMB about 
 its Axis AC ; then (putting p for the Area of the Circle 
 whofe Pvadius is Unity) A will become ipy" * ; and 
 
 Flu. -I py^xx Flu. y 1 xx 
 co&fequentjy AO = -rr , . . r , a . 
 r lit. -i pyx riu*y x 
 
 173. Cafe
 
 m fading the Centers cf Gravity, &c. 
 
 173. Cafe 3. If the Figure propofed be the Curve-line 
 AM 8; then, the Force of a Particle at M being expreflfed 
 by AN or MQ^(x) we fliall (putting AM = z) have 
 
 ^J = AO. 
 
 205. 
 
 "' 
 
 174. Cafe 4.. But if the Figure given be the Superficies 
 generated by the Rotation of AMB about AC. 
 
 Then, the Periphery of the Circle generated by the 
 Point M being 2py. it follows that 
 
 Flu. yxz 
 
 7.. .- = AO. 
 
 rlu. yz 
 
 EXAMPLE I. 
 
 Flu. 
 
 175. Let the Figure propofed be the ifofceles Triangle ABC. 
 
 A 
 
 M, 
 
 O 
 
 B 
 
 It is evident the Center 
 of Gravity (O) will be 
 fomewhere in the Per- 
 pendicular AQ_: And, 
 if AQ=*, BC:=, AN 
 =*, and MM^yj then 
 
 bx 
 y being r: , we mall 
 
 kave, by Cafe I, AO ( = 
 Flu. yxx\ Flu. x*x 
 
 Flu. yx ) Flu. xx 
 
 iX* 1 V 
 
 = TS = -T = 7 A Qj when * = AQ.; and confe- 
 
 AO 
 
 quently OQ^ = ^ 
 
 In the very fame manner, the Center of Gravity of 
 any other (plane) Triangle will appear to be at | of the 
 Altitude of the Triangle. 
 
 EX-
 
 206 The life of FLUXIONS 
 
 EXAMPLE U. 
 
 . Let the Figure prcpofed be a Parabola of any Kind; 
 
 
 whereof the Equation is y zr -fL_ . 
 
 a 
 
 " 
 
 Flu. y xx Flu. *" * -f i 
 Here. - - r n -- __ x x = 
 
 ' Flu. yx p, n. ' M-J-2 
 
 ' Flu. x x 
 
 the Diftance of the Center of Gravity from the Vertex 
 of the Curve. 
 
 EXAMPLE III. 
 177. Let BAG be a Segment of a Circle. 
 
 Then, if the Radius thereof be put r, we fliall 
 havej> (NM) = Virx xx: Whence the Fluent of 
 yxx (xx^irx xx) will, by Art. 163. be found = 
 
 . __ 3 
 
 irxxx^ +rxAreaAN M ; which divided by ANM, 
 3 
 
 NM 3 
 
 . _.. ,, . 
 = A ' ThlS ' 
 
 .^ Art. 171. 6'" 3 x Area ANM 
 
 BAG is a Semi-circle, 
 
 becomes = x 
 
 10CO 
 
 r, nearly. 
 
 -T-. But, with refpecT: to 
 
 _\ the Center of Gravity 
 
 B Q of the Arch BAG; 
 
 we have, Flu. *i, ( by Cafe 3. ) ~ Fluent of 
 
 rxx 
 
 tr >*x AM MN ; and confequently 
 
 y' irx xx 
 
 r x MN 
 
 EX-
 
 in finding the Centers of Gravity, cc. 207 
 
 EXAMPLE IV. 
 
 178. Let ABC (fee the preceding Figure) reprefent a 
 Segment of a Sphere^ or Spheroid. 
 
 In which Cafe, denoting the Axis of the Sphere, or 
 Spheroid, by a y and the other Axis of the generating 
 
 Curve, when anEllipfis, by , we have y* = xax xx; 
 
 and therefore 
 
 If the Solid be an hyperbolical Conoid, the Diftance 
 (AO) of its Center of Gravity from the Vertex, will 
 alfo be exhibited by the Expreflion here brought out, 
 when the negative Signs are changed topofitiveones. 
 
 1 79. In thofe Cafes where theFigure cannot be divided 
 into two Parts, equal and like to each other (as a Curve 
 is by its Axis, &c.) the Pofition of two Lines EO, eo 
 (fee the enfitlng Figure) muft be determined, as above ; 
 in whofe lma'fetion (G) the Center of Gravity will 
 be found. 
 
 EXAMPLE V. 
 Let ABC be a Semi-parabola of any Kind -, whereof the 
 
 Equation is, y ~ 
 a 
 
 It appears, from Ex. 2. that (AO) the Diftaoce of 
 EGO from the Vertex, is exprefled by - X AC : 
 
 But to find the Pofition of oGc (perpendicular to E.O) 
 let Mtf'be parallel to eo^ or AC ; then, AN being ~ x, 
 
 anU
 
 2 8 T&e W e 0f FLUXIONS 
 
 8 
 
 and NM (y) = *_ > if AC be denoted by , we 
 
 ihall have Mn =. b 
 
 * n I 
 
 71 o 
 
 -x y and M x NM X j = b x X 
 = n! ' x x n* * for t h e 
 
 * C* 
 
 Fluxion of the Sum of the Forces in this Cafe (Vid. 
 Art. 171.) whofe Fluent f nlx~ nx 
 
 zr. z 
 
 2+iXtf 
 
 2B 
 
 xl 
 
 - z 2 
 
 nx 
 
 n * / X _ 
 2-fl 2 2-fl 
 
 BC'x AC , n ,. .. 
 
 or : , when * b) divided 
 
 a 
 
 y x r . - 
 
 by the Area ABC (= -^^ ) gives 
 
 EC tot the true Value of Co, or OG. Which, in 
 cafe of the common Parabola, where n = i, and 
 
 where AO (^T"X AC) ='jAC, will become = *CB. 
 
 Before I leave this Subject it may not be improper 
 to take notice, that, whatever Lire you found your 
 Calculations upon, by fuppofcrig the P igure to reft, in 
 
 Equilibria)
 
 B 
 
 m fading the Centers of Gravity, &c. 209 
 
 Equilibria, upon that Line, the very fame Point, for the 
 Place of the Center of Gravity, will be determined. 
 
 1 80. Thus, let 
 O be the Point in 
 the Axis AC, of 
 a given Curve 
 B A D, deter- 
 mined, as above, 
 by fuppofmg the 
 Figure to reft 
 upon EF per- 
 pendicular to 
 AC ; and let 
 RS be any o- 
 ther Line paffing 
 
 through the Point O ; then I fay the Sum of the Mo- 
 menta of the Particles on each Side of RS will, a.'/o, be 
 equal. For, if from two Points, in any Ordinate A1Q_, 
 equally diftantfrom the middle Point N, two Perpendicu- 
 lars mr and ns be let fall upon RS, the Eff.ca v of thofe 
 two Points, in refpe&toRS,will bereprefented bymr-f ns 9 
 or its Equal 2NH (fuppofmg NH alfo perpendicular to 
 RS.) Whence the Efficacy of all the Particles in MQ^ 
 will be exprefled by their Number multiplied by NH, 
 or by MQ_x N T H : Which is to their Efficacy (MQ^x 
 ON) when referred to the Line EF, in the conftar.t 
 Ratio of NH to ON, or of the Sine of the Angle 
 RON to Radius. Whence it is evident that the Force 
 of all the Ordinates (or the wrHe Curve) in the former 
 Cafe, muft be to that in the latter, in the fame Ratio: 
 But the faid Force, in the one Cafe, is equal to nothing 
 by Hypothecs, therefore it muft be likewife fo in the 
 other : And confequently the Sum of the Momenta of 
 the Particles, on each Side of RS, equal to each other. 
 
 Much after the fame manner the thing may be proved, 
 in a Solid : Whence it will appear that there is actually 
 fuch a (fixed) Point in a Body as the Center of Gravity- 
 is defined to be : Which, however evident from mecha- 
 nical Confederations, is not fo eafy to demonftrate, geo- 
 metrically, from the Refolution of Forces. 
 
 P PRO-
 
 210 
 
 Ufe of FLUXIONS 
 
 */ / 
 
 PROPOSITION II. 
 
 18 1. To determine the Center of PercuJJkn of & Body. 
 
 The Center of Percuflion is that Point, in the Axis of 
 Sufpenfion of a vibrating for revolving) Body, at which 
 it may be ftopt, by an immoveable Obftacle, fo as to 
 reft thereon in Equilibria as it were, without acting upon 
 the Center of Sufpenfion. 
 
 Let O be the 
 Point of Sufpen- 
 fion, G the Center 
 of Gravity, and 
 SLM a Section of 
 the Body, by the 
 Plane wherein the 
 Axis of Sufpenfion 
 OGS performs its 
 Motion ; to which Section let all the Particles of the 
 Body be conceived to be transferred in fuch Parts thereof 
 where they would be projected into (ortbograpbically) 
 by Lines parallel to the Axis of Motion ; which Suppo- 
 fition will neither affect the Place of the Center of Gra- 
 vity nor the angular Motion of the Body. 
 
 Since the angular Velocity of any Particle P is as the 
 Diftance, or Radius, OP, its Force in the Direction, 
 PB, perpendicular to OP, will beexprefled by Px OP. 
 Therefore the Efficacy of that Force upon the Axis, at 
 B, in the perpendicular Direction BN (fuppofmg the 
 Axis ftopt at C the Center of percuffion) will be P x 
 
 OP 
 
 OP X 7=75, whofe Power to turn the Body about the 
 (Jo 
 
 OP 
 
 Point C is therefore as P x OP x ^-^ x BC = P x 
 
 vJrJ 
 
 OP* x BC OP*xOC OB_ p OP*xOC 
 
 OB" OB OB 
 
 PxOP z j which, if PQ^be made Perpendicular to 
 
 OS,
 
 in folding the 'Centers of Gravity y 5cc. 2 1 1 
 
 OP* 
 
 OS, will at laft (becaufe ^ = OQJ be reduced to 
 
 Px OQ^x OC P x OP 1 . By the very fame Argument, 
 
 / 
 the Force of any other Particle P will be denoted by 
 
 P X OQ_x OC - P x OP* fcfc. &V. But, as all thefe 
 Forces muft deftroy one another (by the Nature of the 
 Problem) the Sum of all the Quantities P xOQ^x OC, 
 
 Px OQ_x OC, e$V. muft therefore be = the Sum of all 
 the Quantities P x OP 1 , P x OP* &V. and confequently 
 
 PxOP'-f PxOP-f ear*. Uc. 
 OC = - ; - - - . But (by the 
 
 P x OQ.4- Px OQ.+ #< &( 
 preceding Proportion) the Sum of all the Quantities 
 
 &c. is equal to OG X by the Con- 
 tent of the Body. Therefore OC is likewife = 
 
 PxOP*+PxOP-f 
 
 OG x Body. 
 
 "The fame otberwife. 
 Since the Force of the Particle P, in the perpendicular 
 
 OP 1 
 
 Direction NB, is defined by P x Q]T, or its Equal, 
 
 PxOQ_, the Sum of all the Quantities PxOQ_, l^ 
 fefr. 13 c. will exprefs the Force which, a&ing at C per- 
 pendicular to OS, is fufficient to ftop the Body, without 
 the Center of Sufpenfion O being any way affected: 
 This Sum, therefore, drawn into OC ( = OC x 
 
 P x OQ,+ P x OQ.-f & c . &c.) is as the Efficacy of 
 the faid Force to turn the Body about the Point O. But 
 the Force of the Particle P, in the Diredion BN being 
 
 PxQg-, its Efficacy to turn the Body about the fame 
 P 2 Point
 
 212 the Vje 0/' FLUXIONS 
 
 Point (the contrary way) is as P x OP 1 ; and confe- 
 quently the Efficacy of all the Particles as the Sum of 
 
 all the Quantities PxOP% P/ OP 1 &c. &c. Therefore 
 (Action and Re-a6tion being equal) we have OC X 
 
 PxOQ+PxOQ^I- He. - PxOP^PxOP-J- &c. the 
 
 fame as before. 
 
 For the Center of Ofcillation, it will be requifite to 
 premife the following 
 
 LEMMA. 
 
 182. Suppofe two exceeding Jniall Weights C and P, 
 a fling on each other by means of an inflexible Line (or 
 Wire PC) to vibrate in a vertical Plane ROFCM, 
 about the Center O ; it v. required to determine how rr.uch 
 the Mstion cfthe one is afftfted Ly the other. 
 
 Let CH and PQ. be per- 
 pendicular to the horizontal 
 Line OR; alfoletPB and 
 CS be perpendicular to OP 
 and OC re!pec~lively. 
 
 If the Force of Gravity 
 be denoted by Unity, the 
 Forces acting in the Di- 
 rections CS and PB, where- 
 by the Weights, in their 
 
 Defcent, are accelerated, will, according to the Refo- 
 
 OH OQ_ 
 lution of Forces, be reprefented by 
 
 Moreover, fince the Velocities are always in the Ratio 
 of the Radii OC and OP, if the forefaid Forces were 
 
 OH 
 to be in that Ratio, or that of P was to become ^ 
 
 X TTp-, inftead of gfep. I fay, in that Cafe, it is 
 plain the Weights would continue their Motion with-
 
 in fading the Centers of Gravity, &c. 213 
 
 out affe&ing each other, or a&ing at all on the Line 
 of Communication PC (or PB). Whence, the Excefs 
 
 , OQ OH OP 
 
 or ^y.y above -^r^, x -p^rr mult be the accelerative 
 
 Force whereby the Weight P acls upon the Line (or 
 Wire) OC, in the Direction PB ; which multiply'd by 
 
 the Weight P gives P x ~ -- ^ for the ab- 
 
 folute Force in that Direction : Which therefore, irr the 
 
 OQ OHxOP 
 perpendicular Direction NB, is r X 5 
 
 OP 
 X T=T]J ; whereof the Part ating upon C, being to 
 
 the Whole as OB to OC, is truly defined by P X 
 OQ. OHxOP 
 OC " OC 3 
 
 If P be fuppofed to a6t upon C by means of PC (in- 
 ftead of PB) the Conclufion will be no way different : 
 For, let F (to fhorten the Operation) be put to denpte 
 
 OQ" OH x OP \ 
 the Force (Px --- I in the Direction 
 
 PB, found above, then the Action thereof upon PC 
 (according to the Principles of Mechanics) will be ex- 
 
 pre.fled by F X - ^ ; which therefore in the Di- 
 
 redtion SC, perpendicular to OC, is F x - ^-.pu X 
 
 S. PCO S. PCO S. PCO _ OP 
 
 r C.-/CPB " S. OPC = Fx OC" the 
 
 very fame as before. 
 
 P 3 PRO-
 
 214 T& e W e f FLUXIONS 
 
 PROPOSITION III. 
 183. To determine the Center of Ofcillation of a Body. 
 
 The Center of Ofcillation is that Point, in the Axis 
 (or Line) of Sufpenfion of a vibrating Body, into which 
 if the whole Body was contracted, the angular Velocity 
 and the Time of Vibration would remain unaltered. 
 
 Let LMS be a Section of the Body by a Plane, per- 
 pendicular to the Horizon and the Axis of Motion, 
 paffing thro' the Center of Gravity G and the Point of 
 Sufpenfion O ; and fuppofe all the Particles of the Body 
 to be transferred to this Section, in fuch Places of it, as 
 they would be projected into (crthograpbically) by Per- 
 pendiculars falling thereon. (Which Suppofition will no 
 way affecT: the Conclufion, the angular Motion conti- 
 nuing the fame.) Moreover let C be the Center of Ofcil- 
 lation, or that Point in the Axis OS where a Particle 
 of Matter (or a fmall Weight) may be placed fo as to 
 be neither accelerated nor retarded by the Action of the 
 other Particles of Matter fituate in the Plane. Then, 
 if, from C and any other Point P in the Plane LMS, 
 two Perpendiculars CH and PQ_be let fall upon the ho- 
 rizontal Line OR, the Force of a Particle (or Weight) 
 at P to accelerate the Weight at C, will (according 
 to the foregoing Lemma ) be represented by P X
 
 in finding the Centers of Gravity, &c. 215 
 
 OQ, OH xTjP Which, fuppofing GN per- 
 
 OC OC 3 
 
 pendicular to OR, will alfo be exprefied by P X 
 
 OQ~ ON OP 
 
 OC OG x OC* r lt3 
 
 OQ.x OG x OC ONxOP* .' , , 
 
 ^ - * - ' very 
 
 x 
 
 / 
 manner the Force of any other Particle P will be re- 
 
 ' OQ. x OG x OC ONxOP* 
 prefented by P x - * - OC^OG~ 
 
 Of*. fcfr. 
 
 Therefore the Forces of all the Particles de- 
 ftroying each other (by Hypothecs) the Sum of all 
 
 the Quantities P x OG x OQ_x OC ON x OP 1 
 
 + PxOGxOClxOC ONxOPtfr. &c. nouft be 
 equal to nothing : Whence P x OG X OQ_x OC -f, 
 
 P x OG x OQ_x OC & c . &c. - P x ON x OP 1 + 
 
 P x ON X OP 1 &c. &c. and confequently OC = x 
 
 .. But ^^/. 171. and 172.) the 
 
 / / 
 
 Sum of all the Quantities P x OQjf P x OQ,fcfr. is equal 
 to the Content of the Body multiplied by the Diftance 
 (ON) of the Center of Gravity G from the Line LM 
 
 ON 
 
 (perpendicular to OC) j whence OC is alfo = 
 
 P X OP 1 + P X OF &c. &c . _ P X OP a -r- P X OP*&e.&t. 
 
 ON x Body OU x Body 
 
 Which Expreflion continuing the fame in all Inclina- 
 
 P 4 tions
 
 216 The Ufe of FLUXIONS 
 
 tions of the Axis OS, the Point C, thus determined is 
 a fixed Point, agreeable to the Definition ; and appears 
 to be the fame with the Center of Percuilion ; fee 
 Art. 1 8 1. 
 
 COROLLARY. 
 
 * 
 
 184. If PD, PD &c. be perpendicular to OS, the Nu- 
 jnerator of the Fraction found above, will become P X 
 
 2 OGxGD-f PxOG a +GP z +20Gx 
 
 GD + fcr>. &c. (fmce OP 1 r= OG^GP 1 2 OG X 
 GO &c.) Which, becaufe all the Quantities Px 2OG 
 
 x GD + P x 2OG x GD &c. or Px GD + P x GD &c. 
 
 (by the Nature of the Center of Gravity) deftroy one 
 
 another, will be barely P x OG 1 -f GP Z + P x 
 OG' + GP 1 + W*. Vc. = P + P+ &c. x OG a + Px 
 GP a +PxGP+ &c. = Mafs x OG l + PxGP 1 -!- 
 P xGP 1 + &c. Whence it is evident that OC is, alfo, 
 
 _ Mafs X OG% + P X GP^PxGP + &c. tfr.s 
 Mafs x OG ) 
 
 PxGP l -f-PxGP*-f 
 
 = G + " Mafs X QG ' 
 
 PxGP a +PxGP-i- fafc. 
 
 x OG 
 
 Whence it appears that, if a Body be turned about its 
 Center of Gravity^ in a Direffion^ perpendicular to the 
 Jlxis of the Motion^ the Place of the Center of Ofdllation 
 will remain unaltered \ becaufe the Quantities PxGP% 
 
 / / 
 
 P x GP* are no way affected by fuch a Motion of the 
 Body. 
 
 It
 
 in finding the Centers of Gravity, 6cc. 2 1 7 
 
 It alfo appears thai the Dijiance of the Center of Gra- 
 vity from that of Ofcillatisn (if the Plane of the Body's 
 Motion remains unalter'd) w III be reciprocally as the Di~ 
 
 Jiance of the fo r mir from the Point of Sufpenfun. There- 
 fcrc, if that Dijiance be found when the Point of Stifpcn- 
 
 feon is in the Vertex, or fi pvfited, that the Operation may 
 become the mo ft fimtle, the Value thereof in any other pro- 
 
 pofed p niuon of that Point will likeivife be given, by one 
 
 Jingle Proportion, 
 
 185. But now, to (hew how thefe Conclufions may 
 be reduced to Practice, we muft firft of all obferve, that 
 the Product of any Particle of the Body by the Square of 
 its Dijiance from the Axis of Motion is (here) called the 
 Force thereof ( its Efficacy to turn the Body about the 
 faid Axis being in that Ratio.) According to which, 
 and the firft general Value of OC, it appears that, if 
 the Sum of all the Forces be divided by We Prcdutf of the 
 Bod] int3 the Dijiance of the Center of Gravity from the 
 Point cf Sufpenjion^ the Quotient thence arifing will give 
 the Diliance of the Center of Percujfton, or Ojcillation from 
 tbefaid P&int of Siifpen/ton. 
 
 The Manner of computing the Divifor has been al- 
 ready explained ; it remains therefore to (hew how the 
 Sum of all the Forces in the Numerator may be col- 
 lected : Which will admit of feverp.l Cafes. Wherein, 
 to avoid a Multiplicity of Words, I {hall always exprefs 
 the Diftance of the Center of Gravity from the Point 
 of Sufpenfion by g y and the Diftance of the Center of 
 Perculfion, or Ofcillation, from the lame Point, by C. 
 
 1 86. Cafe I. Let OS be a Line fufpended at one 
 of its Extremes. 
 
 Then, if the Part OP (confidered as variable) be de- 
 noted by *, the Force of x Particles, at P, will (as 
 above) be defined by x x x 1 : Whofe Fluent (- * 3 ) 
 therefore expreffes the Force of a!l the Particles in'oP 
 (or the Sum of all the Produces, under each Particle, 
 and the Square of its Diftance from O the Point of 
 Sufpenfion. This Quantity therefore (when x be- 
 comes
 
 'The Ufe ^/FLUXIONS 
 
 T O comes = OS) being divided by OS X JOS 
 (according to the foregoing Rule or Ob- 
 
 _P fervation) we get (JQ^ =) OS 
 
 for the Value of C, the true Diftance of 
 the Center of Ofciilation (or Percuflion) 
 from the Point of Sufpenfion. 
 
 - C 
 
 * Art, 185, 
 
 () 
 
 187. Cafe 2. Let AB be a Lint, vibrating in a vertical 
 Plane^ having its two Extremes A and B equally dijiant 
 from the Point of Sufpenfion O. 
 
 If OG (perpendi- 
 cular to AB) be put 
 =:a, and GP *, the 
 Force of x Particles 
 at P, will be denoted 
 by x Xa* + x* = x x 
 OP**: Whofe Flu- 
 divided by ax 
 
 
 P\ 
 
 B 
 
 C 
 BG 
 
 ent, 
 
 (or PGxOG) gives 
 
 ( 
 
 ax 
 
 . 4 
 
 ~ = OG + -7c = C, when x becomes = GB, 
 3 3OG 
 
 188. Cafe 3. Let APSQ. be a Circle, vibrating in a 
 vertical Plane. Let PQ_be any Diameter thereof ; then 
 OP 1 + OQ! being = lOG* + aPQ*, the Sum of the 
 Forces of two Particles at P and Q, (putting OG 
 = a, and AG = r) will be = a l + r* X 2; whence it is 
 evident that the Sum of the Forces of all the Particles, 
 in the whole Periphery, will be exprefled by their Num- 
 ber x/TrS orby^ + r 1 X Ptripb. APSQ.: Which,
 
 in finding tie Centers of Gravity, &c. 219 
 
 will 
 
 7*:Q 
 .': 
 
 
 
 /A.! \ 
 
 if p be put 3.141 
 
 be =r a 1 + r l X Zpr = t.pa*r 
 + 2pr 3 Hence the Force of 
 the Circle itfelf is alfo given, 
 being Fluent of 
 
 x Circle APSQ. Now, if the 
 two Expreflions thus found 
 be divided by a x Ptriph. 
 APSQ, and a X Circle APSQ. 
 refpe&ively *, we (hall have 
 
 r 1 r 2 
 
 a + and a + r~> for the two correfponding 
 
 Values of C. 
 
 Art. 185. 
 
 189. Cafe 4. JL*/ AHBE be a Circle having its Plant 
 (always) perpendicular to the Axis of Sufpenfan OG, 
 
 Let AGB be that 
 Diameter of the Cir- 
 cle which is parallel 
 to the Axis of Mo- 
 tion RS ; and let EF 
 be any Chord parallel 
 to AB and RS ; whofe 
 Diftance, GP, from 
 the Center of the 
 Circle, let be denoted 
 by x ; putting OG 
 = a, and AG = r : 
 Then, by the Nature of the Circle, EF r= 2vV ** 
 whofe Diftance OP, from the Axis of Motion RS, is 
 alfo given = vV + **. Hence it appears that the 
 Force of all the Particles in the Line EF (defined in 
 
 Art. 185.) will be reprefented by a*+x* x 2^r* x*~ 
 
 Therefore* x ** + ** x 2V>_^ is the Fluxion of 
 the Force of the Plane ABFE; whofe Fluent (when 
 
 x=r)
 
 220 
 
 FLUXIONS 
 
 x r) is = a 1 -f \ . r* X Area AEFEG\ which, if p 
 be put for the Area of the Circle whofe Radius is Unity, 
 will be = a* + r* X k P r * "> whereof the Double (pa*r* 
 + 5/>r 4 ) is the Force of the whole Circle AEFH: 
 whofe Fluxion 2parr -}-/>r 3 r(fuppofingr variable) being 
 divided by r , we likewife get 2pa*r+pr 3 ( a l -\-r* 
 X Periph. AEFH) for the Force of the Periphery 
 AEFH. But the Center of Gravity, whether we re- 
 gard the Circle itfelf or its Periphery, is in the Center 
 of the Circle; therefore the Diftance of the Center of 
 Ofcillation from the Point of Sufpenfion, will in thefe 
 
 r 1 r 1 
 
 two Cafes be exhibited by a + and a + re- 
 
 4*2 2a 
 
 fpe&ively. 
 
 If the Circle, inftead of being perpendicular to GO, 
 cither coincides, or makes a given Angle with it, the 
 Value of C will come out exactly the fame ; provided 
 the Diameter AB ftill continues parallel to the Axis of 
 Motion RS : This appears from Art. 184.. and may be, 
 otherwife, very eafily demonftrated. 
 
 190. Cafe 5. Let the Figure propofed be a Curve AEF, 
 moving ( flat-way 'j, as it were) fo that the Plane de- 
 fcribed by the Axis OAS may be perpendicular to that 
 of the Curve. 
 
 Here, putting AP =. x, PN =y, 
 AN = z , OA - d, OG = g, and 
 AG^:a, the Force of the Par- 
 ticles in MN will be defined by 
 2y X a + x\ . Therefore the 
 Fluent of 2yx xrf+x] 1 will be 
 as the whole Force of the Plane 
 NAM. (or AEF, when x = 
 AS ) and confequently C = 
 Flu d+ v| a X yx m 
 
 Flu. d+x x yx 
 
 : Which, there- 
 fore,
 
 in fading the Centers of Gravity, &c. 221 
 fore, when the Point of Sufpenfion is in the Vertex A, 
 
 Flt4* V V^ X 
 
 will become C = -jr. : r. Let this Value be de- 
 klu. yxx 
 
 noted by v ; then, the Diftance of the Centers of Gra- 
 vity and Ofcillaiion being v a, we have (by Art. 184.) 
 
 g : a :: v a : ( a x v ~~") the Diftance of the fame 
 
 f 
 Centers, when the Point of Sufpenfion is at O, and con- 
 
 fequently C, in that Cafe, = g + "X^a . Which 
 
 g 
 
 Form will be found more commodious than the fore- 
 going in moft Cafes. 
 
 After the fame iManner the Value of C, with refpecl: 
 
 of the Arch AEF, will appear to be = Flu ' ^+*j* * * 
 
 Flu, d+x X 
 
 It may not be improper to give an Example or two 
 of the Ufe of the foregoing Theorems. 
 
 191. Let therefore EAF be, firft, confider'd as an 
 ifofceles Triangle : In which Cafe AP (x) and PN (y) 
 
 hx 
 being in a conftant Ratio, we have^ = (fuppofing 
 
 SF=* and AS=,.) Hence C (= 
 Flu, f 
 
 Flu. 
 
 -- z 
 
 : Or ( accordin g f o the fecond Form) 
 
 Flu. , 
 
 ~ 4 and is kn wn to 
 
 be
 
 122 he Uft of FLUXIONS 
 
 2X . a X v a \ 
 
 Art. 175. be = *, we have C (= H -- J = g + 
 
 x* 
 
 , where ( =d + a) = d -f \ x. 
 
 Again, becaufe % and x are in a conftant Ratio, we 
 alfohave ^^M^** _ Flu.T+xY x x _ 
 
 X * -K. --A? x x 
 
 - , , T ; whence the Center of Ofcillation of 
 a-j- a X 
 
 the Lines EH and AF is given. 
 
 192. For a fecond Example, let EAF be fuppofed a 
 
 n 
 
 Parabola of any Kind, whofe Equation is y - : 
 
 c i 1 
 
 Then (according to Form 2.) we fha!l firft have v (=. 
 
 F!u.yx*x\ Flu. x x n+2xx T1T , 
 
 g . . I = - z = - ' Whence, 
 
 Flu -y** / _ Flu. x* l x n + 2 
 
 n+ixx. , f ^~ f , 
 
 f Art. 176. being = CT+2 ti we alfo get C ( = ,g- + 
 
 n+i X ** 
 
 = + v = - ; where g=. 
 
 ^+2)'x4-3X^ 
 But, with refpe<a to the Arch of the Curve, v ( = 
 
 -v a _ 2 - 
 
 Flu. XX Y c +nnx 
 
 which Value (found by infinite Series, and even with- 
 
 1 Art. i 3 out in fome ^ afes t) tnat f c wil1 alfo be g' ven< 
 
 193. Cafe. 6. Let the proofed Figure be a Curve vi- 
 brating (edge-ways) fo that the Motion of the Axis may 
 be in the Plane of the Curve. 
 
 Then (by Cafe 2.) the Force of all the Particles in 
 the Line PN (fee the preceding Figure) being defined 
 
 by OP* x PN + j- PN, or </+#!* x> -f |/ ( retaining the 
 
 No-
 
 m finding the Centers of Gravity, &c. 223 
 
 Flu. d+x 
 
 Notation above) we have C =. 
 
 flu. 
 
 Which, when the Point of Sufpenfion is in the Vertex 
 
 Fin. yx^x + T v 3 * _ 
 
 A, will become - rr - r 2 * : Let this (when 
 rlu. yxx 
 
 found) be denoted by v; then, it appears from the 
 preceding Cafe, that the general Value of C will, alfo, 
 
 , r ^ , , i tf x v a 
 
 be reprefented by g + - 
 
 o 
 
 In the fame manner the Value of C, with 
 refpect to the Arch EAF, will be expounded by 
 
 or by g + a* 9 fuppofing - 
 
 S 
 
 Flu. XK 
 
 194. Example. Let the Equation of the given Curve be 
 y = -fL- : Then v ( = ^ >*** +. 
 
 1^I fl/ij i/w- 
 
 .,. i +2 . , 3~3" 3". 
 
 /Y. f A" X+ j XX _ 
 
 + 3 
 
 X X 
 
 = rT x x + ===== X - : From which the 
 
 3x3+i * 
 
 Value of C is alfo given ; and from whence it appears, 
 that if n be expounded by 0, v will become = 
 
 2* if" 2 **-f y* ' 
 
 + =: X - ; in which Cafe the Figure 
 
 ^ <J <J ^ 
 
 will degenerate to a Rectangle : But, if n be inter- 
 preted by i, the Figure EAF will then be an ifofceles 
 
 Triangle,
 
 22 4 
 
 T/je life of FLUXIONS 
 
 Art. 
 
 *}J V" 
 
 Triangle, and v = + - : Laftly, if n be taken 
 
 4 4* 
 1, the Curve will be the common Parabola, and v= 
 
 5f 1. 
 
 7 + 3 
 
 195. G?/* 7. Z>/ /& Figure AEFH <? fl Solid generated 
 by the Rotation of a Curve EAF about its dxi's AS j 
 /fonwir its. Bale HH parallel to the Axis of Motion 
 BOC. 
 
 B 
 
 It appears, from Cafe 4. 
 that the Force of all the 
 Particles in the circular 
 Section hh (parallel to HH) 
 will be exprefTed by 
 OP 2 -f 1PN 1 X Circle M, 
 orOP z x PN 2 -f iPN*xp 
 (p being = 3.14.15 &c.) 
 which, in algebraic Terms, 
 
 is d -f 
 Hence 
 
 X / + . 
 we have 
 
 x p. 
 C = 
 
 Flu. 
 
 Which, therefore, when the point of Sufpenfion is in 
 
 the Vertex A, becomes ^7 a . = v ; and 
 
 Jflu. y xx 
 
 , as in the preceding 
 
 confequently C =. g -\- 
 
 & 
 .Cafes. 
 
 But, with regard to the Superficies of the Solid ; it 
 is found, in Cafe 4. that the Force of the Particles in 
 
 the periphery MNA is expreffed by OP 1 + PN 1 x 
 Peripb. Mb Nb = d + ffi* x ?py 4- Py 3 
 
 5 Hence
 
 in fading the Centers of Gravity, &c. 225 
 
 Hence the Fluent of d+x^ x ipy+py* x , divided 
 
 c 77 , . , .F/K. */+*!* X 2yz -J- y s x\ 
 by that of d+x X 2/>y (=: - r-^ - ? - - ) 
 
 /*/. </+* X 2yz ' 
 will give the true Value of C with refpecl to the curve 
 
 Flu, 2yx*z $- y*z 
 Surface EbAhF. Which, putting v - -p- ; : , 
 
 a X v a 
 is hkewife exprefled by g -f- 
 
 O 
 
 196. Ex. I. LetEAF be confidered as a Cone', then, 
 
 bx 
 putting AS =/", SF = i and AF c, we have^ ~ 
 
 ex . _. Flu. 7f^l*xjrV+ iv** 
 
 z = j and therefore C ( = . _J_ y - 
 
 , 
 
 x f. But, 
 with refpeft to the convex Superficies, C will be found =: 
 
 iqy. Ex. 2. Zf/ EAF I3c. be confidered as a Sphert 
 whofe Center is S, and Radius ASrrr; in which 
 
 F/u.y*x*x+y+x\ 
 
 y* being = zrx JT*, we have v ( = ^ ^-r 2 - I 
 * rlu. y xx J 
 
 Flu. r*x*x + rx 3 xx*x r* + rx x* 
 
 Flu. 2rx*x x*x ^ r ^ x 
 
 whence C is alfo given. But, when * =r ir (or the 
 
 IT 
 
 whole Sphere is taken) v := : Therefore a being =:r, 
 
 and ^ = OS, in this Cafe, we have C ( == ^ -f. 
 
 2r 
 
 198.
 
 226 
 
 The Uft of FLUXIONS 
 R S 
 
 198. Cafe 8. Let the Fi- 
 gure propofed le a Solid, as 
 in the preceding Cafe t but let 
 its Axis AG be^ here^ pa- 
 rallel to the Axis of Motion 
 ORS. 
 
 Then, if RP (OG) be 
 put -g, 
 
 M 
 
 AP = * &c. the Force of 
 the Particles in the Circle 
 NM (parallel to EF) will 
 
 be exhibited by g* + *>* 
 x#% or pgY+ipy* (Vid. 
 Cafe 3.) Hence we have C 
 
 1*5- flu. 
 
 g x Solid 
 Flu. 1 i 
 
 g* 
 
 t = 
 
 Moreover, with refpedt to the Superficies ; the Force 
 of the Particles in the Periphery of the faid Circle MN 
 j Art. 185. being 2pg*y + ipy 3 t> we ^ ave > ' n th ' s ^- a ^" e ^ 
 Fin, ipg-y + 2py 3 X g __ Flu. lpg z ) + 2py 3 ~ _ 
 
 ~ * " 1 
 
 g x Superficies. 
 
 g X Flu, 2pyz 
 
 X Flu.yz 
 
 199. Ex. i. Let EAF be a Segment of a Sphere, 
 vhofe Radius is r ; then/ being = 2rjf A;*, we {hall have 
 
 /?.!, , 
 
 F* + 
 
 **./&.. 
 
 7T^ p" 
 
 X_ 
 
 , 
 
 =,+ 
 
 307- 
 
 Which, when j; is expounded, either, by r or 2r, be- 
 
 i 
 
 , for the true Value of C y when 
 
 comes n 
 
 
 either
 
 in fading the Centers of Gravity, Sec. 227 
 
 either the Hemifphere, or whole Sphere, is taken. But, 
 with refped to the Center of Ofcillation of the Super- 
 ficies thereof, we have z in this Cafe = Art. 14. 
 
 y irx xx 
 
 rx Flu. y 3 z 
 
 = : And therefore g+ ^f - = g + 
 
 y gx.FIu.yz 
 
 Flu. irx xx X rx rx \x~ 
 
 . _ . P -\ = : Which, when 
 
 g* Flu rx g 
 
 x = r, or x r: 2r, becomes g + . 
 
 Off 
 
 200. Ex. 2. Let the S0//WEAF be a Paraboloi^ v.'bofc 
 
 n 
 
 generating Curve is defined by the Equation y = 
 
 c 
 
 44 
 
 .v x c 
 
 ... ._ 
 
 g x tin. x x^.c 
 
 xy* .... 
 
 Wtiere, 
 
 if n be taken = o, the Figure will become a Cylinder, 
 
 > a 
 and C g + : But if n be expounded by i, the 
 
 O 
 
 3/ 
 
 Figure will be a Cone, and C =. g + - . Laftly, if 
 
 be taken = |, the Figare will be the Solid generated 
 
 * 
 from the common Parabola and C~g-\ . 
 
 SEC-
 
 228 
 
 The Ufe of FLUXIONS 
 
 Art 
 
 SECTION XII. 
 
 Of tke Ufe of FLUXIONS in determining 
 the Motion of Bodies affetted by centripetal 
 Forces. 
 
 PROPOSITION I. 
 
 201. f I ^ H E Motion^ or Velocity y acquired by a Body 
 freely defcendtng from Re/f t by the Force of 
 an uniform Gravity, is proportional to the Time of its 
 Defcent ; and the Space gone over> as the Square of that 
 Time. 
 
 The firft Part of the Propofition is almoft felf-evi- 
 dent : For, fmce any Motion is proportional to the Force 
 by which it is generated, that generated by the Force 
 of an uniform Gravity muft be as the Time of Defcent ; 
 becaufe the whole Effect of fuch a Force, a&ing equally 
 every Inftant, is as that Time. 
 
 jj /^ Let, now, the Velocity acquired 
 
 during a Defcent of one Second of 
 Time, be fuch as would carry the Body 
 uniformly over any Diftance b in one 
 Second ; and let AB (x) denote the Di- 
 ftance defcended in any propofed Time 
 / ; which Time let be denoted by PQ_; 
 making B r and QJJ =.t : Then it 
 
 will be, as i : / :: b : (bt} the Diftance 
 that would be uniformly defcribed in i^ 
 with the Velocity at B : Alfo i : / :: 
 
 the faid Diftance (bt) to bit = * *. 
 By taking the Fluent whereof we get 
 
 ..B ^ 
 
 c 
 
 ..cl 
 
 - e 
 
 i/
 
 in Centripetal Forces. 229 
 
 ifo*=*. Therefore the Diftance defcended ({b?) is 
 as the Square of the Time. Q E. D. 
 
 Olherwife, without Fluxions. 
 
 Conceive the Time ( PQJ of falling thro' AB to be 
 divided into an indefinite Number of very fmall equal 
 Particles, reprefented, each, by m ; and let the Diftance 
 defcended in the firft of them be A<r, in the fecond cd y 
 in the third de, S3c. &c. Then, the Velocity being al- 
 ways as the Time from the Beginning of the Defcent, 
 it wifrin the Middle of the firft of the faid Particles be 
 defined by ~ m ; in the Middle of the fecond by I f m ; 
 in the Middle of the third by 2 4 "*> &c. bV. But, 
 fince the Velocity at the Middle of any Particle of 
 Time, is a Mean between thofe at the two Extremes, 
 or betwixt any other two equally remote from it, the 
 correfponding Particle of the Diftance AB may, there- 
 fore, be confidered as defcribed by that mean Velocity. 
 And fo, the Spaces Ac^ cd^ de, ef^ &c. defcribed in equal 
 Times, being refpe&ive'.y as the faid mean Celerities | m, 
 1 4 ;j, 2 i m, 3 { 2, &c. it follows, by Addition, that 
 the Diftances, A^, A^, A*, A/", &c. gone over from 
 
 _, m ^m cm i6m y 
 the Beginning, are to one another as , , , , 
 
 222 2 
 
 &c. or i, 4, 9, 16, 25, &f<r. that is, as the Squares of 
 the Times. j>. . />. 
 
 COROLLARY i. 
 
 202. Since the Diftance that might be uniformly run 
 over in one Second, with the Velocity at B, is ex- 
 prefled by /-/, the Diftance that might be defcribed with 
 the fame Velocity in the Time t will therefore be ex- 
 prefTed by /X/, or bf : Whence it appears, that the 
 Space AB ( i It 1 ) thro' which the Body falls in any 
 given Time t y is juft the half of that which would be 
 uniformly defcribed with the Celerity at B, in the fame 
 Time. 
 
 Therefore 5 fince it is found from Experiment, that a 
 
 Body near the Earth's Surface (where the Gravity may 
 
 0.3 be
 
 230 "The Vfe fl/' FLUXIONS 
 
 be taken as uniform) defcends about i6 T * Feet in the 
 firft Second, it follows that the Value of b (is in this 
 Cafe) = 2 X i6 T \ = 32d : And confequently the Number 
 of Feet dclcc-aded in t Seconds, equal to i6 T Vxf a . 
 
 COROLLARY 2, 
 
 203. It is evident, whatever Force the Body de- 
 fcends by, the Value of b will always be as that Force; 
 iince a double Force, in the fame time, genei^es a 
 double Velocity ; a treble Force, a treble Velocity, &c 
 Therefore, feeing our Equation \ bt 1 *, alfo gives t 
 
 v/ 
 
 and b =: ;, it follows, 
 * -5 
 
 1. That the Diftance defcended is, univerfally, as 
 the Force and the Square of the Time conjunclly. 
 
 2. That the Time is always as the Square-root of the 
 Diftance applied to the Force. 
 
 3. And that the Force is as the Diftance apply'd to 
 
 the^Square of the Time What is above demonftrated 
 
 with refpedl to the Times, holds alfo in the Velocities, 
 when the accelerating Forces are equal. 
 
 PROPOSITION II. 
 
 204. 70 determine the Velocity ', 
 and Time of Defcent^ cf a Body 
 along an inclined Plane AC. 
 
 From any Point F, in AC, 
 draw FE perpendicular to the ver- 
 tical Line AD, and make FB and 
 CD perpendiculartoAC, meeting 
 AD in B and D. Becaufe (by 
 the Principles of Mechanics) the 1 
 Force of Gravity in theDireclion 
 CF, whereby the Body is made to 
 defcend along the Plane, is to the 
 abfolute Force thereof, as AF to 
 
 AB,
 
 in Centripetal Forces. 
 
 AB, or as AC to AD ; and fince (by Cafe I. Art. 203 .) 
 the Diftanccs defcended in equal Times are as the 
 Forces, it follows that the Time of Defcent thro' AF 
 will be equal to the Time of the perpendicular Defcent 
 thro' AB : And confequently the Time of Defcent thro* 
 AC equal to that thro' AD ; which is given by Prop. i. 
 Moreover, btcaufe the Velocities at F and B, acquired 
 in equal Times, are as the Forces, or as AF to AB j 
 and it appears from Prop. i. that the Velocity 
 at E*is to that at B, as v/AE : v/AB, or as 
 
 V/AE x AB (=AF) : /AB x AB (= AB) it fol- 
 lows, by Equality, that the Celerity at F is equal to 
 
 ng 
 . 1. 
 
 that at E ; which is therefore given, by the precedin 
 Propofition. 
 
 COROLLARY. 
 
 205. Hence the Time of Defcent along the Chord 
 AC of a Semi-circle ACD is equal to the Time of De- 
 fcent along the vertical Diameter AD: And, if the Chord 
 DGbeof the fame Length with AC (its Inclination to 
 the Horizon being alfo the fame) the Time of Defcent 
 along i: will alfo be equal to that along the vertical 
 Diameter. 
 
 PROPOSITION III. 
 
 206. Ij r , from two Points 
 A and D, equally remote 
 from the Center of Attrac- 
 tion C, two Bodies move t 
 with equal Celerities^ the 
 one along the Right-line 
 AC, the other in a Curve- 
 line DBQ_, their Celerities 
 at all other equal Dijiances 
 from the Center, will be 
 equal. 
 
 For, let CBandCEbe 
 
 any two Tuch Diftances ; 
 let the Arch. BE be de- 
 
 0.4 
 
 A 
 E
 
 232 ^he Ufe ^FLUXIONS 
 
 fcribed, from the Center C, and alfo tb, indefinitely 
 near to it, cutting CB in n : Let the centripetal Force 
 at the Difhnce of CB, or CE, be reprefented by/, and 
 the Velocity at B, by v. 
 
 By the Refolution of Forces, the Efficacy of the 
 Force (f) in the Direction B, whereby the Velocity 
 
 of the Body is accelerated, will be =^7- x f: Alfo the 
 
 Time of moving over B (being as the Diftance apply'd 
 
 B 
 to the Velocity) is reprefented by : Therefore the 
 
 Increafe of Velocity, in moving thro' B, being as the 
 Force and Time conjunctly, will be defined by T>T */ 
 
 x , or its Equal - X /. In the fame Manner, 
 v "^ 
 
 the Velocity at E being denoted by w 9 the Time of 
 
 E* 
 
 falling thro' E* will be reprefented by , and the Ve- 
 
 E<r 
 
 locity generated in that Time by x/: Which is to that 
 
 (TD 
 - Xf) acquired in falling thro' the Arch B, as 
 
 - to . Therefore, feeing the correfponding Incre- 
 ments of Velocity are always, reciprocally, as the Ve- 
 locities themfelves, it is manifeft, if thofe Velocities are 
 equal, in any two correfponding Pofitions of the Bodies, 
 they will be fo in all others, being always increafed 
 alike. But they are equal at A and D by Suppofition : 
 Therefore, faff. 4 E - D - 
 
 PROPOSITION IV. 
 
 207- To find the Ratio of the Velocities, and Times of 
 Deftent) cf Bodies^ in Curves j the Force of Gravity 
 being confidtrtd as uniform. 
 
 Let ARD reprefent a Curve of any Kind, along 
 which a Body dcfcends, by the Force of its own Gra- 
 vity
 
 in Centripetal Forces. 
 
 vity from A ; let AC, RB, &c. be parallel, and CD 
 perpendicular, to the Horizon ; moreover, let R touch 
 the Curve at R ; and let CB u, AR = a/, and 
 
 Since the Points B 
 and R (as well as C 
 and A) may be looked 
 upon as equally re- 
 mote from the Earth's 
 Center (to which the 
 Gravitation tends), the 
 Velocity acquired in 
 defcending thro' the 
 Arch AR will (by the 
 laft Propofttion ) be 
 equal to that acquired by falling freely through the 
 Right-line CB ; which laft Velocity (by Prop, i.) is 
 
 always as -v/CB (or a ). Therefore the Celerity, 
 whether the Body moves in a Right-line, or a Curve* 
 is always in the fubduplicate Ratio of the perpendicular 
 Defcent ; and fo, the Time in which R (ow) would be 
 uniformly defcribed, with that Celerity, will be univer- 
 
 fally as ; whofe Fluent is as the Time of falling 
 
 233 
 
 through AR. 
 
 E. /. 
 
 EXAMPLE. 
 
 208. Let the Curve ARD be any Portion of the 
 common Cycloid ; whereof the Vertex is D and Axis 
 DC ; and whofe Nature (putting DC =r, and the Ray 
 of Curvature at D ~ a] is defined by the Equation 24 
 X DB:=DR*. Here, we have DR ( v^aTx i/DB) 
 
 _ _ X 
 
 : V/2tf X c wi 1 ; whofe Fluxion / 2a x 
 -it 
 I , with a contrary Sign, is the Value of Rn or ov ; 
 
 7 -")* 
 
 and
 
 The Ufe of FLUXIONS 
 
 therefore ^ = /U x ^~^"~" : Whofe Fluent, 
 
 at the loweft Point D, where u becomes = c, will (by 
 Art. 142.) be equal to Via multiplied by ^ ) 
 
 half the Meafure of the Periphery of the Circle whofe 
 Diameter is Unity. Which Fluent (and confequently 
 the Time of Defcent) will therefore continue the fame, 
 let the Arch DA be what it will. 
 
 PROPOSITION V. 
 
 209. To determine the Paths of Prcjeftiles near the 
 Earth's Surface ; (ncglefting the Refijiance of the 
 Atmofpbcre.) 
 
 Let a Body be pro- 
 jected fiom the Point 
 A, in the Direction 
 of the Line AC, with 
 a Velocity fufficient 
 to carry it uniformly 
 over the Diftance d 
 in the Time t , and 
 let the Space through 
 which it would freely 
 defcend, by its own 
 Gravity, in that time, 
 be denoted by b ; alfo 
 let the Sine of the 
 Angle of Elevation 
 BAG (to the Radius 
 r) be put ~ s, its 
 Co-fine = c, and the Diftance of the Point A from the 
 Ordinate Hm (cor.fidercd as moving parallel to itfelf 
 along with the Body) = x ; then, by Trig. HG (per- 
 pendicular to AB) will be = , and AG = 
 
 Becaufe the Projectile is turned afide, continually, 
 from a rectilinear Path, by the Earth's Attraction, it 
 
 muft
 
 in Centripetal Forces. 23 j 
 
 muft defcribe a Curve -line AmEwB, to which AC is a 
 Tangent at the Point A : But that Attraction, acting 
 always in a Direction (Hw) perpendicular to the Ho- 
 rizon, can have no Effect upon that Part of the Velocity 
 with which the Body approaches the Line BC, parallel 
 to Hm ; therefore the Right-line HG (in which the 
 Body is always found) will continue to move uniformly 
 towards Hu, the fame as if Gravity was not to act; 
 and the D titan ce Gm defcended from the Tangent AC, 
 by means or the Attraction, will be the very lame as if 
 the Bod/ was to defcend from Reft along the Line GH. 
 This being premifcd, it is evident, that as d : AG 
 
 ( ) : : t : ( ; x t j the Time of defcribing Am ; 
 
 r L x' i (br*x 1 \ 
 
 and, as /* : -^ x t 1 : : I : f -TJ*') the Space (Gm} 
 
 through which a Body would freely defcend in that Time 
 (by Prop. i.) 
 
 sx Ir-x 1 csd'-x br** 1 . 
 Hence ~Sd?> or ^^ 1S a g eneral 
 
 Value for the Ordinate Hm: By putting which = 0, 
 we get x =2 -T"! AB =: the Amplitude of the Pro- 
 jection. But, by putting its Fluxion equal to nothing, 
 
 we have x - r-r j which fubftituted for x in the Va- 
 2^r 
 
 i jt 
 
 lue of HOT, gives ji for the Altitude DE of the Pro- 
 jeaion. ^. E. /. 
 
 COROLLARY. 
 
 210. If another Body be projected, with the fame 
 Celerity, in the vertical Direction ASj then, s becoming 
 
 / s*d*\ 
 
 = r, the Altitude of that Projection ^ 70 J will be- 
 come
 
 236 
 
 Ufe of FLUXIONS 
 
 come = AS ; which call h, and let this Value be 
 
 4^ 
 fubftituted in thofe of AB and DE, and they will be- 
 
 come-^ and refpelively. 
 
 Hence, if from the Point Q^ where the Line of Di- 
 rection AC cuts a Semi-circle defcribed upon AS, the 
 Lines SQ_and QP be drawn, the latter perpendicular to 
 AB, the Triangles ASQ_ and AQP being fimilar, we 
 fhall have 
 
 r:*::A(AS): = AQ. 
 
 PROPOSITION VI. 
 
 a 1 1 . To determine the Ratio of the Forces, whereby Bodies, 
 tending to the Centers of given Circle^ are made to re- 
 volve in the Peripheries thereof. 
 
 Let ABH and alb be any two propofed Circles, 
 whereof let AB and ab be fimilar Arcsj in which, let 
 
 the
 
 in Centripetal Forces. 237 
 
 the Velocities of the revolving Bodies be refpectively as 
 /^to v i make DBK and dbk parallel to the Radii AC 
 and ac, putting AC = , ac = r, and the Ratio of the 
 centripetal Force in ABH to that in abh> as F to /. 
 
 It is plain, becaufe the Angles ABD and abd are 
 equal, that the Velocities at B and , in the Directions 
 BK and bk^ with which the Bodies recede from the 
 Tangents AD and ad y are to each other as the abfolute 
 Celerities ^and v *. But thofe Velocities, being the * Art '35 
 Effects of the centripetal Forces acting in correfponding, 
 fimilar, Directions during the Times of defcribing AB 
 and ab, will therefore be as the Forces chemfelves when 
 the Times are equal ; but when unequal, as the Forces 
 and Times conjunctly. Therefore, the Times being 
 
 AB at R r 
 
 univerfally as -77- to , or as -77 to ( becaufe the 
 V v V v ' 
 
 
 
 Arcs AB and at are fimilar) we have, as F x -p : f X 
 
 y- 
 
 : : V '. v\ whence (multiplying the Antecedents by 
 
 V v\ 
 
 -p- and the Confequents by ) it will be, as F :f : : 
 
 V I v* 
 
 -r- : : Therefore the Forces are as the Squares of the 
 
 Velocities directly, and as the Radii inverfely. 
 
 Otberwife. 
 
 Let the indefinitely little Arch AB be the Diftance 
 that the Body moves over in a given, or conftant Par- 
 ticle of Time ; and let the centripetal Force at B be 
 meafured by twice the Subtenfe or Space AE through 
 which the Body is drawn from the Tangent AD in than 
 Time f. 
 
 Then, 
 
 f The Velocity 'which any Force, uniformly continued, is ca- 
 pable of generating, in a given Body, in a given Time, is the 
 proper Meafure of the Intenfety cf that Force *. But this Pe- 
 ftcily is itjelf mtajitrgd by tht Space the Body would move uni- * Art, 203, 
 
 formly
 
 238 ne Vff of FLUXIONS 
 
 Then, by the Nature of the Circle, AB* t=r AH X 
 
 * 
 AE = AC x 2AE, and confsquently 2 AE = 
 
 Therefore, the Force is as the Square of the Velocity ap- 
 plied to the Radius of the Circle (as before). 
 
 COROLLARY I. 
 
 Tf^* ^ 
 
 112. Becaufe, F :/ :: : -, it follows that 
 
 *t f 
 
 V:u :: V~RF : V7Z and 
 
 COROLLARY II. 
 
 113. If the Ratio of the periodic Times be denoted 
 by that of P top ; then the Ratio of the Velocities P, v 
 
 R r 
 
 being as -p to, we fhall have, by Equality V RF: 
 
 R r 
 
 V r f ' ~n ; ~T > whence alfo 
 P p 
 
 R r 
 &'/' ' ~pZ ~i-> and . 
 
 R:r::FP z : fp\ 
 
 firmly over in a given Time ; 'which Space is always the doublt 
 of that through which the Body would freely defcend,from Reft, 
 
 * Art zoa. ** *^ e S ame ttme ** < ?t> ere f ore 2 ^^ ' J *** proper Meafure of 
 the centripetal Force, according as ive haiie ajjumed it. 
 It is true, iv&ea the Forces to be compared are all computed 
 in the fame Manner, from the Nafcent, or indefinitely Jmatt 
 Subtenfes of contemporaneous Arcsy it matters not whether 
 we confider thofe Subtenfes, or their Doubles, as the Meafures 
 of the Forces, the Ratio being the fame in both Cafes. But 
 when the Forces fo found are to be compared 'with others de- 
 rived from a fluxicnal Calculus, it is abfolutely ntcej/ary ta 
 take the double Subtenfe for the Meafure of the Force. 
 *This Note is inferttd, that the Learner may avoid the Errort, 
 which fame 'very confider able Mathematicians ba-~ve fallen intt 
 ly not properly attending is this Particular. 
 
 Co-
 
 in Centripetal Forces. 239 
 
 COROLLARY III. 
 
 214. If the Meafure of the Force, or the Velocity 
 that might be uniformly generated in a given Time (i ) 
 
 be expounded by any Power a of the Radius AC (a] ; 
 then the Diftance through which a Body would freely 
 defcend in the fame Time, by that Force, uniformly 
 
 continued, will be expreflfed by \ a *. Therefore, * Art. ao. 
 the Diftances defcended, by means of the fame Force, 
 uniformly continued, being as the Squares of the 
 Times f, it is evident, if the Time of moving through t Art- ZOI< 
 AB be denoted by /, that the Diftance AE defcended 
 
 in that Time, will be denoted by - x i a" .; And fo 
 
 ' i 
 
 we fliall have AB ( \ / 2AE x AC) = x a~^ ; 
 
 which being the Diftance defcribed by the revolving 
 Body in the Time /, it follows that the Space gone over 
 
 in the given Time (i) will be a : Which, there- 
 fore, is the true Meafure of the Celerity in this Cafe. 
 The fame conclufion might have been derived in much 
 fewer Words from Carol, i. but, as a thorough under- 
 ftanding hereof is abfolutely neceflary in what follows 
 hereafter, I have endeavoured to make it as plain as 
 poilible. 
 
 COROLLARY IV. 
 215. Hence the Time of Revolution is alfo derived; 
 
 for it will be as a ~ : 3.14159 &c. x la (the whole 
 Periphery) : : r : HlJJL^ o, . II ^. x 
 
 I n 
 z 
 
 ia , the true Meafure of the periodic Time. 
 
 Co-
 
 240 7& W e f FLUXIONS 
 
 COROLLARY V. 
 
 2l6. Therefore, if n be expounded by i, o, I, 
 2 and 3 fucceffively, then the Velocity cor- 
 
 i , 
 
 refponding will be as 0, a , i, a 2 , and a l ; and 
 
 JL 2. 
 
 the Time of Revolution, as i, a x , a, a and a* re- 
 fpedtively. 
 
 SCHOLIU M. 
 
 117. From the preceding Propofition, and its fub- 
 fequent Corollaries, the Velocity and periodic Time of 
 a Body revolving in a Circle at any given Diftance from 
 the Earth's Center, by means of its own Gravity, may 
 be deduced: For let d be put for the Space thro* which 
 a heavy Body, at the Surface of the Earth, defcends in 
 the firft Second of Time, then id will be the Mea- 
 fure of the Force of Gravity at the Surface : And there- 
 fore, the Radius of the Earth being denoted by r, the 
 Velocity, />*r Second, in a Circle at its Surface, will be 
 
 , rr-. rT > 3- I 4i5Q& > -X2r 
 
 K/ird't and the Time of Revolution = - 7== - 
 
 v ird 
 
 / 
 3'H T 59 & Ct x V^ ~T (Seconds); which two Ex- 
 
 preffions, becaufe r is = 21000000 Feet and ^=16,^ 
 will therefore be nearly equal to 26000 Feet and 5075 
 Seconds, refpe&ively. Let R be now put for the Radius 
 of any other Circle defcribed by a Projectile about the 
 Earth's Center: Then, becaufe the Force of Gravitation 
 above the Surface is known to vary according to the 
 Square of the Diftance inverfely, we have (by Cafe 4 
 
 J- _ 
 Corel. 5 .) r ""* : R * :: (26000 / the Velocity (per 
 
 Second) at the Surface, to 26000 x \X -, the Ve- 
 
 , 
 
 J\. 
 
 locity
 
 in Centripetal For ces. 241 
 
 a 
 
 locity in the Circle whofe Radius is R. And f* 
 
 :: (575 ' the periodic Time at the Surface : to 5075 X 
 
 R> 
 
 , the Time of Revolution in the Circle R. 
 
 r 3 
 
 Which, if R be aflumed equal to (6or) the Diftance of 
 
 S. D 
 
 the Moon from the Earth, will give 2360000, or 27 . 3 
 nearly, for the periodic Time at that Diftance. 
 
 In like fort the Ratio of the Forces of Gravitation 
 of the Moon, towards the Sun and Earth, may be com- 
 puted. For, the centrifugal Forces in Circles, being 
 univerfally as the Radii apply'd to the Squares of the 
 
 ,,,. /SiooooooN 
 
 I imes of Revolution, it will be as ( - J the 
 
 Semi-diameter of the Magnus Orbis divided by the Square 
 of one Year (the periodic Time of the Earth and Moon 
 about the Sun) is to (240000x178) the Diftance 'of 
 
 the Moon from the Earth divided by r, the Square 
 
 of the periodic Time of the Moon about the Earth, fo 
 is 1,9 to i nearly ; and fo is the Gravitation of the 
 Moon towards the Sun to her Gravitation towards the 
 Earth. 
 
 Alfo, after the fame Manner, the centrifugal Force of 
 a Body at the Equator, arifing from the Earth's Rota- 
 tion, is derived. For fince it is found above, that 5075 
 Seconds is the Time of Revolution, when the centrifugal 
 Force would become equal to the Gravity, and it ap- 
 pears (by Cafe 2. Carol. 2.) that the Forces, in Circles 
 having the fame Radii, are inverfely as the Squares of 
 
 the periodic Times, we therefore have, as bbiboj" (the 
 
 H M 
 Square of the Number of Seconds in (23 56) one 
 
 whole Rotation of the Earth) to 5075! * (the Square of 
 
 the Number of Seconds above given) fo is the Force of 
 
 R Gravity
 
 242 
 
 The Ufe of FLEXIONS 
 
 Gravity (which we will denote by Unity) to -^-, the 
 
 289 
 
 centrifugal Force of a Body at the Equator arifing from 
 the Earth's Rotation. 
 
 But, to determine, in a more general Manner, the 
 Ratio of the Force of a Body revolving in any given 
 Circle^to its Gravity, we have already given 3.14 fcfr. x 
 
 / 2r 
 
 V* ^ for the Time of Revolution at the Surface of 
 
 the Earth, when the Gravity and centrifugal Force are 
 equal : Therefore, if the Time of Revolution in any 
 Circle whole Radius is a, be denoted by /, it follows, 
 
 from Carol. 2. lajl Prop, that, 
 
 2r ' f 3 
 
 :: the Gravity of the Body : to its centrifugal Force 
 in that Circle ; which, therefore, is as Unity to 
 
 3. 14]* eft, x 
 
 : j or as i to 1.228 x ~r vei 7 near - 
 
 Ul t 
 
 ly : where a denotes the Number of Feet in the Ra- 
 dius of the propofed Circle, and t the Number of Se- 
 conds in one intire Revolution. So that, if the Length 
 of a Sling, by which a Stone is whirled about, be two 
 Feet, and the Time of Revolution ^ of a Second, the 
 Force by which the Stone endeavours to fly off, will 
 be to its Weight as 9.824 to Unity. 
 
 From this general Proportion, the centrifugal Force 
 and periodic Time of a Pendulum defcribing a conical 
 Surface may likewife be deduced. 
 
 For let SR, the Length 
 of the Pendulum, be de- 
 noted by g; the Altitude 
 CS of the Cone, by c ; the 
 Semi-diameter CR of the 
 Bafebyfl; and the Time 
 of Revolution by t : Then, 
 the Force of Gravity being 
 re-
 
 in Centripetal Forces. 243 
 
 rcprefented by Unity* the Force with which the re- 
 volving iiody at R, the End of the Pendulum, tends 
 to recede from the Center C, will be defined by 
 
 **'' ~ M , as has been already {hewn. There- 
 
 , 
 
 fore, becaufe the Body is retained in the Circle RR by 
 the Action of three different Powers, i. e. the centri- 
 
 fugal Force ^ - 7 - / in the Direction CR, 
 
 the Force of Gravity ( i) in a Direction parallel to SC, 
 and the Force of the Thread or Wire RS, compounded 
 of the former two ; it follows, from the Principles of 
 Mechanics, that as SC (c) to CR (g) 9 fo is the Weight 
 of the Body at R, to the Force with which it acts upon 
 
 TM_ t*r- T> c 
 
 the Thread or Wire RS ; and as i 
 
 :: CS (c) : CR (a} : Whence d? = 3. t4&c.\* x 
 
 and / = 3. 14 &c- * \/T =i,io8/f nearly. Be- 
 
 d 
 
 caufe <&% or its Equal 3. 14^ r x 2c, exprefles the 
 Space a heavy Body will dcfcend, by its own Gravity, 
 in the Time / *, and fince i 1 : 3. 14 &V.]* 2c : 
 
 3. 14 &c\ l X zc ( dt 1 } it therefore appears that, as 
 the Square of the Diameter of any Circle, is to the 
 Square of its Periphery, fo is twice the perpendicular 
 Altitude of the Cone, to the Diftance a heavy Body will 
 freely defcend in the Time of one whole Gyration of 
 the Pendulum, let the Bafe of the Cone and the Length 
 of the Pendulum be what they wiU. 
 
 PROPOSITION VII. 
 
 218. To determine thi Ratio of the Velocities of Sadies dt- 
 fcending, or amending, in Right-lines^ when accelerated, or 
 ) by Forces^ varying according to a given Law. 
 
 Suppofe a Body to move in the Right-line CH, and 
 let the Force whereby it is urged towards C, or H, 
 
 R 2 b?
 
 244 the Vfe gf FLUXIONS 
 
 be as any variable Quantity F ; Moreover, let the Ve- 
 locity of the . Body be reprefented by v ; putting its 
 Diftance CD, from the Point C :=:.*, and D</=*. 
 
 ' 11 Then, fince the Time wherein the Space 
 Dd (x) would be uniformly defcribed, with 
 
 A * 
 
 the Velocity at D, is known to be as , the 
 
 "D Velocity that would be uniformly generated, or 
 deftroyed, in that Time by the Force F (be- 
 ing as the Time and Force conjunct! y) will 
 
 FJC 
 
 confequently be as : Which therefore muft 
 v 
 
 be equal to, Hh v, the uniform Increafe or 
 Decreafe of Celerity in that Time ; and confequently 
 -\-vv=: Fx. From whence, when the Value of F 
 is given in Terms of x, or v 9 the Value of v will like- 
 wife be known. >. E. L 
 
 COROLLARY I. 
 
 219. Hence, the Law of the Velocity being given, 
 that of the Force is deduced : For, fince Fx = + vv 9 
 
 Vu 
 
 it is evident that F =: -f- -r-. 
 
 x 
 
 COROLLARY II. 
 
 220. Hence, alfo, the Ratio of the Velocity at D 
 to that whereby a Body might revolve in the Periphery 
 of a Circle about the Center C, at the Diftance of CD, 
 will be known : For, if this laft Velocity be denoted by 
 
 w 1 
 w, the Value of F will be rightly exprefTed by * : 
 
 VJ^x 
 
 Whence, by Subflitution, we have + vv = - , or
 
 in Centripetal Forces. 245 
 
 'f X *V X 
 
 f V X w a x : Whence w* : v* : -f -- : -*-, 
 v JT ~ v * ' 
 
 and confequently w; ; v :: v/^ -I -- V^ 
 
 ~ v ' x 
 
 as well as above, the Sign of v mufl be taken -f or 
 according as the Body is urged from, or towards the 
 Center C. 
 
 PROPOSITION VIII. 
 
 2 2 if Suppojing a Body, let go from a given Point A with 
 a given Celerity (c) along a Right-line CH, to be 
 urged, either way, in that Line, by a Force varying as 
 any Power (n) of tbe Dljlance from a given Point C ; 
 to find^ not only, tbe Relation of the Velocities* and Spaces 
 gone over, but a If o tbe Ti?nes of djcent and Defcent. 
 
 The Conduction of the preceding Problem being re- 
 tained, F will here be expounded by x" t and we {hall 
 therefore have -Jrvv (~Fx) =xx; and confequently, 
 
 W *"T~' 
 
 by taking the Fluent thereof, + =- - ; but to 
 
 2 n+ 1 
 
 correct the Fluent thus found, let x be taken = CA 
 (which we will call a) then v being r= r, the Fluent in 
 
 f* a*** 
 
 ^hat Circumftance will become -| -- ~ - _ There- 
 
 " 2 n+i ' 
 
 v 
 
 fore the Fluent duly corrected is + 4. ~ 
 
 2 2 
 
 v* 
 " I 
 
 / X 
 
 = V c * + 
 
 come out = c * + - _ . W here the 
 
 +i 
 
 Signs of v and x" X muft be alike, when both Q_ ; an- 
 increaie, or decreafe, at the farr.e time ; thit is, 
 H- 3 when,
 
 246 7&? Ufe 0f FLUXIONS 
 
 Art. azo. when the Force, from C, is a repulfive one * ; but, un- 
 like, when one increafes while the other decreafes, or 
 the Force, tending to C, is an attractive one. In the for- 
 
 mer Cafe we therefore have v 
 
 y n -H 
 _ S+ 2X ~ 
 
 fl T A " I * 
 
 .y 13 2-V 
 
 and, in the latter, v = ^ c ~ H ^~+~i 
 
 The Value of v being thus obtained, let the required 
 Time of moving over the Space AD be now denoted 
 
 by T j then, fince T is univerfally = ^, we have T 
 
 or f = 
 
 -M 
 === according to the two forefaid 
 
 2* 
 
 n + i 
 
 Cafes refpeclively : From whence, by finding the Fluent, 
 the Time itfelf will be known. %. E. /. 
 
 COROLLARY. 
 
 222. If the Body proceeds from Reft at A, c will be 
 
 j_ 
 
 s o, and we (hall have f = r* + ^ X *--.. or 
 
 T = 
 
 V 2.V 2<2 
 
 t 
 I -f ]* X * 
 
 ^/ fl+I +! 
 
 V 2 2X 
 
 SCHOLIUM. 
 
 223. Although, the Fluents of the Expreflions given 
 above cannot be exhibited, in a general Manner, nei- 
 ther, in finite Terms, nor by means of circular Arcs 
 and Logarithms ; yet, in lome of the moft ufeful 
 
 Cafes
 
 in Centripetal Forces. 
 
 Cafes that occur in Nature, they may be obtained with 
 great Facility. 
 
 247 
 
 Thus, if in 
 
 i + l** 
 
 n-J-i 
 
 (exprefling the Flux- 
 
 ion of the Time of Defcent along AD) n be expounded 
 by i, o, 2, and 3 fuccefTiveJy, the Fluxion itfelf 
 
 X X 
 
 will become equal to ^===, -===, 
 
 ^i**** and -r^=^ refpeaively : Whence, if 
 
 **-** **~* 
 
 ARF be a Quadrant of a Circle whofe Center is C, and 
 J\SC a Semi-circle whofe Diameter is AC, and DSR 
 be perpendicular to AC ; then it will appear, 
 
 i. That, when n I, 
 
 the Velocity (VV S) 
 at D will be repre- 
 fented by DR, and the 
 
 AR 
 Fluent fought by 
 
 Art. 
 
 142. 
 
 2. That, when n = o, and T . =-, the 
 
 r 2a 2^ 
 
 Velocity at D, and the Time of Defcent thro' AD, will 
 
 each be defined by 
 
 3. That, when n 2, and T = . ' ^ 
 
 V ax xx 
 
 .he Vcloc ity 
 
 be as 
 
 and the Time of Defcent thro' AD, as V| ACxAS+DS. 
 
 R 4 4".
 
 248 ^he Ufe of FLUXIONS 
 
 axx 
 
 4. And that, when n 3. and T . . 
 
 V a 1 x 1 
 
 DR 
 
 the Velocity will be as Vvjj x '/ -r\> ar| d the Time as 
 
 ACxDR. 
 
 Hence the Time of the whole Defcent thro' the Ra- 
 dius AC, appears to be as -r^, V/2AC, V . AC X AF, 
 
 or AC 1 . But the Time of one whole Revolution in 
 
 4AF 
 Art.aif. the Periphery ARF &V. was found to be as - -^r^* ; 
 
 AC~ 
 
 AAF 4.AF 
 
 which in the four Cafes above fpecified ia ^TT, - , 
 
 A ^ VAC 
 
 4AF x VAC", and 4AF x AC : Therefore, if the Time 
 of moving over the Quadrant AF be denoted by J-jj>, it 
 follows that the Time of Defcent thro' the Radius AC, 
 
 AC" y/~T . 
 
 will be truly defined by & , jp x J G) v y i. 
 
 } ^ ^ AF J ** a 
 
 AC 
 
 or ^ x -r-p according to the forefaid Cafes refpe&ively, 
 
 LEMMA. 
 
 224. The Areas which a r rushing Body defer ibes t ly 
 Rays drawn to the Center of Force, are proport'tsnal 
 to the Times of their Defcriptisn. 
 
 For, let a Body, 
 in any given Time, 
 defcribe the Right- 
 line AB, with an 
 uninterrupted uni- 
 form Motion ; bu,t 
 k c upon its Arrival at 
 B let it be impelled 
 
 towards the Center S, fo that, inftcad of proceeding 
 
 along
 
 in Centripetal Forces. 
 
 along ABC, it may, after the Impulfe, defcribe the 
 Right-line Be. 
 
 Becaufe the Force, a&ing in the Line SB, can nei- 
 ther add to, nor take from, the Celerity which the Body 
 has in a Direction perpendicular to that Line, the Di- 
 ftance of the Body from the faid Line, at the end of a 
 given Time, will therefore be the very fame as if no 
 Force had a&ed ; and confequently the Area B^S equal 
 to the Area BCS, which would have been defcribed in 
 the fame time, had the Body proceeded uniformly along; 
 BC ; bccaufe Triangles, having the fame Bafe and Al- 
 titude, are equal. 
 
 Therefore feeing no Impulfe, however great, can af- 
 fect the Quantity of the Area defcribed about the Center 
 S, in a given Time, and becaufe the Areas ASB, BSC, 
 defcribed about that Point, when no Force ad~ts, are as 
 the Bafes AB, BC, or the Times of their Defcription, 
 the Proportion is manifeft. 
 
 COROLLARY. 
 
 225. Hence the Ve- 
 locity of a revolving 
 Body, at any Poinc Q_ 
 prR ? is inverftly as the 
 Perpendicular SP or 
 ST, falling from the 
 Center of Force upon 
 the Tangent a: that 
 Point. 
 
 For, let two other 
 Bodies m and n be fup- 
 pofed to move uniform- 
 ly from Q^and R, along 
 the Tangents QP and 
 RT, with Velocities re- 
 
 fpe&ively equal to thofe of the revolving Body at Q_and 
 R ; then the Diftances Q;?/ and R, gone over in the 
 fame Time, will be to each other as thofe Velocities ; 
 and the Areas QSm and RSi will be equal, being equal 
 
 to 
 
 249 
 
 T
 
 250 
 
 to thofe defcribed by the revolving Body in the fame 
 s. time * : Whence Qm x SP being r= R x &T, it follows 
 
 that Qm : R* : : ST : SP : : <~ ~. 
 
 PROPOSITION IX. 
 
 To determine the Law of the centripetal Force t 
 tending to a given Point C, whereby a Body may de~ 
 fcribe a given Curve AQH. 
 
 O 
 
 Let QP be a Tangent to the Curve at any Point Q_; 
 upon which, from the Center C, let fall the Perpendi- 
 cular CP ; putCQ_=j, CP=; and let the Velocity 
 of the Projectile at Q_be denoted by v. 
 
 Therefore, fince v 1 is always as -j (by the Carol, to 
 
 Linana) it is evident, by taking the Fluxions of both 
 
 ii 
 
 Quantities, that vv will alfo be as j- : But the cen- 
 tripetal Force, whether the Body moves in a Right-line 
 
 or
 
 in Centripetal Forces. 2 5 1 
 
 or a Curve, is always as r (by Art. 219. and 206.) 
 
 Therefore the centripetal Force is likewife as -j^'J 
 
 The fame otberivife. 
 
 227. Let the Ray of Curvature QO be denoted by 
 R: Then, becaufe the centripetal Forces in Circles are 
 known to be as the Squares of the Velocities directly and 
 the Radii inverfely *, it follows that the Force, tending * Art.ai*. 
 to the Point O, whereby the Body might be retained in 
 its Orbit at Q_, or in the Circle whofe Radius is QO, 
 
 will be defined by x -fc : Whence (by the Refolution 
 of Forces) it will be CP (u) : CQ^(s) :: ^ ( the 
 
 Force in the Direction QO) : -j-pj, the Force in the 
 
 
 Direaion QC : Which, becaufe R = ~ f will alfo t Art. 73 . 
 
 
 be exprefled by ~ . %. E. I. 
 
