This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world’s books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books are our gateways to the past, representing a wealth of history, culture and knowledge that’s often difficult to discover.
Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book’s long journey from the publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes.
+ Refrain from automated querying Do not send automated queries of any sort to Google’s system: If you are conducting research on machine translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the use of public domain materials for these purposes and may be able to help.
+ Maintain attribution The Google “watermark” you see on each file is essential for informing people about this project and helping them find additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can’t offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book’s appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyright infringement liability can be quite severe.
About Google Book Search
Google’s mission is to organize the world’s information and to make it universally accessible and useful. Google Book Search helps readers discover the world’s books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web athttp://books.google.com/
1
SO?, , S L (3
I 7 76
-
i
DOCTRINE AND
APPLICATION O P
FLUXIONS.
CONTAINING
(Besides what is common on the Subject)
A Number of New Improvements
in the THEORY,
AND
The Solution of a Variety of New, and very Interesting, Problems in different Branches of
the MATHEMATICKS.
PARTI.
By THOMAS SIMPSON, F.R. S.
THE SECOND EDITION.
Revised and carefully corrected.
LONDON:
Printed for John N our se, in the Strand, Bookseller to His MAJESTY.
MDCCLXXVI.
T 0 T H E
Right Honourable
George Earl of MacclesjkU.
MY LORD,
AS I esteem it a very great Honour to be permitted to place the following Sheets under your Lordship's Pro- tection, who are not only an Encourager of, but an Ornament to, Mathematical Learn- ui ing j I have taken more than ordinary Pains,
^ that, What is here ushered into the World, with such Advantage, may not be found al together unworthy of so distinguished a Patron.
I am not vain enough to imagine, that, to One so deeply read in these abstruse and cu rious Speculations, as your Lordfliip is uni-
A z versolly
1
versally allowed to be, this Work will appear without Faults : But then, I have the Satis faction to think, on the other hand, that, whatever is Here to be met with capable of bearing the Test of an exact and solid Judg ment, will also have its due Weight, and not fail of receiving your Lordship's Approba tion : And if, upon the Whole, there is Merit enough found to entitle me to a favourable Reception, it will gratify the highest Am bition of>
My Lord,
Tour Lordship's
#fo/i Obedient Humble Servants
Tho. Simpson,
1
PREFACE.
each succeeding Moment, are greater and greater- Therefore the Fluxion must be less than any Space that can be described, in the given Time, when the Line increases. And, in the some Manner, the Fluxion will appear to be greater than any Space that can be described, in the same Time, when the Line decreases. It must/ therefore, be equal to that Space, which will arise, when the Length of the generatingLine, from the given Position, is supposed neither to increase nor de crease : Agreeable to Art. 4,
Thus much it seem'd proper to offer Here with regard to the First Principles— I shall now proceed to soy something concerning the Order observ'd in treating, and putting together, the several Parts of the Work ; wherein the Ease and Benefit of the younger Beginner have been par ticularly consulted : To load such an One with * Multitude of Rules and Precepts, before giving him any Taste of their Use and Application, would, certainly, be very discouraging ; and like obliging a Traveller to ascend an high Mountain, without allowing him to stop by the Way, to take Breath, and refresh his Spirits with a Prospect of the agreeable and extensive View he has to expect when he arrives at the Summit : I have there fore, after demonstrating the First Principles, proceeded immediately to exemplify their Utility in several entertaining Enquiries, before touching at all upon the Inverse Method, or the more dif
ficult 1
ficult Parts of the Direct. And, since that Branch of-the Inverse Method which treats of the Com parison of Fluents is, naturally, somewhat difficult, it is referred to the Second Part of the Work, to gether with such other Matters in the Theory as might appear, either, too tedious or hard to a Learner at first setting out. The like Care has been taken in the Disposal of the rest of the Work As to the ^several Particulars whereof // is composed, I must refer to the Book itself, They being too many to be here enumerated : One Thing, however, I must not omit to take notice of, relating to that Part which treats of the aforesaid Business of Fluents: To which it may, perhaps, be objected, That, notwithstand ing my having insisted so largely on the Subject, there are a Number of Forms of Fluxions and Fluents to be met with in Authors, that I have not so much as touch'd upon. This is granted ; but then they are most of them such as, I dare pronounce, can never arise in any In quiry into Nature : And it would, doubtless, be Time and Labour mifapply'd, to swell the Work, and embarrass the Learner with a Number of un necessary Difficulties, and empty Speculations ; when what is, really, proper and useful, in the Subject, is sufficient (it is well known) to exer cise his utmost Attention and Resolution.
I Cannot put an End to this Preface without acknowledging my Obligations to a small Tract, in-
PREFACE
HAVING, in the Year 1737, published a Piece, on this fame Subject, under the Title of A "Treatise vf Fluxions (whereof the whole Impression hath been long since fold) it may be proper here, first of all, to assign the Reasons why this Work is sent abroad into the World as a New Book, rather than a Second Edition of the said Treatise. Which, in short, are these two : First, because the present Work is vastly more full and comprehensive •, and, se condly, because the principal Matters in it which are also to be met with in that Treatise, are handled in a different Manner.
Besides the, Press-Errors with which the said Treatise abounds, there are several Obscu rities and Defects (which the Author's Want of Experience, and the many Disadvantages he then laboured under, in his first .Sally, may, it is hoped, in some measure excuse.) But what is
A 3 ' now
now offered to the Publick, being a Performance of more mature Consideration and Judgment, it will, I flatter myself, be found much more correct, and claim a favourable Reception ; es pecially, as particular Care and Pains have been taken to put every Thing in a clear Light, and to oblige the lower, as well as the more expe rienced, Class of Readers.
The Notion and Explication Here given of the first Principles of Fluxions, are not essen tially different from what they are in the above- mentioned Treatise, tho' expressed in other Terms.
The Consideration of Time, which I have in troduced into the General Definition, will, per haps, be disliked by Those who would have Flux ions to be meer Velocities : But the Advantage of considering them otherwise (not as the Velocities Themselves, but the Magnitudes They . would, uniformly, generate in a given finite Time) ap pear to me sufficient 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, insensibly involved in metaphysical Difficulties : But according to our Method of conceiving and explaining the Mat ter, less Caution in the Learner is necessary, and the higher Orders of Fluxions are rendered much more easy and intelligible Besides, tho' Sir
6 Isaac
PREFACE. vii Isaac Newton defines Fluxions to be the Velocities of Motions, yet He hath Recourse to the Incre ments, or Moments, generated in equal Particles of Time, in order to determine those 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 obscure Ideas of the higher Orders of Fluxions : For if Motion in (or at) a Point be so difficult to conceive, that, Some have, even, gone so far as to dispute the very Existence of Motion, how much more perplexing must it be to form a Conception, not only, of the Velocity of a Motion, but also in infinite Changes and Af fections of in one and the fame Point, where all the Orders of Fluxions are to be considered ?
Seeing the Notion of a Fluxion, according to our Manner of defining It, supposes an uni form Motion, it may, perhaps, seem a Matter of Difficulty, at first View, how the Fluxions of Quantities, generated by Means of accelerated and retarded Motions, can be rightly assigned ; since not any, the least, Time can be taken during which the generating Celerity continues the fame : Here, indeed, we cannot express the Fluxion by any Increment or Space, atlually, generated in a given Time (as in uniform Motions.) Bur, then, we can easily determine, what the contem porary Increment, or generated Space would be%
jf the Acceleration, or Retardation, was to cease at
at the proposed Position in which the Fluxion is to be found : Whence the true Fluxion, itself, will be ohtained, without the Assistance of infinitely small Quantities, or any metaphysical Considerations.
Thus, for Example, the Motion of a Ball, de scending by the Force of its own Gravity, is con tinually accelerated ; but to have the Fluxion of the Distance fall'n thro' at any given Position of the Ball, we must find how far the Ball would, uniform ly, descend, from that Point, in a given Time, if the Gravity, or the Earth's Attraction, from thence, was to cease acting. By which Means we (hall have as clear an Idea of the Fluxion and the true Measure of the Velocity of the Ball, at any Point assigned, as in those Cases where the Motion is, actually, uniform.
