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ANALYTICAL INSTITUTIONS.
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V
ANALYTICAL INSTITUTIONS,
IN FOUR BOOKS:
ORIGINALLY WRITTEN IN ITALIAN,
BY
DONNA MARIA GAETANA AGNES I^
PROFESSOR OF THE MATHEMATICKS AND PHILOSOPHY IN
THE UNIVERSITY OF BOLOGNA,
■ TRANSLATED INTO ENGLISH
By THE LATE
REF. JOHN COLSON, M.A.F.R.S.
AND LUCA8IAN PROFESSOR OF THE MATHEMATICKS IN THE UNIVERSITY OF CAMBRIDGE.
NOW FIRST PRINTED, FROM THE TRANSLATOR'S MANUSCRIPT,
UNDER THE INSPECTION OF THE
REF. JOHN HELLINS, B.D.F.R.S,
AND VICAR OF POTTER's-PURY, IN NORTHAMPTONSHIRE.
VOLUME THE FIRST,
CONTAINING THE FIRST BOOK,
To which is prefixed,
AN INTRODUCTION BY THE TRANSLATO
LONDON:
Printed hy Taylor and Wills, Chancery-Jane
AND SOLD BY P. WINGRAVE, IN THE STRAND J F. AND C. RIVINGTON, IN
ST. Paul's church-yard; and by the booksellers
. OF OXFORD AND CAMBRIDGE.
1801.
ADVERTISEMENT
BY
THE EDITOR.
nPHE Analytical Injl'ttutions of the very learned Italian Lady, Maria
Gaetana Agneft^ Profeflbr of the Mathematicks and Philofophy in
the Univerfity of Bologna, which were publifhed in two Volumes,
Quarto, in the year 1748, are well known and juftly valued on the
Continent; and there cannot perhaps be a better recommendation of
them in this Ifland, than that they were tranflated into Englifh by that
eminent judge of Mathematical Learning, the late Reverend John
Colfony M. A. F. R. S. and Lucafian Profeflbr of the Mathematicks in
the Univerfity of Cambridge, That learned and ingenious man, who
had obliged his Country with an Englifli Tranflation of Sir Isaac
Newton's Fluxions, together with a Comment on that profound work,
in the year 1736, — and was well acquainted with what appeared on the
fame fubjed, in the courfe of fourteen years afterward, in the writings
of thofe very ingenious men, Emerfon^ Mac Laurln^ and Simpfon^-^
found, after all, the Analytical Injiitutions of Agneft to be fo excellent,
that he was at the pains of learning the Italian Language, at an advanced
age, for the fole purpofe of tranflating that work Into Englifh ; that the
Britifh Youth might have the benefit of it as well as the Youth of Italy.
This
080
yi ADVERTISEMENT BY THE EDITOR.
This great defign he lived to accomplifh ; and had adually tranfcribed
a fair copy of his Tranflation for the prefs, and begun to draw up propofals
for printing it by fubfcription. And, in order to render it more eafy and
ufeful to the Ladies of this Country, (if indeed they can be prevailed upon
by his perfuafion and encouragement, to fhow to the world, as they eafily
might, that they are not to be excelled by any foreign Ladies whatever,
in any valuable accomplifhment,) he had defigned and begun a popular
account of this work, under the title of Tbe Plan of the Ladys Syjiem
of Analytic ks ; explaining, article by article, what was contained in it.
But this he did not live long enough to finifh, nor indeed to give more
than. a rough draught of it fo far as article 256 of the firft Book.
In this ftate the Manufcript remained many years ; and, confidering
the great expenfe which, in the prefent times, attends the printing of
fuch a work, probably might have remained many more, had it not been
for the adive and liberal fpirit of Mr. Baron Maseres ; who, whether
we confider his own ingenious and extenfive labours in the Mathe-
maticks, or the encouragement which he gives to others who employ
their talents in that way, well deferves what Sir Isaac Newton faid of
Mr. Collins^ the great encourager of Mathematical Learning in his time
— Vir In Rem Mathematicam promovendam natus *. But this commen-
dation is far fhort of the deferts of the Patron of this Work. While he
fets a due value upon Arts and Sciences^ he is highly fenfible of the
much greater importance of Revealed Religion, and well-conjlituted
Government y to the happinefs of mankind ; and is no lefs pious and
loyal than he is learned and liberal. To the truth of thefe aflertions
every one who is acquainted with him will readily bear teftimony ; and
they might be fupported likewife by pafiages from various Books which
• See Comm. Epiftol. Edit. 1722, p. 148.
are
ADVERTISEMENT BY THE EDITOR, VH
are well known to be productions of his pen, although fome of them
bear not his name. But I forbear quotations from his works in this
place, that I may not, on the one hand, hurt the modefty of a Friend,
nor, on the other, give occafion to the captious and malevolent to fay I
offer incenfe to my Patron,
When the Baron had refolved to bear the whole of the expenfe of a
handfome Edition of thefe Injlitutions^ he was pleafed to defire me to
fuperintend the printing of them ; to which I readily confented, ia
confequence of favours received from him, and with the hope that I
tpight render fome little fervice to the readers of this work, by taking
care that it fhould be correctly printed, which is a matter that requires
more time and attention than moft are aware of, who have not experi-
enced it.
But, befides correcting the errors of the prefs, it was neceflary to
corred many little flips of the pen, and inaccuracies, which I found in
the Copy. For, notwithftanding it was fairly tranfcribed for the prefs in
Mr. Colfons own hand-writing, it had evidently been written in hafl:e,
and wanted revifion ; and undoubtedly would have received it from
him, if he had lived to fuperintend the printing of it himfelf. Of
thefe inaccuracies, a few were in the language, but more in the
mathematical part, where, although I feldom found any wrong con-
clufion, 1 found many miftakes in the figns and exponents of quan-
tities, as well as omiffions of numbers and quantities, and fometimes
of whole claufes. Some of thefe miflakes I was enabled to corrcCt by
means of the foul fheets on which the Tranflation was firfl written ;
but finding errors in them alfo, (fome of which, I doubt not, were
occafioned by prefs errors in the original, a copy of which I could never
obtain),
VIII ADVEHTISEMENT BY THE EDITOR.
obtain *,) I faw no way of fatisfying myfelf, but to undertake tbe
labour, great as it was, of examining and recomputing every operation
in which I fufpeded or difcovered any error: and this was frequently
the cafe in the fecond Volume. In fhort, my endeavour has been to
prefent this Tranflation to the Public faithfully as the worthy old Pro-
feflbr made it, and would have rendered it, if he had lived to pubiifh
it; altering nothing in it but the miflakes btfcre mentioned, nor infert-
ing any thing of my own but what is included within thefe marks [ ].
With refpe<Sl to the ftyle of this Tranflation, fome of the fentences,
no doubt, might have been better turned ; yet the meaning is, in
general, plain enough, which is all that is requifite in books of this kind.
It has been mentioned above, that the Introdudion was left unfi-
nifhed by Profeflbr Colfon : I have continued it to the end of the firft
Volume ; diftinguifhing what I have written from what was found in
his Manufcript by putting it in brackets.
It appears by a paflage in the Manufcript of the Introdudion, that
Mr. Col/on intended to make fome additions to this Work j but what
thefe additions were to be is not mentioned. Yet I conjedure that they
were to be fome eafy pleafant Queftions, with their Solutions, in the
manner which he has fhown in Sed. VI. of his Comment on Sir Isaac
Newton's Fluxions ; merely to exercife the learner in the rules given
in thefe Inftitutions, and not to contain any new rules, or additional
matter ; for he has called this Work of ^gneji, A Complete Syjiem of
Analyticks f. And finding a fhort Paper of this kind in his hand-
writing, 1 have inferted it at the end of the fecond Volume,
* In the year 1 7991 1 employed two days in making inquiiies amongft the bookfellers of
Jjondon, from one end of the city to the other, for a Copy of tiie Original, without fuccefs.
f See the Introdudion, p. i.
That
ADVERTISEMENTBYTHEEDITOR. IX
That thefe Injtitutions^ confidering the great quantity of valuable
matter contained in them, the judicious manner in which it is arranged,
and the perfpicuity with which it is explained, will be efleemed, by all
candid judges, as the moft valuable work of the kind that has appeared
in our language, need not be doubted. Inftances of the fuperiour fkill
of the Author may be found in various parts of her Work, more
efpecially in the Fourth Book, where it appears in the cohftrudion of
fome fluxionary equations without a feparation of the variable quantities,
— in the feparation of the variable quantities in others, — and in the
reduction of others in which there are fecond and third- fluxions to
equations having firfl: fluxions only. A Angle infl:ance of her great
flcill may ferve to gratify the reader, and, for the fake of brevity, is all
that I fliall produce in this place. It is taken' from the beginning of
the fifth Article of the firfl: Sedion of the Fourth Book", where fhc
fhows that the equation of the fluents of y^y = x^y -\r yx ^ x is
fny V = ii^y^ ± h ; which, by only writing x for /and y for ^,
is the folution of the equation y^x + xf^^y = /^ ; from which the
folution of the equation — + ^-~ =z ^ ^ is moft: eafily obtained.
This equation is taken from page 289 of the fecond Volume of
Simpfons Fluxions, (publiflied in the year 1750,) who has there
expreflTed his opinion, That the only cafe in which this equation admits
of a folution " by multiplying, or dividing it, by fome power or produd
of the quantities concerned," is, when « = i : whereas Agnefi has given
a general folution by that method *. What is here faid is only to
* I am aware that a folution of this equation has, of late, been given by feveral
ingenious perfons of this Country ; which, however, fome of them may fee reafon to
revife.
Vol. I. » prove
X ADVERTISEMENT BY THE EDITOR.
prove the great Ikill of Signora Agneft^ and not with any intent to
leflen the reputation of Mr. Simpfon ; for whofe memory and abilities I
have the higheft refpedt, efteeming him as one of the greateft Mathe-
matical Geniuses that this Country has produced fmce the time of
Sir Isaac Newton.
It may perhaps be objected to thefe Injlitut'ions^ That there are a
number of Mechanical and Phyfical Problems to be met with, in fome
Treatifes of Fluxions in our language, which are not found here. The
anfwer is, That fuch Problems are properly placed in Treatifes of Ma-
thematical Fhilofophy; but, as the folutions of them require a knowledge
of Mechanicks, and Natural Philofophy, they could not, with any more
propriety, be admitted into an Elementary Treatife of Fluxions, than
the Problems of meafuring Land, or of taking Heights and Diftances,
could be admitted into EiiclicCs Elements of Geometry.
But here I would not be underftood to infmuate that thefe Injlttutlons
are fo perfect as to admit of neither improvement nor addition : on
the contrary, I have obferved that fome of the inveftigations might be
made in a fimpler manner ; and that the Methods ofjifiding the Roots
of numerical Equations by Approximation^ — Of folving literal and
fuxionary Equations by infinite Series^ — and Of comparing together homo-
geneous Fluents^ are wanting in them ; all which might be contained in
a few fheets, and which, if added to this Work, would fave the learner
the expenfe of money and time in procuring and reading a number of
books on thefe fubjeds. Thefe Methods therefore, together with Notes
on feveral parts of the Work, I purpofe to draw up, under the title of
A Sup-
ADVERTISEMENT BY THE EDITOR. XI
A Supplement to Maria JlgJieJts Analytical Injlitutions ; to be printed
with the fame type, and on the fame kind of paper, as this Work ; if
heakh and leifure (hould permit, and if it fhould appear to be defired
by Mathematical Readers.
The wonderful fagacity which appears in thefe Injlltutions, and the
fingular circumftance that fo large a work of this kind was performed
by a Lady, raifed in me a wlfh to obtain a particular account of the
Author ; but the confufion and mifery which have been brought upon a
great part of Europe, and particularly upon Italy, by the French Revo-
lution, have deprived me of the means of getting authentic information
refpeding this Phenomenon of Literature from the Univerfity of
Bologna^ of which fhe was once fo bright an ornament. All the
information I have been abk to get of her, (befidcs what appears in
her excellent Work, and fome juft encomiums on her Ikill which I
have feen in foreign books,) I have inferted in the following page^ ;
fuppofing that the reader would be no lefs defirous than myfelf of any
authentic information refpeding fo amiable and fo extraordinary a
perfon. The account comes, indeed, by way of France ; yet, as there
is no vifible motive for the writers of it to deviate from truth ia
what they have related of her, I fee no reafon for difbelieving it.
I have alfo inferted the Teftimony given by Dr. Saunderfon to the
great genius and fkill of Mr. Colfon ; conceiving that it might prove
tifeful information to the junior readers of thefe Injiitutions.
I have only to requefl of the candid reader that, if, notwithftanding
B 2 the
Xil ADVERTISEMENT BY THE EDITOR.
the care I have taken in correding the prefs for this Work, any errors
have efcaped me, (and in printing a work of this kind it is hardly
poffible but fome will efcape unnoticed,) he will corred them himfelf,
and kindly excufe the omiffion.
John Hellins.
Potter*s-Purjr,
September 29th, i8oi.
SOME
SOME ACCOUNT OF MARIA AGNESIy
THE AUTHOR OF THESE ANALYTICAL INSTITUTIONS.
IN the Appendix to the XXXIIId Volume of the Monthly Review,
pages 516 and 517, is an Account of Maria Agnefi^ taken from one of
M. De Brojfes' Letters on Italy, which is nearly the fame in fubftance,
but not in perfpicuity, with what is here printed.
* Letter X. — The account given by Monfieur De Brofles, in the loth Leuer,
of a kind of literary phasnomenon that he met vviih in this journey, is io
remarkable that we cannot avoid tranfcribing ir. This vi^as a young lady of
Milan, about eighteen or twenty years of age, named la Sigmrina Agnefi^ whom
he calls a walking Polyglott, and who, not content with knowing all the oriental
languages, underiook to maintain a ^hefis in any of the fciences againll any
one who fhould choofe to difpute upon it with her. At a Converfatione to which
our traveller [Monfieur De Brolfes] and his nephew were invited, they found
about thirty perfons, of feveral different nations of Europe, lilting in a circle,
and la Signorina Agneji, with her little fifter, feated under a canopy. She could
hardly be reckoned handfome ; but (he had a fine complexion, and an air of
great fimplicity, foftnefs, and feminine delicacy.*
" I had conceived (fays the Prefident *,) when I went to this converfation-
party, that it was only to converfe with this young lady in the ufual way, though
on learned fubjedlsj but, inftead of this. Count Belloni (who had introduced
me to it,) made a fine harangue to the lady in Laiin, with the formality of a
college-declamation. She anfwered with great readinefs and ability in the fame^t
language ; and they then entered into a difputation, dill in the fame language
on the origin of fountains and on the caufes of the ebbing and flowing which is
obferved in fome of them, like the tides in the fea. She fpokc like an angel
on this fubjeft ; and I never heard it treated in a manner that gave me more
fatisfadion. Count Belloni then defired me to enter with her on the difcuflion
of any other fubjefl I ihould choofe to pitch upon, provided that it related to
Mathematicks or Natural Philofophy. This propofal alarmed me a good deal,
* M. De Brojfes was firft Prefident of the Parliament of Dijon, and Member of the
Royal Academy of Infcriptions and Belles Lettres of Paris. According to the Monthly
Reviewer, he travelled in Italy about the year f 740 : from which It follows that Agneji
was about 28 years of age when her Analytital InJUtutions were publilhed.
as
XIV ^OMU AC C OV N r OT MA RI J JG NES t,
as I found it was expeded that I fhould hold a converfation in the Latin
language, with which I had no longer that familiar acquaintance and readinefs
at fpeaking if, which in the days of my youthful ftudies 1 had formerly poflefled.
However, I made the lady the beft excufes I could for my want of fufficient
fkill in the Latin language to make me worthy of converfing in it with her,
and hoped fhe would over-look the incorred expreffions I might happen to
make ufe of in the courfe of the difcufllon ; and we then entered, firft, into an
inquiry concerning the manner in which the foul receives impreffions from
corporeal objects, and in which thofe impreffions are communicated from the
eyes, and ears, and other parts of the body on which they are firft made, to
the organs of the brain, which is the general Jenjoriuniy or place in which the
foul receives them ; and we afterwards difputed on the propagation of light
and the prifmatick colours. Loppin- then difcourfed with her on tranjparent
bodies, and on curvilinear figures in Geometry, of which lafl fubje6t I did not
underftand a word. Loppin fpoke in French ; and the lady begged to be
permitted to anfwer him in Latin, fearing that fhe Qiould not be able to
recoiled the proper French technical names of the feveral fubjeds which they
(hould have occafion to confider.
" She fpoke wonderfully well on all thefe fubje(^s, though (he could not
have been prepared before-hand to fpeak upon them, any more than we were.
She is much attached to the Philofophy of Sir Isaac Newton : and it is
marvellous to fee a perfon of her age fo converfant with fuch abftrufe fubjeds.
Yet, however much I may have been furprized at the extent and depth of her
knowledge, I have been much more amazed fo hear her fpeak Latin (a language
1/ which (he certainly could not ohen have occafion to make ufe of,) with fuch
I "purity, eafe, and accuracy that I do not recoiled to have ever read any book
in modern Latin that was written in fo claffical a ftyle as that in which (he
pronounced thefe difcourfes. After (he had replied to Loppin, the converfation
/' Ijbecame general, every one fpeaking to her in the language of his own country,
■and (he anfwering him in the fame I'anguage : for her knowledge of languages is
i .' prodigious. She then told me that (he was forry that the converfation at this vifit
had taken that formal turn of an Academical Difputation, declaring that (he very
much difliked fpeaking on fuch fubjeds in numerous companies ; where, for one
perfon who received amufement from the difcuflion of them, there were often
twenty who were tired to death by it ; and that therefore fuch fubjeds were only
fit to be (jntered-upon in fmall companies of two or three perfons, who had all
the
SOME ACCOUNT OF M yl R I A J G N E S 1, XV
the fame tafte for difcufllng them. This obfervation, I thought, was very juft,
and was a proof of the fame good fenfe and difcernment which ha^d appeared
in her former learned difcourfes. I was forry to hear that (he was determined
to go into a Convent, and take the veil : which was not from want of fortune, '^^
(for {lie is rich,) but from a religious and devout turn of mind, which difpofes J
her to tliun the pleafures and vanities of the world. After the converfation was '
finidied, her little fifter played on the harpfichord, with the fkill of a Rameau, -^
firfl:, fome of Rameau's pieces of mufic, and then fome pieces of her own
corapofition, and concluded by finging fome airs and accompanying her voice
on the inftrument,"
M. Montucla fpeaks of Maria Agnefi^ and of her Analytical Injii^
tutions, to the tollowing effed, in his Hijioire des Mathematiquesj Vo-
lume II, page 171.
" Befides the foregoing Authors I ought to mention on this occafion, with
much commendation, the Analytical Injiitutions of a learned Italian lady of the
name of Maria Gaetana Jgneft, which is a work of fuch merit that fome female
mathematician of France (for we alfo have fome ladies of that defcription
among us,) would have done well to give us a French tranflation of it. We
cannot behold without the greateft aftonilhment a perfon of a fex that feems fa
lictle fitted to tread the thorny paths of thefe abftrafl fciences, penetrate fo
deeply as (he has done into all the branches of Algebra, both the common and
the tranfcendental, or infinitefimal. She has fmce retired to a cloifter : and,
though we do not prefume to cenfure her condud in this ftep, (which we muft
fuppofe to proceed from the pureft and fincereft piety,) we cannot but lament
that (lie Ihould have thus deprived the learned world of the ufeful improvements
in Literature which her genius and knowledge would have enabled her to
communicate to it, not only on fubjeds of a mathematical nature, but on many
others of a different kind, in which (he had become eminent."
In the Index to the Volume above mentioned, M. Montucla^ at the
name Agnefi^ refers alfo to the third Volume of his work, which is not
yet publifhed.
Maria Agnefi and her Analytical Injiitutions are mentioned alfo in a
note in page 179 of a work intitled " An EJfay on the Learnings Genius y
and Abilities of the Vair-Sex : proving them Not Inferior to Man, from a
Variety of Examples ^ extraEled from Ant lent and Modern Hiflory. l^ranf-
6 lated
XVI DR. SAUNDERSON's TESTIMONY OF MR. COLSON.
lated from the Span'ijh of ' El Theatro Critko.^ London 1774.'* What
is there faid of her is to the effedt following :
" A learned Italian lady of our own times is Signora Agnefi *, daughter of a
<;reditable tradefman in Milan, famed throughout all Europe for her knowledge
of the learned languages and for being the author of a profound treatife of
Algebra, \um\tdL Analytical ItjflituHons, which, befides many eulogiums beftowed
on her by feveral Scientifical Societies, has gained her a Profefforihip of Mathe-
maticks in the Univerfity of Bologna. Neither her inclination to thefe favourite
inteiledual purfuits, nor a defire of preferving and increafing the fame the had
acquired by her attainments in them, nor the intreaties of her father have been
able to ftifle the call from heaven which (he conceives herfelf to have felt in
her child-hood to dedicate herfelf to a monaftick life amongft the nuns known
by the name of The Blue Nuns, than which there are few orders in the Church
of Rome fubjed: to rules of greater feverity. Since her father's death flie has
given herfelf up to the mod fublime devotion, and has facrificed to chriftian
^If-denial all thofe enjoyments in the fociety of the world to which her fine
qualities and literary attainments had already introduced her amongft the moft
refpedable part of mankind."
DR. SAUNDERSON'S TESTIMONY OF THE GENIUS OF MR. COLSON.
Dr. Nicholas Saunderson, Lucajtan ProfefTor of the Mathematicks
in the Univerfity of Cambridge, and Fellow of the Royal Society, fpeaking
of Mr. Colfon in his Algebra, Vol. II. p. 720, has thefe words :
— " The learned Mr. John Colfon, a gentleman whofe great genius and known
abilities in thefe fciences I fhall always have in the higheft admiration and efteem.'*
Mr. De Moivre alfo has, on feveral occafions, fpoken of the great fkill of
Mr. Colfon ; but, for want of books, I cannot quote his words. However,
Dr. Saunderfon s Teftimony, and the office which Mr. Colfon afterward
held in the Univerfity of Cambridge^ are fufficient vouchers of his ability,
* in the Note above referred to, which feems to be a bad tranflation of a paflage in a
book intitled * Ohfervations fur Vltalie^ 6cc,' her name is erroneoufly printed Anglefe. I
have therefore given the fame Account in better Englifh, as it was communicated to me
by Mr. Baron Maferes ; to whom alfo I am obliged for all the reft that is here printed
concerning this very extraordinary perfon. J. H.
THE
THE AUTHOR'S DEDICATION.
TO
HER SACRED IMPERIAL MAJESTY,
MARIA TERESA OF AUSTRIA,
EMPRESS OF GERMANY, QUEEN OF HUNGARY, BOHEMIA, &c. &c.
A MONG the various arguments I revolved in my mind, in*
ducing me to hope, that Your Sacred Majefty, according to
your great condefcenfion, would vouchfafe to receive favourably
this Work of mine, which is proud to fhelter itfelf under your
auguft name, and humbly to crave your gracious patronage and
proteclion ; among all thefe arguments, I fay, none has encouraged
me fo much as the confideration of your fex, to which Your
Majefly is fo great an ornament, and which, by good fortune,
happens to be mine alfo. It is this conlideration chiefly that has
fupported me in all my labours, and made me infenfible to the
dangers that attended fo hardy an enterprife. For, if at any time
there can be an excufe for the rafhnefs of a Woman, who ventures
Vol. I. c to
XVIII THE AVTHORS DEDICATION,
to afpire to the fublimities of a fcience, which knows no bounds,
not even thofe of infinity itfelf, it certainly fhould be at this
glorious period, in which a Woman reigns, and reigns with uni-
verfal applaufe and admiration. Indeed, I am fully convinced,
that in this age, an age which, from your reign, will be diftinguifhed
to lateft poflerity, every Woman ought to exert herfelf, and
endeavour to promote the glory of her fex, and to contribute her
utmoft to increafe that luftre, which it happily receives from Your
Majefly ; who, having diffufed, on all fides, the fame and admiration
of your anions, have obliged Mankind to apply to you, with much
greater reafon, what has been faid of fome of the antient Csefars > —
that, by the juftice and clemency of your Government, you are an
honour to human nature, and a near refemblance of the divine.
To thofe who, zealous for the glory of our fex, fhall faithfully
tranfmit to pofterity the memory of your deeds; to thofe (I fay)
I muft leave to commemorate, how each accomplifhment of the
mind is united in Your Majefly with the moft engaging gracefulnefs
of perfon ; to thofe I fhall leave the arduous tafk to defcribe, the
Ilrength of your underflanding, the extenfivenefs of your genius,
but, above all, that fignal fortitude, that invincible courage and
conflancy of mind, by which you derived frefh vigour, as it were,
from your perils and perfecutions themfelves ; and, after having
been fo feverely tried by the hand of Providence at the beginning
of your reign, gave at lafl fo happy a reverfe to your affairs.
Neither will they fail tp celebrate the engaging fweetnefs of your
temper, your humane and compaffionate difpofition, nor that gene-
rous condefcenfion with which, amidft the hurry and tumult of
4 arms.
THE author's deification. XlX
arms, you cherifh and protecl the arts and fciences; being duly
fenfible how gready thefe redound to the public welfare ; and that
by thefe the minds of men are forcibly excited to the purfuit and
pradice of every focial virtue. Hence it was, that the Sciences fo
early took poffefTion of your mind, and that you became well
acquainted with the whole circle of them. And though the bufy
cares and interruptions of Empire may have withdrawn you from
your more ftudious applications, (Heaven having thought it too
fmall a commendation for you, to be called the mofl knowing and
learned Woman of your age,) yet flill your love of truth is not the
lefs fervent ; fo that whoever employ themfelves in the fearch of it,
are fure to meet with diflinguiOiing marks of your approbation.
Vouchfafe, therefore, Madam, to cafl: a favourable eye on this
Performance of mine, not only as a Work which comprehends the
higheft attempts of the human underllanding, but alfo as the greatefl
tribute it was in my power to offer, to the glory of your aufpicious
reign ; a reign which feems to revive the memory of former
heroines, only to render your magnanimity, prudence, and good
fortune, the more eminently confpicuous by the comparifon. And
if the Volume of Mufic, which my Sifter has had the honour of
prefenting to Your Majefly, has been fo fortunate as to excite your
voice to melodious accents ; let this be fo happy as to have the
defired effedl:, of employing fometimes the fagacity and penetration
of your underflanding. As nothing more remains, but to implore
of Heaven a long and happy continuance of your glorious reign,
for the felicity of the many nations fubje6t to your command ; I
c 2 therefore
« THE AUTHOR'S DEDICATION,
therefore proftrate myfelf, with all humility, at the foot of your
Throne, and am
Your Majefty's
moft humble,
molt obedient,
and faithful fervant,
Maria Gaetana Agnesi.
THE
THE AUTHOR'S PREFACE
TO
THE READER.
nPHERE are few fo unacquainted with Mathenfiatical Learning, but
are fenfible the Study of Analyticks is very neceflary, efpecially in
our days ; they cannot but be apprized what improvements have already
been made by it's means, what are ftill making every day, and what may
be yet expeded in time to come. For which reafon I fhall not amufe
myfelf with making unneceflary encomiums on this fcience, which (lands
in no need of any fuch recommendations, and much lefs of mine. But,
notwithftanding the neceflity of this fcience appears fo evident as to
excite our youth to the earneft ftudy of it ; yet great are the difficulties
to be overcome in the attainment of it. For it is very well known,
that perfons able and willing to teach it are not to be found in every
city, at leafl: not in our Italy ; and every one that would be glad to learn
has not the means of travelling into diftant countries, in quefl of propisr
matters. This I know by my own experience, as I muft ingenuoufly con-
fefs; for, notwithftanding the ftrong inclination I had to this fcience, and
the great application I made ufe of to acquire it ; I might fllll have 'been
loft in a maze of inextricable difficulties, had I not been affifted by
the fecure guidance and fage dircd:ion of the very learned Father Don
Ramiro Rampinellt^ Monk of the Olivetan Order, and now Profeifor of
3 the
XXII THE AUTHORS PREFACE.
the Mathematicks in the Royal Univerlity of Pavla ; to whom I
acknowledge myfelf indebted for what little progrefs I may poffibly
have made in this kind of ftudy ; on whofe deferved praife I fhall
forbear to infift, it being unneceflary to a perfon of his fame and merit,
and oiFenfive to his known, but perhaps too rigid, modefty. True it is,
the aforefaid inconvenience may, in fome meafure, be removed, by having
recourfe to good books, written with perfpicuity, and (what is above all)
ill a proper method. But though what relates to the fubjed: of Analyticks
may have already been treated of, and is to be found in print; yet as
thefe pieces are fcattered and difperfed in the works of various authors,
and particularly in the Leipjic A£ts, the Memoirs of the Royal Academy
of Sciences at Paris, and in other foreign Journals ; fo that it is impof-
fible for a beginner to methodize the feveral parts, even though he were
furnifhed with all the books necefTary for his purpofc : this confidera-
tion induced the celebrated Father Renau to publifh that moft ufeful
Work, intitled U Analyfe demontree^ a. work deferving the higheft com-
mendation. After which, I am very fenfible, that thefe Inftitutions of
mine may feem, at firft fight, to be needlefs, fo many learned Men having
thus amply provided for the occafions of the Public. But, as to this
point, I defire the candid reader to confider, that, as the Sciences are
daily improving, and, fince the publication of the aforementioned book,
many important and ufeful difcoveries have been made by many inge-
nious writers ; as had happened likewife to thofe who had written before
them : Therefore, to fave ftudents the trouble of fceking for thefe im-
provements, and newly-invented methods, in their feveral authors, 1 was
perfuaded that a new Digeft of Analytical Principles might be ufeful and
acceptable. The late difcoveries have obliged me to follow a new ar-
rangement of the feveral parts ; and whoever has attempted any thing
of this kind muft be convinced, how difficult it is to hit upon fuch a
method as fhall have a fufficient degree of perfpicuity, and fimplicity,
omitting every thing fuperfluous, and yet retaining all that is ufeful and
necefTary ; fuch, in Ihort, as fhall proceed in that natural order, in which
confifls
THE author's preface, XXIII
confifls the clofeft connexion, the ftrongeft conviftion, and the eafieft
inftrudtion. This natural order I have always had in view ; but whether
I have always been fo happy as to attain it, muft be left to the judgment
of others.
In the management of various methods, I think I may venture to fay,
that I have made fome improvements in feveral of them, which I believe
will not be quite devoid of novelty and invention. To thefe the judi-
cious Reader may give what weight he pleafes. It was never my defign
to court applaufe, being fatisfied with having indulged myfelf in a real
and innocent pleafure ; and, at the fame time, with having endeavoured
to be ufeful to the Public.
In the Second Volume, in which I treat of the Integral Calculus, or
what is alfo called the Inverfe Method of Fluxions, the Reader will
meet with a fpeculation entirely new *, and no where before publilhcd,
concerning Multinomials, For this I am indebted to the celebrated
Count James Riccati, a gentleman who has greatly deferved of every
branch of literature, and whofe merit is well known to the learned
world. He was pleafed to communicate this to me, which I take as a
favour beyond my deferts ; and for which both the Public and myfelf
are bound to give him our thanks.
To conclude ; As it was not my intention, at firft, that the following
Work fhould ever appear in public ; a work begun and continued in the
Italian tongue, purely for my own private amufement, or, at moft, for
the inftrudion of one of my younger Brothers, who poffibly might have
a tafte for mathematical (Indies ; and as I had not determined to fend it
abroad till after it was pretty far advanced, and had grown to the fize
* It docs not appear to me, that any thing can be done by this new method,, which may
not be done as well, or better, without it. J. H.
of
XXIV THE author's PREFACE.
of a juft volume ; then I thought I might be excufed the trouble of
tranflating it into Latin, (a language which feme may imagine is more
fuitable to works of this nature,) efpecially as I had the example of fo
many famous Mathematicians, as well Italians as others, who have
publifhed their Mathematical Works in their own mother-tongues. Nor
could I eafily overcome my natural indolence, in fubmitting to the
drudgery of tranflating that into Latin which I had already compofed in
Italian, Far am I therefore from laying the leaft claim to any merit
arifmg from that purity and elegance of flyle, which in fubje(5ls of a
different nature may be laudably attempted; being fully fatisfied if I have
always exprefled myfelf, as I fincerely endeavoured, in a plain, but
clear and intelligible manner.
A TABLE
A TA B L E
OP THE
CONTENTS OF THE WHOLE WORK.
VOLUME THE FIRST.
INTRODUCTION.
BOOK I.
The Analyfts of Finite Quantities.
I. Of the firft Notions and Operations of the Analyfis of Finite
Quantities ■■ ■ i
jr. Of Equations, and of Plane Determinate Problems ■ 40
III. Of the Conflrudion of Loci, or Geometrical Places, not exceeding
the Second Degree ■— — . — — 90
IV. Of Solid Problems and their Equations • ■ 140
V. Of the Conftrudlion of Loci which exceed the Second Degree 207
VI. Of the Method De Muximis et Minimis, of the Tangents of
Curves, 9f Contrary Flexure and Regreffion ; making ufe only y^
of Common Algebra — 244
Vol. I. ' D ' 'VOLUME
XXVI A tABJLE OF THE CONTENTS OP VOLUME It,
FOLUME THE SECOND.
BOOK II.
^e Malyjis of ^antities Infinitely SmalK
^^' Page
I. Of the Notion or Notation oi Differentials [or Fluxions] of feveral
Orders, and the Method of calculating with the fame — — i
II. Of the Method of Tangents _. — — j^
III. Of the Method of the Maxima and Minima of Quantities 58
IV. Of Points of Comtrary Flexure, srfd of JRrgfefili'oift — 74
V. Of Evolutes, and of the Rays of Curvature ■ $y
BOOK III.
Of the Integral CalculuS;
I — — — ill r
Sea. P»|c
1. Of the Rules of Integrations exprefled by Finite Algcbr'aical For-
mulas, or which are reduced to fuppofed Quadratures ^- no
II. Of the Rules of Integration, having recourfe to Infinite Series 15a
III. The Rules of the foregoing Seftions applied to the Reftifieation
of CLvrve-lines, the Quadratures of Curvilinear Spaces, the Com-
planation of Curve Superficies, and the Cubature of their Solids 166
JV. The Calculus of Logarithmic and Exponential <^iatttsties — — a^i
6
»DOK
A TABLE OF THE CONTENTS OF VOLUME II» XXVII
BOOK IV.