 Another Way. 
 
 228. Let nq be the indefinitely fmall Part of the 
 Right-line Cy, intercepted by the Curve and the Tan- 
 gent Qj?, expreffing the Effect of the centripetal Force 
 in the Time of defcribing the Area QCn. Now thefe 
 Effects, or the Diftances defcended by means of Forces 
 uniformly continued, are known to be in the duplicate 
 Ratio of the Times , or of the Areas denoting thofe * 
 Times : Therefore, the centripetal Force at Q_, or the r Art z 
 Diftance defcended by means thereof in a given Time, 
 will be as nq applied to the fecond Power <)f the Area 
 
 , or as <ju>a X Q ^. This Expreffion is the fame 
 
 with
 
 be Ufe 
 
 with that given by Sir Ifasc Nervton, in his Principle!* 
 Book i. Prop. 6. But, to adapt it to a fluxional Cal- 
 culus ; let QE be an Ordinate to the principal Axis AG ; 
 and let (as ufual) AE =: x^ EQ_ = ^, AQ_= z, Ee (or 
 Qf) = x, Q_? K. ; fuppofmg eq (parallel to EQJ to 
 interfeft the Curve and the Tangent in m and q. 
 
 Since Q_y is conceived indefinitely fmall (or in its 
 nafcent State) the Triangle nmq may be taken as recli- 
 t Art. 136. ]j nea i * ; a lf o the Angle = CQP and the Angle m 
 Whence, it will be (by Trigonometry) as 
 
 CQP (n) ' 5. Qjp (m} ::mq:nq; that is, as -. ~? 
 
 :: mq : nq ~rp^7V) - : Which fubftituted above 
 
 CO x Qt x mq . 
 
 gives - - - z for tne Meaiure of the centripetal 
 CP 3 x Q_q 
 
 Force at Q^: But mq (fuppofing x to fio\y uniformly) i,s 
 known to be as j * Therefore the Force at Q_, is as 
 
 CQ X Q/ X jr _ sxy 
 
 CP 3 x Qy* ' ' r lts qu ~v& ' ' e 
 vifor (u 3 z 3 ) is as the Cube of (QCj) the Fluxion of 
 the Area AQC. 
 
 The very fame Theorem may likewife be deduced 
 from that given by our fecond Method : For, fince (R) 
 
 x 3 
 
 f Art.68. the Ray of Curvature at QJs univerfally * =; n.rthe 
 x y 
 
 Value of-j- (there found) will here, by Subftitution, 
 
 sxy 
 
 become =. -yr,-' 
 u ' z 
 
 This Exprefllon, tho' in appearance lefs fimplc than 
 -^- fuft found, is, for the general part, more commo.,- 
 
 W 3 J 
 
 dious in Practice. 
 
 Co-
 
 in Centripetal Forces. 253 
 
 COROLLARY I. 
 
 219. If the Point C be fo remote that all Right-lines 
 drawn from thence to the Curve may be confider'd as 
 parallel to each other, the Force will then (making Qr 
 
 ^~"SXV 
 
 perpendicular to C?) be as '' |3 , or barely as 
 
 C^xQH 
 
 ^j fince s (C?) in this Cafe may be rejected. 
 
 From this Expreflion, which is general, in all Cafes 
 where the Force ab in the Direction of parallel Lines, 
 it appears that the Force, which always acting in the 
 Direction of the Ordinate QE, would retain the Body 
 
 in its Orbit, is every where as rr j becaufe QC here 
 coincides with QE, and Qr becomes x. 
 COROLLARY II. 
 
 230. Becaufe the Force, tending to the Point C, is 
 
 r~*o ^^ 
 
 univerfally as cp* x QO ( 0t !sR ) the Force to an y 
 other Point c t will, by the fame Argument, be as 
 
 =TJ . Hence the Forces, to different Centers 
 fp\ xQO 
 
 C and c (about which equal Areas are defcribcd in the 
 
 CP 3 
 fame time) are to each other in the Ratio of rrr^ to 
 
 ~~13 
 
 CjJinverfely. 
 
 COROLLARY III. 
 
 231. Moreover, the Ratio of the Velocity at Q to 
 the Velocity whereby the Body might revolve in a Circle 
 about the Center at C, at theDiflance CO, is eafily de- 
 duced from henCe : For, fiace the Celerity at Q> that 
 
 whereby
 
 254 *be Ufe of FLUXIONS 
 
 whereby the Body might revolve in a Circle about the 
 Center O, and the Forces tending to the Centers O and 
 C are to each other as u (CP) and s (CQJ; it there- 
 fore follows, if the Ratio fought be aflumed as v to w, 
 
 v~ if* 
 that rr^r : :yp :: u : s (by Art. 212.) Whence alfo 
 
 v* : U? :: xQO (uR) : s x QC (V) and confequently 
 
 (becaufe R = J. 
 
 U' 
 
 The fame Proportion may alfo be derived from 
 2. Prop. 7. For it is' there proved that v : iu :: 
 
 V' -i 
 
 i : V ~ an( ^ ^ a PP ea rs from above, that 
 
 = : Whence the whole is manifcft. 
 v " u 
 
 If OL be made perpendicular to QC, QL will be 
 CPxQON uR ,QL uR 
 - = , W*;= - J and there- 
 
 
 
 fore v : w :: QL a : CQj : Which is another Pro- 
 portion of the propofed Celerities. 
 
 COROLLARY IV. 
 
 232. Laftly, the Law of centripetal Force being gi- 
 ven, the Nature of the Trajectory AQ^may from hence 
 be found j for fmce the Force (F) is univerfally defined 
 
 by r> it is evident that H-i will be =1 the Fluent 
 u 3 s 2.u 
 
 of Fs t which, when F is given in Terms of J, will 
 become known ; and then, the Relation between u and 
 i being given, the Curve itfelf is knowru 
 
 E X-
 
 in Centripetal Forces. 
 
 EXAMPLE I. 
 
 233. Let the given Curve AQH be the logarithmic 
 Spiral, and C the Center thereof: Then u (CP) being 
 
 2 55 
 
 = i'., we have t ( = X 
 
 x - i" Unit y- Hence 
 
 it appears that the Force is in- 
 verfely as the Cube of the Di- 
 ftance; and the Velocity, every 
 where, equal to that whereby the 
 Body might revolve in a Circle at 
 the fame Diftance. 
 
 EXAMPLE II. 
 
 134. Let it be required to fnd the Law of the centripetal 
 Force, whereby a Body, tending to the Focus C, is made 
 to revolve in the Periphery of an Ellipfes AQDB. 
 
 From the other 
 Focus F draw FK 
 parallel to CP meet- 
 ing the Tangent PQ 
 (at Right-angles) in 
 K, join F, Qj put- 
 ting the tranfverfe 
 Axis AB = a, the 
 
 Semi-conjugate OD = * , and the Parameter ( 
 
 \ a f 
 
 -p: Then, CQ, and CP being denoted as above* * Art. 231. 
 we have FQ.(=:AB CQ) =a ,; whence, by rea- 
 fon of the fimilar Triangles CQP and FQK, it will be 
 10
 
 2c6 tfbe Ufe of FLUXIONS 
 
 ~W */ y 
 
 a ~~ S * * 
 
 But FK x CP is 
 
 j 
 
 OD X (by the Nature of the Curve.) Hence wi get 
 
 asXit 1 r I 4* 4 
 
 ' - = 4 ; and confequently -77=- TT-; 
 
 J 7 zr Wr 
 
 whereof the Fluxion being -" A- S . , we obtain 
 
 " 
 
 Y/ - - Hence, it appears that the centripetal 
 
 Force is, in this Cafe, as the Square of theDiftance in- 
 verfely ; and the Velocity at Q_ is to that whereby the 
 Body might defcribe a Circle at the Diftance CQ^, every 
 
 i_ jr^ 
 
 where, in the Ratio of FQ Z to AO\ 
 
 If the Curve had been an Hyperbola; then - X 
 
 <r (inftead of - x it 1 } would have been = | b* ; 
 
 * 
 
 and fo --V- x _ __ ? tne ver y f arae as before, 
 
 But, had it been a Parabola, the Equation would have 
 
 been ^ x * = i*% or^-(rr^) = ;- p-, and 
 
 A 
 
 the Force flilL as r . But, the Meafwe of the Ve- 
 ps 
 
 - = \X 2 ^~ 2; ) 
 
 in this Cafe becoming 
 
 barely = V^T, it follows that the Velocity in a Parabola 
 is to that whereby the Body might defcribe a Circle at the 
 fame Diftance from the Center, in the conftant Ratio of 
 
 V2 to Unity. 
 
 EX-
 
 in Centripetal Forces. 
 
 257 
 
 EXAMPLE III. 
 
 235. Let it be required to find the Law of the centripetal 
 Force, by which a Body^ tending to any given Point C, 
 in the Axis, is made to defcribe a conic Seflion AQH. 
 
 Put the femi-tranfverfe Axis (OAJ a, the femi- 
 conjugate =. , and the given Diftance of the Point 
 C from the Vertex A c: Put alfo the Abfcifla AE, 
 =r JT, the Ordinate EQ=iy, and CQ_= s (as before). 
 
 The Area of the Triangle ECQ, being ( = ^ECxEQJ 
 
 cy xy , , . . , r cy x'y yx 
 
 - - -, its Fluxion is therefore = - ; 
 
 2 2 
 
 which added to yx, the Fluxion of the Area AEQ_, 
 gives - - - for the Fluxion of the whole Area 
 
 ACQ^defcribed about the Center of Force. Whence 
 (by Art. 228.) the required centripetal Force at Q_will 
 
 be as 
 
 c 
 
 Which Expreflion is general, 
 
 let the Curve be of what Kind it will. But in the 
 Cafe above, y being V iax + x 1 i we have j =. 
 
 bx x a -j- x 
 
 * -> y 
 
 alx~ 
 
 a Viax -\- x 1 
 
 lax 
 
 [, and rj- + yx xj =
 
 253 The Ufe of FLUXIONS 
 
 _*'-"~ r "" ... and therefore, by fubftituting theftf 
 a V 2ax + x* 
 
 sxj of' s 
 
 Values, we get ; ; =^ . 3 
 
 Cj + yx Ay! * x ca + tf* ih cx t 
 
 A 
 
 Which, becaufe TT is conftant, will alfo be as 
 
 From whence it follows, 
 
 If c be = ^p rf, or the Center of Force be In 
 the Center of the Section, the Force itfelf will be barely 
 as ( + *) the Diftance. 
 
 2. If it be in the Focus, then ac -\- ax -f- r* be- 
 coming = CQ_x a, the Force will be inverfely as the 
 Square of the Diftance. 
 
 3*. If the given Point be in the Vertex A, the Force 
 
 will be as T : Which therefore in the Circle ( where *= 
 x* 
 
 i* \ i 
 
 ) will be as : , or the fifth Power of the Diftance 
 laj i 5 
 
 reciprocally. 
 
 4. Lafily, if the Point C be at an indefinite Diftance 
 from the Vertex, or the Force be fuppofed to acl in 
 the Direction of Lines parallel to the Axis AO j then 
 the Force will be as the Cube of OE inverfely. 
 
 PROPOSITION X. 
 
 236. To determine the Ratio of the Velocities of Bodies 
 revolving in different Orbits^ abcut the fame y or dif- 
 ferent^ Centers ; the Orbits themfefaes t and the Forces 
 whereby they are dcfcribed y being given. 
 
 Let AQH be any Orbit, defcribed about the Center 
 of Force C, and let the Force itfelf at the principal Ver- 
 tex A be denoted by F; alfo let r ftand for the Semi- 
 parameter, or the Ray of Curvature at the Vertex, and 
 
 let
 
 in Centripetal Forces. 
 let CP be perpendicular to the Tangent QP. 
 
 Q 
 
 259 
 
 o 
 
 or 
 
 Then, the Celerity at A being, always, as 
 (by Art. 212.) we have CP : CA :: V7F (the Ve- 
 
 C A x ^ i'F 
 locity at A) to pp , the Velocity at Q_(y Art. 
 
 225.) Which anfwers in all Cafes, let the Values of AC, 
 t and F be what they will. Q E> ! 
 
 COROLLARY I. 
 
 237. If the centripetal Force be as the Square of the 
 Diftance inverfely, or F be expounded by j^z > the 
 
 AC 
 
 Velocity at Q_ will become p-p X 
 
 Qp~ : Whence the Velocities, in different Orbits, 
 
 about the fame Center, are in the fubduplicate Ratio of 
 the Parameters, and the inverfe Ratio of the Perpen- 
 diculars from the Center of Force to the Tangents, 
 conjunctly. 
 
 COROLLARY II. 
 
 238. Hence, if the Celerity at Q^be denoted by Q^, 
 
 and Cy be drawn j then, Q_y being as ^rp, it follows 
 
 that V7 is as CP x Q^, or as the Triangle QCq: There- 
 S 2 fore
 
 a6o he Vfe of FLUXIONS 
 
 fore the Areas defcribed about a common Center oi 
 Force in a given Time, are in the fubduplicate Ratio of 
 the Parameters. 
 
 COROLLARY III. 
 
 239. Laftly, fince the Area of the Curve AQHB &c. 
 
 * Art. *34. whenanEllipfe*, is known to be as (AO X OD) AO X 
 
 yVx AO (fuppofmg O to be the Center) if the fame 
 
 be apply'd to Vr-> exprefling the Area defcribed in a 
 
 given Part of Time (by the laft Corel.) we fliall thence 
 
 have AO x t/AO, or AO* for the Meafure of the 
 Time of one whole Revolution. From whence it ap- 
 pears, that the periodic Times, let the Species of the 
 Ellipfes be what they will, are in the fefquiplicate Ratio 
 of their principal Axes. 
 
 PROPOSITION XI. 
 
 240. The centripetal Force, lending to a given Point C, 
 being as the Square of the Dijlances reciprocally, and 
 the Dire ft ion and Velocity of a Body at any Point Q_ 
 being given ; to determine the Path in which the Body 
 moves, and the periodic Time> in cafe it returns. 
 
 It is evident from Art. 234. and 235. that the Tra- 
 jectory AQB is a conic Section j whereof the Point C is 
 one of the Foci, 
 
 Let
 
 in Centripetal Forces. 261 
 
 Let F be the other Focus, and upon the Tangent 
 PQK let fall the Perpendiculars CP and FK, and let 
 CQ_ and FQ_ be drawn : Alfo put the femi-tranfverfe 
 Axis AO = <7, the given focal Diftance CQ_= d t and 
 the Sine of the Angle of Direction CQP (to the Ra- 
 dius i) ~m ; and let the given Velocity at Q_ be to 
 that whereby the Body might revolve in a Circle about 
 the Center C, at that Diftance, in any given Ratio of n 
 
 to Unity: Then it will be n : i :: FQj : AO* (by 
 Art. 234.) therefore * : i 2 : FQ_ (2* d] : AO (a) . 
 
 whence AO (a) is given = -__ v Moreover, fmce 
 CP = m x CQ_, and FK ~m x FQ_, we have OD 1 ( = 
 
 whence the fe- 
 
 2 n 
 mi-conjugate Axis (OD) is given likewife. 
 
 Laftly, it will be (by Art. 239.) as CT* : AO* :: 
 (P) the periodic Time in any given Circle, whofe Radius 
 
 f * 
 
 is CT, tof 3 x P) the required Time of one Revo- 
 
 VCT* 
 
 lution when theOrbitisanEllipfis; that is, when**islefs 
 
 than 2 : For, if * be = 2, the Curve fas its Axis 
 
 2 n* 
 
 becomes infinite) will degenerate to a Parabola; and, if 
 n 2 be greater than 2, the Axis being negative, it is then 
 an Hyperbola ; whofe two principal Diameters are equal 
 
 id imnd 
 to and -, 
 
 COROLLARY. 
 
 241. Seeing neither the Value of AO, nor that of 
 
 the periodic Time, is affeled with ;w, it follows that 
 
 the principal Axis, and the periodic Time, will remain 
 
 S 3 in-,
 
 262 
 
 'The Ufe g/* FLUXIONS 
 
 invariable, if the Velocity at Q_be the fame, let the 
 Direction at that Point be what it will. 
 
 The fame Solution may likewife be brought out, from 
 Art. 238. by firft finding the principal Parameter : For, 
 it is evident that the Area defcribed by the Body about 
 the Center C, in any given Time, is to the Area de- 
 fcribed, in the fame Time, by another Body revolving 
 in a Circle at the Diftance CQ_, as mn to Unity : Hence, 
 Art. 238. it will be i 1 : rrfn* :: d : (m*a?d) the Semi-parameter* : 
 From which (proceeding as above) we get ff X tf/Vtf 
 (rzOD 2 ) = m* X 2.Gcl>d*', and confcqucndy a =: 
 
 :, the fame as before. 
 
 PROPOSITION XII. 
 
 242. The centripetal Force being as any Power (n) of 
 the Dijlance^ and the DlreElion and Velocity of a Body 
 at any Point A being given y to determine the Orbit or 
 Trajectory. 
 
 From the Cen- 
 ter of Force C, 
 to any Point B in 
 the required Tra- 
 jeaory ABD, let 
 CB be drawn ; 
 join C, A, and 
 let Ab be the gi- 
 ven Direction of 
 the Body at the 
 Point A, and 
 Cb perpendicular 
 thereto ; alfo let 
 the Velocity at 
 A be to that 
 whereby a Body 
 might defcribe a 
 Circle AEF, about the Center C, in any given Ratio 
 of p to Unity j putting CA=<?, and CB=*; Then 
 
 be
 
 in Cenfr'vef a/ Forces. 263 
 
 fceeaufe this laft Velocity (the centripetal Force being as 
 
 -H 
 
 *" (or/) is rightly defined by a * *. the Velocity* Art. 214. 
 of the Body at A will be truly expreffed by 
 5+1 
 
 fa * . 
 
 Moreover, it is proved in Art. 221. and 206. that if the 
 Celerity, at any given Diftance a from the Center, be 
 denoted by <:, the Celerity at any other Diftance x will 
 
 S 
 
 be truly reprefented by v' .* 
 
 _L 
 
 Whence, /> * being fubftituted for c, we have 
 
 
 >*+- x a"" 1 - for the Celerity at B. 
 
 + + i 
 
 But nowj to determine the Curve itfelf from hence, 
 let BP be a Tangent to it at B, and CP perpendicular 
 to BP ; alfo let CB, produced, meet the Periphery of 
 the Circle in E ; putting the Arch AE=rz, the forelaid 
 Velocity at B (to fhorten the Operation) =v, and 
 Cbb : Then it will be (by Art* 225.) v : c (the Ve- 
 
 locity at A) :: b : CP= - Whence BP ( =} 
 
 Moreover (by Art. 35.) we have, as CB : CP :: v : 
 CP 
 
 ~ x v l ^ e Velocit of the Bod y at B in a ^~ 
 
 redion perpendicular to CE; and confequently, as CB : 
 
 CP CPxCE 
 
 CE :: - X v (the faid Velocity) to - B r- ><^ the 
 
 angular Velocity of the Point E (revolving with the 
 
 Body.) By the fame Article* the Velocity at B in the 
 
 S 4 D.i-
 
 264 T/je U/e of FLUXIONS 
 
 BP 
 
 Direction CBE will be ^ x v : Therefore, the Ve- 
 locity of E being to the Velocity of B, in the faid Di- 
 
 CPxCE BP 
 rection, as jp to ^r, the Fluxions of AE (z) 
 
 and CB (x) muft confequently be in that Ratio; that is, 
 CPxCE BP CPxCE 
 
 lc a vx abac 
 
 X X 
 
 x Vx'v* b'c" ' */* V b*S 
 
 Which Equation is general, let the 
 
 Law of the centripetal Force be what it will : But in 
 
 2 ""T* 1 
 
 the Cafe above propofed, v' 2 ' being =r / a H 
 
 r.+i 
 2.V n l~i 
 
 - . and f = p*a : it becomes 
 
 ; whole 
 
 Fluent is the Meafure of the angular Motion ; from 
 which, when found, the Orbit may be conftru&ed : 
 Becaufe, when AE, or the Angle ACE is given, as 
 well as CB, the Pofition of the Point B is alfo given. 
 But this Value of z is indeed too complex to admit of 
 a Fluent in algebraic Terms, or even by circular Arcs 
 and Logarithms, except in certain particular Cafes ; 
 as when the Exponent n is equal to i, 2, 3, or 
 5 ; befides Ibme others wherein the Values of p and 
 n are related in a particular Manner. ^ E. /. 
 
 Co-
 
 in Centripetal Forces. 265 
 
 COROLLARY I. 
 
 243. If the given Velocity at A be fuch that />* -f 
 
 - = o, or/> = \/ 2 (which is always poffible 
 
 when the Value of + i is negative) our Equation will 
 
 become X a P* __ . . Which, by put- 
 
 / 
 
 ting H+ 3=777, CJV. is reduced to ir 
 
 Whereof the Fluent will be found (by the fecond Part 
 of this Work (equal to + multiply'd by the Dif- 
 ference of the two circular Arcs, whofe Secants are 
 
 X & 
 
 and - to the Radius Unity/ From this Va- 
 
 4? 
 
 lue of the Arch AE thePofition of the Point B, in the 
 
 Orbit, is given. 
 
 But if the Angle of Direction CA be a right one, 
 
 the Fluent will become barely n + : ; X Arch whofe 
 
 t 
 
 "X 
 
 Secant is ^-7- (becaufe then I a, and the Arch whofe 
 
 Secant is -7-, =: o) which therefore when x a becomes 
 
 in-
 
 266 Ike life 0f FLUXIONS 
 
 infinite, will be truly defined by -f- x whole PerU 
 
 - 2m 
 
 phery AF, &V. Whence ft is evident that the Body 
 muft either fly intirely off, or' fall to the Center C, in 
 
 a Number of Revolutions exprefled by -} ; accord- 
 
 2m 
 
 ing as the Value of m is pofitive or negative. 
 
 This, if n 2, and m = I, the Body will fly 
 intirely off in half a Revolution: And, if n = 4, 
 and m = I, it will fail to the Center in half a Re- 
 volution. 
 
 COROLLARY II. 
 
 244. Moreover, tho' the Fluent expreffing the Angle 
 at the Center cannot be exhibited in a general Manner 
 yet there are certain Cafes of the Exponent (n) where 
 its refpective Values may be derived from each other. 
 
 For let (as above) tf+3 be put ;/:, and (to 
 fhorten the Operation) let CA (a] be taken as Unity : 
 Then onr Equation will be transformed to i 
 
 '\/ I +-. 
 
 Make 
 
 x ,*_*_ ^ 
 
 y = X * , and it will be farther transformed to x = 
 2 by 
 
 m A 
 
 y 
 
 Put r 
 
 V 1 + 
 
 4 
 ' 
 
 2 4 2 ^ 
 
 ^Z./, 1 ^> w 2 ./> z 
 
 and it will become s zr x 
 
 
 
 ft 
 
 : T nfttrr 
 
 
 ' ry* 
 
 juaniyj 
 
 r 2-r 
 2 
 
 ^ + i - - a x y 
 let
 
 in Centripetal Forces. 267 
 
 2 2/> \ 
 
 P , or a* = - =- J 
 r 2.* r a xr 2/ 
 
 and then we fliall 
 
 , . 2 by 
 
 pave * v 
 m 
 
 Which Expreffion (excepting the general Multiplicator 
 2 \ 
 J being exactly of the fame Form with the firfl 
 
 above given, muft therefore be the Fluxion of the Angle 
 at the Center, when the Index of the Force is r 3 ; 
 for the very fame Reafons that the former appears to OQ 
 the Fluxion thereof when the Index is m 3 (or n.) 
 Hence, if the Fluent of 
 
 by 
 
 = , or the 
 
 Angle at the Center, when the Exponent is r 3 (or 
 
 - -- 3) be denoted by w, the Value 
 T"3 
 ofz, (the Meafure of the faid Angle, when the Ex- 
 
 2M/ 
 
 ponent is m 3 (or n) will be truly defined by 
 From which we collect that, if the Indices of the 
 
 A 
 
 Force, in any two Cafes, be reprefented by n and - 
 
 3, and the refpe&ive Diftances from the Center by 
 + 3 
 
 x and x , then the Angles themfelves correfponding 
 to thofe Diftances will be every where in the conftant 
 Ratio of 2 to + 3. Therefore, when the Orbit can 
 
 be
 
 Ibe Ufe ^FLUXIONS 
 
 be conftru&ed in the one Cafe it alfo may in the other, 
 
 2/> 4 \ 
 
 provided the above Equation a 1 ( = ) 
 
 r p i Kr 2/ 
 
 * _1_ i 4\^" 
 
 . "*~ 3*? , for the Relation of the Celerities at A, 
 
 _ v -- I _ J 
 
 does not become impoflible, as it will, fometirnes, when 
 n is a negative Number. 
 
 COROLLARY III. 
 
 245. If the Body be fuppofed to move in a ver- 
 tical Direction AH ; then, putting the Velocity 
 
 >* -f -^- x a" + l = o, we get at 
 
 (CH) Ip 1 X n-f i + il" X a the Height 
 
 i-f-i 
 _ 
 
 Xa a ( AH) is the Diftance through which itmuft 
 freely defcend to acquire the given Celerity at A : This 
 Diftance, in cafe of an uniform Force, when n = o, 
 will become \ p z a : And, when the Force is in- 
 verfely as the Square of the Diftance, it wiil then be =; 
 
 But, when/* i, or the Velocity at A is juft fyffi- 
 cient to retain a Body in the Circle AEF, AH becomes 
 
 = 1 X a a: Which in the two Cafes 
 
 2 
 
 aforefaid will be equal to i<7, and a refpedively ; but, 
 infinite, when n is = 3. 
 
 Co-
 
 in Centripetal forces. 269 
 
 COROLLARY IV. 
 
 246. When the Value of n -j- I is pofitive, the Ve- 
 locity at the Center, where x = 0, will be barely = 
 
 
 ** -f- -- x a*"*" 1 } but if the Value of + I 
 
 be negative, the Velocity at the Center will be infinite j 
 becaufe, then o * is infinite. 
 
 COROLLARY V. 
 
 247. Moreover, when + I is negative and x in- 
 
 J~~ 
 
 finite, the Velocity alfo becomes \s />*H xa *; 
 
 n + i 
 
 becaufe then x = o. 
 
 Hence, if the centripetal Force be inverfely as fome 
 Power of the Diftance greater than the firft, the Body 
 may afcend, ad infinitum^ and have a Velocity always 
 
 greater than V' p* + - X a * j which is to, 
 
 /~, ~ 
 
 pa a , the given Velocity, at A, as V' P h ^~^ l 
 p. And this will actually be the Cafe when the Value 
 
 2 2 
 
 of p* -{ ; is pofitive, or p* greater than > 
 
 n + I n i 
 
 but not otherwife, the fquare Root of a negative Quan- 
 tity being impoflible. 
 
 Thus, if n = 2, or the Force be inverfely as the 
 Square of the Diftance, and p\ at the fame time, greater 
 
 than 2 ( ' j the Body will not only continue to 
 
 V""" 71 ' - 1 * 
 
 afcend in infinitum, but have a Velocity always greater 
 
 than that defined by Vp* 2, which is its Limit. 
 
 Co-
 
 2 7 'The Ufe ^FLUXIONS 
 
 COROLLARY VI. 
 
 248. Hence the leaft Celerity fufficient to caufc the 
 Body to afcend for ever in a Right-line is given. For, 
 
 putting V p h xa = o, we have /> = 
 
 \/ - - -- Therefore the leaft Celerity by which 
 n i J 
 
 the Body might afcend for ever, is to that whereby it 
 
 may revolve in a Circle AEF, as \s - ? . , to 
 
 i 
 
 Unity. From which it appears that, if the Force be 
 inverfely as any Power of the Diftance greater than the 
 third, a lefs Velocity will caufe a Body to afcend ad tn- 
 frutum than would retain it in a Circle. 
 
 SCHOLIUM. 
 249. From the Ratio of the Velocity 
 
 wherewith the 
 
 _^ x/* 1 2^ ' | 
 +i n+i J 
 
 Body arrives at any Diftance x from the Center, to that 
 
 ( ^ 
 Art 2*4. which it ought to have to revolve in a Circle 
 
 at the fame Diftance, it will not be difficult to determine 
 in what Cafes the Body will be forced to the Center, and 
 in what others it will continue to fly from it ad infinitum. 
 For, firft, if the Angle C Ab be acute, or the Body 
 from A begins to defcend, it will continue to do fo till 
 it actually arrives at the Center, if the former Velocity, 
 during the Defcent, be not fomewhere greater than the 
 
 ,= === fl + ! 
 
 latter, or the Quotient \r p" + ; X rr 
 
 ; J 
 
 X 
 
 greater than Unity j becaufe, if it ever begins to afcend, 
 
 it
 
 in Centripetal Forces. 271 
 
 it muft have an dpfe, as D (where a Right-line drawn 
 from the Center cuts the Orbit at Right-angles) and 
 there the Celerity muft evidently be greater than that 
 fufficient to caufe the Body to revolve in a Circle. 
 Secondly, but if the Quantity 
 
 V' p* -f x - - 2 > in the Accefs of 
 
 the Body towards the Center, increafes fo as to become 
 greater than Unity, or be every where fo ; then the Ve- 
 locity at all inferior Diftances being more than fufficient 
 to retain a Body in a Circle at any fuchr Diftance, the 
 Projectile cannot be forced to the Center. 
 
 After the fame Manner, if the Angle CA be ob- 
 tufe, or the Body from A begins to afcend, it will con- 
 tinue to do fo for ever, when the forefaid Quantity is 
 always greater than Unity, or, which is the fame, when 
 the Body, in its Recefs from the Center, has in every 
 Place through which it pafTeth^a Velocity greater than 
 fufficient to retain it in a Circle at that Diftance. 
 
 It therefore now remains to find in what Laws of the 
 centripetal Force thefe different Cafes obtain : And, firft, 
 k is eafy to perceive that when the Value of n+ i is pofi- 
 
 V 
 
 - tive > that of V f + x - will,- 
 
 by increafing *, become equal to nothing. Therefore 
 the Body cannot afcend forever in this Cafe : Neither 
 can it dcfcend to the Center (except in a Right-line) 
 becaufe the forefaid Quantity, by diminifhing AT, be-< 
 comes greater than Unity (or any other alfignable 
 Magnitude.) 
 
 13 ut, if the Value of n be betwixt I, and 3, 
 the faid general ExprelTion, taking x infinite, will allo 
 
 2 
 
 become infinite, provided the Value of p* -f ; be 
 
 n+ i 
 
 2 \ 
 
 pcfitive (or /> z greater than - J. Therefore the 
 
 Body
 
 272 be Ufe of FLUXIONS 
 
 Body, in this Cafe, may afcend adinfimtum^ but cannot 
 poffibly fall to the Center (except in a Right-lineJ fince, 
 
 \/ ? , the Value of the general Expreflion ? 
 
 n -j- i 
 
 when A- = o, is greater than Unity. 
 
 Laftly, if n be exprefied by any negative Number 
 greater than 3, or the Law of the Force be inverfely 
 as any Power of the Diftance greater than the third, the 
 
 s~ a"^ 1 2 
 
 two extreme Values of V' ** -j x 
 
 + i /+' +i 
 
 will, Jiill, be denoted as in the preceeding Cafe ; but 
 
 here the latter of them, \/ , is lefs than Unity. 
 
 + i . 
 
 Therefore the Body muft, in this Cafe, either afcend for 
 ever, or be forced to the Center ; except in one parti- 
 cular Circumftance, hereafter to be taken notice of. 
 Now, from thefe Obfervations we gather, 
 1. That, when the centripetal Force is as any Power 
 of the Diftance directly, or lefs than the firft Power 
 thereof inverfely, the Orbit will always have an higher 
 and a lower Apfe j beyond which the Body cannot 
 afcend or defcend. 
 
 2. That, when the centripetal Force is inverfely 
 as any Power of the Diftance (whole or broken) be- 
 twixt the firft and third, the Orbit will alfo have two 
 
 Apftdcsi if p be lefs than \/ ; but otherwife, 
 
 n+i 
 
 only one; in which laft Cafe the Body, after it has 
 pafled its Apfe, will continue to recede from the Center 
 in infmtum. 
 
 3. That when the Force is inverfely as any Power 
 greater than the third, the Orbit can, at moft, have but 
 one Apfe', but, in fome Cafes, it will have none at all : 
 And it may be worth while to inquire here, under what 
 Keftridtions of the Velocity (p) this will happen ; fince 
 thereby, befides being able to kngw when the Body will 
 
 be
 
 in Centripetal Forces. 273 
 
 be forced to the Center, fcfr. we fhall fall upon a Cir- 
 cumftance fomewhat remarkable and curious. 
 
 Now it appears, that, if the Body from A begins to 
 defcend, it muft, when it comes to an Apfe at D, have 
 a Velocity there greater than is fufficient to retain it 
 in a Circle ; in which Cafe the general Expreflion 
 
 n 
 
 _!_ (fo often mention'd 
 
 above) muft accordingly be greater than Unity. Let 
 it be therefore made equal to Unity, which is the ut- 
 moft Limit thereof, beyond which the Orbit cannot ad- 
 mit of an Apfe ; putting at the fame time A'-, or its Divifor 
 
 />* + ~~ X Ar 1 p*F -^ , in the 
 
 
 - X x 1 p*F -- 
 
 +i." 1 ' 
 
 general Equation of the Orbit, equal to nothing 
 (it being always fo at the Apfides.) Then, from 
 thefe two Equations, duly order'd, we fhaU get x == 
 
 2-fn-f i.p* 
 
 + 3 
 
 X a, and p* ( = - ) - 
 
 x ~. Now, it is evident, if the 
 P 
 
 Value of p be greater than is given from the laft Equa- 
 tion, the Orbit will have an Apfe ; but if lefs, it can 
 have none. In the former Cafe, the Body will there- 
 fore fly quite off; and in the latter, it will be forced to 
 the Center. But we are now, naturally, led to inquire 
 what will be the Confequence when the Value of p is 
 neither greater nor lefs, but exactly the fame as given from 
 the forefaid Equation : This is the Cafe above hinted at j 
 and here the Body will continue to defcend for ever in a 
 Spiral, yet never fo low as to enter within the Circle 
 
 - == - v t I 
 
 whofe Radius CD is = 2 + "+ t / > | x a. For, if 
 
 * + 3 ' 
 T the
 
 274 *Tbc Ufi of FLUXIONS, &c. 
 
 the contrary were poflible, the Body, at its Arrival to the 
 Circumference of that Circle, would (becaufe of the 
 forefaid Equations) not only have a Direction, but alfo 
 Velocity proper to retain it therein ; which cannot be, 
 becaufe the Parts of the Orbit on either Side of an Apfe 
 are always fimilar to each other. 
 
 From the fame Equation, the Value of the Limit 
 will alfo be given when the Angle of Direction C Ab is 
 obtufe, or the Body is projected upwards : 
 
 For that Equation (as is eafy to demonftrate *) ad- 
 mits of two different Roots, or Values of p ; the one 
 greater, the other lefs, than Unity : Whereof the for- 
 mer, giving CD (x) lefs than CA, is to be taken in 
 the preceding Cafe, and the latter (making CD greater 
 than CA) in the prefent. And the Body will, either, 
 continue to afcend for ever, or come to an Apfe> and 
 from thence fall to the Center, according as the given 
 Value of p is greater or lefs than that here fpecified. 
 But if it be neither greater nor lefs, but exactly the 
 fame, then the Body, tho' it will ftill continue to afcend 
 for ever in a Spiral, yet it can never rife fo high as 
 the Circumference of the Circle whofe Radius CD is =r 
 
 X a, forReafons fimilar to thofe already 
 delivered, in refpedr. to the preceding Cafe. 
 
 Mathematical D.ffirf. p. 167. 
 
 END OF VOL. I.
 
 BOOKS printed for J. NOURSB in the Strand t 
 BookieUer to His MAJESTY. 
 
 1. /^YCLOAiATHESIS, or an eafy Introduaion to 
 \^A the feveral Branches of the Mathematics. By 
 
 William Emerfon, n Vol. 8vo^. 3. 131. 6d. 
 N.B. Any of the above Volumes may be had feparate. 
 
 2. The Method of Increments. By Mr. Emerfon, 410 
 
 js. 6d. 
 
 3. A (hort Comment on Sir Ifaac Newton's Principia. 
 
 Ey Mr. Emerfon, 8vo, 3*. 
 
 4. Navigation ; or the An of Sailing upon the Sea. By 
 
 Mr. Emerfon ; the fecond Edition 12010, 4^. 
 
 5. The Elements of Euclid. By R. Simfon, the fourth 
 
 Edition, 8vo, 6s. 
 
 6. A Treadfe of Mathematical Inftruments. By Mr. 
 ' Robertfon, the third Edition, improved, 8vo, 5*. 
 
 7. A general Treatife of Menfuration. By Mr. Robert- 
 
 fon, the third Edition, i2mo, 3*. 
 
 8. The Elements of Navigation. By Mr. Robertfon, 
 
 the third Edition improved, in Two large Volumes 
 8vo, i8j. 
 
 9. The Elements of Aftronomy, tranflated from the 
 
 French of M.de la Caille. By Mr.Robertfon,8vo,6j. 
 
 10. Mathematical Tracts of the late Benj. Robins, Efq; 
 with an Account of his Life, 2 Vol. 8vo, ics. 6d. 
 
 1 1 . An Eafy Introduction to the Theory and Practice of 
 Mechanics. By S. Clarke, 410, 6s. 
 
 12. Dr. Helfham's Courfe of Lectures in Natural Philo- 
 fophy, the fourth Edition, Svo, 5*. 
 
 13. Hydroftatical and Pneumatical Lectures. By Roger 
 
 Cotes, the third Edition, 8vo, 5*. 
 
 14. Dr. Hamilton's Treatife of the Conic Sections, 
 Tranflated from the Latin, 410. 12^. 
 
 15. The fame Book in Latin, 4to, i2s. 
 
 16. PhilofopbicaJ Eflays. By Dr. Hamilton, i2mo, 2s.6d. 
 
 17. Four Introductory Lectures in Natural Philofophy. 
 By Dr. Hamilton, iimo, 2s. 
 
 18. The Natural Hiltory and Philofophy of Earthquakes 
 and Volcanoes. By Dr. Bevis, Svo, 5*. 
 
 19. Maclaurin's Account of Sir Ifaac Newton's Philofo- 
 
 phical Difcoveries, third Edition, Svo, 6s. 
 
 20. A Treatife of Algebra. By C. Maclaurin, the third 
 
 Edition, Svo, 6s. 
 
 21. Sir Ifaac Newton's Method of Fluxions, tranflated 
 by J. Colfon, 410, 14*. 
 
 22. Ths
 
 BOOKS printed for J. NCUSRE, in the Strand. 
 
 22. The Practice of Perfpective, on the Principles of 
 Dr. Brook Taylor. By Jo. Highmore, 4(10, . I. is, 
 
 23. The Mathematical Repofitory. By J. Dodfon, 
 
 3 Vol. i2mo, i2s. 
 
 24. Mathematical Lucubrations. ByMr. Landen, 410, 6*. 
 
 25. A Difcourfe concerning the Refidual Analyfis. By 
 
 Mr. Landen, 4 to, is. bd. 
 
 26. The Refidual Analyfis, Book I. By Mr. Landen, 
 
 4to, 6s. 
 
 27. Harmony of Ancient and Modern Geometry aflerted. 
 By R. raman, 410, 6j. 
 
 28. The Practical Gager. By W . Symons, the third 
 Edition, I2tno, 4*. 
 
 29. The Britifti Gager, or Trader and Officer's In- 
 ftructor in the Excife and Cuiloms, by S. Clark, 
 8vo. 5J. 
 
 30. A voyage towards the North Pole in the Year 1773, 
 by the Hon. Capt. Phipps, 4to. 12s. td. 
 
 31. A Hiftory of the Earth and Animated Nature, by 
 Dr. Goldfmith, 8 vol. 8vo. 2/. 8*. in boards. 
 
 32. Atlas Cceleftis, on twenty eight Copper Plates, by 
 J. Flamftead, Fol. il. is. in Sheets. 
 
 33. An Elementary and Methodical Atlas, on Thirty 
 two Copper Plates, by John Palaviret. Fol. 2!. I2S. bd. 
 
 34. Trails Phyfical and Mathematical. By Dr.Matthew 
 
 Stewart, 8vo, 7^. 6d. 
 
 35. Propofuiones Geometries, more Veterum demon- 
 ftratae. Auclore M. Stewart, 8vo, 6^. 
 
 36. Menelai Sphsericorum Libri tres, cura G. Coftard, 
 8vo. 
 
 37. Theodofii Sphaericorum Libri tres, Gr. & Lat. 8vo. 
 
 38. Elements of the Theory and Practice of Chymiftry, 
 tranflated from the French of M. Macquer, 2 Vol. 
 8vo, third Edition, 101. 
 
 39. A Courfe of Chymiftry, divided into. twenty four 
 Lectures, by Dr. Henry Pemberton, 8vo, 51. 
 
 40. A Courfe of Phyfiology, divided into twenty Lectures, 
 
 by Dr. Pemberton, 8vo, 51. 
 
 41. A Treatife upon Electricity, tranflated from the 
 Italian of Signer Beccarea, 4:0, with Copper 
 Plates. 
 
 42. A Series of Experiments bn the Subject of Phof- 
 phori and their Prifmatic Colours, by Benjamin 
 Wilfon, the Second Edition, improved, 410. 6;.
 
 THE 
 
 DOCTRINE 
 
 AND 
 
 APPLI CATION 
 
 O F 
 
 FLUXIONS. 
 
 CONTAINING 
 
 (Befides what is common on the Subject) 
 
 A Number of NEW IMPROVEMENTS 
 in the THEORY. 
 
 AND 
 
 The SOLUTION of a Variety of New, and very 
 Interefting, Problems in different Branches of 
 the MATHEMATICKS. 
 
 PART II. 
 By THOMAS SIMPSON, F. R. S. 
 
 THE SECOND EDITION. 
 
 Revifed and carefully corre&ed. 
 
 LONDON: 
 
 Printed for JOHN NOURSE, in the Strand, 
 BOOKSELLER TO His MAJESTY. 
 
 MDCCLXXVI.
 
 [*7Sl 
 
 THE 
 
 DOCTRINE and APPLICATION 
 
 O F 
 
 FLUXIONS. 
 
 P A R T the Second. 
 
 ...-.-,.. \ 
 
 SECTION I. 
 
 be Manner of mvejligating the FLUXIONS 
 of Exponentials, with Thofe of the Sides 
 and Angles offpherical triangles. 
 
 TH E Method of deriving the Fluxion of 
 any Power, x", of a flowing Quantity, 
 when the Exponent (v) is given or in- 
 variable, has been already fhewn : But, 
 if the Exponent be variable, that Method fails; in which 
 
 Cafe the Quantity *" is called an Exponential; whofe 
 Fluxion is thus determined. 
 
 Put z=x" 9 and let the hyperbolic Logarithm of x be 
 
 denoted by y, then that of x(z) will, by the Nature 
 of Logarithms, be = vy, and therefore its Fluxion = 
 
 But the Fluxion of the Logarithm of %x / 
 T fc is.
 
 276 Of the FLUXIONS 
 
 * Art. iz6. is alfo exprefied by *j whence we have =vj 
 
 andconfequently zzvj -f zyu : Which Equation, by fub- 
 
 X 
 
 t Ait. 126. ftituting for its Equal y f, becomes z=zzyi>+ 
 
 vx 
 
 ,... _ M . Lc". x 
 
 X 
 
 I 
 
 + VX 
 
 'The fame otherwife^ without introducing the Properties 
 
 of Logarithms. 
 251. Let i-f-s; #, and n + w= v, fuppofing n con- 
 
 v ,.+w 
 
 flant and w variable : Then*- i-fz' = 
 
 1 k" w u> i 
 
 X I+Z) = I+ZJ X I -f WJB+ X 
 
 W TV 1 W 2 
 
 J Art, 99. Y X X - X 
 
 1 -f wz -J- ~ if 1 \ X Z*+tW> |w*+yW X Z 3 + &c. 
 whofe Fluxion, found the common Way, is nz x 
 
 nl - - _ ====r-=rr 
 
 I 4-z] X I+tWB + iW 
 
 X Z 1 Wr. + I + Z| X tlz + !C'i + tftc- Jil; X Z 1 -f A /* j 
 
 X 2%i + HtJ a w ww f r^ X 2 3 -f ; w 3 w~ + jty X 32*2: 
 tfff. which, by fubftituting .v and -z/ for their Equals 
 
 K I - ====. 
 
 and w, becomes^ X j4- 2 | x i 
 
 _ 
 
 x z 1 + fff . -f i-f z) x rjz + K-X- 4 it- ; x z z -f ^r. 
 
 But, if w b^, now, fuppofed to vanifh, we fhall have 
 the true Value of the Fluxion when i>r=: ; which, in 
 
 - . n I 
 
 that Circumftance, appears to berrwA- x i +s]
 
 of Exponentials. 277 
 
 _ * - . - 
 -f I -fz) X z-v-~ Iz a <i> + jz 3 o> z 4 -vfcrV. = VA- X x 
 
 w I 
 
 It is plain, becaufe the Series, z |z a -f f z j &c . 
 here brought out, is known to exprefs the Fluent of 
 
 "- -, or the hyperbolic Logarithm of i -fz *, that the * Art 
 
 two Conclufions agree exactly with each other : From 
 either of which the following Rule, for the Fluxions of 
 Exponentials, is deduced. 
 
 252. fo the Fluxion found by the common Rule (Art. 14.) 
 considering the Exponent as conftant, add the Quantity 
 arifmg by multiplying the Fluxion of the Exponent, the 
 hyperbolic Logarithm of the Rut, and the propofed Quan- 
 tity itfelf, continually, together : Ihe Sum will be the 
 Fluxion when the Exponent is variable. 
 
 Thus, for Example, let the Quantity propofed be 
 
 a 
 
 a 1 + z 1 ! , then the Fluxion thereof will be z X 2Z* X 
 
 V 4- z a \ + * X <?*-{- z a j X hyp. Log. a 1 + z*. 
 
 But, if the Root is conftant, and only the Exponent 
 variable, the Exponential will be more fimple ; and its 
 Fluxion will then be had by barely multiplying the Quan- 
 tity itfelf by the Produtl under the Logarithm of the Root 
 and the Fluxion of the Exponent. 
 
 Thus, the Fluxion of a* will be exprefled by a* x * 
 
 xhyp.Log.a; and that of <? + b* ) by^-f-^l s.nx 
 
 X h\>p. Log. a^ + b 1 . Thefe Kind of Exponentials oftener 
 occur, in Practice, than any other ; but, as it is very 
 rare that we meet with any, I (hall therefore proceed 
 now to the other Confideration propofed in the Head of 
 this Section ; namely, trie Merhod of determining the 
 Fluxions of the Sides and Angles of fpherical Triangles 
 (a Thing very ufeful in practical Aftronomy) which 
 I (hall deliver in the following Propofitions. 
 
 T 3 PRO-
 
 278 
 
 Of the FLUXIONS 
 
 PROPOSITION I. 
 
 253. To determine the Ratio of the Fluxions of the fever al 
 Parts of a right-angled Jpherical Triangle j fuppofwg 
 the Hypcthenufe^ one Leg^ or one Angle, to remain csn~ 
 Jlant-> while, the other Parts vary. 
 
 Let A, F, and G be the 
 Poles of the three Great- 
 Circles DEFG, ABD, and 
 ACE ; whereof the Pofition 
 of each is fuppofed to con- 
 tinue invariable, while ano- 
 ther Great-Circle HFCB is 
 conceived to revolve about 
 the Pole F: Whence, if GH 
 be fuppofed perpendicular to 
 FH, three variable right- 
 angled Triangles, FGH, 
 FCE, and ABC, will be 
 
 formed ; in the firft whereof, the Hypothenufe FG will 
 
 remain conftant ; in the fccond, the Leg EF ; and in 
 
 the third, the Angle A. 
 
 Let Eb (q) be the Fluxion (or indefinitely fmall In- 
 * Art. 134. crement*) of the Bafe AB, or the Angle F; and let 
 
 Cd meet the Great- Circle bh, at Right-angles, in d. 
 
 then it will be (per Spherics) as Sin. FB (Rad.) : Sin] 
 
 Sin. FC _ Co-f. BC 
 
 , /Co-f.EC \ Co-f.EC 
 And, rang. C : Rad. :: Cd ( - ^ X qj : ^ ^ c 
 
 X q = the Fluxion of BC. 
 Moreover, Sin. C : Rad. :: Cd 
 
 C >-f. BC v,, 
 
 ... ,, x q =. the iluxion of AC. 
 !>/. C. 
 
 2 Laftly,
 
 of Spherical Triangles. 279 
 
 Laftly, Sine of FB (Rod.) : Sin. FH (BC) :: B ( ? ) : 
 
 Sin. BC 
 
 -g : x q (=.Hm) =. the Fluxion of GH, or its 
 
 Complement C. 
 
 Now, if the feveral Quantities, in thefe three Equa- 
 tions for the Triangle AC, be expounded by their re- 
 fpcctive Equals in the other two Triangles CEF and 
 FGH, we fhall alfo have 
 
 Sin. CF. 
 
 cr TT- X a Flux. CF. 
 
 Tar.g. L- * 
 
 Sin. CF 
 
 X q = Flux. CE. 
 
 X q - Flux. C. 
 
 Kad. 
 
 And 
 
 Co-f. FH 
 r^ ^ - r x q = Flux. FH. 
 
 La-tang. Lit! z 
 - FH 
 
 COROLLARY I. 
 
 254. Hence, if, in any right-angled Spherical-Tri- 
 angle, the Hypothcnule be denoted by k, the two Legs 
 by L and /, the Angles, refpeclively, adjacent to them 
 by A and #, we ftiall, by fubftituting above, have three 
 Equations for each of the three Cafes. From the Com- 
 parifon and Compofition of which, the three following 
 Tables are deduced; exhibiting all the different Varieties 
 that "can poffibly happen, whether an Angle, a Leg, or 
 the Hypothenufe be fuppofed invariable. 
 
 TA TABLE
 
 280 Of the FLUXIONS 
 
 TABLE I. 
 
 When one Angle A is invariable, 
 
 r _Tang. a Sin. a : Rad. , * 
 J-i x / ^ . X h . X a 
 
 .*__ Co-f. I __ Co-f. a Co-tang. I 
 
 Tang . a R Tang, a 
 
 b=^! X L= Q~* ', Co-tang. I - 
 Sin. a Co-f. a Sin. a 
 
 R Co-tang. I Go-tang. I 
 
 TABLE II. 
 
 When one Leg L is invariable, 
 __ Tang, a x = Sin, a x /_ _ .&... x ' 
 ~~ Sin. b Sin. b Co*f> b 
 
 Co-f.b : Sin. a Tan?, a 
 
 _ - J y Jl^Z. *~ ~ X / ^~ " X H 
 
 < R Tang, h Tang, b 
 
 _ ^- b j _ C*'f' & ,' _ Tang, b 
 
 ~ Tang, a R Tang, a 
 
 y _ Sin. -b _ .K. / _ T^wg 1 . h 
 
 - X vz - x b =f. X a, 
 
 Lo-J. a 6/. a 
 
 TABLE III. 
 When the Hyp. is invariable, 
 
 __ 
 
 ^-/i Z. Co-f.L Sin. L 
 
 T ' ' x - ' X - 
 
 ^^ . ^^^ - . . I ! /^ Ji 
 
 Co-tang. I R. Tang.L 
 
 Where, and alfo in the two preceding Tables, the Leg L 
 is adjacent to the Angle A> and the Leg /to the Angle a. 
 
 Co-
 
 of Spherical 'Triangles. 
 
 281 
 
 COROLLARY II. 
 
 255. From the third original Equation, exprefilng 
 the Fluxion of the Angle C (Vid. Art. 253.) it appears 
 that the Superficies of any Spherical-Triangle ABC, is 
 proportional to the Excefs of its three Angles above 
 two Right-Angles. For (ECdb) the Fluxion of the 
 Triangle ABC, is = Sine BC x B, by Art. 161.) which 
 
 Sin. BC 
 being to, , x Bb> the Fluxion of the Angle C 9 
 
 above fpecified, in the conftant Ratio of Radius to 
 Unity, the Fluents themfelves (properly corrected) muft 
 therefore be in that Ratio ; that is, the Superficies of 
 the Triangle ABC will always be proportional to the 
 Increafe of the Angle C, from its coinciding with >/, 
 or as the Excefs of A and C above two Right- Angles. 
 
 PROPOSITION H. 
 
 136. To determine the Rat to of ths Fluxions, or the in- 
 definitely fmall Increments^ of the different Parts of 
 an oblique Spherical-Triangle ABC ; two Sides thereof 
 AB, AC being invariable ^ in Length. 
 
 Let Cc be an indefinitely 
 fmall Part of the Parallel de- 
 fcribed by the Extreme C of 
 the given Side AC, in its 
 Motion about the given Point 
 A ; moreover, let Cd be Part 
 of another Parallel, whofe 
 Pole is the given Point B; let 
 the Great-Circle EC meet CJ 
 in d\ and let the three Sides, 
 AB, AC, and BC, of the 
 Triangle be denoted by D, , 
 and F refpedively. 
 
 Then,
 
 282 Of tic FLUXIONS 
 
 Then (per Spherics) we (hall have 
 R : S. E :: CAc 4) : Cc = ^-' x 
 
 And, R-.S.F:: CEct(B) : Cd = -~- x B. 
 
 S E x S C 
 Alfo, R : S. dCc (ACB) ::Cc:F= R1 ' 
 
 But S. C : S. D : : S. B : S. E j therefore S. E X 
 
 S D x 
 S.DxS.B, and confequently /", alfo, = ^ 
 
 S. C 
 ? B 
 
 x A. 
 
 Again, : Co-f. dCc (ACB) : : Cc (~ x ^ 
 S. E. x &-/: C 
 
 x A ( (~aj = 
 
 .... 
 
 \Vhence fl = 
 
 Laftly, Cfl-f. cCd:(C) : R:-.Cd~ 
 
 ^, y\ iv. 
 
 CiO-t. G 
 
 Whence, by the very fame Argument (fubftituting 
 ) for E, and for B in the two laft Equations) we 
 
 rt, T u *_S-D * Co-f. B - ,,,_ 
 likcwife have C =: p^ x A, and -r ( = 
 
 S.F 
 
 A >c o. r 
 
 S.F 
 
 
 Now, from the Equations thus found, it is manifeft, 
 
 J. 
 
 A 
 
 : F: 
 
 R*:S.D 
 
 *S.B(: 
 
 : Co-fea 
 
 2. 
 
 A 
 
 : B: 
 
 Rx S.F 
 
 :S.Ex 
 
 Co-f. C 
 
 3". 
 
 A 
 
 : C: 
 
 RxS.F 
 
 : S.Dx 
 
 Co-f. B 
 
 4.0. 
 
 ) 
 
 : F: 
 
 Co-t. C : 
 
 S.F 
 
 
 5- 
 
 L 
 
 : F: 
 
 Ce-t. B : 
 
 S.F 
 
 
 6. 
 
 B 
 
 : C: 
 
 Co-t. C : < 
 
 ~o-t. B (: 
 
 * ^T* 7? ' 
 
 :T.C) Q.E.I.
 
 of Spherical Triangles. 
 
 257. Thefe Proportions, for the Fluxions of the Parts 
 of a Spherical-Triangle, are very ufeful in various Cafes 
 in Practical Ajlroncmy ; whereof I (hail here put down 
 one or two Inftances. 
 
 The firft is ; To determine the annual Alteration of 
 the Declination and Right-Afcenfion of a fixt Star, 
 through the Prcceflion of the Equinox. 
 
 Here A muft denote the Pole of the Ecliptic, B that 
 of the Equinoctial, and C the Place of the Star j and 
 then (by the firft and fourth Proportions) we have 
 
 Co-feca. D : Sin. B : : A : F-, and 
 S. F: Cot.C:: F : B ; 
 
 That is, i, As the Co-fecant of the Obliquity of 
 the Ecliptic is to the Sine of the Star's Right-Afcenfion 
 from the folflitial Colure y fo is the Precj[Jton of the Equi- 
 nox, or Alteration of Longitude, to the Alteration of 
 Declination. 
 
 2. As the Co-fine of the Star's Declination is to the 
 Co-tangent of its Angle of Pofition, fo is the Alteration 
 of Declination (found as above) to the Alteration of 
 Right-Afcenfion correfponding. 
 
 The fecond Example is to find how much the Am- 
 plitude, and the Time of the apparent Rifing and Setting 
 of the Sun, or a Star, are affe&ed by Refraction. 
 
 In this Cafe A muft de- 
 note the Pole of the Equa- 
 tor, and B the Zenith, and 
 the Side BC muft be an 
 Arch of qo Degrees, fo 
 that the Star C may co- 
 incide with the Horizon 
 QC : Then, from the very 
 fame Proportion, we have, 
 
 Sin. B : Co-feca. D : : F : A* 
 And, R \Co-t. C::F:B 
 
 But, R : Co-t. C(T. QCA) : : Sin. B (C9) : Co-tang. D 
 (Tang. ^A) 
 
 Hence 
 
 283
 
 Of the FLUXIONS 
 
 Hence it apppears, 
 
 i. That, as the Co-fine of the true Amplitude 
 (confidered independent of Refradtion) is to the Tangent 
 of the Pole's Elevation, fo is the given horizontal Re- 
 fraclion to the Difference of Amplitudes thence arifing. 
 
 2. And, that, as the Co-fine of the true Amplitude 
 is to the Secant of the Pole's Elevation, fo is the faid 
 horizontal Refraction to the EffecT: thereof in the Time 
 of Rifing, or Setting of the Sun, or Star. 
 
 But this laft Proportion may be otherwife exprefled, 
 without the Amplitude : Thus, 
 S. AB x S. AC x S. A: R 3 :: the horizontal Refradion, 
 
 to the fame Effeft. 
 
 PROPOSITION III. 
 
 258. To determine the fame as in the preceding Problem ; 
 Suppofmg one Side AB and one of its adjacent Angles y 
 B, to continue invariable. 
 
 If from the End of the 
 given Side, oppofite to the 
 given Angle, a Perpendicular 
 AD be let fall, that Perpen- 
 dicular, as well as the Seg- 
 ment BD cut off thereby, 
 \vill be a conftant Quantity, 
 while the other Parts of the 
 Triangle A^D vary, by the 
 Motion of a along the Arch 
 
 cBD. Therefore the Problem is refolved by Cafe 2. of 
 
 right-angled Triangles. Fid. Art. 254. 
 
 259. It may not be amiis to give one Example of the 
 
 Ufe of this laft Propcfition : Which fball be, in finding 
 
 the Parallax of a Planet in Longitude and Latitude j that 
 
 of Altitude being given. 
 
 Here A muft ftand for the Pole of the Ecliptic, B 
 
 the Zenith, and a the Planet : Then, if the Hypo- 
 
 thenufe Aa be denoted by /;, the Leg. Da by /, and the 
 
 given Parallax, in Altitude, by /, it will appear, from 
 
 the
 
 of Spherical Triangles. . 285 
 
 the Place above quoted, that A (the Parallax in Long.) 
 
 Sin. a f Sin. EaA 
 
 will be = TP 7 X / = -z-. j- x /, and b ( the 
 o/n. b o/. A a 
 
 Co-f.a . Co-f.BaA 
 Parallax in Lat.) = p , X / 55-3 X /. 
 Rod. Rod. 
 
 If the Planet be in (or very near) the Ecliptic, and 
 /?Q_be fuppofed a Portion of the Ecliptic, meeting AB, 
 
 Sin. EaA 
 at Right- Angles, in Q^, then (per Spherics) -r: -T 
 
 fCo-f. Ba$\ _ Tang. $a Co-f. Bad (Sin. Ba^\ 
 
 \ Radius ) ~ Tang. Ba* * Rad. \ Rad. / 
 
 rr - r. ' n - whence, by fubftituting thefe Values 
 6/n. DO 
 
 CT^ ^55/7 
 
 above, we fhall, in this Cafe, have A = <r *' K x 
 
 / and >* 1 - i - X j that is, in Words, 
 o/n. /3 
 
 As, the Tangent of the Planet's Zenith Diftance, is 
 to the Tangent of its Longitude from the nonagefimal 
 Degree of the Ecliptic, fo is the Parallax in Altitude to 
 the Parallax in Longitude. 
 
 And, as the Sine of the Zenith Diftance to the Co- 
 fme of the Altitude of the nonagefimal Degree, fo is 
 the Parallax in Altitude to the Parallax in Latitude. 
 
 Becaufe the Parallax in Altitude, the horizontal Pa- 
 
 Sirit Bo 
 rallax (M) being given, is nearly = X M, if 
 
 this Value be fubftituted for /, in the two laft Equations, 
 . Sin.gB ,. . : Tang. $ax Sin. Ea 
 we fhall get h =-j^ x Af, and ^= ^ d 
 
 S'tn.ABxSin.BAa ,. 
 
 Whence,
 
 286 
 
 Of the FLUXIONS 
 
 Whence, we have theis two other Theorems, for 
 finding the required Parallaxes immediately from the ho- 
 rizontal Parallax, without either the Altitude or its 
 Parallax. 
 
 1. As Radius to the Co-fine of the Altitude of the 
 nonageftmal Degree of the Ecliptic, fo is the horizontal 
 Parallax to the Parallax in Latitude. 
 
 2. And as the Square of Radius to the Rectangle un- 
 der the Sines of the Altitude of the nonagefimal Degree 
 and the Planet's Longitude from thence, io is the hori- 
 zontal Parallax to the Parallax in Longitude. 
 
 PROPOSITION IV. 
 
 260. Still, to determine the fame Thing ; fuppofing^ one 
 Angle A, and the Length of its oppofite Side BD (or 
 
 i i 
 
 Let BD ( equal to 
 BD)imerfeaBDin an 
 indefinitely fmall Angle 
 
 at P, and meet AB 
 
 / / 
 
 and AD in B and D ; 
 alfo in BD produced 
 let there be taken PN 
 
 = PD and PM = PB, 
 i i 
 
 and let N, D, and M, B be joined. 
 
 Since, by Hypothecs, DB r: DB = MN, if from the 
 firft and lait or thcfe equal Quantities DM, common, 
 be taken away, there will remain BM = DN. 
 
 Moreover, fmce the Triangles BMB and DND, in 
 
 their ultimate State, may be confidered as rectilineal, 
 
 * Art.i34. and right-angled at 'M and N*, it will therefore be, as 
 
 BM :B ::*-/ B: Radius 
 And DN : DD : : Co-f. D : : Radius. 
 
 From
 
 of Spherical Triangles. 
 
 From whence, the Extremes in both Proportions be- 
 ing the tame, we have BB : DD :: Co-f. D : Co-f. B : 
 And rh^efore, if AB be denoted by H and AD by K, 
 
 it appears that H: K:: Co-f. D : Co-f. B. 
 
 Again, per Spherics, tin. A : Sin. BD (G) :: Sin. 
 D : Sin. H : : Flux. Sin. D : Flux. Sin. H ; becaufe, 
 the Sines themfelves being in a conftant Ratio, their 
 Fluxions mult be in the fame Ratio : But the Fluxion 
 of the Sine of any Arc, or Angle, is to the Fluxion of 
 the Arc or Angle itfelf, as the Co-fine to Radius * : *Art.i4i. 
 
 j-f.D 
 
 Therefore the Flux. Sin. D being = 
 
 Rad. 
 
 x D, and 
 
 Co r H. 
 
 Flux. Sim 11= ~ J ' x H t it follows that, Sin. A 
 Kaa. 
 
 : Sin. G : : Co-f. D x D: Co-f. H x H ; or D : H : : 
 Sin. Ax Co-f. H : Sin. G x Co-f. D : And, by the very 
 
 fame Argument, B : K :: Sin. A x Co-f. K : Sin. G X 
 Co-f. B. Now, by compounding the former of thefe 
 two Proportions with the firft above given, we get, 
 D : K : : Sin. A x Co-f. H : Sin. G x Co-f. B. And, by 
 compounding this laft with K : B : : Sin. G x Co-f. B : 
 Sin. A X Co-f. K (that immediately preceding it) we alfo 
 obtain D : B : : Co-f. H : Co-f K. 
 
 Whence, by collecting thefe feveral Proportions to- 
 gether, we have the following Table, for all the dif- 
 ferent Cafes. 
 
 H 
 D 
 D 
 B 
 K 
 H 
 
 K::Co-f.D: Co-f. B 
 B :: Co-f. H: Co-f. K 
 H:\Tang. D : Tang. H 
 K : : Tang. B : Tang. K 
 D : : Sin. G x Co-f. B : Sin. A x Co-f. H 
 B : : Sin. G * Co-f. D : Sin. Ax Co-f. K 
 
 It
 
 Resolution 
 
 */ 
 
 It may be obferved, that the fourth and the Jaft are no 
 new Cafes, but only the third and fifth repeated : And 
 that, though the former of the two, lafl named, differs 
 from that found above ; yet it is very eafily deduced 
 
 C " ji 
 
 from it : For, fmce it appears that Z> : ti :: ' : 
 
 L>0-J. U 
 
 C' /" 
 
 JlS-Sy-, and becaufe 5/a. A : Sin. G : : Sin. D : Sin. 
 Co-f. ti 
 
 , f^ rr - Sin. D Sin. H 
 H, it follows that D'.H:: . ...... :: 
 
 Co-f. D Co-j. H 
 
 tang. D : Tang. H. Q. E. I. 
 
 There is yet another Problem, when two Angles re- 
 main conftant ; but this, by taking the Triangle formed 
 by the Poles of the three given Circles, b reduced to 
 Problem 2. 
 
 SECTION II. 
 
 Of the Refolution offluxional Equations, or 
 
 the Manner of fading the Relation of the 
 
 flowing Quantities from that of the Fluxions. 
 
 26i.~lT7 r HEN an Equation, exprefling the Re- 
 \ Y lation of the Fluxions of the two va- 
 riable Quantities, contains only one of thofe Fluxions 
 with its refpe<Stive flowing Quantity in each Term, the 
 Relation of the Quantities will be obtained by finding 
 the Fluent of every Term j as has been already taught, 
 
 / Qg* y* 
 
 Thus, if ax'x =y*j, then will 
 
 3 4 
 
 And, if x y xaj; by reducing it firft to x x = 
 ay y (fo that its variable Quantities may be feparated) 
 
 we have - , ~ ~~ i m 
 
 n But,
 
 of Fluxional Equations. 
 
 But, if the given Equation has its indeterminate Quan- 
 tities and their Fluxion? To complicated togeth-r, that 
 it cannot be brought under the Form ther^ orefcribed, 
 the Taflc will become much more difficult; lor is- there 
 any general Method to be given for fuch Kinils of Equa- 
 tions, whereof there are an infinite Variety. 
 
 The Method of Infinite Seriefes (in fome meafure ex- 
 plained already, and more fully confidered hereafter) is 
 indeed very comprehenfive, and may be applied to good 
 Purpofe in various Cafes ; but, being tedious and at- 
 tended with a Number of Inconveniencies, it is a Me- 
 thod we ought: never to have Recourfe to till we have 
 tried what may be, otherways, effected, by help of fuch 
 particular Rules and Obicrvations as we have been, able 
 to collect. 
 
 Accordingly, I fhall, here, firft point out fome of 
 the moft proper Ways to be tried, in order, if poflible, 
 to bring out the Solution without an Infinite Series. 
 
 262. The fir/I Method is, by multiplying^ or 'l'-vi . 
 the given Equation by fome Poiuer or Produfl cj 
 Quantities concerned ; fa as to bring it, if pojfib'f, under 
 the Form of fuch Fluxions, as, tue know, .do wife, if not . 
 from the firji, yet from the ficond, or third t of the three 
 general Rules in the direct Method. 
 
 Thus, if the given Equation be + ' ~ 
 
 x y ' , 
 af 
 
 then, the whole being multiplied by xy, fo tnat the two 
 firft Terms, 7*- 4 xy, may become the (knownj Huxion of 
 
 wil . 
 
 x x 
 
 the Re&angle xy *, there arifes yx + xy = __, : But * 
 
 ay 
 
 ftill we are at a Lofs for the Fluent of the laft Term, 
 unlefs n be taken n i (fo that y may vanifh). In that 
 
 fK-f-2. 
 
 X 
 
 Cafe v/2 have xy = === ; exprefling the Relation 
 m -f 2 X a 
 
 of the Fluents when that of the Fluxions is 4. 
 
 x y 
 
 x"x 
 
 . : Which appears to be the oily Cafe, of the given 
 
 Equation, where this Method is of UL% 
 
 U Agai.i,
 
 200 The Refolution 
 
 J *S 
 
 A^ y* y M^ *, 
 
 Again, ht the Equation + * be pro- 
 
 ** y n 
 
 ay 
 
 pofed. 
 
 Here, multiplying by a? / (where the Exponents 
 
 x y \ 
 
 are the fame as the Coe&cients of and - ) we act 
 
 x y ' 
 
 A V* v- V v 
 
 px x x y 
 
 ay 
 former Part of the Equation is known to exprefs the 
 
 Art. 15. Fluxion of . v P /*. Therefore, when rcrrr, the Relation 
 of the Fluents may be found, and will be exprefTed by 
 
 w-ty+l 
 
 Which, if no Correction by a 
 
 m + p+i X.a 
 conftant Quantity be neceflary, may be reduced to 
 
 m+l 
 
 ~ m+p+ I X a' 
 
 The fame Method may alfo be extended to Fluxions 
 of the higher Orders : Letxxz*=fa? (which Equa- 
 tion occurs hereafter, in the Refolution of a Problem 
 of fome Difficulty). Then, multiplying by ;c,*it be- 
 comes xx atx*x:fat*x} where, being conftant, each 
 Term admits, now, of a perfect Fluent, and we therefore 
 
 have fxx?; From whence, fuppofing no- 
 Correction neceflary, x = rr and z =: hyp. 
 
 V2fx 4- xx 
 
 Log. f+ x + V 2 fx + x* (by Art. 126.) 
 
 263. // may happen that the Solution of an Equation 
 will become more eajy by fi-Jl taking the Fluxion thereof '; 
 when, by that means, jome cf the Term* dejhoy each other. 
 
 The following is an Inftancc of it (which, alfo, occurs 
 
 hereafter). Lety -f- - yx a * x -7-: WhofeFkuc- 
 x y 
 
 2 ion,
 
 of Fhixionat 'Equations. 291 
 
 ion, making * conftant, is y -\ '-- * 
 
 y > ,' -i ^^VV 
 
 .. J : Which, by reafon of the Terms deftroying 
 one another, is reduced to "- - =--~r '. Therefore, 
 
 x 
 
 by expunging y, &c. we get jy~- x x a x { 
 
 1 -- 1 * r\ 
 
 confequently 2y~* 2Xa x\* -\- fame conjl ant g>uan- 
 tlty. 
 
 264. Another Method, chiefly applicable to Equations, of the 
 firjl Order of Fluxions, wherein only one of the two va- 
 riable Quantities (x or y) enters, is, to fukjiitute for the 
 Ratio of the two Fluxions (x and y) : From whence tie 
 Value of that Quantity iviil be hsd, immediately, in Terms 
 of the faidajfumed Ratio : And then, by taking its Fluxion, 
 that of the other Quantity (and from thence the Quantity 
 itfeif) will become known. 
 
 Thus, let axj* y X xx+jy] (being the Equation of 
 the Curve that generates the Solid of the leajl Re- 
 e^ when the Bulk and greateft Diameter are given). 
 
 Then, by putting v, and fubftituting above, we 
 
 get avj* yX ^u L y L +j~\* =: yj* X "o % -|- 7) * ; and con- 
 
 av a-v zmfv 
 
 lequently y ' , > : l nerefore v _- , - 
 7 J z - 
 
 aw lav v 
 
 and confequently x ( vyi) t i : V/hofe 
 
 Fluent may be found, from Art. 84.. or, othcrwife, 
 thus : Put w~ r d i + I ; then v~ zz w * I , and ww n 
 ^/< ; by fubftituting which Values there anlbs x 
 
 x w * ~ r - 5_ 77 ^..- y -^ a nd 
 
 " ^i.* .- W / y AilU 
 
 U 2 there-
 
 292 *fhe Refolution 
 
 f _ 
 
 lore x ^^ 
 
 4 2 2w r 
 
 +JL-=?g - g x ^ + r ; which, corrected 
 
 2 X W 4- 1 2 X 
 
 . , a x -iw -f i a 
 
 (by taking: y, or wzroj becomes # .. ^ . 
 
 2 X wv + j 1 2 
 
 From this Equation, by completing the Square, C5V. 
 v may be found in Terms of x ; whence the correfpond- 
 
 fi V \ 
 
 ing; Value of v ( ; z ) will alfo be known. 
 
 uv + i ) / 
 
 265. 7/fo fourth Method, which chiefly obtains when 
 one of the indeterminate Quantities and its Fluxion, 
 arife but to a fingle Dimenfion each, may be thus : 
 
 Let the Value cf that Quantity, which is hajl involved, 
 be firfl fought , from the fictitious Equation arifing by neg- 
 lefting all the Therms in the given Equation, where neither 
 that Quantity, nor its Fluxion, are found : Then, to that 
 Value, lit feme Power, or Powers, of the other Quan- 
 tity, with unkniicn Coefficients, be added ( according to 
 th Dimenfions of the "ferms ntgleffed) and let the Sum 
 be fubjlituted in the given Equation, as the true Value of 
 the firjl mentioned Quantity : By which means a new 
 Equation will rejult ; from whence the aj/iimed Coefficients 
 y, fometimcs, be determined. 
 
 Ex. Let the given Equation be cx*x+yx = aj. 
 
 By neglecting cx~x } or feigning yx uy, we get 
 
 x y .v 
 
 : and confequently ; n hyp. Log. y hyp. 
 
 y 
 
 Art. ia6, Log. ^* =hyp. Log. -7 ; d being any conftant Quan- 
 
 tity, which the Nature of the Problem may require. 
 
 y 
 
 Hence -7 =the Nurr.ber whofe hyperbolical Logarithm 
 
 is : Which Number, if Mbe put for (2,71828 
 
 the
 
 of I* luxiona! Equations. 293 
 
 the Number whofe hyp. Log. is Unity, will be ex- 
 
 X 
 
 prefied by M] a (fmce it is evident that the hyp. Log. 
 
 X X \ V 
 
 hereof is x Lo. M = ) Therefore = 
 a a / a 
 
 X X 
 
 JWPand^ = </x ~M\ a . Now, to the Value thus 
 found, let there be added A^ + B^ + C, in order to get 
 
 the true Value ; and then, y being =. 2A**-f B*-f 
 
 x 
 X M\ a *, we fhall, by fubftituting in the given Equa-* Art< 
 
 tion, 
 
 + dxM a 9 and confequently c+A X x*x 4- 
 B 2 Art x xx + C Brt X x o. Whence A = tf, -{Art. 84. 
 B = 2ac,C=. ^aac t and confequently y . f x 
 
 X 
 
 x 1 " + i^ + ?- aa -\- d M a . By the very fame Way, th e 
 Value of y, in 'the Equation ex x~\-yx=.^^ will come 
 out = c X x -i-ax +njt i.a'x " -J-w. n j. 
 
 266. But, what is a little remarkable, in thcfe Equa- 
 
 X 
 
 tions, is, that the Exponential dM a , tho* a variable 
 Quantity, fhould only ferve, as it were, to corredt the 
 Fluent, or perform the Office of a conftant Quantity. 
 What I here mean will plainly appear, if it be con- 
 
 fidered, that the Equation v fx x 3 " -f lax -\- zaa, 
 v/here the faid Exponential is wanting, anfwers all the 
 (Conditions jpf the fluxional Equation firft propofed ; 
 \vhich, upon Trial, will be found ; and muft needs be 
 . U 3 the
 
 294 Me Refohttion 
 
 the Cafe, feeing d may be, cither, taken Nothing at 
 all, or any Quantity at Pleafure. 
 