Again, if a Right-line be supposed to move parallel to itself with an equable Motion, and to increase in Length, at the same Time ; the Area generated thereby, will increase with an accele rated Velocity : But the Fluxion thereof, at any given Position of the Line, will be had by taking that Part of the Increment which would., uni formly, arise, was the Length (as well as the Velocity) of the Line to continue invariable from the proposed Position. For, if the Length be supposed to increase, from the said Position, the Area generated, from thence, will be, evidently, greater than That which would uniformly arise in the same Time ; since the new Parts, produced
10 each
PREFACE.
in titled, An Explanation of Fluxions in a Short Efay en the Theory-, printed for W. Innys : Wrote by a worthy Friend of mine (who was too modest to put his Name to that, his first, Attempt) whose Manner of determining the Fluxion of a Rectangle, and illustrating the higher Order of Fluxions, I have, in particular, follows, with little qr no Variation.
Thomas Simp/ox, P. R. S> and printed for J.Nourfe.
. I. rT^HE Elements of Geometry ; with their Ap- J. - plication to fhe MenTurafio'tf 6f Superficies and Sofldi, td the DeterHrthaHtfh of the M axirna and j Mitiimo of Geometrical Quantities, and to the Coh- Itrudtion of a great Variety of Geometrical Prob- hffns, 8Vb. the third Edition, Jr.
2. Trigonometry Plain and Spherica], with the Con- ftruction and Application to Logarithms, the Second
Edition, 8vo, is. bd.
3. A Treatife of Algebra ; 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. bd.
5. The Doctrine of Annuities and Reverfions, deduced m from General and Evident Principles : with ufeful Tables, {hewing 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 molt difficult Problems of the Firft and Second Books of Newton's Principia are explained, 410, bs.
7. Mathematical Differtations on a Variety of Phyfical and Analytical Subjects, 410. 7s.
8. Mifcellaneous Tracts on fome curious and very in- terefting Subjects in Mechanics, Phyfical Aftro- nomy, and Speculative Mathematics, 4to. js.
THE
[ 1 ]
THE
Doctrine and Application o F
FLUXIONS.
P A R T the Firft.
SECTION I.
• Of the Nature, and Invejiigation, of Fluxions.
i.^^N order to form a proper Idea of the Nature of I Fluxions, all Kinds of Magnitudes are to be I confidered as generated by the continual Motion
|| of fome of their Bounds or Extremes ; as a . Line by the Motion of a Point; a Surface by the Motion of a Line ; 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 flawing Quantity would be uniformly increafcd in a given Portion of Time, with the generating Celerity at any propofed Pofition, or Inftant ( was it from thence to con tinue invariable) is the Fluxion of the faid Quantity at that Pofition, or Inftant.
B Thuf,
Thus, let the Point m be conceived to move from A, and generate the
A 7^ Tn r variable Right-
A 1 i" " line Aw, by a
Motion any how regulated j and let the Celerity thereof, when it arrives at any propofed Pofition R, be fuch as would, was it to* continue uni form from that Point, be fufficient 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.
TheFl uxion of a plane Surface js conceived in like Manner, S G by fuppofing a
I given Right-
] line mn to
j move parallel
...X. — * to itfelf, in the Plane of the parallel, and immoveable Lines AF and BG : For, if (as above) Rr be taken to exprefs the Fluxion of the Line Am, and the Rectangle RrsS be completed ; then that Rect angle, being the Space which would be uniformly de- fcribed by the generating Line mn, in the Time that Am would be uniformly increafed by wir, is therefore the Fluxion of the generated Rectangle Bm, 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 ft\U be expounded by a Rectangle (under that Line and the' Fluxion of the Abfcifla, or Bafe:) For let the cur- vilineal Space Amn be generated btf 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 alfo, be the Fluxion of the generated Space Amn: Becaufe, if the Length and Velocity of the generating Line mn were to
of FLUXIONS.
to continue invari able from the Pofi- tion RS, the Rect angle Rr.S 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 A. Pv 7*
that Pofition.
5. From what has been hitherto faid it will appear, that the Fluxions of Quantities are, always, as the Celerities By which the quantities themftlves increafe in Magnitude: Whence it will not be difficult to form a Notion of the Fluxions of Quantities otherwife generated j 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 fhew 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. Teat the Fluxion of a Quantity rcprefented by a fir.gle Letter, is ujually exprrjfed by the fame Letter with , a Dot or Full-point over it : Thus the Fluxion of * is
rcprefented by x, and that of y by j.
III. That the Fluxion of a Quantity which dtcreafes, in/lead of increafmg, is to be confidered as negative.
PRO
PROPOSITION I.
6. the Fluxion of a Quantity being given, it is proposed to find the Fluxion of any Power of that Quantity.
As a clear understanding of this Problem will be of great Importance throughout the whole Work, it may not be improper to consider it first in one or two of its most simple Cafes.
Case I. Let x express the Fluxion of x, (according to the foregoing Notation) and let the Fluxion of x%
be required.
Conceive two Points m and n to proceed, at the fame time, from two other Points A and C, along the Right-lines AB and CD, in such sort, that the Mea sure of the Distance CS (y), described by the latter, may be, always, equal to the Square of that AR (x), described by the former moving uniformly.
a : : & B
c — ' TV? «
• n, .
X f y [
Furthermore, let r, s, and R, S, be any contem porary Positions of the generating Points, and let the Lines x and y represent the respective Distances that would be uniformly described, in the same time, with the Celerities of those Points at R and S, then thole Lines will express the Fluxions of Aw and C« in this Position, (by the Definition, Art. 2 and 5).
Moreover, since C s = A ra and C S = A R1 (by Hypothesis), if Rr be denoted by v, we shall have CS (j) — *■% and C s ( = x — v]1) = x% — 2xv + v\ and consequently Ss ( = CS — Cj) = 2xv — v*; from whence we gather, that, while the Point m moves over the Distance v, the Point n moves over the Distance SMTP
^/FLUXIONS.
2xv—o*. But this last Distance (since the Square of any Quantity is known to increase faster, in Propor tion, than the Root) is not described with an uniform Motion (like the former), but an accelerated one ; and therefore is equal to, and may be taken to express, the uniform Space thai might be described with the mean Celerity at some intermediate Point e, in the same time.
Therefore, seeing the Distances that might be described, in equal times, with the uniform Celerity of w, and the mean Celerity at e, are to each other as v to ixv
—v*, or as i to 2x—v, or, lastly, as x to ixx — vxy (all which are in the fame Proportion) it is evident, that, in the time the Point m would move uniformly over the Distance x, the other Point », with its Cele rity at /, would move uniformly over the Distance ixx
—vx. , This being the Cafe, let r, R, and /, S, be now supposed to coincide, by the Arrival of the gene rating Points at R and S, then e (being always between j and S, will likewise coincide with S; and the Distance, 2xx — _i, which might be uniformly described in the aforesaid time, with the Velocity at /, (now at S), will become barely equal to ixi ; which (by the Dtfin.) is equal to (j)t the true Fluxion of Cn or x* *.
• It may, perhaps, stem inaccurate, that the Fluxions ofx and xx are compared together, and expressed both by Lines,
•when thestewing Quantities themselves, considered as a Right Line and a Square, admit of no Comparison. —~Tbis Objection
•would, indeed, be offorce, viere the Expressions refrained to a geometrical Signification ; but here our Notions are more ab- straQed and universal, not obliging us to regard •what Kind of
Extension, may be defined by this or that Exprejfion, but only the Values of the algebraic Quantities thereby signified ; to
•which the Measures of all other Quantities •whatever are ulti mately referred. And, though Quant ities of different Kindt cannot be compared viith each other, their Measures, in Num bers, may. Thus, for Instance, though it vjould be wrong to affirm, that a Square •whose Area is 9 Inches is equal to a Line of 9 Inches long, yet it is no Impropriety at all to fay the Numbers expressing their Measures, in Inches, are equal.