^be Inverfe Method of Tangents*
StSt* Page
I, Of the Con{lru<5lIon of Differential Equations of the Firft Degree,
without any previous Separation of the Indeterminates — — — 249
II. Of the Conftrudion of Differential Equations, by a Separation of
the Indeterminates ■ . — 257
III. Of the Conftrudion of more Limited Equations, by the Help of
various Subftitutions ■ 285
rV. Of the Redudion of Fluxional Equations of the Second De-
gree, &c. — 306
An Addition to the foregoing Injiitutions ■ 341
mNmmam Mm
THE PLAN
OF
THE LADT'S SYSTEM OF ANALTTICKS,
INTRODUCTION.
npHAT we (hould receive from Italy, the Mother of Arts, a complete Syftem
of Analyticks, is not fo much to be wondered at ; knowing we have often
had from that quarter very excellent produdions in the fublimer Mathematicks.
But, that we (hould receive fuch a prefent from the hands of a Lady j from that
fex which, however capable, yet hardly ever amufe themfelves with thefe feverer
ftudies; is, indeed, very wonderful and furprifing. Yet fo :•■ is in fadl : a very
learned, ingenious, and celebrated Lady of Milan, by name Donna Maria
Gaetana Agnefi, a member of the Univerfity of Bolonia, and lately advanced by
the Pope to a Profefforfhip in Mathematicks and Philofophy in the fame Uni-
verfity, has publifhed a Treatife in Italian, in two volumes quarto, which (lie
calls Analytical Injlitutions for the Ufe of the Touth of Italy ; of which (he was
pleafed to prefent a Copy to the Royal Society of London. This Copy I had
the curiofity to infped, and thought it might be a proper way of returning the
Author's compliment, to have an Account of the work drawn up and read to
the Society, and perhaps printed in the Philofophical Tranfaftions, as has often
been the practice on fuch occafions. This Account, therefore, I undertook to
draw up, having the confent and approbation of our worthy Prefident. But
when I came to look into the work more clofely, I foon enlarged my fcheme j
Vol. I. a and.
11 JNTRODUCTION.
and, inftead of barely taking the Plan, or giving an Account of it, I thought
it highly deferved to be tranllated into our own language, that the Youth of
England might likewife enjoy the benefit of it. This determined me then to
attempt it*s tranflation, though I well knew how unequal I was to the tafk. I
confefs I alfo entertained fome diftant hopes, that it might excite the curiofity
of fome of our Englijh Ladies ; that it might raife an emulation in them, a
laudable ambition to promote the glory of their country, with a generous
refolution not to be outdone by any foreign ladies whatever. They want no
genius or capacity for the fciences, and have undoubtedly as good abilities as
the Ladies of Italy^ They feem only to want to be properly introduced into
thefe ftudies, to be convinced of their ufefulnefs and agreeablenefs, and to
prevail on themfelves to ufe the neceffary application and perfeverance. They
have here a noble inftance before them, of what the fex is capable to perform,
when their faculties are exerted the right way. And they may be fully per-
fuaded, that what cwie lady is able to write, other ladies are able to imitate, or,,
at leaft, to read and underftand. With not much more pains and induftry than
what they muft be at, to be expert at Whift or Qnadrille, they may become
miftreffes of this fcience ; which they will find to be much more innocent, more
diverting and agreeable, and to have infinitely more amufing variety than thofe,.
or any other games whatever. Indeed, this is rather to be efteemed a game, or
a diverlion, than a ftudy ; but then it is a game of ikill, without any mixture
of chance, like Chefs and fome other ingenious games : and parties of two, or
more, may play at it together, by propofing curious queftions to one another
alternately, to their great diverfion and improvement. The games of Whift,
Quadrille, Back-gammon, Sec and all other games in which chance predomi-
nates, but fkill is alfo required to convert the events of chance to the beft
advantage ; thefe are only particular cafes of this general game or art, and
ought always to be regulated by it. For, in all inftances, Analy ticks may
be ufed to difcover the odds^ or degrees of probability, which are for, or
againfl, the happening of any particular event, and fo the chance may be made
equal on all fides, notwithftanding a fuperiority of fkill on one fide. And thus
all games of chance may be made fair and equal ; and the well-meaning
gamefler will not be impofed on by (harpers, who, by much obfervation, rather
than by fkill in Analyticks, always know what they call the beft of the lay, or
always have the odds on their fide.
4 But
INTRODUCTION. «!
But this Is the lead recommendation of this fciencc. The improvement of
their minds and underftandings, which will neceffarily arife from hence, is
of much greater importance. They will be inured to think clearly, clofely,
and juftly ; to reafon and argue confequentially, to inveftigate and purfuc
truths which are certain and demonftrative, and to ftrengthen and improve,
their rational faculties. Now that thefe, and all other readers, may attain thefc
advantages with as little trouble as poflible, I (hall endeavour to draw out the
Plan of this Work at full length, and in a popular manner, inferting fome
ufeful Obfervations to explain the Art iifelf; fo that the Work, when publilhed,
may be eafily read and apprehended, by fuch as will perufe it with the neceflary
diligence and attention.
The fubjedt of the Work is Analyticks, or the general Science of Compu-
tation or Calculation. That is, the Art of refolving all kinds of Mathematical
Queflions, by finding or computing unknown numbers, or quantities, by the
means of others that are known or given. Thefe computations are performed
either by common numbers, and then the fcience is called Arithmetick : or by
general numbers or arbitrary fymbols of quantities, which arc commonly the
letters of the alphabet, and then it is ufually called Jlgebra : or by lines and
geometrical figures, which are likewife the fymbols of quantities, and then it is
called Geometry : or, laftly, by all thefe conjundly and indifferently, and then
it will properly be called jinalyticks. All thefe fciences our Author teaches
and explains promifcuoufly, but in good order and method, at leaft the higher
and more difficult parts of them ; for (he requires, as very reafonably flie mayj.
that the learner fhould come prepared with a pretty good flock of common
Arithmetick, with a competent knowledge of the firfl elements of Geometry,
and with fome infight into the fimpler properties of the Conic Sedions. Thefe
arc acquifitions with which they may be eafily furnifhed out of the common
mathematical books on thefe fubjeds ; which will then prepare the way for art
cafy accefs to her fublimer fpeculations. Now, to enter upon our intended
Plan. The Author divides her fubjed into two Tomes, or Volumes j in the
firfl of which fhe treats of the common, ordinary, and finite quantities, and
their reprefentatives, whether numbers, general fymbols, or lines. In the
fecond Volume fhe explains the nature of what (he calls Infinitefmals, or infinitely
iinall Quantities; proves their comparative exifleiice, and (hows their ufe and
a St, applicatiofi.
ir THE PLAN OF THE LADV*S
application. This is the grand dlvifion of the whole Work, which is again
divided into four Books y and every Book is fubdivided into it's number of
SeSfions, according to the nature of the feveral fubjeds they treat of. Laftly,
there is a further fubdivifion of the Sections into Jrikles, which are numbered
without interruption from the beginning to the end of each Book, and which
we fhall alfo obferve and enumerate in our explications of them.
PLAN.
The firft Sedion of the firft Book is concerning the primary Notions and
Operations of the Analyfis of finite Quantities ; in which are contained the
following Articles. After a (hort Preface concerning the nature of Analyfis>
the Author obferves,
1. That it's operations are the fame as thofe of common Arithmetick ; this
operating with numbers, and that with fpecies, that is, with fymbolical numbers
or quantities. By which means Algebra has great advantages over Arithmetick ^
for, in this, the fleps of the operations will be confounded and loft by the fubfe^
quent ones, but in Algebra they may be preferved, as they are often not actually
performed, but only infinuated by proper fymbols j it is alfo more univerfal,
and works indifferently with known or unknown quantities.
2. Here the diftindlion of pofitive and negative numbers, or quantities, is
explained. Negative quantities are not in nature, but depend only on the
manner of conceiving them. They are merely artificial, and introduced to fave
needlefs repetitions and diftinftions, by which we can confider the oppofite
operations of Addition and Subtradion under one general view and compre-
henfive idea. In Geometry, they are reprefented by lines drawn oppofite ways.
If pofitive lines proceed to the right-hand, then negative ones wilt be to the
left, with the fame dircdion ; or if pofitive ones are upwards, then negative
will be downwards.
Then
SYSTEM OF ANALYTICKS. V
3. Then different afFeftions of quantities are diflinguifhed, or denoted, by
the figns 4- or — , plus or minus, placed before them j whether the quantities
are reprefented arithmetically, or by common numbers j or elfe algebraically,
by reprefentative numbers, that is, by the letters of the alphabet ; plui being
the mark of Addition, and minus of Subtraftion. And the fign Hh and 4= are
ambiguous, but contrary to each other. The equality of quantities is denoted
by the mark zz, and majority or minority by the marks > or <, Proportion,
or equality of ratios, by :;, and infinitely great by 00 .
4. (^lantities 2XQ fimple that are not connefled by the figns 4- or — , and
compound when they are : of which examples are propofed by the Author.
5, Then is taught the addition of fimple quantities being integers, and
explained by a fufficient number of examples : alfo, the ufe of numeral. co-efE-
cients is (hown,
6, Likewife, the fubtradion of fimple integral quantities is taught, in which
it is fhown that the fign of the quantity to be fubtraded muft always be
changed,, and the reafon of it, together with examples.
7; l^ext the Author proceeds to- the multiplication of fimple quantities,
being integers, whether they are pofitive or negative. Xhen the produd will
be reprefented by the connedion of the feveral fadors, and their co -efficients
\vithout any fign between them. And if the fadors are pofitive and negative
promifcuoufly, like figns will always produce -f-, and unlike figns — . This
(he demonftrates from the nature of proportion.
8. And whereas raifing of powers is a cafe of multiplication ; (he (bows how
fimple powers are formed, and conveniently exprefifed by their indices, or
exponents y annexed to the roots.
9. Thefe powers are diftributed mto Jqu ares ^ cubes, hiquadraiss^ &c. ; that
is, into fecond, third, fourth, &c. powers, of which the given number, or root,
is always the firft power; and they are marked by the exponents i, 2, 3, 4, &c,
refpedively. Their ifigns are always known by the general rule aforegoing*
10. Then
91 T H E P L A N O !F THE L A B X S
10. Then comes divifion of fimple quantities, being integers, which is juft
the reverfe of multiplication, and refoives, or decompounds, that which the
other had compounded ; as by the examples.
11. When common letters or quantities are rejedled, and the diviiion can
proceed no further, it muft be infinuated, by making a fraction of what fliall
remain.
12. When the figns of the dividend and divifor are the fame, the fign of the
quotient muft be pofitive; but when thofe figns are different, the fign of the
.quotient muft be negative. This proved from the nature of proportion,
13. Whence, in fradions, it is indifferent how the figns are changed In th«
numerator and denominator, provided the fign of each is changed into it's
contrary.
14. The roots of fimple quantities will be extracted, by dividing their
'exponents by the number which denominates the root to be extraded. As, by
A for the fquare-root, by 3 for the cube-root, and fo on.
15. If any even root is to be extraded, the fign of that root will be ambi-
guous; but if an odd root is to be extradted, the fign of that root will be the
fame as of the given power^
16. When roots are furd, and cannot be extracted, they are to be infinuated
by radical figns or charafters.
17. From thefe operations belonging to fimple quantities, the Author pro*
ceeds to thofe of compound quantities, or fuch as confift of feveral fimple
quantities, conneded by the figns + and — . Thus, Addition will be per-
formed by fetting down all the given quantities together promifcuouily, and
then abbreviating the fum as much as may be, and expunging equivalents with
contrary figns,
iS. In Subtradion, all the iigns are changed of the quantity to be fubtraded,
And the remainder, or difference, fo found is to be abbreviated as much as may
ke done«
19. Mul-
SYSTEM OP A N* A L y T I C K S, vli
19. Multiplication of compound quantities, being integers, depends on the
multiplication of fimple quantities ; and the procefs is mulch like the fame
©peration in common Arithmetick, as the examples fliow,
ao. But it is often convenient only to infinuate this multiplication, without
adually performing it. And that is done by drawing a line, or vinculum^ over
the feveral fadors, and conneding them by putting the mark x, fignifying
Multiplied by, between them.
21. The powers of compound quantities, as well as of fimple, need not always
be adually formed, but may often be conveniently infinuated, by a vinculum
placed over the root, and a proper index annexed to it. How thefe powens
may be adually formed, when occafion requires, is here fhown.
22. The Author prefents us with a general Canon, (being Sir Jfaac Newton's
Binomial Theorem,) for raifing any binomial quantity, or even multinomial,.
to any power required ; which fhe exemplifies by a fufficient number of
examples,
23. The Author proceeds to divifion of compound quantities, being integers,
of which (he makes three cafes. The firft is, when the divifor is fimple and the
dividend' compound,, and the fecond is on the contrary. Thefe are eafily
reduced to the foregoing rules,.
24. The third cafe is, when both the dividend and divifor are multinomials^
and therefore requires a more prolix procefs. In order to which, the terms of
each are to be difpofed according to the dimenfions (or powers) of fome parti-
cular letter contained in them ; that is, they are to form numbers belonging to a
fcale, of which that letter is the root, jufl as we do in our common Arithmetick,
the root of which is ten, and the numbers are difpofed according to the
dimenfions of that root. Then the procefs of divifion muft be performed
much after the manner of the like procefs in numbers, and which is fuiEciently
explained by the examples produced. When the dividend cannot be intirely
exhaufted, the quotient mufl be completed by adding a fradion to it, as in
common Arithmetick,
25. The Author proceeds to the extradian of the roots of compound quan»
titles, being integers, and firfl of the fquare-root. The terms of the given
quantity
Vlll THE PtAN OF THE
quantity are to be difpofed, as before, in Divifion; and the procefs of extraftion
will be nearly as the fame operation in numbers. Indeed, her procefs is fomc-
thing different in form from the common one, but is very intelligible, and
comes to the fame thing. Her examples make it very clear. When the root
is furd, and therefore cannot be extraded, it mult be infinuated by a quadratick
vinculum.
' 26. The procefs of the extraction of the cube-root is much after the fame
manner, only more operofe, as being a more complicate operation. The
examples render it as plain as the nature of the thing will admit,
27. The biquadratick, or fourth root, is extraded in the fame manner.
28. The fifth root, and all higher roots, may be extraded, by forming rules
for tbem, which are found by raifing a binomial to the fame power. For the
lilce was done in forming rules, by which the fquare and cube-roots have been
extraded.
29. The Author then proceeds to the algorithm of fradions fimple and
compound j obferving that any quantity may be converted into a fradion with
a denominator given, if it be multiplied into that denominator : of which fhe
produces feveral inftances. For this fee the Examples.
30. Then comes the redudion of fradions to more fimple expreffions, when
;that can be done, which it is not always eafy to perceive. When the numerator
and denominator are each multiplied by the fame quantity, whether fimple or
compound, they may each be divided by it again, and a new fradion will arife
equivalent to the former. And fo toiies quoties. This will be a very ufeful
redudion ; for, in all our calculations, we fhould always ftudy to abbreviate as
much as poflible. See the Examples. How thefe common divifors may be
found we fhall be taught afterwards.
31. Then is taught redudion of fradions to a common denominator, which
in two fradions is performed by the crofs multiplication of each numerator into
the denominator of the other, as by the examples. And fo two by two, if
there are more, till all are reduced.
32. This
SYSTEM OF ANALYIICJCS. ^
32, This prepares the way for the addition and fubtradlion of fradions 5 for,
if they have not a common denominator, thofe operations can onlybe infinu-
ated, by writing them after one another with their proper figns. But, when
reduced to a common denominator, their numerators may then be added or
fubtradted, to compleat thefe operations ; as by the examples,
33. The multiplication of fraflions requires no fuch preparation, but is
performed diredly, by multiplying the numerators together for a new nume-
rator, and the denominators together for a new denominator. The prodii<5l,
or fradion thence arifing, may often be reduced by fome of the foregoing
methods. •
34. Divifion of fradlions is reduced to multiplication, by multiplying the
dividend by the reciprocal of the divifor 5 which reciprocal is, when the nume-
rator and the denominator change places. The quotient thus found will often
have occafion for fome redudion, as by the examples may be feen.
35. As for the extraftion of the roots of fradions, whether it be the fquarc-
rodt, the cube-root, &c. the faid roots muft be extraded feverally out of the
numerator and denominator, and the fradion thence arifing will be the root of
the fradion given. But when fuch root cannot be extraded, it muft be infi-
nuated by placing a radical vinculum before the given frad:ion, as by the
examples.
36. To conclude the Dodrine of Fradions, the Author proceeds to a very
curious and ufeful operation, which is, to find the greateft common divifor of
two quantities or formulas given. Where it may be obferved, that a formula
is a combination of quantities, which may ferve as a paradigm, or pattern, for
all combinations of the like kind. Then, by a procefs not unlike that in
Arithmetick, which is, by fubtrading one from the other continually and
interchangeably as often as can be done, the laft quantity fo found will be the
greateft common divifor of the two given quantities. Now, if thofe two
quantities form a fradion, and the numerator and denominator are each di-
vided by the greateft common divifor fo found, a fradion will thence arife
equai to the other, but reduced to the fmalleft terms. Of this redudion (he
gives us the procefs at large, in three feveral inftances.
Vol. I. b 37. The
Jt THEPLANOFTHELADY*!
37. The Author goes on then to the Docftrine of Sards or Radicals, which
are fuch quantities whofe roots cannot be extrafted, yet may often admit of a
partial extradion, or may be reduced to fimpler expreflions; as by the examples
may appear.
38. The redu6lion of different radicals to radicals with the fame index, will
be performed by finding the lead number for a common exponent,- by which
the given exponents may be divided. Then each radical muft be raifed, if
neceffary, till it arrives at that exponent. The examples make it plain.
39. Addition and fubtraftion of radicals is eafily performed, by writing them
one after another with their proper fi^s, and then abbreviating when it may
be done.
40. Radical quantities are multiplied by thofe that are rational, by prefixing
the rational to the radical, with fuch fign as the Rule of Multiplication requires.
And when they are compUcate, their produd will be found by the fame rule.
41. Radicals of the fame denomination, or reduced to fuch, are multiplied
by putting their produd under the fame radical vinculum.
42. If the radicals are affeded by rational co-efficients, their produd mufl:
be put before the radical fo found.
43. "When like quadratick radicals are multiplied into each other, the
radical fign will be taken away, and the produd will often become rational.
Several examples of this are exhibited. ^
44. A rational co-efficient to a radical may at any time be made to pafs under
the radical vinculum.
45. The multipHcation of radicals of different kinds may be infinuated, c*
they may be reduced to the fame kind.
46. Divifion of radicals of the fame kind is performed by leaving out the
radical quantity, and dividing the co-efficients only,
47. If the radicals are of the fame kind, but not of the fame quantity, the
quantities under the vinculum may be divided, and the quotient put under the
fame vinculum.
4S. But
SYSTEM OF ANALYTICKS* Xl
4S. But if the radicals are different, they may be reduced to the fame expo-
nent, and then divided as before. And thus complicate quantities may be
divided as in common Divifion.
49. Then the A.uthor gives us a Rule for extrading the fquare-root of
quantities any how compounded of rational and irrational quantities, and thofc
either numeral or algebraical ; which fhe applies to feveral examples.
50. In order to the calculation of powers, which are exprefled by integer
exponents j from any root (he forms a geometrical progreffion of it's powers,
beginning from unity, and afcending one way by pofitive exponents, and de-
fcending the other way by negative exponents, to fhow the correfpondencc
there is between the increafing powers and their affirmative exponents, and the
decreafing powers and their negative exponents. Then obferves, that when any
power is in the denominator of a fradion, it may be made to pafs into the
numerator, and vice verfd, by only changing the fign of the index.
51. Then, as fraftional powers, or roots, are certain intermediate termsji
between the integral powers in the foregoing geometrical progrefilon ; fo their
exponents muft be correfponding intermediate terms in the arithmetical pro-
greffion. And this will obtain in the defcending progreffion as well as fn the
afcending, and whether the terms are (imple or compound.
52. Hence the multiplication or divifion of powers will eaflly be performed
by their exponents. For, to multiply them, we muft add their exponents ;
and to divide them, we fubtrad the exponent of the divifor from that of the
dividend. This fhe proves from the nature of proportion.
c^^. Hence the raifing of powers, or extradling the roots of any powers, will
eaflly be performed by their exponents. For the index of any power muft be
multiplied by the index of the power to which it is to be raifed; and the index
of the given power is to be divided by the index of the root to be extracted.
54. And this obtains as well in compound quantities as in fimplc. For aJl
which redudions fee the Examples.
55. Another ufeful operation follows, which is that of finding all the linear
©r fimple divifors of any given number or formula j or to refolve a .compound
b 2 quantity
Xa THE PLANOP THE lady's
quantity Into the feveral quantities of which it is, or may be, compounded by
multiplication. The procefs is exemplified and illuftrated both in numbers and
fpecies. Indeed, if this could always be done in numbers, it would amount to
a very valuable difcovery, or delideratum in Analyticks, which is, a method of
refolving a given compound number into the prime numbers of which it is
compounded ; but though it is only a tentative method, yet, however, it is
very ufeful.
e,6. This is extended to any compound formula, or to a number exprefled
by an indefinite root in an arithmetical fcale, which may have been formed by
the multiplication of feveral binomial fadlors. By this method fuch a number
may again be refolved into it's fadors, by the help of the foregoing operation.
And if the number of trials to be made fhould happen to be too great, the
Author fhows a method of reducing them to a fmaller number, which is, by
changing the root, and fo exhibiting the given formula by another fcale.
57. Now, if the firfl term of the given formula fhould happen to have a
numeral co-efficient, it may be convenient (by fubftitution) to change it into
another formula, or to exprefs it by an equivalent root of another fcale, the
co-ef^cient of the firft term of which fliall be unity.
B O O K I. S E C T. II.
Of Equations, and of Plane Determinate Problems,
58. Having explained the firft principles or operations of Analyticks in the
foregoing Sediion, our Author proceeds to the grand inftrument of the art of
computation, which is equation. This is either when fome of the terms placed
before the mark of equality, are colledively equal to all the terms on the other
fide, called the hcmogeneum eomparationis ; or when the whole are one fide, and
equal to nothing on the other fide ; infinuating that the affirmative and negative
a^e equal, and fo deftroy one another. She explains hkewife what is meant by
the law of bomogemity,,
59. She
SYSTEM OF ANALYTIC KS, Xlil
59. She tells us what a Problem is, and what is the diflindion between the
data and qua/it a of a problem.
60. Problems are divided into determinate and indeterminate, of which (he
gives inftances from Geometry. But in this Seflion (he treats only of fuch as
are determinate.
61. Here it is explained how equations are formed, from the dependance of
quantities upon one another, whether they are known and given quantities, or
unknown and required. The inftances are taken from the properties of lines
and figures.
62. How we are to argue from the giverr conditions of the queftion till we
come to an equation between the quantities given and required. ' This is ex-
plained geometrically, and by an abftraft arithmetical queftion.
6^, No more given quantities are to be afTumed than are necefTary, whea
they can be exprefTed by the known properties of the figure,
64; It will often happen, that the lines given in a figure atfe not fufficient
for forming the equations; then fuch other lines muft be drawn as may complete
the figure, and bring us to a determination. A problem is propofed to illuftrate
this ; and the Proportions of Euclid are enumerated, which will be of ufe for
fuch purpofes.
65. Here the Author propofes and folves three or four geometrical problems^
to fhow the method of arguing from one condition to another, in order to
obtain a final equation.
66. When the conditions of a problem involve the properties of angles,
they mufl fomehow be reduced to the properties of lines. This is exemplified
in the problem of finding an equicrural triangle, in which either of the angles
at the bafe is double to the angle at the vertex : which is reduced to the linear
problem, of dividing a line in extreme and mean proportion*
67. Having thus fhown how to find equations from the given circumftances
of a problem, Ihe proceeds to the refblution of thefe equations, or to the
finding the unknown quantity, by means of various redudions. For this end
"^•. ■ ■ fhe
xiv THE PLAN OF THE I.ADYS
llie gives us four axioms. By the firft, (he (hows the ufe of tranfpofing quan-
tities at pleafure from one fide of an equation to the other; which may always be
done without deftroying the equation, only by changing the (igns of the terms
fo tranfpofed.
68. By the fecond axiom (he (liows how we may take away any fra<5tions that
arife in an equation, and fo reduce the whole to integral terms.
69. And how, by the fame, any term may be freed from it's co-efHcient.
70. By the third and fourth axiom (he fliows how equations may be freed
from furds and radicals; and of all thefe redudions gives us si variety of examples.
71. Equations prepared for folution, and diftributed into their terms.
72. Equations further prepared, by which the unknown quantity will be found
equal to a combination of known quantities, and a fimple equation will be
folved entirely.
73. If any power of the unknown quantity is found equal to known quan-
tities, then the root may be extracted on both fides.
74. If the equation is an affeded quadratick, it may be folved by completing
the fquare on one fide, and then extrading the fquare-root on each fide.
yc^. In quadratick equations the ambiguity of the (igns will fupply two values
of the unknown quantity, which may therefore be both pofitive, both negative,
or one pofitive and the other negative, or both imaginar)^ according to the
values of the known quantities. What is analogous to this difference of figns
in geometrical figures, is here (hown, and all is illuflrated by examples.
76. The Author (hows us here the ufe of impoffible or imaginary roots of
equations. ^For they are a fure indication, that the queftion (as now propofed)
is impoflTible, either by chance or defign. And the fame thing is to be con-
cluded, when the final equation brings us to any abfurdity or contradidion.
This (he (hows in feveral indances.
77. And fometimes we may be brought to an identical equation ; which only
(hows that the point required may be any where in the given line, as by the
example.
78. Equations
SYSTEM OF ANALYflCKS, %V
78. Equations and problems are diftinguifhed into degrees, according to the
dimenfions of the unknown quantity contained in them. Alfo, thofe problems
are called Plane, the refolution of which requires only the ordinary Elements of
Geometry. But if they require the defcription of the Conic Sedions, or other
curves, they are Solid Problems,
79. Equations are not always of that degree which their higher powers feem
to infinuate, but may often be brought to a lower degree by an eafy reduilion t
As by the examples.
80. Sometimes neceffity, and fometimes conveniency, will require, that more
than one unknown quantity may be introduced in a problem; in which cafe
(if the problem is determinate,) as many equations miift be found as there are
unknown quantities affumed. Then thefe are to be eliminated one by one, till
we finally arrive at an equation, in which there is only one unknown quantity*
The way of doing this (he lliows by an example.
81. This method of elimination maybe made ufe of, not only in fim>ple
equations, but alfo in afFe(5ted quadraticks.
82. Higher equations may fometimes be reduced, by eliminating their
greateft powers. And when thofe powers have not the fame index, they may
be reduced to fuch as have. Of both thefe reduflions the Author produces
feveral examples.
83. If there be feveral fimple equations including as many unknown quan-
tities, they may be feverally ehminated, and reduced to one equation including
only one unknown quantity, though the calculation will often be tedious.
84. If there are not as many equations to be found as there are unknown
quantities, the problem will become indeterminate^ and will allow an infinite:
number of anfwers. Of this {he produces examples.
85. But if the conditions to be fulfilled, or the equations, are more than
»eceffary, they may be inconfiftent with each other, and fo the problem will
become impoffible ; or fome of the conditions may coincide with others,, and
fo be fuperfluous.
Z6, Havingr
XVi THE PtA$f OF THE LADY'«
86. paving laid this, foundation far calculating with arithmetical or alge-
braical quantities J (he now does the fame for calculating with geometrical
quantities, or with Hnes and figures. She begins with the operations of Multi-
plication and Divifion, or, what is the fame thing, with finding fuch fimple
proportions,- or conftrudling fuch fimple equations, as will give the values of
ihe quantities required exprefTed by lines.
87. The operations of addition and fubtradion of hnes, when thus found,
will be very eafy and familiar.
88. Hence, by fubftitution, any given letter, or letters, may be introduced ;
or a plane may be transformed into another with a given fide, or a folid into
another with one or two given fides, &c. by which the conftrudion of fimple
equations will be much facilitated. \
S9. This redudion is eafily extended to fradions, the numerators or deno-
minators of which are complicate terms.
90. But, without dividing a fraftion into feveral fradions, the method of
transformation may often be preferable, as is fliown by a variety of examples,
91. Here it is (liown how lines may be found, that fliall exprefs the value of
any quadratick radical, by only finding geometrically a mean proportional
between two given quantities : excepting the cafe when that value is imaginary
or impoflible,
92. But, to reduce radical quantities to this rule, there will often be occafion
to have recourfe to the method of transformation, as appears by the examples^
93. Any quadratick radicals may be conftruded by a right-angled triangle,
e;ither alone or combined, with a circle, without transformation; though fome
transformation, will oftea be found convenient. This illuftrated by various
examples,
94. The foregoing rules, may eaifily be applied to the confl:rudion of any
a^Teded quadratick equation i^ byt. they may all be conftruded after a more
general manner., For this purpofe the Author affumes a general afFeded
quadratick equation, which fhe diftinguilhes into four, according to the variety
of
9
SYSTEM OP ANALYTICKS. XVll
of their figns. Thefe (he conflrudls, one after another, by right-angled triangles
and a circle, and exhibits the roots, both affirmative and negative, by right
lines.
95. The fame equations may be otherwife, and more eafily, conftruded,
when the laft term is not a fquare, but a re6langle.
96. Hitherto the learned Author has been laying down the pr.incipal rules of
the Art of Computation, whether arithmetical, algebraical, or geometrical ; (he
now proceeds, as (he tells us, to (how their ufe in the folution of fome particular
Problems, to the number of 15, wuh which (he concludes this Seftion. The
firft is purely arithmetical, and to be found in moil Books of Algebra.
97. The fecond Problem is alfo very common, and is about the motion ot
two bodies with given velocities, in various circumftances, general and par-
ticular.
98. The next is the famous Problem of King Hiero^s crown, in which Archi'
medes difcovered the quantity of bafer metal mixed with the gold, and which
gave the occafion to his celebrated svpYjKx*
99. The next Problem is concerning the relation of two weights to each
other, and is purely arithmetical. And thefe Problems hitherto have produced
only fimple equations.
100. Then we have a Geometrical Problem, which amounts only to a fimple
equation, and is therefore eafily refolved and conftrufted.
1 01. The next Problem is geometrical, which arifes to a fimple quadratick
equation, which is there conftruded, or refolved, geometrically.
102. Then a Geometrical Problem, teaching to infcribe a cube in a given
fphere; which amounts only to a fimple quadratick equation, and is there
conftrudted, and the conftrudion proved by a fynthetical demonftration.
103. A Geometrical Problem, or rather Theorem, concerning a fecant drawn
through two concentrical circles, fo that the parts intercepted by the circumfe-
rences (hall be equal. This being the property of every fuch fecant, the
Vol. I. c folution
Xviii 1 THE PLAN OF THE LADY's
folution brings to an identical equation, which is a proper caution how to
manage fuch Problems, and what conclufions we are to derive from them.
104. Another Geometrical, or rather Algebraical, Problem.
105. A Geometrical Problem.
106. A Geometrical Problem, in which the magnitude of angles enters the
calculation.
107. A Geometrical Problem, with a fynthetical demonftration.
108. The Author gives us here a very notable Geometrical Problem, which
is, two contiguous arches of a circle being given, and alfo their tangents, to
find the tangent of their fum. And this fhe extends very artfully to the folution
of a much higher and more general Problem, which is, any number of arches
and their tangents being given, to find the tangent of their fum. By the way
fhe gives us a general Theorem, for finding all the poflible combinations of any
number of quantities given. She concludes with giving a general canon, or
formula, for finding the tangent of any multiple or fubmultiple arch ; as alfo,
fhows the converfe of thi§ Theorem.
109. Then we have a Geometrical Problem, which is, to find a triangle, the
fides of which and the perpendicular are in continued geometrical proportion.
This amounts to a high equation, but is reduced to an afFeded quadratick :
which is geometrically conftruded.
no. The laft Problem is that famous geometrical one, of trifeding a given
angle. This fhe divides into three cafes, according as the given angle is
right, obtufe, or acute. The firft cafe flie folves by a fimple quadratick
equation, of which fhe alfo gives us the conftrudion. The fecond and third
cafes arife to cubic equations, which (he referves till Ihe comes to treat of thofe
equations.
BOOK
SYSTEMOP ANALYTICKS, XJX
BOOK I. SECT. III.
Of the ConJiruSlion of Geometrical Places^ and of Indeterminate Problems not
exceeding the Jecond Degree,
III. In this article the Author explains the nature of variable quantities;
that there muft always be two of them, at lead, in an indeterminate Problem,
which are varied according to a conftant law, which is exprefled by a given
equation.
112.' A Locus Geometricus is a right line, or a curve, the ahfcijs and ordinate
(or the co-ordinates) of which are variable right lines, which in all cafes exprefs
the variables of the equation. The abfcifs begins from fome certain point taken
at pleafure in an indefinite right line, and the ordinate is placed at the end of
the abfcifs, at a given angle. When a definite value is affigtled to one of thefe
lines, the curve, or locus, will give the definite and relative value of the other,
agreeably to the equation : as by the inftances may be feen.
113. Different equations will require different loci, and vice ver/d* And as
the equations are of different degrees, fo will the loci be alfo.
114. Of a fimple equation the locus will always be a right line.
115. When any combination of the variables, in any one term, does not
exceed the fecond degree, the equation will always require a conic fedion for
it's locus.
1 16. Thefe loci are here diftributed into their feveral orders.
117. All equations of the firft order, or which can belong to a right line, are
here conftruded.
118. In fimple equations, fometimes a determinate problem may be propofed
as an indeterminate, in which cafe one of the variables will vanifti out of the
equation, or not at all appear in it. Then the locus of the equation will be a
c 2 right
XX THE PLAN OF THE LADY's
right line, either perpendicular or parallel to the abfcifs. Of this the Author
produces an inftance or two, with their conftrudion.
119. The Author goes on to the circle, as the fimpleft curve, of which (he
exhibits the firft and fimpleft equations, whether vv«-take the beginning ot the
abfcifs from the centre, or from the end of the diameter ; and ftiows what the
radius muft be, in cafes not fo fimple : and tells us likewife when the circle
will be only imaginary,
1 20. She proceeds then to the parabola, as the next fimpleft curve, of which
fhe exhibits the primary equations, whether the parameter be fimple or com-
plicate, whether the parabola be internal or external.