 But the Equation y c x x z 4 lax -f 2a z (when 
 * 
 
 dM 6 'is wanting) cannot be corrected, in the ufual Way, 
 fo as to give_yrro, when * o; fince, if any other con- 
 ftan: Quantity, befides 2a i c be introduced, the firft 
 Conditions Will not be anfwer'd : The Correction rnufl, 
 
 X 
 
 therefore, be by the Exponential dA4 a ; and is thus. 
 
 X 
 
 Since y rr cx^ icax lca~ -{- dM a , if y be 
 taken o and x o, then 2ca~ -f- dM = o, or d~ 
 lea'; and fo the Equation, truly corrected, is y c x 
 
 X 
 
 + 2a'~cM * . 
 
 267. We come now to the lad Method; namely, 
 ^hat of Infinite Seriefes ; which, tho' lefs accurate, is 
 vaftly more comprehenfive, than any yet explained : 
 The Manner of it is thus : 
 
 For the Quantity whofe Value you would find^ let an 
 I-ifinite Series, c r ^;jiing of tbz Poivcrs of the ether Qiian- 
 tiiy with unknoicn .>, 1; ajjltmtd\ which Series t 
 
 ttgethtr with its Fluxlzn, or Fluxions, mujl be fubfti- 
 tiitsd injlead of thtlr Equals in the given Equation j 
 whence a new Equation will arife, from which, by com- 
 paring the hzmilogout "Terms, the ajjumcd Coefficients^ and 
 ccnfcquently the Value fought, will be determined. 
 
 Thus, let the Equation - - j (reducible to x- 
 
 1 "T~ * 
 
 #v=o) he propofed ; to find x in Terms of y. 
 'Then, affuming *= A; + li/ + Cy 3 + Dj 4 + Ey* &c. 
 We have x = A/-f ittyy + $Cy~y + 4-D>- 3 j + c E/j -f 6fr. 
 Which Values being fubflituted in x j X) 1 O, we get 
 A -; + 2 Bn -f sC/y -f 4.Dy 3 j -t- C5V. ? _ 
 
 >' 
 
 There-
 
 of Fluxiona! Equations. 
 
 Therefore A i = o, cr Am ; aB A=c, or B 
 ~~i sC-Bno.orC^rz^j 4 D-C 
 
 C i 
 
 =3 o, or D =. =s= - &<: . 
 
 4 2-3-4 
 And confequent/y x ( Ay + B/ -f C/ &c ) = y + 
 
 L* , 2l , JL. , ^ + ^v 
 a ^2.3 "*" 2.3.4 2.34.5 
 
 Again, let it be required to find the Value of y, in 
 the Equation c x*x -f- yx ay, or ay yx c x^x n o. 
 Here, afluming jr=A.v + B* 1 + CAT S + D^ + E* J + F* 6 
 ^zff. and proceeding as before, we fhall have 
 aAx+2aBx + 2aCx*x + 4aDx 3 x + 5aEx+x+ &i. ) 
 o A** B*V C.v 3 r D*** esfc. f 
 o p V J 
 
 Whence A =: o 2^B A =r o C = B-- = 
 
 Whence A =: o ; 2^B A =r o ; 3C = B-\-c = c, or 
 C = 1, 
 
 c 
 
 -, or E rrr - - - &c. and confequently 
 
 +-^- +.- 
 
 2* 3-4*" 3-4-5 3 
 
 /-. 4- 
 
 /. 
 
 3-4-5-6^ 
 
 268. It appears from this Example, that the Quantity 
 to be found, will not always require all the Terms of the 
 Series Ax + Bx~ -f C* 3 &c. And it may happen, in 
 innumerable Cafes, that the Series to beaflumed will de- 
 mand a very different Law from that where the Exponents 
 proceed according to the Terms of an arithmetical Pro- 
 grcflion having Dnity for the common Difference. And, 
 indeed, the greateft Difficulty we have here to en- 
 counter, is, to know what Kind of Series, with regard 
 to JLS Exponents, ought to be aflumcd, fo as to anfwer 
 the Conditions of the Equation, without introducing 
 more Terms than arc actually ncceiTary. 
 
 U4 For
 
 296 7/5<? Refolution 
 
 The following Rules will be found very ufeful upon 
 this Occaiion : Which, though they may become im- 
 practicable in certain particular Cafes, never take in any 
 fuperf.uous Tern:". 
 
 i. Having (if neccffary ) freed your Equation from 
 FraFisns and Surds, let the Quantity, whofe Vu<--' is 
 fought i be fup.pofed equal to feme Power cf the other Quan- 
 tity with an Unknown Exponent (n) ; and let that Power, 
 together ^vith its Fluxion, or Fluxions, be fubftitutcd for 
 their (Juppojed) Equals in the given Equation. 
 
 2. Let the Itajl Exponents of the variable, or inde- 
 terminate, Quantity, in the new Equation, thence arifmg, 
 bz put equal to each other : Whence the Value of the un- 
 known Exponent n will be fcund, 
 
 3. Suhjlitute the Value of n, fi found, in all the Ex- 
 ponents where n is concerned j and then take the Dif- 
 ference bitiveen cr.e of the equal ones, above mentioned, 
 and every other Exponent, of the variable Quantity, in 
 the whole Equation. 
 
 4. To thefe Differences, -write down all the leajl Num- 
 bers that can be compzfed cut of them, by continual Addi- 
 tion, either to tkemfelves, or to one ansiber ; //// you have, 
 by that means, get, in the ichole, as many different Terms, 
 GS you would have the required Series continued to. 
 
 5. Lajlly, let each of thoje Terms be increafed by the 
 Value of n (found by Rule 2.) and you will then have tk 
 Exponents of the Scries to be ajjumed. 
 
 E X A M P L E. I. 
 
 269. Let the Value cf x, in the Equation a^x*' -f x~ 
 a*z~ =: C, be required. 
 
 Firft, by \vritir.g 2 for x, and r.z K for x, the 
 Indices of z will be 2 2, 2n, and o (which are deter- 
 mined by Infpe&ion, without regarding the Coefficients) 
 whereof the two leaft (in 2 and o) being put equal 
 to each other, we here find nr=i : Therefore, the Ex- 
 ponents being 0, 2, o, the Differences (according to 
 Rule 3.) are alfo c, 2 ; from whence, by adding 2 con- 
 tinually, we get o, 2, 4, 6, 8 &c. which (being each 
 
 in-
 
 of 'Fluxional Equations. 297 
 
 increafed by the Value of w) give I, 3, 5, 7, q &c. for 
 the Exponents in this Cafe. 
 
 Let, therefore, x Az + Bz 3 + Cz 5 -f Dz 7 + &c. 
 Then, putting z~ i, in order to facilitate the Operation, 
 we fli.i.l have x = A -}- 382;* + 5Cz 4 + 7Dz 6 + We. 
 which two Values being fquared, and fubftituted in the 
 given Equation, it will become 
 rf*A a -f 6*ABz* + icr^ACz* -f- 
 
 + AV 
 
 f +-BV + &c. m 
 
 Whence, a i A 1 =:a' i , and therefore A = i ; 6a z E = 
 
 A, and therefore B = ^r = a > i oa a AC 
 
 + 2A = B x 
 
 - a a B' 2 AB = B x 
 
 1, and therefore C 
 ' 
 
 + 2 = -- = 
 
 2.3-4-5* 4 *~" 2 -3-4-5- 6 '7* 6 
 
 EXAMPLE II. 
 
 270. Z-? the given Equation be afxjr 20 1 xy + oxx*" 
 + X 3 j 0; to find y. 
 
 Here, fubftituting A; "for y, the Exponents will be 
 _!: i, and -J-ii where, making j = i, 
 
 we
 
 The 
 
 we get n2 : Whence, the Differences being O, 2, the 
 Series to be aflumed for^- wili be Ax'- -f B.v* -f C,v 6 -f 
 D# 8 + E.r 10 -f f<r. From which, making x i^ we 
 have j' = 2A# +4B.v 3 + 6C,v 5 + SD# 7 cfc. and 
 
 j ; rr 2 A -f 1 2B:<r -f joC* 4 + 560 v c 
 And, thefe Values being iubftituted, the Equation be- 
 comes 
 
 2*A.r -f- i2fl l B* 3 + 30-C* s + 56<rD.*: 7 -f fjfr. 7 
 4^ Ax 8^'B.r 3 i 2 ^CAr 5 i^D^- 7 + CSV. > =o 
 +2A.v 3 +i2B.v s +3oC^ 7 + Cifc. 3 
 
 Therefore A- = ',E = - -- ^ ; 
 
 2a 4^ 4^ 3 J 
 
 T2B 
 
 atix 
 
 Whkh Series is known to exprefs the Fluent of 
 
 ar + x 
 
 j 
 
 a 1 -f x' L 
 or, k a X hyp. Log. : Confequently _y is alfo 
 
 fl*-f.v= 
 1^ X hyp. Log. - . In this manner, it comes to 
 
 pafs, tbaty though we are obliged, in very complicated 
 Cafes, to have recourfe to Infinite Seriefes, we are 
 fometimes able, at laft, to give the Solution in finite 
 Terms, or, at leaft, by help of Logarithms, Sines and 
 Tangents : Which will always happen when the Series 
 can be fummed, or is found to agree with that arifing 
 from fome known Quantity. 
 
 271. Sometimes it happens, in Equations involving 
 the higher Orders of Fluxions, that the Exponents, 
 mention'd in Rule 2. whereof the leaft ought to be 
 made equal to each other, are fo exnrefTed, as to render 
 fuch an Equality impofTible. When this is the Cafe, 
 the Value of n, and the hrft Term of the required Se- 
 be determined from the Nature of the 
 Prcblcra to which the Equation belongs. We know, 
 
 in-
 
 of Fluxioiia! Equations. 290 
 
 indeed, from the Equation itfelf, that n muft be either 
 equal to Nothing, or to fome pofitive Integer, lefs than 
 that expreffing the Order of the higheft Fluxion in the 
 Equation : Becaufe the Term that has the lead Ex- 
 ponent, and which therefore cannot be compared v/lth 
 any other (being always affected by two or more of the 
 Factors, , n i, n 2, V 5V. will then (one of thofe 
 Factors bein:: ~c) var.ifli intirejy out of the Equation ; 
 which, thereby, is render'd pofuble. 
 
 When and A are known, the reft of the Terms will 
 be found in the common Way, as in 
 
 EXAMPLE III. 
 
 Where the Equation fropofed is yx* + axj a 1 )' r= o ; 
 to fad y. 
 
 By fuppofmg jr=r, and writing x forj, nx for 
 
 - n 2 n n i 
 
 j, and n X n i X * for j', we get x + rtax - 
 
 n 
 
 n X n i Xf?V : But it is plain that no two of the 
 'Indices of A- can, Zvnf, be equal : The Value of n muft 
 therefore be either rro or Unity (in both which Cafes 
 
 n 2 
 
 the Term n X i X a"x vanifhes) but I fhaU 
 take the latter Value, and fuppofe the nrft Term of the 
 Series to be A*- ; then, the Differences of the forefaid 
 Exponents being i and 2, the Law of the Series will be 
 expreffed by i, 2, 3, 4 &c. Whence, affumingj = 
 Ax + B.V 1 + C,v 3 -f- D x* &c. and proceeding as in the 
 former Examples, y will be found n A into x -f- 
 
 x* x 3 x+ x 5 * # 6 
 
 - + -- + 7.-: + r + 7 &c t or = A into x + 
 
 2a ~ -$<? r 8^ 3 r 2+a* QOrt 5 
 
 i 2 ^ 3 3* 4 SAT S _ 8A;6 ._ 
 
 2^ + 2.3^ + 2.3 4 3 + 2.3.4.5^* + 2.3.4.5.6^ 
 c/f. where the Law of Continuation is manifeft, the 
 Coefficient of every Numerator being compofed by the 
 Addition of the two preceding ones. 
 
 272. It
 
 300 The Refslution 
 
 272. It will be proper to obferve here, that, in Equa- 
 tions like the two laft propofed, where the higher Or- 
 ders of Fluxions are concerned, the Series exprefiing; 
 the Relation of the two Quantities muft always be found 
 in Terms of the Quantity flowing unifor.nly. And, 
 that, if the Number of Dimensions of the Fluxion of 
 the {"aid Quantity, after Substitution, be not the fame in 
 every Term, the Equation itlelf, put down to be re- 
 folved, is abfurd and impoffible, and fuch as never can 
 arife in the Solution of any Problem. In all proper 
 Equations the Number of fluxional Points (fuppofin^ 
 the Powers of the Fluxions to be wrote without Indices) 
 will be the fame in every Term. 
 
 EXAMPLE- IV. 
 
 273. Inhere let the given Equation be a 3 y ay'x -f x*yy 
 x 3 x ; to find y. 
 
 By proceeding as ufual the Indices will here be n i, 
 2, 2+i and 3; whereof the leaft (which can be no 
 other than n i and 3) being compared, n will be given 
 4,: And the Differences will therefore be o, 5, 6 j to 
 which the Double of the Second and the Sum of the 
 fecond and third, &c. being put down, and then every 
 Term increafed by 4, tbere arifes 4, 9, i o, 14, 15, 1 6, 19 
 &c for the Exponents of the Series to be affiimed f or y. 
 
 Let therefore j = A;c*-f B* 9 + C* 10 -f D* 1 * &c. then, 
 making* = i, y is = 4& 
 
 + . 
 
 And, by fubftituting thefe Values above, we have 
 
 ts'c.l _ 
 
 Whence A = ,, B = 7 , fcfo 
 
 v' J- 1 * 
 
 &. 
 
 ; j 
 
 * If for J> *be Series Ax*-\rKx s +Cx 6 +Dx 7 fcfr. cw-V r- 
 ponents ate in arithmetical Progrefan* had been aflumed, ac- 
 tor ding to the Method of feme 'very good Authors, ny lefs than 
 fcven Juperfluous Term; muft have been introduced to obtain the 
 four above givtn. 
 
 274-
 
 of F/uxiona! Equations. 301 
 
 274. Before I quit this Subject, it may not be amifs 
 to fubjoin the following Remarks. 
 
 i w . If the indeterminate Quantities are great in re- 
 fpefr, to the given ones, a defcending Series will, in moft 
 Cafes (where it is practicable) converge better than an 
 afcendingone. To obtain fuch a Scries, compare the 
 greateft Exponents, mention'd in Rule 2 inftead of the 
 leaft, and proceed according to the third and fourth 
 Rules *, whence a Series of Numbers will be found j Art. 268, 
 which, being fucceflively fubtracled from the Value of 
 , vou will have the Exponents of a defcending Series. 
 
 Thus, let the common-algebraic Equation a 3 x+ax 3 
 a*y j> 4 o be propounded j to find ^, when x is great 
 in cornparifon of a. 
 
 Then, proceeding as ufual, the Exponents of the 
 four Terms of the Equation will be i, 3, n, \n ; whereof 
 the two greateft (4/7 and 3) being made equal, we get 
 wrr*-; therefore the Differences are c, 2 and 2^; and 
 n~ -^; therefore the Differences are o, 2 and 2^; and 
 the Numbers to be fubtracted from n, are o, 2, -J, 4, 
 ' 4 7 , &c. Confequently the Scries to be affumed for ^ is 
 
 3 -! -.<5 -'3 
 
 A.v 4 + B.r 4 -f Cx *-\-D.v -} c5Y. From whence 
 
 9 > O 
 
 A l ff * a '* 
 
 y will be found =r * + ^ . 
 
 4** 4^ 
 
 2. Butj if the Quantity (x) in whofe Terms the 
 other is to be exprefied, be neither much greater nor 
 much fmaller than the given Quantity (a) 9 it will be 
 proper to fubftitute for the Excefs, or Defect, of the 
 (aid Quantity (x) above, or below, fome given Quan- 
 tity ; Ib that, having, by this means, exterminated *, 
 the Series arifing from the new Equation (wherein the 
 faid Excefs, or Defect, is the con verging Quantity) will 
 have a due Rate of Convergency. 
 
 The Ufeof this is fo obvious that it needs no Example, 
 or farther Explanation. 
 
 3. Laftly, it will be proper to obferve, that, if the 
 Equation for the Value of A, arifing from the firft Co- 
 lumn of homologous Terms, admits of two or more, 
 
 equal
 
 302 The Refolution 
 
 equal Roots (which is a Cafe that may, perhaps, nevef 
 happen in practice) all the foregoing Precepts will be in- 
 fufficient ; unlefs the Equation alfo admits of fome other 
 Root, bcfidcs the equal ones,- whereby A may be more 
 commodiouny exprefled. To determine the Exponents, 
 in that particular Cafe, divide each of the Differences 
 mention'd in Rule 3. "by the Number of the equal 
 Roots ; and then proceed as ufual. The Reafons of 
 which, as well as of the Rules themfelves, I have long 
 ago given elfewhere, and have not Room to repeat 
 them here. 
 
 SCHOLIUM. 
 
 275. Although the Bufinefs of reverting Seriefes is 
 not a Branch of the Doctrine of Fluxions, but, more 
 properly, belongs to common: Algebra ; yet, as it is 
 often ufeful where Fluxions are concerned, and falls 
 under the general Rules iilufirated in the foregoing 
 Pages, I mall here add an Example or two on that 
 Head. 
 
 Let, then, ax + bi~ + ex* + ^ 4 -f ex"" 5"r. j>; tore- 
 vert the Series, or, to find x in an Infinite Series ex- 
 prefled in the Powers ofy. 
 
 Here, by writing y for #, the Indices of the Powers 
 of j, in the Equation, will be , 2, 3^, &c. and I ; 
 therefore i. and the Differences are o, i, 2, 3, 4, 5, 
 C3V. and fo the Series to be aflumed, in this Cafe, is 
 Aj> + B/ -h Cy 3 -{- D/ bV. Which being involved and fub- 
 ftituted for the refpective Powers of x (neglecting, every 
 where, all fuch Powers of x and y as exceed the higheit 
 you would have the Series carry 'd to) there arifes 
 
 + <sC> 3 4-rtD/ 1 - 
 i -f2MB/-|-2/;AC/ 
 
 7 . 
 l&t, 
 
 -y 
 
 Whence,
 
 of Fluxional Equations. 303 
 
 Whence, 'oy comparing the homologous Terms, A=: 
 _L Jt'.'r-/- 2/>AB-fM 3 \ 2^^ 
 
 a ; b -~~ 3i C ( r * ^ : ~ ; 
 
 -f 3c- A 8 B + d A* ^ _ ^c^a^d 
 ~~ ~~ 
 
 f _, v 2/^^ at 
 
 c-c. and coniequently y -j- -f - ; - X v ! 
 
 a a 3 a 5 
 
 X V 4 Cffr. 
 
 ' 
 
 For an Inftance of the Ufe of this Conclufion, Ic'.v 
 
 .r* # J x* 
 
 + fcfir. y: Then, a being, in thi 
 ^34 
 
 Cafe, rri, b ^, =. }, </= J, 3?r. we fha{l 
 
 y a y j 
 by fubftituting thefe Values, have #r=_y + +r~ + 
 
 2 O 
 
 ; 4 
 
 feV. From whence, when y is given, A 1 will alfo be 
 24 
 
 given ; provided the Value of j> be fufliciently fmall *. * Art. jz, 
 Example 2. Let there be given ax-\-by+cy? + dyy + 
 
 e f +J** +g i y + hxy 1 * /> 3 -f kx++lx*y &c. = o j to 
 
 lind y. 
 
 By aflumins ^^A.v+B^+Cv' + D** fcfc. and pro- 
 
 ceeding as above, A will be found =: -7-, B rr . 
 
 </B + 2g A B +/+^ A + 6 A* + : A 3 ^_ 
 ,_- ^ ^,DS 
 
 2e AC 4- ^ B 1 + ^B -f- 2 AA B + 3? A a B + k + /A + 
 
 ~ 
 
 Example
 
 204 *Fbe Reflation 
 
 3. Laftly, let *" + bx m * p + ex 
 
 .. 
 Here, in order to determine the Form of the Series 
 
 to be aflumed, let z be wrote for A- in the given Equa- 
 tion, according to the ufual Method ; and then the Ex- 
 ponents, fuppofmg z tranfpofed, will be i, WOT, nm 4 
 np, nm+2np, nm 4- 3/>, &V. refpectively ; whereof the 
 two leaft (i and nm) being made equal to each other, 
 
 n is found n ; and the Differences are , , 
 m 
 
 op 
 
 =-. &c. Whence the Series to be aflumed for x is 
 
 m ' 
 
 x + Bz w 4 Cz w 4- Dz w -I- CsTr. (for it is evi- 
 dent, by Infpe&ion, that the Coefficient (A) of the 
 firft Term muft here be an Unit.) This Series being 
 therefore raifed to the feveral Powers of .*, in the given 
 Equation, by Art. ic8. and the Coefficients of the ho- 
 mologous Terms in the new Equation compared together, 
 
 T, b i+m+.2Pxf>!>2mc 
 
 it will be found that, B = -- , C ^ ^ * - - . 
 
 2m 
 
 in? 4 qmp 4- 9/>* + yn + 6p 4- i x b 3 
 - -- ~~ 
 
 From the general Value of x, found above, innu- 
 merable Theorems, for reverting particular Forms of 
 Seriefes, may be deduced. 
 
 Thus, if x+i>x* + ex 3 4. dx\ & c . = z ; then (m 
 being = r and p i ) x is = z bz* + ibb c X z 3 
 X z 4 &c. 
 
 And
 
 of Pluxional Equations. 305 
 
 And, if x -f bx* + ex* + dx 1 -f fefc. = z ; f / being 
 ~ I, and p = 2) xz bz 3 + ^bb cXZ 5 12^ 3 8<-M 
 x z 7 fcrY. 
 
 i l |- - 
 
 Alfo, if ** + &** -f / + <&* sfc. rr . z ; then 
 
 being k and />=i) x=z* ibz^+ibb zc xz 6 
 
 18^+2^ x z 8 fcfr. &ff. 
 276. It may be cbferved that, in all thefe Forms of 
 Seriefes, the nrft Term is without a Coefficient (which 
 renders the Conclufion much morefimple.) There- 
 fore, when the Series to be reverted has a Co-efficient 
 in its firft Term, the whole Equation muft be firft of 
 all divided thereby : Thus, if the Equation was 3* 
 6* 1 + Sx 3 1 3# 4 &c, y j by dividing the whole 
 
 by 3 it will become x 2#* -I -- -^ &c. = \y : 
 Where, putting z ^y, we have, by Form. I. #=:z + 
 
 SECTION III. 
 
 Of the Comparlfon of Fluents^ cr the Manner 
 of jinding one Fluent from another. 
 
 277. "\1I7"E have, already, pointed out the moft 
 VV remarkable Forms of Fluxions whofe 
 Fluents are explicable in finite Terms * j and alfo * Art. 77. 
 fliewn the Ufe of Infinite Seriefes in approximating the 7 3 -|3' 8 4 
 Values of iuch Fluents as do not come under any of ar 
 thofe Forms f : But this laft Method (as is before t Art< 99- 
 hinted) being troublefome, and attended with many. 
 Obftacles ; Mathematicians have therefore invented, 
 and {hewn, the Way of deriving one Fluent from 
 another: Which is of good Advantage when the Fluent 
 X fought
 
 306 Of the Comparifon 
 
 fought can be referred to one, like thofe in Art. 126 
 and 142. exprefling the Logarithm of a Number, or the 
 Arch of a Circle ; fince the Trouble of an infinite 
 Series is, then, avoided. 
 
 As the Subject here propofed is of fuch a Nature, 
 that it would be very tedious and difficult, if not 
 altogether impracticable, to lay down Rules and Pre- 
 cepts for all the various Cafes j I fhall deliver, what I 
 have to offer thereon, by way of Problems j beginning 
 with fome very eafy ones, for the Sake of the young 
 Proficient. 
 
 P R O B. I. 
 
 X 
 
 278. The Fluent of. ; being given (by Art. 126.) 
 
 'tisprcpofedto find, from thence , the Fluent of - 
 
 Let both the Numerator and Denominator of 
 ==, be multiply 'd by x t fo that the Quantity 
 
 without the Vinculuin, in the Fluxion, 
 
 thus transformed, may become fome ccnftantPart of the 
 Fluxion of the higheft Term under the Vinculum : 
 XVhich Part, in this Cafe, being -, let ^ of the Fluxion 
 of the firfr, Term tinder the Vinculum (or -!- cfxx] be 
 therefore added to the Numerator, in order to have the 
 
 Whole, ^7= " -> a complete Fluxion ; and then the 
 V a*x + x* 
 
 Art, 77. Fluent thereof, by the common Rule *, will be 
 V^V-J-** = -i x vV-f* 1 : But, from ibis, we are 
 
 now to dedu& the Fluent of the Quantity -: =. 
 
 1 v'v -f x* 
 
 \d i x \ 
 
 ( ~ a j that was added : Which Fluent, aj 
 
 v a* -f- jr/ 
 
 that
 
 of Fluents. 307 
 
 that of Pr= is given = *#. Log. *+ VV- 
 
 *, 
 
 will be = i a* x hyp. Leg. x -f V? -f ^ ; and con- 
 fequently the Fluent fought = I x^a 1 + x* i- a" x 
 /y/>. Z^. AT -f- V 'a 1 -$- x . g). E, I. 
 
 P R O B. II. 
 
 tc-x 
 
 279. Let it be propofed to find the Fluent of ;, 
 
 d*"x~ 
 x 
 from thai of v, - a - ; f(v0i ^y Art. 142. 
 
 ' i? . ^ 
 
 By proceeding as above, and adding \ a 1 xx to 
 
 4 a 1 xx x 3 * 
 
 the Numerator, we have *>- ; = ; whereof 
 
 V -A- ** 
 
 the Fluent, by the common Ride, is { 1/aV 1 ** 
 ( v * v/<r A- 1 : From which deducting the 
 
 7 g"".y^ i^ 1 ^ 
 
 Fluent of =, or y- --- - ('given 
 
 I/a * x* V a 1 x 1 
 
 -;- d~ X Arc (A) whofe Radius is Unity and Sine 
 
 rz f J there comes out a~ A \x V If #* . f An. 143, 
 
 280. In the fame Manner, if the Power without 
 the Vinculum, in the Exprefllon whofe Fluent is fought, 
 exceeds that in the other Exprefiion given, by the Ex- 
 ponent under the Vinculum, or by any Multiple of it, 
 the required Fluent may be determined, by one, or by 
 feveral Operations, according to the Value of the faid 
 Multiple. 
 
 XX 
 
 Thus, if the Fluent of = was fought; then, 
 
 1/V x* 
 
 becaufe the Index of .v, without the Vinculum^ exceeds 
 X 2 'that
 
 Of the Comparifon 
 
 X 
 
 thai in ;_ by twice the Exponent under the 
 
 V a" x 
 
 Vinculum^ the required Fluent may be had from that of 
 
 X 
 
 v, ~ a , at two Operations j by the firft whereof, 
 
 X'Jf 
 
 we have already found the Fluent of , ; to be = 
 
 ' Q jc 
 
 k<?A ~xV a ^ X 1 " : Whence, putting this Value 
 = 5, and proceeding as before, we alfo get ^ yVx 6 x* 
 
 O ' A//7* v> 2 | 
 
 8 
 
 ' x =. the true 
 
 Fluent of 
 
 P R O B. III. 
 
 281. Suppofog the Fluent of a + cz"- X z f "' " ! 2 to be 
 
 given = ^/, to find the Fluent of a + cz n ( X z l ~ 
 
 r= B (where the Exponent of z, without the Vinculum 
 is incrtafed by the Exponent under the Vinculum). 
 
 Let the Part affected by the Vinculum be multiplied 
 
 by z W? , and the Part without be divided by the fame 
 Quantity ; then our Fluxion will be transformed to 
 
 ~o Vtf| W f n \" w ? 1 
 az* + ez I xz K B : Where let ^ be, now, 
 
 fo taken that the Exponent (n + q) of the higheft Power 
 of z under the Vinculum may be equal to (pn + n mq) 
 that of the Power without the Vinculum -f- I ; that 
 
 is, let q =-^-~: Then (by Art. 77.) if the firft Term 
 
 ttl T 1 
 
 under
 
 of Fluents. 309 
 
 under the Vinculum was conftant, the Fluent of the 
 
 "i ^r "t*- 1 
 
 faid Expreflion, or its Equal az + cz x, 
 
 J + cz+* ? . But the 
 
 would be had 
 
 Fluxion hereof, fuppofing both Terms to be variable 
 (as they actually are) is ez* + " t? l x ~ 
 
 /7/T " " \*W 
 
 = x az* + fz" 1 ^! x **"*' (by the common 
 
 ? L ^ 
 
 T> i \ TI r flz + fz n 
 Rule.) Therefore == = - 
 
 w+ ixn-H X<r 
 
 Flu. of az* 4- <:z" Jf l v ^""'i = B ; that is, 
 
 wtl 
 
 +" ^ ? ^ . x Flu'a + aF x 
 
 x + ^ Xr w-f ? X r 
 "'i = 5 ; or, by fubftituting for y, 
 
 xz 
 
 __ p a 
 
 i = B : But the /7. of 
 
 given = A therefore, laftly, a ^ c *___X_ 
 
 m -f /> + i x nc 
 
 s 
 
 = B. Z.E.I. 
 
 x r 
 282- If the Quantity under the Vinculum be a Mul- 
 
 tinomial, <7 + <rz -f dz + ez &c- Then, fmce 
 
 i OT 'f i 
 
 the Fluxion cf a + cz" 4- Jz^" + fz 3 * 6ffj x z^" is 
 
 n i . ; i . 
 
 xm-z x + 2n^z x -r-
 
 .310 Of the Comparifon 
 
 .-m > f I 
 
 w 2 pi " za 
 
 a + cz + dk l3'c\ X a + 
 
 ^ i 
 X /* 2 = 
 
 * 
 
 , it is evident, that, if the 
 
 r pn I . pn f n I . *-f- 2 I . , 
 
 fluents of s sr, 2 , z r 55c. drawn 
 
 into the general Multiplicator a-\-cz -f d* n I3c. , be 
 denoted by A, B, C 1 , Z>, &c. the Fluent of the 
 Whole-Quantity exhibited above ( which Fluent is 
 
 1 - ~ n - - - "\ m ^ * \ 
 
 + f*+ dz " -f gg 5 Vf.i x !/" ) will alfo be ex- 
 
 preffed by pnajf+p + m+ i x <:+/> + 2*; + 2 x 
 
 x /D e?V. Therefore, if there be given 
 as many of the Fluents d, B, C, D fcf t -. as there are 
 
 Terms in rf + cs" -f~ 2 * -f ^ 3 " fcf f . OT / W O ne, that 
 other Fluent, be it which it will, will alfo be given from 
 hence. Thus if <sfc=o, e=o 9 & c . and the Value of 
 
 -- 1 m f I 
 
 /be given, we fhal] have a -f "' x /" p 
 
 - '.wf 
 
 _ i "I 
 
 j and confequently 5z; g 
 p 
 
 tne very fame as before. 
 P R O B. IV. 
 
 283. The Fluent of a+cx* Xx^"** being given (as 
 in the preceding Problem) to determine, from thence, 
 
 the Fluent of a + cs?\ X K ; fuppo/tng v to 
 
 denote a whole pofoivs Number, 
 
 Let
 
 of Fluents. 
 
 Let a + cz"' be denoted by M-, alfo put/-f i 
 ' / H H tn 
 
 P, />-*-! (/" + 2 ) /*/*+ * (P + 3) P & c ' a nd let the 
 
 Fluents of fl-ffs" 1 x , a-\-cz ' x s:^" 
 
 + ' x a/ 5 '- J i, rf + fz"' x s^"" 1 * > Gfr. be re- 
 prefented by 4, j?, C 1 , D, & c . refpeaiveJy. Then, fmce 
 
 , ?-' ,. r= B (by the preceding 
 
 rn+p+i x nc m+p+ixc 
 
 Prob.) it follows, from the very fame Argument, that 
 
 ' i 
 
 M*F _ PB =( , 
 
 m + 'p + I X nc m -\-p + I X/r 
 
 n 
 
 M f paC _ 
 
 = D 
 
 n 
 
 X c 
 
 Hence, by writing the Value of B in the fecond Equa- 
 tion, we have : 
 
 m 
 i 
 
 . . 1 A 
 
 j. - LL -' -- - r= C. In the fame Manner, 
 
 f i 
 
 m+p+ i Xm + p+l X ? 
 by fubftituting this Value for Cin the 3d Equation, we get 
 
 u 
 
 m+p+ i x w w -f /> -f i x w + ^ + i x w* 
 X 4
 
 3 1 a Of the Comparlfon 
 
 =D 
 
 m+p+i Xm+p+i XOT+/I+I X; 3 
 Where the Law of Continuation is manifeft ; and from 
 whence it appears that the Value of any of the Quan- 
 tities B, C, D, E, I3c. or the Fluent exprefled in a ge- 
 
 ncral Manner, will be 
 
 m + q + I X nc 
 
 - 
 
 X. ncc 
 x/-f 2X/> + 3 (v) x a" 'A 
 
 / v ; x _ : Where, ^ = Fluent of 
 l 
 
 X 2, q=p + v r, sq + m, t=p + m+ii and 
 where the Sign of the laft Term (m which y/ is 
 found) muft be taken + or according as v is an 
 even or odd Number : Note, alfo, that the Parenthefis 
 (v) is put to exprefs the Number of Terms, or Factors, 
 to which the Series, or Product, preceding it, is to be 
 continued. The like Notation is to be understood in 
 other Cafes of the fame Kind, when they hereafter 
 occur. 
 
 Tb*
 
 of Fluents. 
 
 *f be fame oiberwlfe-. 
 284. Let q p + v i, and let a -j- c*"\ X 
 
 P" 
 
 ^-. 
 
 Rz + Sz +Tz ...... -FA* -f M be 
 
 affumed for the Fluent fought : Then, by taking the 
 
 Fluxion thereof, you will have m+ i x ncz z x 
 
 X Rz + Sz , + A -f~ + f2 r x 
 
 . 
 
 qnzRx + ^ n x *oz ...... -{- p 
 
 _ m 
 
 f ^ X a + cz"] y.z f>a ~ I z' t which muft be = 
 
 X 
 
 ] or -f K i . x 
 
 z (or *+ >< ? V) the Fluxion 
 
 propofed : Whence, dividing the whole Equation by 
 x 2""^, and tranfpofmg, there comes out 
 
 o ? 
 
 S-z 1 
 
 9" - i 
 
 P" 
 
 qn-n 
 
 jn-z 
 
 .+/A Z f "^ 
 
 Which, reduced, and the homologous Terms united, 
 becomes , 
 
 ? XJ8 f 
 
 S 
 
 n X 
 
 x 
 
 "* 
 
 = o: Where, by making m -f q + i X ncR I = o, 
 
 r 
 
 Q &c. we 
 
 qaR. 
 
 //z-f?xc 
 
 m + q + I X c n 
 ~ i - or (putting
 
 Of the Comparifon 
 
 Hfi 
 
 Vc. 
 
 i I Xt 5-f-lXJXJ I X we 
 
 Where, becaufe the Exponent of the firft Term of the 
 Equation is qn (pn -\-vtt n) and that of the laft Term 
 (in which A and 3 are concerned) = />, it follows that 
 the Number of Coefficients to be taken as above (where- 
 of A is the laft) is exprefled by v : From which laft, 
 the Value of is given = pna&. 
 
 But, from the Law of the faid Coefficients, R, S, 
 . . . .A, it appears that the Value of A (whofe Place 
 from the Beginning is denoted by v) will be, = Hh 
 
 y.f^i. < j^2 qv+2 a"- J _ 
 
 S -J- I.J-J I.....J V + 2 
 
 X 
 
 g-r-'-g-' *+'-.. ""I': And therefore 
 
 c 
 
 s.s I p +m + 
 
 rf" P-P~^~ l 'P~^~ 2'P~{~ 3 ff^ " 
 
 a; ^ 1 X -7 ( putting 
 
 c" A/+I./+ 2. /+ 3 ('t/; / 
 
 77i-f-i=fj as before.) Now, if the feveral Values of 
 
 S. T and 3> thus found, be fubftituted in 
 
 the aflumed Exprelfion, you will have the very fame 
 Conclufion as in the preceding Article. 
 
 COROLLARY i. 
 285. Since q is =:/>-}- v I, the Fluent -f f**i X 
 
 1" r> ?""" "~ P" f i 
 
 J?2 + o* .... -fA^ 4- 3 given above, may 
 
 be exprefled by N X Rz +Sz"' +Tz 
 
 j+i 
 
 (vj 4- p/f; where A r = + "/ x 3 ", R =
 
 of Fluents. 
 
 : And, where the Coefficient (/J) of the 
 m +p + v 2 . c 
 
 given Fluent (A) will always be exprefled by the laft of 
 the Quantities R, <S, T . . . A multiplied bypna: 
 This is evident, becaufe it is found that #r: /># A 
 And the fame thing will alfo appear from the feveral par- 
 ticular Cafes (in Art. 283.) for the Values of B, C and 
 D : In each of which the Coefficient of the laft Term 
 (where A is concerned) is to that of the Term imme- 
 diately preceding it, in the conftant Ratio of i>a to 
 
 , or of pna to Unity. 
 
 COROLLARY II. 
 
 286. If the Value of c be negative, the general Fluent 
 (in Art. 283.) when a + cz ' = o (provided m+ 1, , and 
 
 p be pofitive) will become barely = + x - - x 
 
 " t t +i 
 . , f jj 
 
 (v) X -^j- } becaufe, in this Circumftance, all 
 
 * T 2 
 
 the Terms multiplied by a Z intirely vanifli. 
 
 If, therefore, b be wrote for c (to render the Ex- 
 
 preffion more commodious) we fliall have +- x - 
 
 ^+2 a" A . - , 
 
 X j C-y) x - for the true Fluent of a h?\ X 
 
 a * , generated while *, from Nothing, be- 
 
 comes = a : Where A denotes the Fluent of a z"' 
 
 /mi j 
 
 x * ^j generated in the fame time ; and where 
 
 t "==, 
 10
 
 Of the Comparifon 
 t =. p -f- m -f i. Hence it follows that the Fluent of 
 
 , nl m pn l . , - n ^n , -\n 
 
 a t>z\ xz x X e+fz -f gz -f hz y &c. 
 (where *,/, , are any given Quantities) will be = A X 
 
 paf p.p+i.a*g 
 
 - 
 
 < m the 
 
 forementioned Circumftance. 
 
 P R O B. V. 
 
 287. The Fluent (A) of a + cx"i x z p " " T x. being gi~ 
 
 . ~ m-\-r 
 
 vcn, to find the Fluent of a + cz"\ x z f *' * z ; f u p- 
 pojing r to denote a whole pofitive Number. 
 
 Since a + czfl =r a +cz"l x a 4- ^iz", it is evident 
 
 . . v m-f-l j _ ~~1 W A 
 
 % a + fz I x az *-f- 
 
 ,^+ i i: Whofe Fluent ^y 7>* 7 ^ 
 
 Q+ CZ 
 
 . t: 
 
 a+c* 
 
 + ' 
 
 \m+i 
 
 j 
 
 In like Manner, if 
 
 this Fluent, of a -f ex" I > be denoted by 
 
 B, that of a + ? X z p " * % by C, ^fr. it will ap- 
 
 <7 + ftB*' X Z " , 
 
 pear that 
 a + cz"\ 
 
 m -f 2 X aB 
 
 p + m + 2 X n 
 
 'X z^" 
 
 _ Z)> ^ Whence, 
 
 by fubftituting thefe Values, one by one, as in the pre- 
 ceding
 
 of Fluents. 317 
 
 ceding Problem, and putting Q = a -f cz^ we get 
 *+**? j^TIxg"^Vl_ 
 
 m + . 
 
 r. Whence it is 
 
 -.m + r 
 
 evident, by Infpedion, that the Fluent of 
 
 x a*" 1 " x z, exprefled in a general Manner, will be 
 
 r.n p + m + rxp-f m-\-r i.n 
 by putting m+rf, p + m-{-r=g, and making J^*" 1 '! X 
 z fn a general Multiplicator, will be reduced to J^*^ 
 
 2 X ' * * -f- ~~'~ ", " I- -"""-* ~~ " _ _.'_~ ( T j "T" 
 
 m+i m -f 2 m -f 3 
 7 X TT T~" x r+ T~ r ^ a -^ where it 
 
 appears (from the foregoing Values of 5, C, and DJ 
 that the Coefficient of A is always equal tothelaft Term 
 
 of the preceding Series, multiplied by m -\- j x na (in- 
 ftead of ^ z^"). ^. . /. 
 
 COROLLARY. 
 
 288. If c be negative, fo that j^, or its Equal, 
 a + cz", may become equal to Nothing, the Fluent will, 
 
 in 
 9
 
 31 8 Of the Comparifon 
 
 in that Circumftance, be barely rz X ' 
 
 p + m + i p + m + 2 
 
 X ~ (r) X tf r ^J provided the Values of m+i, 
 
 />, and n are pofitive : Or, if c, p, and n be pofitive, and 
 m + r+p negative, the fame Exprefiion will exhibit the 
 true Value of the whole Fluent, generated while z, 
 from Nothing, becomes infinite. 
 
 P R O B. VI. 
 
 289. The fame being given as in the preceding Problems ; 
 
 . w r 
 
 it is propofed to fnd the Fluent of a -f- cz"' X 
 
 />-!. 
 
 a x. 
 
 If r be wrote inftead of r, in the laft Article, 
 we fhall have m r=zf, p + m r=zg, and ^"+ I z f * 
 
 - (-r) X a"~ r ^, exprefling the required Fluent in 
 p + rn + 2 
 
 this Cafe. 
 
 m+i rn + 2 
 
 But - x . - C5<r. continued to r 
 i + m+z 
 
 Faftors, fignifies the fame thing as the Product con- 
 tinued downwards, or the contrary way, to r Factors, 
 according to the fame Law : And therefore is rr 
 
 , \ A r* u r 
 W. After the fame 
 
 , . 
 
 ( -
 
 of Fluents. 319 
 
 (r) and confequently the Fluent itfelf = j^z x 
 
 -gT r j$T *+'.jr+.g 
 
 290. It appears from hence that the Coefficient of A^ 
 the given Fluent, will always be equal to that of the 
 laft Term of the preceding Series, multiplied byp + mxn : 
 For, feeing the Coefficient of the faid laft Term (whofe 
 Diftance from the tirft, inclufive, is denoted by r) muft be 
 
 x ~ r (by the Law of 
 
 the Series) where f+rm and ^-fr i = /> + w i (as 
 appears from above) it follows, by inverting the Order 
 
 of both Progreffions, that t + >*-i.p + >-~*.(r-i) 
 
 m.m i.m 2 (r) 
 
 x JL w ill alfo exprefs the fame Coefficient : Which, 
 
 na 
 
 - . p + m.p + m-i.p + m-z (r) 
 multiplied by p + X, gives w . M _ Itffl , 2 (r; 
 
 , the very Coefficient of ^, above determined. The 
 
 a 
 
 Ufe of this Conclufion will be feen in what follows. 
 
 P.R O-
 
 320 Of the Companfon 
 
 P R O B. VII. 
 291. The fame being, ft ill, given j to find the Fluent of 
 
 m 
 
 ~"T\ pn -vni 
 
 a + cz i x z z. 
 
 By proceeding as in the laft Problem, the required 
 Fluent of a + cz"\ X z'" is derived from that of 
 
 ' ^ fft -fif-l-^TJit *1 
 
 a + cz n \ Xz -x ( given by Prol. 4.) and comes out 
 
 vn 
 -.m+1 pn 2 
 
 i 4L -'* x : 
 
 i ' 2 / 
 
 , i . , . 
 
 (v) - x - x * (v) 
 
 l 2 
 
 c 
 
 . - : Where, G> = 
 
 ' And where, the Coefficient of A\s equal 
 to that of the laft of the preceding Terms, multiplied 
 by m + p x we. If the Manner of deducing the re- 
 quired Fluent, in this, and the laft, Problem, {hould not 
 appear fufficiently plain and fatis factory to the Beginner j 
 the fame Conclufions may be, otherwife, brought out ; 
 by finding A^ in Terms of B, C, or Z>, from the fe- 
 veral particular Equations in Art. 283. or, by afluming 
 a defcending Series, inftead of an afcending one. Vid. 
 Art. 284. 
 
 P R O B. VIII. 
 
 292. The fame being, ftill, given ; to find the Fluent of 
 
 - ?l n + r pn+vnl . 
 
 a + cz l xz ? x. 
 
 Let the Fluent of a -f cz" I xz x. (given by 
 
 Trob. 4.) be denoted by B, and that required by F: 
 
 Then,
 
 of Fluents. 321 
 
 HI ' 
 
 Then, if p + v be put = />, the Value of F (the Fluent 
 
 of a + cz"\ x^"~ T x} will be given from that of B 
 (the Fluent of a + cz"l x yf n ~ l ) by writing 5 for 
 A and /> for />, in A/. 287. Whence we get F=.> 
 
 Z ** ~ ~t~ i - . -- 1 -j- ~" ^ ~- " " ^ ^ /*. 1 i 
 
 ^ g.gi.n g.gi-gw ( } 
 m-\-i m-\- r i ?K+3 . r o . TVd 
 
 i>-}-;7z-f-i p -\-tn-\- 7. / ) +'W-h3 
 
 i y 
 
 , and 
 
 Which Fluent, by fubftituting the Value of B (in 
 
 Prob. 4.) becomes F=.^ t+1 z t "' X 5 + ^^ - 
 
 t.eiM 
 
 + . v : j^_ rr ; + - x 7- - , ; 
 
 < g r -^ i-g i' n p+to+i p+m + 2 ( 
 
 x 
 
 ?+2 /> 
 
 (*) i x / ( r ) x a x T x 
 
 p+m+l p+m+2 
 
 \i x ~V : Where q=p+vi, s=m+q= 
 
 and tp+m+i ; and where the Sign of the laft Term 
 j s + or according as v is an even or odd Number. 
 3> F L 
 
 +y .i-/ 1 * * 
 
 Co-
 
 322 Of the Cot/jpanjbn 
 
 292. If the Jaft Term of the firft Scries, exdulive 
 
 i - 
 
 of the general Multiplicator j^ , be denoted by 
 
 r 
 0, the Multiphcator, -, - * - - (r) x a , to 
 
 p + m+i p+ m+2. 
 
 Art. 287, t h e f econc j Series will be m+i X na$ * ; and there- 
 fore the firft Term of this Series, including its Mul-* 
 
 .. .tf . Wh : ch :r n 
 
 tiplicators, is = - i^r -- vvmcii, it A. 
 
 be put to denote the laft Term &3*+* x f*+ v * of the 
 firft Series (with its Multiplicator) will be expounded by 
 
 _L TO 
 
 ' - ; Hence it follows, that the Fluent of 
 
 S+l .CZ n 
 
 +r Xx pa+ ' vtt ~ 1 x, given above, will alfo be truly 
 
 *& f"*~I *"* / 
 
 exprefled by r- ZZTj x ~jf ^7^. X 
 
 fTW o *^ ** 
 
 ,/ / 2 f^ / x . m + ' aR __ 9_ 
 
 f2l ?"" a aV f \ f 
 
 X """"1 "* "*" T" X ~~"T" ( ^V 
 
 p + m+i .p + 7n + 2 (r) X /./> 
 
 Where H, I y K, L . ^, 5, T t V, &t. repre- 
 
 fent the Terms immediately preceding thofe where they 
 ftand, under their proper Signs : R being the laft Term 
 of the firft Series i alfo /= m + r, g = m 4- r + /> + -, 
 
 qp+v i, f=w+f, t=?n+p+i> and ^= 
 
 Co-
 
 effluents. A 23 
 
 COROLLARY II. 
 
 i 
 
 293. Since the Divifor, p + m + i ,p+ m+2 (r) x 
 l.t + i.t (v) y of the laft Term of the Fluent (by 
 
 fubftituting for / and p fcff.) is = p+m+i.p+m+z 
 
 (y) x />+V+/B+I . />+^+;tf+2 (r) : Where, the laft 
 Factor (p+m+v) of the firft Progreflion, is lefs by 
 Unity than the firft Fadtor of the Second; it is evident 
 that the faid fecond Progreflion is only a Continuation 
 of the firft to r more Factors : And fo, the laft Term 
 of the Fluent, where A is found, is truly exprefled by -f~ 
 
 m + 2 . ffl+3 (r) a 1 ^' 'A 
 
 . m+p+2 . 
 Hence it follows, that the Fluent of a + ex" 
 
 fn + vni . .. - T"7r + r 
 
 x v *, or that of a bz*\ X z 
 
 ( making <rrr ij will, when a bz n becomes equal 
 to Nothing, be barely zr 
 
 p.p + i . p+2 (v] x m + i . m+2 . m+3 (r) a"*' A 
 
 m+p+i . m+p+2 . m+p+s (v + r] 
 
 - m ^ _ l 
 
 A being the Fluent of a z"! x tf , in that Cir- 
 cumftance, v and r whole pofitive Number?, and p 
 and m + 1 any pofitive Numbers, either whole or broken. 
 
 SCHOLIUM. 
 
 ~tm-fr * f 
 
 294. If the Fluent of #-j-: n l x 2 (given 
 by Prc^. 5.) be denoted by Cj then (F) the Fluent of 
 
 . . 
 
 r^" X ^" "'a: (where m = w-f r) will be had, 
 from C (by Prob. 4.) according to a new Form, dif- 
 Y 2 ferent
 
 324 Of th 
 
 ferent from thofe already given. And, by following 
 the fame Method, the Fluents of a + "- ! * 
 
 fr.+vi . ~~rr+ r '/ i. rr"' 
 
 js z, a-tczl x a K, and a+cz* \ 
 
 X sr x. may alfo be found, each, according to 
 
 two different Forms, from a Combination of the cor- 
 refponding Cafes in the foregoing Problems. 
 
 But, as it is extremely tirefome to repeat the fame 
 thing, again and again, where fuch a Number of Sym- 
 bols are neceflarily concerned, I fhall here put down 
 one Solution to each Cafe (becaufe of their Ufe) leavig 
 the Procefs and the other Forms (which contain no new 
 Difficulty) toTbofe v?ho will be at the Trouble to fet 
 about them. 
 
 , TL fl r 
 
 i. The rluent of a+c* ' x s 
 
 z is 
 
 X set** g-*~J ^f 1 + * %/ N 
 i _L ~~z~\ X - 4~ ^ . X \r} 
 
 f+ i.na 
 
 * ^ w^,,. _ V^ ^^^ , TX ^ 
 
 _ ^" .. ... ^^ ^\ ^ 
 
 n f ^__. T f ^ O 
 
 . j rz A cz 2 
 
 XVhere //, /, K, L, R, 5, T 9 &c. denote the 
 
 Terms immediately preceding thofe where they ftand, 
 under their proper Signs ; R being the laft Term of the 
 
 firft Series, alfo ^=fl-f -ex.", f=mr 9 g=p + m~^vr, 
 qp-^v i, sm-\- p-\~v j, / /-f m + r, and A ~ 
 
 J" tn i 
 
 tfie given Fluent of ff-f "/ x z * 
 
 2. Th
 
 of Fluents, 
 
 325 
 
 2. The Fluent 
 
 x* 
 
 s = 
 
 w 
 
 Where qpvi^ s~m+r+p fm-\-r y gp-{- 
 +r, and the reft as in the preceding Cafe. 
 
 3. The Fluent of a-\-c~'' x 
 
 m 
 
 pn vn I . 
 
 Z IS = 
 
 '+* P 
 X z 
 
 ) 
 
 . 
 
 I . m 2 ( r ) X pl^2.p 3 ( v ) /+* 
 
 In which y=w r, g-=m+p r -v, qp v?l 
 j ^ -f ?, and the reft as before. 
 
 205. From what has been delivered in this SeHon, 
 the Fluents of various Forms of Fluxions may be ex- 
 hibited, by means of circular Arcs and Logarithms. 
 
 For, fince the Fluents of "' X 
 
 ^X^"'" 1 ^ and +^T~' X* i (which 
 I call original Ones) are all of them explicable by one, 
 or the other, of thefe two Kinds of Quantities (as will 
 
 - **""' ' 
 
 appear farther on) thofe of 
 
 
 ^ w ~ r i will alfo be given from thence, by the fore- 
 ' Y 3 going
 
 326 Of the Comparifon 
 
 going Theorems. Whence the moil ufeful Forms of 
 Fluents in Csttis's Hannonia Menfiirarum will be ob- 
 tained, befides feme others, more general than any, or 
 the fame Kind, put down by that fagacious Author. 
 
 Here follow a few Examples of fome of the mod 
 ufeful Cafes, 
 
 EXAMPLE I. 
 
 296. Let the Fluxion given be 
 
 X Z 2V ) v being any whole pcfitive Number. 
 Then, the Fluent of d x -z-\ x a, or 77 TJ~T 
 
 being = hyp. Log. w -3 ; or, equal to the 
 
 t Art. 126. Arch whofe ine is r and Radius Unity * j according 
 
 as the fecond Term, in <5P+*% is pofitive or negative ; 
 let A be, therefore, taken to denote the faid Arch, or 
 
 Logarithm j and let a*+z\ * X * be compared with 
 
 X 2 2 (whofe Fluent is, all along, fup- 
 pofed to be given /I] and you will have <3zr^% <:= 
 ~j-i 9 n=2, m~ -|, 2/> 1=0, and therefore /> 4 5 
 Whence, by fubftitutjng thofe Values in Art* 283. we 
 
 "J^J T 
 
 ] ik e wife get q (p + v i) = , f (^-f-?) = 
 
 I, / (m+p+ i) ni ; and, confequently, the Fluent 
 
 fought - F^T x + "!!. _ jEE^jgH! + 
 
 2V 2V.2Z; 2 
 
 5 -- ' - ' ' 2-u 7 
 
 , ,. 
 'IV 1.11) - 3-fl Z 2V 1. 2V 3- 2V 
 
 2V. 21; i-2V 4
 
 cf Fluents. 327 
 
 {) + x x |-x ~r (v)xd iv : In which the 
 ~ 2 4 6 8 
 
 laft Term is negative, when the given Fluxion is 
 
 z a<1 ' " 
 . and v, at the fame time, an odd Dumber ; 
 
 but in all other Cafes, affirmative. 
 
 EXAMPLE II. 
 
 ?97- Let z~~'z Vfij'S (or d 1 +2;'M X z v z) be 
 
 propounded. 
 
 4. 
 
 Hre, denoting the Fluent aTfir\ l * by A (as 
 
 above) and comparing d*-j-z?\, x z 2 ^, with 
 
 pn-t-vrtr l . .ft..' 
 
 a -f cz ' X z 2 ( ^/a, fyob. 8. ) we have r = r, 
 
 and the reft as in the laft Example : Whence alfo 
 
 t 
 
 p 
 
 z% and the Fluent itfelf = 
 
 2"J -f 
 
 x .( w ; x' r^ 5, r, eifr. being the pre- t A rt. 292 . 
 
 2V-|-2 
 
 ceding Terms with their Signs) rr 
 
 2V+2 
 
 ,7, 1*V 1 O7 . x/4- 2a; 3 
 
 - + 
 
 2-L".2-y 2 2-y.2-z; 2.2^ 4 
 
 the Sign of the laft Term muft be regulated as in th^ 
 Y 4 pre
 
 3 28 Of the Comparlfon 
 
 ~WII ' 
 
 preceding Example If the Fluent of . K ~ or 
 
 _ y?*? 
 
 of z "** Vd z -j^K L (in which the Exponent is ne- 
 gative) be required j the Anfwer will be had in finite 
 Terms, independent of A> by Art. 85. 
 
 EXAMPLE III. 
 
 - \ I+ r 
 298. Wherein the Fluxion propofed is d" K "\ x 
 
 s*V? ~*x J r and v being any whole pofitive Numbers. 
 
 __ _ _^ j j 
 
 Since the Fluent of ;/" z" ( ~ r X^je* * (as will 
 
 2 
 
 appear hereafter) is truly exprefled by x Arch y whofo 
 
 > 
 
 Sine is -r- and Radius Unity, let this Value be de- 
 d* 
 
 noted by A\ and then, by writing d n for a, i for r, 
 * for w, and i for />, / y/r/. 292. we fhall have f 
 
 j ____ T 
 
 
 2V- I 
 
 *) = < z ' l T-''j and the Fluent, itfelf, equal to 
 ~r <L- ' r 
 
 2r I 
 
 J- . - ^- V 
 
 _L - v - 
 
 ^^ J 
 
 . i fr) _ : l 
 
 ' U 
 
 - - f . .. 
 
 -T" + 1^2 X ^ 2.4.6.8.10.12 (r+v) 
 
 t* 
 
 Art, 293. x d rn + vtt J: In which //, I, K . . . R, S, T, &c. 
 denote the preceding Terms with their Signs ; R being
 
 Of Fluents. 329 
 
 the laft Term of the firft Series. Hence, becaufe all the 
 Terms, but the laft, vanifli, when 4L~ il follows that 
 
 the whole Fluent of d" z r ] * x z z, generated 
 
 while z, from Nothing, becomes equal to d^ is truly 
 
 expreffed by 
 7 
 
 .4.6.8.10.12 (r + v) 
 L 3.5.7 (r) x ,,.fr 7 f; ^G G bei 
 
 2.4 6.8.10.12 (r + v) ff 
 
 the Semi-Periphery of the Circle whofe Radius is Unity. 
 
 EXAMPLE IV. 
 
 299. Let it be required to find the whole Fluent of 
 - -, n i 
 a bz ' X 
 
 . .,,/. >T 
 , generated while bz , /J-^/n .ZV0- 
 
 m 
 
 -. i 
 
 thing) becomes a ; that of a bz''\ X s 
 given ( A.} 
 
 o (=^) 
 
 Here, by expanding <f+lxn , our given Fluxion 
 
 m 
 
 becomes ~ a bz'"\ x z ~ into d * 
 
 d 1.2.</* I.2.3.^ 3 
 
 Which Series being compared with +/*" -t- ^* 2fl f^ 4 
 
 Q 7 
 
 fFVi/. ^r/. 286.) we have <? = i, / rr , ^ 
 
 t-^ , &c. and confequently the Fluent fought 
 
 A 
 (by fubflituting thefe Values) equal to into i 
 
 d 
 
 " 
 * 3 ak p p+i fi+i ak 
 
 - V/ - NX 4- V . - V 
 
 t X I X ^ f ' /+! I 2 X
 
 33 Of the Companfon of Fluents, 
 
 L PL til * L I!I _ 2 , 3 3 fcr,. (< 
 
 S\ ^^ 1 if I \ 
 
 / /+I- f+2 12 3 Mf 
 
 being =r />+CT+ i.) 
 
 Here the Values of w+i, n and p are fuppofed po- 
 
 / 
 * Art> 2 g6. Titive ; * and it is requifite that I + 7^ fiiould alfo be 
 
 pofitive; otherwife the Fluent will fail. Although the 
 Series brought out above runs on to Infinity, yet it may 
 be- fum'd, in many Cafes : Thus, if the given Fluxion 
 
 i then, the forefaid Series be- 
 
 _ 
 coming I I X + i x | x &?' its Sura 
 
 confequently x f + 
 
 ^ 
 p: the Fluent fought : Where, /f (the tw&<?& Fluent of 
 
 a=:3z n < Xz* being =: ^ x Semi-Peri- 
 
 nVb 
 phery of the Circle whofe Radius is Unity, the Fluent 
 
 given above will, therefore, be rz 
 
 * + adk 
 
 X by the fame Semi-Periphery. If the Reader is de- 
 firous to fee a further Application of the Summation of 
 Scriefes, to the finding of Fluents, I muft refer him to 
 niy Differ tations (where it is handled in a general Man- 
 ner) having neither Room nor Inclination to treat of 
 it here. 
 
 SEC-
 
 33* 
 SECTION IV. 
 
 Of the Transformation of Fluxions. 
 
 301. T) Y the Transformation of Fluxions may be 
 j underftood, the reducing any fluxional Quan- 
 tity to a different, or more commodious, Form ; ac- 
 cording to which Senfe, a great Part of the fecond 
 Section would properly fall under this Head. But, what 
 is here propofed, and what is commonly meant by the 
 Transformation of Fluxions, ;V, the Method of or- 
 dering thofc Kinds of Expreflions which involve one va- 
 riable Quantity only with its Fluxion ; which, yet, are 
 fo affedted by radical Signs, that the Fluent, without 
 an Infinite Series, would be impracticable, were it not 
 for a new Subftitution, or fome other Kind of Tranf- 
 formation, whereby the given Fluxion is render'd more 
 manageable. 
 
 Something of this Sort has been already touch'd upon 
 in Art. 83. And in what follows I (hall farther point 
 out and exemplify the principal Cafes wherein fuch 2 
 Procedure will be of Service. 
 
 302. If the Number of Dimenjions of the variable Quan- 
 tity , without the Vinculum, increafedby Unity ^ be fome ali- 
 quot Part) or Parts, of the Dimenjions of the fame hian- 
 tity, under the Vinculum, the Fluxion will be reduced to 
 a better Form by fubjlituting for that Power of the va- 
 riable Quantity, which arifes by dividing its Exponent, 
 under the Vinculum, by the Denominator of the Fraftion 
 txpr ejfmg the faid aliquot Part, or Parts. 
 
 n I 
 
 I 
 
 Thus, if the Fluxion propounded be ~; by 
 
 liibftituting *=s a , and taking the Fluxion of both Sides 
 of the Equation, we have A =.\nz i z ; and there- 
 fore z a x 7- ' Which Value, with that of K , 
 
 being wrote for their Equals, in the given Fluxion, it 
 10 will
 
 332 Of the transformation 
 
 X 
 
 will be transformed to , -, -^-=-\ Which, putting; 
 *"v c" + x* 
 
 * 
 
 a-=c^ (to make the Terms homologous), is alfo ex- 
 
 9f 
 
 prefled by - ;. : Whereof the Fluent will be 
 
 ~nv a -v_x 
 
 given by Art. 126. or Art. 142. according as the Sign 
 of x is pofitive or negative. 
 
 303. If the Power of the variable Quantity under 
 the Vinculum has a Coefficient, it will be bell to bring 
 that Coefficient without the Vinculum. 
 
 i,: i . 
 
 z* ~ 
 
 Ex. i. Where let the Fluxion given be -== : 
 
 V a + *" 
 
 Which, by bringing c without the Vinculum^ becomes 
 l*-i . in 
 
 * 
 
 : From whence, by putting x r= z. 
 
 and proceeding as above, we get 
 
 Whofe Fluent, by Art. 126. is 7- x kyp. Log. x 4- 
 
 2. 
 
 f- **. This, by reftoring z, becomes --, x 
 
 f JlC'i 
 
 fyp. Log. z? + \ - + ^". Which > correded (by 
 
 2 
 
 fuppofing it o when % = o) gives, at length, -^ x 
 
 1 " 
 
 ' " typ- Log. \/ ~ 
 
 Fluent of the Quantity propofed.
 
 of Fluxions. 
 
 But, when c is a negative Quantity, this Fluent fails, 
 becaufe the fquare Root of c is to be extracted. In 
 
 * X 
 
 this Cafe ; > - muft be transformed to 
 
 ia/\/- + x* 
 
 And then its Fluent (by 
 
 , / a 
 
 * V ex \S x 1 
 
 1 C 
 
 Art. 142.) will be had . X the Arch of a 
 
 Circle whofe Radius is Unity, and Right-Sine rr 
 x 
 J~ '" 
 
 Ex. 3. Let the given Fluxion be . : 
 
 * V a -}- czf 
 
 Which, by bringing c without the Vinculum t and put- 
 ting x =. z, 1 , is transformed to -j- 
 
 Whereof the Fluent, by Art. 126. is -= x 
 
 ny a 
 
 v/- J + ^ + 
 
 C 
 
 V~7-~ V a + " 
 
 = ~= X ^. LV. 
 n v a 
 
 But here, when c is pofitive, 
 
 V a + V a + c *r 
 
 the Numerator will be negative; in which Cafe it will 
 
 be proper to change its Signs, andexprefs the Fluent by 
 
 h= X hyp. Log. "*" __. 
 
 V -f " + V ____ 
 
 an
 
 334 Of the Transformation 
 
 an Alteration of the Signs can make no Difference i.t 
 the Fluxion, is evident from the Nature of Logarithms ; 
 
 X x \ 
 
 becaufe the Fluxion of the Lb. of * ("=. / 
 
 v x X ' 
 
 is the fame with that of the byp. Log. of x. It will be 
 proper to obferve farther, that, iriftead of the Logarithm 
 above derived, any one of the following, equal, Quan- 
 
 r V a -4- cz n v/ a\ 
 titles may be taken ; viz. hyp. Lo*. ; : - - 
 
 (found by multiplying both the Numerator and Deno- 
 minator of the forefaid Logarithm by V 'a -{- cz" VV/) 
 
 y' a _j_ c ~ y^~a~ 
 
 = 2 X fyp. Logi ~ (by the Nature 
 
 V cz? 
 
 of Logarithms) =: 2 x hyp. Log. 
 
 V a + cz n + V 
 
 a 
 
 ( by multiplying, equally, by V a + cz" -}- V a ) 
 But, take which of thefe Forms you will, the Fluent 
 fails when a is negative j becaufe the general Multiplicator 
 
 = is then impoflible. In this Cafe the Fluent of 
 
 n V a 
 
 or its Equal ~ , w il! 
 
 be given- by Art. 142. and is expounded by -, 
 
 ' , 
 c 
 
 2 
 
 X A 7 ; where A denotes the Arch whofe 
 n Y a 
 
 *9 J \ 
 
 Radius is Unity, and Secant j=-( A/lfl ]. 
 
 /-^ aj 
 
 V c 
 
 In
 
 tf Fluxions. 
 
 t B j . 
 
 In the fame Manner the Fluent of fll , i$ found 
 
 + CZ* 
 
 I 
 
 = / x Arch, whofe Radius is Unity and Tan- 
 
 V <7 
 
 J . I 
 
 eent V' 1 > or equal to , X hyp. Log, 
 a "V ca 
 
 *\, according; as the Value of c is affir- 
 
 V a V - c ^ 
 
 mative or negative ; a being fuppofed affirmative. 
 
 304. When the Power 9 or Powers, of the variable 
 Quantity without the Vinculum, or radical Sign, fall y 
 mcflly, in the Denominator, it may be of Ufe to fub- 
 Jlitute for the Reciprocal of the faid Quantity, cr for 
 the Quotient which arifes by dividing feme known Quan- 
 tity, either, by it, or by fame Compound of it in the De- 
 nominator* 
 
 33 
 
 Ex. r. Let the propofed Fluxion be ~i ; , ; 
 
 z \/ a* -f 2* ' 
 
 then, putting x = , we have z = ~, and % 
 a ~ ; and confequently ^5- 
 
 Whereof the Fluent is 
 
 Ex. 2. Let the given Fluxion be 
 
 Here, putting A: = , we Have % = 
 - 
 
 1 = Va* ax + x* ; and therefore the 
 
 Quantity
 
 33 6 Of the 
 
 Quantity propofed is transformed to T-T. ===== 
 
 Whofe Fluent may be found from a Table of Loga- 
 rithms j as will appear farther on. 
 
 305. If the Fluxion given is ajfefted by two dlf- 
 fertnt Surds, and the rational Fatter, or the Quantity 
 without the Vinculum, be in a conjlant Ratio to the 
 Fluxion of the Quantity tinder the Vinculum of either 
 Surd, or be related to it as in Art. 83. the given Fluxion 
 will be reduced to a more f:mple Form, by fubjlituting for 
 that Surd, 
 
 y~ \^ / z \ z. 1 
 Ex. I. Let - be propounded. 
 
 ^ZEE 7 
 
 Then, putting x V f -f ~% we have z?.x* 
 ^ = xx > and Vc * T = vV + f x* - V ^ 
 
 23 y p- _j_ x * 
 
 (by making a V '<? + 2 ) Whence / 
 
 Or, if x be put ~ VV x 1 (inftead of >/* -f- z 1 ) ; 
 then x* = c* *% g^ = ' xx, V b~ -f- * = 
 V i z -I- r 1 ^ ^r 2 n V a 2 x 1 ', and confequently 
 
 . = x Va 1 x* : Whofe Fluent is 
 
 given by Art. 297. or 131. 
 
 Ex. 2. Let the given Fluxion be a -f f"l x e -f /* I X 
 f Art* 83. zf"~ ~ 1 K ; fuppofing p to denote any whole pofitive Number fu 
 In this Cafe, let that of the two Quantities, a -f ex." 
 and ^ + /z", whofe Index (m or r) is the mftl com- 
 plex (which we will fuppofe the latter) be put = x ; 
 
 
 then we (hall have s" --jr--, z n -* z = ~ F J
 
 of Fluxions. 
 
 337 
 
 d + by putting d =. a 
 
 __\ and confequently d + - \ x 
 
 = the Fluxion propofed : Where, p i being a whole 
 
 pofitive Number, the Value of x e\ will therefore 
 be exprefled, in finite Terms ; whence, if m be alfo a 
 whole pofitive Number, the Fluent itfelf will be had in 
 finite Terms : But, if m and r be the Halves of odd 
 Numbers, then the Fluent will be found (from Art. 
 298 or 294.) by means of circular Arcs and Logarithms. 
 
 306. If the given Exprejfion be ajfefted by two Surds 
 wherein the Powers of the variable Quantity are the 
 fame, and the rational Quantity , without the Vinculums, 
 be related to the Fluxion of either Surd, as in Art. 83. 
 it may be of Ufe to fubjlitute fir the Quotient , or Ratio, 
 of the two Quantities under the radical Signs ; especially, 
 if the Sum of the faid radical Signs, or Exponents 
 (fuppofing both Surds to be reduced to the Denominator) 
 is a whole Number. 
 
 Ex. I. Let the given Fluxion be 
 
 z'x 
 
 b 3 + z* Sxtf 
 
 Then, writing x -, --, we have z 3 
 
 r 3 x* i +# 
 
 and
 
 Of the Transformation 
 and confequently 
 
 z z 
 
 Whofe Fluent is 
 
 x 
 
 Ex. 2. 
 
 J_ A" 
 Here, putting x = > you will have " = 
 
 ax e ni. afce X* 
 
 fcx 
 
 _ i^ x _ ax 
 i*\ 
 
 . 
 
 pn _ n 
 
 , a 
 
 n X fcx\ 
 
 
 tr ,+/v _ ^rf 
 
 ' ^^" 
 
 confequently the 
 
 
 - 7 - ~ r 
 
 \ _ a f ce \ 
 ~ / cx\ 
 
 Fluxion given = 
 
 Where, if ffl + r be a whole pofitjye Number, greater 
 than ralfo a vhole pofitive Number) the Fluent w.ll 
 
 becaufebolh the Seriefe. 
 
 tM+r -p -I 
 
 and / r.J do 
 
 for the Values of ** 7| 
 
 Art. 99. j nthat Cafe t?rn -; :r ,ate *. But, if r and m-fr / I 
 be the Hakes of whole Numbers, pofitive or negative, 
 theifthe Fluent will be given by the laft Setion. 
 
 307. A Trinomial is reduced ta a Binomial by taking 
 away its middle Term ; that is, by ful/lituting for the 
 Sum or Difference of the Power of the variable Quantity
 
 of Fluxions. 2 39 
 
 n that Term and half i's Coefficient ; according as the 
 Signs of the two Terms, where the fald Quantity is found, 
 are like, or unlike. 
 
 Ex. I. Let the given FLixion be == 
 
 V t>~ -j- cz -f- *~ 
 
 then, putting X z-J- Ic, or 2; x if, we have z x, 
 
 vV i r -f ** ; whence (making a* = b l ^c 1 ) 
 
 there refults x = /- . : Whofe Flu- 
 
 t//' -f^z + z Va- + x- 
 cm is given, iy ^'/. 126. 
 
 /" * T ^. 
 Ex. 2. Z,*/ //;^ Fluxion given be - .... .' . z - r -^-. 
 
 V+te" + f 2 a "' 
 
 Firft, by bringing c without the Vinculum^ according 
 to Art. 303. we have V* 4- tz* + cx~" V7" x 
 
 5 -L + z 2 " : And, by putting x = z* + 
 c f 
 
 b -\r n ~ l * 
 
 , or z " = A: -, we alfo get 2 * = _, and 
 
 c 
 
 5 + fi } = V^ 7 - ^ + * : Therefore the 
 Fluxion, transformed, is - 
 
 . a bb 
 
 nV c x / --- 
 
 Whofc Fluent is given by drt. 126. when c is a pofitivc 
 Quantity : But, when c is negative, the FJuxion muft be 
 
 _ fie _ ^ 
 exprefled thus, - > -,, ' =>: 
 
 
 
 Anfwcring to -Psr/n 2. ^r/. 142. 
 
 Z 2 Ex.
 
 340 Of the Transformation 
 
 /* 
 Ex, 3. .*/ 
 
 
 
 be propofed. 
 Then, following the Steps of the laft Example, 
 
 f) wi 
 
 m v a . b * . J4 Will 
 
 C v ~" f- " 
 
 c ' c 
 
 be transformed to c m X + * a | : More- 
 
 c 41 r 
 
 over, 2" being =* =^ </(by putting </ =: 
 
 ^ \ ^ ir L 2 I 
 
 ) and *"-' = -. we alfo have * z / = 
 ic / n * 
 
 * \ ' j' 
 
 K T/ ~T~ ; s * 
 
 - X 
 
 
 fcfr. Wf. From whence, by fubftituting thefe fc- 
 veral Values in the given Fluxion, and putting 
 
 _ __ - = f * there comes out 
 
 x xx dx -H h X * a * 
 
 Whofe Fluent, when the Exponent m is the Half of 
 any Integer* pofitive or negative, will be found, by 
 means of circular Arcs and Logarithms, from Art. 
 
 295- 
 
 308. When the Denominator is a rational Trinomial^ 
 er Multinomial (that , when it is without a Vinculum) 
 the beft Way of proceeding^ for the general Part, is, to 
 refolve the given Fraftion into binomial Ones. In or- 
 dtr to this, let its Denominator be ftigned = o ; by 
 
 means
 
 of Fluxions. 341 
 
 means of which Equation , whofe Roots muft be found \ you 
 will, by fubtraftir.g each Root from the indeterminate 
 Quantity (.*), have the binomial D criminatory of the re- 
 quired Frafliom into which the gjbin Orte may be rt- 
 jolved : If^hofe correfponding Numerator s, let be denited 
 Ax, Bx, Cx &c. then, by putting the Sum of the Frac- 
 tion*, thus a-ifing, equal to the given FrmEhtn* and re- 
 dv.::ng the whole Equation to the fame Denominator, the 
 ajjiimed Quantities A, B, C &c> by comparing the ho- 
 mologous 7<?rmj, will be determined. 
 
 x 
 
 Ex. T. Let the given Fraction le -5 - 7; then, 
 feigning x 1 -f ax -j- 1 o, the two Roots of the Equation 
 
 be la V ^ ^ and -ia + V S b: 
 Which being denoted by /> and y, we have x p and 
 x q for the two binomial Fadors whereby x"' -f ax + b 
 may be refolved, or by whofe Multiplication (x p 
 X x q) the faid Quant'uy is produced. 
 
 Let therefore -- + -- be now affumcd ( = 
 x p ^ x-q 
 
 whole Equation to one Denomination &c. we get 
 A -i- B X xx qA -f pB -f i x x o : Whence A 
 
 is found = - , B = - ; and, confer uemlr. 
 ~~ '* 
 
 p yXAr f 4 p x x q x*+ ax + b 
 
 X X 
 
 Ex. 2. Let the Qttanfity prrtefed&t , : . 
 