B 3 j.Cafi
7. Cast a. Let the Fluxion of x3 be required.
Suppose every Thing to remain as in the preceding Cafe ; only let Cn be here equal to the Cube of Am (instead of the Square).
Then, in the very fame manner, we have Si ( =CS
—Cj=*3— x—v\l) = 2x~'v— 3*v*+v* : From whence it appears, that the Distances which might be described, in the fame time, with the uniform Celerity of tn, and the mean Celerity at e, will, in this Cafe, be to each other as v to 3**© — ^xv1 + vz, or as x to 3*** —
%xvx-i-v*x : Which last Expression, when s, e, and S coincide (as before) will become 3***, the true Fluxion of x1 required.
8. Universally. Let Cn be, always, equal to Am, " i also let x— SI* (or x—v raised to the Power whose Ex ponent is n) be represented by*"— ax" 'v + tx" *vl
— ex" v*t &c. and let every Thing else be supposed as above.
Then, since Sr (*■" — x — is — ax"~ 1 v— bx" *v*
+ ex" 3wJ, &(. it is plain that the Spaces which might be described, in the same time, with the uniform Ce lerity of m, and the mean Celerity at e, will, here, be to each other as v to ax" 'v —bx" 1v*-\-cx* 3t>', fsV.
or as x to ax x —bx vx + cx v x, isc.
Therefore, all the Terms, wherein v is found, vanish ing, when s, e, and S coincide, we have ax Jx for the required Fluxion of Cn, or ar ; which Fluxion, because the numeral Co-efficient of the second Term of a Binomial involved is known to be, universally, equal to the Exponent of '.he Power, will also be truly ex pressed by t;x" 1 x. Q. E. I.
9. If the Quantity Am (or *) be generated with an accelerated, or a retarded Motic.i, instead of an uni
form
os FLUXIONS.
g
form one, the Fluxion of * (or Cn) will come out exactly the same :
For the Spaces rR and *S, actually described in the same time, being always, to each other, in the Ratio of x to ax'~'1 x — bx V*i is'c. the mean Celerities, at certain intermediate Points between r, R and j, S must, also, be in that Ratio: Which, when v vanishes (as above) will become that of x to ax" lx,[ornx* '*) the very fame as before .
PROPOSITION II.
10. To find the Fluxion of the Product or Re angle of two vat table 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 selves, and thereby gene rate the Rect angle DF. Let
the Path of their A
/ F
E
• T~
Q '/
'1 I 1
X xl
B
Intersection, or the Loci of the Angle H, be the Line BHR; also let Dd (x) and F f (y) be the Fluxions of the Sides BD (x) and BF (y), and let dm and fj, parallel to DH and FH, be drawn. Therefore, be cause the Fluxion of the Space or Area BDH is truly expreflld by the Rectangle Dm (=zy.i*) 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 the Rectangle xy = DF ( = BDH + BFH) is truly expressed by yx + xy. Q. E. I,
B 4 Ut
the same otherwise,
11. Let xy be the given Rectangle (as before) ; and put z = x + y, then z1 being = ** + 2xy + y*t we have izT — k*1— 4/ = xy. But the Fluxion of {z*— ±x%
— iy*t (and consequently that of its Equal xy) is zi
— xx —yy (by Art. 6) : Which, because z — x+y and z=x+y, is also equal to x+yXx+y—xx—y}=yx + xy.
Q. £. I.
s. Corollary r.
12. Hence the Fluxion of the Product of three va riable Quantities (put) may be derived : For, if x be put =: zu ; then yzu will become ~yx, and its Fluxion
— yi + xy (as above :) But x being = zu, and, there fore, x — zu + »i, if these Values be substituted \nyx
+ xy, it will become jr x zu + uz+zity—yzu+yuz + zuy the Fluxion of ^zk required. In like Manner the Fluxion of jryztt will appear to be xyiu + jry*a + xyzu + *j'Z«, and that of xyzuw = xyzuw + xyzuw + xyiuw + *yzaw + xyzuw.
Corollary 2. . „ . u 13. Hence, also, the Fluxion of a Fraction — may be determined. For, putting x =: — , we have w, and therefore + zx = « (<7s above) ; whence, by Transposition and Division, * = — ~=~ — ^i(by
u r xu — t-.
writing — for x) = > j— j which is the true Huxi-
1 z z
on of x, or its Equal — , the Fraction proposed.
14. Now, from the foregoing Propositions, and their subsequent Corollaries, the following practical Rules, for
1
^FLUXIONS.
for determining the Fluxions of algebraic Quantities, are obtained.
R U L E I.
To find the Fluxion of any given Power of a vari able Quantity.
Multiply the Fluxion of the Root by the Exponent of the Power, and the Product by that Power of the same Roet whose Exponent is less by Unity than the given Exponent.
This Rule is investigated in Prop. 1, and is nothing more than nx* 1 i (the Fluxion of x") expressed in
Words.
Hence the F/uxion of x3 is %x*x j that of xs is $x*x ; and that of a~+y~)7 is jj X a + y\6, (because, a being constant,,; is the true Fluxion of the Root a+y, in this Cafe).
Moreover the Fluxion of «* + ?)*, will be Ixizi X a1 -f z1!?, or 3«2 V <? + For here, * being put
■
= a1 + **, we have x = ixk, and therefore |*Ti, the Fluxion of *» (or a* + ss1]') is = 322 vV + s% as above.
RULE IL
15. To find the Fluxion of the Product of several variable Quantities multiplied together.
Multiply the Fluxion ofeach, by the Product of the rest
»f the Quantities, and the Sum of the Products thus ari sing will be the Fluxion sought *. ^
Thus the Fluxion of xy, is xy -j- yi ; that of *yz, is xyz + xzj+yzx ; and that of xyzu, is xyzu + xyuz+xzu)
+ yzux.
RULE
RULE in.
16. Tq find the Fluxion of a Fraction.
From the Fluxion of the Numerator drawn into the De nominator, subtract the Fluxion os the Denominator drawn into the Numerator, and divide the Remainder by
♦Art. i j. the Square of the Denominator *.
x yx—x'y x
Thus, the Fluxion of — is ; that of —;— , is
y j * + y
t+j\ *+;
0s 1 + ~T~> - m : 5 an<1 f° of others.
*+>]
17. In the Examples hitherto given, each is resolved by its own particular Rule ; but in those that follow, the Use of two, and sometimes of all the three, Rules is requisite.
Thus (by Rule t. and 2.) the Fluxion of is 2 V1 XX —~ %9t 99
*r*jf + ifxx ; that of -5 is — — —, (by Rult I. and 3.) and that of & is 2^+»/***«-=jg«
where all the three Rules are necessary.
When the proposed Quantity is affected by a Co-effi cient, or constant Multiplicator, the Fluxion found as above, must be multiplied by that Co-efficient or Mul tiplicator.
Thus, the Fluxion of 5*1 is 15*1*. For, the Flu xion of x3 being 3*1*, that of 5*', which is 5 times as great, must cofltequently be 5X3***, or 15***.
And, in the very fam« Manner the Fluxion of ax* will appear to be nax"~1x. Moreover, the Fluxion of
a — *
.,, or a X x1 -f f\ , will be expressed by a x
fl/FLUXIONS. ti
l aXxx +yy
«X — ixiw+jjjx/i;1] , or — 7==—,, ;
* +y I*
that of v/ J+7b or *+.yjT*, by ^ + iX±i^ X >7 _
; which , „ , n. 2*X -v — a1)1 — JMrX*— rt' ^X^ + tf
by Reduction, is zz 1 —'
x—a
_?.rx xz —az— xix x + a _2xxx — a X x + a— ix y x 4 «
* —a X *x—az] '- *• —a x V'jt1— a1
* + a x xr— 2ax cVV—a1
Having explained the Manner of confidering and de termining the firft Fluxions of variable or flowing Quan tities, it will be proper to fay fomething, now, con cerning the higher Orders, as Second, Third, Fourth,
&c. Fluxions.