121. The next conic fedion is the hyperbohj or rather the two oppofite
hyperbolas, of which (he exhibits the fimpleft equations, when the ordinates are
referred to the axis ; whether the abfcifs commences from the centre, or from
cither of the vertices ; or whether the equation is exprelTed by the axes, or by
the parameter. She finds the equation when the hyperbola is equilateral ; and
reduces complicate parameters, or diameters, to fimple ones.
122. She fliows likewife what will be the fimpleft equation belonging to the
.hyperbola between it's afymptotes.
123. The fimpleft equations are alfo derived for the ellipfis, whatever is the
angle of ordination j and whether the abfcifs begins from the centre, or from
either of the vertices; or whether the equation is exprefled by the diameters, or
the parameter. And what will be the equation, when the diameters and para-
meter are equal. In this laft cafe, if the angle of ordination is a right angle,
the ellipfis will degenerate into a circle. Complicate diameters and parameters
are reduced to fimple ones, as before in the hyperbola, from the equations of
which thofe of the ellipfis will differ only in their figns ; fo that they will eafily
pafs into each other.
124. When the fimple equations to the diameters of the hyperbola, or
ellipfis, are not given exadly in the terms of the diameters, but rather in
difguifed terms j the Author (hows how, by the Rule of Proportion, thofe
diameters may be found. Of which redudion fhe gives Examples.
125. Or
SYSTEM OP ANALYTICKS. XXI
125. Or when the fame equations arc exprefled by parameters, though
fomething obfcurely ; flie fhows us how to find thofe parameters, and gives
Examples of it.
126. Having thus exhibited the fimpleft equations belonging to the Conic
Sedlions, and fliown how we may find the diameters or parameters when in-
volved, by which thefe fedions may be defcribed ; the Author proceeds to
conftruft any complicate equations that may be given, belonging to thefe
fedlions or curves; in order to which, fhe diflributes all fuch equations into
three fpecies or clafTes. The firft are thofe that contain the fquare of one of the
variables, and the redangle of the other into a conllant quantity. The fecond
fpecies contains the reftangle of the two variables, with other fimple terms.
The third contains the redangle and both the fquares of the variables, with any
other fimple 'terms.
127. She then proceeds to conftrud equations of the firft fpecies, however
complicate they may be, and reduces them to a fimple form, by one or two
fubflitutions of new variables. And of this fhe gives us two Examples. In the
firft, by one fubftitution, (he reduces the given equation to the fimpleft form
belonging to the parabola, which fhe then conftruds. In the fecond, fhe
reduces the given equation, by two fubftituiions, to the fimpleft form belonging
to the hyperbola between the afymptotes, which (lie then conftruds, and
purfues it through all it's varieties. When the conftant quantities are fuch, as
not to admit of thefe fubftitutions, fhe changes them, by the tranfmutations ftie
had taught before, into fuch as will be fit for ihofe fubftitutions.
128. Then fhe reduces equations of the fecond fpecies to the firft, by a
method not unlike that of extrading the fquare-root of an affeded quadratick
equation. By which means, and by a lubLtitution, flie introduces a new
variable. Of this (he gives an Example in an equation to the parabola, which
(he reduces and conftruds. Alfo, another to the hyperbola, reduced by two
fubftitutions.
129. Then fhe fhows, by an example, how an equation of the third fpecies
may be reduced to the firft, and fo conftruded.
130, Here
XXll THEFLAN OF THE LAD Y *S
130. Here (he propofes various complicate examples, of which fome are to
the parabola, fome to the hyperbola, and fome to the eHipfis, which require
feveral fubftitutions and transformations; but are all reduced to limple equations,
and conftruded with great art and ingenuity.
131. All the variety of equations to the hyperbola between the afymptotes,
are reduced to four general equations, which are here conftruded, by one, two,
or more fubftitutions," or changing of the variables •, and that according to all
the variety of their figns. To illuftrate thefe conftrudions, and to (how their
application in particular cafes, (he propofes and refolves the feveral Problems
following,
132. The equation of the firft Problem is found to belong to the parabola,
being the property of the focus of the parabola in refped of the diredrix,
which is therefore eafily conftruded by one fubftitution.
133. The equation of the next Problem is found to be a locus to the
hyperbola between the afymptotes, and is conftruded by means of two eafy
fubftitutions.
134. This Pro'biem is propofed concerning the properties of two circles and
their tangents, but the general folution and conftrudion of the equation require
all the three conic ifedions, according to the three cafes included in it. Thefe
cafes are conftruded feparately, by the help of feveral fubftitutions and tranf-
mutations.
135. A Problem to the three Conic Sedions, according to it's three difTercnt
cafes.
136. A general Problem folved by a canonical equation, and illuftrated by
three Examples of particular curves, of which the laft arifes to a cubical
equation, and therefore goes beyond the Conic Sedions.
137. A Problem concerning two equal interfeding circles, which arifes to an
equation to an ellip(is, which is here conftruded by means of one fubftitution.
138. A Problem, or rather two Problems to the circle, with fynthetical
demonftrations of the folution.
139. A
6
8 Y STEM OP ANALYTICK8. Xxili
139. A Problem of a normal Aiding between the fides of a right angle, and
with one end defcribing a curve. This curve, by it's equation, is found to be
an ellipfis, and is here conftruded.
140. The equation of this Problem is either to the pai-abola, the hyperbola,
or the ellipfis, according to different circumftances, and is refolved by various
fubftitutions, or changes of the indeterminate quantities, and is here conftrufted.
141. The Method of Majority and Minority is here occafionally explained,
which proceeds in the fame manner as the redudion of equations. For, by a
feries of comparifons duly made, we may know which of two quantities is the
greater or leffer.
142. A Problem producing an equation to the hyperbola between the
afymptotes, which is very artfully refolved and conftrudled, by three fubfti-
tutions, or changes of the variable quantities.
143. Here the Author concludes her Problems, and recommends the proving
the folution, after it is finifhed, by tracing back the fcveral fubftitutions, and
fo returning to the original equation. Of this (lie gives us two Examples in
the foregoing Problems.
BOOK I. SECT. IV.
Of Solid Prohkms and their Equations.
144. The Author having thus difpatched what are called Plane Problems, or
fuch as require only equations of two dimenfions ; (lie proceeds to thofe called
Solid Problems, which require equations of more than two dimenfions, and
therefore higher and more difficult conftrudlions. She begins by informing us
what are the roots of fuch affedled equations, or what are the values of the
unknown and indeterminate quantities, which are to be extrafted out of thefe
equations. That they are fuch numbers or quantities, that, if they were to be
fubftituied in the equation given, inftead of the root, they would reduce the
whole to nothing -, which would be a full proof, when the root, or roots, are
extradted, that they are the true roots of the equation.
145. Or,
xxiv ' theplanofthelady's
145. Or, in another acceptation, thofe fimple equations are often called the
Roots of a compound equation, which, being multiplied into each other conti-
nually, will produce the equation given. Confequently that equation maybe
rcfolved into it's components by continual divifion. Hence every equation will
have fo many roots as it has dimenfions. Of this flie gives us inftances in
equations of two, three, or four dimenfions, or of quadratick, cubick, and
biquadratick equations, which are formed by the multiplication of fimple, but
general equations, and which therefore will be the roots of the equations fo
formed.
146. Hence, when any of the roots of a compound equation happen to be
known, we have a method, by divifion, of depreffing that equation, and re-
ducing it to a fimpler, which (hall include only the unknown roots.
147. From this way of raifing compound equations by multiplication, we
may know the conflitution of every fingle term, when the whole equation is
difpofed in a proper and regular order, and made equal to nothing. For the
higheft term muft always be pofitive, and have no other co-efficient but unity,
which can always be efFeded. The co- efficient of the fecond term will be the
fum of all the roots, under their proper figns. The co-efficient of the third
term will be the fum of the produds of every pair of roots, &c. And the laft
term will be the product of all the roots, affefted by their proper figns.
148. It follows from hence, that, if the fecond term is wanting in any
equation, then the fum of the pofitive roots will be equal to the fum of the
negative ; therefore, when that term is prefent and affirmative, the fum of the
pofitive roots will be lefs than the fum of the negative 3 but the contrary, if that
term be negative.
149. When any term is wanting in an equation, it's abfence is commonly
indicated by putting an aflerifm in it's place.
150. If no imaginary root appears in the equation, yet it may have them,
two by two, always in pairs, and with contrary figns. If the degree of the
equation is odd, it will have, at leaft, one real root; and if it's degree is even,
it may have all it's roots imaginary. The like may be obferved of furd roots.
151. Many
SYSTEM OF ANALYTICKS. XXV
151. Many- ufeful indications, concerning the roots of an affeded equation,
may be had from the iigns of the feveral terms.
152. A proof that, in cubick and biquadratick equations, if the fecond term
is wanting, and the third term is pofitive, there will neceflarily be imaginary
roots.
153. In any equation the affirmative roots may be made negative, an3 the
negadve affirmative, only by changing the figns of thofe terms which are in
even places. Here the afterifm, or vacant place, muft always be reckoned for
one. This proved by Examples,
154. The roots of affedted equations may be increafed or diminiflied by any
quantity at pleafure, by refolving the root into two parts, one unknown, and
the other known ; and that only by a fubftitution of equivalents. The new
equation fo found will have the fame roots as the given equation, only they will
be increafed or diminiflied by a known quantity. See the Author's Examples.
155. By a like fubflitution of equivalents, the roots of any equation, though
unknown, may be multiplied or divided by a given quantity, and undergo
many other changes at pleafure.
156. The reafon of thefe feveral procefles is, that, as equals are always
fubftituted for equals, fo the refults muft always come out equal.
157. The ufes of thefe fubftitutions are many. One of which is, that,
though the roots of an equation are unknown, yet, by fuch a transformation,
they may often become known.
158. Another ufe is, the freeing an equation from fradlions or furds. Of
this the Author produces feveral Examples.
159. Some neceflary conditions in the equation, in order to it's being freed
from furds or radicals.
160. But the chief ufe of this tranfmutation of equations, is intirely to take
away the fecond term from any equation by an eafy fubftitution ; of which the
Author gives feveral inftances.
Vol. I. d 161. Or
XXvi , THE PLAN OFTHE LADY *S
i6i. Or the third term may be taken away, by folving a qnadratick equation-,
the fourth by a cubic, he. ; as may appear from the Author's general procefs.
. 162. In an equation wanting the fecond term, the penultimate term may be
taken away ; but it will be by reftoring the fecond term,
163. Thus, in an equation wanting the third term, the ante-penultimate term
may be taken away ; and fo on,
164. Or any equation, in which any term or terms are wanting, may be
• made complete by a new fubftitution.
165. If equations have divifors of one, two, or more dimenfions, they are
^ properly of that order, to which they may be reduced by divifion.
166. Divifion ought firft to be tried by a divifor of one dimenfion, then by
thofe of two, &c.
1 67. Equations of the third degree, if reducible, may be reduced by a linear
or fimple divifor, which is to be found in the manner taught in the 56th Article
before. If an equation of the fourth degree cannot be reduced by a divifor of
one dimenfion, to be found in the fame manner, the reduction muft be at-
tempted by a divifor of two dimenfions. To perform which, the Author throws-
out the fecond term of the equation j as Qiown before, and then afllimes two
general equations of two dimenfions, and multiplies them together, and com-
pares the terms of the produced equation with thofe of the equation given. By
this comparifon (he determines the co-efficients of the affbmed equations,, the
laft comparifon of which amounts to an equation, which in effed is no mora
than cubical. This cubic equation is refolved by the Method of Divifors,. and
it's roots, being fubftituted in the aiTumed "equations, will make them become
divifors of the biquadratick equation propofed. Of this method of folution fhe^
gives us two Examples.
168. Here is the fame procefs as before, but after a more general manner,
and applied-«o'a-{)articular biquadratick equation, which is refolved by it.
169. Sometimes this method will fucceed only by taking awiy the fecond
term of the equation, which will deprefs it to a quadratick.
170. The
SYSTEM OF ANALYTICKS. XXVlt
170. The fame method is purfued, but without taking away the fecond term
of the given biquadratick equation. Two general quadratick equations are
alTumed, and multiplied together, and the general co-efficients oF the product
are determined and eliminated, as far as may be, by a comparifon with thofe of
the given equation. The laft co-efficient in thefe comparifons muft be deter-,
mined by the foregoing Method of Divifors. But this way of refolution feems
to be too tentative to be of any general ufe. It is illuftrated by threp Ex-
amples.
171. The fame method is carried on to equations of five dimenfions, in
which the two affumed general equations are, one of two and another of three
dimenfions. When, by comparifon, the general co-efficients are determined,
they are fubftituted in the fimpleft of the afi'umed equations, which then becomes
a divifor of the given equation ; as by two Examples.
172. The Author extends this method to equations of fix dimenfions, which
(lie manages with great fagacity and fuccefs, though it muft be owned to be
very tedious, precarious, and tentative; but, however, is the bed that can be
had in thefe high equations. She afTumes two general and fubfidiary equations,'
one of two, and another of four dimenfions, which are multiplied together to
produce a general formula for equations of fix dimenfions, that may be refolved
into two fuch equations. Then the general co-efficients are determined as
before, and fubflituted in the fimplefl of the afTumed equations, which will then
become a divifor of the given equation. Of this redudion ihe gives us an
Example.
But an equation of the fixth degree may poffibly be refolved into two cubic
equations, and not otherwife. She i)ierefore affumes t]^o general cubic equa-
tions, and multiplies them together, to conflitute a general formula for thefe
equations. Then, a particular equation of fix dimenfions being given, the
-general co-efficients are determined by comparifon, as far as that can be done,
and their values are finally fubftituted in one pf the afTumed equations, in order
to form a divifor to the given equation.
173. The Author affures us, that the fame method might be applied to the
folution of higher equations, if it -was not for the exceffive tedioufnefs of the
operations. It may very well be fuppoied, that the calculation will becom^j
d 2 , very
XXVill THE PLAN OF THE
very laborious in thofe equations, by what we fee in thcfe of a lower prtler^
And as the method is but tentative at beft, it can hardly deferve to be profe-
cuted any further ; efpecially as we have an exegejis niimerofa to recur to in thefe
cafes, which, though only an approximation to the root, yet will anfwer all real
occafions that can offer. The Author now proceeds to propofe and refolve
feme particular Problems, in order to (how the ufe and application of what is
now delivered.
174. The firft Problem is purely arithmetical, and is elegant enough : To find
four numbers which exceed one another by unity, and their -produSi is 100. I'he
equation of this Problem amounts to a biquadratick equation with all it's terms;
but, by throwing out the fecond term, it is reduced to a quadratick with four
roots. Thefe are irrational, of which two only are real, one pofitive, the other
negative, either of which will folve the Problem. The firft and lead of the four
numbers required, when reduced to a decimal, will be the negative number.
175. The next is a Geometrical Problem, relating to a right-angled triangle.
It's equation is a biquadratick with all it's terms, but when the fecond term is
taken away, it degenerates into a quadratick with a plane root, but irrational.
176. A Geometrical Problem producing a biquadratick equation, the four
roots of which are irrational, and may be all real, and are exhibited by the
figure.
177. An equation may often appear of a higher order than the Problem
really requires, if a prudent choice is made of the unknown quantity, by which
ihe Problem is determined. This is illuftrated in feveral appofite Examples,
178. Another artifice that often prevents Problems from rifing to too high
equations, is finding two values of the fame unknown quantity, and making
them equal. An inftance of this is feen in the next Problem.
179. This Problem is, in a given circle to injcribe a regular heptagon. Thc
Author gives feveral folutions of this Problem, which amount to high equa-
tions ; but, by being compared with each other, are reduced lower. At laft
flie brings it to a cubic equation with a plane root. This is performed by
finding two different expreffions for the fame quantity, and comparing them
together.
x8o. Whea
SYSTEM OF ANALVTICKS. XXIX
x8o. When cubic (or higher) equations cannot be thus reduced, their roots
may be found analytically, but involved in furds, by what are called Cardan's
Rules. But the geometrical method will be more univerfal, by conflru6ting
them, and finding their roots by the interfedion of curve-lines.
i8i. She begins with the analytical folution, or with finding Cardan's Rules;
All cubic equations, that want the fecond term, are rcprefented by four general
formula, differing only in the feveral changes of the figns. To refolve the
firfl general formula, fhe divides the unknown root into two parts, which, after
fubflitution, gives room for fplitting the equation into two, fuch as may cafily
be refolved feparately. This finds commodious values for the two alTumed
parts of the root, and brings us two cubical radicals for the value of the root.
See the Philofophical Tranfadtions, Number 309.
182. The folution of the fecond formula does not differ from the firfl, but
only in the figns.
183. The fame may be faid of the third.
184. And likewife of the fourth;
185. All the four formulse are folved fomething differently, in which the two
parts of the root have only one cubic radical ; but which coincide with the
foregoing folution, and are eafily reduced to it.
186. The limits of thefe roots are here affigned, and it is fhown when they
will be real, and when two of them will be only imaginary,
187. When one root is found of a cubic equation, the other two may be
found without divifion. For, as unity itfelf has three cubic roots, fo any other
quantity has the fame. Therefore, multiplying the root found by the three
roots of unity fucceflively, we fhall have the three roots of the given equation.
This is proved here fynthetically, by returning to the original equation. See
Phil, Tranf. No. 309.
188. This method of folution is illuflrated, by applying it to a given cubic
equation, of which the three roots are thence found.
9 189. Or,
XXX THE PLAN OF THE LADY*ft
; 189. Or, without recurring to the general folution, any particular cubic
equation may be folved, by purfuing the method of that folution. Of thefc
here are given feveral Examples.
190. The Author proceeds to the folution of biquadratick equations, of
which (he takes a general formula, with the fecond term abfent. Then affumes
two general quadratick formulae, which, multiplied together, produce a general
biquadratick equation; and, by comparifon with the firfi: general equation, flie
determines tli€ affumed co- efficients. This will bring her to a transformed
cubic equation, in the manner taught in Article 167 aforegoing. And thus fhe
proceeds to determine the four roots of the affumed biquadratick equation.
See Phil. Tranf.
This folution fhe applies to an iExamplc.
i^i. From the algebraical .refolution of th cfe equations, (lie proceeds to the,
more general (as (be calls it), or to the geometrical folution, which is, by con-
ftruding the feveral loci geometrici^ oV curve-lines, adapted to every equation
confiding of two indeterminates. Every determinate equation may be refolved
into two indeterminate equations, by introducing a quantity into it at pleafure.
Thefe two equations muft confift of the fame two variable quantities, and the
fame conftant quantities, and may be conftruded by two curves. If thofe two
curves are combined in fuch manner, as that they fhall have a common abfcifs,
they will alfo have fome common ordinates at their common points, that is,
their points of interfedion. Thefe common ordinates will be the roots of the
determinate equation, if the quantity reprefenting thofe roots is made one of
» the variable quantities. To exemplify this,, llie affumcs a determinate biqua-
dratick equation, and alfo an equation to the parabola. This flie introduces,
by fubftitution, into ihe given biquadratick, which will then be an indetermi-
nate equation to the hyperbola. She then conllruds thefe two curves upon a
common axis, and draws four ordinates from the four points of interfedion- of,
the curves, which will be the roots required.
192. From this conftruclion thefe notable circumftances will evidently follow;
tkai: the pofitive and negative roots will be on different fides of the common
abfcifs ; that, when two ordinates become equal, or when the two curves do not'
• . • cut
SYSTEM OF ANALVTICKS. KxH
cut but touch each other, two roots of the equation will be equal ; or, when the
two curves cut each other in the vertex, one of the roots will be equal to
nothing ; and where the curves neither touch nor cut, the roots will be ini-
pofljble. *
' 193. It is here Ihown, that, as there may be great variety in reducing a
determinate equation given, to two indeterminate equations, in order to be
conftruded ; fo fuch a choice is to be made of the two loci, that the con-
flrudion may be as fimple as poflible. According as the equation is in degree,
fo each /ocuj fhould be taken, as together to make up nearly the dimenfions of
the given equation.
• 194, Here it is (hown, by an Example, how the feveral loci to the Conic
Sections are to be diftinguiQied from one another.
195. Other cautions to be obferved, in adapting the loci to their equations.
196. Here follow fome Examples, to illuftrate the foregoing dodrine. The
firft is, a determinate cubic equation wanting the fecond term, which is reduced
to a biquadratick, by multiplying the whole by the root, and a fimple equation
to the parabola is aflumed. This is introduced into the given equation by
fubftitution, by which it becomes an indeterminate equation to the circle.
Theri thefe two loci are combined, or conftru^ted to a common abfcifs, and
from their interfedion a common ordinate is drawn, which will therefore
reprefent the root of the given equation. Their other interfedion is at the
vertex, and therefore it's root will be nothing, which was introduced into the
equation. The truth of this conftrudUon is confirmed by a demonftration,
197. The fame equation is again conftrudled by combining two parabolas,
and the conftrudion demonflrated.
1^98. Or, to conftrud the fame equation> the equilateral hyperbola might be
introduced, only by fubtrading one of the equations to the parabola from the
other.
J 99. Or, laftly, by a fmall alteration, one of the loci might have been to
the circle, the other to the hyperbola.
2 GO. Butj
6
XXxIi THE PLAN OF THE LADY*S
aoo. But, without increafing the dunenfions of the cubic equation, it may be
condrudted by an hyperbola between the afymptotes, combined with a parabola;
as is here performed, and the conftrudion demonftrated. And fo may all other
equations be conftruded, that do not exceed the third degree.
In her next Example fhe takes a determinate equation of the fourth degree,
which (he changes into an indeterminate, by the fubftitution of an equation to
the parabola. Into this (lie introduces an equation to the circle, and then
conftruds it by means of thefe two loci : which conftrudion the then demon-
itratcs.
For another Example (he takes a determinate cubic equation, into which (he
introduces a known root by multiplication, which raifes it to a biquadratick.
Then taking an equation to the parabola, by the fubftitution of this after
various manners, (lie produces feveral indeterminate equations ; the laft of
which, being to the circle, fhe makes choice of for conftruding the biqua-
dratick equation. One of it's roots is the known root that was introduced, two
are imaginary, and the fourth is a real but negative root. Then flie demon-
ftrates the conftrudion.
Another Example is, an equation of fix dimenfions, but, being divifible by
a divifor of two dimenfions, it is reduced to a biquadratick equation. By
various fubftitutions of an equation to the parabola, various loci are formed, of
which (he conftruds one, which is to the equilateral hyperbola. But thefe two
loci, being combined as their equations require, will no where interfed each
other, or will have no common ordinates. Which proves, tliat all the roots of
the givfen equation are imaginary and impofTible.
201. In this Example a biquadratick, or cubic, equation Is propofed, to be
conftruded by two conical lociy not to be found (as before) from the given lines
of the equation, but fuch as are already known and defcribed, or otherwife by
fuch as (hall be like to thefe. . This is performed by deriving the two loci in
general (as before), and then introducing new quantities, which are to be
determined from the known lines of the given lociy according to their various
circnmftances. This equation, therefore, is conftruded by means of a given
parabola, combined with a given hyperbola.
It
SYSTEM OF ANALYTICKS. XXXHl
If it Ihould be required to conftruft a biqnadratlck equation with a given
parabola, and with an eilipfis that is of the fame fpecies with an ellipfis given ;
here is an inftance of it, by means of introducing new quantities into the
equation ; which are afterwards to be determined as occafion fhall require.
And the truth of the conftrudion is demonftrated at length.
202. The Author here, by way of anticipation, gives us fome conflrudions
of equations that exceed the fourth degree, though fhe referves the fuller
treating of fuch conftrudions to her next Sedion. She afliimes a determinate
equation of the fifth degree, and likewife an indeterminate equation to the ■
parabola, and, by fubftitution, forms an equation, or locus, to a line of the
third degree, which, combined with the parabola, will conftrud the given
equation. Or, (he (hows how it may be done with the fame locus combined
with an hyperbola. Or, with an hyperbola, and the firft cubic parabola.
Likewife, (lie conilruds an equation of the fixth degree, by a parabola com-
bined with a line, or locus, of the third degree : of which equation (he finds
two real roots, one affirmative and the other negative, and the other four are
imaginary.
203. Then (he tells in what order the loci muft rife, by which we would
conftrud higher equations ; and conftruds (for example) an equation of eight
dimenfions by means of a parabola, combined with another locus of four
dimenfions.
ao4. She then obferves, how equations of the ninth degree (and therefore
thofe of the eighth degree, reduced to the ninth by multiplying by the unknown
root,) may be conftruded by combining two loci of the third degree : which rule
ftie makes general.
205. The mod natural way of conftruding an equation of any degree, is by
a right line for one of the loci, and a curve of the fame degree for the other.
As an example of this method, the Author a(rumes a definite equation of the
fifth degree, makes one of the divifors of the laft term to become indefinite,
that is, aflfumes a locus to a right line, and, fubftituting it in the given equa-
tion, makes it become an indefinite equation of the fame degree as the equation
given. This being conftruded, and the right line drawn as it ought to be by
Vol. I. e ' ttie
XXXiv THE PLAN OF THE LADY's
the nature of the equation, the common ordinates will determine fo many
abfcifles, which will reprefent the roots of the given equation. Thofe roots
will be impoffible, where the right line does not meet the curve.
206. She tells us this method may be of ufe in verifying other conftruclions ;
then proceeds to particular Problems, with their conftrudions.
207. The firft is a Geometrical, or rather Analytical Problem ; between two
given quantities f to find as many mean proportionals as we pleafe. This is applied
to finding two mean proportionals, and arifes to a fimple cubic equation, which
(he raifes to an afFedled biquadratick, by multiplying it by the unknown root.
Then affumes a locus to the parabola, and, by fubftituting it various ways in
the given equation, (he forms feveral other lociy one to a parabola, one to an
hyperbola, and one to a circle. This laft (he combines with the aflumed locus
to the parabola, and conflru<5ls the equation given ♦, finding one real affirmative
root, and the root that was introduced which is equal to nothing, and the other
two roots will be imaginary.
208; Or, without introducing a new root equal to nothing, (he con{lru(^s it
by a parabola, and an hyperbola between the afymptotes.
209. To find three mean proportionals is a plane Problem.
210. To find four mean proportionals amounts to a fimple equation of the
fifth degree, which (he conftruds by means of a parabola combined with an
hyperboloid of the third degree.
211. Or, by the common hyperbola between it's afymptotes, and the fecond
cubical parabola.
212. To find five mean proportionals amounts only to a cubical equation.
Then (he obferves, by what loci fix, feven, or any other number of mean
proportionals may be found.
213. The next is a Geometrical Problem, of three contiguous chords being given,
terminating at the diameter of a circle, to find that diameter ; which Problem has
two cafes. For the middle chord may cut the diameter, either within the
circle or (produced) without.. The equation tliat arifes for the folution of this
4 Problem
SYSTEM OF ANALYTICKS. XXXV
Problem is cubical, which (he multiplies by the root to make it a biquadratick.
Then, aflbming a locus to the parabola, by fubftitution (he finds another locus,
which is to the circle ; by the combination of which two loci (he finds the three
roots, and then determines which of them will folve the prefent Problem.
After which (he proceeds to the other cafe, which, with little variation, requires
the fame conftruftion.
214. A Geometrical Problem, by which the Problem of § 176 is made more
general, the equation afcending to the fourth degree. It is conftrucled by a
parabola combined with an hyperbola.
215. This Problem is, to trifeSi a given angle, (fee § no.) and amounts to a
cubic equation, which is conftruded by two loci, the parabola, and the hyper-
bola between it*s afymptotes. The conftrudlion is demonftraied, and extended
to all the cafes.
216. A further explanation of the trifedtion of an angle, (howing how the
three roots of the equation ferve for all the three feveral cafes, which are
implied in the trifedbion of any angle.
217. The fame otherwife conftruded, by combining two other loci, one to
the parabola, and the other to the circle.
218. This Problem of dividing a given arch into any given number of parts,
is here extended to five equal parts, and arifes to an equation of the fifth
degree. It is conftruded by alTuming a locus to the parabola, and thence
forming an indeterminate equation of the third degree, which is conftrufted by
a curve proper to it, 1 hefe two, being combined, give all the five roots of
the equation,
219. And this may be extended to the dividing any angle into any greater
odd number of equal parts.
e2 BOOK
ZXXVl
THE PLAN OF THS LAD^'s
B O O K I. S E C T. V.
^hf Conjiru^iion of Loci exceeding the Jecond Degree^
220. Having difcourfed at large of the nfe of the Conic Sedions, as
geometrical loci for the conflru6lion of equations ; the Author proceeds now to
higher curves, and their defcription, as the proper loci for conflrudling equations
of more dimenfions. Thefe curves, (he fays, may be defcribed in two different
manners ; one is, by finding as many points as we pleafe in each curve, and
tracing regular curves through them. The other is, by taking a curve already
defcribed of a lower order, and finding by that the points of the other curve,
or locus.
221. In order to defcribe a curve by an infinite number of points, from it's
equation we muft derive the value of one of it*s unknown quantities, and
fuppofe it the ordinate of a curve. Then we muft affume a fucceffion of
values of the other unknown quantity, or the abfcifs, and then the corre-
fponding ordinate will become known, and fo give us a fucceffion of points in
the curve, through which we may trace a regular curve, which will be one
locus. Of this (he propofes an Example in an equation of three dimenfions.
222. This ordinate may be drawn at any conftant angle to it's refpedive
abfcifs.
22J. As an example of this defcription of a curve by points, the Author
alTumes the equation to an equilateral hyperbola ; and, interpreting the 'abfci(s
by fmall numbers continually, (he finds the correfponding ordinates, which give
fo many points in the curve.
224.. And the fame thing will obtain if the abfcifs is interpreted by negative
numbers, beginning from the centre of the hvperbola; fo that the fame,
hyperbola will arife, but only in an inverted pofition.
225. And when the ordinate is made nothing, the value of the abfcifs will
ihow when the curve cuts the axis*.
226, Alfo,
SYSTEM OF ANALYTICKS. XXXVU
226. Alfo, intermediate points may be found, by intermediate values of the
abfcifs and ordinate.
227. A Rule to find whether a curve has afymptotes or no, and where they
are if it has any.
228. But this Rale holds only when the afymptotes are parallel to the-
co-ordinates ; for the hyperbola has it's afymptotes, which may be found from,
another equation to the fame curve, and by the fame rule.
229. The affair, of finding the afymptotes of curves, properly belongs to thcc
Method of Infinitefimals, to which therefore it is referred.
230. Other circumftances of the propofed curve are here inquired into, as,
whether it is convex or concave towards it*s axis. This is eaiily determined by
the Rule of Proportion. For, if a triangle is infcribed in the curve, and an
ordinate is drawn which is in common both to the curve and the triangle; if
the ordinate to the triangle is lefs than that to the curve, the curve will be
concave to it's axis; otherwifc not.
231. But this Rule will not always obtain in all curves; for, in fome, parti-
cular methods miifl: be ufed, as will be feen hereafter. The Author proceeds
to give another Example of defiribing curves by points, which is the firft ,
cubical parabola. Of this flie determines a fufficient number of points, to fhow
it's progrefs, that it cuts the axis cnly in one point, that it goes on ad infinitum,
that it has no af)mptotrs, that it is concave towards it's axis, and that it has a
negative branch like the pofitive, but contrarily pofited.
The next Example is of the firft cubical hyperbol id, the form of which fhe
determines by finuifg it's points ; as alio it*s alympiotes, and other circum^
dances.
She then gives an Example of a curve of the fourth degree, the form of
which (he determines by finding the fevtial points.
232. She further profecutes the fame equation through all it's varieties^ of
pofitive, negative, and imagmary roots; (ho^^ing the ojllertni circumftances of
the curve, and of it's ieveral branches, whidi reluk from thole roots*
Q Another^
XXXviii THE PLAN OF THE LADY*S
Another Example of an equation of three dimenfions, from the roots of
which, and finding the mofl material points, the form and other circumftances
of the curve belonging to it are determined : as it's afymptote, it*s conjugate
oval, &c.
Another Example of a curve of three dimenfions, in which the principal
points are determined by the feveral roots of the equation,
233. The fame equation and the fame curve is further profecuted, and other
of it's properties difcovered : as it's two parts extending to infinity, their
common afymptote, the convexity towards it's axis, &c.
234. The fame method, of defcribing the curve by points, may be extended
to equations in which the indeterminates are involved together, and not eafily
feparable. The points required may ftill be found, though the trouble will be
increafed.
235. The Author makes an apology, for fceming to depart from the method
(he had prefcribed to herfelf, in treating of thefe high equations and their
curves ; and then illuftrates what (he has delivered, by propofing and folving
feveral Problems,
2.^6. The conftrudion of the firft Problem produces a well-known curve
called the CiJJbid of Diodes, and arifes to an equation of the third degree.
This locus the Author defcribes, by finding leveral of the principal points, and
determines it*s afymptote.
237. In this Problem the Author finds another curve by it's points, the
equation of which arifes to four dimenfions.
238. A Problem in which the Author conftrufts a curve, which (he calls the
Witch» It's equation arifes to three dimenfions, and (he determines it's
afymptote and other circumftances.
239. The curve of the next Problem will be the Conchoid of Nicomedes,
t'he equation of which arifes to the fourth degree. This (he conftruds by
finding h*s principal points, it's two diftind parts feparated by a common
afymptote, it's concavity and convexity, and that it has points of contrary
flexure
SYSTEM OF ANALYTICKS, XXXlX
flexure and regreffion. This is in the firft cafe ; for (lie diftinguifhes the
Problem into three cafes^ which (he purfues feparately.
240. As the firft cafe depended upon the equality of two certain lines, fa
this requires that one of them (hall be bigger than the ocher, and fo will
produce a different figure with fomething different properties. The point of
regreffion m the former cafe now becomes a node, where the curve croffes itfelf,
and forms a foliate. The afymptote remains as before, and the curve will have
a like concavity and convexity towards it,
241. The other cafe is, when that line, which before was the greateft of the
two, is now the leaft. This produces a great alteration in the curve of the
former cafe ; for now the foliate entirely vanifhes, and makes the curve have a
continued curvature at it^s vertex, not much unlike that part on the other fide
of the afymptote.
242. The Author propofes a way here, of improving this method of de-
fcribing curves by points ; which is by geometrical conftruflion. In this her
firft Example of it, fhe refumes the Cijfoid oi Diodes and it's equation, § 236,
which (he conftrudts an eafier way by geometrical efFedion.
In her fecond Example (he refumes the curve of § 237^ which fhe conftru(5ts
after a like manner.
Then (he does the fame by the curve called the Witch, § 238'.
And by the Conchoid of Nicomedes, of §, 239, which (he conftruds geome-
trically in all it's varieties. .