 J * " J x* + ax~ + fix + c 
 
 Here, if the binomial Favors whereby .r 3 -f ax 1 -{- ^.i- 
 -f c is produced be reprefcnted by x />, x q, and 
 
 . Ax Bx CJt 
 
 xy. and there be aliumed -I- -(- 
 
 x p ' x q x r 
 
 Z3
 
 34- Of the transformation 
 
 ; then > 
 
 this Cafe, we ihall have Ax x qxx r +Bxx 
 
 r r + Cxx px^x q x*= o; that is, by Redudion, 
 
 X x -f- 
 
 i 
 
 Whence 
 
 =o, and Aqr+Bpr+Cpq=Q. Now', from the firft 
 
 of thefe Equations, mukiply'd by p-t-q, fubtracl the 
 
 fecond, and you will have A*p r+Bxq r p-\-q : 
 Alfo, from the firft, mukiply'd by />f, fubcracl the third ; 
 
 then^x/>j rq -j- Bxpq pr pq : I.aftly, from the 
 former of tne two Equations thus arifin?, mukiply'd by 
 
 />, fubtracl: the latter, then Ay^pp^r ; 
 
 that is, A*p jxp r p* j and confequen'Iy A = 
 ii 
 
 =^= =- : Whence, by the very fame Argument. 
 P <l x pr 
 
 B =. - -^- , and C = - . 
 
 q p xq-r rp x rq 
 
 309. After the fame Manner you may proceed in other 
 Cafes: But there is an Artifice, or Compendium, for 
 more readily determining the aiiumed Quantities A^B^C 
 &c. by which a great deal of Trouble is avoided : And 
 that is, by coniiiiering the Equation in fuch Circum- 
 ftances of the indeterminate Quantity jf, when it be- 
 comes moft fimple, or when moft of its Terms vanifh. 
 
 Thus, in the preceding Example, becaufe AX x q 
 X x r-f B x.r pKX r-f-Cx xpxx q x 1 is=o 
 (in all Circumftances of .r whatever) let xbc taken =p ; 
 then, all the Terms vani&ing, except the firft and laft, 
 
 we
 
 of Fluxions. . , 
 
 v/c have.f x J -/>*:=o ; and confequently^/ = 
 
 ^/0rr. 
 
 /> q X j> r 
 
 Msre unfaerfally* let the given Fraction be 
 
 . x n x ... 
 
 " i "I , i n 1 n i. c_ 
 
 * f ex + fa -r cx * oV. 
 
 1 
 
 -_ - (where 2 and n may 
 A- r /> X xq X * r X AT J ^f. 
 
 reprefent any whole pofitive Numbers whatever, pro- 
 vidwd the kilter be greater than the former.) Then, 
 
 Ax Bx C.': Dx 
 
 affuming + + -f ^^ feTr. = 
 
 . 
 __ fLf _ &c. we fhall have ^/ x 
 
 " i - 1 i 
 
 x + ax -f 
 A -_ ?x .r r X .v^7 &fr. + 5 x * ^X -r r x ^ f Sfr. 
 4- C X # p x x ^X r s &c. &c. x m =zo: From 
 whence, by expounding x by p 9 q, r &c. fucceffively, 
 
 p m 
 we obtain A = , y c> , S = 
 
 ^ 
 
 ^ p , q r . qs &c. r p . r q . r s &c. ' 
 
 fcfa. Whence the Fradions themfelves, whereof thefe 
 Quantities are the Coefficient, or Numerators, will like- 
 wife be given. 
 
 But the Numerators thus found may, fometimes, be 
 more commodioufly exprefTcd by Help of the given 
 Coefficients, a, , c, d &e. fo as to involve only one of 
 the Roots /, y, r (ffr. in each Fraction. For, fmce 
 
 x pxx q*x r tff. is fuppofed, unherfally> = x* 
 
 ^. av ""~' I _i- ^,v n "~*-|- ex" " 3 &c. if both Sides of the 
 'Z, 4 Equation
 
 344 Pf th e Transformation 
 
 Equation be divided by x />, we {hall have x q x 
 
 , . x" -f ax* l -f bx' 2 + cx"~* &c. 
 
 x r XX i i3V. 
 
 Which laft Expreflion, when x is ~ />, that is, when 
 both the Nume;ator and the Denominator become equal 
 
 to Nothing, will, manifeftly, be equal to (p q X p r 
 
 X /> s &V.) the Divifor of A. Therefore, if the 
 Fluxion of the Numerator be taken and divided by that 
 cf the Denominator, and p be wrote inftead of x (vid. 
 
 Page 155.) we fhail have np"~' 1 + n i x ap"~* -{- 
 h 2 X lp n ~ 3 &'c. pq x p r x ps &c. and there- 
 
 p m 
 
 fore A ( 
 
 p q .p r . p s & 
 
 h m 
 
 ^. .__- -^-- t g _ 3 . By the, 
 
 very fame Reafoning 5 rz 
 
 m 
 ? /, _ 
 
 nr r J + I . ffr- + 2 . r c. 
 
 Hence it appears^ that, if tie Numerator cf tie given 
 Fratiion be divided by the Fluxion of the Denominator 
 (nrgler.iing x) and tie [ei'eral Roots p, q^ r &c. (found 
 by feigning the Denominator o) be, fuccejjivel^ fubfli- 
 tuicd in tne Quotient ^ injhad ofx\ I fay, it is evident, 
 ihui the Quantities fj rejfuiting, divided by x />, x y, 
 x r &c. will be the required, binomial^ Fraftisns into 
 the propojed multinomial One may be refolvcd. 
 
 If fome of the Rco f s />, y, r &c. are impof- 
 fjble, which is often the CaiV, the Fractions thus 
 found, where the impoffible Roots are concerned, muft 
 
 be
 
 of Fluxions. 
 
 be united in Pairs, and fo reduced to trinomial Ones, in 
 order to take away the imaginary Terms. 
 
 Thus, let the Fraction propofed be 
 
 and Jet two of the Roots, p and q, of the Equation 
 
 x* -j. ax* + bx + c:=o be impoffible : Then, - - + 
 
 x p 
 
 Bx Cx xx 
 
 -- ! -- being = -y- - - , we (hall, by u- 
 x x r * J 4-fl*r-fp*4-r * 
 
 q 
 niting the imaginary Terms, have ** 
 
 Cx xx _ . 
 
 -I -- , alfo, = -7 - 1 ; - j where the impof- 
 * 
 
 fible Quantities deftroy one another. But, to render 
 this more obvious, k-t a be taken = o, b rro, and c = 
 
 Vjf 
 
 I, fo that the given Fraction may become -5 - j 
 then the three Roots f/>, y, r) of the Equation, ** I 
 
 =0, will here be -- - ++/ "^, \/ 3^, 
 
 2 42 4 J 
 
 and i ; whereof the two former are impoflible. More- 
 over, by dividing the Numerator (x) by the Fluxion of 
 the Denominator (3**) (according to the Prefm'pt) we 
 
 have i which, by writing p, q, r fucceffively, in- 
 3* 
 
 {lead of *, becomes , and for the Values of A, 
 
 Of 3i O 
 
 B, and C, refpeaively. Whence y/+gx T^^_^ 
 
 x _ r * *'_!/ x' + X + I A I 
 
 1 jj ut t h e f ame ma y b e other- 
 
 3 VI OVI 1> OV -- _._ t - T 
 
 * + 3* + 3 i*~3 
 
 wife,
 
 Of the Transformation of Fluxions:. 
 
 wife, inveftigated, in a more general Manner; by z. 
 fuming ~ H ; and proceeding as 
 
 3 A" + X + 1 X 1 X 3 1 
 
 in the firft and fecond Examples j whence the very fame 
 Conclufu.n will be derived. 
 
 If the Fra&ion propofed be of this Form, viz. 
 , the Method of Refo- 
 
 ~ mn + ax - + bz 
 
 luttcn will,////, be the fame: Since, by patting xK n y 
 the given Expreffion is reduced to 
 
 I 
 
 X 
 n 
 
 m i 
 
 + ax 
 
 It may alfo be proper to obferve, ifjet^ in very 
 complicated Csfes, the Application of two, or more, 
 cf the fix foregoing Rules, may become necefiary. 
 Thus, for Example, if the Fluxion given be 
 
 " f n ~*" 
 
 ajf.cz I Xe + fe + gz 
 
 A B 
 
 into two Binomial Fractions, -v- (according 
 
 to.- Art. 308.) we flaall have 
 
 ~^ - : Where s 
 
 i\" 1 X b+z" a+cz> X k+z n 
 if m be a whole pofitive Number, greater than />, the 
 Fluent will be had in finite Terms (by Art. 306. Ex. ^.} 
 
 SEC-
 
 SECTION V. 
 
 Inveftigation of Fluents of Rational 
 Fractions, of fever al Dtmenjions, according 
 to the Forms in Cotes's HARMONIA 
 MENSURARUM. 
 
 311. A S the Subjeft here propofed is a Matter of 
 f\, confiderable Difficulty, and has exercifed the 
 Attention of fome of the moft celebrated Mathemati- 
 cians (who, yet, feem to have condefcended very little 
 to the Information of their lefs experienced Readers) I 
 fhall endeavour to fet it in the cleareft Light poffible : 
 In order to which, it will be requifite to premife the 
 following Lemmas. 
 
 LEMMA I. 
 
 If the Sine of the Mean of three equi-different Arcs, 
 fuppofing Radius Unity, be multiplied by the Double of 
 the Co-fine of the common Difference, and from the Pro- 
 jufl, the Sine of the hj/'er Extreme be fubtrafied, the 
 Remainder will be the Sine of the greater Extreme. 
 
 LEMMA II. 
 
 312. If G be taken to denote the gr eater 9 and L the 
 lejjtr, of two unequal Arcs, and their Difference be ex- 
 prejjed by D ; then will, 
 
 Sin. G. x Co-f. D Sin. L. x Rad. 
 -~ 
 
 Co-f. L y Rad. Co-f. GxCo-f.D 
 - =-: - =r - ' z o. Cr 
 iz. D 
 
 Sin. G. X Rad. Sin. L X Co-f. D . T 
 
 -^TD = Co 'f- ** 
 
 o The
 
 3 4 8 Of the Fluents of Rational F raft ions, 
 
 The former of thefe tv/o Lemmas maybe met with 
 in moft Authors upon Trigonometry ; and the latter is 
 nothing more than a Corollary to the co?nmon Theorems 
 for finding the Sine and Co-fine of the Sum and Dif- 
 ference of two given Arcs ; for which Reafons I (hail 
 not flop here to give their Demonftration. 
 
 COROLLARY. 
 
 313. If any Arch of the Circle, whofe Radius is 
 Unity, he denoted by j^, its Sine by 5, and its Co-ftne 
 by a ; and there be taken A ~ia, B ~ laA i , C 
 2*5 A, =D=2aCB, E=2aDC, F- 
 &c. it follows ( from Lemma I.) that, 
 Sin. 2^ (Sin. J^x 2a $* n - ) 2sa 
 Sin. 3.^' (Sin. 2$< 2a Sin. 
 Sin. 4^" (Sin. 3 Jx 2Sin. 
 Sin. 5^ (Sat. 4^ X 2aSin. 3^ ) - 
 Sin. 6 Sin. 'x 2aSin. 
 
 LEMMA IIL 
 
 314. To refefae the Trinomial r zr> 2kr"x* -f A- 2 ", where 
 , n is any whde Number, into fimple trinomial Faflors. 
 
 Since the firft Term of the given Quantity r 2 " 
 
 2^"x" -f x 1 " is divifible, only, by the Powers of r, 
 and the laft, only, by thofe of x - y and it appears that 
 r and x are concerned, exactly, alike ; let therefore 
 r* larx+x 1 (where r and x are, alfo^ alike concerned) 
 be allumcd for one of the r_c;uired trinomial Fadlors, 
 
 whereby r~" -*-2k "x n 4- x~" may be refolved : And let 
 
 Cr 3 x- + ^>W 4- H-.x'-\- x* (where r and * are, _/?///, af- 
 fefled alike) K affumcd = r ?JrV~}-*' (the Va- 
 lue of.w, to ren !er the Operation more perfpjcuous^ 
 being firft exprefled by 5.) 
 
 Then,
 
 by refolding them into mere fimpk ones. 349 
 
 Then, by Multiplication and Tranfpofition, we {hall 
 have 
 
 O = 
 
 Whence
 
 Of the Fluents of Rational Fractions, 
 
 Whence, Aia, E~iAai, C=2aB/f, D~ 
 -.5, and iC 2aD+ik=o. But, if ^ be taken to 
 denote the Arch (EF) of a Circle EHK, whofe Radius 
 EO is Unity, and Co-fine (Of) ; and j be put for 
 (Yf) the Sine of the fame Arch j then (by Coral, ts 
 Lem. i.) sA Sin. 2^>, sB = Sin. J, sC = Sin. 
 
 Sin. 2$ D 5i. 3. 
 fcfV. and confequently A = - , 5 = - == 
 
 Sin. 4.$. 
 
 
 Sin, 
 
 
 over, becaufe, aC laD+ik o, or 
 where (as appears from above) D x a 
 
 
 Cafe i. Lem. 2.) we therefore have 
 k* Whence this Conftruclion. 
 
 r Ty/i Take R to denote 
 
 -j the Arch (EM) whofe 
 
 Co-fine (ON) is the 
 given Co-efficient^, and 
 let 4>. (EF) be taken 
 to EM as I to n ; then 
 the Co-fine (Of) of 
 this laft Arch will be the 
 true Value of a. But 
 this is only one of the 
 Values that a will admit 
 of: Fer it is v/ell known, 
 that the Co-fine of any Arch, is a!fo the Co- fine of the 
 fame Arch increafed by any Number of Times the whole 
 Periphery (P). Therefore, feeing the Co- fine of n% 
 (= Co- fine of R} is likewife = Co-fine P+R = Co-f. 
 ~ 2 p + R = Cof. 3^ + R &c. it follows that ^( whofe 
 Co-fine is a} will be exprefied by any one of the Arcs, 
 P+R iP+R tf+R Wf ^ ($)r by Fj EG> EH> EJj
 
 ly refohmg tlem into mortjimpte cms. 
 
 &c. fuppofing the whole Periphery to be divided ino u 
 equal Parts, from the Point F). Hence, if the Co- 
 fines of thefe feveral Arcs, expreffing all the different 
 Values of a, be represented by /, c and d^ &c. refpeo 
 tively, we fhall have r 1 2r#-f *% r 1 icrx + .v% r a 
 % &. for the feveral required Factors, by which 
 
 n X 
 
 ikr x -\- x may be refolved ; and confequently 
 
 r* llrx -f x* X r 2 icrx -f x* x r a idrx -f x 1 ' () = 
 
 -" L " " i z " r?> E> r 
 
 r 2*r j? + A: . Q L. I, 
 
 Note, If the Sign of the middle Term -ikr*x n be po- 
 fitive, the Diftance (or Co-line) ON muft be taken on 
 the contrary Side of the Center : But when k is greater 
 than Unity, this Method of Solution fails ; finee no Co- 
 fine can be greater than the Radius. 
 
 COROLLARY J. 
 
 315. If /= i, the Arch R (whofe Co-fine is k] be- 
 
 ing O, the Values of i, c, d, '&c. will be exprefled 
 
 o P iP %P 
 
 bv the Co-fines of the Arcs , , , - &V. re- 
 J n n n n 
 
 fpeciively : And our general Equation will here become 
 r *" _ 2 rV -f- x' n ~r' L 2t>rx -f x* X r 1 icrx -f ** 
 r* idrx -v x* (n}. From whence, by extracting the 
 Square-Root, on both Sides, we alfo have r" 03 x* == 
 
 ,1 
 
 r * . zbrx -i- A- 1 ) X r" 2crx + X* 1 
 
 COROLLARY II. 
 
 316. But, if k r= i (or the middle Term be 
 
 p 
 4- 2rV) then the Arch R being r: , the Values cf 
 
 i, <:, d y &c. will, ^r^, be defined by the Cofines of 
 
 the
 
 352 . Of the Fluents of Rational Fractions, 
 
 P 
 
 the Arcs - , 
 
 ,p 
 
 , , & c . and our Equation, by 
 taking the Root, as above, will become r" + x" = 
 
 r t ^brx 
 
 X r 2 
 
 (n). 
 
 SCHOLIUM. 
 
 317. From the two preceding Corollaries, the De- 
 monftration of that remarkable Property of the Circle 
 given, and applied to finding a vaft Number of Fluents, 
 in Cotes's Harmonia Menfurarum^ is very eafily, and 
 naturally, deduced. 
 
 For, let the 
 Periphery of the 
 
 whofe Radius is 
 exprefled by r, 
 be divided into 
 as many equal 
 
 Parts AB, BB, 
 
 / // 
 
 BB, &V. as 
 there are Units 
 in the given In- 
 teger n ; fo that 
 
 AB, AB, AB, 
 &c. may refpe&ively exhibit the Values of the forefaid 
 
 P iP 3P 
 
 Arcs , , &c. (vid. Carol, i.) Moreover, let 
 n ' n n 
 
 OQ_be the Co- fine of the firft of them ; and, in the 
 Radius OA (produced if necefTary) let there be taken 
 OP - x> and let OB, QB, PB, OV. & c . be drawn: 
 
 P \ 
 
 Then, the Cc-fme of the Angle AOB (- J to 
 
 the
 
 by refoh'mg them into morejimple ones. 
 
 the Radius I, being exprefled by c (yid. Corel. I.) it 
 will be i : c :: r (OB) : OQ_= cr: Whence PB* ( = 
 OB' + OP* 2OQ.XOP) =r* + ** 2ax=r* 2crx 
 
 By the very fame Argument PB* is =: r* idrx + *% 
 fc"r. &c. Therefore, becaufe r" ^ ** rr r 1 . 
 
 1 
 
 X r 1 2ty*-f A- 1 } x r 1 2^r^ + A- 1 ] ^j, by Carol, f. 
 it follows that their Equals, AO o> OF" and PA x PB x 
 
 
 PBxPB &ff. muft be equal likewife : ^/^ is the 
 Part of the Theorem above hinted at. 
 
 , * 
 After the fame Manner, if the Arcs AC^ AC, AC, 
 
 '" P ^P qP 
 
 A C be taken refpedlively equal to , - - &c. 
 
 2 2 ' 2 
 
 it will appear (from Coral. 2.) that AO" + PO* is 
 
 rrPC xPCxPC () Which is the latter Part of tht fan.e 
 theorem. 
 
 Hence (by the Bye) all the Roots of the Equa- 
 tion ** = r" are very readily found : For, fince 
 
 AO" co PO" =t PA x PB X PB &c. where the fecond 
 Factor and the laft, the third and the laft but one, &c. 
 
 are refpedively equal to each other, it is evident that 
 PO" r*<r.x n is alfo = 
 
 r COAT x r 1 2<rr* -f x 1 " X r 1 idrx + x* &c. 
 Whence, *" ^ r" being Oj it follows that r co * x 
 
 r 1 icrx -4- x' &c. is o : From which, by extra&ing 
 the Roots out of the Equations r en * O, r 1 icrx 
 -4- :r l o, r* 7.drx--x*~ CS'f. we get r, r X 
 
 ^ -t- VV - j, rX <: /^- r, rX </ -f /^~ 7, 
 A a fcfc.
 
 354 Of the Fluents of Rational Fractions, 
 
 &c. for the feveral Roots of the Equation x" r ; 
 whereof the firft, only, is poflible when n is odd} and 
 the firft and laft when n is even. 
 
 By the fame Way of proceeding all the Roots of the 
 Equation, x" -f r" = O, will alfo be found : For, feeing 
 
 r s = r 
 
 X r* 
 
 (PCX PCX PC faff.) where the firft Faftor and the 
 laft, the fecond and the laft but one, &c. are refpeo 
 
 lively equal to each other, it is plain that x" + r" is 
 likewife r a ibrx -j- ** X r* - 2crx + ** & c . and 
 
 confcquently x = r X b + V b* i &c. &c. Where 
 the Roots are all impofliblej except the laft, when their 
 Number () is odd. 
 
 LEMMA IV. 
 
 318. Suppeftng every thing to remain as in the pre- 
 
 iiii 
 ceding Lemma, and that , , f, d &c. denote the 
 
 D p i t> p i T> 
 
 Sines ff the Arcs R, , - , : &c. (whofe 
 n n n 
 
 Cc-fines are t, b> c, d y &c.) then, I fay, the Fraftion 
 
 In n 
 
 nkr x brx 
 
 a 
 
 X 
 
 - 2cr.v -f- . 
 For, 
 
 . _ A' 1 -f 6V 5 .v 3 -f Dr*x+ 
 
 "-f Cr*** -f >' V -f y^rA- 7 4- # 8 C^ '*' firefaid Lemma) 
 
 5m. 2 
 
 and it is alfo proved that A = -- - 
 
 , 
 
 y">
 
 fy refbhtng them into morejimple ones. 355 
 
 C rr ' ^ &c> it is evident, therefore, that 
 
 + ^ r ^ + ^r^ fcfr .) is = 
 v 
 
 Sin. 2.9 . 
 
 - X r 7 x&c. and confequently 
 
 r 2arx + x 
 
 . ^ Xr* + Sin- 2$>xr 7 x + Sm. 3 ^ x r 6 ^ x + Sin. 
 r 5 A? 3 -f- Sin. 5^> X r 4 ^ 4 + Sin. 4^ x r j ^ s *Tf. I n 
 
 which Equation, for a and j, let their feveral refpec- 
 
 / / / 
 
 live Values b y c, d &c. and , c^ d y &c. be, fuccef- 
 
 n 
 fively, fubftituted ; and let the correfponding Arcs , 
 
 t be repre f ented by ^ 4 
 
 72 ?i 
 
 then we fhall have 
 
 r 
 i 
 
 x r 
 
 Which Equations, added all together, give 
 
 A a 2
 
 356 Oftbt Fluents of Rational Fractions* 
 
 cp ga I go a 3 
 S' 5' S" 5" a" 
 
 Co 
 
 a a s a a 
 
 X 
 
 I 
 
 But the Sines of the firft Column, being thofe of 
 an arithmetical Progreffion (whofe common Difference is 
 p 
 ) by which the whole Periphery is divided into n 
 
 (5) equal Parts, their Sum will therefore, it is well 
 known, be equal to Nothing - t or all the negative ones- 
 equal to all the pofitiv ones. 
 
 9 The
 
 by refohing them into morefanple ones. 357 
 
 The fame is alfo true with regard to the Sines of th$ 
 
 fccond Column j whole Arcs , 
 
 
 Q.P 
 
 &c. (having - for their common Difference) divide 
 
 the Periphery (twice taken) into the fame Number (n) 
 of equal Parts. But the Sines of the middle Column 
 (which is the laft above exhibited) will not vaeifh, as 
 
 all the reft do : For, nQ being = , n 
 iP -f , sV. the common Difference will here be 
 equal to (P) the whole Periphery ; and therefore, every 
 Arch terminating in the fame Point with the firft, the 
 
 Circle will, in this Cafe, remain undivided, and the 
 
 / 
 Sine of each be equal to (k) the Sine of the firft. 
 
 Hence, our Equation is reduced to r IO -2/$r 5 # 5 -f-;c IC X 
 
 which 
 
 divided by r' 2^r s ^ lS +* iI& , and multiplied by rr, gives 
 / / / 
 
 brx crx drx 
 
 * 
 
 r 1 2ir.v + of* r 1 2<r* + .r r a idrx + x* 
 S*r*x* "&'' x $> R L 
 
 . ' ...... ~ "" * -x; "* A- 
 
 _ _ 
 
 otberwife. 
 
 319. Since r' -2lr AT -f x is rz r z a/r^-f- * a x 
 r* 2<:rA'-r-x x X r a 2^r^r-f .v* (n) by Lemma 3. it is 
 
 evident that, Log. r 20 2-rV + .v** = Log. 
 r 1 zbrx + x" 1 -f Log, r 1 icrx -f ** + Log. r 1 
 
 And, as this Equation holds univerfaily, let k and 
 be what they will (which two Quantities may be fyp- 
 pofed to flow independently of each other) let the 
 A a 3 Fluxion
 
 3 5 8 Of the Fluents of Rational Fractions, 
 
 Fluxion of the whole Equation be taken, making k va- 
 
 riable (and Arconflant) ; which gives ~ 
 ibrx icrx idrx 
 
 r* 2brx + x~ ' r z ^crx\-x" ~ r~ idrx + x 1 " 
 * Art - 6 - (n) *. But, k 9 b, c, d> & c . are the Co-fines of the Arcs R, 
 
 R R-t-P R+iP : ... 
 
 , - , - &c. (whereof the correfponding 
 n n n 
 
 Sines are k 9 i>, /, &c.) therefore, the Fluxion of the 
 
 firft of thefc Arcs being denoted by R, the Fluxion of 
 
 ij 
 
 each of the reft will be exprefled by : And fo (the 
 
 n 
 
 Fluxion of the Co- fine of an Arch bein? equal to the 
 Fluxion of the Aich itfelf drawn into its Sine, applied to 
 
 . / . J? / . 
 
 t Art. 142. Radius f) it follows that k Rk, b x , = 
 
 n 
 
 T) / 
 
 l x c, &c. Which Values being fubflituted in the 
 n 
 
 P 
 foregoing Equation, and the whole divided by - > 
 
 we have _ ? _ = _._ -f 
 
 i i 
 
 crx drx ,\ 
 
 * ( 
 
 LEMMA V. 
 
 3?O. To determine the Series, artfeng from the Divijion 
 ff Unity by a Trinomial^ x 1 2arx -f r* } and to exhibit 
 the Remainder after any given Number (v) of Terms in 
 the Quotient. 
 
 Let x~ z + Jrx~* + Br*x'~*+Cr 3 x'~ s + D r <x'~ 6 re- 
 prefent the required Quotient continued to 5 Terms
 
 by refilling them into morejimple ones. 359 
 
 (v, to render the Procefs the more obvious, being firft 
 expounded by that Number) and let Er*x * -f Fr 6 x 6 
 
 be the Remainder. Then, becaufe - t is =: 
 
 x 2arx -f r* 
 
 + BSx* -f Cr*x* + Dr+x 6 + 
 
 , we IhalJ, by reducing the whole 
 
 x J 
 
 Equation to one Denomination, have 
 
 M 
 I + 
 
 1 1 
 
 + I + 
 
 Z" ? 5? 
 
 } i: 
 
 i m & 
 
 $ I * 
 
 1 | i 
 
 + } : 
 
 **> % b 
 
 1 1 1 
 
 o i 
 
 I f 1 
 
 + t -f 
 
 t t 
 
 Whence Aia^E 
 
 B, E=2aDC, and F-D. 
 
 A a 4 There
 
 360 Of the "Fluents of Rati 
 
 Therefore, if j^ be now put for the Arch whofe 
 Radius is i and Co-fine a, and there be taken SSin. 
 
 ^, S = SJn. 2 ^, S=Sla. 3^, &c. we fhall, alfo, have 
 
 / // /// /;/ T 
 
 j, ^ s *- s r - $ n S S 
 
 A (l) - y, B - -j, C - -j, /)_,- j 
 
 ''~ 
 
 Lem. i.) And confequently ^ 
 
 2arx -f r' 
 
 i // in . //// 6 
 
 S 
 
 Sin. 6 X Sx 5 5;. X 
 
 
 S X ^* larx 
 
 . Whence, umvtr- 
 
 Sr 3 x~ 5 &c. (to v 
 
 Sin. v + i . Q X r x Sin. v& X r x 
 
 ! ^- . Which 
 
 S X x* 2arx + r 1 
 
 laft Equation (though obvious enough from the preceding 
 one) may be inveftigated in a general Manner (if re- 
 quired) by afluming x *+ Arx 3 + Br l x + Cr*x 
 
 V 2 v vI * \ 
 
 -f ..... dr x -f er x + 
 
 ~ 
 
 V - ~J 
 
 fr x +gr x _ - an d pr oceed- 
 
 *- 2r* + r* ' * wx + r 
 
 ing as above: By which Means you will find A '=2a, 
 
 B=2*J-i, V*. /= 2-rf= ^'^x^. and g 
 
 Sin v Q 
 
 F^ -^ n( * t ^ lus ma y l 
 
 Lemma
 
 by refolding them into more Jtmple ones. 361 
 
 Lemma be made out, if any Objection, or Difficulty, 
 (bould arife about its being general. 
 
 COROLLARY. 
 
 321. If, in the given Trinomial ** zarx -f- r% we 
 fuppofe r% inftead of x a , to be the leading Term 
 whereby the Quotient is produced ; then, fince r and 
 * are affe&ed exa&ly alike ; we {hall, by writing r for 
 
 *, and x for r, have 
 
 
 4 (v) 
 
 
 
 s 
 
 oz/i. z>^X * 
 
 + 1 x r^ 1 
 
 Sin. 
 
 v + I X ^ 
 
 V ' 1 
 
 ' x * r 
 
 
 
 Sxr 1 - 
 
 2axr + #* 
 
 I 
 
 P R O B. I. 
 
 X 
 
 322. To find the Fluent of p , together with 
 
 xx 
 
 that of ; . 
 
 ' rr 2arx-\-xx 
 
 Let ABM &V. be a Circle whofe Radius OA (or 
 OM) is r, and let the Angle AOB be fuch, that its 
 Co-fine, to the Radius I, 
 may be equal to a ; or 
 fo, that OQ_ (fuppofing 
 BQ_perpendicular to OA) 
 may be zr ar : Moreover 
 let s denote the Sine of Q. P O 
 
 the faid Angle AOB, cor- 
 
 refponding to the Co-fine a, and let OP (confulered 
 as variable by the Motion of P along OA) exprtis the 
 Value of x: Then, PB 1 (OB a -fOP a 2OQ_x OP) 
 = rr 2arx + xx : And the Fluxion of the Meafure of 
 the Angle QBP (Radius being Unity) will be repre- 
 
 fented
 
 362 Of the Fluents of Rational Fra5lions y 
 
 B> x Flux. P 
 fented by ^pi ("uid. Art. 142.) or by 
 
 n x ~~\SF 
 
 ; and confequently that of OBP. by 
 
 rr 2arx + xx' 
 
 p : Whence it is evident that the Fluent of 
 
 rr 2arx -f xx 
 
 - (contemporaneous with x) is truly ex-. 
 
 prefled by x OBP. 
 
 
 XX 
 
 Again, fince ' ; may be transformed to 
 
 rr 2arx-\-xx J 
 
 arx-\-xx arx 
 
 rr-2arx+xx + rr-2arx + XX '> where the Fluent of 
 
 r t> T rr 2arx -f xx ^ 
 *Art.6. the former Part is r: I hyp. Log. : * rr 
 
 PB 1 -PB 
 
 | hyp. Log. prjpj =. hyp. Log. -rj-~ i and that of the latter 
 
 Part = X OBP ; it appears that the Fluent of 
 
 xx . , , ,-- PB 
 
 expounded 
 
 
 y x OBP. $. E. /. 
 
 COROLLARY. 
 323. Since, PB-PO:: Sin. BOP (s) : Sin. OBP =. 
 
 ; h follows > if the hyperbolical Lo- 
 
 sx 
 
 f Vr 1 - , . , ,. 
 
 ganthm of - , be reprefented by M, and 
 
 sx 
 
 the Arch, whofe Sine is . " and Radius 
 
 V rr ~ larx -f xx 
 
 Unity,
 
 ly refolding tlem into morejimple ones, 363 
 
 Unit;-, by N, that the Fluents of rr _ J^ + ^ a d 
 
 N aN 
 
 will be exprefied by and M+ re- 
 
 rr iarx-\-xx 
 fpedlively. 
 
 P R O B. II. 
 
 X X 
 
 324. To determine the Fluent of -: ----- r ; 
 
 1 arx - r* 
 
 m tfwy ;/>(?/* pojitive Number, and a //} 
 Ifo/Vy. 
 
 Let every thing remain as in Lemma 5. and then, if 
 the Equation there brought out be multiplied by x"x^ 
 and v at the fame time be expounded by m i, we fhall 
 
 J" x n m 2 - . /' m ? .,01 m 4- 
 
 cet "^ "*" 
 
 fa x 1 
 
 g/a. /fl^.X V"* xx Sin, m i x 
 5 X ^A: 
 
 rr 
 
 Whofe Fluent will therefore be given by the preceding 
 Proportion : For, fuppofing the Values of Mznd N to 
 be as there fpecitied, the Fluent of the laft Term 
 
 (Sin, m^* r m - l xx Sin. m^i .x $ x r^^ 
 V - - ; ^* - - - = ^ - ) will, it 
 
 S x xx zarx -f- rr 
 
 is manifeft *, be exprefled by -=- into Sin. m^x r"* X * Art. 3*3. 
 
 r 
 
 . . . /x* wa 
 
 M + -z tin. mi X ^.X r X TO- = 
 
 o ro o 
 
 
 
 _ 
 
 -r- into Sin. mQx M + Co-f. m^ N (by Lem. 2. 
 a
 
 Of tie Fluents of "Rational Fraftwns y 
 
 Cafe i. )To which adding the Fluent pf the preceding Series, 
 
 I - m I " m 2 /' - 
 
 there refuits - x . . , Srx Sr~x J , 
 
 rn-l 
 
 ^-r- X A. ///^.x -Af-j- Cfl-y. w^ x AT. ^. E. /. 
 *> 
 
 COROLLARY. 
 
 WT I . 
 
 325. Hence, the Fluent of ***+'"*'" x 
 
 xx 2arx + rr 
 be deduced : For, by writing mi, inilead of w, the 
 
 Fluent of 
 
 i *%X.M+Ca-f. mi X J^X^V; Which Flu- 
 
 OS 
 
 ent being multiplied by r s and that of * 
 
 xx 2arx -if-rr 
 
 (given above) by a, we fliall, when the homologous 
 
 1 mi 
 
 Terms are united, have -~- x aS x * '. aS S x 
 
 m " a i VH^^~\ ffl* "! 
 
 ^^ // ' *" 3tf ^ f 
 
 r Jl S 5x ( i) +^5- into ~ 
 
 TO 2 > 3 6 
 
 OT j X .x Cc-/ w X a 
 JV, for the true Fluent of the Quan- 
 
 tity propounded. 
 
 / 
 
 P ^ o 
 
 But (ly Cafe I. I. 2.) ^- ( = 
 
 gfa 1 2g_X < "-^.
 
 by refolding them into morefimple ones. 
 
 a i' / . ?5 xa . a x 7</. 1 
 
 -y- ^ - j = 
 
 fcV. And, by Cafe 2. of the fame Ltmma y 
 
 Co-f.m \x9Co- 
 
 ' 
 
 c . ^ ,. r , 
 -- Sw. iW* Whence, 
 o 
 
 by fiibftkuting thefe Values, our Fluent is reduced to 
 
 m i m 
 
 Co-f. ,x - - Co-f. 2^,x -- Co-f. 
 m i m 2 J 
 
 ? 1" 4 
 
 Co-f. ^9 x - C~iJ r * 
 
 m 4. 
 
 ^, x M Sin. M^ X AT. 
 P R O B. III. 
 
 326. To determine the Fluent of -$ 
 
 the Re/in ft Ions fpeclfed In the preceding Problem. 
 
 If the Equation in Art. 321. be multiply'd by x~*** 
 and v at the fame time be expounded by m t we 
 
 {hall have 
 
 r 1 iarx + ^r 1 
 -f Sr-V '* +^r" 
 
 r"" 1 Sin. CT-f i X 4> .X rx Sin, m^ X xx 
 
 5 x ~ r 1 larx + x* 
 
 W 1 
 
 Where, the Fluent of the laft Term being ~ X 
 
 S into Sin. rnQ. x M 4-
 
 366 Of the Fluents of Rational Fratftons, 
 
 Sin. w-f i X^ Sin. m 
 
 tin. m^ X M+ Co J, m^ X N (by Cafe 3. Lem. 2.) 
 it follows that the Fluent of the whole Expreflion, or 
 the Quantity fought, will be truly exprefled by 
 / 
 
 ~ Z *~ 
 
 i m 
 
 * + fcfr. or its Equal 
 
 
 
 ^H^ m 2.r 3 
 
 X m. X A" 
 
 . 
 Sr ^ 
 
 P R O B IV. 
 
 327. To/*/ the Fluent f ~ n > m and n 
 
 any whole pofit'ive Numbers, whereof the former does 
 
 not exceed the latter. 
 
 L-t bed, &c. denote the Co-fines of the Arc* 
 
 3x360 sxrto" ^ (Radius being Unity) 
 ~~~ 
 
 n n 
 Then (by Carol. 2. Lor. 3.) we {hall have r + - 
 
 "rr 2Ar*.-H^f X rr 2crx + xx\* X rr 
 . Whence 
 
 xx -f " ^' ^- 
 
 . 
 and, confequently, by taking the Fluxion, on 
 
 };x r '~~ I x xtlrx xxcrx 
 
 Sides> * ~ xx2brx + rr + xx 2c r x+ rr 
 
 XX , "* * (n)', which lad Equation, multiply'd by 
 
 XX 2&rx+rr 
 
 xxlrx XYcrv 
 
 xxlbrx + rr xx2.crx + >
 
 by refolding them into morefimpk ones. 367 
 
 XX dfx 
 
 -\ (n). Let each Side hereof be now 
 
 XX idrx -f rr v 
 
 fubtra&ed from n (or, which comes to the fame thing, 
 
 ^ ct n , T b e ^ken from , and each of the (n) 
 
 Terms on the other Side, from Unity) then w 
 nr brx + rr 
 
 ft all have 
 
 xxlbrx + rr " xx 2Crx + rr 
 
 () - w nich multiply'd by f 
 l 
 
 I 
 
 gives - 
 
 m . 
 cx x 
 
 r" + # 
 
 IB 1 - 
 
 xx 2crx 
 
 But now, to determine the Fluent hereof, let the 
 
 fi 80 3x180" 5Xi8o 9 
 
 feveral Arcs I - , - , - &c.) above 
 \ n n n 
 
 fpecified, be denoted by ^ ,, ., ^, &f r . refpec- 
 / * 
 
 lively ; alfo let 2V, A 7 , A 7 , 6fc. exprefs the Meafures of 
 
 , r c . x*Sin. ^ 
 
 the Angles whole bines are - 
 
 V rr ibrx -f xx* 
 i a 
 
 x x Sin. 9 x X Sin. 9 ' 
 
 c. and M, M t 
 
 V xx 2Crx + rr -Jxx 2drx -t rr 
 
 
 , Gft. the hyperbolic Logarithms of 
 
 r 
 
 ;/** 2<.TAC-J-rr V/A-A: idrx + rr ,. 
 - > - - - - err. Then (bv 
 r r 
 
 Carol to Prob. 2.) the Fluent of the firft Term, 
 ^ ex P undin S a b 7 *) comes out 

 
 368 Of the Fluents of Rational Fractions, 
 
 X - ( + r" 1 " 1 into 
 
 /. m^ x ivi. . 
 
 .n the fame Manner, by writing c for a y Q. ^ or -> 
 
 Mj and N for A r ) the Fluent of the fecond Term, 
 -^-^ ^- *, is found = C -f. ^ x - 
 
 xx 2crx + rr m I 
 
 . 01 a 
 
 ' rx 
 
 Co-f. 19 x - We. 
 ^ m 2 
 
 Therefore the Fluent of the whole Expreffion, by 
 collecting the homologous Terms, appears to be 
 
 O n o O 
 
 ^ s> <s <a -a 
 
 5 ^ ^ ^ ^ 
 
 ho K> K> N 
 
 &*&*&?* 
 
 v- 
 
 X 
 
 ! 3 
 
 CC 
 
 v" 
 X
 
 oy refolding them into more Jimpk ones. 369 
 
 X 
 
 Sin. 
 
 &'. 
 
 x N Co-f. mQ X M 
 
 X N Co-f. 
 
 X AT C0-/. 
 
 x 2V 
 
 k xM 
 
 But the Co-fines of the firft Column being thofe of an 
 
 /i 80 3x180 5x180 
 
 arithmetical Proa-reffion , -^ . 
 
 \ n ' n n 
 
 260 
 csVJ whofe common Difference is - , whereby the 
 
 n ' 
 
 whole Periphery is divided into n equal Parts (v'td. Art. 
 317.) they will therefore deftroy one another ; fince it is 
 well known that, if the Periphery of any Circle be di- 
 vided into any Number (n) of equal Parts, the negative 
 Sines and Co-fines will be equal to the pofitive ones j 
 which is felf-evident when their Number is even. 
 
 Hence the Co-fines in the fecond and third Columns, 
 I3c. will alfo deftroy one another (vid. Art. 318.) But 
 thofe of the laft Column of all, as well as the Sines, having 
 unequal Multiplicators, muft remain as above, and 
 
 that Column, alone, (drawn into r m ~ l ) will be the 
 " * X x m T JC 
 
 . Whence, putting m$^ 
 
 true Fluent of 
 
 (= mx. - J =. R, and dividing by flr 
 hall (becaufe 4= 
 
 we 
 
 B b
 
 37 Of the Fluents of Rational Fr actions > 
 
 = Fluent of 
 
 COROLLARY. 
 
 328. Since the firft and the laft, the fecond and 
 the laft but one, &c. of the foregoing Quantities 
 x* 2brx -f rr, xx 2crx + rr, xx 2drx 4 rr 
 &c. are refpeclively equal to each other (vid. Art. 
 317.) the correfponding Fluents, found above, will 
 likewife be equal : And therefore the Fluent of 
 
 n , 
 
 r -f 
 
 will, alfo, be exprefled by 
 
 Sin. R x 2NCo-f. R x 
 
 Sin. 
 
 x 7.N Co-f. 
 n 
 
 X 2M 
 
 Sin. $R x 2NCo-f. $R x o.M 
 &c. 
 
 The Number of Lines to be thus taken being 
 rr i TZ, when n is even ; but, otherwife, r= ; in 
 
 which laft Cafe, the Logarithm, &c. in the laft Line, 
 muft be taken only once, inftead of twice j being that 
 
 . e r + r 
 of 
 
 PR OB,
 
 refolding them into more fimpk ones. 371 
 
 P R O B. V. 
 
 329. To find the Fluent of 1 * ; m and n 
 
 r"--x n 
 
 in the preceding Problem. 
 
 If by r, d, l$c. be taken to denote the Co-fines of 
 
 theArcs 
 
 . to Terms, it will 
 
 n ' n ' n 
 appear (from Carol, i. to Lem. 3.) that r" *" is = 
 
 rr 2brx+xx\ X rr 
 
 (n}. From whence, by following the Method of the 
 
 laft Problem, we alfo have 
 
 n i m i . 
 
 nr X x x 
 
 7 fW . . 7W^~J IW , W ' " I . 
 
 fl* * -f- TAT x cx x + rx x 
 
 xx ibrx -\- rr xx 2crx-^- rr 
 
 Which Fluxion having exactly the fame Form with that 
 in the preceding Problem, its Fluent will alfo be ex- 
 preflcd in the very fame Manner ; that is, by 
 
 rSin. m$i x N Co-f. mQ x M 
 
 Sin. mQ x N" Co-f. m<^ X til 
 
 Sin. mQ X N Cc-f. m'% x til 
 (6-ff. to n Lines.) 
 
 muft here 
 
 for 
 
 o 360 
 
 180 .3 x 180 
 . (inftead of - , - - - , 
 
 
 Bb 2
 
 37 2 Qf tie Fluents of Rational Fractions, 
 
 3 6o 
 
 360 
 2m X ^ , 
 
 Therefore, fmce the multiple Arcs 
 are, in this Cafe, equal to o, m X 
 
 '* t 
 
 360 
 
 7w x T & (whereof the Sine of the firft is = o, 
 
 n 
 
 and its Co-fine = Unity) we (hall, by putting RmX. 
 
 , and dividing the forefaid Fluent by nr"~ I ) have 
 
 n 
 
 r* - M 
 
 Sin. RxNCo-f. RxM 
 
 m T .. 
 
 =Flu- * * 
 
 T fu- 
 
 -^ X ^ Sm.iRxNCo-f.iRxM (" cnt of ~ 
 
 Sin.^R x NCo-f.^R x M 
 (&c. to n Lines.) 
 
 COROLLARY. 
 
 330. Since, in the Fluent here given, the fecond 
 Line and the laft, the third and the laft but one, &c. 
 are refpe&ively equal (vid. Art. 317.) the fame may alfo 
 be exhibited, thus ; 
 
 * . . . M 
 
 X 
 
 Sin. 
 
 Co-f. R x T.M 
 
 Sin. 2R x zN Co-f. 2R x 2M 
 
 ff J 
 
 (&c. to Lines.) 
 
 2 
 
 SCHOLIUM. 
 
 331. If the Semi-Periphery ABCH of the Circle 
 whofe Diameter AH is 2r, be divided into as many 
 
 equal
 
 by refohing them into mcrejimpk ones. 
 
 equal Parts AB, BC, 
 
 CB, B C fafr. as there 
 are Units in (fo that 
 
 373 
 
 1 80 
 
 = 2. 
 
 *3c. vld. Art. 317. and 327.) and in the Radius OA 
 (produced, if'necefiary) there be taken OPzr*, and 
 PB, OB &c. be drawn, it will appear (from the faid 
 Articles, and from Prep, i.) that the Quantities 
 
 &c. in the former 
 
 of the two preceding Problems, will here be expounded 
 
 by PB, PB &c. refpeclively : From whence it is al- 
 
 / 
 fo plain, that the Meafures A r , N &c. of the Angles 
 
 whofe Sines are 
 
 x Sin. 
 
 x x Sin. 
 
 fak. * will here be expounded by OBP, OBP, far* 
 
 - * Art. _ 
 m I . and 
 
 XX 
 
 Therefore the Fluent of , given in the Co- 
 
 n , n 
 
 r + x 
 rollary to the forefaid Proposition, may be thus exhibited, 
 
 Shi. R x 2 (OBP) Co-f. R*2(OA:PB) 
 
 Sin^R x 2 (OBP) Co-f. Rx2 (OA: PB) 
 
 y^. ~v. 
 
 Where the Arch R is ( tn X - -J = m x //, and 
 where (OA:PB) is put (after the Manner of Cotes) 
 
 pn 
 
 to exprefs the hyperbolical Logarithm of ^-7. It is 
 
 alfo to be obferved, that, when the laft of the Points B, 
 Bb 3
 
 374 Of the Fluents of Rational Fractions, 
 
 i n 
 
 B, B &c. faflls upqn H (which will always happen 
 when is an odd Number) the Angle, in the laft Line 
 of the Fluent, vrill vanifh, and the correfponding 
 
 PH\ 
 
 Logarithm (which is that of -~ muft then be 
 
 taken, inftead of twice, only once. 
 
 In the very fame Manner it will appear, that, the 
 
 / // 
 Arcs ,, ^ &V. in the fecond Cafe, where the Fluent 
 
 mi . 
 X X 
 
 pf is fought, will be, refpe&ively, expounded 
 
 r x" 
 
 i i a 
 
 by AC, AC &c. alfo the correfponding Angles N, N 
 
 &c. by OCP, OCP &c. and the Fluent itfelf by 
 
 f * (OA : PC) 
 
 X 
 
 n 
 
 Sin. R*2 (OCP) Co-f. R x 2 (OA : PC) 
 Sw.2R x 2 (OCP) Co-f.2R x 2 (Ovf : PCj 
 
 Where the Arch ( = m x ) = m x y#7 ; and 
 
 where, as well as in the preceding Cafe, all the Arcs, 
 Sines and Co-fines are fuppofed to have Unity for their 
 Radius, 
 
 v. T . ; i . 
 
 XX X x, 
 
 o?2. From the Fluents of and ; 
 
 JJ n . n n n 
 
 r -f- x r x 
 
 vn+tn I .. vn+m i . 
 
 thus given, tbofe of -^ > 7~"~^~> 
 
 r + x r + x 
 
 and , where v denotes any 
 
 n ** *" ji-u-i-i *^ 
 
 whole Number, may be very eafily deduced j either from 
 Art. 283. and 291. or (more readily) by dividing the 
 Numerator by the Denominators and continuing the 
 
 Quo-
 
 by refohing them into more fimpk ones. 
 
 Quotient to as many Terms as there are Units in v*. By Art. 150. 
 which means, if p be put =: vn + m y q =. vn m, and 
 
 I . mi . 
 
 the Fluents of- - and - be denoted by V 
 
 n n n n 
 
 r +* r x 
 
 and ^"refpeftively, the Fluents, in the four Cafes fpe- 
 dried above, will be exprefTed by 
 
 p*n n t an z 
 
 * _ _ J_ , r * (v] , 
 
 p u p 2n p yi 
 
 ~ f " ? x lr> 1 , * , V 
 
 f_ _ _ _5 __^5 (<y) + "77- 
 
 2 " " ' "" 
 
 \ , <vn -rtf 
 
 V) -f r iV) 
 
 pn pin p 3 
 
 x -9 x 9 ~1 W_ 
 
 and, + = JT + =-? ^ v) + -' 
 
 ? r ^ . r 2 ^ . r J ^ 
 
 refpedlively. 
 
 Moreover, from the fame Fluents, tbofe of 't 
 
 e+faq 
 
 e fz? 
 
 will likewife become known : 
 
 For (having transformed the Fluxions here pro- 
 
 _y I 
 n 
 
 pofed to x r-, ^'fj let ^j- be put = *% 
 
 fi* 
 
 1 ^ ^. * i " 
 
 _ J-l" ; then will ^ = -" X *", and 
 t I /I 
 
 W I . 
 
 IT * 
 
 confequently X =y X 
 
 B b 4 Whence
 
 376 Of the fluents of Rational Fractions, 
 
 m m 
 
 T ? ~ I n ^~\~Z m 
 
 Whence z* * = T x "7 " x **" * and i -f- 
 
 T y I 
 
 =iz" ; and therefore 
 
 e\ n if 1 _ _ I 
 
 y I I i* 
 
 / _. x f. 
 A * 1 
 
 Whofe Fluent is given, by Pr<j. 4. or 5. But, r being 
 
 
 
 here = I, the general Multiplicator , there gi- 
 ven, will be barely = ~ : Which, drawn into X 
 
 m "* 
 
 s r> ~* 
 
 ~ , gives X I , for the general Multiplicator 
 
 in this Cafe. 
 
 One thing more, though well known to Mathemati- 
 cians, it may be proper here to take notice of; and that 
 relates to the Sines and Co-fines of the fore-men tion'd 
 Arcs, R, 2/2, 3#, &c. &c. (multiplying the feveral 
 Angles and Ratios) fome of which Arcs do frequently 
 exceed the whok Periphery : When this happens to be 
 the Cafe, the Periphery, or 360, muft be fubtra&ed 
 as often as poflible, and the Sine and Co-fine of the 
 Remainder be taken. If the Remainder be greater 
 than 1 80, the Sine, falling in the lower Semi-Circle, 
 \vill be negative; if, between 90 and 270, the Co-: 
 fine, falling beyond the Center, will be negative. 
 
 P R O B. VI. 
 333. To find the Fluent of 
 
 r 
 
 n and m denote any whole pofitive Numbers, and where 
 the given ' ExpreJ/ion cannot be refolved into two Bino- 
 mials (k being lejs than Unity. Art, 308. and 310.) 
 
 Let
 
 by refolding them into more fimpk ones. 377 
 
 Let R be the Arch whofe Co-fine is k and Radius 
 
 Unity, and let k be the Sine of the fame Arch ; more- 
 
 R R+i6o + 2 x 760 
 over, let the Arcs , , -I * 
 
 . be denoted by , 
 
 
 fcfc. and let b, f, ^ &Y. and , f, </ sfV. exprefs the 
 Sines, and the Co-fines of the fame Arcs refpe&ively. 
 / / 
 
 T-U .,, nkr"x n brx 
 
 Then will - - = , - . 
 
 crx drx 
 
 ~t - ; H- ~i -- -j ; i *3c> (n) by Lemma A.. ) 
 r* - r 2c rx 4- ** ' r 2drx + x 
 
 From whence, multiplying the whole Equation by 
 
 x m ~ I x x nJrm ~~' l x i 
 
 / '- we have ., = / -i into 
 
 ,n r 2AT x -TX Mr 
 
 * ~ r * 
 
 *" 
 
 bx m x 
 
 Now, the Fluent of the firft Term hereof . 
 
 * 
 
 (if M be put for the hyp. Log. of **"" 2brx + 
 
 and N for the Arch whofe Radius is Unity, and Sine 
 
 ~r--^=y=^' ) will appear (from Prop. 2.) to be = 
 yr *brx + x 1 / 
 
 m 2 _ 
 
 Sin. 2^ X - + Sin. 3.^ X 
 
 Af* " " ~ 2t 
 
 x S/. w^x yW + 6V_//. 
 From
 
 37^ Of tie Fluents of Rational Fractions, 
 Erom whence, if the Arcs whofe Sines are 
 x X Sin. 9 xX Sin. 
 
 -" ; ;> ;/ , 
 
 2<r* -f **' VV 
 
 =; &c, be repre- 
 
 / // 
 
 fented by M, M &c. and the Logarithms whofe Num- 
 
 A/r' 2crx + x i Vr 1 idrx + x* ,, 
 bers are , &c. by 
 
 r r 
 
 N, N &c. refpedlively, the Fluent of the whole Ex- 
 preffion, omitting the general Multiplicator 
 
 will be 
 Sin. 
 
 i \ 
 l 
 
 r-V 
 
 i^i^ysi, 
 
 Sin. 
 
 fSin. 
 Sin. 
 
 & V 
 
 jij** 
 
 ., y<:. to w i Terms) 
 
 ' 3 
 
 Sin. m^X M+ Co-f. m^X. N 
 
 II II II I! 
 
 ,* M+ Co-f. m^X N 
 
 a/ 
 Sin. ^X M + Co-f. 
 
 But, the Sines of the firft Column being thofe of an 
 
 arithmetical Progreflion (whofe common Differecce is 
 
 5 360
 
 by refolding them Into morefimple ones. 
 
 360 \ 
 
 - J which arifes by dividing the whole Periphery into 
 
 n equal Parts, their Sum will, therefore, be equal to 
 Nothing. 
 
 Moreover, the Sines of the fecond Column, having 
 
 2 X O^C 
 
 - for the common Difference of their refpeSive 
 
 Arcs do, alfo, divide the whole Periphery (twice taken) 
 into n equal Parts, and therefore deltroy each other. 
 
 The fame is likewife true, with regard to the Sines 
 of every other Column (except the laft of all) when 
 m i is lefs than n. But, if m be greater than , the 
 Arcs, in the Column, whofe Place from the firlt, in-> 
 
 clufive, is denoted by H, being exprefled by j^, j^, 
 
 n&c.(or ^,# + 360, + 2x360 &V.) whereof 
 the common Difference is the whole Periphery ; the 
 Sines of that Column do not deftroy one another, but 
 each is equal to that of the nrft Arc R (Vid. Art. 314, 
 and 318.) and confequcntly their Sum equal to?iXtin.R. 
 
 In like Manner, if m be greater than 2, the Series, 
 continued to m I Terms, will take in the Column, 
 
 / // 
 
 where the Arcs are inQ, 277 -*L 2 ^ & c > (r 2.R 9 
 2# + 2x360, 2/2 + 4x300 &c.) whereof the Sine 
 of each is, alfo, equal to the .oine of the firft (2^) and 
 therefore their Sum rr n x Sin. 1 R. 
 
 Thus, alfo, it will appear that the Sines of the Column 
 whofe Diftance from the firft, inclufive, is 3 (when m 
 is greater than 3^) will be each equal to Sin, 3 R ; C5V. 
 
 &e. 
 
 Therefore, feeing all the Columns do actually vanifh, 
 except thof; above fpecined ; whofe Places from the 
 Beginning are deno'.ed by , 2, yi &c. and whofe 
 correfponcing Terms, or Multiplicators are, therefore, 
 
 n l^m n ^Zr.J^rn a 
 
 r 
 
 reprefented by m _ n 
 
 csV. it is evident that the whole Expreflion will be re- 
 duced to
 
 380 Of the Fluents of Rational Fractions, 
 
 Sin. R x - + Sin. 2 /c 
 
 I Pl- T> 
 
 -f Sin. ?R x 
 
 n m 2* 
 
 5 i min 
 
 nr x 
 
 rSin. w^x M+ Co-f.m^X N 
 I Sin. mQxM+Co.f.m^xN 
 
 into X &' 
 Sin. 
 
 i i 
 
 IXN 
 
 n n 
 
 ni in 
 
 Which, multiply'd by , the forefaid, general, 
 
 ' i 
 nkr 
 
 m n 
 
 Multiplicator, gives Sin. R X ; + Sin. zR X 
 
 m n.k 
 
 v m 2 an m : 
 
 r x r x * 
 
 ~ + Sin. 3 -R x 
 
 OT 2W . k 
 
 rSin. mSi* M+Co-f. 
 
 i i 
 
 I Sin. m& X M+ Co-f. 
 r" " > 
 
 4- x <( &. m $xM+Co-f. 
 
 $*N 
 
 n n 
 QxN 
 
 ni HI. 
 
 Sin. 
 
 for the true Fluent of ^ , x : Where 
 
 the former Part of the Expreffion muft be continued to 
 
 as many Terms as there are Units in (the Re- 
 
 n 
 
 mainder, if any, being neglected.) ^. E. I. 
 
 COROLLARY
 
 fy refolding them Into morefimpk ones. 381 
 
 COROLLARY. 
 
 334. If the Quotient arifmg from the Divifion of m 
 by n (when the former exceeds) be denoted by i>, and 
 the Remainder by t ; or, which is the fame, if vn -M = 
 
 m, it is evident the Arcs m^ m^ m^ &c. which 
 
 360 
 
 are refpeclively equal to m^ -f m X - - , m^_ -f 2m x 
 
 n 
 
 , &c. (by Conflruaion) will 
 
 260 
 alfo be equal to m<^+ v x 360 + / X - - 
 
 360 + 2t x - &c. whereof the Sines and Co-fines 
 
 (omitting v X 360, 2^X 360 fcfr. the Multiples of the 
 whole Periphery) are the fame with thofe of m^.-\-t x 
 360 360 
 
 > ^.Hh 2/ x - "* refpedtively. 
 w 
 
 Therefore, if the Arcs of the Progreffion, whereof 
 the firft Term is 01^,, and the common Difference / x 
 
 3^., be reprefented by T t c f y r &c. refpedively ; it 
 n 
 
 follows that the Fluent of ( or 
 
 2 _ n n 2,n % 
 
 r TJar-M +* 
 
 X Jf -- i 
 
 - I will, alfo t be truly exprefled by 
 
 2 , n n . 21 / '' i l * 
 
 r ikr x + x / 
 
 * "t in 
 
 Sin. R X - - -f Sin. iR X - , -f Sin..^R X 
 m n . k m 2n.k
 
 382 Of the Fluents of Rational Prati'ions* 
 
 rs; n .r x M+CO-/.TXN 
 
 \ Sin. TxM + Co-f. fxN 
 
 // _// ft tf 
 
 Sin. T x M+ Co-f. 1' x JV 
 
 /// j'// 
 
 5m. r x A/ 
 
 In the very fame Manner the Fluent of 
 
 (where the Sign of the fecond Terrtl 
 
 n n in 
 
 TS pofitive) will be exhibited ; if R be taken to denote 
 the Arch whofe Co fine is k\ which will, in this 
 Cafe, be greater than a Quadrant. 
 
 PROPOSITION VII. 
 
 335. To find the Fluent of 
 
 M 1 . 
 
 X 
 
 n n 
 2kr X + X 
 
 : under 
 
 zn * 
 
 the Reftriflions mentioned in the lajl Problem. 
 Let every thing remain as before : Then we {hall 
 
 nm I. I I _ . 
 
 have -^ zn = "71=1 into ^ x , 
 
 f n) Whereof the Fluent (by Prob. 3.) 
 
 r* 2crx -f * a 
 appears to be ~ 
 
 mto 
 
 'Sin. Q ' 
 
 
 Sin. 2J^ 
 
 
 Sin. ^ 
 
 I m 
 
 X 
 
 Sin. 2^, 
 
 t w 
 
 X 
 
 x .. 
 
 Sin. j 
 
 mi.r* 
 
 Sin. 2^. 
 
 OT 2 X r*
 
 refolding them into morejimpk ones* 
 
 Sin. 
 
 5m. 
 
 7 
 
 = ; () 
 
 #z 7 . r* 
 
 . w^x M + Co-f.mQx N 
 
 Sin. wix M + Co-f, m>j(N 
 
 Which, by Reafoning as above, will be reduced to 
 
 m 2 m 
 
 ______ x ________ 
 
 Sin. R X - Sin. zR X ' :; 
 
 OT n 
 
 . kr 
 
 - y3 / \ 
 
 5/. 3 K X (to- Termsj 
 
 z 3 . kr 
 
 ^ 5/. rx M+ Co-f. T x AT 
 
 SCHOLIUM. 
 
 336. If, from the 
 Center O, of the 
 Circle ABCD, whofe 
 Radius OA, or OV, 
 is r, there be taken 
 OL equal to k and 
 OP x ; and if the 
 Arch AB be to the 
 Arch AK, whofe 
 Co-fine is + J, as 
 I to n > and each 
 of 
 
 _ K 
 
 B 

 
 384 Of the Fluents of Rational F rations* 
 
 of the Arcs BC, CD, DE tic. be taken equal to 
 
 ^-^ tic. tic. Then the Angles R> ^,, ^ &V. fpe- 
 
 cified (in the two preceding Problems) being here ex- 
 pounded bv AK, AB, AC &c. refpe&iveJy, we have 
 PB ~ V r * 2^rx+^% PC ~ r r~ icrx -\- x* tic* 
 (Vid. Art. 317. and 323.) Whence, alfo, the Angles 
 
 f ' /< X X 5" "* 
 
 AT, N, $ &c. whofe Sines are 
 
 _ c. will here be 
 yr~ idrx -f * 
 
 equal to B, C, D tic. Therefore the Fluents of 
 
 and 
 
 given) will, alfo, be truly defined by 
 Sin. iR. r ' I x w ~" Ztt Sin. 
 r Sin. R m *j n "~ ~*~ bin, 
 
 (there 
 
 (to Terms J 
 
 : PB) +Co-f. rx 
 S/n.T'x (OC-.PC] +Co-f.T x 
 5m. f X ^O/) : PZ>; + Co-f. f X 
 Sin/f x (OE'.PE) +Co-f"fx(E) 
 Sin."fx (OF-.PF) +Co-f. f (-F; 
 
 And
 
 by rcfohing them into morejimpk ones. 385 
 
 And by 
 
 mn.r" 
 
 . r 
 
 /> 
 
 -x^ 
 
 refpedively. 
 
 S;n.Tx(OC:PC) + Co-f.Tx(C) 
 
 .E 
 
 Where the Arc AK (or R} will be greater than a 
 Quadrant when the Sign of k is pofitive ; but lefs, when 
 
 ' // 
 
 negative ; and where the Arcs 2~, 7", T &c. denote an 
 arithmetical Progreflion, whofe firft Term (T) is equal 
 to mxAB) and whereof the common Difference is 
 
 360 
 equal to (or EC) multiplied by w, when m is lefs 
 
 than j butotherwife by the Remainder, of m divided 
 by n. 
 
 C c 337.
 
 386 Of the Fluents of Rational Fractions, 
 
 337. Hence the FJuent of ' - ~, where f 
 
 e -f- fv + gz * 
 
 is any Number, either whole or broken, may be very 
 eafily deduced : For, having transformed the Denomi- 
 
 nator to g X - qr & + 2 2 f, put = r a % < - 
 
 S Z S g 
 
 2*r", and z ? := x" j and then it will become g X 
 
 , 
 ? ? 
 
 4: 2*rV + A: 2 " : Moreover, s " being = 
 
 , tn _ 
 
 1 ~ w . w f f I . 
 
 l x y and ^j: ?xz 
 
 n + /// X # -I *> the Numerator will be reduced to 
 - X jr"**-' 1 * : And fo, we have -^- 
 
 L x ; ~ n In which A- = z " , r = 
 
 ~ X r 2 " + 2 *rV + a;"" 
 
 i 
 Tl 2 ", and *(= pj = -^y=- But, it may be 
 
 obferved, that the Fluent hereof is, only, given when 
 
 if 
 
 Art. 3j3. *_ ^ or j ts Eq ua l k) is lefs than Unity *. Therefore, 
 
 V eg 
 
 jf '/ be greater than V fg \ or if the Values of e and g 
 a re unlike, with regard to pofitive and negative, fo that 
 
 v 7r is impoflible, the above Solution fails. But, here, 
 the given Trinomial may be refolved into two Bino- 
 mials (by Art. 310.) and, from thence, the Fluent 
 may be found at two Operations (by Prab. 4. and $.) 
 
 For,
 
 by refolding them into morefimple ones. 387 
 For, by feigning e+ fy + gy* = y in order to fuch a 
 
 Refolution, we get j * ^ and 
 
 __ 
 ~ ? -^^ 4/f ~~ eg for the Roots of that Equation, 
 
 o 
 
 or the two firft Terms of the required Binomials : 
 Which therefore are always poflible when ^f* eg is 
 pofitive, or when the foregoing Solution fails. 
 
 By denoting the faid Roots by H and K, the Trino- 
 
 mial e +f z ?+gz i t is refolved into^ x H z* x K z?> 
 
 ry e~f 
 
 from whence - - is reduced to 
 
 whofe 
 
 Fluent is given by y/r/. 332. 
 
 338. By proceeding the fame Way the Fluent of 
 
 - rr: may likewife be found : For. 
 + g jl + kz 3 * 
 
 fmce one, at leaft, of the three Roots of the Equation 
 e -\- fy -f gy*+ hy 3 = o, muft be poflible, the propofed 
 Fluxion, if it cannot be refolved into three Binomials, 
 may, however, be reduced to one Binomial and one 
 Trinomial ; and fo, be brought under the foregoing 
 Forms : But this being a Speculation too much out of 
 the Way of common Ufe to be farther purfued, I fhall 
 here conclude this Section, with obfcrvirg, that, when , 
 in the original Trinomial, above fpeciried, is neither 
 lefs, nor greater than Unity, the Fluent cannot then be 
 had directly, from either of the preceding Methods ; but 
 muft be found by Comparifon from the Fluent of 
 
 n-^m I . 
 
 - - -. Fid. Art. 289. 
 '"+*" 
 
 C c 2 SEC-
 
 3,88 Of the Fluents of Exprejjions, 
 
 SECTION VI. 
 
 The Manner of invejiigating Fluents, when 
 Quantities, and their Logarithms ; Arcs 
 and their Sines, &c. are involved together : 
 With other Cafes of the like Nature. 
 
 P R O B. I. 
 
 339- CUP POSING Q and n to denote given 
 Quantities ; it is propofed to find the Fluent of 
 
 Let $* x fa* + x - + Cx*~ * fcfr. be afTumed 
 
 for the Fluent required : Then the Fluxion thereof, 
 which is 
 
 x x hyp. Log. g.* x Ax* -f 
 
 muft confequently be rr x"xf : And therefore, by 
 putting m for the hyp. Log. of ^, we have 
 
 . 7 
 
 &V. $ 
 
 x -f nAx-~ +i. * + 2.C 
 Whence, comparing the Coefficients of the homologous 
 
 I nA n 
 
 Terms, we get A = , B = -- = -- ; , C = 
 z mm 
 
 n 1.5 W.K i 
 
 V. and confequently r x 
 
 Series,
 
 involving the Fluents of other given Fluxions. 389 
 
 Series, it is plain, will always terminate when n is a 
 whole pofitire Number. j^. E. 7. 
 
 340. In the preceding Problem the Coefficients A t 
 5, C, &c. of the -affumed Series were taken, in the 
 common Way, as conftant Quantities ; which, becauf* 
 of the general Multiplicator Qx, was fufficient. 
 
 But, in other Cafes, where a proper Multiplicator, 
 to exprefs the mechanical., or logarithmic, &c. Part of 
 the required Fluent, cannot readily be known, it will be 
 convenient to afiume a Series for the Whole (independent 
 of any general Multiplicator) wherein the Quantities 
 A) B, C } D, &c. muft be considered as variable. 
 
 P R O B. II. 
 
 341. 1o find the Fluent of z m x*~ t x ; z being the 
 Hyperbolic-Logarithm of x ; and m and n any givm 
 Numbers. 
 
 Let there be aflumed Az n + Bz m ~ l + Cz*~ z + 
 
 Dz m ^ fcfc. = the Fluent of z m x n - J x : Then, in 
 Fluxions, we fhall have 
 
 -f B 
 
 ill 
 I 
 *: 
 
 But i = whence, by ordering the Equation, there 
 arifes 
 
 A 1 B 1 _ j+ c 1 M ,e*. 
 >x z w ^>x 2 : CT " Mi.g*> z _ 
 
 x*""" 1 * J x J A 1 J 
 
 Now, by making the Coefficients of the like Powers 
 
 of z, equal to Nothing, we have J=x"~~ I x i A 
 
 - ' mj **\ - M**""':* B = - - ^1- 
 
 n ' ~~ * x ' n * n** ' 
 
 Cc 3
 
 39 Of the Fluents of Expreffions, 
 
 m i . Bx m . m i . x 
 
 _ 
 
 n I . 
 
 C. jjl 4 , ux nt . ,it * Jf X 
 { ~ / 2. C. = 
 
 X 
 
 m.tn i . x" . 
 
 . and confequently the Fluent fought 
 
 ft 
 
 x n . m mz m.m r.z 
 
 into z h 1 
 
 " n * n 
 
 m . m i . m 2 . z* * m . m I . m 2 nt 3 . z m ~~ 
 
 &c. Which, when m is a whole pofitive Number* 
 will terminate in m+i Terms. j^ ; E. 7* 
 
 P R O B. III. 
 
 342. To find the Fluent of z*y ; z being the Arch of a 
 given Circle^ and y the Sine correfponding. 
 
 Let there be afTumed Az n +Bz n ~ l + Cz~* + 
 
 Dz"~~ 3 = Fluent of z'jj then, by taking the Fluxion, 
 we fhall have 
 
 / 
 Az 
 
 > = o 
 
 z H y + nAz r! ~ I z-i- n i . 
 
 Whence, putting A y = o, B + nAz = o, C 
 
 >7 i . JBirro, /> + 2 .Cx=o, &c. we get^ j. 
 wyi, C = n I . Bz &c. 
 
 But, if a and * be taken to denote the Radius and 
 Co-fme of the Arch z, it will appear, from Art. 142. 
 
 that^z ax and xz = ay : Therefore B = nax, 
 
 and g = nax; alfo C (n i . Bx) = 
 
 n.n i -nxz =. n . n I.*j, and Crr n.ni.a^y- 
 likewife D (= w 2 . Cz) n . n i . n 2 . tfyz 
 
 n . n i . n 2-.a 3 x t and >:= n .n i . n 2-a 3 x 
 
 Vc.
 
 ittwhmg the Fluents of other given Fluxions. 3 9 1 
 
 &c. &c. and confequently v/z" -f Bz"" 1 + Cz"~ z & f . 
 
 = yz* -f naxz""~ l n.n 
 2 . tf 3 A-z"~ 3 __ . 
 
 4 2 
 
 I . a yz n . n j . 
 
 ^ / 
 
 In the very fame Manner the Fluent of z*w, or 
 z* X x (w being the Verfed-Sine of the Arch z ) 
 will be found =: xz" + nyaz n ~* -f . I . J 
 
 tf. n i. ?z 2 .. 
 -f &c. 
 
 5 . l.n2, 
 
 Cc4 
 
 PROS.
 
 392 Of the Fluents of Exprejfions, 
 
 P R O B. IV. 
 
 343. The Quantities, x, y and z being the fame as in the 
 preceding Problem ; to find the Fluent ofz"x r f'j. 
 
 By afluming Az* -}- Bz"~ l + Cz"~ 2 -f Dz"~ 3 &c. 
 and proceeding as above, we have A x r y m y, B = 
 jiAz> C = n i . Bz, D 2 . Cz &c. or 
 
 (becaufe = B = - ?, C = - 
 
 - 
 
 D = " 2 ' aC J & f . Therefore, if the Fluent 
 x 
 
 of x r fj (found from Art.- 142. and 291.) be denoted 
 
 Qy Ry c 
 
 by \ that of by Ri that of -, by o j that of 
 .* * 
 
 ~, by T &c. it follows that the Fluent of z"x r y m j 
 x 
 
 will be truly reprefented by Qz" naR*" 1 " 1 + n . n i . 
 a*5a rt ~~ a n . n i . n 2 . a 3 2V*~~3 tffr. 
 
 COROLLARY. 
 
 344. Since y '(V id. Art. 142.) it fol- 
 lows that z*'fj is c V ' l y*~*x ~ ^ : 
 
 Therefore the Fluents of thefe two laft Expreflions are, 
 alfo, exhibited in the foregoing Scries. 
 
 345. As the Values of J^, R, 5, fcrV. in the preceding 
 Articles, are too complex to be purfued in a general 
 Manner, it may not be amifs to illuftrate the Method 
 of proceeding by an Example or two. 
 
 7 Let
 
 involving the Fluents of other given Fluxions. 393 
 
 zy*i> 
 Let, then, the Fluxion propofed be - - : Where n 
 
 y*y 
 being =i, m=2, and r = i, we have j^ ** = 
 
 i (becaufe </a l f - *.) Whence .= 
 
 = ij* + i*z*,and therefore (= Art.. 79. 
 
 ? = i>j + iz* (becaufe = 
 * , x 
 
 ) and confequently R = * / + x 2 *j an j fo> 
 
 ^~>'^ x z . a x ^- gz or tfg a ~2jryz + ay * . 
 xz-ftfX ,or - - - ls 
 
 z 4 4 
 
 the true Fluent of ~ (= *y* = ~^) fArt. 344. 
 
 A^ain, let the Fluent of p* X z+^J 1 (exprefling 
 the Content of the Solid generated by the Revolution 
 of the Cycloid) be required. 
 
 Here, the given Expreflion, in fimple TermS, will 
 become pz*x ipzyx py*x : Whereof the Fluent 
 of the firftTerm />zV, will be had, by making n 2, 
 m i =:o, and r + i =. o (Vid. Form. 2. in Corol.} 
 
 1'y 
 
 Where, we therefore, have ^ = ~ = * ; whence 
 ^ x ; alfo ~ = J* and * = ~ y J 
 
 likewifeS (--^) = - = *, 5 = x ; and 
 confequently the Fluent of *** ($z* naRz n ~~ l 
 
 4- . n i . a 1 Sz"~ &c.) =: ** 
 
 To which, adding the Fluent f - : 2U 1 of the 
 
 fecond
 
 394 Of the Fluents of Expreffions, 
 
 fecond Term izyx (found in the preceding Exam- 
 ple) and alfo that of y*x (or a*x + ***, found 
 the common Way) we get, in the Whole, a x x a 1 
 -f zay y x x x -f I ay* -f a*x -f ^ * 3 ; which, multi- 
 ply'd by p t and corrected, gives, p into i a x x z 2 
 + 2ayyx X z -{- ? ay* + a*x -f j* 3 7+, for the 
 true Fluent that was to be determined. 
 
 P R O B. V. 
 
 346. Suppofmg H to denote the Fluent of 
 
 z vn ~ l i ; to find the whole Fluent of Hx abz"T X 
 a*"" 1 *;, (when a bz" becomes equal to Nothing.) 
 
 By refolving k + lz"\ Xz*"*" 1 * into fimple Terms, 
 and taking the Fluent, the ordinary Way, we get H 
 rlz , r . r-l . / a z^ Which 
 
 rvn ' 
 
 x 
 
 Value being fubftituted above, and p wrote inftead of 
 
 m i r 
 
 q + v, we {hall have H X a bz"} Xz f K= x 
 
 ; n 2 " 
 
 / i. . r r/z r.r j./~ 
 
 2 . V+l.k 
 
 \ +,n-^~i 
 
 Let, now, the Fluent of a bz\ X x (in the 
 propofed Circumftance) be denoted by A^ and put t= 
 p + m+i ; then it follows, from Art. 286. (by writing 
 
 ~ for r, , for/, ^.) that L * A into + 
 
 v-f I . * a
 
 involving the Fluents of other given Fluxions . 395 
 P-T+1 . r.7=! 
 
 ~~~~ ~^=~~~~~~"=^:^^ 
 f.f+I -2.^+2 
 
 P . p+ i . p + 2 . r . r i . r 2 al\ 3 
 
 = = T == ' x r& + fcfo will be 
 
 / . t+i .t + 2. 2.3 -y + 3 
 
 the true Value of the Fluent. Q /. 
 
 JVfl/^, p and z+ i muft here be pofitive Quantities * } Art.agfi. 
 and it is alfo requifite that -j (hould be greater than 
 
 : otherwife the Fluent will fail. 
 a 
 
 t 
 
 Ex. i. Let H \y~\ X j; and let the whole 
 
 j_ 
 
 /"// of Hx i j*) a j, ^ demanded. 
 
 Then, -f being = j, / i, r:^, w = 2, r r= 
 
 4 * =^ j alfo a = i, ^ = i,m= 4, yr=i ; 
 
 =f, and yf(=the 
 
 _ 
 Fluent of i /) a >>J i ; we {hall, by fub- 
 
 ftituting thefe feveral Values above, get i -J --- f. 
 