1 8. The Second Fluxion r.f a Quantity is the Fluxion of the variable or algebraic Quantity exprcjjing the Firft Fluxion already defined *. By the Third Fluxion is »Art.
meant the Fluxion of the variable Quantity exprcjjing the Second: And by the Fourth, the Fluxion of the variable Quantity exprejjing ths Third Fluxion: Andfo on.
Thus, for Example, let the Line A 13 reprefent a va riable Quantity, generated by the Motion of the Point B, and l:t the (tir(t) Fluxion thereof (or the Space that might be uniformly defcribed in a given Time, with the Ceierity of B) be always exprelTcd by the Dilianct
6 • of
of the Point D from a given, or fixed Point C : Then, if the Celerity of B
g be not every where
A "—h the same; the Dis-
D tance CD, expres-
O — ——+~ — sing the Measure of
F that Celerity, must
E ■ 1 also vary, by the
G.
II Motion of D, from,
-+ or towards C, ac
cording as the Cele rity of B is an increasingor a decreasing one : And the Fluxion of the Line CD, so varying (or the Space (EF) that might be uniformly described in the aforesaid fiven Time, with the Celerity of D) is the second luxion of AB. Again, if the Motion of B be such that neither it, nor that of D, (which depends upon it) be equable, then EF, expressing the Celerity of D, will also have its Fluxion GH; which is the third Fluxion of AB, and the second Fluxion of CD.
And thus are the Fluxions of every other Order to be considered, being the Measures of the Felicities by wbieb their respective flowing Quantities, the Fluxions of the
•Ait z. precldi"g Order, are generated *.
19. Hence it appears, that a second Fluxion always shews the rate of the Increase, or Decrease, of the first Fluxion ; and that Third, Fourth, We. Fluxions, dif fer in Nothing (except their Order and Notation) from First Fluxions, being actually such to the Quantities from whence they are immediately derived ; and there fore are also determinable, in the very fame Manner, by the general Rules already delivered.
Thus, by Rule 3. the (first) Fluxion of x% is 3*** : And, if x be supposed constant, that is, if the Root x be generated with an equable Celerity, the Fluxion of 3*\r (or 3XX*1) again taken, by the fame Rule, will be ycY.ixxy or bxx% \ which therefore is the second Fluxion of xl : Whose Fluxion, found in like Sort, will be tx3, the third Fluxion of x1. Farther than which
of FLUXIONS.
which we cannot go in this Cafe, because the last Fluxion 6-*1 is here a constant Quantity.
20. In the preceding Example the Root * is supposed to be generated with an equable Celerity : But, if the Celerity be an increasing or a decreasing one, then xy expressing the Measure thereof, being variable, will also have its Fluxion ; which is usually denoted by x : Whose Fluxion, according to the same Method of No tation, is again designed by i; ; and so on, with respect to the higher Orders.
2t. Here follow a few Examples, wherein the Root r, {ot y) is supposed to be generated with a variable Celerity.
Thus, the first Fluxion of x* is 3*1* (or 3**x*).
And, if the Fluxion of 3*Ix* (considered as a Rect angle) be, again, found (by Rule 2.) we shall have 6jsjrXx+3*1xi'=6AsiI + 3*1>, for the second Fluxion of xK
Moreover, from the Fluxion last found we shall in like manner get 6*X** + 6xx.lxx + 6xxx.x -\- 3*1Xj*
(or 6*3 + iSxxx+yfji) for the third Fluxion of **.
Thus also, if y — nx* ' x, then will >=»xa— 1 x
*" x* + nxxn 1 ; and if then will 2&z~
ijr+i* '• And so of others. But, in the Solution of Problems, it will be convenient to make the first Fluxion of some one of the simple Quantities (x or y) invariable, not only to avoid Trouble, but that it may serve as a Standard to which the variable Fluxions of the other Quantities, depending thereon, may be always referred. The Reader is also desired here (once for all) to take particular Notice, that the Fluxions of all Kinds and Orders, -whatever, are contemporaneous, or such at may be generated together, with their respective Celeri ties, in one and the fame Time.
SECTION II.
On the Application of Fluxions to the Solu tion of Problems de Maximis et Mi-
NIMIS.
aa. T F a Quantity, conceived to be generated by Mo- J_ tion, increases, or decreases, till it arrives at a certain Magnitude or Position, and then, on the con trary, grows lesser or greater, and it be required to de termine the said Magnitude or Position, the Question is called a Problem de Alaximis W Minimis.
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 increasing, or decreasing, but so that it may, at a certain Position, D, become equal to that of the former Point m, moving uniformly.
This being premised, Jet the Motion of » be first considered as an in-
A Ti C V. creasirig one 5 in
<* T. , . 7 . ■ i which Cafe the Di-
7t 771 stance of n behind
m will continually increase, til! the two Points arrive at the cotemporaiy PositiorjS C and D ; but afterwards it will, again, de crease; for the Motion of », till then, being flower than at D, it is also flower than that of the preceding Point m (by Hypothesis) but becoming quicker, afterwards, than that of m, the Distance mn {as has been already said) will again decrease : And therefore is a Maximum, or the greatest of all, when the Celerities of the two Points are equal to each other.
•But, if n arrives at D with a decreasing Celerity ; then its Motion being first swifter, and afterwards flower, than that of m, the Distance mn will first decrease and then
de Maximis et Minimis.
then increase ; and therefore is a Minimum., or the least of all, in the forementioned Circumstance.
Since then the Distance mn is a Maximum or a Mi nimum, when the Velocities of m and n are equal, or when that Distance increases as fast through the Mo tion of m, as it decreases by that of n, its Fluxion at that Instant is evidently equal to Nothing *.»Art.
Therefore, as the Motion of the Points m and n may S«
be conceived such that their Distance mn may express the Measure 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 L
23. To divide a given Right-line AB into two such Parts, AC, BC, that their Product, or Rectangle, may be the greatest possible.
Put the gi
ven Line AB C
= a, and Jet A' 1 lB
the Part AC,
considered as variable (by the Motion of C from A to wards B) be denoted by x : Then BC being — a — xt we have AC X BC=<w — x* : Whose Fluxion ax —2a*
being put = o, according to the prescript, we get ax
= 2*vr, and consequently x — {a. Therefore AC and BC, in the required Circumstance, are equal to each other : Which we also know from other Principles.
EXAMPLE II.
24. To find the Fraction which Jhall exceed its Cube by the greatest Quantity possible.
Let x denote a variable Quantity, expressing Number in general ; then the Excess of x above *' being uni versally represented by x—x3, if the Fluxion thereof be taken,__£sV. we shall have* — 3**i=o; and therefore x—V\, the Fraction required.
E X-
EXAMPLE III.
25. To dttermint the greatest Rectangle that can be in*
scribed in a given Triangle.
Put the Base AC of the gi ven Triangle =
b, and its Alti tude BD = a ; and let the Alti tude (BS) of the inscribed Rect angle me (consi dered as variable) be denoted by x : Then, because of the parallel Lines AC, and ac, it will be as BD (a) : AC (£) :: DS {a—x) :
—at : Whence the Area of the Rectangle, or ac x BS
•11 i_ box—bx1 . bai— zbxx .
will be = ■ : Whose t luxion — being
a a "
(as before) put = o, we shall get x= [a. "Whence the greatest inscribed Rectangle is that whose Altitude is just half the Altitude of the Triangle.
26. It will be proper to observe here, that the Value of a Quantity, when a Maximum or Minimum, may oftentimes be determined with more Facility by taking the Fluxion of some given Part, Multiple, or Power, thereof, than from the Fluxion of the Quantity itself.
Thus, in the preceding Example, where the genera]
bax—bx% b
Expression is • ■ = — x ax—**> if the constant Multiplicator— be rejected, we shall have ax—x* ; whose Fluxion ai—2xx being put = o, we get the wry same at befort.