. 243. The foregoing conftrudlions are eafily performed by the affiftance of a
circle j others may be made by the help of other fimple curves. As, here an
equation of four dimenfions is conftruded by means of a parabola j but that
parabola muft be varied for every new ordinate. However, every new parabola
gives four points in the curve.
244. Parabolas are here enumerated, and diftributed into orders, according
to their dimenfions. There is only one of the firft order, which is the ^polUnian,
or common parabola : two cubic parabolas, or of the fecond order; three of the
third order, or of four dimenfions ; &c.
245. In
^\ THE PLAN 0*F THE LADY*S
245. In thefe feveral orders of parabolas, thofe are called firfl: parabolas in
xvhofe equation the abfcifs afcends no higher than to the roof, or firft power.
She begins with the conftrudion of the firft cubic parabola, the equation of
which Ihe changes (by fubditution) into that of the common parabola, which
ihe conftruds ; then, by means of this (he eafily finds the points of the other
parabola : and that both for the pofitive and negative branch.
246. The Author proceeds to conftruA the firft parabola of the fourth
degree, by changing it's equation of four dimenfions (by fubftitution) into the
equation of the firft cubical parabola, which has been conftrufted. Then, by
the help of fimilar triangles, for every ordinate of the alTumed parabola (he
determines a point of the curve required, in each branch affirmative and
negative.
247. By the fame method, from the firft parabola of the fourth degree the
Author conftru6ls the firft parabola of the fifth degree, as to both it*s branches
affirmative and negative.
248. She then fhows, in general, that we may always conftruft a firft
parabola of any degree, by means of a triangle, and of the firft parabola of
the next lower degree.
249. The Author then proceeds to conftru6l other parabolas befides the firft,
and that of any degree, by means of the firft, which (he fuppofes already
defcribed. As, here flie defcribes the fecond cubic parabola, by finding it*s
ordinate from that of the firft, being reduced to a common abfcifs. And, in
like manner, flie conftruds the third parabola of the fourth degree, by reducing
the value of one ordinate to that of another.
250. She adds here a ufeful Remark concerning any of thefe parabolas, or
paraboloids ; which is, that the fecond parabola of the fourth degree is no other
than the ^common parabola, only redoubled on the negative fide : and fo in all
other, in which the index of the power of the ordinate is double to that of the
abfcifs, and both even numbers. But if the index of the power of the abfcifs
is an odd number, the curve will be no other than the common parabola,
without fuch reduplication. And this holds good of all hyperbolas as well as
parabolas.
251. She
SYSTEM OF A N A L Y T I C K S. xfi
251. She goes on to the conilrn(5lion of hyperbolas (or hyperboloids) of any
degree. There are only two of the third degree ; the fiiil has it's ordinates
reciprocally proportional to the fquares of the abfcifles, in the fecond the fquare
of the ordinate is reciprocally as the abfcifs. The firft of thefe ihe conftrufts
by the help of a common parabola and hyperbola, by means of which flie finds
it*s points. The other will be the fame curve in efFed, and may be conftru6ted
the fame way, only by changing the co-ordinates into each other.
252. The Author proceeds to conftrufl hyperboloids of the fourth degree,
or fuch wherein the ordinate is reciprocally as the cube of the abfcifs ; or the
fquare of the ordinate is reciprocally as the fquare of the abfcifs ; or the cube
of the ordinate is reciprocally as the abfcifs. The firft: (he conflrufts by the
help of the common hyperbola and the firft cubical parabola j the fecond is no
other than the common hyperbola itfclf i and the third is the fame as the firft:,
if the co-ordinates change places.
■253. She goes on to conftruft hyperboloids of the fifth degree ; and, firft:,
that in which the ordinate is reciprocally proportional to the fourth power of
the abfcifs. She finds the points of this, by firft conftruding a common
hyperbola, and then, in proper circumftances, a firft paraboloid of the fourth
<legree. She alfo-conftruds another hypcrboloid of the fifth degree, in which
the fquare of the ordinate is reciprocally as the cube of the abfcifs, by affuming
two other curves of an inferior degree. In all thefe confl.rudions fhe deter-
mines the afymptotes of the curves, and their other afFedions. And the fame
method might be purfued in Im of higher degrees.
254. She obferves that all firft parabolas, defcribed about the fame axis,
will cut one another in the fame point. This point will be diftind from their
common vertex ; and, befides, they muft all have the fame parameter, x
255. Likewife, that thefe firft parabolas, in tending to this common inter-
fedion, the higher their dimenfions are, the nearer they approach to the
tangent ; and, after they are paft it, the nearer they approach to the axis.
And the firft hyperboloids have alfo a like property.
256. Having conftruded thefe paraboloids and hyperboloids, or curves of
two terms only ; the Author proceeds to fuch as have feveral terms, which (he
Vo£. I. f diftinguilhes
Xlii THE PLAN OP THE LADY^S
diftingurflies into three cafes. The firft cafe is of thofe curves, or their equa-^
tions, in which the ordinate is but of one dimenfion only, and is found only m.
one term. In the fecond,. the ordinate arifes to any power, but is found in one-
term only.
In the third, the ordinate is found in more terms than one, and of any
number of dimenfions.
[257. She gives here an Example of the firft cafe. The equation of the
curve to be conftrudled is of the fourth degree, and has three terms. By a
convenient fubilitution this equation is refolved into two others, one of which
contains only conftant quantities, and the other belongs to a firft parabola of
the fourth degree, which is here conftruded, and the co-ordinates of the other
curve are eafily derived from it ; which curve,, it is obferved, will be a portion
of a parabola of the fame degree.
258. Another Example of the fame cafe, in which the equation of the curve
to be conftruded has three terms and four dimenfions. Here again, firft, the^
equation is refolved into two others by a fubftitution, and then the curve is
conftructed by means of two firft parabolas, one of three, and the other of
four dimenfions.
259. A third Example of the fame cafe. The equation of the curve is of
four dimenfions, and has. four terms. This likewife is refolved into two other
equations by a fubftitution, of which one is fimilar to that which was con-'
ftrufted in the preceeding example, and the other is to the Apollonian parabola ;.
and by means of thefe two curves the required one is eafily conftruded.
Here the Author remarks that, if an equation fhould more- abound in terms,
the fame artifice might be ufed ; and that, although the conftrudion in this
cafe might become more compounded and perplexed, yet the fame method
would obtain.
She obferves alfo that the equation in this example might have been'
refolved, by another fubftitution, into three equations belonging to as many
parabolas of different orders ; and then, by means of thefe auxiliary curves,,
the principal curve might have been conftruded.-
060.. It
■-.V
SYSTEM OF ANALYTICKS. xUn
160. It is here fhown, that the co-ordinates of thefe curves may make
any angle.
261. The Author gives here an Example of the fecond cafe, in the con-
firuflion of a general equation of many terms, which, by a convenient fubfti»
tution, fhe reduces to cafe the firfl. See the Example.
262. An Example of conftruding a curve of the third cafe. The equation
here propofed is general, and is refolved, by a proper fubftiiution, into others
which belong to the firft cafe ; fo that, by the conftrudtion of thefe curves, i;he
co-ordinates of the propofed curve are obtained.
263. Hitherto the Author has confidered only thofe equations which have
their indeterminate quantities feparate ; (he here obferves that, when the inde-
terminates are involved with each other, the foregoing rules cannot take place,
hm that a feparation of the variable quantities muft be made, either by common
divifion, or by the extraction of roots, or by a congruous fubftitution, or by
other expedients. She then gives two examples of the feparation of the
indeterminate quantities : in the firft, it is performed by common divifion ; in
the fecond, by a convenient fubftitution.
264. Having (hown how to prepare equations of that kind for conftrudlion,
fhe proceeds to the a6lual conftruflion of them, taking here the firft equation
in the preceding article, and conftrudting it by means of equations which
come under cafe the firft of article 256.
265. The conftru<5tion of the other equation in § 263 : which, it is (liown,
may be performed by cafe the third of § 256.
266. A remark. That a convenient fubftitution may be of ufe even in thofe
cafes in which the indeterminate quantities are already feparated ; and may
fuggeft a conftrudlion which is more eafy and elegant.
267. An inftance of the truth of the foregoing remark appears here, where
the conftrudiion of a curve, the equation of which has four dimenfions, is
f 2 facilitated
Xliv THE PLAN OF THE t A D TT ''s
facilitated by a fubftitiition, although the variable quantities in that equation
were feparate. With this the Author ends her examples of the conftruclioH'
of curves.
BOOK I. SECT. VI.
Of the Method De Maximis et Minimis, of the Tangents of Curves, of Contrary-
Flexure and RegreffiQH ; making iife only of the Common Algebra..
268. The Author here obferves that, although the Calculus of Infinitefimals*^
be the fimpleft and (horteft method, and the mod univerfal, for managing fuch'
(peculations, yet the folution of fuch queftions may be performed by common^
Algebra^ in curves that are expreffed by finite algebraical equations.
She begins with the Maxima and Minima ; that is, to find in geometricaf
curves the greateft or the leaft ordinates ; and (hows that, in either cafe, two-
ordinates coincide, and confequently two abfciffes become equal ; and thence
two roots of the equation belonging to the curve, taken either in terms of the
letter which expreffes the abfcifs, or of the letter which exprefles the ordinate^,
become equal to each other.
Her firft Example is. To find the greateft or leaft ordinate when thit curve
is an EUipfis ; which flie does by forming a quadratick equation which has.
equal roots, and comparing it, term by term, with the equation of the curve.-
She then Qiows how to perform the fame thing when the equation of the curve-
is of the third, fourth or higher degree; which is, by forming an equation of
the fame degree, that has two equal roots, and comparing it,, term by term>,
with the equation of the curve. See the Examplesr^
259. A fhorter and eafier way of doing the fame thing; which is, by
multiplying the terms of the given equation by the terms of an arithmetical'
* Rather Flwdons* Editor.
progreflion.^
SYSTEM OF ANALYTICKS. xIt
progrelTion, For, if an equation has two equal roots, (which is the cafe of a
maximum or minimum,) one of thefe roots will, of neceffity, be included in the
pFodud of that equation multiplied by the arithmetical progreffion. This is
demonflrated j and the two preceding examples are refumed, and the fame /
refulis obtained, although different progreffions are ufed.
270. The Author proceeds to find tangents and perpendiculars to curves b^
a like method ; previoufly fhowing that the queflion is reduced to this : To find
a circle that Ihall touch the curve in any given point. This alfo is performed
by means of equations that have two equal roots : which (lie explains, and
illuitrates by an example of drawing a tangent to the JpoUonian parabola. Tiie
equation which thus arifes is folved, firft, by comparing it with another qua-
dratick having two equal roots; fecondly, by multiplying the terms of it by
the terms of the arithmetic progreffion 3, 2, i ; and, laftly, by multiplying
the terms by the progreflion 2, i, o.
271. Another Example of drawing a tangent to a curve of which the
equation is cubical, worked both by comparing it with an equation of the fame
degree which has two equal roots, and by multiplying the terms of it by the
arithmetical progreflion 3, 2, i, o.
272. It is obferved, that, in general, the moft convenient progreflion will
be that which forms the exponents of the letter according to which the
equation is ordered.
273. The Problem of drawing tangents is folved in a way fomewhat dif-
ferent, but more fimple ; and the formula here derived are of ufe alfo in
finding points of contrary flexure and regreflion.
274. Points of Contrary Flexure and Regreflion are here defined 5 and it is
fhown that, as the nature of maxima and minima, and of tangents, requires
equations that have two equal roots, fo in contrary flexures and regreflions
three equal roots are required. An example of finding the point of contrary
flexure is given, by way of illuftration.
275. The Author obferves that the operation is the fame for finding the
points of regrefl[ion in curves, as for finding points of contrary flexure ; fo
6 that.
Xln THE PLAN OF THE LADY*S SYSTEM OF ANALYTICKS,
diat, to didinguifh them, there is no other way, but to find, by means of a
conftru(5tion, the figure and proceeding of the curve.
She fays that tlie fame ambiguity arifes in queflions de maximis et mtn'imis^
which can only be removed by acquiring fome knowledge of the difpofition of
the curve. She then obferves that, by the fame condition of three equal roots,
we may find the radii of curvature ; but, intending to treat of thefe things in
the fecond Volume, (he here puts an end to the firft.]
N. B. It being my intention to deliver what I have to offer on the fecond
Volume in Notes, as is mentioned in my Advertifement prefixed to this
Work, the reader will fee the propriety of my continuing the I^lan of the
Lady's Syjlem of Aualyiicks no further. J. H,
ANA-
ANALYTICAL INSTITUTIONS
BOOK I.
ANALYTICAL INSTITUTIONS.
BOOK I. '" , V
THE ^NJ LYSIS OF FINITE qUANTTflES.
THE Analyfis of Finite Quantities, which is commonly called the Algebra of Introduaioai
Cartefius, is a method of folving Problems by the life and management ^y^\ "
or finite quantities : that is, from certain quantities and conditions, which are j^iahthks.
given and known, we may come to the knowledge of others which are unknown
and required ; and that by means of certain operations and methods, which I
propofe to explain by degrees in the following Sections.
SECT. I.
Of the Firji Notions and Operations of the Analyfis of Finite ^antities.
I. The primary operations of this Algebra, or Analyticks, are the fame asTheopera-
thofe of common Arithmetick; which are, Addition, Subtradion, Mulripiica- tions of Al-
tion, Divifion, and Exiradion of Roots. But with this difference, that whereas S^^'"''' '^^^^'
in Arithmetick thofe operations are performed with numbers, in Algebra they
are performed (or perhaps only infinuated) with fpecies, or the letters of the
alphabet ; by which quantities are denominated and calculated in the abftrad,
of whatever kind they may be, whether Geometrical or Phyfical, as Lines,
Surfaces,, Solids, Forces, Refiftances, Velocities, &c. And therefore this kind
B of
t ANALYTICAL INSTITUTIONS. BOOK I.
of Aritbmetick is often C2i\\t(^.^he Algorithm of ^antities, or Specious ^rithmetick.
And indeed this is of a much more excellent and general nature than that can
be, though all it's operations are the fame; as well becaufe thefe quantities are
not confounded one among another in their operations, as numbers are ; as
becaufe in this Calculus known and unknown quantities are treated indifferently,
and with the fame facility ; and lallly, becaufe Analytical demonftrations are
general, and therefore applicable alike to all cafes; whereas in Arithmetick
they are particular, and in every different cafe require a new determination.
Pofitiveand 2. Now of thefe quantities fome are pofttive, or faid -to be greater than
negative nothing ; Others are lefs than nothing, anci therefore are called mgativs» To
dS"-u^ni€d ^xp^^'" t^^s t)y an example. The goods in our own poffeflion may be called
^ ' pofitive, but thofe which we owe to others are negative, becaufe they muft be
fubtraded from the pofitive, and therefore will diminiOi their fum total.
Wherefore, as the capitals in our poffeflion are pofitive, and are anfwerable for
our debts ; fo the debts we owe will be negative quantities. In like manner,
if a body or point in motion is diredled towards a certain mark, and in it's
paffage defcribes a fpace, this fpace may be called pofitive; but afterwards if
jt receives an oppofite diretlion, it will indeed defcribe a fpace, but this fpace
will be negative in refped: of the mark to which it ought to go. Wherefore,
in Geometry, if a line drjiwn one way is affumed as pofitive, (for this is quite
arbitrary,) a line dtawn tlie contrary way will be negative.
Thefignsof 3. Pofitive and negative quantities in Algebra are diftinguilhed by means of
pofitive and certain marks, or figns, which are prefixed to them. To pofitive quantities
quantufes, ^^^ ^^^^ +» °^ P^^^^ ^^ prefixed : to negative quantities the fign — , or minus,
with other And when a quantity has no fign prefixed, as when it ftands alone, or is the
marks, ex- firft among Others, it is then always fuppofed to be affeded by the pofitive
plained. ^g^^^ 'pj^g ^^^^ _^^ jj^g Contrary of which is hF» is an ambiguous fign, and
fignifies either plus or minus. So, for example, ± a infinuates, that the quan-
tity or number reprefented by a may be taken either affirmative (that is, pofi-
tive) or negative. The mark = fignifies equality, and therefore a ::=: b informs
us, that the two quantities expreffed by a and b are equal to each other. So
a > b means, that a is greater than b, Alfo, a < b tells us, that a is lefs
than b. The equality of ratios, or the geometrical proportion of three or four
terms, is thus expreffed : a . b :'. b . c, when there are three terms ; that is,
the ratio of ^ to ^ is equal to the ratio of b to c. Alfo, a . b :: c » d means,
that a hto b as c is to d. Laftly, the fign 00 denotes infinite, and therefore
a :=: 00 fignifies, that a is equal to infinite, or is an infinite quantity.
Quantities 4. A quantity is fimple, incomplex, or of one term only, when it is ex-
are divided preffed by one or more letters, but thofe are not feparated or diftinguiffied from
mto fimple or ^^^ another by the fign either of addition or fubtradion. Such are a, ab, aac,
corapoun . ^^^ ^^^^ ^.^^^ ^^ ^^ ^^^^ contrary, quantities are compound, or of feveral
terms.
SECT,!. ANALYTlCAt INSTITirTIOMS. fjf
terms, when they are exprefled by fgveral letters, feparated from one another
by the figns -4- or — . Such are ^ + ^, ^^2 — ff-^ ^^^ s"d ^^^^ ^i^^- ^^^
therefore a + If will be a quantity of two terms, or a binomial; aa — /f + hh
will be one of three terms, or a trinomial, &c.
Addition of Simple ^antities^ being Integers,
5. Simple quantities are added to one another by writing one after another, Addition of
prefixing to each it's proper fign. As if we were to add a io b and c, the i'v^P^'^ ^"^«-
fum would be reprefented by a -\- b -\- c. If we were to add a to — b, the ^^^"^^*
fum would ht a — b. To add « to ^ to <z to b^ the fum would bQ a + b-\- a -{-b.
But here it muft be obferved, that a + a Is the fame as 2^, and ^ -f- ^ is the
fame as a/' ; therefore the fum will be la 4- ib. Therefore, to add the fame
quantities, or fuch as are exprefled by the fame letters, it will fuffice to prefix
to the fame letter fuch a number as fliall contain fo many units, as are the times
that the letter Ihould be repeated. Thus, the fum of ac to ac to ac, that is,
ac -{- ac -{• ac, will make 3*?^, and this number is called ihe Numeral Co-efficient
of the quantity. And if the quantities to be added, being denominated by
the fame letter, fliall have different co-efficients, thofe co-efficients are to be
added by the ordinary rules of Arithmetick. Thus the fum of la and ^a^ to-
gether with b and 4^, will be 7a + 5^. And thus the fum of a and 3^, and
— 2r, and yr, and 5^, will be ^ + 3^ — 2^ + 7^ + 5^. But ^ -f ^^ will,
make 6a, and — 2c ■\- jc make fr. Therefore the fum will be 6^ -f- 3^ 4- ^c.
S^]ftra6lion of Simple ^antities, being Integers,
6. To fubtrad one quantity from «nother, the fign muft be changed of that Subtraaion
quantity which is to be lubtraded, and then with it's fign fo changed it rnuft °^^*'"p!'-'
be wrote with the other. Thus to fubtrad b from a, we muft write a ■— b ; ^l""*"*^""-
where it may be obferved, that if a is a quantity greater than b, the remainder
of the fubtradion, or the difference, will be pofitive. But if ^ is greater th'an a
m that cafe the difference will be a negative quantity. To fubtra^fl aff from bbc,
it will make bbc — aff. To fublraft 2a from ^a, it will make 5^ — la ; but
^a leffened by 2a make 3^2, fo that the remainder will be ^a. And to fubtrad
. — b from a, it muft be written a + b. Nor fliould it feem ftrange, that to
fubtrad the negative quantity — ^ it muft become pofitive, that the remainder
" 2, niay
^ ANALYTICAL INSTnUTIONS, BOOK!,
may be « + ^ ; for as much as to fubtrad one quantity from another is the
fame thing as to find the difference betueen ihofe quantities. Now the dif-
ference between a and — ^ is a A- b^ juft in the fime manner- as the difference
between a capital of loo crowns and a debt of 50 is 150 crowns. For frpm
having an hundred and having none, the difference is aa hundred ; and from
liaving none to having a debt of fifty, tlie diifcrcnce is fifty ; therefore, from
having an hundred to having a debt of fifty, the difference mud be an hundred
and fifty. Thus, for the fame reafon, to fubtrad b from — «, it muft be written
— ^ — h \ and to fubtrad — h from — ^, it muft be written — a ^ h^
Multiplication of Shnple ^anfities, hei^^g Integers,
Multiplica- 7. Simple quantities are multiplied by writing them one after another, without
lion of fimple any fign between, (unlefs fometim.es the mark x,) and the refuking quantity is
jjuacuties. called the Prcdi'My as alfo the quantities fo multiplied are called the Favors or
MultipItJrs, But as to the fign which is to be prefixed to ihefe produds, the
general rule is this ; that if the quantities to be multiplied are both pofitive or
both negative, then the pofitive fign muft always be prefixed to the produd :
but it one of thofe quantities, whichever it is, is pofitive, and the other ne-
gative, then the negative fign muft always be prefixed to the produd. The
reafon of this is, becaufe multiplication is nothing elfe but a geometrical pro-
portion, of which the firft term is unity, the fecond and third terms are the two
quantities which are to be multiplied together, and the fourth term is the pro-
dud. And therefore being placed in a row, unity for the firft term, one of the
multipliers for the fecond, and the other multiplier for the third ,• therefore, by
the nature of geometrical proportion, the fourth muft be fuch a multiple of
the third, as the fecond is a multiple of the firft. If the fecond and third
terms are pofitive, for example, if it is i . ^ :: /^ . to a fourth -, the firft term
or unity being pofitive, the fourth muft therefore be pofitive. *But if the fecond
is negative, and the third pofitive, that is, if i . — ^ :: ^ . to a fourth; whereas
this fourth muft be fuch a multiple of the third as the fecond is of the firft,
and the fecond being negative, therefore the fourth muff be negative. Let the
fecond be pofitive and the third negative, that is, let it be 1 . a \\ — h . to ^
fourth. Now, whereas this fourth muft be fuch a multiple of the third, as the
fecond is of the firft, and the fecond and firft being pofitive and the third ne-
gative, the fourth cannot be otherwife than negative. Laftly, let both the
fecond and third be negative, that is, let it be i . — a '.', — b . to a fourth.
Now the fecond being here a negative multiple of the firft, it follows that the
fourth muft be a negative multiple of the third. But the third is already ne-
gative, and therefore the fourth muft be pofitive. Wherefore the produd of
^ into b will be ab. That of a into — b will be — ab. That of —• « into ^
8 will
SECTr !♦ ANALYTICAL INSTITUTIONS. $
will alTo be — nl>. That of — a into — l> will be al^. That of a into ^
into £ will be ai^c. That of a into — (^ into c, will be •— ^bc ; bceaufe « into
— i? will be — aif, and — ^/^ into c will be — «/'<:. And the produdl of
-— a into — ^ into c will be ^/'<r.
If the quantities to be multiplied Qiall have numeral co-efficients, thofe co-
efficients muft be multiplied together by the common rules of numbers, and
the produd muft be prefixed to'that of the letters. Hence the produd of 6a
into — Sk will be ^ — 48«^c. And the produd of 2a into — 2^. into — 3^
will be iiabc. And the like of all odiers.
8. Now becaufe the produd of a into a is aa, that of a into a into ^, or of Notation
aa into a, is aaa, that of a into a into a into *?, or of aaa into ^z, is aaaa, and °^ ^"^P'e
fo on fucceffively ; to prevent the repetition of the fame letter fo often, it is ^°
ufual to write a^ inflead of aa, a^ inftead of aaa, a* inftead of aaa:i, and To of
others. That is, we may write a little above the letter fuch a number as (liews
the number of times the letter ought to be repeated, which number is called an
Jfidex or Exponent. We may write indifferently aa or «*, but higher produds
or powers are more commonly exprelfcd by their exponents.
9. As the produd of a number multiplied by it fe If is called the Sjuare ofNamesof ths
that number, or it*s fecond power j fo if this produd is again multiplied by the po^'^'"^! ^."^
fame number, the new produd is called the Cube or the third power of the JjI^^jJ'^j^^Jq'"'^!
fame number. And the produd of the cube by the fame number is called thepofulve and
Biquadrate or fourth power of the fatne number; and fo on. Thus the quan- negative,
tity a multiplied by a, or a*, is called the Square of a^ or the fecond power of
/z ; «' is it's cube or third power ; a^ it's fourth po-ver, &c. Therefore za and
«* will be very different from each other, the firll: being the fum of a and ^, or
a + a, the other their produd, or aa. The fame is to be underftood of 3^1
and a^j of 4^ and a* y and fo of others. Now as the produd of + into 4-,
or of — into — , is always pofitive ; it proceeds from thence, that as well the-
fquare of ^ as of — a will be always pofitive, or aa. So on the o;her hand
the cube of a, or a^, will always be pofitive, but the cube of — a will always
be negative, or — a^. For — a into — a makes aa^ and aa into — a
makes — a*. Thus the fourth power as well of — 'a as of a will be pofitive.
And in general, when the exponent of the power, to which we would raife the,
given quantity, is an even number, whether the quantity itfelf is pofitive or
negative, that which refults will always be pofitive. And when the exponent
is an odd number, if the quantity is pofitive, the refult wil] be pofitive; and if.
k be negative, the refult or power will be negative. .
Htvlfion.
ANM/YTICAL INSTITUTIONS*
BOOK If
JOivifion cf Simple ^aiititieSt being Integers,
Divlfton,
what.
10, Djvifion is an operation diredlly contrary to Multiplication; and what
this compounds, that again refolves. Thus, becaufci?,^ is the produft oi a into ^,
therefore if we divide ^^ by <2, we (hall have b for the quotient. And if wc
divide it by b, the quotient will be a. So dividing abc by c, we (hall have
the quotient ab. And fo on. The quantity to be divided is called the Divi-
dend, that by which the divifion is performed is called the Bivijor^ and that
which refuhs from the divifion is called the ^uoliertt, as in common Arithmetick.
Therefore whenever in the dividend and the divifor the fame quantities are
found, they iTiay be taken out of both, or as it were cancelled, and what is
left will give the quotient. Thus, if we are to divide- aa by a, the quotient will
be a. If we divide ti^ by a, the quotient will be aa. If we divide a^b^ by tf*^\
the quotient will be ab. If the dividend and divifor (hall have numeral co-ef-
ficients befides, they are alfo to be divided by the common rules of Arithmetick,
and the refuking numeral co-efficient muft be prefixed to the hteral quotient.
Thus, if we divide 3a'<^^ by 3^% the quotient will be a^. Dividing s6a^b^ by
Saby the quotient will be 'jab\ And here it may be obferved, that whenever
the quantity to be divided is the fame as the divifor, then the quotient will be
unity; as dividing b by b, ya^ by 7^', and fuch like. The reafon of which is
plain, becaufe to divide is to find how often the divifor is contained in the
dividend, the anfwer of which queftion is the quotient.
When quo- ^^' Wherefore, in the divifor and dividend, when no common quantities or
ticnts are to letters are found, by means of which the divifion may be performed in the
be repre- foregoing manner, the quotients will receive the form of fradions. Thus, to
divide a by b, a^ by be, gaabb by zcCy &c. the quotients are to be wrote thus :
a' t^aabb
IT '
fented as
iraftions.
"T"* TT > i^^ , &c ; that Is, place the dividend above, and the divifor
2CC
under it, with a little line between them ; and It Is to be underRood, that a
ought to be divided by b, a^ by bc^ &c ; and thefe are called Fractions, in which
the quantity above the line is called the Numerator, and thd.t below is the De-
nominator. Thus if any of the letters of the divifor, but not all, ihall be in
common with the letters of the dividend, thofe that are common are to be taken
away from each, and of thofe that remain a fradion is to be formed. Thus, if
wc were to dWi^ca^bb by ^abcct the quotient will be -J^r > ^^ "~ — • And if
^abcc
5'<=
2abb
WC divide loab^ by j^bcc^ the quotient will be -^ — . And fo of all others.
12. Now,
SECT. I. ANALYTICAL INSTITUTIONS, 7
12. Now, becaufe both the dividend and divifor maybe either pofitive orThefijrnof
negative, it is neceffary in every combination of cafes to fix a rule, for the fign the quotient,
which is to be prefixed to the quotient. This rule is the fame as that which ^^'^^'
ferves for multiplication. That is to fay, that if the dividend and the divifor
h:ive both the fame fign, whether pofitive or negative, the quotient will be
always pofitive. But if they have contrary figns, the quotient muft be nega-
tive. The demonilration depends on that of multiplication. For as multipli-
cation is a proportion, of which the firft term is unity, the fecond and third arc
the two multipliers, and the fourth is the produft ; fo divifion is the fame pro-
portion, but inverted. Of this the firft term is the dividend, the fecond the
divifor, the third is the quotient, and the fourth is unity. Let it be required
to divide d: ^^ by + If. Then the proportion will be Hh <2^ . Hh ^ :: ma . i.
Here 1 place the mark * before the third term or quotient, as not yet knowing
whether it ought to be pofitive or negative. Now, confidering this proportion
to be like that of multiplication, but the terms placed inverfely, it is known
that when the fecond term l> is pofitive, the firil term ai^ cannot be pofitive,
unlefs the third term a is pofitive alfo ; and the fecond i> being negative, the
firft ah cannot be negative, unlefs the third a be pofitive. Wherefore, in divifion,
when the two firft terms, or the dividend and divifor, are both pofitive or both
negative, the third term, or quotient, muft neceflarily be pofitive. In like
manner, in this proportion, the fecond term ^ cannot be pofitive and the firft
a!f negative, or the fecond if negative and the firft alf pofitive, unlefs the third a
be negative. So that in divifion, the dividend being pofitive and the divifor
negative, or on the contrary, the quotient of neceflity muft be negative.
1 2' ^or this reafon it will be the fame thing whether we write (for example) signs rcci-
a —a . • . . ■ -r procal in
■-^ , or ~-y-i becaufe if a pofitive is to be divided by If negative, or if a ne- fim pie frac-
tions*
gative is to be divided by If pofitive, in both cafes the quotient muft be nega-
tive. Thus it will be the fame to write -^ , or -r- •
Eittra5Jion of the Roots of Simple ^antities, being Integers.
14. As quantities have their feveral powers, the fquare, the cube, the bi- Roots of
quadrate, the fourth power, &c, fo among the roots of fuch powers there is ^."?P'^ ^l'^*""
the fquare-root or fecond root, the cube-root or third root, the fourth root, &c. j^g^^,^ "
The denomination of roots is the fame as that of the exponents of powers.
Therefore the index or exponent of the fquare-root is 2, of the cube-root is 3,
&c. And to extradl the root of a given quantity, we muft find fuch another
quantity, as being multiplied into itfelf as many limes, all but one, as are the
units
6
ANALYTICAL INSTITUTIONS.
BOOK I.
- units in the index of the root, fhall have for the produfl the (^uaniity whofc
root is propofed to be extra6led. Thus a will be the fqnare-root of aa^ the
cube-root of ^% the biquadratick-root of ^*, &c. In the 'fame manner the
fquare-root oi aaid will be aby of iGcabbcc will be j^abc ; the cube-root of
2 7«V will be ^ax i and fo of others.
Signsofroots. 15. And fince the produd of minus into viinus is always plus^ as above-,
imponible thence it follows that the fquare-root of <?^ will be either d! or — ^z, that is ±: a.
roots. Yi is not fo with the cube-root, which will always be pofitive if the given cube
is politive, and will be negative if this be negative; for the cube of ^ will be a},
and the cube of — ^ will be — a^. But the fourth root will be either pofitive
or negative. And to fpeak in general, the roots whofe index is an even num-
ber will always be either pofitive or negative ; but thofe whofe index is an odd
number will be pofitive if the power propofed be pofitive, and negative if that
be negative. And hence it is, from the fame property of the figns in multi-
plication, that no pofitive or negative quantity can ever produce a negative
power having an even exponent. So that it is impoffible to find the root of a
negative power with an even exponent. Such roots as thefe, of a negative
quantity with an even index, are therefore called Impoffible or Imaginary. Thus
the fquare-root of »— - aa will be imaginary, as aifo the fourth root of — <«*,
the fixth root of — ' a^, he. But fuch as thefe will be true and real roots, the
cube-root of — a^
'% the fifth root of — ^^, &c.
Roots ex-
t rafted of
imperfeft
powers.
16. But for the generality the quantities propofed, of which we are to extra<n:
the loors, will not be true fquares, cubes, or other powers, which are produced
by the mvihiplicaiion of rational quantities into themfelves, but will be the
products of another kind ; as ab^ abc, &c : in which cafe we make ufe of the
mark v/, called the Radical Sign or Vinculum. Hence \/ab, or fimply */ ab,
denotes the fquare-root of ab ; \/ abc denotes the cube- root of abc. And thus
\/ ftands for the fourth or biquadratick-root, 5^ flands for the fifth root, &c,
And fuch quantities as thefe, affefted by a radical fign or vinculum, are called
Swds, or Irrational ^iantities.
Addition of Compound ^antitieSy being Integers,
Compound
quantities
added.
17. By the addition or fubtradion of fimple quantities, compound quantities
are produced. Therefore, to add thefe together, it is fufficient to write them
one after another with their proper figns. So to add a -[ b to c — ^, we may
write a -\- b -{■ c — d. To add laa — ax to o^cc -^ lyy^ the fum will be
laa — XX -\- o^cc + lyy. To add aa — xx to bb -f xx -f yy, we fhall have
aa — XX + bb '\- XX -^ yy -, but here it is to be obferved, that — xx and + xx
remove
SECT. I. ANALVTICALIN5TITUTI0NS. 9
remove or deftroy each other, and therefore may both be cancelled or omitted,
and then the fum will be aa -{- bl^ -^ yy. To add laa — c^hh to aa ->r ihh -^r yy ^
the Turn will be %aa — 5^^ + ^^ + ihb 4- yy ; but here Q.aa -f aa make
tJ^z^, and khb + 2^^ make — 3^^, and thcictorc the fum will be 3^^ —
Subtra^ion of Compound ^iantliies^ being Integers,
18. The figns mnft be changed of that quantity which Is to be fubtracted, Compound
and then with the figns fo changed it is to be wrote after that, from which the ^"^"'^'o*^^
fubtraftion is to be made. Thus to fubtraft c — d from a ■{• b^ we mufl " "^ ^ •
write them thus, a + b — c '\- d ; and the reafon is plain. For if we were to
fubtraft only the quantity c, we fliould write a ■\- b -• c. And now having
fubtraded too much, (for we ought to have fubrrafled only c — d, or the dif-
ference between c and d,) having fubtrafted, I fay, more than we ought by the
quantity d, to make amends we muft add d^ and fo write the remainder
a -{■ b — c + d. The fame is to be done for quantities more compounded.