 3-3 
 
 J~S + Tl + ^ + iT L 7I^- -Fluent of //x 
 
 (or ///; when ^=i. Which Fluent 
 
 //^ 
 
 , it follows that = ~ + 
 
 Wl!ere ^ is * of the Pe - 
 
 //^ 
 
 being alfo expreffed by , it follows that = ~ 
 
 9 w i 
 
 liphery of the Circle whofe Radius is Unity. 
 
 Ex.
 
 3 96 T^be Manner of making fluents converge. 
 
 Ex. 2. Let H c~ -J- z 1 ] * X x -, to find tht 
 
 ofHxFz^xfz. 
 Here, k f*, / I, w rr 2, r -|, v i ; alfo 
 
 
 (p + m+ i) = |, and ^ ( w&/* Fluent of / l z a | 
 X z) r= h : Whence, by Subftitution, we have c * 
 
 ~~1T~ ~V W 
 
 AXI TX-^-fTX"; TX-^- y<r. which, 
 
 ^ C C 
 
 multiplied by c* (the Coefficient of z) gives X 
 
 - j + ~ &c* for the true Fluent in this 
 
 3 C 5 C 7 C 
 
 Cafe : Where the Series is that exprefling the Arch of 
 
 Art. 141. the Circle whofe Tangent is h and Radius c * ; and is 
 
 therefore equal to c x Arch, whofe Radius is Unity and 
 
 Tangent = : Whence this laft Arch (taken without 
 the multiplicator c) is the true Value of the Fluent. 
 
 SECTION VII. 
 
 Shewing bow Fluents, found by Means of Infinite 
 Seriefes, are made to converge. 
 
 347. TT is found, in Art. 85. that the Fluent of 
 
 m 
 
 a -f- cz"\ x dz* r ' , in an infinite Series, 
 
 T+ 1 on 
 
 ... . . /r j u +<*"' XT 
 
 (makings 4- q*) w expreiied by - x
 
 *Ihe Manner of making Fluents converge. 397 
 
 i 
 
 . . .. rf TX7L 
 
 - + - -- ; - -- tfc . Whence 
 f+I .a q + l .q+^.a 
 
 it follows (and is evident by bare Infpeclion) that the 
 
 Fluent of a cy n \ X /" ~* y (where the fecond Term 
 under the Vintulum is negative) will be truly defined by 
 
 into i H- 
 
 qna 
 
 4- & fuppofing J ~ 
 
 But, befides the Series here given, and Thofe, / 
 y/r/. 83. 84. exprefling the fame Value, the Fluent of 
 
 a C y n l x y? tt ~~ l y will, yet, admit of another Form, 
 different from all of them ; by means whereof and that 
 above, we (hall be enabled to draw out fome very ufeful 
 Conclufions. 
 
 X z 
 
 Which Fluxion, fo tranf- 
 
 formed, being compared with a + cz"\ X d-"~~ l z.\ we 
 
 have m r q I, d=a 2r+ ?+ I J and j (q + m) 
 = r I ; whence, by fubftituting thefe Values 
 jn the firft Series, above given, the Fluent fought will 
 
 
 r . r r . r 2 .
 
 398 The Manner of making Fluents converge. 
 
 a~ ay" 
 
 Which, by reftoring y (or writing and 
 
 a tf a cy* 
 
 for their Equals a -f- (z" 9 and z") becomes 
 
 qn q+i a cy" q+l.q+2. 
 
 1 ~ ->r 
 
 fcfr. the true Fluent, of a cy'' 
 
 349. This Fluent may be otherwife found, inde- 
 pendent of that above, in the following Manner : 
 
 a c f\ x /" 
 It is evident, by taking the Fluxion of 
 
 qn 
 (which Quantity would be the Fluent fought, if 
 
 Tf 1" 
 
 ' > r a cy I X y . , 
 
 a f/| was conftant) that - ; 
 
 Fluent of a cy"f X /""' } Fluent of ~ x 
 
 'a^fff 1 x /""*"'~ I t y : This Equation, by tranfpofmg 
 the laftTerm, and writing x in the room of a cy" 
 (for the Sake of Brevity) will become Flu. x'f^jt = 
 
 x y + If x F/ a . /~ I / fl+B ~" I >. From the very fame 
 
 Argument (if, inftead of r, we fubftitute r i, r--2 
 ^c. fucceflively ; and, for y. write q-\-i, q+ ? 
 &c. refpeaively) we fhall, alfo, have 
 
 T-,, r Z a + 2)i 1 . 
 
 Ftu. x y* y >
 
 The Manner of making Fluents converge. 399 
 
 Whence, by fubftituting thefe Values, one by one, in 
 
 that of, Flu. x r y**~ J j, we get 
 
 re x- r .r-i.c* 
 
 ? 1 q-ri.n q-q+i 
 
 p. r 2 jn+zni. _ x 'y__ i rcx r '~ l j qn ' 
 
 1" q.fTl.n 
 
 i T"" 2 on"\-j.n ** ~~~~ "*"" 
 
 rf v v ' r o f 
 
 '** ^ y / I ' J& * 
 
 q.q+l.tt 
 
 ( 
 
 ^.y+i.^4-2.w ^ . y+ j . q-\-2 . y+3 . n 
 
 &c. Where the Law of Continuation is manifeft j and 
 
 x r y?" 
 
 where, by making _ - a general Multiplicator, we 
 fhall have the very Series above exhibited. 
 
 350. From the Equality of the two foregoing Ex- 
 
 preffions, for the Fluent of a cy"\ x y j 9 (or 
 
 x r y }n ~ l y} the Bufinefs of finding Fluents, by infinite 
 Seriefes, will, in many Cafes, be very much facilitated. 
 For, in the firft Place, it follows (by .dividing both by 
 
 tf +l ?" r+i \ 
 
 V ' xy x or f L_ ) that the Seriefes i -h 
 
 qna qna J 
 
 &c. and X 
 
 . a 
 
 * 
 
 3 
 
 muft alfo be equal to each other, let the feveral 
 
 Quan-
 
 4 The Manner of making Fluents converge. 
 
 Quantities, therein concerned, be what they will (which 
 may be otherwife proved, independent of Fluxions.) 
 Therefore, if in the room of q and s we write any other 
 Quantities p and /, the Equation will, /A'//, hold, and 
 
 will then become i -f * + * '/ + 
 P+l .a 
 
 _ 
 
 p+l .X p+l.p+2.X~ 
 
 (t being = p+r.) 
 
 Moreover, if as many Terms of the firft Series i 
 
 J +V* , J+"i H-2 . f ' 
 
 q+l.a q+i.q + 2.a* q + I q+ 2 . ? + 3 . a 
 &c. be taken as are denoted by any given Number v, 
 and the laft of them be reprefented by j^, it is evident, 
 from the Law of the Series, that the rirft of the re- 
 maining Terms will be exprefied by 
 
 the fecond, of them, by j| 
 
 LL. GV. and therefore the Sum of all of them (putting 
 q + v=p and s-^-v (zrr-f y-f -y) /) will be = ^ X 
 
 p+l .a p+l .p + 
 
 (by writing the Series found above in the room of its 
 Equal) and confequently the whole Series (including 
 
 the v firft Terms) i + '-LI' ^
 
 Manner of making Fluents converge. 401 
 
 P* 
 
 { 
 
 r "^" 1 y f* 
 
 Which, drawn into the general Multiplicator * JL_ 
 
 y/?rt 
 
 (vid. Art. 317.) will give the Fluent of a cf\ r X.y q "~~ l y 
 
 (or x r y <i *~*y} according to a new Form, compounded 
 out of the two preceding ones ; where the fecond Series 
 (the Value of p being large in refpett of r j will al- 
 ways converge much fafter than the remaining Part 
 of the firft, for which it is iubftituted : But this will, 
 more fully, appear from what follows hereafter. It will 
 be proper to take notice here that the Fluent of 
 
 a-\-cz n \ Xz x (the Fluxion firft propofed, where 
 the fecond Term under the Vinculum is pofitive) will 
 alio be had from hence (by writing z for y, m for r, 
 
 
 an d c for c) and is therefore equal to 
 
 qna 
 
 drawn into the Sum of the two following Seriefes, 
 
 ^ l.Cx Jl.f2.f __ 
 
 Where, 5=^ + ?, p^v + y, t=s + % xa -f 
 and ^= the laft Term of the firft Series continued 
 to v Terms, v being any whole Number, at pleaiure. 
 A few Examples will (hew the Ufe of what is above 
 
 delivered. 
 
 D d Ex.
 
 42 The Manner of making fluents converge. 
 
 351. Ex. ,. Let ~, or P^f"^ be propounded . 
 
 ->m 
 
 Which being compared with a + cz"* X * 
 
 ? " 
 
 *, we 
 
 havea=i, f I, ~i, A-rri-J-a;, 7~ i, yn I o, 
 ory~i; whence alfo s (m-\-q} o, /> (f-f-^) 'y-J-i, 
 f (j-f v) zr v, and consequently the Fluent itfclf (by 
 fubftituting thefe feveral Values in thelaft general The- 
 
 . x z? K* v i & 
 
 orem ) =. z into i. -f ( i>] ^' 
 
 2 3 4 t'+i.x 
 
 z 
 * I-f 
 
 &c. Where (Q) the laft Term of the firft Series 
 
 ^-1 / v*& \ 
 
 -f - the Multiplicatorf ' ) to the 
 
 v \v -f i . x / 
 
 Second, will be = ^ . ~ ; and fo the Fluent itfelf 
 
 tvill be reduced toz -- -f --- (v] 
 _ 23 4 v -y--.A 
 
 <x ^ J 
 
 _L j== -- + . -- ----- - ~ + &c. In which 
 
 X V + 2.V+3. X 
 
 the Signs -- and -f-> before **^*, obtain alternately, 
 according as -y is an odd or even Number. But, to 
 (hew the Advantage of expreffing the Fluent in this 
 Manner, by two different Seriefes, letzrrj, and let 
 ti be taken =. 8 ; then the Value of the firft Series (con- 
 tinued to 8 Terms) being = 0,6345238 &c. and That 
 
 i A iB 3C 4/) 
 
 of the fecund Scries = -$ + - -f- + :d +- L -z 
 
 L8 20 ^ 22 r 24 26 
 
 r 
 
 ^^r-^c. (where A, #, C, D y<:. denote the Terms 
 
 preceding thofe where they ftand) = 0,0555555 + 
 0,0027778 -f 0,0002525 -f 0,0000316 -f- 0,0000048 
 -j--o,occcoo94-o,OOOC!002=:c,C5b6233; it is evident 
 
 that-
 
 The Manner of making Fluents converge. 403 
 
 IX 
 
 that the Fluent of - >, when becomes = i, will 
 
 be rr 0,634.5238+0,0586233 = 0,6931471 : Which 
 is true to the very laft Decimal Place ; and would have 
 required, at leaft, looooo Terms of the firft, or com- 
 mon, Series. 
 
 x 
 
 352. Ex. 2. Let the Fluent of ^ (ex^reffing the Arch 
 whofe Radius is I and Tangent z) be required. 
 
 In this Cafe we have azri, r=i, = 2, xm-fz*, 
 jw= I, qn 1=0, or q = !, j = l, />=-y-|-i, 
 
 3 2 s 2 7 
 and the Fluent itfelf =. z -f --- (v) -f- 
 
 X I + 
 
 2t/+ 3 
 
 2.4.6.x 6 
 
 - bV. Where, if z be taken 
 3 
 
 x 
 
 rri. and v = 6. we fhall have i -- + + 
 
 3 5 7 
 
 III 1 I 2 I 2~ 
 
 ]T~n+;6 X] - rs + ii x ^ + 75 X i7 x 
 
 ^. = 0,785398 = the Fluent of ~ when z 
 19 i +z 
 
 i ( = -J of the Periphery of the forefaid Circle) 
 Which Number, brought out by taking, only, 8 Terms 
 of the fecond Series, is more exact than if 100000 
 
 Terms of the common Series i -- -f -- fcjV. 
 
 had been ufed. And, if z be taken = y/ ^_L (r= 
 
 3 
 
 Tangent of 30) and ^y=r6, as before, the fame Num- 
 ber of Terms, will be fufficient to give the Anfwer, true 
 lo twice the Dtcimal Places above exhibited. 
 
 D d 2 353. Ex.
 
 404 The Manner of making Fluents converge 
 
 ji 
 
 353. Ex. 3. Let the Fluxion prapofcd be *+_y*|* x y. 
 
 Here we have, a=e* 9 f I, zy y n^, xe^+y 4 , 
 
 =, q=, s (ro-f?) = *, p (v + q) - + | j / 
 
 (s + v) =. v+i; and therefore the Fluent fought (by 
 
 4^ + 5 .. 
 
 2 . 2y s 2 . 6 . 2y'* 
 
 - 4-y+9 . A' 1 4^-1-5.4^ + 9.4^+13 . x 3 
 
 te'c. in which (as in all other Cafes) j denotes the laft 
 Term of the firft Series. This Fluent approximates 
 equally faft with thofe in the foregoing Examples : And 
 it may be obferved farther, that the Fluent will always 
 converge, however great the Value of z is taken, if 
 
 both a and c^ in the general Fluxion a + tz r \ xz** ~ x z, 
 are pofitive Quantities. But, if the fecond Term under 
 the Vinculum be negative, the Cafe will be otherwife, 
 when that Term becomes greater than half the Firft j 
 
 fince the Powers of , in the latter Part of the 
 
 Fluent, will then form an increafing Geometrical Pro- 
 grelfion. It may, therefore, be of ufe to (hew how the 
 Theorem may be varied fo as to anfwer in this Cafe. 
 In order thereto, if in the Equations j=y-ff, and i -f 
 
 . a q+ I . 
 
 rcf r. r i . c*y . . 
 
 I -f == r- ^rr- =-^ fate, (given in Art. 350.) 
 
 q+l.X f+I.f+2.** 
 
 you write k for r, and p for f, and multiply by , 
 5
 
 The Manner of making Fluents converge. 405 
 
 key" 
 
 you will have szzk+p, and i 4- = 4- 
 P+J.* 
 
 /f - ki. C*V M x T^\ . cv" , 
 c. = X i 4- == 4 
 
 -P+2.X 1 a 
 
 . 7+2 . C*y 
 
 ' 
 
 + i.p + 2 . a 
 
 Moreover, if the v firft Terms of the above Series i -{- 
 
 - r 2 
 
 rr r . r i . f ) 
 
 f- ^=: --- &c. be taken, and the laft 
 
 J-r-i .JT 
 
 of them be denoted by j^, it is plain the firft of the 
 
 remaining Terms will be = .9 x X 2 , 
 
 q + v x 
 
 the fccond =; 9 x t ~ v ^ 1 x -. r ~ V x ' w ' 
 
 y4^ ^4^4 l x 
 and the Sum of them all (putting q 4 ^ P ant l 
 
 r- V =^) equal to lUl^L x t 4 J^" + 
 ^ ^+i.x 
 
 J . J^Ti . t -y" ~i . j^:/ x , J4l.f/ 
 
 = &r<;. = - = ^ L - x - x 1 4 = 
 
 p+i.p + 2.x* P x a p+i.a 
 
 . (by the Equation above) and 
 
 confequently the Sum of the whole Series (i -f 
 
 _ 
 
 Vc.) = i -f ^U 4. r ' r ""-^ + 
 
 ^-j-l . x q+i . q + 2 .x* 
 
 .y + 2.^+3. ^ _ 
 
 '" + ^ Whichf 
 
 a p + i .p + l a" 
 
 D d 3 rnul-
 
 40 6 he Manner of making "Fluents converge. 
 
 x r y ?n i r 
 
 multiply'd by , gives the Fluent of a tf\ 
 
 * Art. 34 3, X/"~"j (* or //"""' j) where k = r v, p = v 
 
 349< +f, s (=&+p) = r + q and x=acy". I fhall put 
 
 down one Example of the Ufe of this laft general Ex- 
 
 preflion ; where we will take y T/2y-y* or 2 >1* * 
 
 y*y (being the Fluxion of the Area of the Circle whofe 
 Radius is Unity and verfed Sine y) In which Cafe, 
 a 2, c~\) r=i, rrri, yn I^l* r ? = t> ^ = 
 y + i, =z; + |, s=2 y A-=2 7; and therefore the 
 
 Fluent fought r: - 
 
 into i + 
 3 5 s 
 
 ; j 
 
 TP + 
 
 r^; T 
 
 5-7 
 
 V 1 
 
 .9*' 7 9 
 
 I !** Q . II . 
 
 1 3- v$ 
 
 
 3V 
 
 ? 4.y 
 
 i 
 
 2V 
 
 + 3 - 2 
 
 2V -f 5 2V -f 
 
 5 . 2v 
 
 if 7 
 
 H^- 
 5 
 
 + 7 
 be taken 
 
 . B C 
 "73 
 
 
 3 . 4 5/ 
 
 - ^. Which, 
 
 2-y+ 
 
 "5.2^+7 w + ' 
 and w=5 wil 
 
 5 
 1 become = - 
 
 5 . _7- + + + ^. =0,785398 
 "77 2x13 15 '7 ^9 
 (where y/, .6, C Cffr. denote the feveral Terms, re- 
 fpectively, without their Signs.) In bringing out which 
 Conclufion, fix Terms of the fecond Series are required : 
 But if y be taken =T the Radius of the forefaid Circle, 
 then four Terms of each Series will be more than fuffi- 
 cient to give the fame Number of Decimal Places. And 
 it may likewife be obferved, that, although no general 
 Rule can be laid down for affigning the Value of v, fo 
 as to anfvver the bsft in all Cafes, yet the Conclufion 
 6 will,
 
 The Manner of making Fluents converge. 407 
 
 will, for the general Part, require the feweit Terms, 
 when the Number of thofe, taken in each Series, is 
 nearly the fame. 
 
 354. But, after all, another Theorem or Series, ftill, 
 feems wanting, to exprefs the Value of the whole Flu- 
 ent, when the Quantity under the Vinculum becomes 
 equal to Nothing ( which, in the Refolution of Problems, 
 is, commonly, what is required.) For, it is plain the 
 laft, above given, anfwers no better, here, than that 
 preceding it ; becaufe (the Divifor (x) being Nothing) 
 the former Part of it fails. 
 
 In order, therefore, to determine a proper Form, to 
 obtain in this Circumftance, it will be requifite to ob- 
 ferve, firft of all, from Article 286. that the whole 
 
 Fluent of a bz n \ X ~ T *, fuppofmg that of 
 
 -.m 6 . 
 
 a 2 "1 X yf x to be denoted by A y will be truly 
 
 /r i u P P+* P + 2 / ) *' 
 exprefled by - x 7^ X (v) X _ : In 
 
 which t~m + p+ j ; and where it is requifite that the 
 Values of w-t-i and p (hould be pofitive, otherwife, A 
 being infinite, the Fluent (or Comparifon) fails. Hence, 
 
 becaufe the whole Fluent of a bz\ x a:*""" 1 *, (when 
 
 w-f-i 
 
 a bz n zr o) is found ^r - , by the common 
 
 m -f i X nb 
 
 Way *, it follows, by writing this Value in the Room of Art. 77. 
 A, and expounding p by i, that the ivhole Fluent of and 7 8 - 
 
 bz"\ m X 2"+* ^ is rightly exprefled by - 
 
 D d 4
 
 4 8 The Manner of making Fluents converge. 
 
 That of abz'"x a:"" 1 *, by fubftituting r inflead of 
 
 fc'-^i, will confequently be equal to - x 
 
 rti + 1 jti -\- 2 
 
 ^ W "J"/* 
 
 X -_^- T r ^ x . Let this Quantity be denoted 
 
 rnb r 
 by B' t then, ^y ^ fime Article^ the Fluents of the 
 
 fcveral Terms of the Series I, -, 
 
 drawn into the general Mukiplicator abz* 
 
 will be, refpeftively, expounded by thofe of the Series j, 
 
 &c.. drawn into B ; t be- 
 
 t . t+i t.t+i 
 ing nw + rH- 1, 
 
 If now the Differences of the Quantities i, , 
 *j r ^" ; & f , be, continually, taken * ; and for r t its 
 
 t . 
 
 Equal m I be fubftituted, the Value of any Term 
 of the Series, whofe Diitance from the firft, exclufive, 
 is denoted by s, or whofe correfponding Term, in the 
 
 b'z" 
 preceding Series, is - , will be univerfally ex- 
 
 a s 
 
 1.2.3 x / . t+i . t+^ 
 jf j be interpreted by o, i, 2, sj^r. fucceffively, you 
 
 - ^bove exhi- 
 
 ill have the Values x, -, 
 
 bited But, if s be taken as a Fraction, then the Value 
 pf fuch an intermediate Term will be found as will give 
 
 tnc 
 
 * See my Mathematical E/ayt, P' 94-
 
 Manner of making Fluents converge. 409 
 
 , J S _ 
 
 b Z "\M rn I . . 
 
 the Fluent of - x c bz n \ x z z, m any pro- 
 
 a 
 pofed Circumftance of s ; which Fluent, it is evident, 
 
 J w "h_? _L 
 
 will therefore be exprefled by B x i - 1 f 
 
 or its E al _ 
 
 2 
 
 ! . m + 2, 
 
 into i- 
 
 2 - I S-'^ 2 4 
 
 G fcf. (where , F, G fcfr. denote the Terms im- 
 mediately preceding thole where they ftand, under their 
 
 j 
 proper Signs.) Whence, dividing by __, we have 
 
 (r \ . _ X 1 
 
 r 
 
 , tff. for the true Fluent of a bzf\ X 
 
 21 
 
 From the laft Fluent /Aa/ of a n X z 
 fin which /> denotes any pofitive Fraaion, proper 
 or improper) is very readily obtained : *or, U 
 fame (when a bz" = c) be denoted by A\ then the 
 
 Fluent of 7^1^^^^- will (according to the 
 
 p 
 Article above quoted) be exprefled by - +fl| + > 
 
 
 11- / v ) x 3 fuppofing ^ any 
 ' V 
 
 pofitive
 
 410 he Manner of making Fluents converge. 
 
 ~ -- "i w 
 pofitive Integer. Therefore, by making a lz n \ x 
 
 z r*+>-^ - a -i z \ X z /> " +t ' r ~ I i, or r + s =p+ v, 
 the correfponding Fluents muft, alfo, be equal j that is, 
 
 P I* 1 * I 2 
 
 m-f-2 
 
 x 1 fv) x *1A _ I 
 
 +2 l ' f m+i 
 
 s-m+i 
 
 5V. And 
 
 confequently A (the whole Fluent of a bz"i x 
 
 2 7 <? 
 
 x - X :2 frj X - X into the Series i 
 m + 2 01 -f- 3 l wr / 
 
 I _ j i . m-j-2 j 2 . w + 3 
 
 1 ' 2.r+i 3.^ + 2 
 
 and ^r= 
 
 ^ r ; v and r being any whole pofitive Numbers at 
 pleafure. 
 
 355. An Example^ or two, of the Ufe of this Con- 
 
 clufion, may be proper. 
 
 i 
 1. Let the whole Fluent of i Vj" "* x ( exprefimg 
 
 the Length of i of the Periphery of the Circle whofe 
 Radius is Unity) be demanded. In which Cafe, a be- 
 ing + I, *=J, ! = .-, = 2, p ==i, r=r+^ 
 
 2r-f i 2V 2r + i 
 
 - , and j~i/ r-f z r: --- , the Fluent 
 
 2 2 
 
 fought will, therefore, (by fubflituting thefe Values) be 
 
 had = i x x A MX 2-x ^ x 1 W 
 1 3 5 ' 3 S
 
 The Manner of making Fluents converge. 411 
 
 1 . I 
 
 into i _ 
 2r 
 
 2r -f- I 4 . 2r -f 3 
 
 5 . M -ir- f _ , ^^^ 
 6 . 2r -f 5 8 . 2r + 7 
 
 by expounding -y by 5 and r by 3, will become =r 
 
 2,16710 Csfc. into I ^- 1 -^_ 2-^ 
 
 2.7 4.9 6.11 
 
 JLl_L G -f -21 # + _ii_5 7 + ^ ? 
 
 8.13 jo . 15 12.17 
 
 In the bringing out of which Value, all the Terms above 
 exhibited are requifue : But, of the common Series, i -f- 
 i , 1.3 1.3.5 
 
 rr3 2.4.5 + 2.4^6.7 + r* more than I0 
 
 times tha-t Number of Terms would be neceflary to an- 
 fwer with the fame Degree of Exacinefs. 
 
 dx 
 Ex. 2. Let the Fluxion frcpifed be , 
 
 (whofe whole Fluent, when x d, exprefTes the Time 
 of Defcent of a heavy Body in half the Arch of a Semi- 
 circle, whole Radius is d*.} * Art. 207. 
 
 Here, by comparing d z A*] L x A- ** with 
 
 -. 
 
 a bz Xz z y we have a d'^ p=ri, n~ 2, fn i 
 
 = ^, or p = -fi alfo s (p + v r) = -y r + ;, 
 /{'r + w + i) =r + i: Whence, by taking r and v, 
 each, equal to 4, the Fluent, itfclf, comes out := 
 
 ? 7 ii 15 . 246 8 
 X X X into X X X X 
 i 5 9 13 '357 
 
 4-9 ii 12. 13 
 
 1 1 . i . i qti. 
 
 - 4-
 
 4 1 2 The Manner of making Fluents converge. 
 
 _ 
 
 2,6215^* : Which is to 2 v/^', the Time of De- 
 fccnt along the vertical Diameter of the ferrfaid Circle, 
 as 2,6215 to 2,8184, or as 100 to 108? ; 
 
 After the fame Manner the Fluent will .ind in 
 
 other Cafes : But, with regard to the affigning of the 
 Values of r and z;, it may be obierved, that the An- 
 fwer will, common<y be brought out wkh the leaft 
 Trouble when v is taken greater by an Unit or two than 
 r ; which laft Quantity muft be greater ,r Icfs, ac- 
 cording as a greater or lefs Decree of F ,s ne- 
 
 ceflary. - From the foregoing Expn. fiion-, by varying 
 the Values of v and r, a great Number of Theorems, 
 for the Summation of Serieies, may be deduced. But 
 this being foreign to my preient Purpoie, I am not at 
 Leifure to purfue it here. 
 
 356. Hitherto Regard has been had to Fluxions of 
 the Binomial-Kind : But, from thence, the Fluents of 
 Trinomials may alfo be found ; when theie laft can be' 
 reduced to Binomials (by Art. 307.) without introducing 
 new Radical Quantities. - -Befides which Method, I 
 {hall, hers, give another, which will anfwer where that 
 fails, and is alfo applicable to Multinomials. 
 
 m 
 
 X 
 
 z i) be denoted by A\ and let it be required to 
 find, from thence, the Fluent of the Radical Multino- 
 mial^ or Infinite Series, a -f- <r.v" + dx*" + ex 3 " -f/* 4 " fc? f . T" 
 
 pnl . 
 X * x. 
 
 Make "= ex" + dx -f ex*" -f &f f . and y - x*" i 
 i 
 
 then, x n being y f > if this Value be fubftituted for 
 
 i 
 
 #% in the firfl Equation, it will become cz" =z cy f -f 
 
 JL J. 
 
 dy* -f ey f &< Whence, by reverting the Series, (by 
 
 Art*
 
 The Manner of making Fluents converge. 4*3 
 
 Art. 275.) y (/") is found = J* + Rf"*" 1 4. 
 
 Where R = '-, S = P-P+* v 1 ^ r - 
 
 2 X ' ~ 7 ' ^ ~ 
 
 6 ?- 4X 7 7- 
 
 Moreover, by taking the Fluxion of the Equation 
 thus brought out, and dividing by />, we have x *x 
 
 J+ I *+!. . / + 2 p />n4-ani. 
 
 x Kz r % -j. i - x Sz x. 
 
 . 
 
 Now let this Value, with that of cx n + dx z " -f /* 3 * 
 + t^c. (given above) be fubftituted in the proposed 
 
 Fluxion, and it will become a + cz"}"' x z fK " l x + 
 
 Alfo, let v denote the Place, or Diftance, of any 
 Term of this Series from the firft, cxclufivc ; then the 
 Term itfelf, drawn into the general Multiplicator, will 
 
 + V 
 
 /T J I , A Pt+VM~l . f A 
 
 be expreiied by a -f cz ' X - - A * * (A 
 
 being the correfponding Coefficient /?, S, 7", 2V.) and 
 
 y> 4. v - 
 
 the Fluent thereof by - - A X a -f cz 
 
 t-n 2 /i^.v" 1 '" 3" 
 
 f+I.Mf* 7-f I . i - S I HC * 
 
 Where,
 
 414 *Tbe Manner of making Fluents converge. 
 
 Where, qp + v I, j=/w + y, tp + ni+ I, and the 
 Sign of the laft Term is -f or , according as v is an 
 even or odd Number. Now, if in the Fluent thus gi- 
 ven, v be expounded by j, 2, 3,4, t3c. fucceflively, it 
 is evident the Fluent of the whole Expreflion will, in 
 all Circumftances of z, be obtained. But, if the Co- 
 efficient c be negative, fo that a + cz n may (by increafing 
 z) become equal to Nothing ; then, in that Circum-' 
 
 .ITT 
 
 ftance, the Fluent of the forefaid general Term a -f cz"\ 
 
 % ~*z, making c b) being, barely, = x 
 
 X (v) x X *, it follows that 
 
 r -f i / + 2 l p h" J 
 
 the whole Fluent of the given Expreflion, or its Equal, 
 
 be truly 
 jf 
 
 A I. T>~ V>_Lr A. 
 
 reprcfented by 
 
 t.t + i. 
 
 pd 
 In which, R 
 
 ^. and A - the Fluent 
 
 _ 
 
 x J i, when <2 bz" =r o. 
 
 357. Hence, if the Fluxion given be of the Trino- 
 iial Kind (then, f, f, &c, vanifliing the ifhole Fluent 
 
 of
 
 The Manner of making Fluents converge. 415 
 
 of a bx n + dx*"} X /" '* (when a bx n -f <& 2fl 
 = o) will, by fubftituting for R, 5, T, &c. be = A X 
 
 v i . ,. 
 
 ' * 77 T - . X 77 T 
 
 I * bb i . 2 . t .t+i 
 
 P-p+i 
 
 J . 2 . 3 . r . /-M . /+a W 
 358. If m+ i and /> are the Halves of any odd Affir- 
 
 mative-Numbers, the Fluent of a bz \ x a /> *~ I a:, 
 
 when lx" =. o, will be equal to 
 
 i 3-5- 7 (M-f-Qx 1.3.5-7 (P-O 
 
 2. 4 . 0.8. 10. 12 ( 
 
 G being the Periphery of the Circle whofe Diameter is 
 
 Unity. Therefore the Fluent ofa^bx" + dx 
 
 pn I . _ . ~~^\ *B I . 
 
 X * A-, or its Equal, a bz Jx + " ' 
 
 X Rz K, &c. is found, in this Cafe, by multiplying 
 
 the Expreffion here given, into the. foregoing Series, i -f 
 
 -~* Ra 
 
 359. An Example or two will help to (hew the Ufe 
 f what is above delivered. 
 
 Firft, let the Fluent of 
 
 v/ >_,>- -^ 
 
 r4 
 
 (when the Divifor becomes equal to Nothing} be re 
 quired. , 
 
 Then, by comparing a 1 ** ~I with. 
 
 the.
 
 41 6 The Manner of making Fluent* converge. 
 
 the general Trinomial a bx" + </***{ x *r*"> x, it 
 appears that a 1 muft be, here, wrote in the room of a, 
 and that n, m t p, b and rf, will be interpreted by 2 
 
 , . i. and refpe&ively : Whence we 
 
 22 raa 
 
 = _, and the Fluent fought = x 
 
 JLll . T -3-5-7 _. l .-3-5-7-9-" . ^. 
 
 ~ 2 . 2r + 2 . 2 . 4 . \r~ ' " 2 . 2 . 4 . 4 . 6 . 6r 3 ' 
 
 360. The fecond Example fhall be, to find the Fluent 
 expreffing. the Apfide Angle in an Orbit defcribed by 
 means ofa centripetal Force varying according to any 
 Power of the Diftance. 
 
 In which Cafe the given Fluxion being 
 
 ===== * PX W' *t. 242- 
 
 where A is fuppofed the higher Apfe, and CA (and 
 confequently Cb) equal to Unity) we fhall, by putting 
 
 j p* p, = v, and I x 1 = jr, reduce it to 
 
 I v x V 3y + ~ ?._ . 
 
 yy V.V-2 ^.^-2.^-3 , 
 
 " 2 H 2.3 ' 7 2 -3-4 
 
 the Q- uan - 
 
 tity
 
 The Manner of making Pluents converge. 417 
 
 tity under the Radical Sign (now anfwering to the 
 Form above prefcribed) being compared with 
 
 abx* -f dx 1 * + ex*" bV.r, we have m = i, 
 
 2 3 3.4 
 
 &>V. Alfo the Value of/> with regard to the firft 
 
 Term (y~~*j) will be = * (becaufe pn i = t) 
 likewife its Value in the fecond Term (y*y) is = -^-i in 
 
 the third = -|- & c . In the firft of thefe Cafes we> 
 therefore, have t (m+p+ i) = i, & (p x -T-) = 
 
 4.V s . _ V 2 .16^-- 3^t; +-22 
 
 6 ' 72 16X45 
 
 Whence it follows, that the Fluent of the firft Term 
 
 2 / tf x> when the 
 
 2 2 . 3 
 
 Quantity under the Radical Sign becomes equal to No- 
 thing (or the Body arrives at its lower Apfe) will be 
 
 G V2 
 
 truly exprefled by . into i + 
 
 
 S.v2 -4^S a * + 7 . P 2 . Ibv 1 37V + 22 j 
 
 4 8v a 6x48^ 
 
 In the fame Manner it will appear^ that the Fluent 
 of the fecond Term, in that Circumftance, is = 
 
 Ee
 
 4*8 be Manner of making Fluents converge. 
 
 that of the Third = =. x 
 
 V *v 
 
 3S ' V 3 2 . P &c. that of the Fourth - ~j . s 
 fcV. Uc. 
 
 Whence, the Fluent of the whole Series, by col- 
 liding thefe feveral Values together, will come out 
 
 G 9V 1 < 
 
 1 1 2v 3 63^42^ 8 
 
 - - A Q 3 - P + &' Which, drawn into 
 o x ^.o'y 
 
 X~i i iP l -r l s P 3 fcrV. (the Value of 
 
 _ G 
 
 the general Multiplicator \ V i &) gives - x 
 
 V 2v 
 
 * ^ 2 . 2V I 8* ?; 2 . 2V J . 2V - 1 
 
 I * + - - i _ - 
 
 48 v* ' 72 
 
 fl3 
 
 X j &c. for the true Meafure of the Angle required, 
 
 in Parts of the Radius, or Unity : From whence, by 
 writing 180 inftead of G, we (hall have the fame in 
 Degrees : Which, laft of all, by reftorina; -, becomes 
 
 Where n is the Exponent of the Law of the Force, 
 whereby the Orbit is defcribed ; and j3, the Defect of 
 the Square of the Meafure of the Celerity, at the higher 
 jfpfe, below That which the Body ought to have to re- 
 volve in a Circle, this laft being denoted by Unity. 
 
 The
 
 *he Manner of making Fluents converge. 419 
 
 The fame Concluflon may be otherwife derived, by 
 bringing i y, in the transformed Fluxion, under the 
 Vinculum ; but this Way of going to work, though 
 we have but one Series to manage, will prove rather 
 more troublefome than the foregoing. 
 
 It will appear from the two preceding Examples, ef- 
 pecially the firft of them, that this laft Method of find- 
 ing Fluents is, chiefly, ufeful when all the Terms of 
 the given Expreflion, after the two firft, in refpeft of 
 thefe, are but fmall. Which is a Circumftance that 
 frequently occurs in the Refolution of phyfical Pro- 
 blems ; fuch as determining the Effect of the Atmo- 
 fphere's Refiftance upon the Vibration of Pendulums ; 
 and the Inequalities of the Planets arifing from their 
 
 Action on each other. In fhort, wherever the Fluent, 
 
 or the Quantity it exprefTes, would belong to the Circle, 
 or fome other of the Conic-Sections, were it not for 
 the Interpofition of fome fmall perturbating Force 
 (whereby new Terms, fmall in Comparifon of the two 
 firft, are introduced) the faid Method will be found of 
 very great Service; 
 
 SEC-
 
 420 Of the Motion of Bodies 
 
 SECTION VIII. 
 
 life of Fluxions in determining the Motion 
 of Bodies in rejijiing Mediums. 
 
 P R O B. I. 
 
 361. Suppofing that a Eody^ let go from a given Point 
 A, with a given Celerity, in a Right-line AQ^, is 
 ref:ficd ly a Medium (or any Force) afting according 
 to a givtn Power of the Velocity : To determine the 
 Velocity, and alfo the Sface run ever, at the End of a 
 given 'Time. 
 
 LE T the given Celerity at A (meafur'd by the 
 Space which would be uniformly defcribed in any 
 propofed Time r) be put c, and that at any other 
 Point B, = v j moreover put AB = #, and the Time 
 
 B 
 
 of its Defcription z ; and let the Refiftance, or Force, 
 adling upon the Body at A, be fuch, that, if the fame 
 was to be uniformly continued, the Body would have 
 all its Motion deftroyed thereby, in the Time wherein 
 it might move, uniformly, over a given Diftance d 
 (CD) with its firft Velocity c : Which Time, let be de- 
 noted, by /. 
 
 Then, fmce the whole Celerity c would bedeftroy'd in 
 the Time/, that Part of it which would be uniformly ta- 
 ken away in the Time r, above propofed, will be truly re- 
 
 T tC 
 
 prefented by - X c ; or by -\ > which is equal to it, 
 becaufe the Spaces (c and d) defcribed with the fame
 
 in rejlftmg Mediums. 421 
 
 Celerity are always as the Times (r and t) of their 
 Defcription j and therefore . 
 
 Hence, the Refiftance at B being to that at A (by 
 
 Hypothecs) as v" to c" it follows that the Velocity 
 
 which might be deftroyed in the given Time r, by a 
 
 ' Force equal to the Refiftance at B, will be exprefled by 
 
 cc i) v 
 
 -7 X ~ or its Equal n j_ a ' ' : Which Expreifion is, 
 
 therefore, the true Meafure of the Force of the faid 
 Refiftance. 
 
 Now, it appears, from Art. 218. that, if the Force 
 with which the Body is adted on (or the Velocity it 
 would generate in the given Time r) be reprefented by 
 jp, the Relation of the Meafures of the Velocity and 
 Space gone over, will be exprefied by the Equation -j~vv 
 
 v n 
 Fx : From whence, by writing - ^^ inftead of 
 
 dc ~ 
 
 F t we have vv ~ B __ 2 " ( J he Sign of wu be- 
 ing negative, becaufe v decreafes while x increafes *.) *Art. 5. 
 From this Equation, we get x rr dc"~~ 2 v l ~'''u ; 
 
 whofe Fluent is x ^ Lt Xv -f ; which, 
 
 2 n 
 
 corrected (by taking x o, and v c) becomes x = 
 
 , n z 2 n 
 
 it x v 
 
 ' 
 
 X - I. 
 
 ;; 2 i- j 
 
 2 
 
 Moreover, fmce the Time (~) is to the Time r, a 
 the Diitance x to the Diftance f, \ve alfo have * (:= 
 
 ) = rflt* "" z v~~ *-v j and confeiiuently z ~ 
 * /
 
 4 22 Of the Motion of Bodies 
 
 rd _ I= _^_ X _I _ x (b . 
 
 writing t for its Equal - J : From which Equation 
 
 c ^__ sV" 
 
 we get = i + if i x -| : Likewife, 
 
 from the preceding Equation, we get r= 
 
 i + 2 x -jl : Which two equal Values be- 
 ing compared together, there, at length, refults * == 
 
 into i + n i x -7- 
 n 2 
 
 j, for the required Re- 
 lation of x and z. . . L 
 
 COROLLARY. 
 
 362. If =s 2, or, the Refiftance be in the Duplicate 
 Ratio of the Velocity, the Equation exhibiting the Re- 
 
 c z 
 
 lation of % and v, will be = i -f , or v =: 
 
 - : But the other Equation (the Fluent failing) 
 
 B 
 
 becomes impracticable. Here x, the Fluent of 
 t Art 6 * W ^ ^ e explicable by d X hyp. Log. - *, or by d x 
 
 C 
 
 
 
 #>. lo^. i + j becaufe v = 
 
 In
 
 in rejlftmg Medium*. 
 
 In the like Manner, when =i, or the Refiftance is 
 as the Velocity, the Relation of v, x and z, will be 
 
 d x 
 exhibited by the Equations v = CX ~~T~ and z ~ ' x 
 
 423 
 
 that above, are the only two wherein the general So 
 lution fails. 
 
 P R O B. II. 
 
 363. If a Body, let go from a given Point 
 A with a given Celerity, in a vertical Line 
 CAQ^ is atted on by an uniform Gravity, 
 and alfo by a Medium, refiftinj according to 
 any given Power of the Velocity, 'tis pro- 
 pojed to determine the Relation of the 'Times, 
 the Velocities, and the Spaces gone over. 
 
 Let the Notation in the preceding Pro- 
 blem be retained ; and let the Force of Gra- 
 vity, in the given Medium (meafured by the 
 Velocity it might generate in the propofed 
 Time r *) be reprefented by b. Then, Art. 561, 
 this Value being added to, or fub tracked 
 
 A 
 
 C 
 
 the Meafure of the Re- 
 
 fiftance t according as the Body is in its Afcent, or f Art. 361. 
 
 n 
 1) 
 
 Defcent, we thence get - - -f b for the whole 
 
 J * - 
 
 dc 
 Force (F) whereby the Motion, at B, is affe&ed : 
 
 Whence (by Art. 218) * (= =^) - dc n -*w 
 
 - 
 
 E c 4 
 
 may
 
 4 2 4 Of the Motion of Bodies' 
 
 may be had, by the Means of circular Arcs, and Loga- 
 rithms,//<wj Art* 331. Q . /. 
 
 COROLLARY I. 
 
 / v- N 
 
 364. It appears that the Force f " a _ a J of the Re- 
 
 \<st S. 
 
 fiftance is to (b), that of Gravity, in the given Me- 
 dium, as v n to bdc~ * \ Therefore, if this Ratio be 
 expounded by that of v* to o", or a" be put = bdc t ~~ z - t 
 it follows that a will exprefs the Celerity with which the 
 Refiftance would be equal to the Gravity (fmce, when 
 v~ a, the faid Ratio becomes that of Equality.) Hence, 
 
 
 alfo, by fubftituting -r for its Equal dc , we get 
 
 a <w . ra <v 
 
 x > and x = 
 
 b x v" a n b x v"a n 
 
 COROLLARY II. 
 
 365. If the Refiftance be in the Duplicate Ratio of 
 the Celerity, cur two Jail Equations will become * 
 
 vv -f- aa * 
 
 - * TTT^" 
 
 a* r<: + aa d tc -\- bd 
 
 = r x /^/. Z.5P-. =7- 3= x w*. Z.OP-. ^-r, 
 
 20 ^y + ca 2 yr 6 wrU 
 
 (becaufe, here, a* bd.} From whence, when vz=o t 
 (fuppofing the Body to afcend) there comes out x = 
 
 x hyp. Log. i -\ , for the Height (A$J of the 
 
 whole Afcent. But, if t be taken ~ o, or the Body 
 9 be
 
 in refilling Mediums. 425 
 
 be fuppofed to defcend from Reft, we {hall then have 
 
 X hyp. Log. i = the Diftance AE defcend- 
 
 2 /r aa 
 
 ed. Whence, if N be put for the Number whofe Hyper- 
 
 2X 
 
 bolical Logarithm is -r, it follows, (becaufe^ Log. I 
 
 VV. 2K W I 
 
 - = 7- = Log. ff) that i = -^7, and 
 
 aa d aa N 
 
 SJT~^ 
 
 confequcntly v =: a \S From which, the Di- 
 ftance AB being given, the Velocity acquired in the Fall 
 will be deiermintrd. Bur, if the Body, firft, afcends 
 from a given Point A, with a given Celerity c, and the 
 Celerity, acquired in falling, when it arrives, again, at 
 that Point, be required ; the fame may he exhibited in 
 a more commodious Form, independent of Logarithms, 
 
 c 
 
 and will be equal to "' * - ; betaufe N, in this 
 
 Cafe, is found above to be = i -f . Furthermore, 
 
 aa 
 
 with regard to the Time (), we have already found 
 
 ( = 
 
 ) according as the Motion of the Body 
 
 b x aa vv/ 
 
 is from, or towards the Center of Force. Therefore 
 
 ra 
 
 the Time itfelf, in the former Cafe, will be rr - 
 
 b 
 
 drawn into the Difference of the two circular Arcs 
 
 c v 
 
 whofe Tangents are and , and whereof the com- 
 a a' 
 
 mon Radius is Unity * : Whence it follows that the 
 
 Time
 
 Of tbe Motion of Bodies 
 
 Time of the whole Afcent will be denoted by mul- 
 tiplied into the former of the faid Arcs. 
 
 But, in the other Cafe, the Fluent, exhibiting the 
 Time of Defcent, is not explicable by the Arcs of a 
 Circle, but by the Difference of the hyperbolical Lo- 
 
 r a-\-v a + c ra 
 
 Art. 1*6. ganthms of and - - drawn into * There- 
 
 a v a c zb ' 
 
 fore, when c o, or the Body falls from Reft, the 
 
 Time z will be barely r: x bvi>. Lo<r. ^ ~ 
 
 2 a v b 
 
 t_ jr^ 
 
 X hyp. Log. N* + N iT (by fubftituting the Value 
 of v found above, and ordering the Logarithm as in 
 This Equation, in the forementioned -Cir-. 
 
 cumftance, where N =. I H -- , and v = 
 
 aa 
 
 aa 
 
 becomes * = x hyp. Log. ^ 1 , i , L. 
 aa a 
 
 SCHOLIUM. 
 
 366. If, according to Sir Ifaac Newton, we fuppofe 
 the Refiftance of the Air, to Bodies moving in it, to 
 be in the Duplicate Ratio of the Celerities * j and that 
 
 * That the Refijlance is at tbe Square of the Celerity, the 
 Learner may, in fame meafure, ccncei've, by conjidtring that 
 tbe fame Body, with a double Velocity, not only puts twice the 
 Number of refijling Particles in Motion, in tbe fame time, but 
 alfo als upon each with a double Force; and therefore muft fujf'er 
 a four- fold Refinance, or a Refijlance proportional to thi Square 
 of tie Velocity. 'This would be ftrittly true, were it not that 
 the Particles fo put in Motion impel others lying before them* 
 and thereby prevent, as it were, the Action of tbe Body. What 
 Deviation from tbe foregoing Law may hence arife, is not eajy 
 to determine, 'fhis, however, feems plain, that tbe Refijlance 
 at the Beginning of any very fwift Motion (till tbe Air in tbe 
 Way of tbe Body cames July to participate of that Motion) will 
 l.e greater than 'That fuftaintd by another equal oay, moving 
 with the fame Celerity, that has been in Motion fame time. 
 
 a Ball,
 
 in rejijling Mediums. 427 
 
 a Ball, in the Time it might move, unifcrmly, over a 
 Space ( d] which is to | of its Diameter as the Denfity 
 of the Ball to that of the Medium, would have all its 
 Motion taken away by a Force equal to that of the Re- 
 fiftance, uniformly continued : Then, from thefe Data^ 
 applied to the Theorems in the preceding Article, we 
 fhall be able to determine the Velocities, and the Times 
 of the perpendicular Afcent and Uefcent of Bodies near 
 the Earth's Surface j allowing for the Refiftance of the 
 Atmofphere. 
 
 Thus, for Inftance, let a Cannon Bali, of four Inches 
 Diameter (whereof the Denfity, or fpecific Gravity, is 
 to that of Air as 6000 to i, nearly) be fuppofed to be 
 projected, perpendicular to the Horizon, with a Velocity 
 fufficient to caufe it to afcend to the Height of half a 
 Mile, or 2640 Feet, in vacua j which Velocity (by Art. 
 203 j will be found to anfwer to the Rate of about 412 
 Feet/^r Second: Then, according to the Proportion juft 
 now mentioned, it will be as i : booo : : | x 4 : 64000 
 Inches, or 5333 Feet ; which is the Value of d in this 
 Cafe. Therefore, if the Time r, in the preceding Ar- 
 ticle (which may be aflumed at pleafure) be here inter- 
 preted by one Second^ the correfponding Values of </, c 
 and b will be expounded by 5333 F. 412 F. and 32^ 
 F. * refpectively. Which Values being fubftituted in * An. ao*. 
 the feveral Equations in the laft Article, we fhall get 
 
 i. a(VTd} = 414 F. the Velocity, per Se- 
 cend, wherewith the Refiftance would be equal to the 
 (Gravity, or Weight, of the Ball. 
 
 2. X hyp. Log. i -\ =: 1835 Feet, the whole 
 
 2 r aa 
 
 Height of the Afcent. 
 
 3. -r x Arch, whofe Tang, is =: 10,08 Seconds^ 
 
 the whole Time of the Afcent (which is lefs than the 
 
 s 
 Time, in vacuo t by 2>73.)
 
 428 Of the Motion of Bodies 
 
 c v F 
 
 4. v ( . A = 292, the Velocity, per 
 
 ^ I+ 7j 
 
 Second^ acquired in the Defcent. 
 
 
 5 . Laftly, y x hyp. Log. ^ , + 1 + _1 _, 
 
 aa a 
 1 1,30 Second?, the Time of the Defcent. 
 
 Note, In this Example the Meafure of the abfolute 
 Gravity of the Body, in vicuo, is taken, inftead of its 
 Gravity in Air (the Difference, there* being too incon- 
 fiderable to he-regarded.) But, in Cafes where the fpe- 
 ciric Gravity of the Medium bears a fenfible Proportion 
 to that of the Body, the Force of Gravity (I) muft 
 
 n _ n* 
 
 be expounded by 32^ x - ^ (inftead of 32 7 V) 
 
 Where E is to J/as the fpecific Gravity of the Body to 
 that of the Medium. 
 
 P R O B. III. 
 
 367. To determine the Refinance, by means tuber eof a 
 Body> gravitating uniformly in the Direction of parallel 
 Line s^ may defer ibe a given Curve 
 
 Let ABC be the given Curve, and BQ_, parallel 
 to the Axis (or any given Line) AH, be the Direction 
 of Gravitation at any Point B : Make PER perpendi- 
 cular to AH and BQ^; and let AP=*, PB=_y, AB-s, 
 BM (N) = *, MN (B^) =>, BN = i, and the 
 Velocity of the Body at B in the Direction PBR = v. 
 Then, the Decreafe of Velocity in the faid Dire6tion, 
 which is wholly owing to the Reuftance *, being re- 
 
 prefented by -u, it follows that the correfponding De- 
 creafe of Motion in the Direction BN, arifing from the 
 
 rf~l ^* " ^iw 
 
 fame Caufe, will be exprefled by X v=z -- ; 
 
 y y 
 
 *ux 
 
 and, that in the Direction BM, by -- -. But, the 
 
 Celerity
 
 in refijling Mediums. 429 
 
 Celerity in this laft Diredion being, every where, re- 
 
 x 
 
 prefented by v X , its Fluxion 
 
 -f 
 
 will be the 
 
 whole Alteration of Motion in the faid Dire&ion, arifmg 
 from the Refiftance and the Force of Gravity, con- 
 jun&ly : From which deducting the Part owing to the 
 
 vx vx 
 
 Refiftance, found above to be -r, the Remainder 
 
 will be the EfFeft of the Gravity. Which being to 
 
 ( ) the EffecT: of the abfolute Refiftance in the 
 
 3' 
 
 Direction BN, as i to r, the Force of Gravity, 
 
 muft therefore be to that of the required Refiftance, in 
 
 vx 
 
 the fame Ratio of i to . 
 
 Moreover, the Force of Gravity, meafured by the 
 Velocity it would generate in a given Part of Time ( i), 
 being denoted by Unity, the Velocity generated thereby, 
 
 in the Time ( J of defcribing B/>, with the Celerity v 9 
 
 y 
 
 will likewife be truly exprefed by, , the Meafure of 
 
 the
 
 43 Of *- JC Motion of Bodies 
 
 the faid Time : Which being put rr to ( ~ J the Va- 
 
 lue of the fame Quantity, given above, we thence have 
 
 yy 
 v* =. - : From whence, not only the Velocity, but 
 
 the Refiftance will be found. But, if you would have 
 the Refiftance exprefled independent of v j then let the 
 
 i*\ 
 Fluxion (zvu = -- . J of the laft Equation be di- 
 
 XX ' 
 
 *u ~ jc 
 
 vided by the Fluent, which will give = - : 
 
 v x 
 
 1JZ 
 
 And then, by fubftituting this Value in , you will 
 
 vx 
 
 eet-^-, for the true Force of the Refiftance, that of 
 
 * 2xx 
 
 Gravity (or the Weight of the Body) being expounded 
 by Unity. 
 
 The fame otberwife 
 
 Let BO be the Radius of Curvature at B, and let 
 OQ_ be parallel to PB> meeting BM, produced, in Q_: 
 Then, if the abfolute Gravity, acting in the Direction 
 BQ., be denoted by Unity, its Force in the Direction 
 BO, whereby the Body is retained in the Curve, will 
 
 BO 
 
 be reprefented by rrrr . Therefore, fmce the Veloci- 
 
 ties in Circles are known to be in the Subduplicate Ratio 
 Art.au, of the Radii and of ths Forces conjundly *, the Ve- 
 
 \/ 
 
 locity at B will be rightly exprefted by \BO x 
 
 or its Equal -V/BQ. (For the Curve at, and inde- 
 finitely near, B may be taken as an Arch of a Circle 
 whofe Radius is BO : And it is evident that the Re- 
 fiftance has nothing to do in forcing the Body from the 
 
 Tangent,
 
 in rejijlmg 'Mediums. 431 
 
 Tangent, but only ferves to retard its Motion fo, that 
 it may, every where, bear a due Proportion to the 
 given Force of Gravity ac~ling in the Direction BO.) 
 Hence, putting BQ_= s, the Increafe of the Celerity 
 
 in the Time ( ^ J of defcribing BN, will be ex- 
 prefled by the Fluxion of Vj, or r- Moreover, 
 
 2 Vj 
 
 the Celerity that might be generated by Gravity in the 
 
 
 
 faid Time - =: being meafured thereby, the Increafe, 
 Vj 
 
 in BN, arifing from the fame Caufe, will therefore be 
 
 = -7= x -= : Which, being taken from 
 Vj x / s 
 
 the whole Increafe, found above, the Re- 
 
 (Wf) 
 
 mainder, - _ ^f, will be the Effe& of the Refiftance : 
 
 Which is to the EffecT:, =, of the abfolute Gravity 
 v s 
 
 as f~ 2 * to I. Therefore the Refiftance is to the 
 2* 
 
 
 
 Gravity (or Weight of the Body) as *** to U- 
 
 2% 
 
 nity : Where the Signs are changed, becaufe the two 
 Forces acl: in contrary Directions. 
 
 Becaufe BO = ?:. *, therefore s (BO x 4 ") = t Art. 6f . 
 
 JfX X / 
 
 * 
 
 (= the Square of the Celerity) whence 
 
 confequent i y t he Re- 
 fiftance
 
 43 2 Of the Motion of Bodies 
 
 fiftance . *~~i rr y ~*~ * x x ^-, tie very fame 
 at before. 
 
 COROLLARY. 
 
 368. IF the Refiftance be fuppofed as any given 
 Power of the Velocity drawn into (D) the Denfity of 
 the Medium ; then, from hence, the Denfity of the 
 Medium, at every Point of the Curve, may be deter- 
 mined : For, the abfolute Celerity at B being repre- 
 
 fented by, the Refiftance at that Point will, according 
 
 ~~- VI 
 
 to the faid Hypothefis, be as _ I x D j and therefore 
 
 9 
 
 the Velocity that would be deftroyed thereby, in the 
 
 Time (i-\ ofdefcribingtf, as 3 x ? : Which 
 \v J j\ v 
 
 being put = ( -r J the Effect of the fame Re* 
 
 X _ . X 
 
 fiftance, found above, we thence get D = __, : 
 
 "Usd 
 
 Which, by fubftituting for v and <i>, becomes D = 
 
 In this Corollary, and what, elfewhere, relates to un- 
 equal Denfities, the Gravity of the Body in the Me- 
 dium is fuppofed to continue, every where, the fame, 
 or, that the Attraction increafes with the Denfity, fo 
 that the Difference between the fpecific Gravities of 
 the Body and Medium may, at every Point, be a con- 
 ft ant Quantity. 
 
 E X-
 
 in refifting Mediums. 
 
 EXAMPLE I. 
 
 369. Let the propped Curve ABC be the ctmmon 
 
 Parabola : 
 
 2 
 
 V ^ W 
 
 Then, x beiog here = , we have x = -^, x 
 d a, 
 
 - and x =: oj and therefore * is alfo = 
 
 <* 2i> 
 
 Whence it appears that a Body, to defcribe this Curve, 
 muft move in Spaces intirely void of Refiftance. 
 
 EXAMPLE II. 
 
 370. Let the Curve 
 ABC be taken as a Qua- 
 drant cf a Circle, whofe 
 Radius BO is a. 
 
 In this Cafe we have s 
 
 (BCD f = * (= AO 
 AP) whence }=. x, 
 
 Q. X - -'- S 
 
 and therefore : =s O 
 
 433 
 
 F fom which it is evident, that ^ e tArt, 
 
 Velocity is, every where, as /BQ, and the Refinance 
 to the Gravity (or Weight of the Body) as 3?B to 
 2 OB. 
 
 P R O . IV. 
 
 371. 'lie Centripetal Force (F) being given-, to find 
 the Refinance and Velocity whereby a Body may defcribe a 
 given Spiral (or any other, po/tble, Curve) about the 
 Center of Force. 
 
 Let P be the Center of Force, and BO the Radius 
 
 of Curvature at any Point B in the propofed Curve, 
 
 F f and
 
 434 
 
 Art.5' 
 
 Of the Motion of Bodies 
 B 
 
 and let OQ.be per- 
 pendicular to BPQ; 
 alfo let BP = y, BQ, 
 = s, AB = z, BM 
 = y *, and BN 
 = x. Then, it is 
 evident from Art. 367. 
 ^ that the Velocity at B 
 will be exprefled by 
 
 V/BOx 
 
 or, its Equal, v^.* And therefore its increafe in the 
 
 f _*_\ sF + Fs 
 
 Time l ) of defcribing BN will be - ,- : 
 
 == x -^- j| the Ef- 
 
 From which, deduding (^ X -f== 
 
 feft of the centripetal Force, in the fame Time and 
 
 Direction, the Remainder, - 7= - > is theEf- 
 
 2 v sr 
 
 of the Refiftance. Therefore the Refiftance is to 
 
 the centripetal Force as 
 
 SF + F*-k~2Fj . TTniftr 
 
 as : yr. to Unity 
 
 EXAMPLE. 
 
 372. Let the Meafure (F) of the centripetal Force 
 
 be exoounded by any Power / of the Diftance ; and 
 
 let the Curve be the logarithmic Spiral ; putting the 
 
 fAru 61. Co-fine of the given Angle PBN f (to the RMiu^rJ 
 
 1 Art. 74, = t. Then, s being here = y t, and F = / >, 
 
 we
 
 in refjling Mediums. 
 
 *Lt 
 
 2i 
 
 x - = x - 
 
 a 2 r 
 
 Hence it appears that the Velocity muft be, every 
 
 where, as y * i and the Refiftance, to the centripetal 
 Force, as x - to Unity. But, when n = 3, 
 
 X - becomes = o ; therefore the Body, in this 
 
 Cafe, muft move in Spaces intirely void of Refiftance ; 
 agreeable to An. 233. And, if n-f-3 be negative, an 
 accelerating, inftead of a refifting Force, will be required. 
 
 SCHOLIUM. 
 
 373. If the Denfity of a Medium, wherein a Body 
 moves, be either uniform, or varies according to a 
 given Law, the Nature of the Curve, or Trajectory 
 may be determined from what is delivered in the pre- 
 ceding Pages. 
 
 Thus, for Example, let the Denfity be fuppofed every 
 
 where the fame, and the Refiftance as the Square of the 
 
 1 . / 
 
 Celerity; then, from Art. 368. we have -^- =r D; 
 
 zx 
 
 which, in order to exterminate , may be transformed 
 
 to ## = yy + xx x D*xx : Where, D being a conftant 
 Quantity (depending upon the given Denfity of the 
 Medium) the Value of x will be found, as is taught in 
 
 y* Dv* 
 
 Stfl. 2. Art. 268. 271. and comes out = + 
 
 P IP 
 
 ~~ r fisfr. In which p is put to denote the Para- 
 F f 2 meter
 
 436 Of the Motion of Bodies 
 
 meter of the Curve at the Vertex, or higheft Point A, 
 (to be determin'd from the Force of Gravity and the 
 given Velocity of the Body at that Point.) This So- 
 lution anfwers near enough when the Refinance is but 
 fmall in Proportion to the Gravity ; in other Circum- 
 ftances, the Series not converging, it becomes ufelefs : 
 For which Reafon, and becaufe the Cafe above fpe- 
 cified is That fuppofed to obtain, in refpec~l to the Air 
 near the Earth's Surface, and its Refiftance to Bodies 
 moving therein, I fhall {hew, by a different Method, 
 how the Nature of the Curve may be inveftigated. 
 o In order thereto, let the Celerity at the higheft Point, 
 A, above the Plane of the Horizon EC, be denoted by 
 c ; and let a be the Celerity with which the Refiftance 
 is equal to the Gravity (vid. Art. 365. and 366.) 
 
 Moreover, let d be put for the Diftance over which the 
 Ball might uniformly move in the Time that the Me- 
 dium would deftroy all its Motion, was the Refiftance 
 to continue the fame, all along, as at the firft Inftant 
 (Which Diftance, according to Sir Ifaac Newton, is, al- 
 ways, in Proportion to \ of the Ball's Diameter, as 
 the Denfity of the Bali to that of the Medium.) 
 
 Then it will be, as</: a (BN) :: ~, the abfolute 
 
 Celerity at B, to f- r l the Part thereof that would 
 
 be uniformly deftroyed by the Refiftance in the Time 
 of defcribing BN, with the Velocity at B : Which 
 
 Value being alfo expreffed by- - (wV. Art. 367.) we 
 
 there-
 
 in refifting Mediums. 437 
 
 liz 1 vz z v 
 
 therefore have -r- n ; whence ~r r= , and 
 
 dj y d v 
 
 rt- 
 
 confequently, by taking the Fluent, -j =. hyp. 
 
 Log. v, which corre&ed (by putting arzo, and v = c) 
 
 z c 
 
 gives -j ( = hyp. Log. c hyp. Log. v) = hyp. Log. -. 
 
 Furthermore, fince (by Hypothefis) the Refiftance 
 with the Celerity -f (at B) is to the Force of Gravity, 
 
 or the Refiftance with the Celerity a, as r- to a* ; 
 
 yy 
 
 and it appears, from the aforefaid Article, that the fame 
 
 T"| f2 
 
 Katio is alfo univerfally exprefled by that of to 
 I, it follows, from the Equality of thefe Ratios, that 
 ~r is == -. But, in order to the Refolution of 
 
 the Equation thus given, let the Tangent of the Angle 
 PBA (or N) which the Ordinate, PB, makes with 
 the Curve (fuppofing Radius Unity) be, every where, 
 
 reprefented by w: Then, becaufe x"=.u>y y z (Y > a + x*} 
 
 = _^ r i -f to 2 , and x = <wy (y being conftant) we 
 fhall, by fubftituting thefe Values in the forefaid Equa- 
 tion, get -^j- = , V i + ;* ; whereof the 
 
 Fluent will be given, ~ = I w Vi + w 1 -f 1 hyp 
 
 Log. w; + ^J + w 1 * : Which corrected (by taking* Art. 126, 
 
 JL .j* i /7 1 an( ^ 2 ^ If 
 
 v=f and wno) becomes ~- f w "/j -f- w* 
 
 -J- i hyp. Log. w + v/j -f w*. But, to fhorten the 
 F f 3 re-
 
 Of the Motion of Bodies 
 
 remaining Part of the Procefs, let the latter Part of 
 the Equation, or the Fluent of ov Vj + w z be de- 
 
 aa aa 
 
 noted by 9\ then - being = -- \- .9 , we have v = 
 
 2VV ICC 
 
 ^d confequcntly (= hyp. Log. -l 
 
 
 l/aa + 2^.9 2jp 
 
 = hyp. Log. - - = 4 hyp. Log. i -I- -^. 
 
 From which two Equations, the Velocity of the Ball, 
 and the Diftance it has moved, when its Direction 
 makes any given Angle with the Horizon, may be com- 
 puted, let the Medium be as denfe as it will: Alfo, 
 from hence, if the Celerity anfwering to any one given 
 Angle of Direction be known, the Celerity conefpond- 
 ingto any other given Direction may be found, together 
 with the Diftance defcribed between the two Portions. 
 For v (in the Deicent of the Body) being, 
 
 equal to -... ---. =, the Value of r, exprefling 
 Yaa+irQ 
 
 the Celerity at the Vertex A, will be had from that 
 
 av 
 
 Equation, and comes out = - - ; whence 
 
 y aa 2-y j^ t 
 
 alfo z ( d x hyp. Log. j ~ d x hyp. Log. 
 
 Vaa 2i>^ ~ aa 
 
 From which, the Celerity at A being known, the reft 
 
 is obvious. 
 
 But, in the afcending Pnrt of the Curve EA, both 
 % and Q muft be confiticred as negative, or wrote with 
 contrary Signs: And then, from the foregoing Equations, 
 
 ac eru 
 
 we {hall alfo get v = , c ~ _ -, 
 
 V aa 2fc<S> V aa - 
 
 and
 
 in rejijlmg Mediums. 439 
 
 f 
 
 and z=i^x hyp. Log. i --- * = 1 
 
 . -w' 
 
 hyp. Log. i + - ; and, confequently, z r= ^ 
 
 X hyp. Log. i 2U ^ - 1 - d* hyp. Log. I + 
 
 aa oa 
 
 d x hyp. Log. : Anfwenng in this Cafe. 
 
 It flill remains to take fome notice of the Values of 
 AC and y (in order to have the Form, as well as the 
 Length of the Curve.) Thefe, indeed, are not fo 
 eafy to bring out as That of K, given above ; nor 
 can they be exhibited in a general Manner, either by 
 circular Arcs, or Logarithms (that 1 have been able to 
 difcover) but may, however, be approximated to any 
 required Degree of Exa<Slnefs, as will appear from what 
 follows. 
 
 Since z ( = AB) is found * d X hyp. Log. 
 
 O(l "^ 2 ^P 
 
 ', by taking the Fluxion thereof, we get 2 r: 
 
 aa + 2cc^ aa -f 2c 
 z 
 
 Therefore j ( = -* ' = + J and 
 
 (-ivy]- aa + 2fl ^: Which Equations, by taking 
 
 r to i, as a* to c* (or as the Square of the Force of 
 Gravity to the Square of the Refiftance at A) are re- 
 
 Ff 4
 
 440 
 
 Of the Motion of Bodies 
 
 . And * :r */ into 
 
 Thefe E ffions 
 
 (brought out by afTuming 
 
 for the Fluent fought, and proceeding as in Art. 34.0.) 
 converge very faft when r is large in comparifon to Q ; 
 but in other Cafes the required Values will be had, with 
 lefs Trouble, from the following Method. 
 
 ^^ 
 
 M T. 
 
 S S C S B 
 
 TO, 
 
 Let PKTK and AMTM be two Curves, whereof 
 the Ordinates SK and SM, to the common AbfdJJa w 
 
 ~ AS) are expreflcd by 
 
 ref pec- 
 
 tively : Then it is plain, from the foregoing Equations, 
 that the Meafures of the Areas of the laid Curves, mul- 
 tiplied by </, will truly exhibit the Values of y and * ; 
 
 an-
 
 in rcjijling Mediums. 44.1 
 
 anfwering to any given Val ue of w (or AS ) the Tangent 
 of the Angle of Direction ; or, or fpeak more geo- 
 metrically, a Square upon AC (fuppofing AC Radius 
 = Unity) will be to either of the faid Areas ASKP, 
 or ASM as the given Diftance d, to the Value of y or 
 
 x required But now as to a Way for computing 
 
 thefe Areas (without which what has been faid about 
 them would be to very little Purpofe) the Method of 
 Eqtti-diftant Ordinates may here be applied to very good 
 Advantage (when the foregoing Serieies do not converge) 
 By means whereof the required Quantities may, with a 
 little Trouble, be brought out to a fufficient Degree of 
 Kxaftnefc, let the Refiftance be as great as it will. 
 
 According to the fame Way of proceeding, the Va- 
 lues of x and _y, in the Alcent of the Ball, will alfo be 
 found, if the Ordinates sk and :m, generating the re- 
 quired Areas, be taken, every where, equal to 
 
 . n i c l iv \ 
 
 infteau of - and -. ). 
 
 1 r-f 2^ r + 2.:/ 
 
 From what has been thus far delivered, it will not 
 be very difficult to calculate (according to the foregoing 
 Hypothcfis) all the principal Requifites concerning the 
 Motion and Track of a Ball in the Air, projected with 
 a given Velocity, at a given Elevation ; as will be more 
 clearly feen by the Example fubjoined. 
 
 Suppofe a Cannon Ball of 4 Inches Diameter (where- 
 of the Weight is nearly 9 Pounds) to be difcharged at 
 an Elevation of 45 Degrees, with a Velocity fufficient to 
 carry it to the Diftance of one Mile, on the Plane of the 
 
 Horizon, were it not for the Refiftance of the Air. 
 
 Then that Velocity, being the fame as might be freely 
 acquired in a perpendicular Defcent of half a Mile *, * Art. 366. 
 will be found to anfwer to the Rate of 412 Feet, per 
 Second^ according to Art. 202. and 366. From whence 
 it is alfo plain, that the Diftance d (fo often mentioned 
 above) will here be expounded by 5333 Feet ; and that 
 the Celerity (a) with which the Refiftance would be 
 equal to the Gravity (or Weight of the Ball) anfwers 
 to the Rate of about 414 Feet per Second. 
 
 More-
 
 44 2 Of the Motion of Bodies 
 
 Moreover, fmce the Tangent of the Angle of Ele- 
 vation, or the firft Value of w, is given equal to Unity 
 
 (or Radius) we have Q(tw v'w* -f i + ^ hyp. Log. 
 
 w + Viv 1 + i) = 1.1478 : From which, and v ( = 
 
 . _ N 2wQ \ 
 
 412 V '. ;, we gets: (= i d X hyp. Log. i + 7 
 2025 Feet the Arch defcribed in the whole Af- 
 cent. Alfo (c - - V A - 199 i Feet, for 
 
 \ 
 
 7 2VV& ] 
 
 V i + -J 
 
 aa / 
 
 the Rate of the Velocity, per Second., at the higheft 
 
 GO \ 
 
 Point: Whence r ( = j = 4,314; by Means 
 
 whereof the greateft Altitude of the Ball, and the ho- 
 rizontal Diftance correfponding thereto will iikewife be 
 found : For let AF, in the preceding Figure, be taken 
 ~ i (the given Value of w) and let the fame be di- 
 vided into three Parts by equi-diftant Ordinar.es (which 
 Number will anfwer fufficiently exact) then the fuccefiive 
 Values of w> for the Ordinates AP, h, ks and TF, 
 being o, i, 7 and J, tbofe of ^ will be o, 0.3394,0.713, 
 and 1.1478, and the Ordinates themfelves (or the cor- 
 refponding Values of ^ j to 0.2318,0.2751, 
 
 0.3463 and 0.4953, refpeively. From whence, by 
 adding the two Extremes to three times the Sum of 
 the two middle Term?, and dividing the whole by 8, 
 we get 0.3239 for the Value of a mean Ordinate *: 
 Which, as AF is here equal to Unity, is aifo the Mea- 
 fure of the required Area AFTP: Which, therefore, 
 being multiplied by 5333 (d) gives 1727 Feet, for the 
 horixontal Diftance made good in the whole Afcent. In 
 
 p. 117. of my Mathematical Differ tat Ions* 
 
 7 the
 
 in refijling Mediums. ..+ 
 
 the fame Way the Area Am is found 0.1828. 
 Whence the greateft Height of the Ball appears to be 
 (= 0.1828 x 5333) = 975 Feet - 
 
 By taking AC = 1, and repeating the Operation (only 
 changing r 2^, to r + 2^) the Area ACT? will 
 come out = 0.1883, and ATC = 0.0875; which 
 multiplied by 5333 (as above) give 1004 F. and 467 
 F. for the Amplitude, and the Diftance defcended, from 
 the higheft Point, when the Direction of the Ball makes 
 an Angle with the Horizon equal to that in which it 
 was projected. 
 
 But, to have the Direction 'when the Ball ftrikes the 
 Ground, and the whole Amplitude of the Projection, 
 we muft find the Value of the Tangent AB, when the 
 Area ABL is equal to (0.1828) the Area AFm (fo that 
 the Defcent, from the higheft Point, may become equal 
 to the whole Afcent.) In order thereto, let 0.0875 
 (ATC; be deduded from 0.1828 (AFw) and the Re- 
 mainder 0.0953 will be = CTBL ; this, divided by 
 TC (0.1513) quotes 0.63; which would be the Value 
 of CB, if all the Ordinates CT, SM, bV. were equal: 
 But, as it is obvious from the Nature of the Problem, 
 and from the Law of the Ordinates already computed, 
 that BL will be Cometh ing greater than CT, and con- 
 
 fequentlyCB leis than 0.63 1 therefore fuppofe the 
 
 Value of CB may be about 0.56 ; and, accordingly, 
 proceed to compute the Area of CBLT anfwering to this 
 Number; by means of CT (0.1513) and BL (0.1852) 
 and one intermediate Ordinate SM (0.1715) and find 
 
 . . CT + BL + 4 SM 
 it (rrom the Approximation - ? xCB) 
 
 to come out 0.0955 : Which is fo near the required 
 Value 0.0953, that it will be altogether needlefs to re- 
 peat the Operation. It is evident from hence, that the 
 Tangent (AB) of the Angle of Direction, when the 
 Ball ftrikes the Ground, is 1.56 ; anfwering to 57 : 20'; 
 From whence, CBKT being found rr 0.0752, the 
 whole Area ABKP will be had =r 0.2635, and confe- 
 quendy 0.2635x5333=1405 F. = the Amplitude in 
 the whole Defcent. 
 
 Fur.
 
 444 Of f k e Motion of Bodies 
 
 Furthermore, from the faid Value of w and that of 
 (= 199 y) given above, we get z (=: -5 d X hyp. 
 
 Lac. i + 
 
 aa 
 
 1788 Feet, for the Arch defcribed 
 
 in the Defcent ; and alfo v ~ 142 |F. which multi- 
 plied by 1.8527, the Secant of 57 : 20', gives 264 F. 
 for the Celerity of the Ball, per Second, at the End of 
 its Flight. 
 
 Now, by collecting the principal of the foregoing 
 Conclufions, it appears, 
 
 i. That the Velocity at the higheft Point A of the 
 Trajectory will be at the Rate of 199 $ Feet, per Se- 
 cond : Which is to the Velocity at the higheft Point a 
 of the Parabola (E<?<:) that would be defcribed, were it 
 not for the Refiftance, as 2 to 3, nearly. 
 
 2. EA = 2025 and Ea = 3030-* 
 
 3. EF 1727 and E/~'= 2640 I 
 
 4. AF = 975 and af = 1320 > Feet 
 
 5, AC = 1788 and ac r= 3030 I 
 
 6. FC ~ 1405 and fc == 2640 J 
 
 7. Angle C = 57: 20' and *(=)= 45. 
 
 8. Velocity at C to that at E, as 264 
 to 412, or as 2 to 3, nearly. 
 
 Thefe Proportions, between the Diftances, in 
 Air and in vacus, hold at an Elevation of 45, when 
 the Refiftance, at going off, is nearly equal to the Gra- 
 vity, or Weight, of the Ball. If the Velocity be greater 
 than that above fpecified, or the Body, projected, be, 
 
 either,
 
 in refilling Mediums. 
 
 either, lefs, or lefs denfe, the Curve will differ, Jllll, 
 more from a Parabola. 
 