10 The
de Maximis & Minimis. J7 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 R is alio the greateft or leaft poflible.
EXAMPLE IV.
27. Of all right-angled plain Triangles having the fume given Hypoibenufey t» find that (ABC) whoft Area is
the greattjl.
Let AC = a, AB=*, and BC = j : Then,
** +y* being = a\ we fhaJi have y = /a1—*1, and consequently — — - y/ a1 — *x = the 2
Area of the Triangle ;
vrhofe Square — — — being, aljb, a Maximum *, •Art.iB.
4 4 » •
a xx 1
the Fluxion thereof x1* muft therefore be = 0, t ! Whence x is found = a V i , and y +Art.i«.
7fo otherwife.
Since is a Maximum, and*1 +Jf* — «*, let the Fluxions of both be taken, and you will have t*j + i.y*
=0, and lxx + iyy — o; from the former of which y
VX XX
will be =s —1 — : and from the latter, it will be = :
x y
Therefore — and — are equal to each other, and con-
* y n
fequently * S jr, (the fame as before.)
C EX-
EXAMPLE V.
*8. Of all right-angled plain Triangles containing the fame given Area, to find that whereof the Sum of the twa Legs AB + BC is the least pojftble. (See the pre ceding Figure.)
Let one Leg, AB, be denoted by x, and the Area of the Triangle by a ; then the other Leg will be dc-
2a
noted by —, and the Sum of the two Legs will be x -j- 2a
31
i whereof the Fluxion is x — ; which, put = 0, gives * (AB) = v/2«: Whence BC (— ) is also =:
V fy. Therefore the two Legs are equal to each y other.
EXAMPLE VI.
29. To determine the Dimensions of the least Isosceles Tri angle ACD that can circumscribe a given Circle.
Let the Distance (OD) of the Vertex of the Triangle from the Center of the Cir cle, be denoted by and let the remaining Part of the Perpendi cular, wbich is the Radius of the Circle, be represented by a:
Then, if OS, perpen dicular to DC, be drawn, we shall have DS = \/x%— a1 ; and therefore, since DS : OS : : DB : BC, we likewise have BC = ; which multiplied by 7+~a (BD)
7 gives
de Maxlmis & Minimis. 1 9
>*
— a for the Area of the Triangle : Which being a Minimum, its Square rhuft be a Minimum, and consequently —■—-, or its Equal —I , a Mtm-
x — a x — a
mum *Hb •» Whofe Fluxion, therefore, which is *Art.»6.
•2 —i betng put =r 0, and
the Whofe divided by — we alfo get 3 x x— a
— x + a r= o ; whence x—7a: Therefore, OD being
= 2OS, and the Triangles OX)S and BDC equiangular, it is evident that DC is Iikevrife s 2BC St AC ; and fo
^ the Triangle ACD, when the lead poflible, is equila teral.
EXAMPLE VII.
30. To determine the grtatejl Cylinder, dg, that can be mfor&ed in a given Cone ADB.
Let e~BC, the Altitude of the Cone ; t—AD, the Diameter of its Bafe ;
x=fg (dh) the Diameter of the Cylinder, con- fidered as variable ;
g-ftdgftJ*:) the Area of the Circle whole Diameter is Unity.
Then, the Areas of Circles being to one another as the Squares of their Diameters, we have, il : x* : : P '■ (P**) *c ^rea °* *ne Circle fsgr : Moreover, from the Similarity of the Triangles ABC and Adf, we have J*(AC) : tf(BC) :: 44— i* {Ad) : df="i~<1Xi which multiplied by the Area pxx (found above) gives
C 2 fabx1
20
pabx1 — pax1
' b
for the solid Content of the Cylin der: Which
being a
Maximum , its Fluxion
ipabxx
b ~
322- must
ib a
•Att.j2.be = o *, consequently and df — — ; From whence it appears, that the inscribed Cylinder will be the greatest possible, when the Altitude thereof is just
| of the Altitude of the whole Cone.
*•■• 4^ c A
EXAMPLE VIII.
31. To determine the Dimensions of a cylindric Measure ABCD, open at the Top, which Jhall contain a given Quantity (of Liquor, Grain, &c.) under the least in ternal Superficies pojfible.
Let the Diameter AB=x, and the Alti tude AD ~y ; moreover let p (3>I4I59>
denote the Periphery of the Circle whose Dia meter is Unity, and let c be the given Content of the Cylinder. Then it will be 1 :p::x : (px) the Circumference of the Base; which, multiplied by
de Maximis & Minimis.
hy the Altitude y, gives pxy for the concave Superficies of the Cylinder. In like Manner, the Area of the Base, by multiplying* the same Expression into £ of the Dia meter x, will be sound = — ; which drawn into the
4 px*y
Altitude y, gives for the solid Content of the Cy linder ; which being made = c, the concave Surface
4*
pxy will be found = —, and consequently the whole Surface = 1- — : Whereof the Fluxion, which is,
* 4
Aex pxx . .
— -—s -f —, being put — o, we shall get — Se + px*
SO} and therefore * = 2 \/T= Further, because/*1
=8s, and/**y = 4f, it follows, that *•= zy ; whence y is also known, and from which it appears, that the Dia meter of the Base must be just the Double of the Alti tude.
EXAMPLE IX.
32. Of all Cones under the fame given Superficies (s) to find that (ABD) whose Solidity is the greatest.
Let the Semi- diameter of the Base, AC = x, and the Length of the slant Side AB =y ; and let * ( as in the preceding Ex amples) denote the Periphery of the Circle whose Dia meter is Unity.
Then
Then the Circumference of the Base will be = %pxy the Area of the Base. and the convex Superficies of the Cone zz pxy, (which last is found by multiplying . half the Periphery of the Base by the Length of the slant Side) : Wherefore,, since the whole Superficies is zzpx* + pxy — it we have y zz — —• x ; whence the Alti- tkude CB (/AB1— AC1) = y/~ — ^ ; which multiplied by i of the Area, of the Base, givea
1 V
IS for the solid Content of the Cone.
3 |V P
s*x* ipsx*
Which being a Maximum, its Square — -— ~ must
... , , , 2s*xi 9psxtx
aJso be a Maximum : and therefore — — zza ;
9 9
whence j — 4/>**=o,andconsequently xzzy/fjL. From which y ( = ; x zz - ^- zz y) will like-
* \ px px px Jr'
\vise be known; and from whence it will appear that the greatest Cone under a given Surface, (pr a given Cone under the least Surface; will be when the Length of the slant Side uv, to the Semi-diameter of the Base 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 whole Diameter in the Ratio of 2 to 1.
EX-
de Maxim is & Minimi's.
EXAMPLE X.
33. T» dttermint the Position if a Right-line DE, which*
faffing through a given Point P, stall cut two Right- lints AR and AS, given by Position, in such sort that the Sam of the Segments* AD and AE, made thereby, may be the Itaji pcjjible.
Make PB, parallel to AS, — a, and PC, parallel to AR, = £ ; and let BD = * ; Then, by reason of the parallel Lines, it will be, * : a : : h' : CE — — z x Therefore AD + AE = £+*+a+— , and its Fluxion,
abx
i — , which, in the required Circumstance, being
= 0, we have x1 — ab also — o, and consequently xrz V ab ; whence the Position of DE is known^But the fame Thing may be otherwise determined, independent of Fluxions, from the general Solution of the Problem for finding the Position of DE, when the Sum of the Segments AD and AE (instead of being a Minimum) shall be equal to a given Quantity. Of which Problem, the geometrical Construction may be as follows.