To fubtraCl « +• 3^ from 3^ + 2^, it will be wrote 3^ + lb — a — 3^ ;
but by a redudion of fmiilar terms, becaufe 3^31 — a is 2^, and ^b — 3^ is
— bi the remainder will become 2a — b. To fubtrad ^ab — 2bc 4- 2cd
fi-om c^ab — ^bc + 2cdy after a proper redudion the remainder will be
<iab — 2hc,
Multiplication of Compound ^antitieSy being Integers,
19. The rule for the multiplication of fimple quantities being underftood, Compound
that for compound quantities will be very eafy. Let one of the fadors be wrote quantities
under the other, as is ufual in the vulgar Arithmetick, and then all the terms ™"^'"'P^^*^'^'
of the multiplicand muft be multiplied fucceflively by every term of the mul-
tiplier, according to the rules already given for the multiplication of fimple
quantities ; and what refults, after the ufual redudion of fimilar terms, will be
the produd required. Thus if we are to multiply a -\-b — c
^ -f ^ — . f by a:, let them be wrote as in the margin. Then let every
X term of the multiplicand, placed above, be multiplied by
■ the multiplier placed under it, and the produd will be
ax 4- bx — ex ax -^ bx '^ ex, as by the operation. Thus if we were ro
C multiply
90
ANALYTICAL INSTITUTIONS.
BOOK I.
2a +
Multiplica-
tion how
infill iiated.
3^
c multiply 2^ + 3/^ — f by 3 a? — 2^, let ihetn be
2y placed as in the margin. Then multiply all the
terms above by 3^;, and do the fame by the other
term* — 2y, and fo if there were more terms in the
multiplier. The product will be as is here to be
feen. It is no matter whether the operation begins
from the right hand or from the left, in regard to either of the
factors; or which of them is wrote above, and the other below;
or in what order the terms are placed. Suppofe we were to
multiply aa + xx by aa -^ xx ; proceed as in the margin,
where becaufe aaxx and — aaxx deftroy each other, the pro-
duft will be reduced to a* — x*.
In long multiplications, in order to reduce fimllar terms with greater eafe,
it will be convenient to write thofe fimilar terms, which will arife from the multi-
plication, one under another as in the foregoing and following example. Let it
be propofed to multiply 4^' -}- 3^*^ — 2^^* + P by a"" — 5^^ + 6^*. The
— 4/iy — 6l?y + 2cy
aa + XX
aa — XX
a'' + a'-x'
— a^x" — x*"
A^} +
\a^b ^
2^^* + h^
c,ab + 61"-
—Q^oa^b —i^a^b^+ loa^P — ^ab^
4- 24^^^' + I Sa'b' - I lab* + 6b'
work will lland as in the margin.
Here it is eafily perceived, that
4. na*b — 20^^^^ make — 17^*^,
And that — aa^b"- — 1 ^a^b^ +
24.7'^* make jaH*. And that
+ a^b^ + loa'-b^ + iBa^b^ make
2ga^bK And that — 5a b* —
iiab* make — lyab*. There-
fore the produd is 4«^ — 174'*^ -f ya^b'' + o.^a'^b^ — 17^?^* -f- 6^^
20. Sometimes it will be unneceffary adually to perform the multiplication
in this manner, but it may be fufficient to infinuate it only, which is commonly
done by inferting this mark X, and drawing a line or vinculum over each of the
multipliers, extended over all the terms which enter the multiplication. Thus
aa + XX X aa — ax denotes the produft of aa + xx by aa — xx. But in the
quantity aa + xx X aa — XX ± a\ the term ± a*^ not being included in the
vinculum, is not intended to be comprehended in the multiplication. So that
being wrote in this manner it muft be underftood, that to or from the produdt
of aa 4- xx into aa — xx, muft be further added or fubtraded the term «*.
Powers of -I. After the fame manner that in fimple quantities the produd of a into a
compound is called the fquare of a, the produd of aa into a is called the cube of a, the
quantities piodud of a^ into a is called the biquadrate of a. &c. So in compound quan-
how iniinuat- '.. , inr- i ,r ?%.• ? —
ed: how ac- ^^^^^^ the produd ot a + b (tor example) into a -i- b, or a + b X a + *, is
tuallyformed. called the fquare of ^ 4- ^, which is wrote thus, a + i?X> when we would not
adually form it by multiplication. In the fame manner a + bY X a + b will
be the cube, which may be wrote thus, a~+T)^
SECT, I, ANALYTICAL INSTITUTIONS. tl
rt~+~7)* X «~+7)% or a -^ hY ^vlll be the fourth power of ^ + <^. And this is
to be underftood of quantities of any number of terms.
Aflually to form thefe powers, the quantity given muft be multiplied into
icfelf, and the produdt by the fame quantity fucceffively, as many times, fave
one, as the exponent of the power required contains unity. But for the fecond
power, or the fquare, the operation may be thus abbreviated. If the quantity
given is a binomial, or confifls only of two terms, fuppofe a ±: b, write down
the fquare of the firft: term, then the two rectangles, or twice the produd of
the firft term by the fecond, with fuch a fign as the rule of multiplication re-
quires ; and lafily the fquare of the fecond term muft be added. Thus a + h\^
will be aa + 2ab + bb ; and a - 3j* will be aa — 2ab + bb, Alfo - ^ - i^*
will be aa + ^ab 4- bb, if the quantity given is a trinomial, or conlifts of
three terms ; befides the fquare of the two firft terms found as before, muft be
wrote two redangles of the firft into the third, and alfo of the fecond into the
third, (taking care that thefe redtangles may have their proper figns, according
to the rules of multiplication,) and laftly the fquare of the third term. Thus
rt 4- ^ — c\^ will be aa + 2ab -\- bb — 7.ac — ^bc + cc. If the quantity is a
quadrinomial, or of four terms, there muft be wrote befides, twice the rectangles
of the three firft terms into the fourth, and alfo the fquare of the fourth term.
And fo on to other multinomials.
22. But as to all binomial quantities, the following general canon will be of Powers raifed
good ufe, not only to raife it to the fquare, but to any power denoted by w, ^7 the Bi/io-
where m ftands for any number whatever. Therefore let /> + ^ be to be raifed of 5;^ /^JJ'*
to the power /» ; this power will' be p*^ + mp*" y + w x ^ ~ ■ /)"*"" j* +
m X — — ■ X p q' + m X X X — - p q^, vcc. ; which
fcries of terms may be contihued as far as we pleafe, obfcrving the fame law.
From hence let us derive the fquare of /> -}- ^. In this cafe m will be 2, and
therefore in this canon, by fubflrituting 1 inftead of w, the firft term will be p* »
the fecond 2p "" ^, that is 2pq ; the third will be 2 X ^^^*'"^^*, that is q*,
(Here we do not admit the quantity p, becaufe being raifed to no power, it is
equal to unity, as will be Ihown afterwards. And the fourth term will be
2 X X p q^» But 2 — 2 in the co»efHcient is equal to nothing,
and therefore this term being multiplied by nothing will be nothing, or will
vanifh. And thus fince all the following terms are multiplied by nothing, they
will all vanifli, and the canon will terminate after three terms. So then the
fquare required will be />p -f 2p.q + qq.
Ca If
22
ANALYTICAL INSTITUTIONS.
BOOK I.
If we would have the cube or third power of /> -f- ^, then w ~ 3 ; whence
the fifth term of the canon, and all the following ones, will be equvil to nothing.
So that the power requited, by fubftituting 3 inftead of ;w, will be p^ -f ^p^q
+ .V?^ + ^f' ^f ^^^^ quantity to be raifed is /> — q, it will be fufficient to
place the fign minus before all the terms, in which the index of q is an odd
number.
The foregoing canon will not only ferve for the binomial p ± q, but for any
other whatever. So that if we would have the third power of zax — xv, we
muft fuppofe p =: ia\\ and ^ zz — .v^, as alfo tn zz 3. Then in the canon,
inftead of p and the powers of />, we muft fubftitutc lax and it's powers ; which
muft alfo be done by putting — xx inftead of q and it's correfponding powers^
Then inftead of m put 3, and the cube will be 8i^^v' — i2aax^ + dax^ — x^.
It may likewife ferve for any polynome, or for any quantity confifting of
more terms than two. Let there be a trinomial a + b — £• to be raifed to the
third power, and then it will be w iz 3. Jf we make p zz a and q zz b — f,
and fubftitute a and it's powers inftead of p and it's powers, and alfo b — c
and it's powers inft'ead of q and it's powers ; the cube will be a} + 'i^aa x b — q
+ 3^ X b - ?\* + h - cl ' J that is, a^ 4- ja*^ — ^a^c + 35^' — ^abc -\- 3<?c*
-i- ^' - 3^V -t- 2fic- — c\
JDivifton of Compound ^amities j being Integers,
Corapound
quantities
divided.
23. There may be three different. cafes, or combinations, in the divifion oft
compound quantities i the firft is, when the quantity to be divided is com-
pound, and the divifor is fimple ; the fecond is on the contrary,, when the di*-
vifor is compound, and the dividend fimple j the third is when they are both
compound quantities. A^s to the two firft cafesy it will fu^ce to make ufeof
the rule for fimple quantities. In the firft cafe every term of the quantity pro-,
pofed is divided by the divifor, and there will arife either integers or fradions,.
as follows from the nature of divifton of fimple quantities. Thus if we arc to-
divide aa + ab — ac by a, we (hall have for the quotient a -\- b — c. If.
we are ro divide 4^^— 6^f-|- xx by 2j^, we (hall have 2<3
we are to divide \ah -^ cc -\- ^xx by 2^> we fliall have ^^—^
«3^ +
IT
le
y
or elfe
' ^^'^ , _^ I
In the fecond cafe the divifor is wrot-e under the dividend.
4£^ _
3*^ 3
as is ufual in fradions j and if; in every term of the numerator and denomi--
f«CT. I, ANALYTICAL I M S T 1 T t7 T I O N S» I 3
nator there fhall b^ any common quantity, it may he cancelled ; then what
remains will always be a fradion. Thus dividing ^a^^ by ^la — nx + a/y^ the
quotient will be ^""^ . And if we divide 6a^ by ma — lax -f zv.v, the
quotient will be • —~ .
* aa — ax -J- xx
24. In the third cafe it is neceflary, fird to put the terms of the dividend inFrocifsof:
order, and likewife of the divifor, in refpedl to f( aie certain letter which fliall ^ivihon,
be thought the mod proper for that purpofe. This is done by writing that for
the firfl: term of the dividend, and alio of the divilor, in which that letter is
found of the higheft power, or of mod dimenfions. Then making that the
fecond term, in which that letter is of the next greatefl: power. And fo luc-
eeflively till we come to ihofe terms, which are not affeded by that letter at all,
which therefore mud be made the lad. Thus the quantity a^ + 2«V — a^^
— ^al^c J^ bbc will be ordered in rcfpeft of the letter a, and alfo the divifor
a — b. If we would difpole this in order, in refped of the letter b, it mud be
done thusi ^V -—' ^abc — a^b + a^ + za'c ; and the divifor thus, — b + a.
This fuppofcd, the divifion mud be performed after this manner. The fird
term of the dividend mud be divided by the fird term of the divifor, and the
quotient mud be written on one fide. By this quotient the whole divifor mud:
be multiplied, and the produd fubtraded from the dividend. When the fub-
tra(5tion is made, and the terms reduced, in the fame manner the fird term of
the remainder mud be divided by the fird term of the divifor, and this term of
the quotient mud be wrote after the other, wiili-fuch fign as it ought to have.
Then the whole divifor mud be multiplied by this fecond quotient, and the
produdl fubtraded from the dividend,^ that is trom the fird remainder. And
proceeding in this manner, the calculation mud be repeated, till at lad there
is no remainder. Then the fum of all thefe quotients, thus found" by parts, ,
will be the whole quotient of the divifion...
Let it be required to divide a^ + 2;iV — a^b — ^^bc + b^c by a — b. Let
the quantity to be divided be wrote at A, the divifor at B. Now dividing a^
by a, the quotient will be «*, which is written at D. Then finding the produdt
of the quotient into the divifor, and fubtrafling it from the dividend, there
will be left the fiift remainder,, as at M. Then dividing the fird term 2C!ac in
this remainder M by the laid fird term of the divifor ^, and writing the quotient
aac after the other at D, we mud fubtrad the produd of lac into the divifor B,
and we diall have the fecond remainaer N. Divide the fird term — abc of this
fecond remainder by the lame term a of the divifor, and write the quotient
--^ be at D after the otl.er. The produd of — be into the divifor mud be.-
fubtraded from the fecond remainder, and nothing will now remain. There-
fore thexompleat quotient will be aa + 2ac — be*.
A. ^^'
14 ANALYTICAL INSTITUTIONS. BOOK!,
A. a^ + 2«V — a'b — 'i^ahc -^ Pc B. a — ^
M. za-c — 3^^^ + i^^'i^ l). ^a + lac — be,
N. — abc -{■ b'^c
Let a} — "^a^b + 3«^* — ^'' be to be divided by a — b. Let the dividend
be wrote at A, and the divifor at B. Let the firft term a^ be divided by a, and
the quotient aa be wrote at D. Then finding the produdl of the quotient into
the divifor, and fubtrading it from the dividend, there will be left the firft
remainder M. Let the firft term of this remainder, that is — la^b, be di-
divided by the fame firft term of the divifor <2, and let the quotient — lab be
wrote after the other at D. Then let the produd of — • 2ab into the divifor be
fubtraded from the firft remainder M, and we ftiall have the fecond remainder
N. If we divide the firft term ab'^ of this fecond remainder by the fame firft
term of the divifor a^ the quotient bb muft be wrote at D after the other. Then
let the produd of bb into the divifor B be fubtraded from the fecond remainder
!N, and nothing will remain j fo that the whole quotient will be aa — 2ab 4- bb,
A. a? — ^a'-b + 3^^* -— ^» B. /? — ^
M. — ^a'b + 3<?3* — b^ D. aa — lab + bb
N. + ^^* — b'
Another Example,
A. 7.aa 4- $ab + ibb — ac — zbc B. a -\- 2b
M. + <7^ -i- 2bb — ac — 2bc V>, la -{- b — c
N. — ac — Q.bc
Another,
A. 9^^ + i2d'e — J^de^ — c* B. 3^* — tf*
M. i2d'e + 3^V — 4^/^ — <♦ D. 3^* + 4^^ + «»
M. 3^/V — ^*
Another,
A. 4a* + /^b — lac 4- ^* — c^ B. 2<J + ^
M. 2ab — 1 2tff + ^* — c^ D. 2« + ^ — e
N. — lac '^c^
Now here it is to be obferved, that the laft remainder at O is not divifibic
by 2a, and confequently the operation cannot proceed, but it muft remain as a
fradion - " ^f * That is to fay, that the quantity propofed is not entirely
7
divifibic
S£qT. r. ANALYTICAL INSTITUTIONS. IQ
divifible by 2a + (^, but only in part, and therefore the quotient will be partly
an integer, and partly a fradtion, as 2a + If — c + — — r . Or the whole
, r CL' t. 4^^ + 4^^ — 2^^ + [>i) — cc
may be wrote as a fraction thus, 2a + b ' *
Extras ion of the Roots cf Ccmpound ^lantities, being Integers,
25. As in fimple quantities, fo in compound ; the root of any quantity is Roots how
that, which being multiplied into itfelf, if once produces- the given fquare, if to be ex-
twice produces the given cube, and fo on. traaed ; par-
*■ o 3 ticularly the
The manner of extrafling the fquare-root in compound quantities is as fol- ^"^'^^''^^ *
lows: It being firft underftood, that the terms mufl: be difpofed in order, ac-
cording to fome one of it's letters, agreeably to the caution before given, § 24,
Let the given quantity be «* + ^ab + b^, whofe root is to be extraded,
and let it be wrote down as at A. Extrad: the fquare-root of the firft term a^,
which will be ^, and let it be wrote as at B. The fquare of this, or ^% mufl
be fubtraded from the quantity propofed. A, and the remainder wrote down
at D. Then the quantity a^ wrote down at B, muft be doubled, and wrote as
at M, which will be la. By this quantity la the firfl term at D muft be di-
vided, and the quotient b wrote at B. Then the divifor ia muft be multiplied
by the quotient b, and the produd fubtraded from the quantity D -, and more-
over the fquare of b muft be fubtraded from the fame j and as there is no
remainder, the root required will be ^ + ^,
A. «' ^ nab -\- b"- 'B, a + b
D. lab + b"- M. 2a
Let the quantity given be a* + 6a^b + 5^'^* — iiaP -|- 4^*; let it be
wrote at A, and lei the fquare root of the firft term be cxtraded, which is a*,
and let this root be wrote at B. Let the fquare of «* be fubtraded from the
quantity A, and there will remain the quantity D. Let a^ be doubled and
wrote at M, and by this double, that is by 2^*, let the firft term be divided of
the firft remainder D, and the quotient ^ab be wrote at B. Then fubtrading
the produd of 2>ab into the divifor 2aa, as alfo the fquare of 'T^ab, from the firft
remainder D, there will be left the fecond remainder H. Let the whole quan-
tity B be doubled, and wrote at G. By it's firft term let the firft term of H
be divided, and the quotient — 2^* be wrote at B. Then fubtrading the
produd of the quotient into the divifor G, and alfo the fquare of the fame
quotient^
a[6 rAXALYTlCAL I N" S T1 T U T I N S. BOOK i;
•quotient, From the quantity H ; and, as there is no remainder, the quantity
written at B, that is, aa -f- ^ab — ibby will be the root required.
A. a*' + (iaH + 5^"'^* — viaP 4- 4^* B. a" + n^ab — 2^*
H. — 4^"^* — 12^^^ + 4^* G. 2<j* 4- 6^^
^he Operation of another Example,
A. _>•* 4- 4-^7' — Sa^y + 4(2* B. y 4- 2^J — 2^'
,D. 4^}^ — 8^^_y 4- 4«* M. 2)'*
H. — 4«''y — 8«'_>' + 4^* G. 2j'* 4- 4^_y
Another Example.
A. i6tf* — 24^\Y^ — i6<2*^* + 1 23V 4- 9.V* B. 4«' — 3.V' - zb*
D. — 24^ V — i6a^b^ + i2i'V 4 ^x* M. 8«*
H. — lea'b'' 4- i2/^V G. 8^* — 6x'
K. - — 4^*
In this laft operation there is a remainder of — 4^*, which cannot be di-
vided by 8^/'', as the method requires, which in this cafe cannot take place.
That is to (liy, that the fquare-root of the propofcd quantity cannot be adlually
extrafted, and therefore we muft make ufe of the radical fign, as above at § 16;
which expedient muft alfo be applied in other extraftions, as the cube-root, the
■biquadr atick-ro ot, &c. Thus i^aa + l>& reprefents the fquare-root o^ aa -{- bb^
and {/a.i/fcM will ftand for the cubic root of aab — abb j and the like for
jother roots.
The cube- 26. As to the cube-root, let- It be required to extract the root of the quan-
root ex- tity a^ 4 3^1^ b + ^aP 4- b^^ as is written below at A. Extrafl the cube-root
traced. ^f j}^g ^,.£^ ^^^^^ ^3^ which is a, and is written at B. Let the cube of this,
or a'% be fubtrac\ed from the given quantity A, and let the remainder be
written at D. Then take the triple of the fquare of a, which is ^aa, and let
it be wrote at Mj by which divide the firft term of the remainder D, and let
the quotient b be wrote at B. By this multiply the divifor ^aa, and the pro-
dud, together with the triple of the fquare of b into <?, and the cube of by
muft be ilibtradted from the remainder D. And as nothing remains, a + k
will be the root required.
A. a^ 4- 3^'^ 4- 3^^* 4-^3 B, a + b
D. 3^*/^ 4. 3^// 4- P M. ^aa
Let it be required to extrad the cube-root of the quantity z* 4- 6bz*
- 4oPz' 4- 9^^b'-'^ — 64b\
Extraft
SECT. 1. ANALYTICAL INSTITUTIONS, 17
Extrad the root of the firft term z^, which is 2*, and let it be wrote at B.
Let the cube of B be fubtraded from the propofed quantity A, and let the-
remainder be wrote at D. Take the triple of the fquare of B, and write it
at M, and by that divide the firft term of the remainder D, and write the
quotient ibz at B. Then fubtraft the produft of 2bz into the quantity M,
and moreover the triple of the fquare of zbz multiplied into zz, with the cube
of 2^z, from the remainder D, and write the remainder at H. Then find the
triple of the fquare of B, which write in G, and by the firft term divide the
firft term of the remainder H, and write the quotient ~ a^hb in B. Then
mvikiply this quotient by the quantity G, and the produdl, together with the
triple of the fquare of — ^bb into 22 + 2bz, and the cube of — ^bb muft
be fubtradled from the quantity H, and nothing will remain. Whence the
cube-root of the quantity propofed will be the whole quantity B, that is,
az -f 2bz — j^lb.
A. z' + ebz' — 40^^z* + ^db'z — 64^' B. z^" 4- 2bz — 4^*
D. 6bz^ — ^ob^z" -f 96^*z — 64^* M. jz*
H. - i2^V — 48/-^ 4- s^b'z — 64^' G. 32* 4- I2^z^ 4- i2^V
After the fame manner is extraAed the cube-root of the following quantity.
A. 27/ — 549' 4- i44^V — i52f'j' 4- iL)ic*y'- — ijSc^y + 646*
D. — 349** + i^'^c'^y^ — 132^'^ 4- igzc*^ — ^Sc^y 4- 64/'
H. io8^*j4 — I44f'j^^ 4- i92cy — ^(u^y 4- 64^"
B. 3^ -T- 2cy 4- 4^*
~ M. 27/
G. ^-jy^ — 2>^iiy^ 4- i2t*^*
27. For the fourth root. Let the quantity propofed be a* + ^a^b 4- 6^'^^ The fourth
4- A^ab'^ 4- b^, of which we would extra6t the biquadratick or fourth root. Let root ex-
it be wrote at A, and extra(5t the fourth root of the firft term, which is a, and ti'a'^^^*
write it at B. Subrrad the fourth power of B from the quantity A, and write
-the remainder at D. Then find the quadruple of the cube of ^, and write it '
at M. By this muft be divided the firft term of the quantity D, and the quo-
tient b muft be wrote at B. From tiic quantity D mult be fubtrac5ted the pro-
dud of the quotient b into the divifor 4.^', and moreover the fextuple of the
fquare of b into the fquare of ^, and the produdt.of the quadruple of the cube
of b into the quantity ^, and laftly the biquadrate of b. And as there Is no
remainder, the root required will be a -\- b,
A. a" 4- /^'b 4- ^a'-b'- + 4^^' + ^* B. ^ 4- <^
D. 4^'/^ 4- 6^^//^ 4- ^aP 4- b" M. d,a'
D '28. As
iS ANALYTICAL INSTITUTIONS. BOOK!,
The fifth and 28. As to the fifth root ; in order to difcover in what manner the operations
^'S^^!, '"^"'^^ proceed, which are to be made in the extraftion, it will be fufficient to foim
the fifth power of a binomial, fuppofe of a -f- b, which will give a rule here j
as the fecond, third, and fourth powers of the fame binomial have fupplied us
with rules for the extiT.elion of the fecond, third, and fotirth roots. The like
obtains in the fixth, feventfi, and other roots.
Of Fra5fions, Simple and Compound,
Notation of 29. We have feen before, how fradions or broken numbers arife from the
fradions. divifion of quantities. Therefore a fraction infinuates a divilion that is to be
made, of the numerator by the denominator. Whence it proceeds, that if the
numerator is the fame as the denominator, as — , or ^ -rr , and fuch like^
thofe fractions can fignify nothing elfe but unity ; becaufe in fa(^, if we divide
a by a, or aa — bh by aa — bb^ the quotient will be unity. And becaufe
multiplication is an operation contrary to divifion, it is plain, that any integer
whatever may be reduced to a fradion with what denominator we pleafe, if it
is multiplied by the quantity which is to be the denominator, and then divided
by it again. Thus to reduce the integer ^ to a fradion with the denominotor b,
we muft write -7- , To reduce a •- b to a fraflion with the denominator d,
we muft write 7 — - . To reduce a 4- b to a fradion whofe denominator
fl ,, , , n • a-\-l> X c—d ac + ic — ad — hd
mall be f — ^, we mult write ; , or .
' c-'d c ~- d
Redu5iion of Fra51ions to more ftmple ExpreJJions,
How frac- 30. When fractions have the fame letter or letters in every term of tlie nu-^
tions are to merator and denominator, it will be fufficient to expunge the common letters
e re uce . j^ \^q[\^ . having regard to their powers, as is faid in Divifion, at § 10. Thus
— Will become ; —r- will be — j — i — -n- will be j— • But
ac c abc c ab — bb a--b
though there are not the fame letters in both the numerator and denominator,
yet if each of them is multiplied by the fame compound q,uantity, they may be
9 ' divided
SECT. T. ANALYTICAL INSTITUTIONS. t^
divided by it again, and coniequently the fradion may be reduced. Thus
, that IS ■ — — , Will be reduced to —7- . So — 7- — 5—, that is
•^-'^^ d X c-d
■X J ^_ <Z'* "t ^'^"^ o cac — aad — acd -\- add
or
€ia-\-2ah X J J -\- lab .,, , , j ^ <Z'« "t 2*7(5 aac — aad — acd + add
— ^ — , Will be reduced to j . So -; — ~— I- —
b X aa + ^ai> ^ cd - dd
'^^^^^ ^ ^^, will be reduced to ^^-^^^
d X c—d
Therefore in general, as often as the fradlion is fuch, that ii*s numerator and
denominator are both divifible by one and the fame quantity, which in this cafe
is called their common divifor, by adually dividing both, the two quotients wilt
give the fradtion reduced. But it muft be obferved, that, if that common divifor
is not the greatefl: that can be, the fra6tion indeed will be reduced, but not to
^* - o^^ ^1 ^^ ' a y^ a + b X a^b
a X c X a+b
the fimpleft expreffion. Thus the fradion ^~ ^^ - , diat is
may be divided, both as to it*s numerator and denominatoi, by a, by a + b^
and by aa + ah, the greateft of which divifors is aa + ah. And as the fraction
fliould be reduced to it's leaft terms, we muft divide it by aa >{- ab, and the
quotient or fradion reduced will be . But very often it will be difficult
to know if there is a common divifor, and what it is ; and therefore we (liall
give a rule to find it, at § 36. afrervyards. At prefent we Ihall omit it, that we
may not too much difcourage young learners, as yet not fufficiently confirmed,
and fliall proceed to other operations ; making ufe of fradions that jfre any
how reduced to lower and fimpler expreffions.
Redu^ion of Fra5lms to a Commcn Dsnominatcr.
31. If two fradions are given, let the numerator of the firft be multiplied Fraaions
by the denominator of the lecond, and the numerator of the fecond be multi- ^^^^^^^'^ '°
plied by the denominator of the firft, and each produd be divided by the pro- denomh°a'tor.
dud of the two denominators. Thus 4- + ~ will be "lAJf. : and ^ - 2il
^ y h J* 36
will be ldLzJ£t, Alfo "-^-±^^ - ^ will be ffl+i!f!_=_^£ji.f^ ^^at
ih m + n m mm -f- mn '
Js ^^^^ ^ ^^ . But here we muft take notice, that as often as the two denomi-
nators of the fradions have a greateft common divifor, in this cafe the multi-
jD 2 plication
2(3 ANALYTICAL 1NSTITUTI01?S. BOOK I.
plication of the numerators into thai common divifor is fuperfluous, and alfo
-of thofe common tijvifors into each other, for forming a new denominator} for
then it may be necelfary to reduce the fractions to more fimple expreffions.
Wherefore the faid numerators (hould be multiphed, not by the denominators,
but by the quotients which will refult by dividing the faid denominators by their
common divifors : and the denominator will be the product of thofe quotients,
and of the faid common divifor. For example, let there be eiven — V ^ — .
*-* mil mx
Being reduced as ufual to a common denominator, it will be "''^ ^ '""^ ;
" mtnnx
that is — ~ — . Therefore it was needlefs to multiply the numerators
by m^ the common divifor of the denominators, as it was fuperfluous to mul-
tiply the denominators together. It was fufficient to multiply a^ into a", and abb
into «, to form the numerators, and to multiply m into n into x, to form the
common denominator. Thus to reduce to a common denominator the fractions
— IL ^ , it will be enough to multiply -7 into a -^ h. and it will
be , that is — — — . In like manner to reduce to a,
common denominator the fraflions — — , + % — — 1 becaufe c — ^ is a
common divifor of both the denominators, it will fuffice to multiply h^ by d^
and d^ + b^ by <s*, as to the numerators ; and to multiply d^ into d into c -^ dj
as to the denominator, and therefore it will be ' / f ,! — •
If three fradions are to be reduced to a common denominator, let the twofirft
be reduced, then that which refults from thefe with the third ; and fo on fuc-
ceffively if there are more. So to reduce thefe to a common denominator,
-^ + -^j ~ , let the two firft be reduced, and we (hall have "^ i^, ' . Let
ban ' bd
this be reduced with the third, and we (hall have —-^~jj-~ — " • "^^'^ "^^y
alfo be done in refpe6t to integers ; for whereas any integer may be confidered
as a fradlion, having unity for it's denominator, we may proceed after the fame
manner as before. Thus laa + ^"^ , " y , that is -^^ + ^*,"";f-^ , will be
■^Xfg — Sax
Addition
SECT. I. ANALYTICAL INSTITUTIONS*
Addition and SubtraSfion of Fra^ions,
zi
32. Fradions are added by writing them one after another with the fame figns. Fraaions
And on the contrary they are fubtra(5ted by changing the figns of the quantities ^ow addec
to be fubtraded. And the fame things muft be done, if there are integers with j^^^^^Jj*'
the fradions. Thus to add — to — , they are wrote . To add ~
to ~ y, it muft be wrote — 4- — — y \ which afterwards (if we pleafe)
m "^ c m ^ ^ *■ ' .
may be reduced to a common denominator, and then it will be Zi^^
. ■^ ' cm
rp J J ^^i* aahb i r -ii t ^^^^^ ,. ^*'^* i • i
10 add -7 -7 — - to Tj « the fum will be rrr-rn + rr > which
if we would further reduce to a common denominator, we may obferve, that
the denominator of the firft is the fquare of aa — hb » therefore the two deno-
minators have a greateft common divifor aa — bby by which being divided, the
quotients will be aa — bb in the firft, and unity in the fecond. Wherefore it
will be enough to multiply the numerator of the fecond fradion by aa — bb,
and to divide the whole by a^ — 2a^b^ + b*, and the fum required will be
— T TT-m — i that IS _ , . To fubtract — trom — , it will be wrote
^^—- — . To fubtrad a — ~ from -^^ , it will be wrote — « + — ,
which being reduced to a common denominator, if we think fit, will be
myy — amm + amn + mxx — nxx >-*-• r i n b* f a^ ■\- b^ . ^
-^ . To fubtrad — --. from — j 7-, , it muft
mm — ma /\arc — /^ahi zed — 2aa
be wrote — ; — — -^^ • ; and to reduce it to a common denominator,
2cd—zdd s^a^c—AtO^d '
we muft multiply a^ -\- b^ by 2aa, and — b* by d, and the whole muft be
divided by Aaacd — Aaadd 1 then it will be — P-- — ^^^77- •
Multiplication of Fraaions,
33. The numerators muft be multiplied into one another, and alfo the deno- Fraftlons
minatorsj and the new fradion will be the produd of the fradions to be mul- how multl-
liplied. Thus to muhiply -^ into ~, the produd will be ^y which is^'^ *
reduced
41 ANALYTICAL INSTITUTIONS. 300K I,
reduced to ~P- . To multiply 7^^ into ^"^^ "^ , it will be wrote thus,
a *■ •' b + c 5c' '
— , 7 ^ . The fame mud be done if there are integers with them, by con-
fidering an integer as a fraftion, the denominator of which is unity. Thus to
multiply zay or — , mto ^^ , the product will be < ^ .
Let it be required to multiply — -— into a — i?. In this and the like
cafes, becaufe the quantity which ought to multiply is the fame as the denomi-
nator of the fraftion, it will be fufficient to expunge the denominator, and then
the produfl will be aa -j- i^L If aa — l>^ is to be multiplied into '^'^ "" f , it
may be obferved, that aa — ^^ is the fame as a •{■ b X a — by and therefore
fince it would be required to multiply aa — ab into a ■{■ b into a — b, and
afterwards to divide by ^? + ^ ; and becaufe a -\- b would be a common divifor
both of the numerator and the denominator which would thence arife j the mul-
tiplication and divifion by the fame a •{• b may be omitted, and it would be
fufficient to multiply the numerator by a — ^, and the produd will be
jc' — laab 4- abb. Thus the produdt of — — — into rr will be
*■ XX — vv aa — ha
XX —yy aa — bb xx —yy
Divifion of FraSiions,
Fraaions 34. The Divifion of Fraflions is performed by multiplying crofs-wife, that
how divided, js^ by multiplying the numerator of the dividend by the denominator of the
divifor, which produfl muft be the numerator of the fraction which is to be the
quotient : and then multiplying the denominator of the dividend into the nu-
merator of the divifor, which produd will be the denominator of the quotient.
This quotient, if there is occafion, muft afterwards be reduced to the moft
fimple expreflion. Let it be required to divide -i- by — j the quotient will
be ^. Divide -fi. by :^; the quotient will be _f^, or :i^i which
cm c ■' n ^ — cm cm
.. , -ir . . • 1 T • 1 «^ — ^^ u aa—ah ■\-hb • -ii
IS all one by § 13. Let it be required to divide ^ ^ by ; it will
DC ■ 3 , ,3 •
fl^ + p'
It
SECT I. ANALYTICAL INSTITUTIONS, 23
It is eafy to perceive, that if the two fraftions, the dividend and divifor, (hall
have the fame denominator, it would be needlefs to multiply them crofs-wife.