 Hence it evidently appears, that the Effect of the 
 Air's Refiftance upon very fwift Modons, is too con- 
 fiderable to be intirely di (regarded in the Art of Gun- 
 nery. 'Tis true the Method s;iven above is, by 
 
 much, too intricate for common Practice ; but when 
 the Law of the Refinance to very fwift Motions is once 
 Sufficiently eftablifhed (which, according to fome lace 
 Experiments, feems to be in a Ratio greater than thac 
 of the Square of the Celerity) it will be no very difficult 
 Matter to find out proper Approximations to correct 
 the Proportions in common Ule. 
 
 445 
 
 SECTION IX. 
 
 Tie Ufe of Fluxions in determining the At- 
 traftion of Bodies under different Forms. 
 
 P R O B I. 
 
 374. OUppofing AC perpendicular to AB, and that a. 
 ^ Corpufcle at C is attracted toivards every Point 
 or Particle of the Line AB, by Forces in the reciprocal 
 duplicate Ratio of th^ Dijlanccs ; to determine the Ratio 
 of the whole Force wbertby tht Corpufcle u urged in the 
 Direflion CA. 
 
 Put AC=fl, and 
 let AD (confidered 
 as variable by the 
 Motion of D to- 
 wards B) be de- 
 noted by x : Then, 
 the Force of a Par- 
 ticie at D being as 
 
 ~ (byHypothe- 
 fis) its Efficacy in
 
 44^ ?%? W e f F 
 
 the propofed Direction AC will (by the Refolution of 
 
 AC 
 
 AC 
 
 T? \ t ^^ J A <- 
 
 torces) be as prrj x TTT =r ^3 
 
 CD 
 
 r~, 
 
 ; : There- 
 
 fore 
 
 Ox 
 
 is the Fluxion of the whole Force ; 
 
 whofe Fluent, which (y Art. 85.) is = 
 
 AD 
 
 ~ CA CD* 
 
 itfclf. 
 
 AD=AB, be as the Force 
 
 ^. E. I. 
 P R O B. II. 
 
 ^fcrB = 
 
 375- Suppofing BCDE to reprefent a circular Plane, 
 and that a Corpufde H, in the Axis thereof AH, is at- 
 tracted by every Point or Particle of the Plane by Forces 
 in the reciprocal duplicate Ratio of the Dijlances ; to find 
 the whole Force by which the Corpufde is urged towards 
 the Plane. 
 
 Let AH = a, and 
 Hbx; then A* 
 <z* ; which 
 multiply 'd by (p= 
 3,14159 &c.) the 
 Area of the Circle 
 whofe Radius is U- 
 
 nity, gives^ x x z a" 
 for the Area of the 
 Circle hcdbe: whofe 
 Fluxion is = 2pxx. 
 But the Force of a 
 fingle Particle at b, 
 
 AH a 
 in the Direction HA, is as -jjTj-f or -^ (fee the laft 
 
 Problem) therefore the Fluxion of the whole Force is 
 
 truly
 
 in determining the Attraction of Bodies. 447 
 
 a ipx 
 
 truly defined by ^pxx x -5 or its Equal -~ and the 
 
 Force itfelf by the Fluent of -^ ; which (properly 
 
 zpa zpa a 
 corrected) is -f = 2/> X I = ip x 
 
 AH 
 
 I TU , when x = HB. Q. . /. 
 
 376. In the preceding Problems, we have fuppofed 
 the Attraction of each Particle, to be as the Square of 
 the Diftance inverfely ; that being the Law which is 
 found to obtain in Nature: But if the Force, according 
 to any other Law of Attraction, be required, the Pro- 
 cefs will be very little different. 
 
 Thus, let the Attraction be as any Power (n) of the 
 Diftance : Then (in the laft Prcb .) the Force of a 
 
 Particle at b (upon H) being as *", its Force in the 
 
 g 
 
 Direction HA will be as X x" or ax"~~ l j which 
 
 x 
 
 multiply 'd by 2pxx (as before) gives 2pax x : whereof 
 
 n-f-I . -f-l 
 
 ax T 2 
 the 
 
 AH x BH" + ' AH""*"*) will be as the Force required. 
 
 P R O B. III. 
 
 377. To determine the Atirafi'ion of a Cane DHF at 
 its Vertex ; the Attraflion of each Particle being a: the 
 Square of the Diftance inverfely. 
 
 Put the Axis EHrr0, the Length of the Slant-Side 
 HD (or HF) r= b, and AH (confidered as variable) 
 = x: Then (by fim. Triangle*) a (HE) : b (HF)
 
 Vfr'J 
 ::* (HA) : HB = -. But, by the laft Problem, 
 
 H 
 
 the Attradion of all the Particles in the Circle 
 BC will be meafured by ip X I ^75 = *P x 
 
 J - (becaufe HB = ~) : Which therefore being 
 multiply 'd by*, and the Fluent taken, we thence have 
 
 * __ ^ for the Attraction of ACHB : And this, when 
 a 
 
 L- LJi 
 
 x-c, will be, 2 /> X EH-, the Force of the 
 
 whole Cone DEHF : Which, if HK be made = HE, 
 and KG perpendicular to HE, will likewife be truly de- 
 
 FH*\ 
 fined by 2/xEG (becaufe HG = ^j- ^ E. I. 
 
 COROLLARY. 
 378. Seeing the Attradion of ACHB is, every 
 
 where, as x j , or ~^ X x, it follows that the 
 Forces of fimilar Cones, at their Vertexes, are diredly 
 as their Altitudes. P R O B.
 
 in determining the Attraction of Bodies. 449 
 
 P R O B. IV. 
 
 379. To find the Force of a Cylinder CBRF, at any 
 Point A in the produced Axis ; the Law of Attraction being 
 Jiill as in the preceding Problems. 
 
 Put BG ( = CG = 
 RH) = 6-, and let AS 
 (taken as variable) =: x: 
 
 Therefore AT = Vf + x\ 
 
 and 
 
 AS 
 AT 
 
 i 
 
 Which 
 
 (by Prob. 2.) exprefles the 
 Force of all the Particles 
 in the circular Surface 1ST. 
 
 B 
 
 XX 
 
 Therefore 2J x*.- y -- a is the Fluxion of the 
 
 required Force : Whofe Fluent (ip x * vV -f **) 
 when x = AG, will be = 2p x_AU Atf ; but when 
 A- = AH, it will be = 2p x AH AF : Hence, by 
 taking the former of thefe Values from the latter, we 
 have 2/> X AB + BF AF for the Meafure of the true 
 Force by which a Corpufcle at A is urged towards the 
 Cylin'de'r. 
 
 G 
 
 PROB,
 
 The Uje ^FLUXIONS 
 
 P R O B. v. 
 
 380. The Law of the Force being ftill fuppofed the fami ; 
 to determine the Attraction cf a Sphere OABGS, at any 
 givtn Point H above its Surface. 
 
 Let BS be perpendicular to HG, and let HB be 
 drawn ; alfo put the Radius AO=*, OH=, AH (b a) 
 = f, Hn = y, and HB = c + x; then An=yc> Gn 
 
 =2.ay + c, and confequently y^c K?a y 4- c ( = 
 An x G = BTZ I = BH* Hn 1 ) = TTTj* / : From 
 
 , . , ^ 2<?f -f 2C* + 1CX + Af* 
 
 which Equation we get y == 
 
 2a 
 
 (becaufe a + cb.) Whence alfo ip x 
 
 375- 
 
 ' " 
 
 X ZtfA- . 
 
 x 2ax * *** 
 
 Which multiplyM by => gives 
 
 D I) 
 
 for the Fluxion of the required Forcej whereof the Fluent
 
 in determining the Attraction of Bodies. 451 
 
 * p - will be the Attraction of the Segment 
 
 ABS : Which therefore, when B coincides with G and 
 x is rz 2*, becomes -^rr, for the Meafure of the 
 Attraction of the whole Sphere. >. . /. 
 
 COROLLARY I. 
 
 381. Hence the Attraction f^rr) at the Surface 
 
 of the Sphere, where b is =r a y will be \ and 
 therefore is directly as the Radius of the Sphere. 
 
 COROLLARY II. 
 
 382. Since 3d j s known to exprefs the Content of 
 Sphere whofe Radius is a *, it is evident that the At- * Art - H'- 
 traction ( /r) of any Sphere is, univerfally, as its 
 
 Quantity of Matter directly, and the Square of the Di- 
 ftance from its Center inverfely ; and is, moreover, the 
 very fame as it would be, was all ths Matter in the 
 Sphere to be united in a Point at the Center. 
 
 COROLLARY III. 
 
 383. If inftead of a Corpufcle, or a fingle Particle 
 of Matter, at H, we fuppofe another Sphere, having its 
 Center at H: Then, fince the two Spheres, at O and 
 H, act upon each other with the very fame Forces, as 
 if each Mafs was contracted into its Center, it follows 
 that the abfolute Force, with which two fpherical Bo- 
 dies tend towards each other, is as the Produdt of their 
 Mafles directly, and the Square of the Diftance of their 
 
 G g 2 Centers
 
 452 Ttie life of FLUXIONS 
 
 Centers inverfely: And therefore, if the MafTes are 
 given, will be barely as the Square of the Diftance. 
 
 P R O B. VI. 
 
 384. To determine the fame as in the loft Problem, the 
 Force of each Particle being as any Power (n) of the 
 Dijiance. 
 
 Let HB ~ *, and let every thing elfe remain as 
 
 above ; then we fiiall have y = - 7 - -r 
 
 x 1 - c~ + 2ac\ _ xx 
 
 r (by putting a = - - 1 and confequentlyj . 
 
 Now the Attra&ion of all the Particles in the circular 
 
 Surface BS, is as -^- x H x HB" 4 " 1 Un"^ (by 
 fl-f I 
 
 Art. 376.) -2L x y/" 1 " 1 /"*"*: Which, multi- 
 - 
 
 ply'd by j, gives X/ jjy / *y for the Flux- 
 
 ion of the required Force : Which, becaufe yy is = 
 
 T x~ xx d x X 3 x 
 + -^ X -7- rr -T- + 7i will hkewife be exprefled 
 
 L^f^j; Whereof the 
 
 ~. i7 + 3 
 
 Fluent is 
 
 + 5 x 2A 
 
 Which, when B coincides with A, or x-yf, willbes 
 ,"f3 "-I-S "-t-3 
 
 L x == ^ - - + ^-i- ___ - : But, when 
 -f 3 x^ H + 5X2* 1 +3 
 
 B co-
 
 in determining the Attraction of Bodies. 453 
 B coincides with G, or * = y = 2a + c (=f) it will 
 
 * 
 become = 
 
 +l 
 
 Therefore the Difference of thefe two, which is =: 
 2/-/"" +3 n -t-5X?3.-? 2.^-f n+ 3 X /* 2/>f" + 
 +l + 3 X /: + 5 X 23* +l 
 
 n + $X 2bd 2 * ~ + 3 X ? 
 
 + 3 ' - - 5 X 23* ~ 
 
 I + n x 03 " A 2pf^l + S + ' 
 
 -t i x + 3 x + 5 x 3* 
 
 (bccaufe /= + 3 , and 2^3 - , + 2 ^) will be the 
 Attradtion of the whole bphere. >. E I 
 
 COROLLARY. 
 
 385. Hence, the Attraction at the Surface of 
 the Sphere ( where c o ) will be -^- x 
 
 TTT, - 
 
 : Which, if 
 ~2a\ 
 
 pofitive, will be : _^ ^==-j but, otherwife, in- 
 
 finite. 
 
 386. Suppoftng ADB^A to be a Cuneus of uniformly 
 denfe Matter, comprizd by two equal and ftmllar elliptic 
 Planes AD BE A and AdbeA, indin'd to each other, at 
 the common Vertex A., of either their firft or fecond Axes^ 
 in an indefinitely fn.all Angle BAi ; To determine the At- 
 traftion thereof at the Point A, fuppofmg the Force of 
 each Particle of Matter to be as the Square of the Di/iancc 
 
 g 3
 
 454 'The Ufe of FLUXIONS 
 
 Let DE be any Ordinate to the Axis AB, and let 
 AD be drawn ; alfo put AB=r<7, BC=*, CDzry, and 
 the Sine of the Angle BA, formed by the two Planes 
 
 (to the Radius i) = d\ and let the Equation of either 
 Curve be y* fx AT* gx 1 : Which will anfwer, 
 to the Conjugate, or Tranfver.e Axis thereof, according 
 as the Value of g is pofitive or negative. 
 
 Now it will be, i (Radius) : d:\ ax (AC) : Cc 
 = d X a x, the Thicknefs of the Cuneus at the Or- 
 dinate (or Section) DE : Moreover, becaufe AD* = 
 
 AC*+CD*, we have AD = V^Zj'+Jx *' gx*: 
 Whence, expreffing (by Art. 374.) the Attrac- 
 
 tion of the Particles in the indefinitely narrow Rectangle 
 
 J. 
 
 DE x Cc, will be defined by ~r- 
 
 Which therefore, multiply'd by x, will give the Fluxion 
 of the Force to be found. But when fx x 1 gx* 
 
 be-
 
 in determining the Attraction of Bodies. 455 
 
 f 
 
 becomes = o, x will be = -~- (=AB) a\ there- 
 fore, by fubftituting for /, our Fluxion will be tranf- 
 
 formed to 
 
 ax i + X x 
 
 i+gXax i+g X ** 
 
 idx V i + g X ax x * 2dxV i -f g X x 
 
 Fluent, when x = <?, will be i -f ^| z x 2ad x 
 
 f.__A x J: , 3L x Jll_ 1 x i!5l y 
 
 3 i>. 7 X 2. 4 9 2.4.6 
 j. i 
 
 Which, becaufe i +^1* x a is =/X k -f ^| * r= / X 
 
 1 - f + ^ - ITT^ ^ wil1 (by muIti P ! y>"ng 
 
 the two Seriefes together &c.) be reduced to idf x 
 T 2 . 4g- 2 . 4 . 6^ 3 2.4.6.8^' 
 
 3 " ' 3-5 3-5-7 3-5.7.9 
 
 ^.^./. 
 
 It may be obferved, that the Fluent given above 
 may be brought out without an Infinite Series (by Art. 
 126. and 2/8.) But the Solution here exhibited is beft 
 adapted to what follows hereafter j to which the Pro- 
 pofition itfelf is premifed as a L,emma. 
 
 G g4 PROB.
 
 45$ The Ufe gf FLUXIONS 
 
 P R O B. VIII. 
 
 387. To determine the Attrafilon at any Point Q_ in the 
 Surface of a given Spheroid OAPES. 
 
 Let QRL be perpendicular to the Axis PS of the Sphe- 
 roid, and QT perpendicular to the Tangent F/of the 
 generating tllipfis at Q_, meeting PS in T : Moreover, 
 let Qalib be a Section of the Spheroid by a Plane per- 
 pendicular to that of the Ellipfis APES, and thro' any 
 Point r, in the Axis thereof, draw CBr and rL parallel 
 to AE and PS : And make the Abfcifla Qr=x, its cor- 
 refponding Semi-Ordinate ra (or rb) =z y> QR ~ ft) 
 
 F W 
 
 a n d RT = b ; alfo let the Sine (NG) of the Angle 
 HQp (to the Radius NQ= i ) = />, its Co-fine QG 
 == q. and the Ratio of OA* to OP% as any given 
 Quantity h to Unity. Now, by reafon of the fimilar 
 Triangles QrL and QNG, we have rL (BR) = px, 
 and QL = qx, and therefore Br (RL) ax a : 
 Alfo, from the Nature of the Ellipfis, AO 1 : PO* 
 
 (bii) :: RT (b) : OR = ~: Likewife AO 1 : PO 1
 
 /;; determining the Attraction of Bodies. 457 
 
 (h: i) :: QR* : OP a OR*; and PO^ : AO* (i : h) 
 ::OP Z OB*:BC 4 = x OP 1 OB 1 = /., x 
 
 QP* UK.+ Kbl l =A x OP a OR 1 2 OR x RB RB* 
 
 aOKxRb KB*; becaufe (by the former 
 
 Proportion) Qtr^xOP* OR X : Whence, by the Pro- 
 petty of the Circle, tacb, we get/ (BC 1 Br^rrQR 1 
 
 Br 2 A x 2 OR x RB-fRB* =* 7^l l Ax 
 
 -jr X />* H- ^ aq bp x 1x f + hp* X A: 1 : 
 Which Equation, by making i -f B k^ becomes > z = 
 
 2* q 1 + 
 
 ^/>V (becaufe q* + p*= I =QN a : Which being only 
 of two Dimenhons, the Curve QaH3, whereto it be- 
 longs, is an Ellipfis. 
 
 The Equation of the Curve QaHb being now ob- 
 tained, let its Axis QH be fuppofed to revolve about Q_, 
 as a Center (the Plane of the Curve being always perpen- 
 dicular to that of the Ellipfis APES) and let the Fluxion 
 of the Arch MN (exprefling the Angle defcribed from 
 the time the faid Axis begins its Motion at the Pofition 
 
 ALD) be denoted by A: Then, it is evident from 
 the preceding Problem, that, 2aq 2bp x lA X 
 
 2 2 . 4.d z 2.4. 6 2 /> + c . .,, i a tr 
 _ - * ^.. . -t -- - &c. will be the Fluxion 
 
 3 3-5 3-5-7 
 
 of the Attraction of the correfponding Part DQH of 
 the Solid, upon a Corpufcle at Q_, confulered as aHng 
 in the Direction HQ_ (which Expreffion is found, by, 
 
 barely, writing laq zfy, ^, and Bp* y in the faid 
 problem, for f, d, and g rcfpedtively.) 
 
 Hence,
 
 45 8 Me Ufe of FLUXIONS 
 
 Hence, by the Resolution of Forces, the Fluxion of 
 the Attraction, in the Directions QR and Qw (per- 
 
 pendicular to QR) will be truly exhibited by 2aq 2bp 
 
 2.4. bJl'p 4 - ._, 
 
 * -- - fcfc and 
 
 Let now another Plane Q/; be fuppofed to revolve 
 about the Point Q_, the contrary Way to the former, from 
 QP towards Q/; and let (ng) the Sine of the Angle 
 RQ/j be denoted by P, and its Co-fine (Q^; by ^: 
 Then the Fluxion of the Attraction of the Part DQ/, 
 in the forefaid Directions QR and Qw (by writing P 
 inftead of p and ^ inftead of q) will appear to be 
 
 2 ' 
 
 and 2a -f 2t>P X lAP X 
 
 3 3-5 
 
 ' ^ &c. Which being added to thofe of 
 
 3-5-7 
 
 the former Part, in the fame Directions, and 2. and 
 
 9 
 
 n . 
 
 J + 2i _ reflectively fubftituted inftead of A * we have 
 3. 
 
 into x0'+/- - x ^ + 
 ^ 3 5 
 
 - tf, into xPPpp ' x P 3 Pp 3 p& . 
 
 3 3-5 
 
 And 
 
 4* into x pj>PP 2-l_ x fp P 3 P &c. 
 
 
 2 />> P 1 P 2 . 4B p+'p P*P 
 
 -4J .ato - x 'J + ^--TV X + -T ^ 
 
 for
 
 in determining the Attraction of Bodies. 459 
 
 for the Fluxion of the Attraction of both Parts together 
 in the forefaid Directions : Whereof the Fluents, when 
 N coincides with F, and n withy, will be the Attrac- 
 tion of the whole Spheroid in thofe Directions. But 
 now, in order to determine thefe Fluents with as little 
 Trouble as poffible, let m be aflumed to denote any 
 
 P tm P 
 whole pofitive Number ; then the Fluent of L > ( 
 
 I . mm, /t --- 
 
 i will be univerfally = - X p 2 " 1 "" 1 
 
 Vip* 
 
 2m 
 
 zm 3 2m I . 2m 3 im 5 
 
 2 . 4 . 6 ... 2m 
 
 nzmn pzmp 
 
 is p * : And that of &" or y''~H"^* ( in the*Art.49. 
 
 Q r 2w X . 2WJ - r ^* 3 
 
 fame Manner) = - X P + x P 
 
 T .2.3....2M--I 
 
 2 . 4 . 6 . . . 2m 
 is P. But when N coincides with F, and n with f, 
 the Sines p and P, of the Arches MF and Mf, be- 
 coming equal, and (the Co-fine) ^= (Co-fine) ?, 
 
 .2T 
 
 it is evident that the Sum of the Fluents of -- and 
 
 P wp w ai in that Cafe, be truly exhibited by 
 "^7* _ _ 
 
 i ?_ L S..*-2i i x MF + y-3-5*---*"-' x 
 ^T 4 . 6 . . . 2w _. 2 ' 4 ' ' ' 2m 
 
 JV1/, or its Equal L^ll|Iil|^=I x FM/j be- 
 
 2.4.0... 2/B 
 
 caufe,
 
 460 'Ibe Ufe of FLUXIONS 
 
 caufe, then all the reft of the Terms (by reafon of the 
 equal Quantities P, p and j^, q) deftroy one another. 
 After the fame Manner the Sum of the Fluents of 
 
 qp'p and 4>P 2W P, in the forefaid Circumftance, will 
 
 Art. * 97 . appear to be = T ' 3 5 7 ?m--i x FM f *. 
 
 2 . 4. . 6 . 8 . . . 2w + 3 
 
 Now, to apply this to the Matter in hand, let the 
 Exponent of ,8, in any Term of either of the above 
 found Fluxions be, univerfally, exprefled by n ; then 
 the numeral Coefficient (annexed to B) will be defined 
 
 by .?J_: U_!_ 2 " + 2 , and the variable Quantities 
 
 i . 3 . 5 . . . 2 + 3 
 multiplied thereby, in the firft Line of the former 
 
 Fluxion, will be qp ta p + QP- n P : Therefore 
 
 neral Term, (from whence, if n be expounded by i, 
 2, 3 &V. fucceffively, that whole Line will be pro- 
 duced.) But, the Fluent of gp**p + gP**P 9 in the 
 Circumftance above fpecified, (putting m=:n and FM/ 
 
 =s /O appears to be = T 3 ' $ 7 
 
 2 . 4 6 . 8 . . . in -t- 2 
 
 Which, therefore, multiplied by 2 . 4 6 . . . 2 + 2 
 
 3 . 5 . . . a -f 3 
 
 2 .4. 6 . 8 . . 
 
 D" i 
 
 X B*k - -, for the true Fluent of the 
 
 2n + i X 2 -I- 3 
 
 faid General Term : Which, if n be expounded by
 
 in determining the Attraction of Bodies. 461 
 
 0, I, 2, 3 &V. fuccefilvely, will become equal to , 
 
 Bk B*k B*k : 
 
 - . - . - &c. refpecuvely ; and therefore the 
 3-5 5-77-9 
 Fluent of the whole Line (drawn into the general 
 
 i B ~B* 
 
 Multiplicator 40 ) is \ak X - - 4. - _ 
 
 i-3 3-5^5-7 
 
 V. But now, for the Fluent of the fecond 
 7-9 
 
 n 
 
 Line : This, it is plain, will be = 4^ into x 
 _ _ 3 
 
 P* P 1 2 . 4# P 4 6+ 
 
 ~ - 7- - -J7J x -.- r ~ Gfc. Which, in the 
 
 forefaid Circumftance, when P p, intirely vanifhes. 
 Therefore it appears, that the Attraction of the whole 
 Spheroid, in the Direction QR, is truly exprefled by 
 
 ~T~ B & & 
 
 tak x -- - + - -- > r <*s Equal 
 1.3 3-5 5-7 7-9 
 
 x 
 
 i-3 3-55-7 
 
 After the fame Manner the Fluent of the firft Line, 
 in the latter of our two Fluxions, will be found to 
 
 A ,2.4.6... 2-f 2 fin * 2n + 2 ft 
 
 vamfli : And - - - == X o x - _, 
 _ i . 3 5--- 2 + 3 9 
 
 P^_ ^ w iH be a General Term to the fecond Line. 
 
 Whereof the Fluent (by expounding zm by 2 + 2) 
 
 2 . 4 6 
 
 
 
 3-5-7 
 
 , 2 . 4 6 . . . . in + ? 
 
 appears, from above, to be r: - X 
 
 I . ? . < . . . . 2W+ I B n k T T 7 , , 
 
 X - ^ 2 - === . - : Which, when 
 2.4.6... 2n -f 2 2 + 3 
 
 is
 
 he life of FLUXIONS 
 
 n is interpreted by o, I, 2, 3 &c. fucceflively, comes 
 
 z PL D* 
 
 out equal to , , - fc? c . refpeclively : There- 
 
 m 9 -f 
 
 fore the Attraction of the Spheroid, in tlie Direction 
 
 D Ol Z?3 
 
 Qw, is exhibited by Lbk x 4. . 
 
 3 579 
 &c. and confequently, That in the oppofite Direction 
 
 Qv, 
 
 1 B & F 
 
 J. =4^x 
 
 3579 3 
 
 
 - &V. x RT = 4* x i - 
 
 x OR (becaufe i~+lf x OR = RT.) 
 
 x ~ 4 
 35 
 
 From which and the Force in the Dire&ion QR 
 (found above) not only the Direction of the abfolute 
 
 H 
 
 Attraction, but that Attraction itfelf will be known : 
 For, let RI be taken to QR, as the Force in the Di- 
 rection Qv to that in the Direction QR i and then, by 
 10 the
 
 in determining the Attraction of Bodies* 463 
 
 the Ccmpofition of Forces, QI will be the Direlion of 
 the Attraction, or the Line in which a Corpufcle at Q_ 
 tends to defcend : And the Attraction itfelf, in that Di- 
 rection, (being to that in QR, as QI to QR) will be 
 
 _ ~D ZJ1 
 
 defined by 4* x 4. &c. x QI ; 
 
 i-3 3 5 5 7 ^ 
 
 which, fince $k is conftant, will alfo be as L_ _ 
 i -3 
 
 F7 + 5^7 
 
 COROLLARY. 
 
 388. Since, by Conftruftion, RI : QR :: i + B x 
 
 JL ^ 5! & c OR ' 
 3 " " 5 7 ' " 9 " i 3 "" 3 5 
 
 i B 
 
 . x QR, it follows that h 
 
 _- + ~ &c. ::RO : RI ; 
 3 57 
 
 . .. n . . r , i B E 1 3 B 
 
 whence (by Divihon) + cflr. : * 
 
 i 3 3-5^5-7 3-5 
 
 i B B* 
 
 confequently, + &V. ! ? x 
 
 i-3 3-5 5-7 J . 
 
 + ^. : : OT : Ol. 
 5-7 7 9 
 
 3'S 
 
 Hence it appears that the Direaion QJ, of the 
 abfolute Attraction, divides the Part of the Axis 
 OT, intercepted by the Center and Normal, in a 
 given Ratio : And that the Attraftion itfelf (being de- 
 
 fined
 
 4 6 4 The W e gf FLUXIONS 
 
 i 7? /" 
 fined by _i_ &c. x QJ) is every 
 
 1-3 3-55-7 
 where as the faid Line of Direction QT. 
 
 SCHOLIUM. 
 
 389. Although the foregoing Conclufions are ex- 
 hibited by infinite Seriefes, yet the Sums of thofe Se- 
 riefes are explicable by means of the Arch of a Circle. 
 
 i B K l 
 Thus, let the Series + &c. (which is 
 
 *y c M 
 
 3 5 / 
 
 one of the two original ones above found) be put rr 5, 
 and let B rr /*; then by Subftitution, and multiplying 
 
 t 3 t* 
 the whole Equation by f 3 , we fhall have -f 
 
 t 7 t 3 S f 
 
 &fc =: t*S> and confequently t -I- - 
 7 357 
 
 sY. rr / - t 3 S : Where, the former Part of the Equa- 
 tion is known to exprefs the Arch of a Circle, whofe 
 
 Art. 142. Tangent is / (B*) and Radius Unity * : Wherefore, 
 putting that Arch 4, we have A t t 3 S, and con- 
 
 t A i B B 1 r 
 fequently 8 j3 =- -+ fff* 
 
 J / 
 
 Moreover, fince it appears that 
 
 7> D^ 733 
 
 Jj & JJ 
 
 T "" T + ~7 
 
 - _ 
 
 B B* B>, 3-55-77-9 
 
 _r_ + & 
 
 5 7 9 
 
 5 * 5 J CJ 
 (where the Sum of ~ + y ^- ls alread y 
 
 t ___ ^ t . J 
 
 found :r -= X B ~
 
 in determining the Attraction of Bodies. 465 
 
 D 751 
 
 of -f &V. by the fame Method will come 
 
 out = 5 - ] it is evident that 
 
 * ' 3-55-7 
 
 7 -_ 
 
 / 3 -f f // x i + t* c r 
 
 - - i and consequently - __ 
 
 JL , _?1 . , _ A 
 3-5 5-7 ' l ~ 3 
 
 '; Which is the Value of the other 
 
 original Series found above : From whence that of 
 w iH a lf be had = 
 
 3-5 5-7 7 -9 
 
 3* + 2/ 3 $A x i + f* 
 "iT^" 
 
 Hence, if 
 / A f i 5 
 
 And 
 
 2/ s 3-5 5-77-9 
 
 it is evident that OT will be to OI, in the conftant Ra- 
 tio of g to b-y and that the Forces in the Directions 
 QI, QR, and Qy, will be as g x QJ, g X QR, and / x 
 
 i + E x OR refpetfively : Where i +5 is = ^ . 
 
 H h P R O B.
 
 Ufe of FLUXIONS 
 
 P R O B. IX. 
 
 3QO- to determine the Attraftion at any Point D within 
 a given Spheroid OAVES. 
 
 i 
 
 Let Oapes be another Spheroid, concentric with, and 
 fimilar to, the given one; whofe Surface D*M fcfr. 
 pafles through the given Point D ; alfolet FD/and HD 
 be taken as two oppofite, indefinitely {lender, Cones 
 (or Pyramids) conceived to be formed by drawing innu- 
 merable Lines HDF, hDf &c. through the common 
 Vertex D) which Cones (or Pyramids) having the 
 fame Angl?, may be confidered as fnnilar; and fo their 
 Art. 378. Forces, at D, will be as the Altitudes DF and DH * ; 
 And, therefore, the Excefs of the former, above the 
 latter, or the Force whereby a Corpufcle at D, tends 
 towards F, through the, contrary, Action of the two op- 
 pofite Cones, will be as DF DH, or as DM ; becaufe 
 (ly the Property of the Ellipfts) MF is, in all Portions, 
 equal to DH 
 
 Hence it appears that the Parts of Matter FMmf 
 and HD, without the Spheroid apes (adling equally, 
 in contrary Directions) can have no Effedt at D : 
 And this, being every where the Cafe, the whole, effi- 
 cacious, Force at D muft therefore be that of the 
 Spheroid Qapes. 
 
 Hence, if the Ratio of (V to Op 1 (or of OA a to OP 1 ) 
 
 be denoted by that of I +B to I, as in the laft Problem, 
 
 10 it
 
 in determining the Attraction of Bodies. 467 
 
 it follows, from thence, that the Attraction at D, in the 
 Directions DM and DN (perpendicular to PS and AE ; 
 
 ~~i B fi*~ 
 
 fee the next Ftp-.) will be expounded by _i_ 
 
 1-3 3-5^5-7 
 
 &V. X DM, and 
 
 3-5-7 9 
 
 (5V. x DN refpe&ively, or by their Equals g x DM 
 
 and / x i -f B x DN : Where the Values off and g 
 are the fame as given in the preceding Article. 
 
 COROL L AR Y. 
 
 391. Hence the Force wherewith a Corpufcle, any 
 where within a given Spheroid, is attracted, either, 
 towards the Axis, or the Plane of its Equator, is d,i- 
 redlly as the Diftance therefrom. 
 
 P R O B. X. 
 
 392. Suppofing every Particle of Matter in a Spheroid 
 to have a Tendency to recede, both, from the Axis PS, and 
 from the Plane cf the greatejl Circle, by Means of Forces 
 that are as the Di/lances from the [aid Axis, and Plane, 
 refpeflively j to find the Direftion DI wherein a Corpufcle, 
 at any Point D, tends to move through the Aclion ef the 
 fold Forces and the Attraction conjtinftly j and likewije the 
 
 whole compound Force in that Direction. 
 
 Let DM and DN 
 
 be perpendicular to 
 PS and AE, and let 
 the given Forces, in 
 the Direction of thofe 
 Lines (independent of 
 the Attraction) be ex- 
 preffed by m x DM and 
 n X DN refpe&ivcly. 
 
 Hh a 
 
 There-
 
 Ufe of FLUXIONS 
 
 Therefore, fmce (by the laft Problem) the Force 
 of Attraction in the faid Directions is defined by g x DM 
 
 and / x I + B x DN, the whole refulting Forces 
 
 will be truly denoted by g m X DM, and f x i + B n 
 X DN : Whence (by the Compofuion of Forces) it 
 
 will be, g m:fx \ + B n :: DN (OM) : MI ; 
 whence the Point I is given : 
 
 Alfo DM : DI :: gmx DM (the Force in the 
 Direaion DM) : f^m x DI, the Force in DI. QE. I. 
 
 P R O B. XI. 
 
 393. Every thing being fuppofed as in the preceding 
 Problems, it is required to determine the Force of all the 
 Particles in the Line (or Column} QDO tending to the 
 Center O of the Spheroid. 
 
 Let IH be perpendicular to QO produced (fee the 
 laft Fig.) then the abfolute Force, in the Direction DI, 
 
 being ~g-m * DI,that in the Direction DH, whereby 
 a Corpufcle at D is urged towards the Center, will be 
 
 g m x DH. Let now OD (confidered as variable) 
 be denoted by x ; then becaufe the Ratio of OM to MI 
 
 is given (being every where as g m to /"x i -f B #, 
 by the Precedent) and the Triangles ODM and IOH are 
 fimilar, it follows that the Ratio of OD to OH will be 
 given, or conftant ; and confequently that of DH to 
 OH, likewife : Let therefore this Ratio of DH to OH 
 be exprefled by that of r to j, and we fhall have DH 
 
 f and confequently (g wxDH) the Force at D, 
 
 rx 
 
 equal to g m X : Which therefore being multi- 
 plied
 
 in determining the Attraction of Bodies. 469 
 
 plied by x y and the Fluent taken, there comes out 
 - -- for h 
 
 2S 2 
 
 Force of the Line or Column OD at the Center. 
 
 COROLLARY. 
 
 394. If the given Forces m and n be fuch that the 
 Ratio of OM to MI, (which is found to be univerfally 
 
 as g m to f x i + 5 n ) may become as i : i + E 
 (or as /O X : aO z ) it is evident (from the Property of 
 the Ellipfis) that the Line of Direction DI will be al- 
 ways perpendicular to the Surface of the Spheroid Oapes. 
 In which Cafe OD x DH is alfo (by the Nature of 
 
 the Ellipfis) = Ofi 1 : And therefore the Force 
 
 x OD x DH) of OD is = ^^ x O* 1 : Which, 
 
 p m 
 when D coincides with Q_, will become x AO*; 
 
 and is, therefore, a conftant Quantity. 
 
 Moreovti fmce in this Cafe, g m:fx i + B n 
 
 :: i : i +B (by Hypothefis) we have m ^ = g 
 
 1 T O 
 
 f : Which Equation, if be taken = o, gives 
 
 * 
 
 , a f + &(.* ; i * Alt. 389. 
 
 g J 3-5 5-77-9 K* ' 
 
 But, if m be taken = o, it will then give n = i -f B 
 
 TS ?F 6P",. .... 
 
 x _/=-- 1 + l?x - + <* Where, 
 
 i 
 
 / r= B*, and A = the Arch whofe Tangent is /, and 
 Radius Unity. 
 
 Hh 3 PROP.
 
 47<> 
 
 PROP. XII. 
 
 395. If an oblate Spheroid OAPES, whereof th 
 Square of the Equatoreal Diameter AE, is to that of the 
 Axis PS, in any given Ratio of r -j- B to I, revolves 
 about its Axis, in fuch a Time, that the centrifugal Force, 
 at the Equator A, is to the Attraction at the Surface of 
 
 a Sphere whofe Radius is O A, in the Ratio of 
 
 7J1 753 * 
 
 -f &'c. to : / fay, in that Cafe, every 
 
 5-7 7-9 3 
 
 Particle of the Spheroid will be in Equilibrio ; fo that, 
 though the Cohejion of the Parts was to ccafe, the Figure itftlf 
 would remain unchanged. 
 
 For, the Attraction of the Spheroid, at A, being de- 
 fined by __ + &c. x AO (Art. 387.) 
 i-3 3-55-7 
 
 AO 
 it is evident (by conceiving B o) that will re- 
 
 O 
 
 prefent the Attralion at the Surface of the Sphere 
 whole Radius is AO : Whence (by Hypothefis) the 
 
 centrifugal Forcq at A (putting m n -f. 
 
 O J -J " / 
 
 6B 3 
 
 &c.) will be truly defined by rn x AOj and con- 
 
 79 
 
 fequently
 
 in determining i/je Attraction of Bodies. 47 1 
 
 fequently That, at any other Point D, by /w.xDM (be- 
 caufe the centrifugal Forces of Bodies describing unequal 
 Circles, in equal Times, are known to be directly as 
 the Radii *.) Hence, and from the Corollary to the laft * Art. 213. 
 Problem, it appears that the Direction of Gravitation 
 DI i c always perpendicular to the Surface apes; and 
 that the P'orce of all the Particles in the Line (or Canal) 
 OD or OQ_, towards the Center O, v/ill continue in- 
 variable, take the Point Q_ in what Part of the Arch 
 APE you viil : From which laft Confederation, it fol- 
 lows that the Force, or preffure of every Canal QO, 
 at the Center O, (confiJering the Body in a fluid State) 
 will be the fame : Whence (by the Principles of Hy- 
 droftatics) a Corpufcle at D has no^Tendency to move, 
 either Way, in the Line OQ_: And therefore, as it 
 hath no Tendency to move in the Direction of the Sur- 
 face D/>? (the Gravitation being perpendicular thereto) 
 it is evident, from Mechanic^ that no Motion at all 
 can enfue, in any Direction. j^. E. D. 
 
 COROLLARY I. 
 
 iB 48* 6B 3 c 
 
 ?qo. Since m is -J- oY. the 
 
 3-5 5-77-9 
 
 Gravitation (g m x DI) at any Point D in the 
 
 ~"i ~B B* 
 
 Spheroid will therefore be as 4. &c. 
 
 357 
 
 X DI = -~ r - X DI (fee Art. 389. 
 
 COROLLARY II. 
 
 397. If the Time of Revolution be given r: ^, and 
 q be put to denote the Time wherein a (folid) Sphere, 
 of the fame Denfity with the Spheroid, muft revolve ; 
 fo that the centrifugal Force, at the Equator thereof, 
 may be equal to the Gravity : Then, as this laft Time 
 is known to continue the fame, whatever the Magnitude 
 of that Sphere is f > and the centrifugal Forces, in equal f Art. ai-. 
 H h 4 Circles, ani !*
 
 Ufe of FLUXIONS 
 
 Circles, are alfo known to be inverfely as the Squares 
 of the periodic Times -- it follows, that p* : q* : : \ AO 
 (the Attraction, or centrifugal Force, refpecling the 
 
 Sphere OA, revolving in the Time a) : - _ 
 __ _ _ 3-5 5-7 
 
 f. D3 
 
 f - &c. X AO, the centrifugal Force of the 
 
 Spheroid at A, revolving in the Time p. From which 
 
 ?* 25 4J3 1 6 3 
 
 Proportion we get ^ = - - + - & c . 
 _ _ 3P 3 5 5.77.9 
 
 3 -M*x 3* 
 
 
 x 
 
 Trigonometrical- Canon, the Value of / ( = 5 1 ) and, 
 confequently, the Ratio of the two principal Diameters, 
 will be found j fo that all the Parts of the Spheroid 
 
 - 
 
 may remain in Equilibria. But, when -*-; is fmall, 
 
 the Solution by an Infinite Series is preferable : For, then 
 
 28 4B* a* 
 
 the Series - - - - Oft. ( -?-J converging fuf- 
 
 j " ^ J / <J* 
 
 ficiently fwift, we {hall, by the Reverfion thereof, find 
 ca 1 25 x 6fl 4 125 x 37? 6 , 
 
 B = w + ^n# + "S^F n which 
 
 Cafe the Ratio of the Equatoreal Diameter to the Axis, 
 if we take only the firft Term of the Series, will be, as 
 
 l:!, orasi + nearly. 
 
 Which, if ^ = 289, or the centrifugal Force at 
 
 the Equator be to the Gravity as i to 28q (that beino; 
 At ^ e P r P rtlon at tne Equator of the Earth*) wiil 
 7 * come out as 231 to 230. 
 
 Co.*
 
 in determining the Attraction of Bodies. 47? 
 COROLLARY III. 
 
 398. Becaufe, 3 ~ the latter Part of 
 
 our foregoing Equation will be equal to Nothing, both 
 when / is Nothing and Infinite, it is evident that the 
 Value thereof cannot, in any intermediate Circumftance 
 of /, exceed a certain affignable Quantity. 
 
 Wherefore, to determine this Limit of the Value of 
 t 
 
 (beyond which the Problem becomes impoflible) 
 o* * 
 
 let the Fluxion of 3 + rx yf 3^ or hs Double 
 
 1 + / X yi *? 
 
 ~ be taken and put rr o, and you will 
 
 Which, becaufe A -- , * will be reduced to at 
 
 _ I + 1 v * Art. 14*, 
 
 + Jt 3 i + f x 9 -J- t~ x A O ; where / is founJ 
 2,5293, from whence the correfponding Values of 
 
 V'l+r*, and -* come out -=. 2,7198, and 0.5805 
 
 &c. refpeclively. Hence it appears that it is impof- 
 fible for the Parts of the Spheroid, in a fluid State, to 
 continue at Refl among themfelves, when the Time of 
 
 q 
 Revolution is fo great that exceeds 0,5805 &c. 
 
 And thar, of all the Spheroids which can beaflumed by 
 a Fluid revolving about an Axis, That whofe Equatoreal 
 Diameter is to its Axis as 2,7198 to Unity, will per- 
 form its Revolutions in the fhorteft Time. 
 
 Thus, for Example, if a (folid) Sphere of the fame 
 common Denfity with the Earth was to revolve about 
 jts Axis in the Time of 841 Minutes, the centrifugal 
 
 Force
 
 474 y& e ^V e f FLUXIONS 
 
 Force at the Equator thereof would, it is known, be 
 
 equal to the Gravity*: Therefore, by taking ~( } 
 Art. 417. = p \ p/ 
 
 := 0,5805 &c. the Time p will come out =: 
 M H M 
 
 746 or 2 26. Which Time is the leaft, poflible, 
 wherein a Fluid, of the fame common Denfity with the 
 Earth, can revolve, fo as to preferve its fpheroidal Fi- 
 gure. And this holds univerfally, let the Magnitude of 
 the Body, or Fluid, be what it will. 
 
 COROLLARY IV. 
 
 399. Hence alfo may be determined the Spheroid, 
 which a fpherical Body (of Ice or any other Matter) 
 revolving in a given Time 5, will converge to, when 
 reduced to a fluid State *. 
 
 For, fmcethe Momenta of Rotation, in equal Spheres 
 and Spheroids, are to one another, in a Ratio com- 
 pounded of the diret Ratio of their Equatoreal Dia- 
 meters, and the inverfe Ratio of the Times of their 
 Rotation, it follows, if d be put = the Diameter of 
 the given Sphere, and E = the Equatoreal Diameter of 
 
 d E 
 
 the required Spheroid, that rr (becaufe the Quan- 
 tity of Motion about the Axis is not affe&ed by the 
 A&ion of the Particles one upon another, while the 
 Figure of the Fluid is changing.) Moreover, fince 
 the Maffes of the Sphere and Spheroid are alfo equal to 
 each other (by Hypothefis) we have d 3 (r:AK-xPS) 
 
 PJ 
 .^ .. : From which two Equations, exterminating 
 
 r+? 
 
 i 
 
 4, there arifes p i + t 1 } 5 X J, for the Time of Re- 
 volution of the required Spheroid : Whence, by fub- 
 
 2 
 
 ftituting this Value of p in the general Equation -** 
 
 /* 
 
 * The Author in a Note, page 135 of his Mifcellaneous 
 Trotls in MO, has correed an Overfight in this Corollary, 
 
 i \ 
 
 by taking here -j x , (infteadof x s j whereby 
 
 the remaining Part of this Article is rendered erroneous.
 
 in determining the Attraction of Bodies. 47 r 
 
 ; from the Solution of which the 
 
 Value of f, and the Spheroid itfelf, will be given. 
 _ But, fmce the Value of the latter Part of the Equa- 
 tion can never exceed a certain affigeable Quantity, the 
 Matter propofed can therefore be only poflible under 
 certain Limitations : In order to determine thefe Limi- 
 tations, let the FJ- Mon of 
 
 >' 
 
 be taken and put o, and it will be found that 
 
 ** + 24**+27 X A ist 3 27* = o : Whence t 
 comes out =r 7.5, and the correfpondlng Value of 
 
 q . 
 
 = 05927, nearly. 
 
 Hence the Parts of the Fluid cannot poffibly come 
 to an Equilibrium among themfelves, when the Time 
 
 s is lefs than , but will continue to recede from 
 
 0,927 
 
 the Axis, in Infiniium. 
 
 M 
 
 If q be taken 84.1 (as in the Example to the 
 
 M H M 
 
 preceding Corollary) s will be equal 91 = 1:31. 
 From which it appears, that, if the Earth (or a 
 fpherical Body of the fame Denfity) was to revolve 
 
 H M 
 
 about its Axis in lefs than i : 31 ; and, in the mean 
 time, be reduced to a State of Fluidity, the Parts thereof 
 towards the Equator would afcend, and continue to re- 
 cede from the Axis, in Ir.finitum. 
 
 COROLLARY V. 
 
 400. Seeing the Values of/ and A are given when 
 the Spheroid is given, it follows that the Gravi- 
 tation
 
 476 The life of FLUXIONS 
 
 ft A 
 tation ( 5 X QI) at any Point in the Surface of 
 
 a Spheroid, whereof the Parts are kept in Equilibria* 
 by their Rotation about the Axis, will be accurately as 
 a Perpendicular to the Surface at that Point, continued 
 to the Axis of the Figure. Therefore the Gravitation 
 at the Equator is to that at either of the Poles, as the 
 Equatoreal Diameter to the Axis inverfly. 
 
 COROLLARY VI. 
 
 401. But, if the Spheroid differ-but little from a 
 Sphere, the Excefs of QI above AO will (by the Pro- 
 perty of theEllipfis) be neatly as OR*. Whence it 
 appears that the Increafe of Gravitation, in going from 
 the Equator to the Pole, is as the Square of the Sine of 
 Latitude, nearly. 
 
 COROLLARY VII. 
 
 402. Moreover, fince the Ratio of the Equatoreal 
 Diameter to the Axis is found, in this Cafe, to be that 
 
 CO* 
 
 Art ,, of I + *?*to i f, the Excefs of that Diameter above the 
 
 Art - J97- 4^> 
 
 _ 
 
 Axis will be to the Axis as ~j to Unity; that is, as -- 
 
 4P 4 
 
 of the centrifugal Force at the Equator to the mean 
 Force of Gravity. Whence, as the centrifugal Forces, 
 in unequal Circles, are univerfally as the Radii diredly, 
 and the Squares of the periodic Times inverfly, it fol- 
 lows that the forefaid Excefs (in Figures nearly fpherical) 
 will be as the Radii directly, and as the Denfity and the 
 Square of the Time of Rotation inverfly : From which 
 Proportions, the Ratios of the greateftand leaft Diameters 
 of the Planets may be inferred from each other ; fup- 
 pofmg the Times of their Rotation, about their Axes, 
 to be known. 
 
 PRO,
 
 in determining the Attraction of Bodies* 
 P R O B. XIII. 
 
 403. To determine the Figure which a Fluid will ac- 
 quire when, bcfides the mutual Gravitation of the Parts 
 thereof, it is attracted by ,another Body, fo remote^ that 
 all Lines drawn from it to the Surface of the Fluid, may 
 be taken as Parallels. 
 
 477 
 
 H M 
 
 Let OAPES be the 
 
 propofed Fluid, and let 
 
 MPS and MQ^ be Right- 
 
 lines, drawn from the re- 
 
 mote Body M\ whereof 
 
 the former MPS paffes 
 
 thro' the Center of Gra- 
 
 vity O : Moreover, let 
 
 the Plane AE be perpen- 
 
 dicular to the Axis MOS ; 
 
 and put NQjr:0 and OM 
 
 (the Diftance of the re- 
 
 mote Body) == d ; alfo 
 
 put the Semi-diameter of 
 
 the Body (at M) = r, 
 
 and let its Denfity be to 
 
 that of the Fluid APES, 
 
 as any Quantity v to 
 
 Unity. Then, fmce, according to the foregoing Cal- 
 
 culations, the Attraction at the Surface of a Sphere (of 
 
 a given Denfity) is exprefled by | of the Radius, it fol- 
 
 lows that the Attraction of the Body M y at its Surface, 
 
 will be explicable by : And therefore, the Force 
 varying according to the Square of the Diftance in- * Art ' 3 8 * 
 verfly *, it will be, d* (MN) 
 
 r :: - 
 
 Attraction of M, at the Diftance MN 
 
 VT* 
 
 its Attraction at 
 
 Diftance
 
 47$ The W e gf FLUXIONS 
 
 Diflance MQ.. Whence the Difference of thefe t\vo, 
 vr 5 _ vr* ^vr* ^ ^s 
 
 ' r JlT;^r ~~ 3 > ( ~ ^ x 2 -+ T + ^ 
 
 (s'c.) will be as the Force whereby a Corpufcle at Q_ 
 endeavours to recede from the Plane AE: Which bc- 
 caufe (by Hypothecs) d is very great in refpecT: of <?, 
 will (by rejecting all the Terms after the firft) be ex- 
 
 , 
 prefled by ^ x a, or its Equal - x 
 
 In the very fame Manner, the Force whereby a 
 Corpufcle at q t below the Plane AE, tends to recede 
 
 2 IT' 
 therefrom, will be defined by x Ny. 
 
 Now, therefore, feeing thefe Forces are, every where, 
 as the Diftances NQ, Ny, from the Plane AE, it appears 
 (by Art. 393. and 394.) that the Figure OAFE6 will be 
 a Spheroid j whereof the Equation, for the Relation of 
 
 2 Vf 3 \ 
 its two principal Diameters (putting n = ~~j~i) n 
 
 
 X __ , fefr. (In which, 
 
 3-5 5-7 7-9 
 the Ratio of PS 1 to AE a is denoted by that of i to 
 i + B.) Hence, by reverting the Series, we have B 
 
 LL"__ liil & Ct an d confequendy PS : AE :: i : 
 
 2 2b 
 
 Which, by reftoring the Value of , becomes PS : AE 
 ::.:.- 5 - * * ' 
 
 Co-
 
 in determining the Attraction of Bodies. 
 
 COROLLARY. 
 
 404. Becaufe -7- exprefles the Sine of the apparent 
 
 Semi-diameter of the Body M, to the Radius i) feen at 
 the Diftance OM, it follows, if the faid Sine be de- 
 noted by c, that PS : AE :: I : i ^ x <r 3 ; and 
 
 2 
 
 CfcJ 
 
 confequently, by Divifton, PS : PS AE :: i : ~ Xc J . 
 
 Hence it appears, that the Forces of the Planets, to 
 produce Tides at the Earth's Surface, are to one an- 
 other as their Denudes, and the Cubes of their apparent 
 Diameters conj unduly. (For the Sines of fmall Arcs are 
 nearly as the Arcs themfelves.) 
 
 EXAMPLE. 
 
 405." If c be taken the Sine of 16' (expreifing the 
 mean Apparent Semi-diameter of the Moon) and v = 
 
 - (the Ratio of her Denfity with refpeft to that of 
 
 the Earth) our lail Proportion will become PS : PS 
 AE :: i : 0,000000315 : Whence, if PS be taken =. 
 42000000 Feet (the Meafure of the Earth's Diameter) 
 
 F 
 
 PS AE will come out = 1 3,23. 
 
 SECT.
 
 Of Problems De Maximis & Minimis 
 
 SECTION X. 
 
 Of the Application of FLUXIONS to the Re- 
 
 folution of fuch Kinds of Problems DE 
 
 MAXIMIS ET MINIMIS, as depend upon 
 
 a particular Curve, whofe Nature is to be 
 
 determined. 
 
 I SHALL begin this Section with premifing the 
 following ufeful 
 
 THEOREM. 
 
 406. If the Relation of two flowing Quantities y and 
 u be required; fo that, when the Fluent of y it become: 
 
 equal to a given Value, that of 2. ~ : may be a 
 
 Maximum or a Minimum ; / fay, their Relation 
 
 V ti X tttt~\ VVJ 
 
 mujl be fuch that J - J may be, every 
 
 /"" 
 where, the fame, or equal to a conjlant Quantity. 
 
 The Demonstration hereof depends upon the fub- 
 fequent 
 
 LEMMA. 
 
 407. If aa. -r ty == ^., wherein A and are inde- 
 terminate, the Value of Ax**j$ + B*W+pp}* 
 will be a Maximum or Minimum, when X 
 
 are equal to each 
 other
 
 depending upon a particular Curve. 481 
 
 other. For, by taking the Fluxions of both Expref- 
 fions we have aa, + 1$ o, and 2n^Aat x * + pp]"~ 
 
 xn I 
 
 /53 + pp] = O : From whence, a. and 
 
 jf A . _ 
 being exterminated, there refults x *+/>/') "~~ I 
 
 
 Hence, if * + + cy + ^ fefr. = ^ (where 
 ', <A Wf. are indeterminate) it follows that A 
 
 + D X 7J~^pp\" &c. will be a Maximum or ^f/- 
 nimum, when all the Quantities x ** + / > / > l"~ I > 
 
 ^ are equal 
 
 to each other. For that Expreffion is a Maximum (or 
 Minimum) when it cannot be increafed (or decreafed) 
 by altering the Values of the indeterminate Quantities 
 involved therein ; but it may be increafed (or decreafed) 
 by altering only two of them (as A and ft) whilft the 
 
 J A _ ^ 
 reft remain unchanged j unlefs x * i PPt""* an ^ 
 
 "' are equal to each other. (This is 
 proved above.) Therefore, wheny/ X a.& + pp * + 
 
 " + C X il" + fcff. is a Maximum or 
 
 Minimum, the Quantities - +/ > p|""~ I an d "T 
 
 " x ~ 
 
 X ^3 + Pf\"~* cannot be unequal : And, by the very 
 fame Argument, no other two of the Quantities above 
 (pearled can be unequal. 
 
 li If,
 
 482 Of Problems De Maximis & Minimis 
 
 If, in the Right-line PR, there be now aflumed 
 NN = tt t NN = 0, &c. and upon thefe, as Safes, 
 
 Re&angles l^K, NK be fuppofed, whofe Altitudes NK, 
 
 NK &V. are denoted by a, , c, d&c. it is evident that a& 
 -f ^ + ty + ^ &?f. f=: ^) will be exprefled by the 
 Sum of all the faid Re&angles, or the whole Polygon 
 
 Hi 
 
 Moreover, if, in the Right-line PL (perpendicular 
 
 to PR) there be taken MM, 
 
 &c. each equal to 
 
 />, and, upon thefe equal Bafes, Re&angles MV, MV" 
 &c. be confHtuted, whofe Altitudes are denoted by 
 
 A X 
 
 B X 
 
 Z 
 
 Ut . it 
 
 Iik ewife 
 
 Tr , c 
 plain that the Value of 
 
 Jgxgg_j 
 
 ^-x 
 
 C x yy pp\* w iH be truly reprefented by the 
 
 i _ 
 
 whole
 
 depending upon a particular Curve. 483 
 
 whole Polygon Mb. Which Polygon (as p is con- 
 ftant) will be a Maximum or Minimum, when A X 
 
 **~^pp\ + Bx. 03 ~4~ W 1 ] ~1~ & c * i s a Maximum or 
 
 Minimum ; that is when all the Quantities x 
 
 a 
 
 &c. are equal to 
 
 each other (as has been proved above.) 
 
 Let now, A^ B, C, D &c. be expounded by any 
 
 Powers, (MP r ,Mp r , MP' , &f f .) of the refpedive 
 Diftances from a given Point P; and let, at the fame 
 time, the correfponding Values of a, b y f, d &c. be 
 
 interpreted by any other propofed Powers MP" 1 , MP , 
 a 
 MP* csV. of the fame given Diftances : Then the 
 
 Area of the Polygon N/ will be exprefled by MP* X * 
 
 + MP" 1 X p + MP" X y&c. ( i}\ and that of the 
 ; 1" ; i* 
 
 rtet-f- Apt f Rfl -f-fip| 
 
 Polygon Mb, by MP r X ~^~ + MP r x -^~ 
 
 And the forefaid equal 
 
 ^ refpeaively. 
 
 Now let the Number of the Rectangles be fuppofed 
 indefinitely great, and their Breadths indefinitely fmall, 
 
 I i 2 fo
 
 484 Of Problems De Maximis & Minimis 
 
 fo that the Area of each of the two Polygons N/ and 
 MA may be taken for that of its circumfcribing Curve : 
 Moreover, let u and y be put to reprefent the Diftances 
 of any two correfponding Ordinates EF and GI from 
 the given Point P; and let j be every where exprefled 
 
 i i 11 
 
 byp (=MM=MM= faV.) Then, u being a general 
 
 Value for any of the Quantities a, 0, y, / &c. (orNN, 
 
 NN &c.) it follows; Firft, that the Fluxion of the 
 Area of the Curve NEFK (the Ordinate being, every 
 
 where, = y") will be truly defined by/*a; Second- 
 ly, that the Fluxion of the Area MG'lV (by fubfti- 
 tuting y, u and y inftead of their Equals) will be 
 
 and, laftly, that the Value of each 
 
 r m et X 
 
 of the equal Quantities, Mr 
 
 r 
 
 . above fpecified, will 
 
 be expreff.d by 
 
 the Theorem Is manifejl. 
 
 408. If R and S be affumed to denote any Funclions 
 of v (that is, any two Quantities exprefled in Terms of 
 y and given Coefficients j then, in order to have the 
 
 ~ 
 
 Fluent of 5 x *"~ a Maximum or Minimum, 
 
 IKl 
 
 y 
 when that of Ru becomes equal to a given Value, it is 
 
 ~ - r-\ c l 
 rcquifite that% X ""^ Ihould be a conftant 
 
 " y ' 
 
 Quan-
 
 depending upon a particular Curve. 
 
 Quantity : This, alfo, is evident from the preceding 
 Demonftration ; and may be of Ule when the above 
 premifed Theorem is not fufficiently general. 
 
 P R O B. I, 
 
 409. To determine the Nature $f the Curve ACE ; fe 
 /, the Length of the Arch AE being given , the Area 
 ABE/W/ be a Maximum. 
 
 Calling (as ufual) the Ab- 
 fcifla AD, A- ; the Ordina:e 
 DC, y; and the Arch AC, 
 
 2, we have x \/&* j** ; 
 and therefore yx f y x 
 
 K.%. jjrj 1 rz the Fluxion of 
 the Area ADC. Now, 
 fmce, by the Queftion, the 
 
 Fluent of y x zz jjl 1 is to be a Maximum^ when 
 That of z becomes equal to a given Quantity (ACE) let 
 thefe two Fluxions be, refpeclively, compared with 
 
 485 
 
 y x 
 
 .in I 
 
 y 
 
 and y m u (as given in the foregoing 
 
 Theorem t) By which means, n = ', r = i, u z 
 
 i J Art. 406, 
 
 and m =. o ; and confequently 
 
 
 
 =: yz X Kxyy\ * : Which (according to the faid 
 Theorem) brine, every where, equal to a conftant 
 Quantity, we (hall, by putting that Quantity = a, 
 
 and ordering the Equation, get a 1 = fl __ ., and * 
 
 J> 
 
 and, confequently, (by 
 taking
 
 486 Of Problems De Maximis Sc Minimis 
 
 taking the Fluent) * ~ a v a * >% or lax xx 
 y 1 i which is the common Equation of a Circle. 
 
 COROLLARY. 
 
 410. It follows from hence, that the greateft Area 
 that can poffibly be contain'd by a Right-line f AE) 
 joining two given Points, and any Curve-line ACE of 
 a given Lengthy terminating in the fame Points, will 
 be when the faid Curve-line is an Arch of a Circle. 
 
 P R O B. II. 
 
 411. The Length of the Arch AE (fee the preceding 
 Figure) being given, to determine the Nature of the 
 Curve, ft that the Solid generated by the Rotation thereof 
 may be a Maximum. 
 
 Art.i45. Since the Fluent of y* X * -y^ - ( y*x *) 
 is required to be a Maximum, when that of z has a 
 given Value ACE, every thing will remain as in the 
 laft Problem j only, r muft here be 2 : And there- 
 
 = a. 
 
 fore (by the 'Theorem) we have y*z x j$ 
 
 ft y 
 
 Whence * =. , ~ ; and coafequently x ( 
 v a" y* 
 
 _ 
 
 Vz* - i'O : -~=\ Which Values, if F be 
 V a y*- 
 
 put = a (in order to have the Powers homologous) 
 
 b*y ^ - 
 
 will become z rz ' and x =: 
 
 Vb* y* 
 
 Whence z and x will be known. 
 
 P R O B. III. 
 
 412.' The Super fetes generated by the drch of a Curve, 
 in its Rotation* about its Jx'n, being given ; to determine 
 the Curve, Jo tkatthe Solid, itjelf, way be a Maximum. 
 
 f Art i . B ecau *e the Fluent of / X s" /i| t is to be a 
 Maximum, when that of yx becomes equal to a given 
 
 Quan-
 
 depending upon a particular Curve. 487 
 
 Quantity ; let the Fluxions here exhibited be therefore 
 
 ' "' "V 
 compared with > x uu IE yy an d y m u (given in the 
 
 .zn I 
 
 y 
 
 Theorem.) By means whereof (r being 2, u = , 
 
 ^ 
 
 n = -, ar.d JTZ = I ) we have yz x a j*j (a 
 c:mftant Quantity * ;) which is the very Equation found * Art - 4 6 - 
 in Prob. i. belonging to a Circle. 
 
 If the Solid be fuppofed given, and the Superficies a 
 Minimum^ we fhall come at the very fame Concluiion : 
 
 i 
 For, y^x and y x xx -f jj'| l (which are refpeftively as 
 
 their Fluxions) being compared with y m u and 2 ^L_ 
 
 ^i 
 
 we have m 2,u ^, r = i, and n f ; and there- 
 
 Jf 
 
 fore : equal to a conftant Quantity : Which 
 
 yVS + y" 
 
 being denoted by (fo that the Terms may be ho- 
 
 mologous) there comes out ax y V x 1 -\- y~ j whence 
 lax x* y~ (as before.) 
 
 PROB. IV. 
 
 413. To determine the Curve HFB, from wbcfe Re- 
 volution a Solid B K jhall be generated ; which , myving 
 forward, in a Medium, in the Direftion of its Axis DA, 
 will be Ifji reft/ltd than any other Solid of the fame given 
 Length DA and Baft BC. 
 
 If AE JT, EFzr^, Ff>=x ts'c. it is evident, from 
 the Principles of Mechanics, that the refifting Force of 
 a Particle of the Medium at F (being as the Square or 
 the Sine of the Angle of Inclination pq) will be truly 
 
 reprefented by ^ ( ' JK\ 
 ** +* \ Fy) V '
 
 488 Of Problems De Maximis & Minimis 
 
 the whole Number of Par- 
 ticles acting upon FHKG 
 is proportional to the 
 Area of the Circle FG, 
 or as y 1 ; the Fluxion 
 hereof (2yy) drawn into 
 
 yy 
 
 will therefore 
 
 give for the 
 
 xx + yy 
 
 Fluxion of the Refinance 
 upon FHKG. 
 
 Now, fince it is required (by the Queflion) to have 
 
 the Fluent of 
 
 XX -f- 
 
 Maximum^ when That of x becomes equal to a given 
 Quantity (AD), let thefe two Fluxions be therefore 
 
 Art. 406. comoared with 
 
 (r being I, u = x, n i 
 
 '*. Whence 
 
 m o) we get 
 
 Art. 
 
 confequently ^j 3 ^ r: a x xx + jjj" : Whereof the Flu- 
 ent will be found, by A? t. 264. That the Curve does not 
 meet its Axis in the extreme Point A, but has an Or- 
 <3inate AH at that Point (as reprefented in the Figure) is 
 
 evident from the foregoing Equation. For xx yyi" 
 (Fy) 4 ^ being, always, greater than j 3 .v (pg\ 3 x Yp) y 
 y muft therefore be greater than <?, in the fame Propor- 
 tion ; and fo, can never be equal to Nothing. 
 
 Now, as it is demonftrable that the Angle AHF muft 
 be I of a Right-Angle, AH (the leaft Value of y) will 
 therefore be =. 40 (fince x and y are, in this Circum- 
 
 ftance,
 
 depending upon a particular Curve. 489 
 
 fiance, equal to each other.) But, what a, itfelf, 
 ought to be, muft be determined from the given Values 
 of AD and BD, and the Refolution of the forefaid 
 Equation. 
 
 P R O B. V. 
 
 414. To determine the Solid of the leqfl Refiftance^ fup- 
 pofing the Area of the generating Plane AHBD, and its 
 greatejl Ordinal e DB to be given ; (fee the preceding 
 Figure,) 
 
 Since (by the laft Article) the Fluxion of the Re- 
 
 fiftance is exprefied by - ** _JP' , and that of the 
 
 y ' 
 
 Area AEFH by j*, it is plain (from the premifed Theo- 
 
 rem *) that fLfi_ is a conftant Quantity. * Art. 406. 
 Whence, rL=& or its Equal BL^ll , being 
 
 every where the fame, the Angle pYq muft alfo be in- 
 variable j and confequently HFB a Right-line. There- 
 fore the Solid of the leaft Refiftance is (in this Cafe) 
 either a whole Cone, or the Fruftrum of a, greater, 
 Cone. But it is eafy to fliew, that, when the Area of 
 the generating Plane AB is given fo fmall, that the 
 Angle B may be taken equal to the Half ef a Right- 
 angle j I fay, it is demonftrable, in this Cafe, that the 
 Fruftrum fo arifing will be lefs refilled than a whole 
 Cone, or any other Fruftrum, whereof the Bafe and the 
 Area of the generating Plane are the fame. 
 
 In like manner the Solid of haft Refiftance, when its 
 Bulk and greateft Diameter are given, may be deter- 
 mined : The Equation of the generating Curve being 
 
 y~ I x X xx + yy}~~' I , 
 
 7 - m > or axy ~ y x xx -J- yj \ ' 
 
 Whereof the Solution is given in Art. 264. 
 
 PR OB.
 
 49 Of Problems DC Maximis & Minimis. 
 
 P R O B. VI. 
 
 415. To determine the Line, along which a Body, fy 
 its own Gravity, wilt, dcfcend^ from one given Point A 
 to another 'By inthejhortfft Time poflible. 
 
 Let AD be parallel, and BC perpendicular, to the 
 Horizon, interfering each other in C ; and let QP be 
 any Ordinate to the Curve parallel to BC : Then (calling 
 AP, x ; PQ_,.y fcfcj the Celerity at Q^will be exprefled 
 
 L 
 
 by y* j alfo the Fluxion of the Time of Defcent thro' 
 An. 204. A Q^ will be truly defined by -7 *, or its Equal y i 
 
 ~ 
 X xx -f jy\ . Here, therefore, the Fluent of y~~^ x 
 
 ** + yfi" is to ^ e a Minimum, when that of x arrives to 
 t Arties- a given Value (AC). Whence, by the Theorem f, 
 
 y 3 x x xx -\-yj\ * muft be a conftant Quantity : 
 Which (to have the Terms homologous) let be denoted 
 
 by a" 5 "* (or =. \ Then a r x rr y* x xx -f j$ T ; 
 V/^ 
 
 _i^ 
 
 . . y*y rr 
 
 whence x ~ ^--^ = J - - t % = ^ ^ , -i\ 
 
 9 =
 
 depending upon a particular Curve. 
 
 491 
 
 - - ; and confequently z=2<z 
 
 a y. 
 
 Therefore, when ya, z is =: 20 ; which two cor- 
 refponding Values let be denoted by DV and A V ; and 
 let QE, parallel to AD, meet DV in E ; then VE 
 <VD ED) being = a y> and VQ_ (AV AQJ 
 
 id 1 v a y, it follows that 
 
 VD (af: VE (a-y) :: VA* (4*') : VQ_* (4* x^J 
 Which is one of the moft remarkable Properties of the 
 Cycloid ; the Curve which, therefore, anfwers the Con- 
 ditions of the Problem. 
 
 If the Celerity be fuppofed as any Function (S) of 
 the Quantity y> the Problem will be refolved in the 
 fame manner: The Equation of the Curve being 
 
 Art. 408. 
 
 P R O B. VII. 
 
 416. To find the Nature of the Curve AQE, along 
 u-hich a heavy Body mujl defend from an horizontal Line 
 RC to a vertical Line CD, fo that the Area CAE may 
 be given, and the Time of the Defcent a Minimum. 
 
 If the Ordinate PQ, 
 ( parallel to CD ) be 
 called .y, and the Velo- 
 city at Q_be denoted by 
 
 /" ; it is evident that 
 the Fluent of y~~" X 
 
 muft be a Minimum 
 
 when that of yx amounts to a given Value. 
 
 Therefore
 
 492 Of Problems De Maximis & Minimis 
 
 Therefore (by the Theorem already mention'd fo 
 
 i 
 often) we have y~' ~*x X xx + _v>l * a~~ > ~~ 1 j and 
 
 confequently = ' ..; which, by wri- 
 
 y. y 
 
 ting i inftead of , becomes x ~ \ Whence 
 
 V a 3 y 3 
 
 x will be known. But, if the Celerity was to be every 
 where uniform, then ( being o) we fhould have 
 
 VV r _r .- 
 
 * = ~r -^; and therefore # = a Vcfy*: 
 * a y 
 
 Which anfwers to a Circle. 
 
 LEMMA. 
 
 417. If, upon a Tangent EP, from any Point C in 
 the Circumference of a Circle FEC, a Perpendicular 
 C? be let fall, the Chord (CE) joining that Point and 
 the Point of Contafl, -will be a Mean- Proportional be- 
 tween the faid Perpendicular CP and the Diameter CF 
 tf the Circle* 
 
 For, the Angles P and 
 CEF being both Right ; 
 and alfo CEP ir F, the 
 Triangles CPE and CEF 
 are fimilar : And there- 
 fore CP : CE :: CE : CF. 
 
 P R O B. VIII. 
 
 418. In themixt-lind Triangle ACB, the Lengths of 
 all the Sides (whereof C&andCB are Right-lines) are 
 fuppofed given ; 'tis required to find the Nature of the 
 Curve-fide AEB, fa that the Ana may be a Maximum.
 
 depending upon a particular Curvf. 
 
 Put the Arch AE rr z, 
 and the Ordinate CE = y 
 then, the Fluxion of the Area 
 
 ACE bein /^ j\ * 
 
 the Fluent of y X z* jj] z > 
 generated in the Time where- 
 in y y from CA, increafes to 
 CB, muft be a Maximum : 
 Therefore, by the TJ?eorem f, 
 
 F 
 
 493 
 
 * Art. 113. 
 
 f Art. 406. 
 
 we have yz X zz yy 
 
 * a 
 
 = t> or 77 
 
 r* c 
 
 yy 
 
 = . But, if CP be fup- 
 
 pofed perpendicular to the Tangent EP, then will 
 
 yy 
 
 (Art. 35.; - 
 
 CE y 
 
 CP ~ d? 1 con 
 
 quently j = j . or , C P : CE (y) :: CE (y) : a : 
 
 Which Proportion, by the Lemma^ anfwers to a Circle ; 
 whereof the Quantity a is the Diameter. 
 
 Now, that AEB muft be an Arch of a Circle is alfo 
 evident from Prob. j. but, that the fame Arch, con- 
 tinu'd out, will pafs thro' the Angle C, does not appear 
 from thence. This is known from above ; and is re- 
 quifite in finding the particular Circle anfwering to any 
 propofed Data. 
 
 PROB. IX. 
 
 4.19. To find the Path AEB which a Body mujl de- 
 fer the in moving uniformly from one given Point A to 
 another B ; fo that, being every where acled on by a Force, 
 cr Virtue, which varies according to the Inverfe- Duplicate- 
 Ratio of the Di/lances from a given Center C, the whole 
 Aclion upon the Body frail be a Minimum. 
 
 Putting
 
 494 
 
 Art. 134. 
 
 Of Problems De Maximis & Minimis 
 
 Putting AE = z, 
 CE ~ y> de (indefinite- 
 ly fmaU) = j*\ Ee = z, 
 
 j l ) 
 
 and 
 
 = j 
 Ed ( y 
 
 for the Meafure of the 
 Force which a&s upon 
 the Body in defcribing 
 the Particle Ee (zf: 
 Moreover, if from the 
 Center C, with any gi- 
 ven Radius (r) an Arch KT/S of a Circle be defcribed, 
 interfering CE in T, we (hall have Tt (the Meafure 
 
 
 
 of the Angle ECe) = . Therefore, fince the 
 
 _ 
 
 Fluent of y~* X uu -f yy ' is required to be a Ml- 
 ximum, and the cotemporary Fluent of >*~ x u (between 
 CAandCB) a given Quantity; it follows, from the 
 Theorem premifed at the Beginning of the Section, 
 
 
 
 muft be equal to a con- 
 and confequently 
 
 that/"" ri * X uu + yy 
 flant Quantity ( J 
 
 C =**=?} -L. 
 
 \. * ) - a 
 
 ti0n found in the preceding Problem. Therefore, if thro' 
 the three given Points A, B, and C, the Circumference 
 of a Circle be defcribed, the Arch thereof terminated 
 by A and B will be the Path of the Body. Jg. . /. 
 
 COROLLARY. 
 
 420. If FR be a Tangent to the Circle, at the Ex- 
 tremity of the Diameter CF, and CA and CE be pro- 
 duced
 
 depending upon a particular Curve. 
 
 duced to meet it in R and Q_, it follows that the whole 
 Action upon the Body, in defcribing the Arch AE, 
 will be proportional to the correfponding Part RQ. of 
 the faid Tangent. For, if Ce be, alfo, produced to 
 meet FR in y, and EF be drawn, it is plain that the 
 Triangles CEF and CFQ^, as alfo CE* and C?Q_, are 
 fimilar : Whence it will be, CE (y) : CF (a) :: CF (a) 
 
 : CQ. (or Cf) = ~ 5 and CE (y) : Ee (*) :: C ? (-) 
 
 : Qj -- .' Which (a being conftant) is as f ) 
 
 the Force that acts upon the Body in defcribing E* (2). 
 And, as this every where holds, the whole Action in 
 defcribing AE mult therefore be proportional to RQ. 
 Which Force (it is eafy to prove) will be to that ex- 
 erted on the Body in moving through the Chord AE, as 
 the Chord to the Arch. 
 
 P R O B. X. 
 
 421. To determine the Path in -which a Body may move 
 from one given Point A to another B, in the Jhortejl Time 
 poj/ibte ; fuppoftng the Velocity to be, every where, propor- 
 
 tional to any Power (y? ) of the Dtftance from a given 
 Center C. (See the laft. Figure. ) 
 
 Here every thing will remain as in the preceding 
 Problem ; only yf muft be wrote inftead of jr-*. 
 
 T-t- - /"+ 1 ~" "1 \ 
 
 1 nerefore we have y x u x uu + yy> ~ a 
 
 conftant Quantity : Which Qiiantity (to have the Terms 
 
 b 
 homologous) let be denoted by ; then 4 by Reduction, 
 
 -f yy 
 
 CP_ CP 
 
 ' Ee CE "" 
 
 kf 
 
 And confequently CP = -^. Hence, if p =.0, or the 
 
 Ve-
 
 496 Of Problems De Maximis & Minimis 
 
 Velocity be conftant ; then CP being every where = b, 
 the Body muft, in this Cafe, defcribe a Right-line. 
 
 by 
 But, if p i, then CP being = j the Curve will 
 
 * An. 74. b e a logarithmic Spiral, whofe Center is C * : Except 
 in that particular Cafe, where CA = CB, when it de- 
 generates to a Circle. 
 