C 4 Compleat
Compleat the Parallelogram ABPC (as before) and, in RA produced, take Ac = AC, and let tF be equal to the given Sum of the two Segments : Also let two Semi-circles be described upon Be and BF, and let AH, perpendicular to Be, intersect the former in H ; like wise let HK, parallel to Fc, intersect the latter in I;
draw ID perpendicular to Fc, and, through P and D draw DE ; which will be the Position required. For AB x Ac being = AH1 = DP = BD X DF, we have BD : AB : : Ac (AC) : DF; also, because of the parallel Lines, we have BD : AB : : AC : CE ; whence DF = CE, and consequently AD + AE (AD + AC + FD) is equal to cF, which Construction is more neat than that in p. 155. of my Geometry. But to shew how far this may conduce to the Matter first proposed ; we arc to observe, that, as the Problem here constructed appears to be impossible, when the Right-line HK (instead of cutting or touching) falls wholly below the Circle BWF, the least possible Value of BF (and consequently of AD + AE) must, 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 Conclusion may also be deduced from the algebraic Solution of the foresaid Problem : For, put ting b + x + a+~ (AD + AE) = s, and solving the
s—a—b > _
Equation, x will be found = + yX a fj .-^j. .
2 4
Which Equation being no longer possible than till s~a-b\
4
— ab \s — 0, we have *, in that Circumstance, = -———- 3= Vab ; still as before. In like Manner the Maxima and Minima may be determined in other Cases, by finding the Position or Circumstance wherein the general Problem begins to be impossible, (supposing the Quantity sjught to be given). But the Operation by Fluxions
de Maximis & Minimis. 25 Fluxions is, for the general Part, much more short and expeditious.
EXAMPLE XI.
34. The same being given as in the preceding Example, to determine the Position, when the Line DE, itself, is tbt least possible.
Upon AF let fall the perpendicular PQj make BQ_
—c, and, the rest, as before : Then DP1 being ( = DB^BP1— 2BQ.X DB) = + 2cx, and DB1:
DP1 :: DA1 : DE% we have ** : x^+a1— 2cx :: b + x\z : DE1 = ^x*-"*+^-I+> x j_ if ,£
#l 1 *
whose Fluxion, which is 2xXb + x X 1 — ~ + — +* x1 T f+^f x 1—— __, being put = 0, and the whole Equation divided by_2£x/ + *, there will come out I —
2c a* c a%
+£ + *x — — -7= 0; whence x1— lex1 +a*x
X X XX
+ b + x*cx—a* = O ; that is, (by Reduction) x*—cxx -\-bcx—d*bzzo : From the Resolution of which Equa tion, the Position of DE is determined.
; * Lemma.
rt, 35. Is a Body or Point ( n) be supposed to move in a Right-line AB, its absolute Celerity, in the Direction os that Line, will be to the relative Celerity, whereby it tends to, tr from, a given Point C, any where out of the Line, as the Distance Cn, is to the Distance D», intercepted by n and the Perpendicular CD ; or, as Radius to the Co-stne of the Angle of Inclination D»C.
For, putting CD — a, Dn = x, and Cn = y, • A we have + = f% and consequently 2Xx = 2yy * : iad 5.S
Whence
Solution of Problems
Art. » and 5.
A D
C
ft B
the Fluxion* of Quantities are as the Celerities of their Increase *, therefore the Truth of the Proposition is manifest.
l/-"V .. LXdUIUS ; Co-sine D«C: But,
Whence * :
CoROLtART.
It follows from hence, that the relative Celerities in any two different Directions nE and »C, are directly as the Co-sines of the corresponding Angles D»E and D»C. Therefore, when nE is perpendicular to Cnt (and the Angle DnE therefore equal to C) the Celerity in the Direction »E, will be to that in the Direction xC, as the Sine of DaC is to its Co-sine. From whence it appears, that the Celerities in the Directions D», Cn, and En (perpendicular to mC) are to each other as Cn, D«, and CD respectively.
if three Right-lines be dratvn to Jo many given Points A, Bt C, their Sum Jhall be the least pojstbU.
Let HPG be the Periphery <jf a Circle described about the Point A, as a Center, at any Distance AG ; in which let the Point P be conceived to move with an uniform Celerity, from G towards H. Then, because the relative Celerity thereof, in the Direction PC, is to that in the Direction BP produced, as the Co-sine of the Angle CPH to the Co-sine of the Angle BPG, (by the preceding Lemma) ; and, since these Celerities, when the
de Maximis & Minimis. 37 the Sum of CP and BP is a Minimum , must be equal *, • Art. »
it follows, therefore, *nd **•
that the said Angles CPH and BPG, as well as their Co-sines, will in that Circum stance become equal to each other ; and consequently A P C also equal to A P B.
From whence it ap pears, that (take AG what you will) the Sum of the three Lines, AP, BPy and CP, cannot be the least possible when the Angles A P B and A P C are unequal.
And, by the fame
Argument, it also appears that then- Sum cannot be the least possible, when the Angles BPA and BPC ace unequal : Therefore, their Sum must be the least possible, when all the three Angles about the Point P are equal to one another ; provided the Cafe will admit of such an Equality, or that no one of the Angles of the Triangle ABC is equal t»,. or greater than 4. of 4 Right Angles (for otherwise, the Point P will fall in the obtuse Angle) : Hence this
Construction.
Describe, upon BC, a Segment of a Circle, to con tain an Angle of 1200, and let the whole Circle BCQ.
be compleated ; and from A, to the Middle (QJ of the- Arch BQC, draw AQ_ intersecting the Circumference of the Circle in P ; which will be the Point required.
For, the Angles BPQ. and CPQ^, standing upon the equal Arches BQ_ and CO , have their Complements APB and APC equal to each other; and therefore, the Angle BPG being iao" (by Construction j each of the said
a8
said Angles APB, APC, will, likewise be 120 De- Aster the same Manner, it will appear that the Sum of all the Lines AP, BP, CP, &c. drawn from any Num ber of given Points A, B, C,
\3c. to meet in another Point P, will be the least possible, when the Co-fines of the Angles RPA, RPB, RPC, Ut. that the said Lines make with any other Line RS, passing through the Point of Concourse, destroy each other : Which will be when all the Angles APB, BPC, CPD, Uc. are equal, in all Cafes where the Position of the given Points will admit of such an Equality. But, if the Number of given Points be four, the required Point will be in the Intersection of the two Right-lines joining the opposite Points : For, supposing APC and BPD to be continued Right-lines, the Co- sine 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 Directions, AP and BQ_, with Celerities in a given Ratio ; it is proposed to find their Position, and how far each has gone, when they are the nearejl possible to each other.
L»t M and N be any two cotemporary Positions of the Bodies, and upon AP let fall the Perpendiculars NE and BD } also let QB be produced to meet AP grees.
in
de Maximis & Minimis.
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, and let AC, BC, and CD, (which are also given) be denoted by a, i, and c respectively, and make the variable Dis tance CN — x : Then, by reason of the parallel Lines NE and BD, we stall have b(CB):x (CN) :: c (CD) : CE = —r- . Also, because the Distances, BN and
b
AM, gone over in the fame Time, are as the Cele rities, we likewise have, n : m :: *—b (BN) : AM __ mx — mb^ ^ consequently CM (AC— AM)=u+
n
mb mx mx ... . **\ tin.
— —d , (by writing d=a+ — 1. Whence
n n n n '
MN1 ( =CM' -f CN1— CM x 2CE) will also be found
2cdx tcmx1 limx im*Xx
T x 7- 4- —— ; whose Fluxion + ——
o no n n
+ ixx-
2cd;x ismxx
^ -r —it- being made — o (because MN is to be a Minimum) we get — bdrnn+mzbx + n1bx—n\d
, , , , mnbd+n\d
+ 2mncx~o; and consequently » = . ,,—
1 ' m b + n b+2mnc gfe^lg from whence BN, AM, and MN bxrn1 + n*+2mnc
aye also given.
tbt
3<>
The fame otherwise.
Because the relative Celerities of the two Bodies, at M and N, in the Direction of the Line MN (pro- duced) are truly expressed by tjgjgf X m, and -jfe- rArt.3s. x», respectively * ; and as these Celerities, when the Distance MN is a Minimum, do become equal to each Art.i2.other f» it follows that, in this Circumstance, m : a : : Co-s. N. : Co-f. M : : Secant of M : Secant of N (by plane Trig.)