As if we were to divide ~ by ^-^^— , in this cafe it would be enough to
mm
divide aa by c — d. For by mulciplying crofs-wife it would be ^ "\ , and
then reducing it to it's ieaft terrn^, it would be -^. Thus dividing ~ ^"' -
J3y ffjLif— i — , the quotient would be ".'"?,;,. ; b^it by redudion, be-
caufe the numerator is ^ X a-{- b X a — bt and the denominator is « + ^ X 7+1,
it will become — ^-r- . After the fame manner we muft proceed when we are
to divide an integer by a fradion, or a fraAion by an integer ,• confidering an
integer as a fradtion whofe denominator is unity. Thus dividing the quantity
aa — XX. or "^ " ^" , by VUSI-^ the quotient will be ^^ "" ^"^"f- . And fo
of Others,.
m r' '
Extra£iion of the Roots of Fra5}ions»
3:5. The root of a fradion is extraded by extrading the root of the nume- Roots of
rator, and then of the denominator, and the new fraction arifing lliall be the fraaions how
,, , extradted.
root of the fradion propofed. So the fquare-root of will be ~, The
fquare-root of "",^'^'', , ,. will be ^^ "" , . The fquare-root of A.aa +
^ aa + ^ab-\-ifbb a + 2b ^ ^
64XX — iGoax , . r looaa — iSoax + 6±xx .,, , loa — ^x n-., ^
-^ , that IS or — , will be . The lame
25 ' 2S ^
is to be underftood of the cube-root, the biquadratick-root, and all others.
But now if the root cannot be extraded out of both the numerator and
denominator, yet poffibly it may be extraded out of one of the two. Let it
be extraded out of which of the two it can, and before the other let the radical
fign be placed. Thus the cube- root of — r will be ^ . . The cube-
root of '^ "^3" ~ will be ^—^-Hi-. And if the root cannot be extraded
neither
24 ANALYTICAL INSTITUTIONS. BOOK I,
neither out of the numerator nor denominator, then the whole fraftion muft
be included under the radical fi2:n. Thus the Iquare-root of wiU
^ -^ xx + tx
be \/ — TT- •
A-.V -f- OX
Of the greatejl Common Divlfor of ^wo ^mntities, or Formulas.
Greateil 36. By a Formula I mean any analytical expreffion whatever, whether com-
■common plicate or not, the letters of which reprefenting indeterminate quantities, may
diviforhow \^Q what vve pleafe ; provided that whatever may be faid of that formula is to
*°"" * be underftood as laid of any other, compounded of other letters, but fimilar to
the firft.
To obtain the greateft common divifor of two quantities or formulas ; in the
rirfl: place it muft be obferved, that if every term of both is multiplied into the
fame quantity or numl^r, in this cafe they muft be divided by that quantity.
Then each of the formulas mud be fct in of^der according to any letter at
pleafure ; that is, that mud be made the firft term, in which that letter arifes to
the moft dimenfions, and then the others in order. Let the two given for-
mulas be i^a'^bx •— ?>a*b — n,ahx^ — %c^bx'^ + bx*^ and (^a^b + bx^ — abx"^
'— Zd'bx ; which becaufe they are divifible by the letter b, let them be fo di-
vided, and then fet in order (if you pleafe) according to the letter x. They
will be thus, x^ — 3<2.v' — 'ia'^x^ + i^a^x — 8^*, and x^ — ax"^ — Ba^'x -\- 6^^
This being done, the firft term, or that wherein the letter is of moft dimenfions
by which the terms are fet in order, muft be divided by the like term in the
fecond, namely x* divided by x^ will give x in the quotient. Then the product
of this quotient into the divifor muft be fubtracled from the dividend, and we
ihall have the firft remainder — la^^ + iia^x — 8^% which muft be reduced
to the moft limple exprelTion, (as ought always to be done,) by dividing by
— 2a •, then the remainder will be x^ — 6a^x -f- 4(2^ And becaufe the di-
menfion of x in this remainder is the fame as in the divifor, by the faid divifor
• this remainder muft be divided ; from whence la like manner muft be f ibtraded
the product of the quotient into the divifor, and we (hall have a fecond re-
mainder ax"" -f 2a'x — 2fl', or dividing by a it will be x- -f- lax — 2^\
Now becaufe in this remainder the dimenfion of x is lefs than in the divifor,
the order muft be inverted, and this remainder muft be made the divifor, and
the firft divifor the dividend. And making the divifion, the produd of the
quotient in:o the fecond divifor muft be iubiraded from the fecond dividend,
that is from x^ — ax^ — Sa-x -f 6a\ and the remainder will be — j^^v^
— 6^\v -f- 6a^, which dividing by — - 3^ is x'- -j- lax — la^. Now whereas
this laft remainder is the fame as the divifor, it will be the greateft common
divifor
SECT, r, ANALYTICAL INSTITUTIONS* 35
divifor of the two formulas x^ — ^ax^ — 8^'.v* + i8«^r — 8a*, and x^ — ax^
— Sa^x + 6a' i which bei'ncr muhiplied into b, or ^x- 4. 2^Av — 2a^b, will
be the greateft common divifor of the two formulas ac firfl propofed.
Let the two formulas be x* — ^.ax^ 4- iia'^x'' — 2oa'x -f 12-^% and
X* — ^ax' + 12a* x"^ — i6a\v -h 24^4, being ordered according to the letter .v.
And as this is of the fam.e dimenfions in both, we are ac liberty to take which
of them we pleafe for the divifor. Let the firft therefore be divided by the
fecond, and "fubtracling the produft of the quotient into the divifor from the
dividend, the firft remainder will be — ax' — a'^x'- — 4a'x — 12a*, which
being divided by —a is x' -\- ax"" -{- ^a^x +» 12^2^. Here inverting the
6rder, let this remainder be taken for the divifor, and the firft divifor for the
dividend. Then making the divifion, and fubtrading the producfl of the
quotient into this fecond divifor from the fecond dividend, the fecond remainder
will be — j^ax' + Sa^x' — 2Sa\x 4- 24^*, which being divided by — 4a will
be x' — zax"" 4- ya^'x — ^aK By the fame fecond divifor let the divifion of
this fecond remainder be continued, and making the fubtraftion as ufual, vve
(hall have a third remainder — ^axx 4- 3^"^: — 18^% or dividing by — 3^
it will be X* — ax -{- 6a''. Let the order be again inverted, and let the fecond
divifor be divided by this third remainder x^ + ax"- 4- 4<7^.v + iia^ y and
making the fubtiadion as ufual, the remainder will be found to be Ian"- — 2a^x
4- i2a^ ; or dividing by 2^, it will be xx — ax -jr 6aay the fame quantity as
that which was a divifor before, and which is therefore the greateft common
divifor of the two propofed quantities.
Let the two formulas be /* — aaff — hoff 4- aahb, and p — aff — labf
-4- 2<2'^, which arc ordered according to the letter y. Let the firft be divided
by the lecond, and the product of the quotient into the divifor being fubtracled
from the dividend, will give the firft remainder af^ — ^Y* 4- tabff — bbff
— 2a'^bf -{- a^b''. And if vve go on to divide by the fame divifor, and the
produd; of the divifor into the quotient being fubtradled from the dividend,
we (hall have a fecond remainder labff — b^Jf — la^b 4- a^b^ , or dividing"
by b it will be laff ~ bff — la' 4- a^b. Then invert the order, and divide
the firft divifor by this fecond remainder, and taking the produd of the quotient
, into the faid remainder, which has now fervcd as a divifor; and then
making the fubtradion, we fhall have a third remainder -^ aff 4- a^f — 2ab/
•4- la'-b^ or dividing by — - tf, \i \s ff -— af ■{• abf — 2ab. The divifion is to
be continued in the fame order, and the produft of the quotient — —r into the
divifor 2aff — bff •\' a^b — 2a' being fubtracfted, we fhall have a fourth re-
mainder — af + 2bf — 2ab 4- <?'•, by which, inverting the order, the third
remainder muft be divided, and the produd of the quotient /^ into the dl-
E ' vifor
36 ANALYTICAL INSTITUTIONS. I06K I.
vifor being fiibtra^led, we fhall have a fifth remainder ihf — lah^ or dividing
by 2^, it is/ — a. Now if this is divided by the fourth remainder — af
+ 2^/ — lah + «% and the produd of the quotient — into the divifor is
fubtraded, nothing will remain. .Whence if by the denominator of the laft
quotient, it being a fradlion, the laft divifor — «/ + '2-^f — 2^3 -f- ^* ^lall be
divided, the quotient will h^ f-^a^ the greateft divifor of the two quantities
propofed. But becaufe it was at pleafure whether we chofe for a divifor that
which was made the dividend, or viceverjd-^ that is, we might have divided
-^ af -\- ihf — ^ab 4- aa by / — a-, let the divifion be adtually made, and
the quotient will be ^b — a without a remainder ; and therefore / — a
will be the greateft common divifor, as found above by means of the other
divifion.
Wherefore two formulas may have a greateft common divifor, though being
ordered according to fome certain letter, it cannot be found in this manner;
in which cafe it muft be fet in order again, according to fome other of it*s
letters. Now if this be tried by fetting it in order according to any other
letter, and if it will not then fucceed, the quantities propofed will have no
greateft common divifor. Thus it would not be found in the laft example, by
fetting them in order according to the letter h ; which however is found by
ordering them according to the letter /.
Now the fraction -. — Ti — tt-x being given, if wc divide the
numerator and denominator by a^ + ^ax — 2^*, we (hall have the fradion
x* — X^ax + 4fl*
» — l«
Alfo the fraction — j — , , —. ^-^ — • 7, by dividing hv x* ^ ax
+ 6^*, will become ^ ~ 3^^ — f^ ^
And the fraaion ^Lf-^^L±^ , by dividing by /- ^, will become
f* --zab - \
Thus thcfe fraftions are reduced to more fimple expreffions, as is faid above
at § 30.
Rcdu^ion
SECT. |4 ANALYTICAL INSTITUTIONS. 17
Redu^ion of Irrational ^dantities to more ftmple Ex'prejpons*
37. It has been obferved already, how irrational quantities arife, which are Surds re-
otherwife called Surds, or Radicals. For when the root required cannot be ^^^"-'^^ ^°^*
actually extracted, then we have rccourfe to a radical vinculunjy which infinuates
it. But it often happens that the quantity under the vinculum is the produd of
two fadtors, one of which is a true power of the fame name as the root required.
As if it were K^aabc, or \^ a'b — a\v ; the firft of which is the produd: o( aa into
h, and the other is the produd of aa into i — x. Thus alfo ^^^.v — a^y is
the cube- root of the produft of a^ into x — y. In this cafe the root may be
cxtraded out of fuch of the factors as will admit it, and wrote vvithout the
radical fign, and the other fa6tor may remain under the fign. And this is called
extrafling the root in part, or reducing the radical to a more fimple expreffion.
Thus >^ aabc will be reduced to aV be. And \/ aab — aax will be the fame as
aV h — x't '^a^x — a^y will bc reduced to a<^x —y ; and fo of others. In like
manner, becaufe \/ ^Saabc is the root of the product of i6aa into ^bc, it will
be reduced to ^a\/ ^bc. Thus, becaufe v^^ — ^^^-^ — is the root of the
produdl of ^ ^ mto ab, and the root of ^— is
L±_
p*z:
zb
the root reduced will be V ab. Thus the root ^ ^ ^ * ^ T " i when re-
duced, will be \/ x* + 4^/. And the root ^u^b+iba* will be o.a'^b + za.
pz
Thus \/ a> — $a^b + 3-7^* — b^, which is the root of the produd of aa — 2ab + bb
into a — b, will be reduced to a — b X \/ a — b' But very often it cannot be
known by infpedion only, what are the fadors from whence the propofed radical
proceeds. In which cafe we muft have recourfe to the method of finding all
the divifors, which I (hall give in it's proper place; and if among thefe fliall be
one, which is exadly a power with the fame exponent as the radical indicates ;
the propofed quantity may then be reduced in the manner now explained.
Redu£lion of Radicals to the fame Deiwminaticn.
38. Thofe are called radicals of a different denomination which have a d if- Radicals how
ferent index or exponent. To reduce them therefore to radicals of the fame ''^^"ced to
index, we mutl proceed thus. If the index of one of the radicals is an ^^ ' 9 'J o^^ nominal
part of the index of the other, the greater index mult be divided by the lelfer,
E 2 and
2$ ANALYTICAL INSTITUTIONS. BOOK U
and the quotient fhows that power, to which the quantities muft be raifed which
are under the radical of the lefl'er index, and to which mud be prefixed the
radical of the greater index. Let it be propofed to reduce to the fame index
the quantities \/' ^/ ax and \/a ; or which is the fame, ^^x and ^a. Becaufe
4 divided by 2 gives 2 for the quotient, therefore the quantity a of the lelTer
index mufl be raifed to it's fquare, which is aa, and it will be \/aa^ and there-
fore is reduced to the fame index or denomination as ^yax. Thus -^ a^b^ -j- ab^
and v/^i" will make ^ aH^ -f- ab^ and -^ a}b^. But if one of the exponents is
not an aliquot part of the other, the leaft number muft be found which is di-
vifible without a fradion by each of the exponents of the given radicals, and
this will be the index of the common radical. Then the quantities mud be
raifed to the next inferior degree of the number, by which the exponents are
increafed of the refpe<5tive radicals, and then to the powers fo raifed let the
common radical now found be prefixed. Let the two quantities \/ aci and ^ aaq^
be given, to be reduced to a common radical. The leaft number divifible by
2 and by 3 will be 6, and therefore \/ will be the common radical. Now,
becaufe the index of the fquare-root is in this cafe increafed by 4, and that of
the cube-root by 3 \ therefore the firft will become ^/a^q^, and the fecond will
be ^a*qq. If the radicals to be reduced are more than two, any two are to be
reduced firft, then the third, and fo on fucceflively.
. The manner of reducing rationals to any radical, is plain of itfelf, without
the- afliftance of rules ; by railing the rational to any power of the fame name
or index of the radical given, and then prefixing to it the fame radical.
Addition and Suhtra£lion of Radical ^aniilies.
fubtraflcd.
Surds how 39. To add them together, the radical quantities are wrote one after another
added or ^-j^h their proper ligns. And to fubtrad them, the figns of thofe to be fub-
traaed are to be changed, as is done m other quantities. Thus to add c^as/^bc
to ib\/bx to — ,fv'z>', they muft be wrote thus, ^aVbc + 2b*/ bx — c*/ zy»
To add c^xs/ ab 10 ^x\/ab to yi/ bx, they mvift be wrote thus, ^x»/ ab -f- ^t^^/ ab
•\- yV bx i and then reducing like terms, which ought always to be done, they
will become 2>x\^ ab + y\/bx. To add a — ^ to V^«a — xx, it muft be wrote
a — b •\- V aa — XX* And the fame is to be done in fubtradion, having regard
to the figns.
Mulls -^
SECT, I, ANALYTICAL INSTITUTIONS. 8^
vluhi^li cation of Irrational ^antittest
^o. To multiply rational quantifies by furds or radicals, the rational is wrote Simb how'
together with the radical, without any fign between, only prefixing to the pro- multiplied*
dudl fuch fign, whether pofitive or negative, as fliall be required by the common -'
rules of multiplication ; and this is to be underftood always to be done. There-
fore the produd of a into \^ aa - xx will be a\^aa - xx > The produ^fl of ab
into — \^al> will be — ahs/ ab. And if the rational quantities or radicals fliall
confift of feveral terms, or if they are complicate, every term of one mufl: be
multipl ied int o every term of the other. Wherefore the produft of aa — xx
intoV^;r ^ yy will be aa.^ xx\/ XX —yyi where it is underftood, that all thofe
terms are multiplied into the radical, which are under the vinculum.
41. To multiply radicals among themfelves, fuppofing them to be of the Surds multi-
fame denomination, or reduced to fuch, the quantities muft be multiplied into plied by furda,
each other which are under the radical figns, and to the produd mufl; be put
the fame radical vinculum, with fuch a fign, either pofitive or negative, as the
common rule requires. Thus to multiply '^ be into V xy, the produft will be
Vhcxy, To multiply v/flll-fi intp — s/ aa + xx, the produd will be
— >/
4 «.
42. Moreover, if the radicals (hall have rational co-eificients, whether nu-whenthejr
meral or literal, thofe co-efficients mufl: be multiplied together, and alfo the have rational
radicals together, and the produd of the co-efficients mufl: be put before the '^^'^^^^^"^^•
radical, without any lign between. Thus a^bbc into a^bxx will be aa^b'^cx'^
which reduced is aab^cxx. So la — \^aa — xx into — »/ aa + xx will be
a
43. According to this rule, to multiply m*/ah into n^/ ah^ the product would Sometimes
be mn»/ aahb. But aabb is a fquare whofe root is ah, and therefore the produd'^^y becoma
will be mnab. So that, to multiply two like quadratick radicals into each other, '"^^'°"* ^'
it will fufSce to take away the radical vinculum, and the quantities which were
under it, multiplied into the produd of the co-efficients, will be the total
produd. Thus -i- ^/7^:^7x into — -^ V ax - xx will be — — X ax - *.v,
*» 3 3«
that IS, T-ffc^ -I . But here it mufl: be obferved, that if the radicals
I la
having
20 ANALYTICAL INSTITUTIONS. BOOK I.
having no co-efficients, or unity only, are affeded by the fame fign, pofitive or
neo-ative, the vinculum being taken away, the quantities muft be left with the
fign they have. And if the radicals have contrary figns, all the figns of the
quantity muft be changed. For example, v mto v , or elle
,aa — XX . , aa — XX .,, , aa — xx . ,/- / aa — xx .
«_ a/ into — V » Will be . Alio s/ into
X X X X
_ v/!l-^^- will be ZLfi±jff , or ^-^^^^ . The reafon of which is, be-
X X — X
caufe \/^.^—^-^, (and fo of any other,) is always undcrftood to have
X
+ I for it's co-efficient, and — y/^^-— ^ to have — i. Therefore the
prodiift ought to be 1 x ^"^ ~ "" in the firft cafe, and — i x "^"^ " '"^ in the
fecond. Here are other examples of thefe multiplications.
\/ab 4- V aa - XX into V ab -{- s/ aa - XX makes the produ6t ab ►{- y'aH-abx^
'^ aa — XX + s/a^b - abx^, or ab -i- a^ — x"" + 2\/a^b - abx^»
XX
, _ J:i^^±f^Zl. into X + Jt^±±Jlj^ „3kes the product
_ ^y^^ZEZ _ :liEZ>l^l>! + ,jl^n^^ that is.
, , >J i,a'^ +y*
9CX + ^yy ,
^ — ^q — ^kqq-^itt >"fo ^ — \(i — '^^qq- ^\pp makes the produd
^in + ^"^An-^SfP + "4?? — •^V/'A that is, ^^(iq - ^\^p + ^ v'^y ^^^pp,
Ritional co- 44. Becaufe a\/ax, a - b X \/ ax — xx , and fach others, are the products
efficients how of a rational quantity into a radical, and we already know how to reduce any
und"? the rational to any radical we pleafe ; we can always make the rational multiplier to
viBculum, pafs under the vinculum without any alteration of the quantity. Thus ay/ a —x
will be the fame as V a^ - a^x ; a ^ b X \/xy will become V a'^xy - 2 abxy -)- u^xy j
ax^ m—n will be ^;/;«3;t3 _ j^^i^i ; and fo of any others.
DifFercnt 45. If ihc radicals to be multiplied are not of the fame name, they may be
fiirdshovv reduced to fuch, and then the multiplication may be made as before. But very
mu ipie . qIj^j^ it will be more commodious to infinuate it only, without adlually per-
forming it, and this by writing one radical after another, without any fign inter-
pofed, except the mark of multiplication. Thus n/«4 - xx X ^ xxy will denote
ihe produd of theie two radicals.
Divifion
SECT, I, ANALYTICAL INSTITUTIONS, 3 1
Divifion of Radical ^an titles ,
46. In every term of the dividend and of the divifor, if the fame radical is How furds
found, omitting this, the rational quantities are to be divided as ufual, and are to be
what refulcs will be the quotient. Thus to divide S^\/ 1 ^^7 Z^s/ 'h-^ ^^^^ 4^^o- divided.
tient will be -^ . To divide d^ a^ + a^d* by 2\^ a^b^ + b^, or 6a \/ a"- + b^ by
2bV a^ -f ^», the quotient will be -y- . To divide aa^ aa + xx — 'i^^'c^ aa-\-xx
+ xx^a^ + x^ by a^aa + xx — x^^aa + xx. Omitting the radical, and di-
viding aa — lax -{• xx by a — x^ the quotient will be a — x. To divide
aa + hb by y/^a + bb, be caute the dividend is \/ aa + bb X v/aa + bb , the
quotient will be s/aa + bb .
47. But when the radicals are not the fame, though they have the fame When the
exponent of the root ; let the quantities under the vinculum ht divided by the i"dex is the
rational quantities in the ufual manner, and to the quotient prefix the common J?"^* ^^^ ^^^^
vinculum. Thus to divide ^ a^b — abi by ^^ _ bbj dividing a^b — ob^ by different.
o' — b^ there arifes ab, and therefore the quotient required is '^^ab.
different.
48. And if the exponents of the roots are different, they may be reduced "When the
to the fame, and then the operation will be as before. Thus to divide ^^idex alfo i»
\/a* + 2aU — iab^ — b* hy a + b, the fquare oi a ■\' b muft be found, and
put under the vinculum, which will be then \/ aa + zab -\- bb. Then by the
quantity under this vinculum the other quantity muft be divided, and the refult
will be aa — bb. Therefore the quotient required will be s/'aa — bb.
By combining thefe rules with thofe of common divifion, quantities ftill more
complicate may be divided. Thus to divide a^b — ab'^c — a^by/ be + b'^c\/ be
by « — \/ be, it may be performed as is ufual in divifion.
Dividend, c^b — ab'^c — a^b-^bc + h^c^/bc, Divifor a — >/bc
Rem. — ab'-c — b\\/bc Quotient a'^b — b^c
Thus dividing a^ — abc + c^s/bc — bc\/bc by ^ — \/^^> the quotient
will be aa -\- be ■\- 2ay/bc, And when the divifion will not fucceed, the
quantities muft be wrote in form of a fradion.
ExtraSlion
3?^ analVtical institutions. book r.
ExtraSficn of the Square-Rcof of Radical ^antities*
The fquare- 49. -When quantities any how compounded of rationals and radicals are
^^'^"^ ft h"''^^ quadratick radicals, the rule for extracting the fquare-root will be this. Taking
^*^'^ ^ * fuch a part of the quantity propoled as is greater than rhe remaining part, irom
the fquare of this greater part let the fquare of the lelier part be fubtraded, and
to the greater part let the fquare-root of the remainder be added, and likewife
be fubtrafted from it. The fquare-root of the half of this fum, and of the
half of this difference, being taken together, and taking the fame fign to this
fecond as belongs to the minor part, ^vill make the fquare-root of the propofed
quantity. Thus let us extraft the fquare-root of the quantity 3 + y'S ; fub-
trading the fquare of v^S from the fquare of 3, there will remain i, the root
of which is alfo i. Adding this therefore to the greater part, or 3, they will
make 4, and fubtrading it from the fame, it will make 2 ; now the fquare-root
of the half of 4 is ^2, and the fquare-root of the half of 2 is 1 j therefore
y^2 -f- I wijl be the root required.
If we would have the fquare-root of 6 -|- y'S — v/i2 — '/24 ; from the
fquare of 6 + y'S fnbtradling the fquare of — ^12 — '•'24, there remains 8,
the root of which y'S being added to 6 + 1/8, the greater parr, will make
6 -1- 2V'8, and fubtraded from the fame greater part will make 6. Therefore
the firft part of the root required will be ^Z ^ — > tJiat is, y/^ + ^8, and the
fecond part will be — •/ — , that is — '/3, (for the leffer part of the pro-
pofed quantity was affeded by the negative fign ;) whence y'3 4 ^8 — y^3 will
be the root required. But by the laft example it may be feen, that v^3 + ^i-
is the fame as i + ^1 ; therefore, laftly, the root of the quantity propofed will
be I 4- v'z — VZ*.
Let us extrad the fquare-root of aa -^- ix\/aa — xx . Taking from the
fquare of aa the fquare of 2x\/ aa - xx , there will remain a* — 4aaxx + 4.x*,
the root of which is aa — 2xx. This added to the greater part aa^ and taking
the half of it, will make aa — xx : and fubtraded trom the fame, and taking
half the difference, will make xx. Therefore the root required is K^aa - xx 4- x.
Let us extrad the fquare-root of the quantity aa 4- ^.ax — lay^ax 4 ^xxo
From the fquare of aa 4- ^ax, the greater part, fubtrading the fquare of
— la^ax 4- ^xxi there will remain a* 4- 6a^x -f 9«V, the root of which is
aa + -^ax. This added to the greater part, and taking it's half, it will be
aa 4- ^ax ; and fubtrading and taking the half, it will be ax. Therefore the
root required will be \/aa~+~^v — ^ax,
■ To
^ECT. t. ANALYTICAL INSTITUTIONS. 33
To extradt the fquare-root of this quantity aVbc + d^bc + itjahcd. From
the fquare of ai/bc + d^bc fubtrading the fquare oF ^^abcd, there remains
aabc — labcd -}- bcddy the root of which is ay/bc — dVbc\ which being added
to the major part, and fiibtradled from the fame, and taking half of the ftim
and difference, the half of the fum will be a\/bc, and half of the difference
ds/bc. Therefore the root required is \/ a^/bc -4- >/ d^b^ that is, >/ Vaabc -f
Whcddy or ^ aabc -i- ^ybcdd. If the root cannot be extraded, the quantity
mud be put under a radical vinculum, as ufual.
The Calculation of Powers,
50. There is nothing now to be obferved concerning the Addition or Sub- Powers how
traftion of Powers ; they are to be written one after another with their proper calculated
figns in the firfl: cafe, and in the fecond by changing the ligns of the quantities ^"j, „!
to be fubtracled. But as to the other operations which belong to their ex- ^re integers,
ponenis, it may be firft obferved, that, taking unity for the firft term, and any
quantity whatever, as a, for the fecond, and then fucceflively the other powers
of the fame quantity a in order, it is plain we fhall form an increafing geome-
trical progreffion, i, a, «*, <2', <?*, &c. ; and that the exponents of this pro-
greflion will form an arithmetical progreffion increafing, which will be o, i, 2,
3, 4, 5, &c. The firft term of this is o, becaufe unity being the firil term iti
the geometrical progreffion, in this the quantity a is raifed to no power ; for
Wherefore, multiplying either ~, or a°, by a, which does
not deftroy the equality, the produd will be a =r <z° , which are magnitudes
plainly identical. And befides, if we continue the fame geometrical progreffion
below unity, it will be i, — , ---, -V, ~, &c. And likewife, conti-
a a* u* a*
niiing the arithmetical progreffion of the exponents, they will become o, — i,
— 2, — 3, — 4, &c. And therefore the exponents of fuch powers will be
III ™" I ** 2 ^" ^
negative. So that — , -^, — ^, &c. will be the fame as a , a y a ,
&c. And in general, will be the fame as a"" ; that is to fay, we may
a
always make a power to pafs into the numerator of a fra6lion out of the deno-
minator, and vice verfd^ only by changing the fign of the index.
51. Moreover, if wc fhould defire to introduce new intermediate terms into when they
the geometrical progreffion, the exponents of thefe would alfo be intermediate are fradions.
F terms
34 ANALTTICAL INSTITUTIONS* 8O0K I.-
terms in the arithmetical progreffion, analogous to the former. So, becaufe
V a is a geometrical mean between unity and a, the exponent of this ought to
be an arithmetical mean between o and unity, and therefore muft be — ; fo
that a'' will be the fame as \/a. If two mean proportionals are interpofed
between i and a, of which the firft will be '^a^ and the fecond >^aa, there mufl:
• I 2 "^
be two arithmetical means between o and i, which are ■ — and — ; fo that a^
3 3
will be the fame as '^^a, and a^ will be the fame as '^aa. If three mean propor-
tionals are introduced, they will be '^a the firft, ^yaa the fecond, and ^aaa
the third, and their exponents will be — , — , ~; therefore ^/a will be the
i. * 1. s
fame as a*, and -^aa the fame as a^, or a'\ and </^^ will be the fame as a^ .
And thus we may proceed to as many mean proportionals as we pleafe j fo that,
n
in general, it will be \/a , the fame as «''.
The fame things obtain in refped of the progreffion produced by defcending
below unity. Thus, as — - is a mean proportional between unity and — , or
between unity and a" j {o it's index fhould be an arithmetical mean between
1 I . •— -^ I
O and — I, that is — — j therefore — r will be the fame as a *, or — — .
2 A/a t
Thus likewife ~ and — r- and a"^ will be the fame. And r^- , — r-, a~^
n
will be the fame. And fo, in general, , , and a "* will be the
_ ^ a m
lame.
And what has been fald concerning integral or fradlonal powers of fimple
quantities, is to be underflood alfo of compound quantities. Thus, for ex-
ample, ■ is the fame as aa + bb)^^. So \/aa + bty will be the fame
as aa-\'bb^ ; and the like of others,
fowers how c^i. From the nature of the two foregoing progreffions, the geometrical and
"l"?^pji«^ o'^ arithmetical, we obtain a method for the multiplication or divifion of any twa
powers of the fame quantity, whatever they may be ; and that is, by adding.
the exponents together when the powers are to be multiplied, and by fubtraft-
wig.
SECT. 1. ANAtYTieAL INSTITUTIONS. 3^
ing the exponent of the divifor from that of the dividend, when the powers are
to be divided. For, as to mLiltiplication, as the produd is the fourth propor-
tional from unity and the two factors, thefe four terms will be in a geometrical
proportion, and their exponents in an arithmetical progreflion. Therefore the
exponent of the fourth, that is of the produdt, muft be greater than the ex-
ponent of the third, by as much as the exponent of the fecond is greater than
the exponent of the firft. Bur the exponent of the fecond is greater than the
exponent of the firft, which is o, by it's whole quantity ; therefore the exponent
of the fourth ought to be greater than the exponent of the third by the whole
exponent of the fecond ; that is, it ought to be equal to the fum of the ex-
ponents of the fecond and third. As to divifion, it has the fame proportion as
multiplication, but only inverted. It*s firft term is the dividend, it's fecond the
divifor, the third the quotient, and the fourth is unity. Therefore as much as
the exponent of the dividend is greater than the exponent of the divifor, fo
much the exponent of the quotient ought to be greater than o. Therefore it
ought to be exadly the difference of the exponents of the dividend and the divifor.
So that to multiply aa by a, or a" by «', the produd will be a*"^* or a^. To
multiply tf' into <»% the produfl is a^ % or d^. To multiply a^ into «"*,
the produd is « , or a*. To multiply a* mto a*, the produd is «* *,that
is d^. To multiply a ^ intofl*, the produd is a ^ *, that is a *^. To
±— ±- ± — ±-
multiply a ^ into a \ the produd is a "* , or a — - — ^ .
And fo to divide a^ by a\ the quotient will be «^~', or «% To divide a^
by a"^, the quotient will be a^ , or a\ To divide a^ by «*, the quotient
will be a *, or a*. To divide a^ by a % the quotient will be a^ "■, or
a^. To divide a "^ by a , the quotient will be a "* , that is,
+ »f T »?r
a ^ — .
mt
53. And becaufe in the progrefTion before confidered, taking any term Powers may
whatever, the fame term with a double exponent will be the fquare of the term be raifed, or
fo taken ; and a term with a treble exponent will be the cube of the affumed ^°^^^ extradt-
term; and a term with a quadruple exponent will be the fourth power; andg^ponems.
fo on. And a term with half the exponent will be the fquare- root of the term
affumed ; a term with a third part, a fourth part, &c. will be the cube-root, the
fourth root, &c, of the term alTumed. It follows therefore ilitat, to reduce one
F a power
3«
ANALYTICAL INSTITUTIONS,
BOOK I»
power to another, it will be fufficient to multiply the exponent of the given
power by the exponent of that power to which we would raife it : and to extract
any root, it will be enough to divide it's index by the index of the given root.
Thus to raife a^ to it's cube, it will be ^ "^j ov a , To raife a^ to the cube,
it will be a^^^, or a*. To raife <z"~* to the fifth power, it will be ^~*^^, or
<?""*. To raife a "" to the power whofe index is + — , it will be a "'^ .
s
Thus to extrad the fquare-root of a\ it will be a^. To extrad the cube-root
±^
±--
of ^*, it will be ^"*.^ To extrad the root r oi a " , it will he a "'^ , &c.
Extended to ^4. What I have here faid concerning the powers or roots of one and the fame
compound finiple quantity, may be understood in like manner concerning the powers or
quanutiea. j-qqj-s of any compound quantities, as is evident. And by this method the
calculus of fractions and radicals will be much facilitated.
Simple dU
vifors how
found ; as
alfo com-
pound di-
vifors.
Of Linear or Simple Divifors of any Formula whatever.
55. Any quantity or formula whatever, whether complicate or not, is faid ta
he prime or Jimple, when it is not exaftly divifible by any other quantity, except
itfelf or unity. And it is called compound when it is exadly divifible by forrre
other quantity. Thus, for example, a -\- b^ aa + xx, x^ — aax + aab, and
fuch others, will be prime or fimple. But ab is compound, becaufe divifible by
a or k So aa — xx is compound, becaufe divifible by a + x or a -^ x.
And fo of others.
Two or more formulas are relative primes, when they have no common di-
vifor, and that the lefler is not a divifor of the greater. Such between them-
felves will be aa and bb. Alfo aa + lab + bb and aa -f bby &c. And on
the contrary, they are abfolutely and relatively compound, between themfelves,
when they have fome common divifor, or that one of them can divide the
other. Such are aa and ab, which are both divifible by a ; fuch are aa — xx
and a -V X, which are divifible by ^ + a^, &c.
In order to have all the fimple divifors of any quantity, either numeral, or
literal, or mixt, it muft be divided by the leaft of it's divifors, and the quotient
again by the leaft of it's divifors, and fo on continually till a quotient arifes,
which
SECT. I» ANALYTICAI, INSTITUTIONS. J7
which can no longer be divided except by Jtfelf. The quantities thus arifing,
unity being comprehended among them, will be all the limple divifors. And if
they are taken two by two, three by three, and fo on ; according to all the com-
binations poffible, they will give likewife all the compound divifors.