 Laftly, if p = 2, the Curve will be a Circle (by the 
 
 preceding Lemma) whofe Diameter is -r, and whofe 
 
 Periphery pafles through the given Point C. 
 
 After the fame manner, the Value of CP (upon 
 
 which the Nature of the Curve depends) may be de- 
 
 termined, when the Velocity is expounded by any given 
 
 Fundion (S) of the Diftance (y) from the Center of 
 
 t Art-^oy. Force : And (by writing S in the room of f \ &c.) 
 
 bS 
 will come out CP = j where b and c reprefent con- 
 
 ftant Quantities. 
 
 When the Velocity is That which the Body may ac- 
 quire, in defcending through BE, by a centripetal Force 
 
 cxprefled by /, then the Value of S (the Meafure of 
 JArt.22i. that Velocity) being interpreted by V d f+I y f * t 
 
 aad 106. 
 
 (where CB= d) we therefore have CP= 
 
 -.v 
 
 for the Equation of the Curve of the fwifteft Defcent, 
 according to this laft Hypothefis of a centripetal Force 
 varying as any Power/* of the Diftance. 
 
 422. Befides the Problems already refolved in this 
 Seclion, there are others of the fame Nature which are 
 confined to more particular Rsftri&ions, and require a 
 different Method of Solution. 
 
 Thus,
 
 depending upon a particular Curve. 497 
 
 Thus, if j^> , R and S be fuppofed to denote any 
 given Powers, or Functions, of the Ordinate (y) of 
 a Curve ANM, and the A p JV 
 
 Nature of the Curve be 
 required, fo that, when, 
 the Fluent of >x be- 
 comes equal to a given 
 Quantity, the Fluent 
 
 of Rz may alfo be- ~j j-jy-j. 
 
 come equal to another D 
 
 given Quantity, and That of S*> a Maximum or Mi- 
 nimum : Then, becaufe there is, in this Cafe, a fecond 
 Equation, or new Condition, beyond what is to be met 
 with in any of the foregoing Problems, the Method of So- 
 lution hitherto explained, will, therefore, be inefficient. 
 But, by a Procefs iimilar to that whereby the faid Method 
 was demonftrated (afiuming,here, three Expreflions, and 
 three indeterminate Quantities, inftead of two*) a ge-* Art< 47 
 neral Anfwer to this Problem (under all its Reftridions) 
 will be obtained : And is exhibited by the Equation, 
 
 * pR + oS 
 
 - ~ - (^~ > wherein p and q denote conuant 
 
 Quantities. 
 
 423. Though it feems unneceflary to put down the 
 Invention of this Equation, after what has been hinted 
 above, yet it may not be improper to obferve, by way 
 
 of Corollary, that, if Q I, R = I, and 5 /, the 
 Equation will then become -r p^r_ qy n ; expreffing 
 
 the Nature of the Curve,when, the whole AbfcifTa (AM) 
 and correfponding Arch( AN ) being both given Quantities, 
 the Fluent of y"z is a Maximum or Minimum^ According 
 as the Value of n is pofitive or negative : In both which 
 Cafes, it is very eafy to perceive, that the Curve mull 
 
 be concave to AM. and that the Value of , or its 
 
 X 
 
 K k Equal
 
 49 3 Of Problems De Maximis & Minimis 
 
 Equal p + qy* y muft, therefore, decreafe as y increafes ; 
 whence we may infer that the Sign of qy n muft be ne- 
 gative in the former Cafe, and pofitive in the latter. 
 
 Ex. Let the Curve ABDE, be the Gatenaria ; 
 formed by a flender Chain, or perfectly flexible Cord, 
 
 B 
 
 fufpended by its two Extremes in the horizontal Line 
 
 AE : Then, fince its Center of Gravity muft be the 
 
 loweft poffible, the Fluent of yz, when AC=AE, muft 
 
 Art. 173. therefore be a Maximum * : Whence (n being here = i) 
 
 our 
 
 Equation f 4- = p + q y" ) becomes -j- = p 
 
 But, in order to reduce ft to a more convenient 
 Form, let the Diftance (DF) of the loweft Point of the 
 Curve from the horizontal-Line AE be put b ; then, 
 when y (BC) becomes rr b, x will be = z ; and 
 therefore the Equation, in that Circumftance, is i =r p 
 
 qb j whence p i -f 
 
 > and confequently -7- = 
 
 i + qb qy I +q X b y : Which, by putting 
 
 I 55 
 
 , y (DH) s and a ~ is reduced to =: i 
 
 f *^^"^^*-. _ - 
 
 -f : From whence a'z* ( rro+7] 1 x x"} ~ a + 7] 1 
 a 
 
 X %? s 1 ; and confequently BD ~ Y2as + ss. 
 
 For another Example (wherein the Exponent n will 
 be negative) Jet the required Curve be That along 
 
 which
 
 depending upon a particular Curve. 499 
 
 which a Body may defcend, by its own Gravity, 
 from one given Point A to another B, in lefs Time 
 than through any other Line of the fame Length. In 
 
 which Cafe, the Fluent of %,y 2 being a Minimum, 
 when A- and z become equal to given Quantities, our E- 
 quation (by writing ^ f r n ) w ^ nere become 
 
 _ 
 = p -j- fX * : From whence exterminating x, or 
 
 , by means of the Equation x 1 + y 1 % the Fluent 
 may alfo be determined. 
 
 SECTION XL 
 
 Tihe Refolution of Problems of various Kinds. 
 
 P R O B. I. 
 
 424. /I N T hyperbolical Logarithm (y) leing given 4 
 -** it is propoffd to find this natural Nlirtiber anfwer- 
 ing thereto. 
 
 If the Number fought be denoted by I + x t we (hall 
 
 X 
 
 (by drt. 126.) have y =. , or y + xy x rr o. 
 
 I T X 
 
 Let Ay + By 1 + Cy* & c . x ; then Ay + 2.Byy 
 4~ 3Ot? & " - *> an< ^ ur Equation will become 
 
 y -f Ayy + By*j + C> 3 j- ^. ? _ 
 y# 2 Byy $Cy*y ^Dy 3 y &. J " 
 Whence, by comparing the homologous Terms, we 
 
 A i B i . 
 
 ret A = i, 5 = , C = - ', D = 
 2 2' 3 2 3 
 
 
 Therefore i + + y 
 
 - -f 
 
 42.3-4 2 2.3 ' 
 
 &V. is ( = i + x) the Num- 
 
 2. 3 
 
 ber fought. 
 
 PR OB.
 
 5 
 
 The Refolution of Problems 
 
 P R O B. II. 
 
 425. The Radius AO and any Arch Aft of a Circle 
 ABD being given j to find the Sine BC, and Co- fine OC 
 of that Arch. 
 
 Let AO (BO) = r, AB = z, AC - *, BC = y y 
 
 c A 
 
 Bb rr: a, B = x, and In -=j> : Becaufe of the fimilar 
 Triangles OBC and Bnb, it will be 
 
 OB (r) : BC (y) : : Bb (x) : Bn (x) 
 And OB (r) : OC (r x) : : Bb () : bn (j) 
 
 From which we have 
 yk =: rx 
 
 And rj = ; xz. 
 
 Let x-Az + Bz- + C7? + Dz* + Ez* &c. 
 And _y ~ ^zz 4- />z z + tz 3 + dz* + ez* &c. 
 Then, by Subftitution and Tranfpofition, our two 
 Equations will become 
 
 azx. 
 
 And 
 
 7. 
 c. S 
 
 5^2 4 i Wr. ? _ 
 ^. J " 
 
 P'rom which, by equating the homologous Terms, we get 
 
 Alfo =i, ^= 
 
 /I 
 
 , 
 2r 
 
 7? 
 
 There
 
 * 
 
 of various Kinds. 501 ' 
 
 Therefore 2rB = i, -irC = , 
 
 ' 
 
 $rE = -- , &c. and confequently 5= , C o, 
 
 * 
 
 
 5 . 6r z "2.3.4.5.6.^ 
 Whence, alfo b (yC) = o, c (= 
 
 2 . 3r 
 
 Hence it is evident that y ( az + bz* -\- > 
 
 x 3 z 5 z 7 
 
 2 3 r 2.3.4.5^ 2.3.4.5.6.7^ 
 
 2;* 
 
 r. And that x (^dz + Bz^ + Cz 3 &c.) rr 
 v 2r 
 
 __ 
 3 " s " 
 
 2.3. 4r 3 " 2 . 3 . 4 . 5 . 6r s 
 
 P R O B. III. 
 
 426. To find the Value of *, when x* is a Minimum. 
 
 The Logarithm of x* is = x X / . *; whofe Fluxion 
 x x /- x -f i being = o, we have /:* = !. But 
 (by P rob. i.) the Number whofe hyp. Log. is > will 
 
 bei + j+T + r-; + Hrr^- Therefore ' b y 
 
 4 i J * " ' 4P 
 
 writing i inftead of y> we have * i I + 
 
 J 7* Subjlance of this Solution (being the mojt r.eat and 
 artful I have feen to that ufeful Problem) I had from a Letter 
 
 florid Needier ; which teas put into my Hands by a 
 
 Friend, who rtcei-iSd it from the late Dr. Halley, to wbnft 
 it e was wrote. 
 
 Kk 3
 
 5 2 <fbe Refolittlon of Problems 
 
 i 
 
 3>4> 5 ^v. =0,367878 
 
 P R O B. IV. 
 
 427. To divide a given Number (a] fo that the con- 
 tinual Produtt of all its Parts may be a iMaximum. 
 
 It is evident (from Art. 23.) that all the Parts "muft 
 be equal : If* therefore, any one of them be denoted 
 
 by *, their Number will be , and we {hall have 
 
 4 
 
 x\* a Maximum: And therefore its Logarithm x 
 
 x 
 
 L . x a Maximum alfo : And its Fluxion ? X L . x 
 
 x 
 
 Art. aa. ~ =0.*: Whence R-L.x=i, and confequently 
 
 Bad 126. 
 
 I I I 
 
 2 2.3 2 3 4- 
 
 &c. Therefore the next inferior, or fuperior, Num- 
 ber to 2,71828 &c. that will exactly meafure the. 
 given Number a, is the required Value of each Part : 
 
 Thus, let a = io: then becaufe ,, ; A 
 
 f . 'Tlk'AX (F=T/- T 
 
 2 . 
 nearly, the Number of Parts, in this Cafe, will be 4, 
 
 and the Value of each == =2.5. 
 
 P R O B. V. 
 
 428. To divide a given Angle A OB into two Parts 
 AUC and BOC, fo that the Produft of any given Powers, 
 AP" X BQ^, of tbeir Sines AP and BQ. may. bt a 
 
 Let
 
 of various Kinds. 
 
 Let AP, produced, cut the Radius OB in D, and 
 the Arch AB in F ; likewife let FE and ALbe perpen- 
 dicular to OB, and join O, F : Putting AOrrr, AP~x 
 and BQjry. Then, becaufe x n y m is to be a Maximum, 
 
 we have nx"~~ J x x /" -|- x" X my m ~ l yQ' t and con- 
 fequently nyx = mxy. 
 
 Moreover, fmce the Fluxion 
 of the Arch AC is = -; 
 
 and that of BC = ^";""T' 
 
 we aifo have 
 
 rx 
 
 t= 0, 
 
 which multiply'd by the 
 
 ny 
 
 former Equation, &c. gives - . " = -7= 
 
 Vr^y Vr 
 
 or X ^ =. mx : Whence, becaufe. OQ_ 
 
 (Vr l /) : QB (y) :: OP (v/r 2 x*) : PD = 
 
 .. . -, we have n x PD (tnx] =: m x AP ; 
 Vr 1 ^* 
 
 and therefore PD : AP :: m : n\ whence (by Com- 
 pofuion and Divifion) AD : DF :: m + n: m n: But 
 (by fan. Triang.) AD : DF :: AL : EF ; confequently 
 r-f n : m n :: AL : FE ; that is, as the Sum of the 
 Indices of the two pn-.pofcd Powers is to their Dif- 
 ference, fo the Sine of the whole given Angle to the 
 Sine of the Difference of its two, required, Parts. 
 This Proportion is given in Words, at length, becaufe 
 it will he found of frequent Ufe in the Solution of "me- 
 chanical Problems. 
 
 Kk4 PR OB.
 
 54 
 
 Refohition of Problems 
 
 P R O B. VI. 
 
 429. To Jkew that the leaft Triangle that can le de- 
 fcribed about, and the greatejt Parallelogram in, a given 
 Curve ABC, concave to its Axis, will be when the Sub' 
 tangent FT is equal to the Safe BF of the Parallelogram^ 
 or half the Bafe BT of the Triangk. 
 
 B 
 
 T A F 
 
 It appears from Art. 25. and is demonftrable by 
 commbn Geometry, that the greatcft Parallelogram that 
 can. be infcrib'ci in the Triangle BTR (fuppofmg the 
 Pofition of TR to remain the fame) will be that whofe 
 Bafe BF is half the Bafe BT of the Triangle : There- 
 fore, as a greater Figure cannot poffibly be infcribed in 
 the Curve BAG than in the Triangle BTR circum- 
 fcribing it, the greateft Parallelogram that can 'be in- 
 fcribed, either in the Triangle or the Curve, muft be 
 That above fpecified. 
 
 But now, to make it alfo appear that the Triangle 
 BTR is a Minimum when FTnBF j let B/r be any 
 other circumfcribing Triangle, and let the two Tan- 
 gents TER and ter interfecl each other in P. Then, 
 ER being ET, it is plain that RP is lefs than PT, 
 and Pr (lefs than PR lefs than PT) lefs than Pt : There- 
 fore, the Sides PR and Pr of the Triangle RPr being 
 lefs than the Sides, PT and Pt of the Triangle TP/, 
 and the oppofite Angles RPr and TPt equal to each 
 other, it follows that the Triangle PRr is lefs than TP* ; 
 and confequently, by adding the Trapezium BTPr to 
 both, it appears that BTR is lefs thanB/n 
 
 9 Co-
 
 of various Kinds. ro c 
 
 COROLLARY. 
 
 430. Hence the greateft infcribed Parallelogram is 
 half the leaft circumfcribing Triangle. 
 
 In the fame Way it may be proved, that the greateft 
 infcribed Cylinder, and the leaft circumfcribing Cone, 
 in, and about, the Solid generated by Revolution of a 
 given Curve, will be when the Sub-tangent is equal to 
 twice the Altitude of the Cylinder, or f of the Altitude 
 of the Cone : And that the two Figures will be to each 
 other in the Ratio of 4 tog. 
 
 P R O B. VII. 
 
 431. Three Points A, B, C being given ', to find the 
 Pcfttion of a fourth Point P, fo that, if Lines be drawn 
 from thence to the three former, the Sum of the Produfts 
 a X AP, b X BP, and c xCP (ivhere a, If and c denote 
 given Numbers) jkall be a Minimum. 
 
 If CP and BP be produced to E and F, it will appear 
 from Art. 35. and 36. that the Sine of BPE muft be to 
 that of APE, as a to b\ and the Sine of CPF (BPE) 
 to that of APF, as a to c. Therefore, the Sines of 
 the three Angles BPE, APE, and APF (which Angles, 
 taken all together, make two Right-ones) being in the 
 given Ratio of a, b and r, it follows, that, if a Tri- 
 angle RST be conftru&ed, whofe Sides RS, ST and 
 RT are in the faid Ratio of a, b and c, the Angles 
 T, R and S oppofite thereto, will be refpe&ively equal 
 
 to
 
 506 72v Refolution of Problems 
 
 to the fore-mention'd Angles BPE, APE, and APF. 
 From whence, alJ the Angles at the Point P being gi- 
 ven, the Pofition of that Point is given by common 
 Geometry. 
 
 But it is obfervable, that, when one of the three 
 given Quantities a, b, c (fuppofe a] is equal to, or 
 greater than, the Sum of the other two, a Triangle 
 cannot then be formed whofe Sides are proportional to 
 the faid Quantities : In that Cafe the Point P will fall 
 in the Point (A) correfponding to the greateft Quan- 
 tity (a}. For, it is plain that b X AB is lefs than b X BP 
 + <?xAP; and that c* AC is lefs than cxCP + r X AP ; 
 whence, by adding the Lefs to the Lefs, and the Greater 
 to the Greater, it alfo appears that x AB + * ACmuft 
 be lefs than l>xBP + c X CP-f-+7 X AP lefs than 
 ^xBP+^xCP + cxAP; becaufe a (by Hypothefis) 
 is equal to, or greater than, -f c. 
 
 P R O B. VIII. 
 
 432. To determine in what Latitude a Right- line per- 
 pendicular to the Surface cf the Earth, and Another 
 drawn, from the fame Point, to the Center, make the 
 greatejl Angle, pojfible, with each other-, the Ratio of the 
 Axis end the Equator eal Diameter being fuppofid given. 
 
 Let AE reprefent the Equa- 
 toreal Diameter, and SP the 
 Axis of the Earth (taken as 
 an oblate Spheroid) alfo let 
 RO and RM reprefent the 
 two Lines fpecilied in the 
 Problem, whereof let the latter 
 (perpendicular to ARS) meet 
 SP in M ; and let RB be per- 
 pendicular to SP. 
 
 It is. evident, from the Property of the Ellipft?, that 
 SP* : AE a :: BO : BM. And (by Trigonometry) BO 
 : BM :: Tang. BRO : Tang. BRM ; whence, by Equa- 
 lity*
 
 of various Kinds. 
 
 lity, SP* : AE 4 :: Tang. BRO : Tang. BRM; there- 
 fore, by Compofition and Divifion, AE*-f SP* : AE X 
 SP Z :: Tang. BRM + Tang. BRO : Tang. BRM 
 Tang. BRO. But, the Sum of the Tangents of any two 
 jingles is to their Difference, as the Sine of the Sum of 
 thofe Angles to the Sine of their Difference * ; whence it 
 follows that AE* + SP 1 : AE a SP* :: Sine. BRM -f 
 BRO : Sine. BKiVi BKO (ORM). 
 
 Now, fince the Ratio of the two firft Terms is 
 conftant, or in every Part of the Ellipfis the fame, it is 
 obvious that the Angle ORM, or its Sine, will be the 
 greateft poflible, when its Antecedent (the Sine of 
 BKM-f ttKUj is the greateft poflible, that is when 
 BRM -f BRO = a Right-Angle and its Sine = Radius. 
 Therefore, in the propofed Circumftance, when ORM 
 is a Maximum^ our laft Proportion wili become AE a + 
 SP Z : AE 1 SP 1 :: Radius : Sine of ORM : And half 
 the Angle, fo found, added 45, will give (BRM) the 
 Complement of the required Latitude; becaufe BRM 
 + BRO (or sBRM ORM) being == 90, it is evi- 
 dent that aBRMrrgc + ORM, and confequently BRM 
 - 45 + j. ORM. 
 
 P R O B. IX. 
 
 433- Of a ^ *^ e Semi-cubical Parabolas^ to determine 
 that) whereof, the Length of the Curve being given y tht 
 rfreajhallbe a Maximum. 
 
 The general Equation is ax* = y 3 : Moreover, the 
 
 37* 
 
 Area is univerfaJly == -, and the Length of the Curve 
 
 (fa Ari ^ , 3?i ) Let the 
 
 be put = f, and, by ordering the Equation, you will 
 
 __ get 
 
 * Vi4, p> 56. of my
 
 5 o8 
 
 The Refolutlon of Problems 
 
 get y 
 
 i 
 a 3 x 
 
 8a] T 
 
 !: Whence, **- ( an d 
 
 confequently 
 
 4 j being 
 &J 
 
 a Maximum^ it is evident tha; 
 
 -, or its Equal a* x 27^ + 80! 
 
 40 5 ' muft likewife be a Maximum : Which, put into 
 
 Fluxions and reduced, gives a c x ? ' 3 ^ . 
 
 3^ 
 Whence AT and j; will alfo be found. 
 
 P R O B. X. 
 
 434. To determine the Ratio of the Periphery of any given 
 Ellipfis to that of its circumfcribmg Circle. 
 
 Call the Semi-tranfverfe Axis CB, a ; the Semi-con- 
 jugate CEjtj any Ordinate Dr, y, and its Diftance 
 
 A C D B 
 
 CD from the Center, x: Then (by the Nature of the 
 Curve) y being = / aa **, we have y = 
 
 . CXX ; and confequently (/ x* -j- f ) = 
 V* ** 
 * */^ 
 
 _5 c x ^ : Which by making d rr
 
 vf various Kinds. 
 
 aa cc ... xV aa dxx 
 
 will be reduced to % = . 
 
 V aa xx 
 
 2 . 40* 2.4. oa 
 (by throwing the Numerator into a Series) whereof the 
 -whole Fluent, when * becomes rz a, will be z (ERB) 
 
 yj X I ~ 
 
 - 
 
 2.4.4 2. 2.4. 4. 6. 6 
 
 - 3 -3; 5-5-7^ ^ ,, ^ 2g6 x where ^ 
 
 2.2.4.4.6.6.8.8 
 
 denotes the Length of the Arch GB, or i of the Pe- 
 
 riphery of the circumfcribing Circle. 
 
 Hence it follows that the Periphery of the Ellipfis is 
 
 d 
 
 to that of its circumfcribing Circle, as i - 
 
 -id' 
 J 
 
 
 2.2.4.4 2.2.4.4.6.6 
 
 B + l x 
 
 & c , or as i 
 
 2.2 4.4 
 
 &c, to Unity : Where A^ B, C 9 D &c. denote the 
 
 preceding Terms, under their proper Signs. 
 
 PROS. XI. 
 
 435. To determine the Difference between the Length 
 of the Arch of a Semi-hyperbola infinitely produced^ and 
 its Afymptote. 
 
 Call the Semi-tranfverfe Axis (AC) a ; the Semi- 
 conjugate (or its Equal AE) ; b the Diftance (CF) of 
 any Ordinate from the Center, #; the Ordinate itfelf, 
 y ; and the Arch correfponding, z : Then, from the 
 
 Nature of the Curve we have y * a - ; whence 
 
 a 
 
 y =
 
 5io 
 
 The Refolution of Problems 
 
 */ */ 
 
 C N 
 
 bxx 
 
 
 V* 1 a 
 
 \ and confequently * (=Vx i + j 1 ) 
 
 bbxx 
 
 V 'xx aa 
 
 f CA a \ a 
 
 \ r= CF 
 
 and u m 
 x 
 
 i 
 
 : Which, making d*= <gp 
 transformed to x 
 
 \ X l a& L_ j ^hereof the upper Surd, ex- 
 
 panded, is = i 
 
 uu 8.6.8/1 
 
 (s'c. Now the 
 
 Fluent of the firft Term hereof, ~-r into 
 
 __ uu 
 
 XX \ 
 
 n: a . == j is univerfally exprefTed by 
 
 v; 
 
 L 
 
 , or its Equal 
 
 BF x CE 
 
 : Which, if BN 
 
 be parallel to the Afymptote EC, will (becaufe AE : 
 
 CE :;
 
 of various Kinds. 511 
 
 CE :: BF : BN) be alfo truly reprefented by BN : And 
 this Line BN, when x or z becomes infinite, will co- 
 incide with the Afymptote. Therefore the Fluent 
 of the remaining Terms is the Difference fought : 
 Which Fluent, when ni, orj=ro (putting A for 
 \ of the Periphery of the Circle whofe Radius is Unity) 
 
 will be =: a A x H 
 
 2 
 
 ^ , 3 3^ 5 
 
 + 
 
 2.2.4' 2.2.4-4.6. 
 
 3-3-5-5^ 7 
 
 3.3.5.5.7.7^' 
 
 w 
 
 2. 2. 4. 4. 6. 6. 82. 2. 4. 4. 6. 6. 8. 8. 
 (by Art. 286.) but m O when u = o (or y is infinite). 
 Therefore the Excefs of the Afymptote above the Curve 
 is truly exhibited by the preceding Series. >. E. I. 
 
 If a be taken r: j, and b = o, then ^ will become 
 r= I : And therefore, the Curve in this Cafe falling 
 
 into its Axis AG, we have A x -4- - _L 
 
 ' 
 
 < 3 3.7.5.5 ^ _ CA 
 
 2.2.4.4.6 " 2.2.4.4.6.6.8 
 or Unity. Whence it appears that the Sum of the Se- 
 
 1 i 3*3 
 
 ries -f -f : 2 r is the Reciprocal 
 
 2 T a . 2. 42. 2. 4. 4. 6 
 
 of | of the Periphery of the Circle whofe Radius is 
 Unity. And, from the Problem preceding the iaft, it 
 will likewife appear, that the Sum of the Series i 
 
 -J__ 3 _ 3.3.5. eft, will be 
 
 2.2 2.2.4.4 2.2.4.4.6.6 
 
 denoted by the fame Quantity ; and confequently that 
 thefe two Seriefes are equal to each other. From the 
 Addition and Subtraction of which and their Mul- 
 tiples, various other Seriefes may be produced, whofe 
 Sums are explicable by means of the Periphery of a 
 Circle. 
 
 PROB.
 
 Refolution of Problems 
 
 P R O B. XII. 
 
 4.36. To determine the Nature of the Curve CDHj 
 which will inter fett any Number of fimilar and concentric 
 Ettipfts A MB, amb &c. at Right- Angles. 
 
 Let the Tangent 
 DT, which is a 
 Normal to the El- 
 lipfis AMB, meet 
 the Axis AB in T j 
 and, fuppofmg AC, 
 CM, aC, Cm &c. 
 to be the principal 
 Semi-diameters of 
 their refpe&ive Ellipfes, let the given Ratio of AC 1 to 
 CM 1 (or of flC 1 to C^&rV.) be that of I ton: Put- 
 ting CE x, ED n jr, Dp (*) *, and dp y. 
 
 It is a known Property of the Ellipfis that AC 1 : 
 CM* :: CE : ET ; therefore ET -nx : Moreover ET 
 (nx) : Dp (x) :: ED (y) : pd (j) by fimilar Triangles) 
 
 whence = , or ; whereof the Fluent 
 nx y x y 
 
 Art. 126. is L : x L : a = L : y L : a * (where a denotes 
 any conftant Quantity at Pleafure.) Hence we alib 
 
 n 
 
 x y y 
 
 have L: n xL: = L : , and consequently 
 
 = , or a 
 a a n 
 
 P R O B. XIII. 
 
 437. To find the Equation of a Curve ERD that wit! 
 rut any Number of Ellipfes^ or Hyperbolas, having the 
 fame Center O and Vertex A, at Right- Angles. 
 
 Let RT be a Tangent to any one of the propofed 
 Conic Sections ARF, at the Interfe&ion R, meeting 
 
 the
 
 of 'various Kinds. 
 
 O T A. B E O 
 
 the Axis AO in T ; and put AO=a, OB=x, BR=y, 
 nr=x, Rn=j: Then (per Conies) BT == 
 
 ax' 
 
 in the Ellipfis, and =r 
 
 tt 
 
 -, in the Hyperbola : 
 
 Whence, by reafon of the fimilar Triangles TBR, 
 
 and Rrn, it will be 
 
 (BT) :y (BR) :: - > 
 
 (Rn) : -f x (rn] : Therefore + yy = 
 
 * 
 
 XT, and confequently -f + d* = a'x L : 
 
 2 a 
 
 i x*. Where d denotes a conftant Quantity, depending 
 on the given Value of AE. 
 
 PROS. XIV. 
 
 438. Let two Points n and m move y at the fame time, 
 from two given Portions B and C, with equal Celerities^ 
 along two Right- lines BA and EC perpendicular to each 
 other: 'Tis proofed to determine the Curve ASC, to 
 which a Right-line joining the faid Points Jhall, always* 
 be a Tangent. 
 
 Let DS and ev be parallel to BA, and Sr perpen- 
 dicular thereto : Putting BCz=rf, CDr=A:, SD = _y, Sr 
 =. X-, and rv ~y. Therefore (by fim. Triangles] y : x 
 LI :: y
 
 Rtfofati<m of Problems 
 
 = Dm, and i : >:: 
 
 bn: Whence Cm (CD Dm) = x - ?, and En (B* 
 -f bn] = v -f **"-? : Which two laft Values, be- 
 
 X 
 
 caufe the Velocities of the Bodies are equal, rouft alfo 
 
 yx_ a xxy 
 be equal to each other, that is, x -- r > + - : - ; 
 
 y x 
 
 Hence, by making X- conftant, and taking the Fluxion 
 
 i*y _. - 
 
 ~ 
 
 of the whole Equation, we get x 
 
 there 
 
 '"s .- .-. j _ _ 
 
 ^xj =yx~, and ,- 
 
 the Fluent on both Sides being taken, we have 2 
 
 _ 2 v x T 2 Va x, and confequently * = 2 V/^- 
 _ ^; Which Equation pertains to the common Pa- 
 rabola.
 
 of various Kind*. 515 
 
 Otberwife more univerfatty, thus. 
 
 > t 439 : Put Cm v and En = w, and let thefe Quan- 
 tities (inftead of being equal) have any given Relation 
 to each other. Then, fince the abfolute Celerity of m 
 isexprefled by <u, its angular Celerity, in a Direction per- 
 pendicular to Sw, by which the Line Sm tends to re- 
 volve about the Point of Contact S as a Center, will 
 
 be tru!y defined by = X * (Art. 35 
 
 In the fame manner the angular Celerity of , about 
 
 the Point S, will be defined by ' , X w. Now, 
 
 Kaa. 
 
 as thefe Celerities muft be to each other as the Di- 
 ftances Sm and Sn from the Center S (or directly as 
 the Radii) we have S; : Sa (:: DS : bn) :: Sin. Bmn X 
 y : Sin. Enm x ow ; whence, becaufe Sin. Bmn : &'#. 
 Enm :: En (w) : Em (a v) we alfo have DS : bn :: 
 
 w X v : a v x ov: Therefore, by Compofltion, DS : 
 (DS -f bn) w :: w>v : wv -f- a v x w, and confe- 
 
 quently DS = = - ' - : Whence bn (w 
 
 wv + a v x w 
 
 __ a v X ww . nr\ r 01 bnxBm\ 
 
 SD) == ; and BD (= S = I 
 
 uj<ij .j. a v X w "* 
 
 , . ... ; From whence the Curve itfelf 
 
 will be given. 
 
 If v and w be taken equal to each other (as above) 
 
 then SD (y) will become = , and BD = 
 
 - - _ a u> 
 
 IV 
 
 2iv -f ; in which laft, if for w its Equal 
 a. 
 
 LI 2
 
 516 The Refolution of Problems 
 
 }/~ay be fubftituted, we fhall have BD = a 
 
 4- y ; and confequently CD (a BD) = 2 Y ay y, 
 the very fame as before. 
 
 P R O B. XV. 
 
 440. Suppofmg a Body T to proceed, uniformly, along 
 a Right-line BC, and another Body S, in purfuit of the 
 fame-, always direttly towards it, with a Celerity which 
 is to that of T, in any given Ratio, of i to H; it is pro- 
 pofed to find the Equation of the Curve ASD defer ibed by 
 the latter. 
 
 Let the Tangent AB, which makes Right-Angle 3 
 with BC, be put = , BR = #, RS y, and AS = z: 
 
 B R T c 
 
 vx 
 
 Then theSubtangent RT being = ^, we have BT 
 = x + L. : Moreover, fince the Diftances BT and 
 
 AS gone over in the fame Time, are as the Celerities 
 and i, we alfo have BT ( = n X AS) = z = x + 
 
 VAT 
 
 . : Whence, in Fluxions (making j conftant) j- 
 
 V \ * S r \/ V "T~ X 
 
 The
 
 of various Kinds. 517 
 
 The Fluent of which (by Art. 126.) is X Log. y 
 
 x 4. VV 4- x 1 
 = Log. I - : But when ya, x is = o, 
 
 y 
 
 and then the Equation becomes n x Log. a =: o ; 
 therefore the Fluent, duly corrected, is n x Log. a n 
 
 x -4- V^y* -4- .* z tf" 
 
 x Log. v = Log. -1- ^ -L , or Log. = 
 
 y y* 
 
 ji 
 Whence it is evident that 
 
 / 
 and f5. ^ = Vv^Tl 1 ; 
 
 from which, by fquaring both Sides, 2^ is found = 
 
 * ' v " ' a v 1 * 
 
 Q*-'| whofe Fluent is 2* = + 
 / a* l 
 
 B -f-I 
 
 But when ^ = a, x is = o, and then, 
 
 a a ina 
 
 o = - -- 1 -- -- ; therefore the 
 l n n + i i 
 
 Fluent corrected is 2* 
 
 l _ n 
 
 Otherwife (without fecond Fluxions.) 
 441. Put ST = P and RT == ^. Then fince the 
 abfolute Velocity of the Body S is denoted by Unity, 
 that with which the Ordinate SR is carry'd towards the 
 
 6) 6) 
 
 Body Twill be denoted by -p X i or jf (by Art. 35 .) 
 
 which fubtradted from n the Velocity of 7", leaves n 
 
 ^ 
 
 ^ for the relative Celerity with which T recedes from 
 
 L 1 3 R :
 
 5 1 8 The Rejolution of Problems 
 
 > 
 R : After the fame Manner, if from x the Celerity 
 
 of Tin the Direction ST produced, there be taken (i) 
 the Celerity of S in the fame Direction, the Remainder, 
 
 n9 
 
 -0 s i, will be the Celerity with which T recedes 
 
 from S : Therefore, the Fluxions of Quantities being as 
 
 jp n> 
 the Celerities of their Increafe, we have n ~7J : "~5~ 
 
 ; and confequently nQ P X <j>=nP<l>x P. 
 But, fmcethe Quantities P and j^are concerned exactly 
 alike, the Equation thus derived will, in all probability, 
 become more fimple, by fubftituting for their Sum and 
 Difference: Let therefore P + ^ J, and P Q=v, 
 
 or, which is the fame, let P =?= , and j^z= : 
 
 2 2 
 
 ns nv s v 
 Then, by Subfhtution, we ihall have 
 
 i v ns 4- nv s -f v s + v 
 X r= X j which con- 
 
 tracted, &c. becomes i+nxv's=zi Xj>i/, or i-J-x 
 
 * 
 
 = i x i whofe Fluent (corre&ed) is i-f 
 
 X Log. s = i wxLog. v + 2nx Log. a t or Log. J 1+ * 
 -= Log. a v 1 -". Whence s'+ n = a 2 " v l ~\ and 
 
 confequently s l+n x T/ I + B =fl 2 " v a : But sv ( = S 
 
 X ST RT = RS 2 ) =y* therefore j*+ x v 1+ " ~ 
 
 2 , y" +I x v* 1 
 
 =a v\ and v = . w hence f ( - -) 
 a V w 

 
 cf various Kinds. 519 
 
 _ But RS : RT 
 
 JL 9 and ix ~ 
 , the very fame as before. 
 
 COROLLARY. 
 
 442. If the Velocity of S be greater than tbat of T 
 (or n be lefs than Unity) the two Bodies will concur 
 when the latter has moved over a Diftance expreffed by 
 
 - ; becaufe, when y becomes = o, ix is bardy = 
 But if the Velocity of S be lefs thaa tbat of 
 
 j 
 
 ina 
 
 j 
 
 T y it is plain that 5 can never come up with T: Bat its 
 
 n l 
 neareft Approach will be when j ^T^j xa.-For, 
 
 +i 
 
 fmce ST is univerfally = r- - > let the FIux- 
 
 2/~ If? 
 
 ion of this Expreflion be taken and put equal to No- 
 thing ; and / will be found as above exhibited. 
 
 It" the Celerities of S and 7, inftead of being uni- 
 form, vary according to a given Law ; then, denoting 
 the former by A and the latter by B> the Equation of 
 
 x By 
 
 the Curve will be , ~ v : And if the 
 
 Yj-t-x "/ 
 
 By 
 
 Fluent of -jr be explicable by a Logarithm, as L. NI 
 
 then, the Fluent of - l being L. -L_1-L_J!!* , Mtt. 
 
 vy +* j 
 
 LI 4 we
 
 co 
 
 The Refolution of Problems 
 
 we fhall have N 
 
 v 
 
 - 
 
 v v 1 -f- x 1 
 
 - 
 
 j which, ordered, 
 
 gives A- rz - : Whence x will be found. 
 
 2 2^V 
 
 P R O B. XVI. 
 
 443. To determine the Frujlum CDEF cf a Trian- 
 gular-Prifm, cf a given Bafe CF and Altitude BA ; 
 uibicb, moving in a Medium^ in the Direfiion of its 
 Length B A, Jhall be refijled the hajl po/ible. 
 
 Draw CH parallel to BA meet- 
 ing ED, produced, in H : More- 
 over, let HP, PQ^and PR be per- 
 pendicular to CD, CH and DH 
 refpeclively. 
 
 Since the Number of refitting 
 Particles acting upon DC is as 
 DH, and the Force of each as 
 
 iTpT/ the Square of the Sine of 
 
 the Angle of Incidence DPR, 
 the whole Refinance fuftained by DC will therefore be 
 
 DH x DR Z 
 
 cxprefied by -- ryrn; , or DR, which is equal to it (by 
 
 the Similarity of the Triangles DHP and DPR) Whence 
 the Refinance upon ADC is truly exprefied by AR (AD 
 -f DR) and is a Minimum when its Defed (PQ.) be- 
 low the given Quantity AH (or BC) is a Maximum : 
 But PQis a Maximum when CQ^ and HQ_ are equal ; 
 becaufe, the Angle CPH being Right, a Semi-circle de- 
 fcribed upon CH will always pafs through the Point P ; 
 and it is well known that the greateft Ordinate in a 
 Semi-circle is That which divides the Diameter into two 
 equal Parts. 
 
 Hence the Angle DCH, when the Refiftance upon 
 ADC is a Minimum, will be juft the Half of a Right- 
 Angle, provided BC be given greater than BA; other- 
 
 wife,
 
 of various Kinds. 
 
 wife, the whole Prifm CAP will be lefs refifted than 
 any Fruftum CDEF of a greater Prifm. 
 
 P R O B. XVII. 
 
 444. To determine the Angle RBE which a Plane EBF 
 mujt make with the Wind blowing in a given Direction 
 RB, fo that the Plane itfelf may be urged in another given 
 Direction BA with the greats/I Force pojjible. 
 
 It is known, from the 
 Refolution of Forces, that 
 theForce whereby the Plane 
 EF is urged in the given 
 Direction BA, by a Par- 
 ticle of Air, acting in the 
 Direction RB, is directly as 
 the Rectangle of the Sines 
 of the Angles (ABE, RBE) 
 which the two given Directions make with the Plane : 
 Therefore, fmce the Number of Particles acting on EF 
 is as the Sine of RBE, it follows that the whole Force, 
 or Effect, of the Wind, in the Direftion BA, will be 
 as 5. ABE X Squ. S. RBE ; which being a Maximum, 
 we have (by Prob. 5.) 3 : I :: Sine of the whole given 
 
 Angle RB A : Sine of RBE ABE. Whence the Angles 
 RBE and ABE arc both given. Q E. I. 
 
 COROLLARY. 
 
 44 <j. If the Angle RBA be a Right one (which is 
 the Cafe with regard to the Sails of a Windmill) then 
 
 the Sine of RBE ABE being ^ ,333 t$c. we 
 fhall have RBE ABE = 19 : 28' ; and confequenily 
 
 = 54 : 44' 
 
 521 
 
 PROB. XVIII. 
 
 44.6. If two Bodies A and B, joined by a String, be 
 urged in oppojite Directions, towards P and Q_, by any 
 given Forces F and f, uniformly continued ; // is propofedty 
 find the Tenfion of the String, or the Force ivhsreby the 
 Bodies endeavour to recede from each other. 
 
 Since
 
 522 The Refohitfari 'of Problems 
 
 Since F /is the abfolute Force by which the two 
 Bodies are, conftantly, urged towards P, the whole 
 Motion, generated in Both, in any Time jT, will there- 
 fore be exprefled by Ff X T: Whence, becaufe both 
 Bodies (by reafon of the String) acquire the fame Ve- 
 locity, the Motion generated in jf, alone, will be 
 
 - -. x Ff x T 9 or that Part of the irbaU defined by 
 
 Ji -f- o 
 
 -3 . But the Motion of jf 9 had it not been 
 
 retarded by the String (or B) would have been F 
 X T-, therefore the Lofs of Motion, by the A&ioa 
 
 B 
 
 _ 
 upon the String,, is FxT -j ^ x Ff x T y 
 
 = 4~T1T X T: Wbich ' divided b y *e Time T 9 
 (wherein that Lofs or Effeft is produced) gives *} ^- , 
 
 for the Tenfion of the Thread, or the Force fufiicient to 
 caufe the faid Lofs or Motion, 
 
 "The fame otberwife. 
 
 447. Becaufe the Force F, was it to aft alone, would 
 communicate, by means of the String, the fame Ve- 
 locity to B as to A, the Part therefore of the Force F 
 employ'd upon B, by which the String is ftretch'd, will 
 
 B ' BF 
 
 be -3 n X F, or >. p.* And, from the very fame 
 
 Argument, if the Force / was to aft alone, the Teniion 
 of the Thread would be /. P : Therefore, when both 
 
 the
 
 of various l&nds. 
 
 fJi 2? p 
 
 the Forces act together, the Tenfion will be ~\~- ' 
 
 For it is very plain that, their acting both at the fame 
 time, no way influences their refpective Effects on the 
 Thread. ^. . /. 
 
 COROLLARY. 
 
 448. If the Forces F and f be refpectively expounded 
 by the Mattes, or Weights, of the Bodies ^and B ; the 
 
 Tenfion of the Thread will then become 
 
 Whence it appears that the Tenfion of a Thread fliding 
 over a Pin or Pulley, by means of two unequal Weights 
 A and B 3 fufpended at the Ends thereof, is equal to 
 
 -r., r>, - . 
 
 ~TT~D : ln e Double whereof, or -j 77, is the Weight 
 
 which the Pin or Pulley fuftains, while the Bodies are 
 in Motion j becaufe the Thread hangs double, or on 
 both Sides the Pulley. 
 
 If feveral Bodies A^ 5, C, D &V. communicating by 
 means of a String or Wire AF, be urged towards a 
 Point P, in the Direction of the String or Wire, by 
 any given Forces^, ^, r, s &c. refpectively, the Tenfion 
 of the Part AB will be 
 
 /> X ~B~+ C + D bV. A X g + r -f s &c. . 
 
 A + B 4- C -f D (3'c. 
 of the Part BC 
 
 X D + E+F 4+B + C X 
 4 + B -f C -f D & f . 
 
 All
 
 524 
 
 The Refolution of Problems 
 
 / y 
 
 All which eafily follows from above ; and will an- 
 fwer alfo in thofe Cafes where fome of the Forces are 
 fuppofed to a6t in the contrary Direction, if every fuch 
 Force be confidered as a negative Quantity. 
 
 P R O B. XIX. 
 
 449. Let it be required to raife a given Weight N, to 
 a given Height BC, along an inclind Plane AC, by means 
 of another given freight M, cmnetfed to the former by a 
 flexible Rope NrM, moving ever a Pulley at C j to find 
 the Tenfion cf the Rope ; a/Jo the Inclination and Length 
 of the Plane, fo that the Time of the whole Afcent may be 
 the lea/1 pojfible. 
 
 It is well known that 
 the Force by which N 
 tends to defcend along 
 the Plane AC, or acts 
 in oppcfition to J</(fup- 
 pofing BCntf, and AC 
 
 r: x ) will be : 
 x 
 
 Therefore M 
 x ' 
 
 A. 
 
 or 
 
 xMaN. 
 
 x 
 
 is the efficacious Force, by which the 
 
 Bodies are accelerated : But it is likewife demonftrable 
 that the Time of defcribing any Line by means of a Ve- 
 locity uniformly accelerated, is in the fubduplicate Ratio 
 of the Length thereof, directly, and the fubduplicate 
 Arti 101. Ratio of the accelerating Force, inverfely * : Whence 
 it follows that the Time of defcribing AC will be 
 
 reprefented by 
 
 x 
 
 Whofe Fluxion ( or 
 
 that of its Square) being made equal to Nothing, x will 
 
 be found = -1*7 , or M : zN :: a : x. Hence the 
 
 Time 
 
 M
 
 of various Kinds. 
 
 Time of the Afcent will be the leaft poffible, when the 
 Sine of the Plane's Inclination is to the Radius, as the 
 Power (M) is to twice the Weight, (N) to be raifed. 
 The Tenfion of the Rope will be determined from 
 
 aN 
 the laft Problem, (by writing AT for A, for F, M 
 
 fylN a -4- x 
 for B, and M for f) and comes out = , -, x . 
 
 . E. L 
 P R O B. XX. 
 
 450. Let AC reprefent a Piece of Timber, movealle 
 about a Center C, making any Angle ACG with the 
 Plane of the Horizon CG ; to determine the Pofttion of a 
 Prop or Supporter OS, of a given Length, which fball 
 fujlain it -with the greatejl Facility, in any given Pofi- 
 tion ; and alfo what Inclination AC will have to the Ho- 
 rizon when the leajl Force that can fujlain it } is greater 
 than the leajl Force in any other Pofttion* 
 
 Let R be the Center of Gra- 
 vity of the Beam AC, and let 
 R, R/n and CD be perpendi- 
 cular to AC, CG and OS re- 
 fpeaively : Putting SO=a, CR 
 r, Cm x, and the Weight 
 of the Beam = w. 
 
 Then, by the Principles of 
 Mechanics, we (hall have, firft, 
 
 as ROT : Rn, or as, r : x :: w : 
 
 C 
 
 7/1 
 
 ( J 
 
 the Force 
 
 which adding at R, in the Direction R*, is fufficient to 
 
 xw 
 
 fuftain the Beam AC j fecondly, as CO : CR ( r ) :: 
 
 xw 
 
 the Force able to 
 
 (the Quantity laft found) 
 
 fupport it, at O, in a perpendicular Direction; and, 
 
 laftly, 
 
 525
 
 526 T'e Rcfolution of Problems 
 
 hftly, as CD : CO :: g : ^~, the Force, or Weight, 
 
 a&ually fuftained by the given Prop SO. Whirh Force 
 will therefore be the leaft poffible when the Perpendi- 
 cular CU is the greateil poffible, let the An^le of In- 
 clination GCA be what it will : But of all Triangles, 
 having the ferwe i> fe (OS) and vertical Angle (SCO) 
 the Ifoiceles one is known to have the greatcft Perpen- 
 dkular: Therefore the Triangle CSO will be Ifofceles, 
 and the Angles S and O equal to each other, when the 
 Weight fuftain'd by the Prop OS is a Minimum. 
 
 But, now, to give a Solution to the latter Part of 
 the Problem, or to find (fuppofing the Angles S and 
 
 V 
 
 O to be equal) when ^rr\ X iv is a Maximum^ let CD 
 
 produced meet ;;.'R in F ; and then, becaufe of the fimi- 
 lar Triangles CDS and CwF, we fliali have CD : x 
 
 (Cm) :: SD (i aj : mF, or TTR = ~ i and confc- 
 
 v #jF 
 
 tiuently TTK X w - X w : But, fmce CF bifeds 
 , CIJ * a 
 
 the Angle ?>;CR, we alfo have, r + x (CR-f CM) : x 
 (Cm) :: V&=^ ' (W : ? = 
 
 x^/ r . -: Whence the Force ^ x w, acting 
 
 upon the Supporter, is likewife truly exprefled by 
 : Whereof the Fluxion being taken and 
 
 put equal to Nothing &c. we get x = 
 
 2 
 
 Therefore CR : Cm (::i:^ S ^ M;; Radius: Co- 
 
 line of RCG = 5i : 50', the Inclination required. 
 
 P R O B.
 
 of various Kinds. 
 
 527 
 
 P R O B. XXI. 
 
 451. To determine the Pofttion of a Beam CD, move- 
 eible about one End C a! a Center^ and fuftained at the 
 other End I) by a given Weight Q_, appended to a Cord 
 QAD faffing over a Pulley at a given Point A. 
 
 A F K TT B Let G be < hc 
 
 Center of Gra- 
 vity of the Beam; 
 alfo let DF, GK 
 and CH be per- 
 pendicular' to the 
 Plane of the Ho- 
 rizon, and CL 
 and AH parallel 
 to the fame : Put- 
 ting AH =a, CH=, CDrrc, CG-d t DL=x, CL 
 = y y and the Weight of the Beam w. Then AF 
 
 = a y, DF -b + x, and AD (/AF' + DF 1 ) = 
 V a* 2ay -f / + b* -f- 2bx -f- x* ; which (becaufe y* + 
 yi 1 " = t 1 ) will alfo be rz 
 
 a 1 + b* + 
 
 zay (by putting/ *= 
 
 
 
 whofe 
 
 is the 
 
 Momentum of the Weight j^, fuppofing the Beam to 
 to be in Motion. Moreover, becaufe DC : DL :: CG ; 
 
 GI, we have GI = j whofe Fluxion, , multi- 
 
 ply'd by w, is the Momentum of the Beam itfelf in a 
 vertical Direction. 
 
 Wherefore making thefe Momenta equal to each other 
 (according to the Principles of Mechanics) we get 
 
 bx ay dx 
 
 r.v x J= X /, and confequently 
 
 bx oj X f^.= dwx 
 
 2ay: But, fmce
 
 5 28 The Refolution of Problems 
 
 y* -}- x* = \ we have 2yy + ixx o, or y =. 
 
 X *"* 
 
 * 
 
 - : And therefore (by Subftitution) * 4- ~ x 
 _ y 
 
 f- + 2bx 2ay t or by + 
 
 Vf* -\-ibx 20y: From whence, and the foregoing 
 Equation -* 1 -}-/*:=% both x and_y may be determined. 
 
 The fame other-wife. 
 
 452. It is evident, from Mechanics, that the Force 
 which, acting in the Direction DF, would fuftain the 
 End D, is to the whole Weight w, as CG to CD ; 
 
 CD 
 
 and therefore is =r TTTT X w : It is likewife Jcnown 
 
 that two Forces acting in the different Directions DF 
 and DA, fo as to have the fame EffecT: in fuftaining 
 DC, or caufmg It to move about the Point C, muft be 
 to each other, inverfely, as the Sines of the Angles of 
 Incidence FDC and ADC. Therefore we have 5. FDC 
 
 CD 
 
 : S. ADC :: j^: : x iu\ from which given Ratio of 
 
 the Sines, the Angles themfelves will be found, by an 
 algebraic Procefs independent of Fluxions. 
 
 COROLLARY. 
 
 453. If the Pofltion of CD be fuppofed given, and 
 the Tenfion of AD (or the Weight J^J be required: 
 Then, from the foregoing Proportion, we fhall have j^p 
 
 9 F1~)C" C(~" 
 
 ..' . ... . X - - x w. Which will alfo exprefs the 
 
 jo. 
 
 Tenfion of AD when the End C is fuftained by a Cord 
 1>C inftead of ?. Pin at C : Whence it follows that the 
 Tenfions of two Cords AD and BC, fuftaining a Beam 
 or Rod CD, at its Extremes D and C, are exprefled by 
 5. FDC CG ,5. HCD DG 
 
 x x ^ and x x w > and 
 
 CD cD CD 
 
 there-
 
 of various Kinds. 
 
 therefore are to each other as 
 
 or 
 
 CG DG 
 
 S. ADC ty S. BCD 
 as S. BCD x CG to S. ADC X DG refpeaively 5 be- 
 caufe the Sine of FDC and that of its Supplement HCD 
 are equal to each other. 
 
 P R O B. XXII. 
 
 454.. To determine the Pofttion of a Beam DC, fuf- 
 pendedat its Extremes by two Cords AD and BC of given 
 Lengths , from two given Points A and B in the fame 
 horizontal Line AB. 
 
 Let G be the Center of Gravity of the Beam, and 
 Jet DF and CH be perpendicular to AB. 
 
 II B 
 
 It appears, from the Corol. to the laft Problem, that 
 the Tenfion of AD is to that of BC, as - 
 
 whence (by the Refolution of Forces) the 
 
 . 
 
 Force of AD, in a Direction parallel to the Horizon, 
 is to the Force of BC, in the oppofite Direction, as 
 
 CG S. ADF DG S. BCH ___,. , 
 
 v tr> * v . Which 
 
 S. ADU 5 Rod. S. BCD x Rad. 
 
 Forces, that the Beam may remain in Equilibria, muft 
 M m con-
 
 53 o *fbe Refolution of Problems 
 
 confequently be equal to each other; and therefore 
 5. BCD S. BCH DG D 
 ADC ~ ODF x TIG' But now ' to detcrmine 
 the Angles themfelves, from this Equation and the given 
 Lengths of AB, BC foV. let AD and EC be produced 
 to meet each other in P, and let PQ_, perpendicular to 
 AB, be drawn ; putting AB a, AD = b> BC = r, 
 DC=<i, DG=/, CG=, AP=*, and EP=y. 
 
 Then, becaufe AB : AP + BP :: AP BP : AQ_ BQ_ 
 
 AP 1 BP^ AP 1 BP* 
 
 = AB "' we have AQ ~ = * AB + 2 Att 
 
 AB a -f AP 1 BP 1 
 = - AE - : ' an C01 " e< l uen "y t " e Co-fine of 
 
 A (= Sine ADF) to the Radius x = 
 
 Whence, from the fame Argument, it is evident that 
 the Co-fine of B (= Sine BCH) will be exprefled by 
 
 _ 
 2 ABxBP iandThatofAPBby 
 
 And alfo by ~~pr) pr* - > which two laft Quan- 
 tities being equal to each other, we have PD x PC x 
 AP a +BP a AB*=AP x BP x PD^ + PC 1 I>U r rthat 
 
 is x b x y c x x 1 +y*a* = xy X 'xff+jHA* /. 
 Moreover, fincePC : PD :: S. ADC (or PDC) : S. BCD 
 
 PD 5. BCD S. BCH 
 (or PCD) we alfo have = - = - x 
 
 (by the firft Equation) ; whence CG x PD x 
 
 S. ADF = DG x PC x 5. BCH ; that is CG x PD x 
 AB*+AP* BP 1 -^ AB* + BP l AP 
 
 or 
 
 2AB X BP 
 
 CG
 
 of various Kinds. 
 
 CGxPD x BP x AB 2 +AP 2 BP^DG x PC x AP x 
 AB 2 -f BP 1 AP% which, in algebraic Terms, 
 
 x t>Xa* + x i / f x xy cxa' + y* x 1 . From 
 whence and the preceding Equation the Values of x and 
 y will be known. 
 
 P R O B. XXIII. 
 
 455. Suppofing a Beam CD, moveable about one End 
 C, as a Center, to be fujiained at the other End D by 
 means of a given Weight P, hanging at a Rope pa/Jin* 
 over a Pulley at a given Point A, vertical to C ; it is pro- 
 pofed to find the Curve APK along which the Weight mujl 
 afcend, or defcend, fo as to be, every where, ajujl Coun- 
 tfrpoife to the Beam. 
 
 v A From the Center C, 
 
 with the Radius CD, 
 let a Semi-circle HDR 
 be defcribed, and let 
 DB and PF be perpen- 
 dicular to the vertical 
 Line AHCR ; alfo let 
 CD= a , CA-b, AH 
 =c, AF = *, PF=j, 
 HBzrz, and the Length 
 oftheRopeDAP=;;2j 
 likewife Jet HQ_ (h) be 
 the given Value of x 
 (AF) when D coincides with H. 
 
 Becaufe the Weight and the Beam are always in 
 Equilibria, by Hypothecs, their Momenta, and con- 
 fequently their Velocities, in a vertical Direction, muft 
 be every where in a conftant Ratio ; and therefore 
 the Diftance QF (h x) afcended by the Weight P 9 
 will be, to the Diftance HB defcended by the End 
 of the Beam D likewife in a conftant Ratio : Let 
 this Ratio be that of b to any given Quantity d, 
 that is, let h x : z :: b : d, and we fhall have dh 
 dx-bz: Moreover, we have AD a (CD* + AC a 2AC 
 X BC) =aHi* -2bXa z-b a*+2t>z = c 2 + 2&z 
 = S idh + ^dx : Whence AP (m AD) = w 
 M m 2
 
 53 2 
 
 Refolution of Problems 
 
 zdh+idx, and therefore, / (AP J AF 1 ) = 
 x\ >. E. I. 
 
 \/cc - 
 
 m 
 
 After the fame manner a Curve may be found, along 
 which a Weight defcending, {hall be every where in 
 Equilibria with another Weight afcending thro' the Arch 
 of a given Curve. 
 
 P R O B. XXIV. 
 
 456. To find the Equation of a Curve ABH, along 
 which a given Weight P, fujpendsd by a String PED 
 faffing over a Pulley E, mri/f defcend, Jo thai the Tenfwn 
 of the String may vary according to any given Laiu. 
 
 Let EC be perpendicular, and 
 CP parallel, to the Plane of the 
 Horizon ; alfo let AE #, ACn*, 
 CBrr:^, EP = v, and let the Ten- 
 fion of the String (or the Force 
 acting at the End D) be denoted 
 by any variable, or conftant, Quan- 
 tity 4; 
 
 Therefore, becaufe the Celerity 
 of the Weight P, in a vertical Di- 
 rection, is to its Celerity, in the 
 Direction EP produced, (or the 
 Celerity of the other End D) as 
 x to -v, it is evident that the Weight 
 
 itfelf muft be to the tending Force j^, inverfely in that 
 
 Ratio, and confequently Px=<j>v. 
 
 Furthermore, becaufe EC =a + x and BC 2 nBE 1 
 
 EC% we have y* r: v* fl+^| l : From which Equa- 
 tions, when the Relation of P and Q. 1S g' ven > the 
 Curve itfelf will alfo be known. 
 
 Thus, for Example, let the Ratio of P to j^, be 
 conftant, or that of m to , then mx being = /, we 
 have (by taking the Fluent) mx -\- na =. nv j whence 
 
 mx a / , 2max , m * x * 
 
 v a -}- j and therefore jr ( = a + - -f- r~
 
 of various Kinds. 
 
 533 
 
 __ i . t . 7 m a 
 
 n n* 
 
 Which is the Equation of an Hyperbola. 
 
 Again, for a fecond Example, let the tending Force 
 ^.bc to the Weight P, as DE" to AC" X f"-" 1 , or as 
 
 ^ 4J : x* c~" m (fuppofing ziPED and c any given 
 Line AF.) Therefore, fince ^ = tll^L x P, and 
 
 K * M 
 C * 
 
 A 1* 
 
 b v\ 
 
 V (=^y) = Px, we have b v\ 
 
 I 1" + 1 . j + 
 
 n m m . . r b a O V\ 
 
 c xx t and fo 
 
 + I 
 
 m+i 
 
 ; whence b -1} 
 
 n + 
 
 -, and v ( EP ) = b 
 
 ,n + 1 ft -+- ' X 
 
 I 
 + ^ 
 
 . From which 
 
 the Relation of x and ^, or the Value of BC, is alfo 
 known. 
 
 But if m o, and = r, (which will be the Cafe 
 when the Force acting at D is equal to that by which a 
 Beam or Rod is made to move about a Center, as in 
 the laft Problem) v will then become, barely, = b 
 
 /, 0)'" 2]*, and therefore y 1 ( = v* a + x] 1 
 
 =. b V^ a\ l . 2<rA a -\- x\ : Therefore 
 ABH is, in this Cafe, a Line of the fourth Order. 
 
 M m 3 
 
 PfcOB.
 
 534 T&e Refolution of Problems 
 
 P R O B. XXV. 
 
 457- Suppoftng a Ray of Light ABCD to be refrafied 
 at the Surface of a given Sphere MQND, and after- 
 wards reflefted any given Number (n) of Times, within 
 the Sphere j to determine the Diftance of the Incident Ray 
 AB from the Axis MN, fe that the Arch MBCDE, in- 
 tercepted by the given Point M and the emerging Ray at 
 E, may be a Minimum. 
 
 Let the Radius 
 
 ^__O OB = i, the Sine 
 
 A \ sB/'" *>. of Incidence BR 
 
 ^ Z " r, and the Sine of 
 
 and let the given 
 Ratio of the two 
 laftbethat of/>to?. 
 Since all the An- 
 gles of Incidence 
 andReflexionBCO 
 OCD, CDO &c. 
 are equal, the Arcs 
 BC, CDandDEmuft alfo be equal; and confequently 
 
 MBCDE = MB + ;;+i x BC= MB + 2 + 2X BQ : 
 
 A*. 22. Whofe Fluxion is to be equal to Nothing*. Now 
 
 the Fluxion of the Arch MB, whofe Sine is * and 
 
 fArt.142. Radius Unity, will be = 
 
 and that of 
 
 the Arch BQ_, whofe Co-fine (OP) is y, = 
 Hence we have 
 fince x : \ 
 
 have 
 
 :?, y is = and y = j and fo 
 
 -f 2 X 
 
 x 
 
 y ^^^. 
 
 = o j whence (putting 
 jV 
 
 m =
 
 of various Kinds* 
 
 535 
 
 m 
 
 = 2* + 2) X is found = -1 v' t * : From 
 which it is obfervable, that, when mq is lefs than />, or 
 2-}-2 lefs than, the Arch MBCD continually m- 
 
 r 
 
 creafes with BM ; and therefore is the leaft poffible, 
 when B coincides with M. 4>. E. I. 
 
 P R O B. XXVI. 
 
 458. If two Rays of Light PR and Pr, from a gfotn 
 Point P, making an indefinitely fmall Angle with each 
 other ) be reflected at a given Curve Surface ARB; '//'; 
 propofed to determine the Concourfe, or Focus, Q^ of the 
 rrfefad Roys RQjand rQ. 
 
 Let RO, perpendi- 
 cular to the Curve, be 
 the Radius of a Circle 
 having the fame Cur- 
 vature with ARB at 
 R } make PH and QM 
 perpendicular to RO, 
 joinQ_,O; andputRO 
 
 and RQ_= z. 
 
 Then, becaufe the Angle of Reflection ORQJs equal 
 to the Angle of Incidence ORP, the Triangles RQ_M 
 and RPH will be fimilar, and therefore y:v:: : RM 
 
 = -: Whence OQ. 1 (RO 1 + RQ^ 2 RO x RM) 
 
 irvz 
 
 But, fmce this Quantity OQ^ continues the fame 
 
 (by Kypothefis) whether we regard one Ray or the 
 
 other (that is, whether y {lands for PR or Pr) its 
 
 Fluxion muft therefore be equal to Nothing; that 
 
 M m 4 is,
 
 536 ffl' Refohition of Problems 
 
 irvzy -f- 2rviy irvzv 
 is, 22z J - p& = o : Whence 
 
 y 
 
 vyz 
 "~ /. 35.)*= -y therefore z= 
 
 vyy 
 
 : Moreover (by Art. 73.) r =r 21. 
 
 vyy <D E I 
 
 i. Let ARE be an Arch of the Logarithmic 
 Spiral : whole Equation is av by f : And then, < 
 
 4x ^ ^)!J' \ 
 
 being =: , we fhall have z ( : . J = y : 
 
 a v - 2)'-y vy/ 
 
 Therefore in this Cafe the Incident and Reflected Rays 
 are equal to each other. 
 
 Ex. 2. Let ARE be fuppofed to degenerate into a 
 Right-line : In which Cafe v being conftant, its Fluxion 
 
 i>yy\ 
 
 v is = o; and therefore z ( , } v; Which 
 
 <uy' 
 
 being negative, indicates that the Rays do not converge 
 after Reflection, but, on the contrary, diverge from a 
 Point on the contrary Side of ARE, at the Diftance;-. 
 Which is very eafy to demonftrate by common Geo- 
 metry. 
 
 P R O B. XXVII. 
 
 4.59. Let two Rays of Light PR and Pr, from a given 
 Point P, be refrafted at a given Curve Surf ate ARE ; to 
 determine the Focus Q_ of the refrafied Rays RQ_and rQ. 
 
 Let the Lines RO, RH &c. be drawn, and denoted 
 as in the preceding Problem: Moreover, let the Sine of 
 Incidence PRH (to the Radius i) be reprefented by s, 
 and let it be to the Sine of Refra&ion ORQ^, in the 
 given Ratio of i to n. 
 
 Then 
 9
 
 of various Kinds. 
 
 Then (by Trigonometry) i :ns (Sine QRM)::z (RQJ 
 ; QM nsz' t and therefore RM r^ y^ 4 wVz* 
 
 rr z y i n~s~. From whence, following the Steps 
 of the preceding Problem, we alfo get QQ^=r*+z t 
 
 2rzv'i- i -V_; and its Fluxion 2zz irzV i nV 
 
 2rzn ss 
 "; " ~ =: c ; or 
 
 Vl V 
 
 ! V* rz x 1 
 
 = o. But (/y ^r/. 35 J z = ny ; there- 
 fore j V i w*f l 4" r j x * " 2 ' $z ~f" wzw =: O : 
 Moreover (by Trig.} i (Radius) : i (Sine of PRH) :: 
 
 y (PR) : Vy L -y 1 (PH) whence we have jy = 
 
 j; r= 
 
 v*j yv-v 
 = -5 ; which Values, of j a and j^, being 
 
 ftituted in the foregoing Equation, it becomes - 
 
 x -. w - + _ + 
 
 x i 
 
 T- 4- H/-3 X 
 
 o, or 
 
 j nn x > z 4~ 7I * V * ' 
 
 4/ l" ^ I 2, 
 
 ^ 4. V + rjtf X 
 > rr o ; or (putting 
 4- r -r 
 
 537
 
 538 
 
 T'be Reflation of Problems 
 
 */ *x 
 
 yj 
 
 nzv*y nyzwv O. But (by Art. 73.) r , 
 therefore zywi> + ufyy -\- nzv^y nyzv-vO, and 
 confequently z = - 1 : r* >. E. I. 
 
 Art. 142. 
 
 uy + z^ x v 
 From this Solution, that of the preceding Problem 
 is eafily derived : Alfo from hence the Cauftic (or the 
 Curve which is the Locus of all the Points Q_ thus 
 found) will likewifc be given. 
 
 P R O B. XXVIII. 
 
 460. To find the Time of the Vibration of a Pendulum 
 in the Arch cf a Circle. 
 
 Let AB denote the Pen- 
 dulum in a vertical Pofition ; 
 and from any Point D in the 
 given Arch CBH, wherein 
 the Vibrations are perform'd, 
 draw D/" parallel to CH ; and 
 
 and BD 2: By the Nature 
 of the Circle we have z 
 
 -i = * : Whence the 
 
 Y 2ax xx 
 
 Fluxion of the Time, being 
 
 * 2* 
 
 4Art.*c 7 . as -p= t will be defined by - 
 
 GX 
 
 ax 
 
 V tx xx X y/2a x 
 J. 
 
 Vex xx 
 
 i 
 
 ia\ 
 
 ifll X x ^ j | x 
 
 
 3 5* 3 
 
 Vex xx 2 . 2fl 
 
 Y ii* ' " * -* 
 
 ^2.4.4^ ' 2. 
 f \Wpr*nf rhp 
 
 Fluent. 
 
 3-S-7* 4 
 
 2.4.6.8. 
 
 when
 
 of various Kinds. 
 
 i 
 when xc 9 (or ex A-"*) =o) is, (by Art. 142. and 286.) 
 
 c 3 . 3c* 
 
 equal to ? V^ x i + 2 . a . M + 2 ^ a> 4 ^ ^ ^ 
 
 j 3 3 5 5' 3 + 3 3 5 5 7 7** ~ 
 2.2.4,4.6.6.2tf) 3 2.2.4.4.6.6.8.8. 2y 
 CSfc. Which therefore is proportional to the Time of 
 half one Vibration; where p ftands for the Semi-Peri- 
 phery of the Circle whofe Radius is Unity. 
 
 COROLLARY I. 
 
 461. Since the Time of the perpendicular Defcent 
 of a Body through any given Right-line #, computed 
 according to the fame Method, is as the Fluent of 
 
 ~. or 2 V u, it follows that the Time of falling 
 
 Y u 
 
 along the Diameter BF (2*), or the Cord CB *, will *Art.io5. 
 
 be truly defined by 2 v/ 2a : Which therefore is to the 
 
 Time of the Defcent thro' the Arch CDB, as -'- to i 
 
 3 . 3^* 
 
 4. H s =r z &c. From whence, 
 
 r 2 . 2 . 20 2 . 2 . 4 . 4 . 2a\ 
 
 as the Time of falling thro' the Diameter BF, is abfo- 
 lutely given, by Art. 202. the true Time of Vibration 
 will alfo be known. 
 
 COROLLARY II. 
 
 462. If the Arch in which the Pendulum vibrates be 
 very fmall, the above Proportion will become, nearly, 
 as 4 to p : From which it appears, that the Time of 
 Defcent thro' any very fmall Arch CB is to that along 
 the Chord CB, as the Periphery of any Circle is to four 
 times its Diameter. 
 
 COROLLARY III. 
 
 463. Hence, we have a Method for determining how 
 far a Body freely defcends in a given Time j by knowing 
 
 the
 
 54-0 *ft>e Refolution of Problems 
 
 the Time of Vibration, of a given Pendulum: For, if 
 BN be afiumcd for the Space thro' which a Body would 
 detcend during the Time of one whole Vibration, in 
 the very fm all Arch CBHj then, the Diftances de- 
 Art. 201. fcended being as the Squares of the * Times, we havr, 
 
 from the laft Corollary, as 4* : 2pl~ :: BF (20} : BN, 
 or i : lp~ : a : BN ; that is, as the Square cf the Dia- 
 meter of a Circ'e is to half the Square of its Periphery, 
 fo is the Length of the Pendulum, to the Diftance a 
 Body will freely defcend, from Reft, in the Time of 
 one Ofciliation. Thus, for inftance (becaufe it is found 
 from Experiment that a Pendulum 39,2 Inches long 
 vibiates Seconds) it will be as I : 4,934 (r=i/> 2 ) :: 392 
 : 193 inches, the Diftance which a heavy Body will 
 fall in ihe firlt Second of Time. 
 
 COROLLARY IV. 
 
 464. Moreover, from the foregoing Series, the Time 
 which a Pendulum, vibrating in an exceeding fmall 
 Arch, will lofs when made to vibrate in a greater Arch 
 of the fame Circle may alfo be deduced : 
 
 For let T be put to denote the Number of Seconds 
 in 24 Hours (or any other given Time) then the Num- 
 ber of Vibrations, performed in that Time will be as 
 
 ; which, therc- 
 
 3 . 3* 
 
 t . ft 14 2.2.4. 2tf | ^ 5 "" f * 
 
 fore, in an exceeding fmall Arch (where c may be taken 
 as Nothing) will be exprefled by T: And fo the Time 
 (t} or Number of Vibrations loft will be T 
 
 4.4 
 
 -L j_ $ c & c , (by dividing by the Denominator.) 
 %a 256** 
 
 Now, if the Number of Degrees defcribed on each 
 Side of the Perpendicular be reprefented by D, tife 
 
 Arch
 
 of various Kinds. 
 
 Arch itfelf, on each Side, will be = 3.14159 &c. x a 
 D . 
 
 x TTT- ; which, if the Value of D be not more than 
 loO 
 
 about 15 or 20 Degrees, will be nearly equal to its 
 Chord, reprefented by \/2ac (~ 1/BF x BE.) From 
 
 which Equation we get T-T ' This Value, fub- 
 fe a 6560 
 
 r^. *. T~\. A 
 
 ilituted above, gives tT/. 
 
 541 
 
 8x0500 
 
 -f 
 
 = T X Q- nearly : Which, when T is interpreted 
 
 by 864.00 Seconds (or one whole Day) becomes rr i x 
 .>% nearly : And fomany are the Seconds which will be 
 loft per Diem in the Arch D. From whence we gather, 
 that if the Pendulum meafures true Time in any fmall 
 Arch, whole Degrees on each Side the Perpendicular 
 are denoted by ^, the Number of Seconds loll per Diem 
 in another Arch whofe Degrees are B t will be nearly 
 
 ? 
 
 reprefented by x B x A* : Thus, if a Pendulum 
 
 meafures true Time, in an Arch of 3 Degrees, it will 
 lofe lOj Seconds a Day in an Arch of 4 Degrees, and 
 24" in an Arch of 5 Degrees. 
 
 P R O B. XXIX. 
 
 465. To determine the Meridional Parts anjwenng to 
 any propofed Latitude, according to Wright'5 Projeflion, 
 applied to the true fpheroidal Figure of the Earth. 
 
 Let DAR be the 
 Axis, AB the Se- 
 mi-equatoreal Dia- 
 meter, and DBR a 
 Meridian, of the 
 Earth j alfo let bn 
 be an Ordinate to 
 the Ellipfis DBR ; 
 putting AD (=AR) A
 
 54 2 The Refolution of Problems 
 
 / */ 
 
 = r, BArrJ, A=rjr, bny, Bn=z, and the Meri- 
 dional Diftance (in Parts of the Semi-Axis AD) u. 
 Then, by the Nature of the Ellipfis, we have y d 
 
 f dxx 
 
 V i **; therefore j ~^ ; ; and confequently 
 
 r i X 
 
 S~~ d*x*x* 
 * = \/ x* + : Which, by putting b* = d* 
 
 I """* XX 
 
 * A/ _ |_ A^y 1 
 
 I, will be reduced to / Whence, 
 
 by the Nature of the Projection, it will be as bn 
 
 x V\ + b x O f the 
 
 required : But we are no.w to get the fame thing ex- 
 prefled in Terms of the Latitude of the Place n: In 
 order thereto, putting the Sine of that Latitude = j, we 
 
 have, by Trigonometry, as %. ( . J y 
 
 ^ V i x* 
 
 (;. ) :: Radius (i) : JJ and confequently 
 y i *v 
 
 *V i + ^ a = dx j from which Equation x is found 
 
 s J*i 
 
 /.a zr^ : Whence x = 
 
 _ - >a 
 
 -^^v k'-^v" (becaufe ^ = ' + * ) and ' 
 
 / ^ \ d 
 
 laftly, V i + ^V ( = 7 j = TTTZTv : ' ch 
 
 feveral Values being fubftituted in that of w, found ^bove, 
 
 / d s d 
 
 it will become f r x , ^ . = x 
 
 ; which refolved
 
 of various Kinds. 
 into two Parts, for the more readily finding the Fluent, 
 
 gives * = _I_ ^f.^,.: Whereof the Fluent 
 being taken, we have 
 
 2 . 302585 12 c. x \d X Log. 
 
 i * 
 
 2 . 302585 fcfr. X | b X Log. - 
 
 But, as 3,14159 fefc. x 2<f (the Meafure of the whole 
 Periphery of the Earth at the Equator, in Parts of the 
 Semi-Axis AD) is to 21600 (the Meafure of the fame 
 Periphery in Geographical Miles) fo is the forefaid Va- 
 lue of u to 
 
 3958 x Log. i-i-i 
 
 the correfponding Value 
 39586 d + bs 
 
 of , in Geographical Miles, or the Meridional Parts 
 required. 
 
 COROLLARY. 
 
 466. If the Earth be confidered as differing but little 
 from a Sphere, d will be nearly rzi, and confequently 
 
 (V d 1 i ) the Value of , very fmall : Therefore, in 
 this Cafe, the latter Part of our Fluent f * * X 
 
 Log. -j ) will become nearly 3440^ (becaufe 
 
 a "" us ^j 
 
 d+bi 2bs*\ I _ 
 
 Loo-. - - - l x -r- *. But if theEarth 
 
 d bs d J 2 . 3025 &c. 
 
 be taken as a perfect Sphere, this laft Expreflion will 
 vanifh, and fo the Value of u will become barelyr=3958 
 
 * There is a Mijlake in p. 43. and 44. of my Dijfirtalions 
 (by forgetting to divide by tht Modulus 2.3025 &C.) ivbicb 
 may from hence be ;
 
 544 y& e Rtfofation -of Problems 
 
 X Log. . Which Logarithm, it is eafy to prove, 
 
 exprefies twice the artificial Tangent of half the given 
 Latitude increafed by 45 Degrees (Radius being Unity.) 
 Wherefore, if the Meridional Parts anfwering to any 
 given Latitude, thus found (from a Table of logarithmic 
 Tangents) when the Earth is confidered as a perfect 
 Sphere, be denoted by M, it follows that the Meridional 
 Parts anfwering to the fame Latitude, when the Earth 
 is taken as a Spheroid, will be nearly equ a l to M 
 3440V- Which, becaufe AD (i) : AB (/i-f M) :: 
 An. 397-230 : 231 *, will (by fubftituting the Value of b hence 
 arifing) be reduced to M 30;. Whence the follow- 
 ing Rule. , 
 
 M 
 
 As Radius i to the Sine of the given Latitude, fo is 30 
 to a Fourth- Proportional ; which fubtrafted from the Me- 
 ridional Parts when the Earth is taken as a Sphere (found 
 as above) gives the Meridional Parts anfwering to the fame 
 Latitude , when it is confidered as an oblate Spheroid. 
 