Whence this Construction. Take CH to CB in the given Ratio of m to it, and draw HB ; upon which
X
r \
\ \
X v-
/ /
//
\jff p
M m
produced (if necessary) let fall the Perpendicular AR ; draw RN parallel to AH, meeting CQ_ in N ; lastly, draw NM parallel to AR, and it will give the Position required. For, first, it is plain, because AM (RN) : BN (: : CH : CB) : : m : », that M and N are cotem- porary Positions : It is likewise plain, that RN and BN will be Secants of the Angles KNR (CMN) and KNB (CNM) to the Radius NK; because the Angle NKR ( = ARK) is a Right-one. Which Lines or Secants are in the proposed Ratio of mton, as has been already shewn.
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 n 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 nr, parallel to AP, be drawn, meeting HB pro duced in r, and let A, r be joined. Then, fince CB : CH : : B» : nr (by fan. Triangles) and CB : CH : : Bn : Am, (by Hyp.) it follows, that nr and Am, (which are parallel) will alfo be equal to each other ; and there fore Ar and mn, likewife equal and parallel. But Ar is th* leaft poflible when perpendicular to Hr. Whence the Solution is manifeft.
EXAMPLE XIV.
3S. Let the Body M move, uniformly, from A towards Qj with the Celerity m, and let another Body N pro ceed from B, at the fame time, with the Celerity n.
Now it is propoftd to find the Diretlion (BD) of the latter, fo that the Dijktnce MN of the two Bodies, when the latter arrives in the Way or Diretlion AQ_e/
the former, may be tie greateft pojjible.
Let BC be perpendicular to AQ_, and make AC r:
», BC=b, and BN=*. Therefore, if the Pofition M be iuppofed cotemporary with N, we (hall have n : m :: x : AM 2: ; whence CM = — —<7,andcon-
n n
fequently
4
Solution of Problems
sequently MN (CN— CM) = ^/~rZf-— + «;
whereof the Fluxion being taken, and made = o, we
* m mb
get , — therefore x — 1 , and CN
(vV— *') = : Whence, «:»(:: BN :
V m — n
CN :: Radius : Co-fine N. The fame Conclusion is otherwise derived, thus,
Let the Right-line BD be supposed to revolve about the Point B, as a Center, with a Motion so regulated, that the intercepted Part thereof BN may increase with the uniform Celerity n : Then, the Celerity with which
fix Radius* . . _
•Art.35.CN is increased being = ggTgj^ N , this Expression, when MN is a Maximum, must, consequently, be equal fAtt.ii.to (m) the Velocity of the other Body f M ; and there
fore m : n :: Radius : Co-sine N, as before.
EXAMPLE XV.'
39. Supposing a Ship to fail from a given Place A, in a given Direction AQ_, at the fame time that a Boat, from another given Place B, sets out in order ( if pos sible) to come up with her, and fupposmg the Rate at
which each Vessel runs to be given; it is required to find in what Direction the latter muff proceed, so that, tf it cannot come up with the former, it may, however, ap proach it as near as poJJibU.
Let the Celerity of the Ship be to that of the Boat in the given Ratio of m to n ; also let D and F be the Places of the two Veflels when nearest possible to each other, and, from the Center B, through F, suppose the Circumference of a Circle to be described. Then (the Distance DF being the least possible), the Point F must be in the Right-line (DB) joining the Point D and the Center
de Maxlmis & Minimia.
33 Center B; be
cause no other Point in the whole Periphe ry, at which the Boat from B might ar rive in the fame time, is so near to D as that wherein the Line DB inter*
sects the said
Periphery. —But now, to get an Expression for DF, in algebraic Terms, let BC be perpendicular to AQ_, and make AC =<?, BC = b, and CD=* ; and then BD (YliC'+CD1) will be = moreover, be cause m: »:: AD (a+x) : BF,youwillhaveBF= "a+n-*_
m and consequently, DF -VT+S — 22±2if.
whose Fluxion,
nb
XX nJL , being made = o, we find m
xrz --== : whence the Direction BD is known : And, if the Value of jc, thus found, be substituted in that of DF, (found above) we shall have DF = bVm1-—n%—na^ whence the Position of F is known.
m
And from which it is observable, that, as DF must be a real, positive Quantity (by the Question) this Method of Solution can only obtain when m is greater than », and b^m'—n%, also greater than na : For in all other Cases the Boat will be able to come up with the Ship.
the same otherwise.
L«t the Radius of the Circle EFH be conceived to increase uniformly, with the Celerity n, whilst the Point
D D moves
TV
D moves uniformly along AQ_, with the Celerity m : Then, the Celerity at D, in the Direction of BD pro-
. • tn.xCo.jmeD , . , .
duced, being = ; Radius— ' tnerelatlvc Celerity with which the Point D recedes from the Periphery of the said variable Circle, will be universally expressed by
— X^°a's^' "— —n> which being — o, when DF is a Minimum, we have in this Cafe mX Co-fin* D~n X Ra dius, and consequently m : n :: Radius : Co-fine D.
Therefore, if, at C, a right-angled Triangle Cbd be constituted, whose Base Cd—n, and its Hypothenuso 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 also have the Position required.
EXAMPLE XVI.
40. To determine the greatest Parabola that can bt formed by cutting a given Cone ACD.
A
>D us
Let m>, parallel to CA, be the Axis of the Parabola rvm, and rm the Base (or Oxdinate) thcresfj putting DC
de Maximis 6c Minimis. 35 DCzza, CA=b, and Dnzzx ; then, because of the parallel Lines, it will be, a : b :: *■ : —bx — nv: More-
a
over, by the Property of the Circle, we have (=«OT*=D«xCn) — ax— x\ and consequently rm -
. 2 bx
*Vax—x*i which multiplied by — x — (because eve ry Parabola is ? of a. Parallelogram of the fame Base and
&bx i
Altitude) gives — vw—* Tor the Content of the 3a
Parabola : Whose Fluxion, or that of ax1 —x* * being *Art.a6, put equal to Nothing ; we find x= — ; Whence nv~
4
| xAC, m=CDx/J, and the Area of the greatest, or required, Parabola = AC X CD X
EXAMPLE XVIL
41. To determine the greatest Ellipsis BTES that tan it formed by tutting a given Cane ABD.
Let BE be the greater, and TS the lefler, Axis of the El lipsis BTES, consider ed as variable by the Motion of (the End of the Transverse) E, along the Line AD;
moreover let Ev be parallel to AC the Axis of the Cone, meeting the Diameter BD in v, and let the Diameters EF and np be parallel to BD ; whereof the latter np is supposed to B
pass through O the Center of the Ellipsis : Then, pufc- ting AC=a, CD=£, and Cv=x, we shall have Bv=
b + x; also, because of the parallel Lines we have CD (k) : CA (a) :: Dv (b-x) : a**—* -£*; whence
b
BE (vW + E*1) = V^1 x b+x]* + «» x Furthermore, since the Triangles EO», EBD, and BOp, BEF are equiangular, and EO (=BO) =i BE, we likewise have On = lBD=*,.and tty = |EF=Cx/
=.x ; and consequently On x 0/> ( =OT% /As Pr«- p/r/y of the Circle) — bx ; whence ST = 2 Vbx, and therefore BE x ST = ggSg 1^155*.
Now the Area of any Ellipsis being in a constant Ratio to the Rectangle of its greater and lesser Axes (namely as 3,14159, £sfr. to 4; the last general Ex pression must therefore be a Maximum, when the Area is so; and therefore its Fluxion, or that of b*x x b + xV + ^xxb^x]1 (= b\v + tb'x* + Fxs + tfFx t. — ,2azbxx + a**3) equal to Nothing*; that is, A4*
+ 4b*xx + -tf-x^x + a%Vti — 4a*bxi-+ 3«Vi = 0:
Whence #l — *?'x3** a+3*, . - — — — ,3 and jr s=
2bXa1—b*+bv/a*—iv>1bI + b*—75—3- ; fromt which. the El- Sa +3"
lipsis is known.