For example, let us find all the divifors, fimple or compound, of the number
300. Let the given number 300 be wrote at A, and at one fide, as at B, fet
down it's leaft divifor, as 2. Then dividing by 2, write the quotient 150 at A
under 300 ; and again divide this number 150 by 2, and over againft it at B
write the divifor 2, and the quotient 75 at A under the firft quotient 150-
Now, becaufe yg is not divifible by 2, let it be divided by 3, and write the
divifor 3 over againft it at B, and under it at A the quotient 25, The leaft
divifor of 25 will be 5, which muft be wrote over againft it at B, and the
quotient 5 under it at A. The laft quotient 5 is not divifible unlefs by itfelf ;.
therefore it muft be wrote afide at B, and we ftiall have all the prime divifors ;
to which we may add unity, becaufe it is always a divifor of any quantity.
Now to have all the compound divifors, according to all the combinations, lee
the firft and fecond divifors be multiplied together, and the produ(fl 4 be wrote:
at B over againft the fecond divifor. By the third divifor let all above it be mul-
tiplied, and let the produ6ts 6, 12, be wrote afide, fetting down but once thofe-
that may chance to be repeated. In like manner, by the fourth let all above it
be multiplied, and the products fet down as before : and fo on fuccefllvely to
the laft. Now the numbers wrote at B will be all the divifors of the propofcd
number 300.
A. B.
I
300 2
150 2 4
75 ^ 6 i2
25 5 10 15 20 30 6a
5 5 ^5 5^ 75 100 150 3,00
I
Let the given formula be iiahb, of which we are to find all the divifors. As-
it is not divifible by 2, let it be divided by 3, which is to be wrote over againft
it at B, and the quotient yabb under it at A. Let yabb be divided by 7, which
is to be wrote over againft it, and the quotient abb underneath. Let abb be
divided by «, which is wrote afide, and the quotient bb under it. Then divide
hb by bi which is wrote afide, and the quotient k underneath. This is tQ be
divided by b, and wrote over againft it ; and then we ftiall have all the prime
divifors i, 3, 7, «, b, b^ of the propofed quantity. To have thofe that are
compound we muft multiply 3 into 7, and the produdt is 21. Multiply 3, 7,
21 into <2, and the produ(5is are 3^2, 7^, 2i«. Multiply all the divifors 3, 7,
21, a^ 3^, 7<?, 2ia into b, and there will arife 3^, 7^, 21^, abj ^ab, '-jaby 2iab;.
andir
3f ANALYTICAL t N $T I T U T t ON S, f OOK I.
end fo proceed. Thus the column B will contain all the divifors of the quantity
propofed, both fimple and compound.
A. B.
1
2iabb 3
yabb 7 21
abb a 3« 713 21a
hb b ^b 'jb lib ab ^ab 'jab iiab
b b bb ^bb "jbb %\bb abb o^abb "jabb 21 abb
I
In like manner, let labb — 6aac be given. Let it firft be divided by 2,
and the quotient abb — ^aac by a, and the new quotient bb — ^ac by itfelf,
as being divifible by no other quantity. And therefore all the divifors will be
as in the column B.
A.
B.
I
2abb — 6aac
2
abb — ^aac
bb — ^ac
I
a, 2a
bb — 2^c, 2bb
6ac', abb — ^aaCy 2abb — 6aac
Compound ^6. But if the laft quotient, or perhaps the formula itfelf at firfl: propofed,
^°I'?"^^'^**''^{hall ftill be compound, and yet is not divifible, after the foregoing manner, by
any fimple quantity, fo that all it's divifors are compound terms; the way of
obtaining them is different, and may be thus. The quantity is to be fet in order
according to fome one of it's letters, as has been already fhown at § 24 ; and
if there are fradions, they muft be reduced to a common denominator. Then
all the divifors of the laft term muft be found, compounded of numeral divifors
if there are any, and of the letter of one dimenfion. And if the greateft term
has a numeral co-efficient, it muft be divided by fome one of thofe divifors, by
which that co-efficient of the greateft term is divifible. By every one of thefe
divifors, firft added and then fubtradled from the letter, by which the formula
is ordered, the divifion muft be tried ; and all thofe by which it fucceeds will be
fo many divifors of the propofed quantity.
Let the formula y^ — 4ay^ + gay — la^ be given. The divifors of one
dimenfion of the laft term are a and 2a. Therefore the divifion muft be tried
by each of thefe added to the letter y, or fubtraded from it, becaufe the co-
efficient of the greateft term y is unity j that is, by y ±: a, or by ^ ± 2a,
Firft let it be divided by ^ -— 2a, and the quotient is yy — lay -f aa, which
alfo is divifible by ^ — a, giving y — a in the quotient. Wherefore the di-
vifors of the formula propofed are y "-^ a, ^ — a, and y — - 2a, from the
produd of which it is derived.
Let
SECT. I. AMAtYTlCAL INsTlTrTIOJTS, 5g>
Let the formula be 6y* — ay^ — 2iaayy + 2^^y + 20^?*. The divifors of
one dimenrion of the Jaft term are a, 2^, ^^a, 5^, lo^, zoa ; and becaufe the
firft term Gy^ is divifible by i, 2, and 3, we muft try the divifion by ^ ± —^y
y ± a, y ± za, y ± -^ a, y ± s^, y ± 10a, y ±-^a, y ±±-a,y ±-^a,
^ 333
y + — «, y± — ^y y •±. — ^. But becaufe it would be too tedious and
3 3 3
troublefome to try all thefe divifors j in order to know among fo many which
are to be feleded, we may make y •=. 7. + a ; and fubftituting this in the place
of y, and alfo it's powers, there will arife another formula, which is this*
6z* + 24^2' + 36^^2:2 + 2/^a^z -f- 6«*
— az^ — ^aazz — ^a^z — a*
— 21^^2:2; — 4.2a^z — 21a*
+ 20a*
Which by colle<5llng the terms will be this,
6z* + 2^az^ + i2«V — i8^'z + ^a\
Now all the divifors of the laft term ya^ of this formula are found to be if
and ya, which divided by 2 and by 3, the numeral divifors of 62% will make
-^ a. — a. -^ a. —a. And becaufe it was made v = z + ^, if thefe
divifors can be made ufe of in the fecond given formula by z, they will alfo be
ufeful in the firll by y, when they are increafed by the quantity a, that i* by
making them — ^, —«, — ^, -^a. Therefore let thefe divifors be com-
pared with the divifors of the firft formula, and choofe only thofe which agree
with them, that is —^^ and — a, by which added to and fubtradled from j'^
the divifion muft be tried ; which will fucceed with y + — a. But notwith-
3
ftanding this operation, if there fliould flill remain too many divifors to be
feledted by this comparifon, we may make jy =: 2; — ^, and another formula.
will arife. From the divifors found by this, the quantity a muft be fubtra(5led,,
and then they are to be compared with thofe which are feleded by means of the
fecond ; and by them which agree, which will be fewer in number, the divifion
is to be tried* And proceeding in the fame way of operation by new fubfti-
tutions, making y =. z + la, y zn z — 2a, &c% the divifors may be reduced
to fuch fmaller numbers as will be fufficient.
57, Wheiii
40
ANALYTICAL INSTITUTIONS.
BOOK r.
Howthcjco- 57. When the propofed formula has it's firfl or greateft term mukipUed by
cfficlentof any number, inftead of applying the rule aforegoing to this cafe, it may be
the firft term jj^q^q convenient to change the formula into another, the firft term of which is
nioved. ^^* multiplied only by uni[y ; and then find the divifors of the fame, from which
you may afterwards pafs to thofe of the propofed formula.
i-<et the formula be, for example,
3j' + 9^yy "-" i2<j(2y — i2aah.
+ shy + 9^4)'
Make 3j? ~ 2, (or, in general, ny -zi Zy putting n to reprefent the numeral
co-efficient of the highefl power,) and thence y =z ■ — z. This being fubfti-
tuted inftead • of jy, and it's powers exprefled in like manner, we (hall have the
formula z^ + (^az^ + 'T^bz^ — '}^6a^z H- i']abz — loZa^by all divided by 9.
Let the divifors of this be found, (at prelent omitting the denominator 9,)
which will be z + 12^, 2; — 3^, z -{• ^b ; and taking account of the deno-
minator 9, one of thele is to be divided by 9, or two of them by 3, and they
will be, for example, 2 + 12a, ^ ~ ^^ , t L.. but it was made 3 y — 2 ;
and fubftituiing this value of 2; in the divifors, they will become ^y + 12a,
y -^ a, y + b, which are the three divifors of the formula propofed.
SECT. II.
Of Equations, and of Plane Determinate Problems,
Equations
and their
affections
58. Equation is a relation of equality, which two or more quantities, whether
numerical, geometrical, or phyfical, have with one another when compared
together ; or which they have with nothing when compared to that. The
aggregate of all thofe terms which are wrote before the mark of equality, is
called the Firft Member of the Equation ; and the aggregate of all thofe which
are wrote after it, is called the Second Member, or the Homogeneum Compara-.
tionis. Thofe terms of the equation are homogeneous, when each of them is
of the fame dimenfion ; and therefore in an equation they are faid to obferve
the law of homogeneity, as in this equation axx — bbx = a^. And thus, on
the contrary, they are faid not to obferve the law of homogeneity, when the
terms are not fuch, as in this equation x* — ax^ zz b,
59' ^
SECT, n.
ANALYTICAL INSTITUTIONS.
41
59. A Problem is a propofition in which it is required to do or to find fome. A problem^
thing, by means of other things which are known, or of certain conditions w^^^^-
which are given, and therefore called the Data of the Problem. So thofe things
which are required are the ^idfita of the Problem.
60. Of Problems fome are Determinate, and others Indeterminate, The deter- When pro-
minate are thofe which have a certain number of folutions, or which can be Blems are
refolved by one or more determinations, but alv/ays in a finite and limited ^^T^'"•f^^*
number. Such it would be if we ftiould inquire, terminate! ^*
Ft~ ' ' '
'f>'
A
B
where we ought to cut the right line AB, fo that
the whole line, to it's greater fegment, fliould have
the fame ratio, as the greater fegment to the leffer.
Becaufe one point only can be afTigned in this line, for example C, which will
have the property required. The fame thing
would be, if in a given circle AED we were
to find a point, luppofe C, in the diameter
AD, from whence raifing a perpendicular
CE, terminated in the periphery ; this per-
pendicular {hould be jnft equal to a third
part of the diameter. For there are only two
points, each at an equal dirtance from the
centre, that can facisfy this, demand.
Now if it were propofed to find, out of the right line AD, fuch a point E, fo
that drawing from it two right lines EA, ED, to it's extremities A and D, the
angle AED (hall be a right angle ; it will be found, that there are infinite fuch
points as will refolve the problem, or the whole periphery AED, as is known
from Euclid. In the fame manner, if a point C is required in the diameter AD,
from whence raifing the perpendicular EC in the circle, it (hall be a mean
proportional between the fegments AC, DC j it will be found, that all the
points of the diameter will Iblve the problem (and therefore luch points are
infinite in number); which is therefore called an Indeterminate PrtbUm,
Determinate problems have occalion for one unknown quantity only, t)ut
indeterminate ones of two at lead, though the manner of forming an equation
is the fame in both. Of thefe I fliall treat particularly in Sedl. III.
61. The given orknown quantities are ufed to be denominated by the firft Known an^
letters of the alphabet, as has been faid already j but the unknown, or fuch as "=^^<^"^wn
are required, by fome one of the laft letters. And here it may be obferved, Jjovv^diftin-
that if the quantity fought is a line, it ought always to have it's origin or be- .niiilied,
ginning at fome determinate fixed point. And as that which is required is
already fuppofed to be done or known, by calling it, for example, x ; fo that
'Uom thefe quantities fiippofcd as known, others that depend on them come to
G 4^c ,
42
ANALYTICAL INS TI TUT TON 5.
BOOK T,
S^ 2.. lyQ known or given, as it were by bypothefis.
Thus, AD — a belong given, and C being
fuppofed the point required, and therefore
calling AC x, it will be CD zz a — x ;
and thus we may argue of feveral others. And
further, though many of the quantities are not
exprefsly given, like as in the line AD ; yet,
33 however, they are given implicitly, and as it
were by conftruclion. Thus, in the right-angled triangle AED, if the hypo-
thenufe AD z= ^ is given, and the fide ED zz b ; the n, by th e 47th proportion
of the firfl: Book of Euclid^ the fide AE — \/ aa - hb will be therefore
given. Thus, in the femicircle AED, the diatiieter AD ~ a being given,
and the fegment AC = h^ it will be CD zz a — hi and therefore, by
Euclid, vi. 8, it will be CE zi '^ ah - bh' ,Or becaufe AC was called at, it will
be CE = V ax ~ XX ^ which is given both by hypothefis and by conftrudion.
Thus, in the right-angled triangle ACB, from the
right angle B letting fall the perpendicular BD,
let be given, for example, the two lines AC z: a^
and AB ~ b\ then in like manner will be given
all the other lines BC, BD, AD, DC. For BC =
s^ aa - bb, by EucUdy i. 47, as faid before. And
by vi. 8, CD will be a third proportional to AC
and CB ; wherefore it will be CD n: — "^^ , by
^^^- 3
the 17th of the fame book.
and therefore AD ~
Mig, 4-
hb
AD will be. a third proportional to AC and AB>
DB will be a mean proportional between AD
and DC j or elfe it will be a fourth proportional to AC, CB, AB, and there-
fore, by 16 of the fame book, it will be DB =
^ ^"^^ — . Thus, in the right-angled triangle ABC,
if DEI is parallel to BC, and are given AB = a,
BC =: b, AD — X ', then, by 4 of vi., will be given
DH = ^, AH = 1^1^. And the fame may
be obferved of infinite others.
Equations 62. Tbus, by fuppofing that alf-eady doncor kno\vn> which is- to be done or
how aaived. known, and by treating given- and fought quantities indifferently, all the con-
ditions may be fulfilled, which are required by the propofition or problem, and
we fhall thus arrive at an equation. Let there be* a
^^S' I- light line AB, which is to be cut in extreme and
A C B ~ niean proportion. Let AB =: a, and let C be tlia
point
SECT. II.
ANALYTICAL INSTITUTIONS,
41
point required. Let AC — x, and therefore CB n= ^ — x. The condition
implied is, that it ought to be AB . AC :: AC . CB; that is^ a . x :: x . a — x.
But by the nature of a geometrical proportion, the red;angle of the means muft
be equal to that of the extremes ;. fo that aa — ■ t7X = xx, and thus we are now
come to an equation. Again, let there be three numbers given, the firft is 4»
the fecond is 5, and the third is 10. A fourth number muft be found, fuch
that, if from the produ(fl of this into the third the firft be fubtraded, and if the
remainder is divided by the firft, the quotient (hall be equal to the fecond
number given. Let the number fought be denoted by x -, then the product of
this into the third will be lo.v, from which fubtradling the firft, the remainder
will be lo.v — 4, and dividing this by the firft, the quotient will be
;vhich by the condition of the problem fliould be 5, that is
I o.v •— 4.
lo*- — 4 .
■5, which
is the equation required.
Fi^. 4.
Again, in the triangle ABC, are given the fides
AC zz a, BC ~ ^, and the bafe AB zr f ; we are
to find in this fuch a point D, that drawing DH
parallel to BC, the fquare of DH may be equal to
the redtangle AD x DB, Make AD = x, whence
DB — c — X ; and becaufe of like triangles ABC,
ADH, it will be DH r: — . Then by complet-
ing what the problem requires, we fhall have the
equation — ^— iz f at — xx.
63. If the given triangle ABC is right-angled at B, we (hall have no need Somelines tc
to denominate AC r: a, but otherwiie = x^If+Te, to exprefs thereby the^^^^"^™^*
condition of a right-angled triangle. Thusf^^fg^j.^jj^.^^
in the femicircle AED is giv^en the diameter
AD — 2a, and the fegment AC rz if ; hence
confequently is given the line CE, and there-
fore it ought not to be exprefTed by a letter
at pleafure, but to be denominated from the
property of the circle, by making it :i=:
V^ 2ab — bb ; thereby exprefsly to indicate, that
it is an ordinate in the circle at the point C.
And in general it is to be underftood, that the
. fame ought to be done in all like cafes.
64. But perhaps it may make fome difiiculty, that very often the lines giyei^ J^^'^^J ^'"^^ to
in a figure, by which the problem is propofed, are not fufficient to obta:n (uc\\ ^ '^^"'"*
quantities or denominations, as are neceftary to arrive at an equation. Such a
G 2 cafe
44
ANALYTICAL INSTITUTIONS.
BOOK r.
^''^' S* Q^ ' ^^^^ wQuld be, if tvio indefifxite right lines
AE, AF, were given in pofition, and a point
C out of thofe lines : and if it were propofed
to draw a line CF in fuch a manner from the
point C, as that it Ihould include a triangle
AEF, equal to a given plane. The expreflion
of the triangle AEF would be half the rectangle
of AF into EG, letting fall EG perpendicular
to AF. Now make AF r: jc ; but yet it will not be poffible to determine the
value of EG from the lines hitherco defcribed. Upon fuch occafions it will be
neceffary to conftruft or complete the figure, by drawing parallels, raifing or
letting fall perpendiculars, forming fimilar triangles, defcribing circles, or by
iifing the like expedients of the common .Geometry j for which it is not poffible
to give any general rules, as they will depend on the various circumftances of
problems, on fagacity, induftry, and pradice, and often upon chance. Buc
commonly thefe propofitions of the firft Book of Euclid are ufed to be of good
fervice, 5, 13, 15, 27, 29, 32, 47; fome of thefecondj thefe of the third>
20, 21, 22, 27, 31, 35, 36; thefe of the fixth, i, 2, 3, 4, 5, 6, 7, 8;
and fome of the nth and 12th when folids are concerned. Therefore, in the
problem now propofed, from the point C draw CD parallel to EA, and EG,
CB, perpendicular to FA produced. Now bccaufe the right lines AE, AF, are
given in pofition, and alfo the point C ; the lines AD, CB, will be given in
magnitude. Therefoie make AD -=. a, CB =z b, AF — y, and let the given
plane be nr cc. And as the triangles FDC, FAE, are fimilar, as alfo the tri-
angles DCB, AEG ; we fball have the analogies DF . AF :: (DC . AE ::)
BC . EG. That is, ^ + .v . ^ :: ^ . EG. Therefore EG = — . And be-
caufe the triangle AEF, that is, half the rejSlangle of AF into EG, ought to
be equal to the given plane cc, we fliall at laft have the equation — - — — cc.
Equations (i^- The propofing of the problems only, which hitherto I have taken for
how formed examples, has brought me immediately and diredly to an equation ; becaufe it
fromdi&rent ^^^^ required that the two quantities fo found fliould be made equal. But this
iame quan- method will not thus fucceed, when from certain quantities given, it Qiall be
tity. propofed to find others, without fuch a condition as will lead us exprefsly to ^an
equation.. Then it may be needful to ufe a little art to obtain it, and that will
be by means of different properties, and compounding the figure if neceffary,
to find two different expreflions of the fame quantity, and fo to make an equa-
tion between them. I faid by means of different properties, becaufe the fame
property, however managed, will always give the fame expreffion. 1 (ball
produce three examples of this,, which I think may fulBce at prefent.
Given
ANALYTICAL INSTITUTIOlJS.
45
Given the ifofceles triangle CDB, the diameter
AB of the circle CADB is required, in which it
may be infciibed. Make CD — a, CB = BD
zz. b, BA = Xf which is the diameter required ;
and draw CA. The two triangles ABC, BCE,
\ will be fimilar, becaufe the angles BCA and CEB
are right ones, and the angle BCE zz BDC zz
BAC. Therefore it will be AB . BC :: BC . BE j
that is, X . h \\ b » BE ; whence BE z= — ■ .
X
JNIoreover CE is the half of CD, whence CE — f^.
aa . l^
And becaufe of the right-angle CEB, it will be CB^ zz — +
But the
fquare of CB is alio = bb. Therefore we fhall have the equation bb zz
+
XX
In the triangle ABC the three fides are given,
and from the angle A letting fldl the perpendicular
AE upon BC ; the two fegments BE, EC are re-
quired. Make AB zz «, AC = b, BC zz r,
BE z: .V ; then it is EC zz ^ — x. By the 47 of
the firft oi Euclid^ the fquare of AE will be equal
to the fquare of AB, fubtrading the fquare of BE ;
that is AE^ zz AB^ — BEj. But by the fame
it will be alfo AE^ = ACq — EC^. There-
fore ABq — BEf = AC^ — ECy. And reducing to an algebraick ex-
preflion, it will be aa — xx-=z bb — cc + icx — xx, that is, aa zz bb — cc
+ 2cx,
Again another way. Let EF be drawn perpendicular to AB ; then, by the 8 of
the fixth o{ Euclid, it will be AB . BE :: BE. BF; and therefore BF=: -^. Thence
a
AF zz ^ ^ . And, by the fame propofuion, it will be AF . AE : : AE . AB;.
and therefore AE^ — aa '^ xx. From the point E drawing the right line
EM perpendicular to AC, by the fame way of arguing it will be found, that
AEy = bb -^ cc -]r 2CX — XX ; and making a comparifon between thefe two.
values, we (hall have the fame equation as before.
The
4«
ANALYTICAL INSTITUTIONS*
BOOK I.
Fig, 8. The quadrant AH M being given, and the tan-
gents AI, HK, of the two arches AH, HD j it is
required to find AB the tangent of the fum of thefe
two arches. Make the radius CA zz a^ A\ zz b,
HK zz Cy and AB zz x. To obtain an equation,
from the point D let be drawn DE perpendicular
upon AC. Then by the fimilar triangles CBA,
CDE, we may find the values of CE and DE. Let
lis examine then if we cannot contrive to denomi-
nate the fame DE in another manner. Therefore
drawing DF perpendicular to CH, by means of
the fimilar triangles CAI, CEO, we may have the
lines EO, CO ; and in like manner, by means of the fimilar triangles CHK,
CFD, we may have the line FD ; and from the fimilar triangles CEO, FOD,
we may obtain OD ; whence we fliall finally arrive at OD in another manner,
independent on the firft, and then ED =z EO -f- OD, which will give us an
analytical e<iuation.
I have here produced the order of arguing only, which might be ufed to
bring us to an equation ; omitting the adual operation, becaufe the problem
will be completely folved in another place.
How we are
to proceed
when angles
are con-
cerned.
66. It will often require fome particular expedients to be made ufe of, in
fuch problems in which angles are concerned ; for by fome artifice we muft pafs
from the properties of angles to thofe of lines, which may enter the problem in
their ftead. 1 will take an example of this from the loth propofition of the
fourth Book of Euclid. Let it be required, upon
the given right line AB, to conftrutSt an ifofceles
triangle ABC, of which the angle at A fiiall be
half of either of the angles ABC, or ACB.
Let the triangle ABC be fuch a triangle, and
therefore the two angles ACB, ABC, will be
equal to each other, and thence the fides AC,
AB, will alfo be equal. Let the right line CD
be drawn in fuch a manner, that it may bileft
the angle ACB. Then the two triangles ACB,
CDB, will be fimilar, from whence we fhall have this analogy, AB . BC ::
BC . BD. But it is BC = DC =: AD, and therefore it will be AB . AD ::
AD . DB. And now fee the problem propofed reduced to another, which is,
to divide the given line AB in extreme and mean proportion. Wherefore this
fecond problem being refolved, the point D will be found, and the problem at
firft propofed will then be folved. For bifeding DB in E, and raifing the
perpendicular EC, it will meet in C an aj"ch BC, which is defcribed with radius
AB from the centre A. Then if from the point found C we draw the lines
CA, CB, the triangle ACB Ihall be fuch as is required.
67. Now
SECT. IX. ANALYTICAL INSTITUTIONS. 47
67. Now when ihe equation of a problem is found, all that remains to be Equations
done is, to derive the value of the unknown quantity from it ; that is, to re- ^°^^ reduced.
duce the unknown quantity to be equal to fonie known and given quantities,
in which confifts the folution of the problem. And this is called the Refolution
of the Equation.
For this purpofe we mud call to our afiiftance the following Axioms.
1. If to tvt'o equal things we (hall add equals, or if we Qiall fubtrad equals'
from them, the fums or the remainders will alfo be equal.
2. If equal things are multiplied or divided by equals, the prodiids or
quotients will alfo be equal.
3. If from equals a root be extraded with an equal index, the roots or
quantities refulting will be equal.
4. [f equals are raifed to a pawer with an equal index, thofe powers or re-
fulting quantities will be equal.
From the firft of thefe axioms we learn, that if we fhould defire that any
term of an equation, which is on one fide of the mark of equality, iLould pafs
to the other fide ; this may always be done without deftroying the equality of
the terms. Let the equation be ax ■\- bb "=1 -^ xx -^ cc i if we add xx to both
the members of this equation, it will be ax + bb -\- xx in xx — a:a; -f- cc, in
which XX — XX expunge one another, and there will remain ax -\- bb -\- xx i= cc,
where the term xx has paffed into the firft member of the equation ; from whence
if bb is to be taken away, it will be ax -\- bb -\- xx — bb = cc — bb ; but
bb — bb expunging one another, the remaining equation will be ^^ + a;^ =:
cc — bb, where the term bb has paffed into the fecond member of the equation.
"Wherefore in general, when we would have any term pafs from one fide of the
equation to the other, it will be enough to expunge it on one fide, and write it
on the other with it's fign changed. In confequence of this, we may at pleafure
make a term pofitive which in the equation is negative, and fo on the contrary >
and that will be by writing it on the oppofite fide, and changing it's fign.
Therefore aa — xx = bb will be the fame as aa — bb z=: xx, or xx zz aa — bb.
Wherefore if there fhall be the fame term on each fide of the equation, and
afFedled with the fame fign-, they may both be expunged without injuring the
equation. As, rf it were ax — xx = bb — xx, it would be reduced to ax = bb.
For, tranfpofing the term -- xx, it would be then ax 4- xx — xx = bb, where-
XX ~ XX deftroy each other. The fame thing would follow, if, inflead of
tranfpofing the term which is common to both members, it were added to both
if in the equation it were negative, or fubtratled from both if affirmative.
68. From the fecond axiom we learn, that if an equation (hould have frac- Reduced bj-
tions in it, it may always be freed from them without prejudice to the equation; multiplica.
by reducing every term to- a comroon. denominator, and. then rejeding that.'^*^"' ,
deno*
-4^ ANALYTICAL INSTITUTIONS. BOOK I.
denominator : becaufe equal quantities multiplied by equals make equal pro-
duds. Let the equation be a -,- = l>. Reducing all to a common deno-
iiiinaror, it will be "7 '^'^ = -j- , and multiplying all by h, or rejedling the
•common denominator, it will be ab — xx ■±. bh. And if befides we would
have the term — xx to be pofitive, it will be ab :=:^ bb + xx^ or othervvife
XX m ab — bb. Let the equation be — zz ab. Reducing to a com-
mon denommator, it will be zz , and multiplymg all by 2^, it
will be aax —^ ibxx z= laab. And if we defire befides, that the term — 2bxx
fhoujd be pofitive, and moreover that all the terms in which the letter x is con-
cerned (hould be on one fide of the equation, make ibxx — aax zz — 2aabi
or reducing the whole equation to one fide, by which it will be equal to no-
thing, it will be 2bxx — aax •-{- laab = o.
Tleduccd by 69. By the fame axiom we may free any letter, or any power of a letter, in
divifion. any equation, from it's co-efficient, or from any quantity in which it happens
to be multiplied ; and that is by dividing every term by that co-efficient. Now
let there be 2bxx — aax zz — 2aab, and let it be required to free the term 2bxx
from it's co-efficient 2b, Then dividing each member of the equation by the
fame quantity zb, the quodents -—--r — 7- fhall Hill be equal, and
therefore xx — -^^ =: — aa. Again, if the equation hax — "^ bb — ^-^
zb o ' T ^ 2a
—-bx, and if it were defired that xx (hould be pofitive, freed from it's fraflion
and co-efficient, and that all the terms which any how contain the letter x
fhould be on one fide of the equation, and known terms on the other; write
then ^-^ -{• bx -{• ax ~ bb + A- , multiply all the terms by 2a, and it will
be ^bxx + ^abx + laax = 2abb + -7— ; then divide every term by 3^, and
the equation will become xx 4- — ax + —^ zz — ab -A rr* which has
all the cbnditions required.
Reduced by yo. From the fourth axiom we learn, that if an equation contains radicals or
raifing furds, it may be freed from them, by writing the furd term or terms on one
powers, fjjg Qf [j^g equation, and the rational quantities on the other, and then fquaring
each member of the equation if the root is quadratick, or cubing if cubick, &c.
Thus if we had '^ aa — xx -\- a zz x, v/e muft write it thus, Vaa — xx zz x — a,
«nd then fquaring, aa — xx zz xx ^ 2ax 4- aa, that is 2ax = 2XXi or x = a.
Thus
3
SECT. JI. ANALYTIC AX, INSTITUTIONS.
4f
Thus if the equation were ^ aax — x^ — « + j^ z: o, write it ^ aax ~~x^ :=. a
— Xy and it will be, by cubing, aax — *^ = «' — ^a^x + ^ax^ — .v'. That is
^a'^x — ;^ax'^ — a^ — o, or by dividing by a, ^ax -J» 3.V' — «* = o.
But if the radical terms be two or more, {o that they will not vanilh at one
operation, it muft be repeated as often as there is occafion. Thus V bx -=. a +
^ ax : write it thus, \/hx — y/ax zz a; then fquaring, it is bx — is^ ahxx 4-
ax n aUy that is hx + ax — /?^ n 2 x/ abxx. And fquaring again, hhxx 4- aaxx
4- ^* + ^abxx — laabx — %a^x zz ^abxxi that is i-V — labx'^ -h a^x'^ — la^bx
— ia}x + ^* iz o. Thus y zz \/ay + jyj — a\^ ay ^ yy by fquaring will be
yy zz ay -V yy '— as/ ay —yyt that is ay zz as/ ay —yyj ov y zz V ay — yy- And
fquaring again, yy zz ay — yy, or iy zz a.
71. Thefe things being premifed, the manner of refolving equations will beHowequa-
eafy, in order to obtain the value of the unknown quantity, in fuch terms as tions are m
are known and given, and which ferve to the folution of the problem, gyj^-l^e refolved.
firft the equations are fuppofed to be freed from ail afymmetry, that is from, ra-
dicals, if the unknown quantity be under a vinculum; and then reduced to the
moft fimple expreffion ; by expunging fuperfluous terms, if fuch there be ; by
dividing of each member that (hall be multiplied by the fame quantity; or by
U'l* Trj-'jj A-rr i ii <i^x — aax •\- aah a^-\-aab
iplying It lo divided. As it, tor example, we had , ~ — 7 — ^
it would be reduced to xx —^ ax zz aa. Further, by the firft term of an equa-
tion is meant the aggregate of all thofe terms, which contain the higheft powet
of the unknown quantity. By the fecond term is meant the aggregate of all
thofe terms which contain the next inferior degree of the fame quantity, and (b
on. By the known term is meant, the aggregate of all thofe terms which
do not at all contain the unknown quantity. Whence in the equation axx — bxx
— bbx — aax zz a^ —' b^, or elfe axx — bxx — bbx — aax — a' + ^^ zz o, the
firft term will be axx — bxx^ or a — b X xx. The fecond will be — bbx — aax,
that is — aa + bb X X. The known term is — «' -f- P. In the equation.
aaxx -— abxx + a* -^ b"^ — a^b zz o, the firft term will be aa — ab X .va; ; tli^
fecond is wanting, and the known term is «* — h* — a^b. In the equation
ax^ + bi<} — aaxx — a* zz o, the firft term will be a ^ b X x^, the fecond
— ^iV, the third is wanting, and the fourth or known term is — a^. And
thus it is to be underftood in all other equations. Here it may be observed,
that a term fuch as aaxx — bbxxy (which is likewife to be underftood of any
other compound term, having contrary figns,) may be either a pofidve or nega-
tive quantity ; it will be pofitive if a be greater than ^, but negative it the
contrary. So that when it (liall be ordered hereafter to make luch a term of an
equation pofitive, we muft have regard to this explanation.
H 72. This
50
ANALYTICAL INSTITUTIONS.
BOOK !►
Equations
further re-
folved, and
fir ft fimple
ones.
72. This being fuppofed, in order to refolve an equation; firft, if it have a
fradion, in the denominator of which the unknown quantity is found, it muft
be reduced to a comni(Jh denominator. Secondly, the term of the higheft
power of the unknown quantity muft be made pofitive, and all the terms con-
taining the unknown quantity muft be wrote in order on one fide of the equa-
tion, and the known terms on the other fide. And thirdly, if the firft term^
or that which contains the higheft power of the unknown quantity, Qiould have
a denominator, it muft be freed from it's fradion by what is faid, § 68. Laftly,
if it have a co-efficient, or be multiplied into any given quantity, it muft be
freed from this, by what has been taught, § 69.
Hence it is eafy to perceive, that by proceeding after this manner, if the
•equation (hall be fimple, or have an unknown quantity of one dimenfion only,.
it will be now intirely refolved, and that unknown quantity will be found equal
to known quantities only, which was the thing propofed to be done. As if the
equation were aa
ff = -^ — - , and aa were greater than bb. Then tp
2m
make that term poiitive which contains the unknown quantity, write it thus,
*" znff-^ aa\ and freeing it from the denominator,, it will be aax — bhx
= imff — imaa-y and then from the co-efEcient, it will be ;v = 2w// -• rmaa
•'"' OA — bb.
in which the value of x is now intirely known. If aa were lefs than bb^ we
might then write it thus, x = "'ff " — ^, which comes to the fame without
° bb -~ aa
any occafion of tranfpofition.
powers.
S;quation5 73- When the unknown quantity is raifed to any power, which power is the
refolved, fame in all the terms in which it is found ; or, which is the fame thing, if all
!!^w"?c ^''"P thofe terms are conceived to make but one term ; then the equation is to be
refolved by the third axiom before, and we ftiall have the unknown quantity
equal to known quantities only, by extracting fuch a root out of both members
of the equation, as is denoted by the index of that power. Let the equation
be bb zz aa — ^'"^ . Now to make the term pofitive in which x is
2(7
found, write ^^^ J -zl aa^-^ bb ; and to free it from it's fradtion and co-
ze x ^m — i^
2c
efficient, write it xx zz
a -^ b'
or by divifion, xx = 2c X a - ^ 5 and
Mly, by extrading the fquare.root, ;^ = ± s/^ac - ibc Here I put the fign
of the root ambiguous, becaufe of what is faid at § 15. For the fame reafon.
if it were^3 — ^3 ^ p^ ^^ (hould have x zz Va' + iM and fo of all others
74,, But
in general.