 Thus, for Example, let the given Latitude be 50 : 
 Then, rirft, for the Meridional Parts in the Sphere ; 
 we muft, according to the foregoing Prefcript, take the 
 Logarithmic Tangent of 25 4-45, or 70: Which, 
 by the Table, is found = 0,43893 &c. This multi- 
 ply'd by the conftant Multiplicator 7916 ( = 2 X 3958) 
 produces 3475 for the Meridional Parts in the Sphere : 
 Then by the Rule above, it will be as Radius to the 
 Sine of 50, fo is 30 to 23 ; which fubtra&ed from 
 3475, leaves 3452 for the Meridional Parts anfwering 
 to 50 Latitude, in the Spheroid. 
 
 P R O B. XXX. 
 
 467. To determine the Paths which Shadows of 01- 
 jefts deferibe, upon the Plane of the Horizon, during thf 
 Suns apparent diurnal Revolution. 
 
 Let CSODT be the Plane of the Horizon, and AV 
 the perpendicular Height of the Object : Then, fmce 
 the Rays, intercepted by the higheft Point V, would, 
 in the Sun's diurnal Revolution, form a conical Sur- 
 face
 
 of various Kinds. 
 
 face VDFEH about that Point as a Vertex ; whofe Axis 
 PV produced pafTes thro' the Pole of the World ; it is 
 evident that the Path of the Shadow, being the Inter- 
 fe&ion of the Plane of the Horizon with that Surface^ 
 muft be a Conic Section. 
 
 545 
 
 Let its two principal Diameters therefore (when an 
 Ellipfis, that is, when the Sun never defcends below the 
 Horizon) be CD and ST ; alfo let DPE and CG be 
 perpendicular to VP the Axis of the Cone, and CQ 
 perpendicular to DV : Putting the Sine of (QVC) twice 
 the Sun's Declination VEP=/; the Sine of (DCV) his 
 greater Meridional Altitude = e, and that of the lefler 
 (CDV) =h: Then (by planfrrig.) g : I (AV) :: i 
 
 (Radius) :CV = 5 and h (Sine of CDV : (CV) 
 
 o o 
 
 ::/(Sine of DVC) : DC = - : Moreover, i (Ra- 
 
 o 
 
 dius):~ (CV) :: p (the Sine of the Comp. Decl. 
 
 o 
 
 GVC) : GC = : And in the very fame Manner it 
 will be found that DP = j : But GC x DP = OS 1 " 
 (vld. Art. 41.) whence we have ST (zOS) = ~r=~- ' 
 
 From which, and the Tranfverfe Axis (DC = , ) thJ 
 Curve itfelf is given. 
 
 Nn 
 
 gh> 
 
 . E. I. 
 
 LEMMA.-
 
 54 6 The Reflation of Problems 
 
 LEMMA. 
 
 468. In any fpherical Triangle, if Radius be fuppofea 
 Unity, the Produtt of the Sines of any two of the Sides 
 drawn into the Co-fine of the Angle th,y include, added to 
 the Produfi of their Co-fines, is equal to the Co-fine of the 
 remaining Side. 
 
 This is demonftrated by the Writers upon Spherics^ 
 
 P R O B. XXXI. 
 
 469. The Elevation of the Pole and the Declination of 
 the Sun being given, to find at what Time of the Day the 
 jfzimutb of the Sun increafes thejkwejl. 
 
 -p It is evident that the 
 
 Time fought will be when 
 the Fluxion of the Hour 
 Angle P, bears the greateft 
 Ratio poflible to That of 
 the Azimuth Z. 
 
 Now the Fluxion of the 
 Angle P is to that of Z, 
 univerfally, as Rad. X S. ZO 
 _ : S. PO X Co-f. O (by Art. 
 
 256. Cafe 2.) Confequently 
 S.POxCe-f. O Co-f. O 
 
 ' 
 
 c rjr\ * v * " "' - * s a Minimum, in this 
 X o. Ziw 
 
 Cafe, becaufe PO may be confidered as conftant. 
 
 Let now the Sine of PO be put />, its 
 the Co-fine of PZ == b, that of ZO = x, and that of 
 
 O = yi then, the Sine of ZO being = YI A-% we 
 have (by the Lemma) p V ix* X y + dx~b; whence 
 
 bdx Co-f.O ( _ y 
 
 v = -/; and therefore - 7 A ( ~ TT^ 1 
 
 pVi x * *> zu V vi- 
 
 ^ dx 
 
 . : Which put into Fluxions, and re- 
 
 p x i X L 
 
 6 duced,
 
 of various Kinds. 547 
 
 duced, gives x = -HI , for the Sine of the 
 
 a 
 
 Sun's Altitude at the Time required: Whence the Time 
 itfelf is given. 
 
 P R O B. XXXII. 
 
 470. To determine the Ratio of the Heat received from 
 ihe Sun in different Latitudes^ during the Time of one 
 whole Day, or any Part thereof. 
 
 Let/> = the Sine of the Sun's Polar-Diftance P (fee 
 
 the loft Fig.) 
 
 d its Co-fine, or the Sine of the Declination, 
 irz the Sine of the Pole's Elevation. 
 c its Co-fine, or the Sine of PZ. 
 z the Angle (P) exprefllng the Time from 
 
 *:= its Sine, and V I x* its Co-fine. 
 Then (by the foregoing Lemma) we fliall have 
 p c V i _ x * + bd Co-fine Zd = Sine of the Sun's 
 Altitude. 
 
 Now, it is known that the Number of Rays falling in 
 any given Particle of Time, upon a given horizontal 
 Plane, is as that Time and the Sine of the Sun's Alti- 
 tude conjundtly : Therefore the Number of Rays falling 
 
 jf 
 
 in the Time 2, or , (ind. Art. 142.) will 
 
 v I xx 
 
 be dtfined by pcx + bdz: Whofe Fluent pcx + bdz is, 
 therefore, as the Heat required. 
 Where it may be obferved, 
 
 1. That when the Latitude and Declination are of 
 different Kinds, or P is greater than 90 Degrees, 
 the Value of d is to be confidered as a negative Quan- 
 tity. 
 
 2. That, if the Expreflion for the Heat found above 
 be divided by the Square of the Sun's Diftance from the 
 Earth, the Quotient will exhibit the Ratio of the Heat, 
 allowing for the Excentricity of the Earth's Orbit. 
 
 N n 2 Co-
 
 54$ The Refolution of Problems 
 
 COROLLARY I. 
 
 471. If the Place propofed be at the Equator, the 
 Heat, received in half one diurnal Revolution, will be 
 barely as/>; becaufe bo, r=i, and *zrj. 
 
 COROLLARY II. 
 
 472. But if the Place be at the Pole, then the Heat 
 
 will be as d X 3,14.159 &c. fmce, in this Cafe, =0j 
 b I, and z (=. Semi-Circle) =3,14159 &( 
 
 LEMMA. 
 
 473. The Number of Particles of Light) ejefied by the 
 Sun* upon the Earthy in a given Time, is proportional 
 to the /Ingle defcribed about his Center in that Time. 
 
 For, let S reprefent the Center 
 of the Sun, AEB the Orbit of the 
 Earth (or That of any other Planet) 
 and let E and r be two Points there- 
 in as near as poffible to each other: 
 Since the Triangle ESr may be 
 taken as rectilineal, its Area, if 
 the Angle ESr be fuppofed given, 
 or every where the fame, will be 
 as SExSr, or SE* : And there- 
 fore the Time of defcribing Er 
 (being always as that Area) is alfo explicable by SE a : 
 But the Intenfity of the Light, or Heat, at the Diftance of 
 
 SE is as =KZ : Therefore the Intenfity compounded 
 oh, 
 
 with the Time (or the whole Number of Particles re- 
 ceived in that Time) will confequently be as g^ x SE* 
 
 ( i ) : Which being every where the fame, the Propo- 
 fition is manifeft. 
 
 P R O B. XXXIII. 
 
 474. To determine the Ratio of the Heat received from 
 the Sun at the Equator and either of the Pales, during 
 the Time of one whale Year^ cr any Part thereof.
 
 of 'various Kinds. 
 
 549 
 
 ^ If the Sine of the 
 Sun's Declination be 
 denoted by d and its 
 Co- fine by p, the 
 Heat received at the 
 Equator.and thePole, 
 during half one di- 
 urnal Revolution of 
 the Sun, will be ; 
 and dx. 3, 1 41 59 &c. 
 refpe&ively (by the 
 Corollaries to the preceding Problem). 
 
 Let the Sun's Longitude, confidered as variable, be 
 now denoted by z, and its Sine by s ; and let / be put 
 for the Sine of the Obliquity of the Ecliptic : Then 
 (per Spherics) we {hall have^^y}, and confequently p 
 
 ( V\ d*} = /i /V : Wherefore, feeing the 
 Ratio of Heat in the two Places, for one Half-Day, is 
 that of ^i /V 10/^x3,14 fcfr. let each of thefe 
 
 Terms be multiplied by 
 
 ( = z) * expreffingArt.i 4 a, 
 
 the Quantity of Heat falling upon the Earth in the 
 Time of describing z (fee the foregoing Lemma) then 
 
 >,_/* 
 
 J - , and 3.1^ 
 
 the Produces 
 
 will 
 
 I - 5 
 
 be the Fluxions of the required Heat, anfwering to %. 
 
 But now to exhibit the Fluents hereof, let ACB be 
 an Ellipfis whofe greater Semi-Axis AO is Unity, 
 and its Excentricity FO /; and, fuppofing ADB to 
 be a Circle defcribed about the Ellipfis, let the Arch DH 
 exprefs the Sun's Longitude from the Equinodlial Point ; 
 whofe Sine (OR) being = s, its Co-fine RH will ber= 
 
 V/i ss. 
 
 But, by the Property of the Ellipfis, OD (i) 
 
 ss ) : RG = 
 
 OC : ( V i 
 
 ( v/ 
 
 , _ /r x "/i ss : Whofe Fluxion being = 
 Nn 3
 
 55. *he Refo/ufion of Problems 
 
 , we have 
 
 y 
 
 = ^-~ f - i - = the Fluxion of CG. Whence it 
 V 1 ss 
 
 appears that the Fluent of ^ **! is truly denned 
 
 V/"T^ 
 
 by CG, or CG x AO*. 
 
 But the Fluent of the other given Fluxion, 3. \^f x 
 ss 
 
 V i ss 
 
 , will be = 3, J4/x i /i w=r ADB X 
 
 FO X OD RH. Therefore the two Fluent?, when 
 H and G coincide with A, will be to each other as 
 CAxAO to ADBxFO: Whereof the Antecedent, 
 multiplied by 4, will be as the Heat received at the 
 Equator during one whole Year j and the Confequent, 
 multiplied by 2, as the Heat at the Pole in the fame 
 Time (becaufe the Sun (nines at the Pole only two 
 Quarters of the Year.) Hence the required Ratio, of 
 the Heat received at the Equator and Pole, in one 
 whole Year, will be That of CA x AO to DA xFO; 
 
 /* tf 4 1 - T 5 f 6 
 
 or, in Species, as i -- - - * - J * , , 
 
 2.2 2.2.4.4 2.2.4.4.6.6 
 
 Art.434- * &c- toy"; which, in Numbers, is as 959 to 396, or 
 as 17 to 7, nearly. 
 
 P R O B. XXXIV. 
 
 475. To find vjhen that Part of the Equation of 
 Time, arijing frsm the Obliquity of the Ecliptic to the 
 l) if a Maximum. 
 
 In the right-angled fpherical Triangle ABC let the 
 Angle A be that made by the Ecliptic AC, and the 
 Equinoctial AB ; then the Problem will be, to find 
 
 when
 
 of various Kinds. 551 
 
 when the Difference be- 
 tween the Bafe AB and 
 the Hypothenufe AC is 
 the greateft poffible (the 
 Angle A remaining inva- 
 riable.) Now (by Art. 
 254.) we have Co-f. BC 
 : Sin. C : : Fluxion of AC 
 Fluxion of AB : Alfo (per Spherics) Sin. C : Co f. A : : 
 
 Co-f. A x Rod. 
 
 Rad. : Cof. BC = - p . ~ -- : Whence, by mul- 
 i>tn. U 
 
 tiplying the two firft Terms of the former Proportion 
 by thele equal Quantities, refpe&ively, we get this new 
 
 Proportion, viz. Co-f. BC\* : Co-f. A x Radius :: fo is 
 the Fluxion of AC to That of AB. But, when AC 
 AB is a Maximum^ thefe Fluxions become equal ; and 
 
 confequently Co-f. BC] % = Co-f. A x Rad. From 
 which Equation BC, and from thence AC, will be 
 known. 
 
 The fame ', without Fluxions. 
 
 476. It will be (per Spherics} Rad. : Co-f. A :: Tang. 
 AC : Tang. AB ; and therefore by Compofition and 
 Divifion, Rad. + Co-f. A : Rad. Co-f. A :: Tang. 
 AC -|- Tang. AB : Tang. AC Tang. AB :: Sin. 
 
 AC + AB : Sin, AC AB, by the Theorem mentioned 
 in Problem 8th : From which, by following the Steps 
 there laid down, it appears that, Radius + Co-f. A : 
 
 Radius Co-f. A :: Radius : Sine of AC AB, when 
 a Maximum : Whence (AC + AB being then =: gO u ) 
 both AC and BC will be given. 
 
 COROLLARY. 
 
 477. Since, Radius -f Co-f. A : Radius Co-f. A 
 : : Co-tang, k A : Tang. ^ A * :: Radius}* : Tang.A\* i 
 
 Fid. f, 70. and 7 1 . of my Trigonometry. 
 
 N n 4 There-
 
 55 2 *Tbe Refolution of Problems 
 
 therefore Radius f : Tang, i AJ * : : Radius : Sine of 
 
 AC AB, Or, Radius : Tang, i A :: Tang. | A : the 
 Sine of the greateft Equation : Which, fuppofing the 
 Angle A to be 23 29', comes out 2 : 28' : 34" : an- 
 fwering, in Time, to 9 Minutes : 54. Seconds. 
 
 P R O B. XXXV. 
 
 478. To determine when the abfoluie Equation of Time, 
 atijwg from the Inequality of the Sun's apparent Mo- 
 tion^ and the Obliquity of the Eclipiic, conjunRly^ is a 
 Maximum. 
 
 Let ABPD be the 
 Ellipfis in which the 
 Earth revolves about 
 the Sun in the Focus 
 S ; let F be the other 
 Focus, and T the 
 Place of the Earth in 
 its Orbit at the Time 
 required. Moreover, 
 about S, as a Center, 
 Jet a Circle GEKI be 
 defer i bed, whofe Dia- 
 meter GK is a Mean 
 Proportional between 
 the two Axes AP and 
 BDofthe Ellipfis; fo 
 that the Area thereof 
 may be equal to That 
 
 of the Ellipfis : And, fuppofing Sm to be indefinitely 
 near to ST, let ESa be a Sedtor of the faid Circle, equal 
 to the Area TSw. 
 
 Then, the Time in which the Earth moves through 
 the Arch Tin being to the Time of one intire Revo- 
 lution, as the Area TS/w, or ESw, is to the whole El- 
 Jipfis, or the equal Circle GEKF ; and thefe Areas 
 Ei>, and GEKI being in the Ratio of the Arch En 
 fo the whole Periphery GEKJ ; it is evident that E, 
 7 vr
 
 of various Kinds. 
 
 or the Angle ESw, will exprefs the Increafe of the Mean 
 Longitude^ in the forefaid Time of defcribingthe ArchT/n.* 
 And that this Angle or Increafe, by reafon of the Equa- 
 lity of the Areas ES and TS/w, will be to the Angle 
 TS//7, exprefiing the correfponding Increafe of the True 
 Longitude, as ST* to SE*. Therefore, if the former 
 
 SE* 
 be denoted by M, the latter will be reprefented by ^=r 2 , 
 
 X M. But now to get a proper Exprefllon for the 
 Value of this Increafe of the True Longitude, in Al- 
 gebraic Terms ; let FT be drawn, and alfo TH, per- 
 pendicular to AP : Putting AC (=CP) a, CB , 
 CS ( = CF) =f, ST-z, and the Co-line of (TSP) 
 the Earth's Diftance from its Perihelion (to the Radius 
 !)=*.- Then FT being (=AP ST) =2a z 
 (by the Property of the Ellipfis) and SHrrjrz (by Trig.) 
 we have FT + ST x F T b T (lax^a^z = FS 
 X zCH (2cX2Xc + xz} by a known Property of Tri- 
 angles : From which Equation z (ST) is found 3= 
 
 ~ . : And this Value, with that of ES* 
 a -f c x a + ex 
 
 (~ab] being fubftituted in the Increafe of the True Lon- 
 gitude, found above, we thence get -r cy ^ ^ ^ 
 
 b* 
 
 for the Meafure of that Increafe; where M denotes 
 the Increment of the Mean Motion correfponding. 
 
 This being obtained, let ^ yf T (in the an- 
 nexed Figure) reprefent the Southern Semi-Circle of 
 the Ecliptic, P the Place of the Perihelion, VS the 
 Tropic of Capricorn, Q the apparent Place of the Sun 
 in the Ecliptic, and QJ? his Declination, at the Time 
 required : Then it appears, (from Art. 475.) that the 
 Increafe of the True Longitude 0, in an indefinitely 
 fmall Particle of Time, will be to That of the Right- 
 Jtjcenfion ^Q_, in the fame Time, as the Square of the 
 Co-fine of Q_ is to a Rectangle under the Radius and 
 the Co-fine of the Angle A : Therefore, the former, 
 
 being;
 
 5.54 y& e Rtfofatton of Problems 
 
 being exprefled by - ' X M y the latter is truly 
 
 a X a + utf v ,. J?g^. X Cof-. A 
 reprefentedby- -^- Co-f. "<tf 
 
 Which, in the required Circumftance, when the pro- 
 pofed Equation (or the Difference between the Sun's 
 
 Mean Motion and Right Afcenfion) is a Maximum^ muft 
 confequently be equal to (M} the correfponding In- 
 
 creafe of Mean Motion ; and therefore 
 
 . x 
 
 But, to obtain the Value of the latter Part of this 
 Equation, alfo, in Algebraic Terms, let the Sine and 
 Co-fine of (tf P) the Diftance of the Perihelion from 
 YT, be denoted by m and refpe&ively ; then, the 
 Co-fine of P being (as above) exprefled by *, and 
 
 its Sine by V i Xify we fhall thence get nx + 
 
 OT/I **=Co-fine of Vf = Sine of A (by the 
 km. of Trig.} But (putting the Sine of the Angle 
 -: =/> and its Co-fine = q] we have (per Spherics] 
 Radius (i) : Sine &-) (nx + mV ixx :: p : pnx + 
 fm\/ 1 _ xx Sine of Q.5; from whence C 
 
 i pnx + pmVi xx : Which Value, with 
 That of the Co-fine of the Angle Zz 9 beins fub- 
 
 ftituted above, we, at length, get a - a cx =
 
 of various Kinds. 555 
 
 I pnx -f pm i xx\ . f rom w hich Equation the 
 
 Value of x may be determined. 
 
 The foregoing Equation, it may be obferved, gives 
 the Time of the Maximum which precedes the Winter 
 Solftice ; but if the Maximum following that Solftice be 
 fought; it is but changing the Sign of m, and then you 
 
 will have * x ^"+^1* = ' pnx-pm Vi X x\\ 
 
 b* a 
 
 anfwering in this Cafe. And from the negative Roots 
 of this, and the preceding, Equation, the Times of the 
 other Maxima after, and before, the Summer Solftice will 
 alfo be obtained. ^. E. /. 
 
 COROLLARY. 
 
 479. It is evident that the Equation of the Earth's 
 Orbit (or that Part of the Equation of Time arifing 
 from the Inequality of the Sun's apparent Motion) will 
 be a Maximum, when the Center of the Earth is in the 
 Interfection I of the Ellipfis and the Circle ; where the 
 Mean Motion and True Longitude increafe with the fame 
 Celerity. 
 
 P R O B. XXXVI. 
 
 480. To determine the Law of the Denfity of a Me- 
 dium and tic Curve defcribed therein, by Means of an 
 uniform Gravity, fo that the Projeftile may, every 
 
 move with the fame Velocity. 
 
 It appears, from Art. 367. that ^/ 21 is a general 
 Expreflion for the Celerity in the Direction of the Or- 
 
 dinate PER; whence X \/ , or its Equal, 
 y x 
 
 ~= , muft be the true Meafure of the abfolute Ce- 
 lerity,
 
 Refolution of Problems 
 
 lerity, in the Dire&ion BN : Which being a conftant 
 Quantity (by Hypothefis) its Square muft alfo be con- 
 
 ftant ; and fo, we have ~ = a, and confequently xx 
 
 -f yy (s:) ax. 
 
 But, in order to the Solution of the Equation thus 
 given, make u : j :: x : y, or xuj ; then x~=- uy^ and, 
 by Subftitution, u*j* + y 1 =. auy: Hence, y being - 
 
 Ou QUU Hi v C 
 
 , and x 7, we get^r:^ X Arch, whole 
 
 uu 
 
 uu 
 
 Art. 142. Tangent is u * (and Secant Vj +uu) j and * : i<7X 
 
 fAtt.126. Hyp. Log. j + uu ' = a X Hyp. Log. V i -f t 
 
 _ . # 
 
 Therefore, as the Hyp. Log. of V i+uu s -^-, 
 
 the Common Logarithm of ^i -f M will be = 
 0,434 2944 ^.XJf. and con f equ entl y jrr x y/r^, whofe 
 
 TT J T C * 
 
 Radius is Unity, and Log. Secant 
 
 Moreover, with refpeft to the Denfity of the Medium ; 
 if the abfolute Force of Gravity, in the Direction QB,
 
 of various Kinds. 
 
 be denoted by Unity, its Efficacy in the Dire&ion BN, 
 whereby the Body is accelerated, will be exprefled by 
 
 * . T, u 
 
 -, or its Equal 7 == == : Which, as the Velocity 
 
 v i + uu 
 
 is fuppofed to remain every where the fame, muft alfo 
 exprcfs the Force of the Refiftance, in the oppofite Di- 
 reclion, or the true Meafure of the required Denfity. 
 This, therefore, if M be put for the abfolute Number 
 whofe Hyperbolical Logarithm is Unity, may'be had in 
 
 Terms of *, and will be i ~M\ a : Becaufe 
 
 X 
 
 T"~ / X \ 
 
 Hyp. Log. J4\" (=-) being = Hyp. Log. V i + U u 9 
 
 we have i +uu M^ whence u M a 1 , and 
 
 confequently 
 
 VI + uu 
 
 P R O B. XXXVII. 
 
 481. Let a Line, or an inflexible Rod OP (conftdered 
 without regard to Thicknefs) be fuppofed to revolve 
 about one of its Extremes O, as a Center, with a Mo- 
 tion regulated according to any given Law j vuhiljl a 
 Ring, or Ball, carried about with it, and tending to the 
 Center O with any given Force, is fuffered to move or 
 Jlide freely along the faid Line or Rod : It is propofed to 
 determine the Velocity of the Ring, and its Prejfure upon 
 the Rod, in any propofed Pofition, together with the Nature 
 of the Curve ADL described by means of that compound 
 Motion. 
 
 Le ODP be any Pofition of the revolving Line, 
 and D the correfponding Pofition of the Body : More- 
 over, fuppofing ACK to be the Circumference of a 
 
 Circle
 
 558 
 
 Re/o/uf/on of Problems 
 / / 
 
 Circle defcribed from the Center O, through the given 
 Point A, let the Meafure of the angular Celerity ot 
 that Line, in the faid Circumference ACK, be repre- 
 
 fented by tt\ alfo let v denote the Celerity of the Ring 
 at D in the Direction DP ; and w the true Meafure of the 
 centripetal Force: Call OA, a-, OD, x ; and AC, 
 2 ; and let the given Values of u and t>, at A, be de- 
 noted by b and c refpe&ively. Then it will be, as a : 
 
 J the paracentric Velocity of the Body, at 
 
 D; whofe Square, divided by the Diftance OD, gives 
 
 A *** 
 
 Art. 2ii.__. > f or tne true Meafure of the Centrifugal Force * 
 
 a 
 
 arifing from the Revolution of the Rod : From which 
 
 the centripetal Force -w being deducted, the Remainder, 
 
 xu* 
 
 ty, is the true Force whereby the Velocity in the 
 
 Line OP is accelerated. Therefore (by Art. 2i8.J we 
 
 xu* u*xx 
 
 have w = w x x zr wx. 
 
 a a* 
 
 Moreover, becaufe the Fluxion of the Time is ex- 
 
 X J 
 
 preffed either by or by > thefe two Values muft, 
 
 therefore,
 
 of various Kinds. 559 
 
 therefore, be equal to each other, and confequently 
 v == : From which, and the preceding Equation 
 
 (when u and ^v are exhibited in Terms of x or z) the re- 
 quired Relation of v, * and z will alfo become known 
 But now, in order to determine the Action of the Rod 
 upon the Ring ; let OdP be indefinitely near to ODP, 
 interfering ADLand ACK in d and c- t and put O</=: 
 
 / 
 
 x + x. Then, becaufe a Body, acted on by no other 
 Force befides That tending to the Center, about which 
 it revolves, defcribes Areas proportional to the Times*, *Art.**4. 
 and the angular Celerity of a Ray revolving with the 
 Body, is, in that Cafe, as the Square of the Diftance 
 of the Body from the Center, inverfely (vid. Art. 47$.) 
 it follows, that, if the Rod was to ceafe to act upon the 
 Ring, at the Pofition ODP, the angular Celerity at c 9 
 
 x* ' 
 
 would then be x x w, inftead of u + u. There- 
 
 ' ** 
 
 fore the Excefs of u -f- u above ^_, -. ^ x , which is 
 
 / i 
 
 I 1UX 3/AT l . % 
 
 =: u + -- lZ -j- &c. is the Increafe of the faid an- 
 x x 
 
 gular Celerity, at the Diftance OC, arifing from the 
 of the Rod. Therefore it will be, as OC (a) : OD 
 
 / J ' \ 
 
 (x} :: the faid Increafe to ( + ^ ^- & c . 
 
 \ a a ax J 
 
 the Alteration of the Ring's paracentric Velocity, arifing 
 from the fame Caufe. Which, divided by ( - } the 
 
 . XVU 2.UV 
 
 Time wherein // is produced, gives 7 
 
 ax
 
 560 The Refolution of Problems 
 
 i 
 
 *lUT}X 
 
 - & c. for the Meafure of the Force, by which It 
 ax 
 
 is produced. From whence, by fubftituting in the 
 
 Room of , and neglecting all the Terms after the two 
 
 x 
 Art. 134- firft (in order to have the limiting Ratio *) we get 
 
 XVU , 1UV ~,, - . ... , XVU 2UV 
 
 - -f Therefore it will be, as - - 4. to 
 ax a ax a 
 
 bb xvu . 
 
 + Art. an. t, or as -- 4. -77- to Unity, fo is the Action of 
 a bbx bb 
 
 the Rod upon the Ring, to the (given) Centrifugal 
 Force at A (or the Force that would retain a Body in the 
 Circle ACK, with the Velocity b.} ^. E. I. 
 
 COROLLARY I. 
 
 482. If the angular Motion be uniform, the Equations 
 
 Pxx 
 found above, will become vv =r j- wx 9 and v = 
 
 bx 
 
 . From the latter of which, by taking the Fluxion, 
 
 we have {; = : whence (by Subftitution) -rr- 
 x zz 
 
 Fxx xz* wz* 
 
 - - wx, and confequently x -- r =. -- ri" > 
 aa a o 
 
 from the Solution of which, the Relation of # and z 
 will be given. And then, the Value of v f J being, 
 
 alfo known, the Action upon the Rod, which in this Cafe 
 
 2bv ( 2b*x\ 
 is barely =. I = J will be given hkewife, 
 
 being
 
 of various Kinds. rfi 
 
 | . V J 
 
 being to ( ) the centrifugal Force in the Circle 
 
 ACK, as to Unity. 
 z 
 
 COROLLARY II. 
 
 483. But if the Angular Celerity be proportional to 
 any Power (x m ] of the Diftance, and the Centripetal 
 Force w be, alfo, fuppofed to vary according to fome 
 
 Power (*") of the fame Diftance: Then, putting p to 
 denote the Centripetal, and q the Centrifugal, Force, at 
 the given Point A, the Value of iv will, here, be ex- 
 
 x" x m 
 
 pounded by X/>, and That of u by x: And there- 
 a a 
 
 fore, the paracentric Velocity of the Ring at D being = 
 
 x m x ( bx m **\ bb Fx+* 
 x t x I . I it will be as ; r- 
 
 V a m * l J * X a+ Z 
 
 x?, the Centrifugal Force at D *. Hence * Art, an. 
 
 * 
 
 am4-i . . * . 
 qx x px x 
 
 vv 
 
 * ; whereof the (corrected) Fluent 
 
 a~ ' a" 
 
 P* 
 
 is t w i cc = 
 
 2m + 2 
 
 ; From whence v is found = 
 
 / qa ipa 
 
 m \>s // - - 4- 
 
 *""~OTI~ f "-M
 
 562 *Ibe Refolution of Problems 
 
 Moreover, by fubftituting for w, and its Fluxion, we 
 
 xvii 2uv - bx m v 
 
 get 4- - rr m + 2 x T-, exprefiinp; the Action 
 * +' 
 
 ax a 
 
 of the Rod upon the Ring : Which, therefore, when 
 m is expounded by 2, will intirely vanifh : And, 
 in that Cafe, z will become = 
 
 expreffing the Nature of the Trajeaory defcribed by 
 means of a Centripetal Force, varying according to any 
 Power (x") of the Diftance. But this Equation will 
 be rendered fomewhat more commodious, by fubftitutino- 
 the Values of b and c : For, if OQ_ (perpendicular to 
 the Tangent at A) be denoted by b t it will be, h: 
 
 \V />*(AQJ :: b (the Celerity in the Direaion AC) 
 
 * Art - 35- to c ~ /;a = the Celerity in the Direaion AH *. 
 
 h 
 
 t Art.aii. Therefore, b being \/ aq f, we have c z rr ay, 
 
 and x 
 
 " n n+i.q n+l.qa :Jf 
 
 Which Equation is the fame, in effea, with that given 
 in Art. 242. by a different Method. 
 
 COROLLARY III. 
 
 484. If the Angular Celerity be fuppofed uniform, and 
 the Ring to have no other Motion along the Rod than 
 what it acquires from its Centrifugal Force; then f, wand 
 p being all of them equal to Nothing, %. will here be-' 
 
 bx ax 
 
 come, barely := * =- , == : And 
 
 / qx* Y x a" 
 
 \S -qa -\ 
 
 a 
 
 there"
 
 of various Kinds. 563 
 
 therefore z- a x Hyp. Log. 1, + v/ **"" aa . Hence 
 
 21 
 
 if the Number whofe Hyp. Log. is be denoted by 
 N, weJhall have x+ ^ xx aa -N: From which 
 
 * is found := a X 1 ; whence x is> alfo, had := 
 
 2 2JV 
 
 jfcAT aN Nz x f , , N x \ ~, 
 
 ~- - (becaufe -.-_). There- 
 
 2 2A/* 2 2.N N a ' 
 
 N i 
 fore, it will be (by Carol, i.) as Unity is to . 
 
 2 2JV* 
 
 fo is the Angular Velocity () in the Arch ACK to the 
 Velocity with which the Body recedes from the Center 
 of Motion : And fo, likewife, is the Centrifugal Force 
 in that Arch to half the Preffure upon the Rod By 
 
 taking z = the whole Periphery, or =. Q. X 3 .145 
 
 I3c. N will come out =. 535-5, and * 267.7 X a - 
 From whence it appears that the Diftance of the Ring 
 from the Center at the End of one intire Revolution 
 will be almoft 268 times as great as at firft. 
 
 COROLLARY IV. 
 
 485. If a Body be fuppofed to defcend from the Point O, 
 (fee the next Fig.) by the Force of its own Gravity, along 
 an inclin'd Plane OCP;'whilft the Plane itfelf moves uni- 
 formly about thatPoint, from an horizontal Pofition OEH; 
 then the Place, and the Preflure of the Body upon the 
 Plane, in any given Pofition QCP, may, alfo, be derived 
 from the Equations in Corollary i. For let CB (perpen- 
 dicular toOH) beputrr; 1 ; and let the Ratio of the Cen- 
 trifugal Force in the Circle ECK, to the Force of Gra- 
 vity (given by Art. 2 1 7.) be as r to Unity : Then, as 
 
 the Meafure of the former Force is exprefled by , 
 O o 2 That
 
 564 The Refolutlon of Problems 
 
 That of the latter muft be reprefented by : and, 
 
 ' ra 
 
 confequently, its Efficacy in the Dire&ion PO, by 
 
 ( = X 7^ ) : Which Value being fubftituted 
 raa \ ra OL' 
 
 for w t in the aforefaid Corollary, we have x 
 
 4 2. 
 
 JC%+ YX 
 
 = - . But now, in order to the Solution of this 
 
 aa raa 
 
 Equation, put the Radius OC (a} = I (that the Ope- 
 ration may be as fimple as pofiible) alfo, inftead of y^ 
 
 Si 
 2 
 
 * A^ r * et i ts Equal z -J- * &c. be fubfti- 
 
 Art. 425. 2.3 2.3.4.5 
 
 tuted, and let x be aflumed = ^a 3 + Bz* + Cz 7 -f 
 
 Then, by proceeding as is taught in Art. 267, the 
 
 i y? z 7 
 
 Value of x will come out r: -into + j-j-^-^y 
 
 &c. Whence 
 
 - 5 
 
 2 . 3 .4. 5 . 6 . 7 . 8 .9 . ip. ii 
 
 the Velocity r-*) n tne P' an e, is, alfo, found 
 
 b y? z 6 
 
 into 1- &( Which, therefore! is 
 
 r 22.3.4.5.0
 
 of various Kinds. 565 
 
 to () the angular Velocity of the Plane, in the Arch 
 
 ECK, as + - r 4- &c. to r. Moreover, 
 a 2.3.4.5.6 
 
 the Centrifugal Force in the faid Arch being denoted by 
 r {the Force of Gravity being Unity) it will likewife be 
 
 (by the above-mentioned Carol.} as I : :rr : ( - - ) 
 
 K \ * / 
 
 s6 3. 4- 5. .. . 9- .0 
 
 Force fufficient to keep the Body upon the Plane. But 
 the Force of Gravity 'in a Direction perpendicular to 
 the Plane (the Weight of the Body being jreprefented 
 
 OB z* z 4 
 
 by Unit)) is 7^, = I + - * & c . From ^ 
 
 OC 2 2.3.4 Art.45 
 
 which deducting the Quantity laft found, there refts I 
 
 7Z 1 Z 4 "?2 6 
 
 ^- + - - - - - &c. for the true Pref- 
 2 r 2.3.4 2.3.4-5-0 
 
 fure of the Body upon the Plane. By putting of Which 
 equal to Nothing, z 1 will be found rr 0.67715 ; an- 
 fwering to an Angle (EOC) of 47 : 9': Which Angle 
 is therefore the Inclination, when the Force of Gravity 
 is no longer fufficient to keep the Body upon the Plane. 
 
 Though the Value of x, given above, is found by 
 an Infinite Series, yet the Sum of that Series is eafily 
 exhibited by the Meafures of Angles and Ratios. For, 
 putting N to denote the Number whofe hyperbolical 
 Logarithm is z, 
 
 Art. 44 
 
 * 
 
 
 i 
 
 z a 
 
 1 
 
 Z J 
 
 z* 
 
 
 XT 
 
 
 
 a 
 
 cr 
 
 + 
 
 z + 
 
 2 ' 
 
 2 . 
 
 * 
 
 3 ? 
 
 3-4 
 
 * 
 
 
 
 
 P 
 
 
 
 ** 
 
 3 
 
 z 4 
 
 
 I 
 
 
 
 o 
 
 H- 
 
 I 
 
 z + 
 
 2 
 
 2 . 
 
 ~ + 
 32 
 
 3-4 
 
 fcfr. = 
 
 ]V : 
 
 
 
 Halfthc 
 
 Difference 
 
 of 
 
 which 
 
 two 
 
 Equations is 
 
 x H 
 
 
 z 3 
 
 
 z 
 
 7 
 
 
 
 A 7 " 
 
 i 
 
 
 2. 3^2.3. 
 
 4-5~ r 2 
 
 .3.4. 
 
 5 .6. 
 
 w C - 
 
 7 
 
 2 
 
 2iV 
 
 
 
 
 
 
 
 Oo 3 
 
 From
 
 566 The Refolutlon of Problems 
 
 t 
 
 From which taking s -j- 2 4. r~6 7 
 
 &V. ~ _y ; and dividing the remainder by 2r, there re- 
 
 /i ~p ? : "\ i 
 
 fultS ( X -f 7 &* i 1 * 
 
 V r 2. 3^2. 3. 4. 5. -6. 7 S ir 
 
 ~L _i_ y, for the true Value of x. Which, if 
 
 2 2N 
 
 required, may be exprefled independent of r ; by put- 
 ting ^for the Diftance through which a Body, freely, de- 
 fcends in the firft Second of Time, arid taking b to de- 
 note the Velocity of the Plane, (per Second) in the 
 Arch EC T v : For then, the Ratio of the Centrifugal 
 Force, in the faid Arch, to the Force of Gravity (or 
 
 bb f bb \ 
 
 Art. 211. That ofrte j) beins; as ( = 7^:77 I to id * , 
 
 i \ L/t-/ 
 
 bb d 
 
 we ihall have r ~ , and confequently x = - x 
 
 * _L 
 2 "" 2N y ' 
 
 By Computations, not very unlike Thofe above, the 
 Motion of the Moon's Sfpogee y and the principal Equa- 
 tions of the Lunar Orbit may be exhibited ; by means 
 of proper Approximations, derived from the general 
 Equations in Art. '481. But this is a Confideration 
 that would require a Volume of itfelf, to treat it, from 
 firft Principles, with all the Attention and Perfpicuity 
 fuitable to the Importance of the Subject. I fhall con- 
 clude this Work with the following fhcrt Table of 
 Hyperbolical Lcgarithtts^ drawn up and communicated 
 by my ingenious Friend Mr. yohn Turner: Whereof 
 the Ufe, in finding Fluents, will fufficiently appear from 
 the foregoing Pages. In the faid Table we have given the 
 Hvper'oolical Logarithms of every whole Number and 
 hundredth Part of an Unit, from i to 10 (which Form 
 is beft adapted to the Purpofes above-mentioned) by 
 Help whereof, and the following Obfervations the Hy- 
 9 perbolical
 
 of various Kinds. 567 
 
 perbolical Logarithm of any Number, not exceeding 
 feven Places of Figures, may be found with very little 
 Trouble. 
 
 1. If the Number given be between I and 10 (fo as 
 to fall within the Limits of the Table.) 
 
 Then take from it the next inferior Number in the x 
 Table, and divide the Remainder by tbfe faid inferior 
 Number increafed by half the Remainder; and let the 
 Quotient be added to the Logarithm of the faid inferior 
 Number, the Sum will be the Logarithm fought. 
 
 Thus, let the Hyperbolical Logarithm of 3.45678 
 be required i then the Operation 
 will ftand thus: 3. 45339). O056y8(. 0016442: Which 
 added to 1.2383742, the Log. of 3.45, gives 1.2400184 
 for the Logarithm fought. 
 
 2. When the Number propofed exceeds 1 0. 
 
 Find the Logarithm thereof, fuppofing all the Fi- 
 gures after the Firft to be Decimals ; then to the Lo- 
 garithm, fo found, let 2.3025851, 4.6051702, or 
 6.9077553 sV. be added, according as the whole Num- 
 ber confifts of 2, 3, or 4 &c. Places : The Sum will 
 be the Logarithm fought. 
 
 Thus, the Hyperbolical Logarithm of 345.678 will 
 be found to be 5.8451886 : For That of 3.45678 being 
 1.2400184; the fame, added to 4.6051702, gives the 
 very Quantity above exhibited. The Reafon of which, 
 as well as of the Operation in the preceding Cafe, is evi- 
 dent from the Nature and Conftruclion of Logarithms. 
 
 004
 
 568 
 
 A Table of 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 .61 
 
 .02 
 
 3 
 
 .04 
 
 .05 
 
 .0099503 
 .0198026 
 .0295588 
 .0392207 
 .0487902 
 
 
 34 
 3? 
 .36 
 
 37 
 38 
 
 .2926696 
 .3001045 
 .3074846 
 .3148107 
 .3220834 
 
 
 .67 
 .68 
 .69 
 
 .70 
 7' 
 
 .5128236 
 
 5 l8 7937 
 .5247285 
 
 ,5306282 
 5364933 
 
 .06 
 .07 
 .08 
 .09 
 
 .10 
 
 .0582689 
 .0676586 
 .0769610 
 .0861777 
 .0953102 
 
 
 39 
 .40 
 
 4 1 
 .42 
 
 43 
 
 3293037 
 
 .3364722 
 
 343^ 8 97 
 .3506568 
 
 357 6 744 
 
 
 .72 
 
 73 
 
 74 
 
 75 
 .76 
 
 .5423242 
 .5481214 
 .5538851 
 .5596157 
 .5653138 
 
 .11 
 
 .12 
 
 3 
 
 M 
 
 15 
 
 .1043600 
 1133287 
 .1222176 
 .1310283 
 .1397619 
 
 
 44 
 
 45 
 .46 
 
 47 
 .48 
 
 .3646431 
 
 37i5 6 35 
 .3784364 
 .3852624 
 ,3920420 
 
 
 77 
 .78 
 
 79 
 .80 
 .81 
 
 5709795 
 .5766133 
 5822156 
 .5877866 
 59332 6 8 
 
 .16 
 
 17 
 .18 
 .19 
 
 .20 
 
 .1484200 
 .1570037 
 .1655144 
 
 "739533 
 .1823215 
 
 
 49 
 .50 
 
 5> 
 
 5' 
 53 
 
 .3987761 
 .4054651 
 .4121096 
 .4187103 
 
 .4252677 
 
 
 ,82 
 
 .8- 
 .84 
 .85 
 
 .86 
 
 .5988365 
 .6043159 
 .6097655 
 .6151856 
 .6205764 
 
 .21 
 .22 
 
 23 
 
 .24 
 .25 
 
 .1906203 
 . 1988508 
 .2070141 
 .2151113 
 
 2231435 
 
 
 54 
 
 55 
 
 .56 
 
 57 
 . 5 b 
 
 .4317824 
 .4382549 
 .4446858 
 .45 10756 
 .4574248 
 
 
 .87 
 .8* 
 89 
 9 C 
 .91 
 
 .6259384 
 .6312717 
 .6365768 
 .6418538 
 .6471032 
 
 .26 
 .27 
 .28 
 .29 
 
 3 
 
 .2311117 
 
 .2390169 
 .2468600 
 .2546422 
 .2623642 
 
 i 
 
 5c 
 6c 
 .61 
 .62 
 .63 
 
 .4637340 
 .4700036 
 .4762341 
 .4824261 
 ,4885800 
 
 
 .92 
 93 
 94 
 95 
 .96 
 
 .6523251 
 .6575200 
 .6626879 
 .6678293 
 .6729444 
 
 3i 
 .32 
 
 33 
 
 34 
 
 .2700271 
 .2776317 
 .2851789 
 .2926696 
 
 
 .64 
 
 .65 
 
 .66 
 .67 
 
 .4946962 
 .5007752 
 .5068175 
 .5128236 
 
 
 97 
 .98 
 
 9*9 
 
 2.00 
 
 .6780335 
 .6830968 
 .6881346 
 .6931472
 
 Hyperbolical Logarithms. 
 
 569 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 2.01 
 2.02 
 2.03 
 2.04 
 2.05 
 
 .698.347 
 
 703 974 
 .7080357 
 .7129497 
 
 7178397 
 
 
 2-34 
 2.3; 
 
 2. 3 e 
 2.37 
 2.38 
 
 .8501509 
 
 8544153 
 .8586616 
 
 .8628899 
 .8671004 
 
 
 2.67 
 2.68 
 2.69 
 2.70 
 2.71 
 
 .9820784 
 .9*58167 
 .9895411 
 .9932517 
 .9969486 
 
 2.06 
 2.07 
 2.08 
 2.09 
 2.IO 
 
 .7227059 
 .7275485 
 .7323678 
 .7371640 
 7419373 
 
 
 2.39 
 
 2.40 
 2.41 
 2.42 
 
 2 -43 
 
 .8712933 
 .8754087 
 .8796267 
 .8837675 
 .8878912 
 
 
 2.72 
 
 2-73 
 2.74 
 
 2.75 
 2.76 
 
 .0006318 
 .0043015 
 .0079579 
 .0116008 
 .0152306 
 
 2. II 
 2.12 
 2.1 3 
 2.1 4 
 2.1 5 
 
 .7466879 
 .7514160 
 .7561219 
 .7608058 
 .7654678 
 
 
 2.44 
 2.45 
 2.46 
 2.47 
 2.48 
 
 .8919980 
 .8960880 
 .9001613 
 .9042181 
 .9082585 
 
 
 2.77 
 2.78 
 2.79 
 2.80 
 2.81 
 
 1.0138473 
 1.0224509 
 1.0260415 
 1.0296194 
 1.0331844 
 
 2.16 
 2.17 
 2.18 
 2.19 
 2.20 
 
 .7701082 
 .7747271 
 .7793248 
 .7839015 
 
 7884S73 
 
 
 2.49 
 2.50 
 2.51 
 2.52 
 
 2-53 
 
 .9122826 
 .9162907 
 .9202827 
 .9242589 
 .9282193 
 
 
 2.82 
 
 2.83 
 
 z. 4 
 
 2.85 
 
 7. ft 
 
 1.0367568 
 r. 0402766 
 1.0438040 
 1.047318.9 
 1.0508216 
 
 2.21 
 2.22 
 2.2 3 
 2.24 
 2.2 S 
 
 .7929925 
 .7275071 
 .8020015 
 .8064758 
 .8109302 
 
 
 2-54 
 2.55 
 2.56 
 2.57 
 2.58 
 
 .9321640 
 .9360933 
 .9460072 
 .9439058 
 
 9477 8 93 
 
 
 2.8 7 
 
 2.88 
 2.89 
 2.90 
 2.91 
 
 1.0543126 
 I.OS7790Z 
 1.061 2564 
 1.0647107 
 1.0681530 
 
 2.26 
 2.27 
 2.28 
 2.29 
 2.30 
 
 .8153648 
 .8197798 
 .8241754 
 .8285518 
 .8329091 
 
 
 2.59 
 2.60 
 
 2.6J 
 
 2.62 
 2.63 
 
 .9516578 
 
 9>5Sii4 
 
 .9593502 
 
 9 6 3!743 
 .9669838 
 
 
 2.92 
 2.93 
 2.94 
 2.95 
 
 2.96 
 
 1.0715836 
 1.0750024 
 I 0784095 
 1.0818051 
 1-0851892 
 
 2.31 
 
 2.32 
 
 2-33 
 2-34 
 
 8372475 
 .8415671 
 
 .8458682 
 .8501509 
 
 
 2.6i 
 
 2.6c 
 2.66 
 2.67 
 
 .9707789 
 
 9745S9 6 
 .9783261 
 .9820784 
 
 
 2.97 
 *. 9 
 
 2-99 
 3-nc 
 
 1.0885619 
 1.0919233 
 1.0952733 
 
 1.0986123
 
 57 
 
 A Table of 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 3.01 
 3.02 
 
 3-3 
 3.04 
 3.05 
 
 i . i o 1 9 \ oo 
 1. 101:2568 
 1.1085626 
 1.1118575 
 1.1151415 
 
 
 334 
 
 3-35 
 - < 
 3-3 G 
 
 3-37 
 
 - ,0 
 
 3-3 s 
 
 1.2059707 
 1.2089603 
 
 1.21 19409 
 1.2149127 
 1.2I78757 
 
 
 3.67 
 
 ;.6S 
 3- 6 9 
 3-7 
 3-7i 
 
 .3001916 
 .3029127 
 .3056264 
 .3083328 
 .3110318 
 
 3.06 
 3.07 
 3.08 
 
 3-09 
 3.10 
 
 1.1184149 
 1.1216775 
 1.1249395 
 1.1281710 
 1.1314021 
 
 
 3-39 
 3 4 
 3-4 
 3-42 
 3-43 
 
 1.2208299 
 1.2237754 
 1.2267122 
 1.2296405 
 1.2325605 
 
 
 3.72 
 3-73 
 3-74 
 3-75 
 3-76 
 
 .3137236 
 .3164082 
 .3190856 
 .3217558 
 .3244189 
 
 3.11 
 3.12 
 3-U 
 3-'4 
 3-'5 
 
 1.1346227 
 
 i-'37833 
 1.1410330 
 1.1442227 
 1.1474024 
 
 
 3-44 
 3-4S 
 3-46 
 3-47 
 3-48 
 
 I - 2 3547'4 
 1.2383742 
 1.2412685 
 1.2441545 
 1.2470322 
 
 
 3-77 
 3-7^ 
 
 3-79 
 3.80 
 
 3-8J 
 
 .3270749 
 .3297240 
 3323660 
 .3350010 
 .3376291 
 
 3.16 
 
 3-17 
 3-18 
 
 3-'9 
 3.20 
 
 1.1505720 
 
 1.1537315 
 1.1568811 
 1. 1600209 
 1.1631508 
 
 
 3-49 
 
 3.50 
 
 3-5 1 
 3-5 2 
 
 3-53 
 
 1.2499017 
 1.2527629 
 1.255616^ 
 1.2^84609 
 1.2612978 
 
 
 3.82 
 3.8. , 
 
 3-84 
 3-85 
 3.86 
 
 .3402504 
 .3428648 
 
 3454723 
 .3480731 
 
 .3506671 
 
 3.21 
 3.22 
 
 3- 2 3 
 3.24 
 
 3- 2 5 
 
 1.1662709 
 1.1693813 
 1.1724821 
 
 1-1755733 
 1.1786549 
 
 
 3-54 
 3-55 
 3-5 6 
 3-57 
 3-58 
 
 1.2641266 
 1.2669475 
 1.2697605 
 1.2725655 
 1.2753627 
 
 
 3.87 
 3.88 
 3-89 
 3-9 
 3 9i 
 
 353 2 544 
 3558351 
 .3584091 
 
 .3609765 
 3 6 35373 
 
 3.26 
 3-2; 
 3-25 
 3- 2 9 
 3-30 
 
 1.1817271 
 1-1847899 
 1.1878434 
 1-1908875 
 1.1939224 
 
 
 3-59 
 3.60 
 3.6, 
 3.62 
 3.65 
 
 1.2781521 
 1.2809338 
 1.2837077 
 1.2864740 
 1.2892326 
 
 
 3.92 
 
 3-93 
 3-94 
 3-95 
 3-9 6 
 
 3660916 
 .3686394 
 .3711807 
 3737156 
 .3762440 
 
 3-31 
 
 3.3? 
 
 3-3- 
 3-34 
 
 1.1969481 
 1.1999647 
 
 1.2029722 
 
 1.2059707 
 
 
 3- 6 4 
 3.65 
 3.66 
 
 3.67 
 
 1.2919836 
 1.2947271 
 (.2974631 
 1.300191^ 
 
 
 3-9? 
 3-9^ 
 3-99 
 4.00 
 
 .3787661 
 .3812818 
 
 38379 12 
 .3862943
 
 Hyperbolical Logarithms. 
 
 571 
 
 N j Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 4.01 
 
 4 .C2 
 
 403 
 
 4.04 
 4.05 
 
 4.06 
 
 4.07 
 4.08 
 
 4.09 
 4,10 
 
 1.3887912 
 11.3912818 
 1.3937663 
 1.3962446 
 i 3987168 
 
 i 401 1829 
 1.4036429 
 i 4060969 
 1.4085449 
 1.4109869 
 
 4-34 
 4-35 
 4-36 
 
 437 
 4-38 
 
 4 6 78743 
 
 .47077,3 
 .4724720 
 
 4-47630 
 -4770487 
 
 4.67 
 4.68 
 
 4- 69 
 4.70 
 471 
 
 1.5411590 
 1.5432981 
 '5454325 
 1.5475625 
 1.5496879 
 
 1.5518087 
 1.5539252 
 1.5560371 
 1.5581446 
 1.5603476 
 
 4-39 
 4-4 
 4.41 
 4.42 
 
 4-43 
 
 2.4/93 2 92 
 1.4816045 
 (.4838746 
 1.4861396 
 1.4883995 
 
 4.72 
 
 4-73 
 4-74 
 
 4-75 
 4.76 
 
 4.11 
 
 4.12 
 
 4-13 
 4.14 
 4.15 
 
 4.16 
 
 4.17 
 
 4 18 
 4.19 
 
 4.20 
 
 4.21 
 4.22 
 4.25 
 4.24 
 4.25 
 
 1.4134230 
 1.4158:31 
 1.4182774 
 1.4206957 
 1.4231083 
 
 1.4255150 
 1.4279160 
 1.4303112 
 
 1.4327007 
 1.4350845 
 
 4.44 
 
 4-45 
 
 5.46 
 
 4-47 
 4.48 
 
 ' -4906543 
 1.4.929040 
 1.4951487 
 
 1-4973*83 
 1.4996230 
 
 4-77 
 4- 7 S 
 
 4-79 
 480 
 4.81 
 
 1.5623462 
 1.5644405 
 1.5665304 
 1.5686159 
 1.5706971 
 
 4-49 
 4.50 
 4.51 
 4.52 
 4-55 
 
 1.5018527 
 1.5040774 
 1.5062971 
 1.50851 19 
 1.5107219 
 
 1.5 129269 
 ..5151272 
 1.5173226 
 1.5195132 
 1.5216990 
 
 i. 5238800 
 1.5260563 
 1.5282278 
 
 -533947 
 
 1.5325568 
 
 -5U743 
 1.5368672 
 
 '539'54 
 1.5411590 
 
 4.82 
 
 4-83 
 4.84 
 4-85 
 4.86 
 
 '5727739 
 1.5748464 
 1.5769147 
 1.5789787 
 1.5810384 
 
 1.5830939 
 1.5851452 
 1.5871923 
 1.5892352 
 1.5912739 
 
 1.4374626 
 1.43.98351 
 1.4422020 
 
 1 -4445 6 3 2 
 1.4469189 
 
 4-54 
 
 4-55 
 4.56 
 
 4->"7 
 4-58 
 
 4.87 
 4 .8 
 4.89 
 4.90 
 4.91 
 
 4.26 
 4.27 
 4.28 
 4.29 
 
 4.30 
 
 4-3* 
 4-32 
 4-33 
 4-34 
 
 1.4492691 
 1.4516138 
 
 I -453953 
 1.4562867 
 1.4586149 
 
 4-59 
 4.60 
 4.61 
 4.62 
 4-63 
 
 4.92 
 4-93 
 4-94 
 
 4-95 
 
 4-96 
 
 '593385 
 '5953389 
 '5973653 
 
 '5993875 
 1.6014057 
 
 1.6.034198 
 1.6054298 
 1.6074358 
 1.6094379 
 
 1.4609379 
 
 '4 6 325S3 
 1.4655675 
 1.4678743 
 
 4.64 
 4.65 
 4.66 
 4-67 
 
 4-97 
 4.98 
 
 4-99 
 5.00
 
 572 
 
 A fable of 
 
 N 
 
 ogarkhm 
 
 
 N 
 
 jOgarithm 
 
 
 N 
 
 logarithm 
 
 5.01 
 5.02 
 
 5-3 
 5.04 
 5.05 
 
 .6114359 
 .6134300 
 .6154200 
 .6174060 
 .6193882 
 
 
 34 
 35 
 
 3 6 
 37 
 3? 
 
 6752256 
 6770965 
 6789639 
 .6808278 
 6826882 
 
 .6845453 
 .6863989 
 .6882491 
 .6900958 
 .6919391 
 
 
 -.67 
 ;.68 
 ;.6 9 
 j. 7 o 
 
 J-7 1 
 
 5.72 
 5-73 
 5-74 
 
 5-75 
 5.76 
 
 .7351891 
 .7369512 
 .7387102 
 1.7404661 
 1.7422189 
 
 1.7439687 
 
 i-7457 I 55 
 1.7474591 
 1.7491998 
 
 i-759374 
 
 5.06 
 5.07 
 5.08 
 5.09 
 5. ic 
 
 .6213664 
 .6233408 
 .6253112 
 
 627277$ 
 .6292405 
 
 39 
 .40 
 .41 
 
 .42 
 
 43 
 
 5.11 
 5.12 
 
 5-'3 
 
 5.14 
 
 5-'5 
 
 6311994 
 
 633 J 544 
 .6351056 
 i 6370530 
 1.6389967 
 
 44 
 45 
 5-46 
 
 5-47 
 5.48 
 
 .6937790 
 .6950155 
 .6974487 
 6992786 
 1.701 1051 
 
 5-77 
 5.78 
 
 5-79 
 j.Fo 
 
 5.8! 
 
 1.7526720 
 
 i-75443 6 
 1.7561323 
 
 I -757 8 579 
 .7595805 
 
 5.16 
 
 5-i7 
 
 5.18 
 
 5.19 
 5.20 
 
 1.6409365 
 1.6428726 
 1.6448050 
 1.6467336 
 1.648658 
 
 5-49 
 5.50 
 
 5-5 
 5-5 
 
 5-5 
 
 1.7029282 
 1.7047481 
 1.7065646 
 1.7083778 
 1.7101878 
 
 5.82 
 
 5-83 
 5.84 
 5.85 
 5.86 
 
 .7613002 
 1,7630170 
 1.7647308 
 1.7664416 
 1.7681496 
 
 5.2 
 5.2 
 
 5-2 
 5.2 
 5.2 
 
 1.650579 
 1.652497 
 1.654411 
 1.656321^ 
 1.658228 
 
 5-5 
 5-5 
 S-S 
 5-5 
 v5 
 
 1.7119944 
 
 1-7*37979 
 1.715598 
 1.7173950 
 1.719188 
 
 5.87 
 5.88 
 5.89 
 5.90 
 5.91 
 
 1.7698546 
 ? -77 155 6 7 
 i-773 2 559 
 i 7749) 2 3 
 1.7766458 
 
 5- 2 
 5? 2 
 5.2 
 5.2 
 5-3C 
 
 5-3 
 5-3' 
 5-3. 
 5-3- 
 
 1.660131 
 1.662030 
 1.663926 
 1.665818 
 > 1.667706 
 
 1,669591 
 11.671473 
 
 5 I - 6 7335 I 
 ^1.67522!; 
 
 5-5 
 
 5.6 
 
 5.6 
 5.6 
 5.6 
 
 5 .6~ 
 5-6 
 5.6 
 
 .5- 6 
 
 1.720979 
 1.722766 
 1.724550 
 1.726331 
 1.728109 
 
 1.729884 
 1:731655 
 
 '733423 
 1.735189 
 
 5.9: 
 593 
 5-94 
 5-95 
 5-9< 
 
 1.7783364 
 1.7800242 
 1.7817091 
 1.7833912 
 1.7850704 
 
 1.7867469 
 1.7884205 
 1.7900914 
 
 1.7917594 
 
 5-9; 
 5 . 9 i 
 
 5-9( 
 !6.o<
 
 Hyperbolical Logarithms. 
 
 573 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 logarithm 
 
 6.01 
 
 6.02 
 
 6.03 
 6.04 
 6.05 
 
 1.7934247 
 1.7950872 
 1.7967470 
 1.7984040 
 1.8000582 
 
 
 6-34 
 6.35 
 6.36 
 6.37 
 6.38 
 
 .8468787 
 .8484547 
 .8500283 
 1.8515994 
 1.8531680 
 
 
 6.6; 
 6.6 
 6.65 
 6.70 
 6.71 
 
 .8976198 
 .8991179 
 .9006138 
 .9021078 
 .9035985 
 
 6.06 
 
 6.07 
 
 6.08 
 
 6.09 
 6.10 
 
 1.8017098 
 1.8033586 
 1.8050647 
 1.8066481 
 1.8082887 
 
 
 6-39 
 6.40 
 
 6.41 
 6.42 
 
 6-43 
 
 1.8547342 
 1.85,62979 
 1.8578592 
 1.8594181 
 1.8609745 
 
 
 6.72 
 
 6 -73 
 6.74 
 6.75 
 6.76 
 
 .9050881 
 .9065751 
 .9080600 
 .9095425 
 .91 10228 
 
 6.ii 
 
 6.12 
 
 6.13 
 6.14 
 
 6.15 
 
 1.8099267 
 1.81 15621 
 1.8131947 
 1.8148247 
 1.8164520 
 
 
 6.44 
 6.45 
 6.46 
 6.47 
 6.48 
 
 1.8625285 
 1.8640801 
 1.8656293 
 
 1.8671761 
 1.8687205 
 
 
 6.77 
 6.78 
 6.79 
 6.80 
 6.81 
 
 .9125011 
 
 .9139771 
 .9154509 
 .9169226 
 .918392*1 
 
 6.16 
 
 6.17 
 6.18 
 6.19 
 
 6.20 
 
 1.8180767 
 1.8196988 
 1.8213182 
 1.8229351 
 1.8245493 
 
 
 6.49 
 6.50 
 6.51 
 6.52 
 6.53 
 
 1.8702625 
 1.8718021 
 
 i- 8 733394 
 1.8748743 
 1.8764069 
 
 
 6.82 
 6.83 
 6.84 
 6.85 
 6.86 
 
 1.9198594 
 1.9213247 
 1.9227877 
 1.9242486 
 1.9257074 
 
 6.21 
 6.22 
 6.23 
 6.24 
 6.25 
 
 1.8261608 
 1.8277699 
 1.8293763 
 1.8309801 
 1.8325814 
 
 
 6.54 
 6.55 
 6.56 
 6.57 
 6.58 
 
 1.8779371 
 1.8794650 
 1.8809906 
 1.8825138 
 1.8840347 
 
 
 6.87 
 6.88 
 6.89 
 6.90 
 6.91 
 
 1.9271641 
 1.9286186 
 1.9300710 
 1.9315214 
 1.9329696 
 
 6.26 
 6.2 7 
 6.28 
 6.29 
 6.30 
 
 1.8341801 
 
 I-8357763 
 1.8373699 
 1.8389610 
 1.8405496 
 
 
 6.59 
 6.60 
 6.6 
 6.62 
 6.63 
 
 I-8855533 
 1.8870696 
 1.8885837 
 1.890095^ 
 1.891604^ 
 
 
 6.92 
 6.93 
 6.9, 
 6.9 
 6.96 
 
 i -934457 
 
 1.9358598 
 
 1.9373017 
 1.9387416 
 1.9401794. 
 
 6.31 
 6.32 
 
 6-33 
 6.34 
 
 1.8421356 
 
 1.8437191 
 
 1.8453002 
 1.8468; 
 
 
 6.64 
 6.65 
 6.66 
 6.6- 
 
 1.8931119 
 1.8946168 
 1.8961 19. 
 1.897,619! 
 
 
 6.97 
 6. 9 L 
 
 6. 9 r 
 
 7.0. 
 
 1.9416152 
 1.9430489 
 1.9444805 
 i.Q4;qioi
 
 574 
 
 A Table of 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 7.01 
 7.02 
 
 7-3 
 7.04 
 7.05 
 
 1-947337 6 
 1.9487632 
 
 1.9501866 
 1.9516080 
 1.9530275 
 
 
 7-34 
 
 7-35 
 7.36 
 
 7-37 
 7-38 
 
 '9933387 
 1.9947002 
 1.9960599 
 
 '9974177 
 1.9987736 
 
 
 7-67 
 7.68 
 7.69 
 7.70 
 
 2.0373166 
 2.0386195 
 2.0399207 
 2.0412203 
 2.0425181 
 
 7.06 
 7.07 
 7.08 
 7.09 
 
 1.9544449 
 1.9558604 
 
 I -957 2 739 
 1.9586853 
 i .9600947 
 
 
 7-39 
 7.40 
 7.41 
 7.42 
 7-43 
 
 2.0001278 
 2.0014800 
 2.0028305 
 2.0041790 
 2.0055258 
 
 
 7.72 
 7-73 
 7-74 
 
 7-75 
 7.76 
 
 2.0438143 
 2.0451088 
 2.0464016 
 2.0476928 
 2.0489823 
 
 7.11 
 
 7.12 
 
 7-iS 
 
 1.0615022 
 1.9629077 
 1.9643112 
 1.9657127 
 1.9671123 
 
 
 7-44 
 
 7-45 
 7.46 
 
 7-47 
 7.48 
 
 2.0068708 
 2.0082140 
 
 2 -95553 
 2.0108949 
 2.0122327 
 
 
 7-77 
 7.78 
 
 7-79 
 7.80 
 
 7.81 
 
 2.0502701 
 2.0515563 
 
 2.0528408 
 2.0541237 
 2.0554049 
 
 7.16 
 
 7.17 
 7.18 
 7.19 
 7.20 
 
 1.9685099 
 1.9699056 
 1.9712993 
 1.9726911 
 1.9740810 
 
 
 7-49 
 7.50 
 
 7-5 1 
 
 7,52 
 
 7-53 
 
 2.0135687 
 2.0149030 
 2.0162354 
 2.0175661 
 2.0188950 
 
 
 7.82 
 
 7.84 
 
 7-5 
 7.86 
 
 2.0566845 
 2.0579624 
 2.0592388 
 2:0605135 
 2.0617866 
 
 7.21 
 7.22 
 
 7- 2 3 
 7.24 
 7.25 
 
 1.9754689 
 1.9768549 
 1.9782390 
 i .9796212 
 1.9810014 
 
 
 7-54 
 
 7-55 
 7.56 
 
 7-57 
 7.58 
 
 2.O2O2221 
 2.0215475 
 2.0228711 
 2.0241929 
 2.0255131 
 
 
 7.87 
 7.88 
 7.89 
 7.90 
 7.91 
 
 2.0630580 
 2.0643278 
 2.0655961 
 2.0668627 
 2.0681277 
 
 7.26 
 7.27 
 7.28 
 7.29 
 7-3 
 
 1-^823798 
 1.9837562 
 1.9851308 
 1.9865035 
 1.9878743 
 
 
 7-59 
 
 7.60 
 7.6, 
 7.62 
 7.63 
 
 2.0268315 
 2.0281482 
 2.0294631 
 
 2.0307763 
 2.0320878 
 
 
 7-9* 
 
 7-93 
 7-94 
 
 7-95 
 7.96 
 
 2.0693911 
 2.0706530 
 2.0719132 
 2.0731719 
 2.0744290 
 
 7-32 
 7-33 
 7-34 
 
 1.9892432 
 1.9906103 
 1,9919754 
 
 i-99333 8 7 
 
 
 7-64 
 
 7.65 
 7.66 
 7.67 
 
 2.0333976 
 2.0347056 
 2.0360119 
 2.0373166 
 
 
 7-97 
 7.98 
 
 7-99 
 
 8. DO 
 
 2.0756845 
 2.0769384 
 2.0781907 
 2.0794415
 
 \Hyperbollcal Logarithms. 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 8.01 
 
 8.02 
 
 8.03 
 8.04 
 8.05 
 
 2.0806907 
 2.0819384 
 2.0831845 
 2.0844290 
 2.0856720 
 
 
 8-34 
 
 8-35 
 8.36 
 
 8-37 
 8.38 
 
 2.1210632 
 2.1222615 
 2.1234584 
 2.1246539 
 
 2.1258479 
 
 
 8.67 
 8.68 
 8.69 
 8.70 
 8.71 
 
 2.1598687 
 2.1610215 
 2.1621729 
 2.1633230 
 2.1644718 
 
 8.06 
 8.07 
 8.08 
 8.09 
 8.10 
 
 2.086.9135 
 2.0881534 
 2.0893918 
 2.0906287 
 2.0918640 
 
 
 8-39 
 8.40 
 
 8.41 
 8.42 
 
 8-43 
 
 2.1270405 
 2.1282317 
 2.1294214 
 2.1306098 
 2.1317967 
 
 
 8.72 
 
 8-73 
 8-74 
 8.75 
 8.76 
 
 2.1656192 
 2.1667653- 
 2.1679101 
 2.1690536 
 2.1701959 
 
 8.11 
 
 8.12 
 
 8.13 
 8.14 
 8.15 
 
 2,0930984 
 2.0943306 
 2.0955613 
 2.0967905 
 2.0980182 
 
 
 8.44 
 8.45 
 8.46 
 8.47 
 8.48 
 
 2.1329822 
 2.1341664 
 2 -'35349i 
 2 - I 3 6 5.34 
 2.1377104 
 
 
 8.77 
 8.78 
 8.79 
 8.80 
 8.81 
 
 2.1713367 
 2.1724763 
 2.1736146 
 2.1747517 
 2.1758874 
 
 8.16 
 
 8.17 
 8.18 
 8.19 
 
 8.20 
 
 2.0992444 
 2.1004691 
 2.1016923 
 2.1029140 
 2.1041341 
 
 
 8.49 
 8.50 
 8.51 
 8.52 
 8-53 
 
 2.1388889 
 2.1400661 
 2.1412419 
 2.1424163 
 2-1435893 
 
 
 
 8.82 
 8.83 
 8.84 
 8.85 
 8.86 
 
 2.1770218 
 2.1781550 
 2.1792868 
 2.1804174 
 2.1815467 
 
 8.21 
 8.22 
 8.2 3 
 8.24 
 8.2 5 
 
 2.1053529 
 2.1065702 
 2.1077861 
 2.1089998 
 2.1102128 
 
 
 8.54 
 8.55 
 8.56 
 8.57 
 8.58 
 
 2.1447609 
 2.1459312 
 2.1471001 
 2.1482676 
 2-H94339 
 
 
 8.87 
 L88 
 3.89 
 8.90 
 8.91 
 
 2.1826747 
 2.1838015 
 2.1849270 
 2.1860512 
 2.1871742 
 
 8.26 
 8.27 
 8.28 
 8.29 
 8.30 
 
 2.1114243 
 2.1126343 
 
 2.1 138428 
 2.1150499 
 2.1162555 
 
 
 8.59 
 8.60 
 8.61 
 8.62 
 8.65 
 
 2:1505987 
 2.1517622 
 2.1529243 
 2.i 5 : r o85! 
 2.1552445 
 
 
 3.92 
 
 8 -93 
 8.94 
 
 3.95 
 8.96 
 
 2.1882959 
 2.1894163 
 2.1905355 
 2.1916535 
 2.1927702 
 
 8. 3 J 
 8.32 
 
 8-33 
 
 8-34 
 
 2.II74596 
 2.1186622 
 2.1198634 
 2.1210632 
 
 
 8.64 
 8.6s 
 8.66 
 8.67 
 
 2.1564026 
 
 2 - I 57559? 
 2.1587147 
 2.1598687 
 
 1 
 
 8.97 
 8.98 
 -9; 
 
 ).OC 
 
 2.1938856 
 2.1949998 
 1. 1061128 
 
 2.1972245
 
 576 
 
 A Table, &c. 
 
 N 
 
 ogarithm 
 
 
 N 
 
 Logarithm 
 
 
 N 
 
 Logarithm 
 
 9.01 
 9.02 
 
 9-3 
 9.04 
 9.05 
 
 198335 
 .1994443 
 .2005523 
 .2016591 
 .2027647 
 
 
 34 
 
 35 
 .36 
 
 37 
 38 
 
 2.2343062 
 
 2 - 2 3537 6 3 
 2.2364452 
 2.2375130. 
 2.2385797 
 
 
 9 .6 7 
 9 .68 
 9 .6 9 
 9-7 
 .9-71 
 
 2.2690282 
 2.2700618 
 
 -2710944 
 2.2721258 
 2.2731562 
 
 9.06 
 9.07 
 9.08 
 9,09 
 9.10 
 
 .2038691 
 .2049722 
 .2060741 
 .2071748 
 .2082744 
 
 
 39 
 .40 
 
 .41 
 
 9.42 
 
 9-43 
 
 2- 239 6 45 2 
 
 2.2407096 
 2.2417729 
 2.2428350 
 2.2438960 
 
 - 
 
 9.72 
 9-73 
 9-74 
 
 9-75 
 9.76 
 
 2.2741856 
 2.2752138 
 2.2762411 
 2.2772673 
 
 2.2782 9 24 
 
 9.11 
 9.12 
 9.13 
 9.14 
 9.15 
 
 .2093727 
 2.2104697 
 
 2.21 15656 
 2.2126603 
 2.2137538 
 
 
 9.44 
 
 9-45 
 9.46 
 
 9-47 
 9.48 
 
 2.2449559 
 2.2460147 
 2.2470723 
 2.2481288 
 2.2491843 
 
 
 9-77 
 9.78 
 
 9-79 
 9.80 
 
 9 .8i 
 
 2.2 793 l65 
 
 2.2803395 
 2.2813614 
 2.2823823 
 2.2834022 
 
 9.16 
 9.17 
 9.18 
 9.19 
 9.20 
 
 2.2148461 
 2.2159372 
 2.2170272 
 2.2l8ll6o 
 2.219203^ 
 
 
 9-49 
 
 9-5 
 9.51 
 
 9.52 
 9-53 
 
 2.2502386 
 2.25I2 9 I7 
 2.2523438 
 
 2-2533948 
 2.2544446 
 
 
 9 .82 
 
 9-83 
 9-84 
 
 9-85 
 9 .86 
 
 2^2844211 
 2.^54389 
 2.2864556 
 
 2.2874714 
 2.2884861 
 
 9.21 
 9.22 
 9.23 
 9.24 
 9.25 
 
 2.2202898 
 2.2213750 
 2.2224590 
 2.2235418 
 2.2246235 
 
 
 9-54 
 
 9-55 
 9.56 
 
 9-57 
 9.58 
 
 2.2554934 
 2.2565411 
 
 2.2575877 
 2.2586332 
 2.2596776 
 
 
 .9.87 
 9 .88 
 
 9- 8 9 
 9.90 
 
 9.91 
 
 2.28 9 4 99 8 
 2.29105124 
 2.25)15241 
 
 2.2925347 
 2.2635443 
 
 9.26 
 9.27 
 9.28 
 9.29 
 9-3 
 
 2.2257040 
 2.2267833 
 2.227861 
 2.228938 
 2.230014- 
 
 
 9-59 
 9 .6c 
 
 9.6 
 9.6 
 9.63 
 
 2.2607209 
 2.2617631 
 2.2628042 
 2.2638442 
 2.2648832 
 
 
 9.97 
 9-93 
 9-94 
 
 9-95 
 9 . 9 6 
 
 2.2945529 
 
 2.2955604 
 2.25)65670 
 
 2-2 9 757 2 5 
 2.2985770 
 
 9-3 
 9-3 
 9-3 
 9-3 
 
 2.231089 
 2.232162 
 2.233235 
 2.234306 
 
 
 
 9.64 
 9.61; 
 9 .66 
 9.6; 
 
 2.2659211 
 
 z. 2669579 
 
 1.2679936 
 2.2690282 
 
 
 9-97 
 9.98 
 
 9-99 
 
 10. OC 
 
 2.25)5)5806 
 
 2. 300583*1 
 2.3015846 
 2.3025851 
 
 F I N I
 
 
 ^jjvi g; 
 
 ^^^^ 5p O 41 
 * % 
 
 University of California 
 
 SOUTHERN REGIONAL LIBRARY FACILITY 
 
 Return this material to the library 
 
 from which it was borrowed. 
 
 EZ 
 
 i 
 
 
 .al Lib- 
 
 OCT 04 1QQO 
 -iCEl VET
 
 SMI | MI P9J 1 
 
 y 0AavaaiB^ %Aavaan-i^ 
 
 
 ^lOS-ANCEtfj^ ^OF-CAUFOB^