But it is observable, that, when a*— HefP+b* is negative, this Solution fails, because the Square Root of a negative Quantity is to be extracted. Therefore, to determine the Limit, put a* — \$aW + b*=.o; then, by ordering the Equation, you will get az = b1 X 7 + V/48, and a=AX2 + /3; and therefore a : b :: 2 + v/3 : I. Hence, if the Ratio of AC to CD be not greater
de Maximis & Minimis.
greater than that of 2. + V Z to i, or (which comes to the fame thing) if the Angle DAC be not less than 15 Degrees, the Fluxion of the Ellipsis can never become equal to Nothing; but the Ellipsis itself will increase continually, from the Vertex till it coincides with the Base of the Cone ; and therefore is greater at the Base than in any other Position.
But it is further to be observed, that this Problem is confined to, yet, narrower Limits. For, either the Ellipsis will increase, continually, from the Vertex, to the Base, of the Cone, (which is (hewn to be the Cafe when the Angle DAC is greater than 15°) or else it will increase till the Point E arrives at a certain Position H, and afterwards decrease to another certain Position hy and then increase again till it coincides with the Base of the Cone, (for it must always increase again before it coincides with the Base, because, after the Point E is sot below the Perpendicular BQ_, both the Axes of the
Ellipsis increase at the same time).
ib X a*— bl +'bVai—iatazf + b* ~
tion x = ' ' 3<**+3^ ' wk°k two Roots express the two Values of x (or Cv) at the Times of the Maximum (at H) and its succeeding Minimum (at h ). Hence it is manifest, that the Ellipsis may ad mit of a Maximum between the Vertex of the Cone and the Perpendicular B Q_, and yet, that Maxi mum be less than the Base of the Cone, unless the fbresaid Angle DAC be so much less than 150 (above found) that the Increase from h to D, be less than the Decrease from H to h. Now therefore, to de termine the exact Limit, let the foresaid Increment and Decrement be supposed equal to^ach other, or.
which is the fame in Effect, set the Ellipsis BTESB
= the Circle BgDm, or BExST=BD», that is, let 44* : From which
P 3 Equa,
*■ a -n * *' 4^— b*X->-~2bx*— #' Equation you will get a = — x ——*i *
* b — x\
b1 4bl+>ibx+x% , , _ ■
= ~ x ~b—x : Moreover, 'rorn t"c Equation i4* + 4A3a-*- + T/fx1*: + —tyfbxx + ^cfx^xzzO) (gi- ven above) you will, again, get a — ,. ■ \—■■—si-—
— tr-f-^bx — yc
= . ., : Whence, by comparing theso b — xx ix —b
j„ , . .f 4**+ 3**+** 3x»
equal Va!ues,tneie arises ~ — —
which, ordered, gives x*-\-2bx — ^ = o, and therefore x—bJ2—-b.
Moreover, being = —J^~^-—» ^ b%—lbxb<;
substituted herein for, its Equal, xl, it will become
** S^ + bx S&+X _ sb + bVl — b __ 4 + b' bx — x"~ ~ 3*—b ~~ 3^/2— ib — i— —'4 + 3y/Z
4 + ^2X4+3/ 2 _22+l6/2 n .
= ■ a 5= =11 + 8/2.
—4+3/2X4+3/2 2
Hence we have, 1 : /i 1 +8 Vz : : 4 (DC) : a (AC) j : Radius to the Tangent of the Angle ADC = 780 3" : Whose Complement DAC ax 11° 57', is the least Li mit possible. Therefore, unless the Angle which the slant Side makes with the Axis be less than 1 1° 57', the greatest Ellipsis will be less than the Base of the Cone,
EXAMPLE XVIII.
42. Of all Triangles, having the fame given Perimeter, and inscribed in the same given Circle ; to determine the greatest*
Let the Diameter DA bisect the Base "BC of the re quired Triangle BEC in H, draw AE, AB and BD ; also draw AF perpendicular to BE, and GE, parallel to BC, 7
de Maximis & Minimis.
BC, meeting AD in G : Then, putting AD = a, half the given Perimeter of the Triangle = b, and DH=r; we have BH = Vay—y*. andanu therefore EF =b—^ay—y1\ More over DH (y) : AD (a)
:: DBl : DA1 EF*
a A
therefore AG (^q-) = ~^ J~^~
(AG— AH=) ^—^y-y^.j whence the Area of the Triangle BEC (BH x HG) = > -a9jf — tin
+ whose Fluxion zby •
y being put - / ucnig pui = cr,
yVey—yy
gives yVay—~yy — \ ba ; whence /, and from thence the Sides of the Triangle may be determined.
EXAMPLE XIX.
43. To determine the greatejl Area that can be contained under sour given Right-lines.
Though it is demonstrable from common Geometry that the Area will be a Maximum^ when the Trape zium ABCD, formed by the given Lines, may be in scribed in a Circle k, yet I shall here give the Solution from the Principle* of Fluxions, (whose Uses I am now
• By Prop. 13. Page 62. Elem. Trig.
* See Page 117. of Elem. Geometry.
D 4 illustrating).
illustrating. In order to which, let the Diagonal AC be drawn, and upon CB and AD let fall the Perpendi culars AE and CF ; putting AB^a, BC=£, CD=c,
DAri, BE=*, and DF = y:
Then AE being
=vV— x\ and Cff = tM—
the Area of the Trapezium (|BC x AE + iAD x CF) will be = ityer—x*
+ fcfrV— i
—jbxx yyy
yv^T* ~ = ° * *
d)'> — — — Moreover,
•Att.*z. and its Fluxion and therefore
Vc*—y
since 2bx (=AC1) =</"-+<?— 2#, by taking the Fluxion thereof, we have zbx = — ldj, or — dy=z bx ; which, substituted for — dy in the foregoing Equa^
bxy bxx y
tion, gives -; = /
■ — / VV—
X
and
and consequently, Vcz ? (CF):, (DF) :: vV — ** (AE) : * (BE) : From which it appears that the Triangles DCF and ABE are similar, and that (D + ABC being = -2 Right-angles) the Trape zium may be inscribed in a Circle ; but this by the Bye.
We are now to get an Expression for the Area in known Terms, and in order thereto we have &* + as+ibx =s dd+c%—2dy,y = —, and CF = ULi
a a
If! (because AB :BE:: DC : D¥,&c.) : Therefore, bySubstitution, i*+
a*+2bx=d1 + c% — ~ , and the Area (JBC x AE +1AD
de Maximis & Minimis.
td .
rf *AD x CF) = |£yV — x% + — / <s Z =- 2sl V^a* — x ; and therefore the Square thereof =?
4f 4sl 4
X i+-f.Xi— — . Butsince + 2 i.r = +
2<:4* , x dr + s—P—a1 x
, we have — rr — , ,—, i H = I + 4d+f-r-hx-r-a'L _ 2ab + 2cd + dd + c* — P — a* _
zab + 2cd ~~ 2ab + 2cd
7ab + 2cd 5 " 1 * ~ 2<^ + 2«/
^—-tl antj consequently, the Square of the 2a£ + 2£</
• ~ 4 2^ + 2^ * 2ab + icd
c ^--pr^~ x »+«r-^'[TwhiCn (becau(e IO
the Difference of the Squares of any two Quantities is equal to a Rectangle under their Sum and Difference) .„ ... , _ d+c+b—axd+c—b + ox b + a+a—c
will also be ' . x
■ 4
b-i a—d + c - id + ic + ib + ia_a x id+ic+ib + ia-t 4
xid+'it-fib+ia—e x if + — d. Whence it appears, that, if from i the Sum of all the four Sides each particular Side be subtracted, the continual Pro duct of the Remainders will be the Square, or second Power, of the Area.
From this Theorem, the Rule in common Practice, for finding the Area of a Triangle, having the three Sides given, is deduced, as a Corollary : For, making