SECT. ri. ANALYTICAL INSTITUTIONS. 5I
74. But if the equation contain the unknown quantity raifed to it's fquare, Affefted
together with the redtangle or produd of the fame into known quantities, which quadraticks
is called the fecond term (and fuch an equation is called an Affsoled ^adratick^^'^^'^^'^^^'
as it is called a Simple ^adratick when this fecond term is wanting) ; this being
prepared as is aforefai3, to both members of the equation mud be added x\\q
fquare of half the co-efficient of the fecond term, (that is to fay, the fquare of
half that quantity, whether integer or fradion, by which the unknown quantity
is multiplied,) and then it is plain that the firft member will always be a fquare,
the root of which will be the aggregate of the unknown quantity, and of the
half co-efficient with it's proper fign. And then extracting the root, this ag-
gregate (hall be equal to the fquare-root of the other member of the equation ;
and tranfpofing the half co-efficient as a known quantity, we fliall finally have
the unknown quantity equal to the fum or difference (according to the nature
of the figns) of the radical and the faid half co-efficient. Thus let the equa-
tion be XX -j- lax — bb \ if we add to each member the fquare of halt" the
co-efficient of the fecond term, that is aa^ the equation will be xx -[- ^ax -4- aa
,=z aa + bb, and extracting the fquare-root, it will be .v + d: = d: \/ aa + W,
and by tranfpofing, it is a; =: ± ^/'aa + ib — a.
Let the equation be bbx — aax — mxx -\ — zio. Making the greateft term
pofitive, and ordering the equation, it will bcmxx + aax — bbx zz ^^, and dividing
by w, and adding on both fides the fquare of half the co-efficient of the fecond
.,, , , aa '— hb , a* — 2aal>b -\- 1>* a* — 2aabb + b* , fl*3* .
term, it will be xx + x H iz --; and ex-
o- ,v r . • • , aa - hb . /a* - 2a*b* + b* ] '^
tradtins; the fquare-root, it is a: + =■ + / rT-^ — H r- >
and reducing the radical to a common denominator, and tranfpofing the known
aa — bb • .,, , bb — aa jO* -f 2^*3* + i* t> ^ 1 c
term , it will be ;? = h s/ — r—^ — . But the root of
this radical may be adually extradted, and is either + '^■^ , or — "-^^ — ,
becaufe of the ambiguous fign +, Therefore there will be two values of x,
bb — aa , aa + bb bb , , , . bb ^ aa aa + bb
one IS X zz. z: — , and the other is at :::
aa
'•— ^^ ' ♦
m
75. Therefore the ambiguity of the fign, which the extraction of the fquare- The ufeof
root always brings with it, fupplies two values of the unknown quantity, which the ambi-
may be both pofitive, or both negative, or one pofitive and the other negative jS"^"^ fign.
and fometimes both imaginary, according to the known quantities of which they
are compofe d. Fo r example, in the final equation x — ■±, V aa ^ bb — a^ one
value or >/ aa + bb ^ a will be pofitive, becaufe, as \/aa ■\- U is greater than a,
H 2 the
52 /ANALYTICAL INSTITUTIOITS. BOOK I.
the difference will be pofitive. The other value — s/ aa -f hh — a will be
negative, as is evident. In the equation x nz a ■±, \/ aa ~ bb, (fuppofing b to
be lefs than a,) both the values will be pofitive, becaufe \/aa — bb is lefs than a.
And for the fame reafon, in the equation x zz. ±_ y/ aa ~ bb — a, both the
roots will be negative. Now, if h were greater than a, both would be ima-
ginary, as I have already obferved at § 15, becaufe then s/ aa — bb would be
the fquare-root of a negative quantity. In the equation ■x'* =: «* — b^^ which
requires twice the extraftion of the fquare-root, that is, ;vx n ± \/ a^ _ b*
and thence x zz ^^ s/ ■±, V'a* _ ^4, there are four values oi x\ two real ones,
of which one is pofitive and the other negative, that is, ^ = ± v/'^+^'v^l+Zp,
fuppofing b to be lefs than a ; the other two are imaginary, that is, x =:
± \/ — V a* — b* I and when b is greater than ^, all the four roots will be
imaginary : and thefe obfervations may eafily be applied to all other equations*
Thefe negative values or roots, which by fome authors are called falfe ones, are
not lefs real than the pofitive, and have only this difference, that if, in the fo-
lution of a problem, the pofitive be taken from a fixed point, or beginning of
the unknown quantity towards one part, the negative are taken from the fame
point towards the contrary part. Let A be the
pi„ iQ. beginning of the unknown quantity a: in a certain
^__^ problem, and let the final equation (for example)
C A B be a; = ± <«. If we take AB zz a, and it be
determined that the poftive values (hall proceed
from A towards B ; then (hall AB zz ^? be the pofitive value of x. And con-
fequently, taking AC z= AB, but on the contrary part from the point A, we
fhall have AC n — ^, or the negative value of x. And the problem (hall
have two folutions, one at the point B, and the other at the point C. But the
practice of all this will be belt undcrftood by the folution of the problems
which are here to follow*
Ufe of ima- 76. Therefore, whenever the equation to which we are led by the conditions
ginary quan- of the problem (hall fupply us with none but imaginary values, this plainly
^^^'^^* declares, that the problem as now propofed does not admit of a real folution,^
but is abfolutely impofiible. The fame thing is to be concluded, when the
final equation brings us to an abfurdity, fuch as if it (hould give us a finite
quantity equal to nothing, or the whole equal to the part, or fuch like. We
(hould come to an abfurdity of this kind, if in the
^^^' ^' right line AB zz a, it were propofed to find fuch
jj[ "" C B a point C, as that the fquare of the whole line
(hould be equal to the two fquares of the two feg-
ments. For, making AC z: Xy it would h& aa zz xx + a-^--x\'^ zz xx + aa
— - lax -^r XX, that is axx = 2ax, or x zz a ; which is as much as to fay, that
the part is equal to the whole. We (hould hkcwife fall into an inconfiftency, if,.
affuming
ANALYTICAL INSTITUTIONS.
53
B
H afluming a right line, as AB, and raifing an
Q indefinite perpendicular upon it BH, we
lliould fcek for a point in this, as C, from
whence we might draw the right line CA to
D the given point A, fo as that the two lines
CB, CA, may be parallel. For, making
BA = ^, BC =1 X, and taking BD zz -^x^
and drawing DM parallel to BA; becaufe
of fimilar triangles CBA, CDM, it would be
DM = -^a. But if CA and CB are parallel, it ought to be DM = BA,
and therefore —a zz a, which is an impoflible equation.
Now if it fhould be pretended, that the firft of the two foregoing equations,
or 2XX — 2ax = o, is no otherwife abfurd, but that it fupplies us with two
values of x, which, though ufelefs, are however real and confiftent ; relying
upon this argument, that if we divide the equation by 2,v — 2<«, there will refuk
a; 1= o, a real value which folves the problem. For taking x zz o, or dividing
the line AB in the point A, one part of it will be o, and the other will be a.
Therefore the fquare of the whole line will be equal to the fquares of the two
fegments ; that is, aa zz o ■■{- aa. Now dividing the fame equation by 2.v,
there will refult x zz a, which is a real value, and refolves the problem, by
dividing the line in the point B. Whoever Qiould argue thus, as I faid before,
I fhould not venture to oppofe him ; but whatever is the true notion of this
and fuch like equations, it is however certain, that they only make us know
what we knew before.
For an example of an equation which brings us to an abfurd conclufion, 1
have taken one which gives us a finite quantity equal to nothing, or the whole
equal to the part. Yet this muft be underftood only when the unknown quan-
tity cannot be of an infinite magnitude, and the problem is no more than a
determinate problem y for otherwife fuch equations may be very true, as will be
feen hereafter.
77. Sometimes we may meet with equations which contain the (ame quantities \vhat we
on both fides the mark of equality, and therefore when reduced bring us finally leam from
to this conclufion, that = 0. Such equations as thcfe (which are called »(ientical
Identical Equativns) inform us only, that the value of the quantity required *^"* ^°"'*
may be what we pleafe, as it vanifhes out of the equation ; and that the propo-
fition is rather a theorem than a problem. Here follows an example of this.
In
8
54
ANALYTICAL INSTITUTIONS,
BOOK I,
In the given reflangle ACDI*^, from a given point
B in the fide AC is drawn BF parallel to the fide
AE ; in BF is required fuch a point H, that
drawing the lines HA, HC, HD, FIE, to the fe-
veral oppofite angles, the ium of the fqoares of
HA, HD, (hall be equal to the fum of the fquares
of HE, HC. Make AB - a, BC =i i^, CO zi: a
and fuppoling H to be the point required, let
BH :=; Xj and therefore HF zz c — x. Now the
fquare of HA zz aa + xx, that of HC — ^^ + xx,
that of HD n: ^(^ 4- rt — 2i:x + xxy and that of HE z: aa -j- cc — 2cx + xx.
And hence the equation aa + xx -{- bb + cc — ^cx + xx zz bb '\- xx + aa
-{- cc — 2CX + XX. Now as it is an identical equation, the fame as o = o,
which is as much as to fay, that in the right line BF, wherever we take the
point H it will always agree to the property required.
Equations 78. Equations which reduced contain the unknown quantity of one dimenfion
and problems only, are called Simple Equations^ or of the frji degree. Thofe vv^hich contain
how divided. ^^^^ unknown quantity raifed to the fquare, whether they are quadraticks fimple
or affefted, are faid to be of the Jecond degree. Thofe which contain the un-
known quantity raifed to the cube, however the other terms may be, are faid
to be of the third degree. And fo accordingly are thofe of the fourth, fifth,
and higher degrees. Moreover, thofe problems which are exprefled by fimple
equations, as alfo thofe of the fecond degree, are called Plane Problems, be-
caufe they may be conftrudled by the common Geometry of Euclid, or by rules
and compaflcs only. All the others are called Solid Problems, becaufe for their
conftrudlion is required the defcription of certain curves, which therefore are
called Solid Places. I fliall fay nothing here of the Refolution and Conftrudion
of Solid Problems, intending to treat of them exprefsly in Sed. IV.
Equations 79* There are many equations, which at firft fight feem to be of that degree
may fome- which IS intimated by the index of the greateft power of the unknown quantity,
times be de- -^yhjch, however, when duly managed maybe brought down to an inferiour
Wr^degvee. degree. Of this kind are all thofe in which, befides the firft term, which is
' that of the higheft power of the unknown quantity, and the term which is
entirely known, one other term is contained, in which the unknown quantity
afcends to a power which is the fquare-root of the power of the firft term. As
if the equation were this, x* — 2aaxx zz b* ; which being m anaged by the Rule
of Affefted Quadraticks, is reduced to this, xx zz aa ^ \/a* + b^t and there-
fore X = ± \/aa ± >/a* + ^4, After the fame manner, this equation
— fl' =t ^/a^ +4b^ ,
x^ + izV — ^* = o, being reduced, becomes x^ = ^ — , and
therefore
SE€Tr II^ ANALYTICAL INSTITUTIONS. ^^
therefore x =A/^ — i ^^'^ infinite others of a like nature. There
are others of the fame kind, which by means of the extradion of a root may
be brought down to an inferiour degree. Thus x* — 2ax^ + aaxx — ibh-x
-I- 2alfl/x -I- ^* iz aabl> + ^*, having it*s firft member a fquare, the root of
which is XX — ax — ^Z", may be reduced to a lower equation, xx — ax — bb
3: ± bs/ aa + hb' Thus, in the equation x^ + 'i^axx + 'i^a.tix = ^', if we add
fl' on both fides, it will be x^ + zaxx + 'i^aax 4- ^' — «' + b^y of which the firft
member is a cube, whofe root \% x -\- a. Therefore the equation reduced lower
will be X '\- a zz ^ a^ + ^5. But it is not always thus eafy, to know what
quantity may be added or fubtraded to or from the firft member of the equa-
.tion, fo that it may become a perfeft power, nor can any method' be affigned
for it ; fo that the induftry and pradice of the analyft can only be of fervice ia
thefe cafes.
80, But, if the propofed problem fhould be of fuch a nature, that one un- Problems will
known quantity being aflumed, would hardly or not at all be fufficient to have °^^^" require
all the denominations that are neceffary for finding the equation; in this ^^fe know" quan-
maybe taken one, two, three, or as many more unknown quantities as are titles than
needful. And if the problem be determinate in it's own nature, it will always one.
fupply conditions for as many equations as are the unknown quantities aflumed.
Then, by means of each, of thefe equations, one of the unknown quantities
will be eliminated, or it's value may be found by the remaining and the given
quantities ; fo that finally we Ihall arrive at the laft equation, which will contain
one unknown quantity only. The manner of performing thefe operations will
be beft underftood by the examples.
Firft, let there be two fimple equations, or of the firft degree ; as, fuppofe
for example a •\' x "=: b -\- y, and 2:v + jy iz 3*^ ; and let us eliminate j, and
retain x,. Now, by means of which we pleafe of the two equations, fuppofe of
the firft,. by the help of proper tranfpofitions of the terms, we may find the
value of jy, which will ht y zn a -\- x — b. This value may be fubftituted
jnftead oi y in the fecond, and we ftiall have a new equation ix ■\- a -\-x — ^ — i^b^
that is a; zz ^- . And this value being; fubftituted inftead of Jf in either of
the two propofed equations, we fhall have the value of j' = -^ . This may
alfo be obtained by deriving two values of ^ from the two equations, and com-
paring them together. For from the firft equation we fliall have y iz a -\- x
— h; and from the fecond, _y =: 3^ — 2^:5 wherefore it will be, by comparifqn,
tf 4- a; — - ^ = 3^ *— 2;?, and thence x =z i - "" "^ , as before. .
%ii After
56 ANALYTICAL INSTITUTIONS. BOOK I.
How they are 8 1. After tbe fame manner we miifl: proceed, when the equations contain the
to be ehmi- unknown quantity, which is to be eliminated, raifed to the fecond dimenfion ;
"^ ^ * if by means of one of the two given equations, or by the tranfpofition of the
terms alone, or by the rule for fimple or affe6led quadraticks, we can have a
value to be fubftituted in the other equation. Let the two equations be
XX + 5^^ ^^ Syyy ^"^ ^^y ~~ 3^^ ■— • 4'^^« Now if we would eliminate v, the
r J ^' '11 • 4^^ + ^-^^ 1 1 r l6a* + ZAaaxx-\-qx*
lecond equation will give ^ = -^^ — , and therefore yy zz 2_ .
This value being fubftituted in the firft equation, it will be xx + ^ax rr:
-^^ • — 5 which, by reduction, will be 23;^* — loax^ + -jiaaxx
-f- 48^'* = o. But if we would eliminate x, finding it's value by either of the
two equations, for example by the fecond, we fhould have x — ^ ■±^^^''^'^"'^
3 3
This being fubftituted in the firft equation, it will become
%yy ~i2aa'=t:zy Vyy ~-izaa
9
_j- w — S^ ^yy - i2aa __ ^^^^ r|-|^jg being freed from radicals, and fet in order,
3
after a long calculation will come out 6^y^ — ^oay^ + jiaayy + /s^oa^y
-f- 316^* — o.
Qiiantitles to 82. Often by two equations, in which the unknown quantity to be eliminated
l3e eliminated is raifed in both to the fame degree, may be found by means of either of them
by compa- jj^^ value of the highcft power of the unknown quantity; and that is by putting
that higheft power alone on one fide of the equation, and all the other terms
on the other fide : then thefe two values being compared to each other, will
give an equation of a lower degree. The fame operation may be repeated
again, and fo on, till we have an equation truly fimple in refped of the un-
known quantities, and confequently it's value expreflTed by the other unknown
quantity, and by fuch as are known. Then this value being fubftituted in
one of the given equations inftead of the unknown quantity and it's powers,
we ftiall have an equation exprefled by the other unknown quantity only, and
fuch as are known.
Let the two equations be y^ -^ aay zz fa, and y^ — Ixx = aaxy out of
which y is to be eliminated. Therefore by the firft it will be y^ = bxx — aay^
and by the fecond, y^ zz aax + bxx. Then by comparifon, bxx — aay =
aax -f bxXi or y =. — x. Then making a due fubftitution in either of the
two equations, we (hall have — x^ — aax = bxx, or x'^ -{- bx = ^ aa. Again,
let the two equations be xx + ^ax = ^yy, and ^xy — o^^x = ji^aa, from which
we are to eliminate x» It will be by the firft xx zz ^yy — 5^^:, and by the
fecond, !<x zz. ^^ ~ 4^^ ^ Therefore the equation will be ^yy — s^x == ^"y^^ ,
3 3
From
SECT. ir. ANALYTICAL INSTITUTIONS. 57
From hence we fhall have x =: ^ — ^fl.; and this value beln^ Tubftituted in
one of the propofed equations, in the firft for inftance, it will be as is found
above.
But if in the two equations the unknown quantity to be eliminated do not
afcend to the fame power in the highell terms, the equation of the lower degree
is to be multiplied by fuch a power of the fame quantity, that it may be of the
fame degree as the other; and then you are to proceed as before. Thus, if we
have y^ zz xyy + 3<7«^, and yy zz xx — xy —• "^aa^ and we are to expunge y ;
multiply the fecond equation by v, and it will be y^ — xxy — xyy — Z^ay,
Therefore xyy + 3^^.v = xxy — xyy — 3<2^/, which, being compared with the
value of yy given by the fecond propofed equation yy :=. xx — xy — 3^2^, will
give —^ ^—- ' — ~ ATA? — xy — 3^i^, or ^xxy — ^^^ay + j^^a: = 2A'%
and therefore y zz — 3^ ^ which being; fubHituted in one of the propofed
AX ■ I Si^dA"^ ^I, O^^A'A*
equations, fuppofe in the fecond, will be - — 5- . . ' — xx '— ^aa
^ ' ir ' gx* — iSaaxx +ga* *'
1 — 2^ — 3^^^-^ Qj. reducing to the fame denominator, x^ + iZa'x* — ac^zV
4- lyaf^ zz o.
In particular cafes particular expedients may often be ufed, and there may be
more expedite methods of coming to a conclufion; but thefe do not fall under any
rule. An example may be feen of this in thefe two equations, x + y -{
= 20^, and XX -i- yy ■+ -^ zz 140^^. If we would eliminate x we muft
tranfpofe y in the firft equation, which will then be a; + -^ rz 20^ — y ; and
fquaring both parts, it will be xx -{• 2yy -\' -^ — 400^^ — 40^^ + yy, that is
XX -h yy + ■—■ = 400^;^ — 40^^. But the firft member of this equation is
the fame as that of the fecond propofed equation, and therefore it will be
400^^ — 40^7 = 140^^, or ^ =: -^ .
83. By a calculation more laborious and long, but performed after the fame When there
manner, if there be three, four, or more equations, and as many unknown are feyeral
quantities, we may reduce them to one only. For by means of one equation ^^'^^'^°"^'
we may exterminate one unknown quantity, the value of which, expreffed by
the others and known quantities, may be fubftituted in every one of the remain-
jing equations. Then by means of another equation we may eliminate another
I unknown
58
ANALYTICAI' INSTITUTIONS.
BOOK I.
unknown quantity, and it's value may be fiibftituted in thofe that remain ; and
fo on to the end. Let there be three equations x-{-y=:c + ZyZ-\-xz:a -{-yf
z + y :=: b + X, and we would have only one equation including z. From
the firft equation take the value of jy, that is >' = ^ -f z — x, and fubilitute
this value in the other two, which are then z^xziLa-\-c-\-z — a:, and
z-\-c + z-'Xz=ih-\-x, OT rather 2x zz a -^ c, and zz zz b — c + 2a',
which will then be in the place of the fecond and third. In this laft, inftead
of zx fubftitute it's value from the other, and then it will be i% zz. b — c
-{• a '\- c, that IS z zz . Alfo, the fame may be done after another
manner, thus. From each of the three equations given take the value of 7, for
example, that \s y z^ c + z — x, y ■=. z -\- x — a, y zz: b ■{■ x — 2. By
the comparifon of two and two of thefe equations, which you pleafe, you will
form two equations which have no y. From one of which equations you may
take the value of one of the unknown quantities, and fubftitute it in the other.
Thus, if you make the two equations c -\- z — x =. z -{• x -^ a, and c + z
— X zz b -{• X — Zy from the firft; take the value of x^ or x zz ^ , and
fubftitute it in the fecond ; then c -\- z
a + h
a + c
— z i that is.
z = , as above. In the fame manner we muft proceed if the given
equations be more in number, and more compounded. The ufe of the rules
here taught will be feen in the folution of the problems.
Sometimes
the number
of equations
may be in-
iuficient.
84. Whenever the conditions, or the data of the problem, do not fupply us
with as many equations as are the unknown quantities aiTumed, but that two of
them will at laft remain ; the problem will always be indeterminate, and we
cannot find the value of one of the unknown quantities but on fuppofing and
determining the value of the other ; in which cafe every indeterminate problem
becomes determinate. To give fome idea of thefe indeterminate problems,
though by way of anticipation ; let it be propofed to feek two numbers, the
fum of which is equal to 30. I call the firft number x, the fecond will be
^o — X by the condition of the problem, nor (hall I then have any means of
forming another equation. Then I will call the fecond y^ and by the condition
of the problem it will be x + y = ^o. Now becaufe it is not poflible to find
matter for another equation, by which to eliminate one of the two unknown
numbers, the problem of it's own nature will be indeterminate. But if I affign
a determinate value to one of the unknown' quantities, and fuppofe, for ex-
ample, that J = 8, then it will be ;v = 30 — y zi 22. But becaufe we may
aflign infinite values to y fucceffively, the values of x will alfo be infinite, and
confequently the problem is capable of an infinite number of folutions. I will
take ajQOther example of this from Geometry. Let it be propofed to find a
redangle
S£CT. II. ANAtYTlCAL INSTITUTIONS. J^
redangle equal to a given fquare. I^t y be the bafe of the rectangle required,
ii*s height a*, and aa the given fquare. Then I (hall have the equation aazz.xy%
and not liaving matter for another equation, the problem remains indeterminate;
there being' in fadt infinite redtangles equal to the given fquare, the bafe may be
varied infinitely, and the height alfo relatively to it. But if I add this condition
to it, that the bafe, for example, (hall be equal to half the height, or — Xy
then it will be j n: — Xj and the equation will be — xx n: aa. And thus
one of the unknown quantities may be varied an infinite variety of ways, and
likewife the other, fo that the problem may have an infinite number of fo-
lutions.
85. On the contrary, if the conditions of the problem, which are to beMoreequa-
fulfilled, (hall fupply us with more equations than there are unknown quan- 'ions may be
tities, the problem will be more than determinate, and by that means n^^y ^^|^"J^^^" .
become impoflible. For, in order to be poffible, the values of the given quan-the problem
titles mufi; be reftrained to a given law, which will often afford innumerable become im-
cafes in which the problem will become pofiible. In the foregoing example, P°^^^^^*
of finding two numbers the fum of which (liall be ap, when nothing more is
required, it will be an indeterminate problem ; but ]t the condition be added,
that bcfides the difference of the fquares of thofe numbers fl)all be given,
fuppofe for example 60, the problem will then be determinate, we having in
this cafe two equations, that is, ^ -|- j' := 30, and xx — yy = 60; fo that,
taking from the firfl: the value of j, and fubftituting it's fquare in the fecond,
it will be ^ = ^, or A? = 16, and confequenlly v = 14. But befides, if we ,
(hould annex a third condition, that the fum of the fquares of thefe numbers
ftiould be equal to a given number, the problem is more than determinate, and
therefore poffible in one cafe only, in which the number given for the fum, of
the fquares is jufl: the fame as thofe fquares, that is 452. Thus, in the other
example of a rectangle equal to a given fquare, if we require that the redangle
ihould be upon a given bafe, the problem will b.e determinate ; but more than
determinate if we (hould alfo require, that it's fides (hould have a given ratio
to each other. It will be poffible only in one cafe, wherein this ratio is exactly
the fame as refults from the other condition of the given bafe, and from the
equality to the given fquare.
86. The equations being refolved, and the values of the unknown quantities How fimple
being found in geometrical problems, it remains to give the conftrudions of ^1"3''0"3
thefe values ; that is, from the given lines of the problem we muft find ^ijch, a^^J^^'^^"'
that may exactly reprefent the unknown quantities, which were propofed to be metrically!^'
found. In the firfl: place, let the value of the unknown quantity be a fimple
I 2 rational
6o
ANAYLTICAt INSTXTUTIONg.
BOOK I,
rational fradlon, fuch as a; =: ~ . If we convert this into an analogy, it will
— . Therefore,
X ', fo that the fourth proportional required is
I
upon the indefinite right line AC taking A B
r=^, and at any angle drawing BD = b, and
through the points A, D, drawing the inde-
finite line AE ; if we make AG zz a, and
draw CE parallel to BD, it will be CE z::^
ah
— = X, Or elfe in any angle EAC draw-
ing the indefinite right lines AE, AC, if we
take AB = c, AD iz b, AC = ^, and
from the point B to the point D draw the
right line BD \ from the point C draw CE
parallel to BD, it will be AE = — . Therefore by thefe or other theorems or
problems of Geometry may be found a fourth proportional to the three given
quantities, or a third if only two be given j and we Ihall have the value of the
unknown quantity exprefled by lines. If it be ^ = -^ , the firft analogy is
had by taking any one of the letters of the denominator, and two of the nu-
merator j for example, m . b :: a , ~ , which is therefore the fourth. Then
let this be found as before, and call it /: therefore it will ht x "=. ~. The
fecond analogy then will be thus, n .f :', c , x =: — -, which will be the fourth
= • ■ . Taking therefore (Fig. 13.) AB = w, AC = ^, BD = b, it will
be CE z= — - =/j whence producing CE indefinitely, take CH = «, CKrzr,
and draw HE; if from the point K the right line KI be drawn parallel to HE,
it will be CH . CE :: CK . CI ; that is, n , — :: c. — = Cl ^ x.
If the dimenfions in the numerator and denominator (hall be more in
number, the analogies muft alfo be more, but always in the fame order.
Ori'fthey 87. Whence if the value of the unknown quantity fiiall be compounded of
confiftoffe- feveral fimple fradtions, or of integers and fractions; find the lines which are
veral terms, equal to each term, and adding or fubtradling them according to their figns.^
they will give the line which exprefles the value of the unknown quantity.
88. From
SECT. II.
ANALVTICAli I NSTI Tlf TIONS. 6l
88. From this rule we may derive a method of transforming: any plane into How the
another with a given fide ; a folid into another with one or two given fides, &c.; ^^''"1°^^"
that is, any term of two, three, or more dimenfions, into another which tliall ^^"^j.^'°"f^^^
include any given letter, if it be of two dimenfions; or one or two given letters, ed at pkafure,.
if it be of three dimenfions. Thus let the term be bb which we defire to tranf-and fo fitted
form into another, which fliall include the letter a. By this letter a let bb be
hh
divided, and it will be — . By the given rule (Fig. 13.) a line may be
found equal to — , which call m. Then is — =: m, and therefore bb z: am,
'■a a
Let ffc be fo transformed as that it may include ab, A line may be found equal
for conllruc-
tion.
J^ xirhinVi roll M. TKpn If wrill Kp r-' . ,^ ^^ ..^,,.
to ^ , which call ». Then it will be =^ n: ;;, or f/c = abn. If it had
been required that it fliould only include ^, we fhould have made
and dierefore — 1:1 fn, or ffc =: afn. This is manifeft, and needs no other
examples.
89. This being fuppofed, let the value of the unknown quantity be a com- How com-
plicate fiadlion, or more than one, that is, let the denominator have feverai P^'^^\« '*^''"^f.
* ' may be traai-
terms ; as x •=. ^ , One of the terms,, fuppofe rr, is to be transformed ^o^"^^^-
into another, which Qiall include the letter b, and let it be bm^ Then we fhall
have 77-xT"> which is refolved into thefe two analogies, b , a :: a , — r- , the
fourth, and b 4- m . -r- \\ a . ,. . , , the other fourth. And making as ufual
the conftrudiion by the help of fimilar triangles, we (hall have the line which
is the value of the unknown quantity x. We might as well have left the term.
cc in the denominator, and have transformed bb into another, which fliould have
included the letter c, for example en ; then the fradion would have been
aa y . aa
, which is refolved into thefe analogies, c , a w a . — , and f -f- « •
cc •'r en " c c
l\ a , —^ — . Let the fradion given ht x zz — r ; in the denominator the.
cc Jf en ° tfS + ^3 '
term b^ may be transformed into^^«, and the quantity to be conftruifled will be.
This may be refolved. into three analogies, a » b w b . — , and
a3 + a* ft
a , b :: — . — -, and a + n . c y, — — . — — — . If the denominator fhould.
baye three terms, then perhaps two of them muft be transformed ; if it fliould'
have:
52 ANALYTICAL INSTITUTIONS; SOOK I.
have four, tlie^i three are to be transformed, &c. Thus, if there were given
be X
3 - 4- ^j — > ^^'^^^ having made P zz aan^ and hcc =: aap^ then it would
a^ + «*K — a^
This, in the fame manner, is refolved into three ana-
log;ies, a , b \\ b , — . a ,b v. — . -^, and a -\- n -^ p , c : : -r . -; — r
fjJ -}- ^3 — ^fC- *
It can make no difficulty if the numerator of the fradlion (hould be com-
phcate, or have feveral terms ; becaufe the fradion will be equivalent to fo
many fradions as are the terms of the numerator. Thus ^^ ~ is the fame as
flJ — C3
hh
, . _!. -^ — ; . Therefore each being; refolved in the manner here explained,
the fum or difference of the lines fo found, according as their figns may re-
quire, will give the line which is the value of the unknown quantity required.
Other frac-
tions con-
90. But without multiplying operations, by reducing a fradion with a com-
plicate numerator to feveral fradions, it will be enough to make ufe of a con-
venient transformation of the terms of the numerator and denominator, after
the fame manner as has already been feen for the denominator. Thus let it be
.V = ^^ , / ; transform the term be into am, and the fradion will be ^"^ '^'" ;
whence it is a + b , a + m :: a . -7- . Let it be — - . ,J ; make b/zz
a + o acf -f bff ^
cm, and the fradion will be — 7- -j- . that is -— r- — -r- ; then f , a :: c »
acf + amf
ac ^ . ac acc — acm
—— , and c-\-m . c-^m :: -p- . — -: 7-
C/+ fft/
But if the numerator and denominator of the fradion be fuch, that without
transforming any term they may be refolved into their linear components; then
no ufe is to be made of transformation, which would only multiply operations
aab
, ^ ; and fuch others^
unneceffarily. Such will be the fradions
■' aa -^ cc ac -f- cc .
The firft of thefe may be refolved into thefe two analogies, a + c , a :: a . - — ,
r aah
and a — c . b : :
a -^ c ' aa —' cc
And the fecond into thefe two, c . a ::
. I aa + ai , . , aa -^ ah a^ — abb r.-., c
a + b * — : — , and a + c . a ^ i :: — :; — . rrrrrr' Thus very often.
ac + cc
without
SECT, ir.
ANALYTICAL INSTITUTIONS.
63
without transforming the terms, it will be more convenient to make ufe of the
Extradion of Roots, for refolving a fradion into analogies. Thus the fradlioti
f^ "^ '^ may be refolved into this analogy, a , \/aa + be :: ^ aa + be . — — ^;
a ^
thoueh more fimply thus, a -\ ^ . The fradion - — -^ is refolved into
\/ aa + cc » ^ • •
Vaa-^cc aa+cc
aa + hh
v aa + cc
. Yet fometimes it may be necefTary to
thefe two analogies, \/ aa ■\- cc • v aa ■\' bb '.'. \/ aa + bd , , , and
^ V aa-\-cc
aa •{■ bb <z' + abb
transform a term ; as in the fradion , which cannot be refolved even
aa — cc
by radicals, unlefs one of the terms of the numerator be transformed, fuppofe
bhc into acnti fo that it may be
tf' 4- '^'^ff*
, For then it may be ^ + ^ . ^2 ::
/ a\ a*- ■{- cm , y aVaa -^ cm a^ ■\- acm
\/ aa 4- cm . . ; and tf — f . \r aa + cm '.'. . =
a -^ c a + c aa — cc
. The fame obtains in fradions more compounded.
aa — cc *
Among the variety of ways here produced, it cannot eafily be determined
which will be beft in particular cafes ', perhaps more than one fhould be tried,,
that we may pitch upon that which will furnilh out the fimplcft conftrudion of
the propofed problem,
91. As to what concerns the finding fuch lines as are cxprefTed by radicals ; Radicals how
in the third place, let the value of the unknown quantity be an integer quadra- conftruftfed.
tick radical^ fuppofe x rr \/alf. That is, x is a mean proportional between
a and /p. Take AB zz a, and diredly to it
BC =^, and bifecling the compofed line AC-
in H, with radius HC defcribe the femicircle
ADC, and from the point B raife the perpen-
dicular BD terminated at the circumference.
The redangle of AB into BC will be equal to
the fquare of BD ', that is, al> := BD^, and:
therefore \/ai; zz BD 3: X' Let it be x zz
^zaa ; taking AB — la^ and BC =: a, it will
be BD = ^/zaa, &c.
And if the radical confided of complex quantities, as x = v^4a« ± ab, or
elfe X ■=. \/ ■i^aa ±ab ± 2ac i in the firft cafe, making AB = 4^ Hh ^ ; and in
the fecond, AB = 3^ ± ^ -|- 2f, and taking BC = ^ ; if a femicircle ADC be
dcfcnbed upon t!ie diameter AC, and a perpendicular BD be raifed, that per-
pendicular
Fig. 14
64
ANALYTICAL INSTITUTIONS,
BOOK r.
pendlcular in the firft cafe will be equal to s/ ^aa ± ab zz x, and in the fecond,
V 2"a ± fl^ ± 2ac — A?.
And, in general, let the terms under the vinculum be as many as you 'pleafe,
and combined with their figns in any manner, it's value may always be con-
flrufted by means of a femicircle, when every term is multiplied into the fame
letter; making, for example, one of the fegmenis CB equal to that letter, and
the other fegment BA equal to the fum or difference of all the terms divided by
that letter, and raifing the perpendicular BD. It is eafy to perceive, that if
the combination of the figns Qiould make the fegment B:V a negative quantity,
that then the quantity under the vinculum would be negative, and therefore that
the value of the unknown quantity would be imaginary. Such would be
X — V ab — ac) fuppofing c to be greater than h.
5iow radicals 92. Now ifevery term be not multiplied by the fame letter, they rliay become
transformed ^^^^ ^1 transforming thofe that are not lb. Thus, if a; iz V aa ±. bb, make
jn order to /»^ — anit and it will be ;«• — \^ \
ironftruclion.
aa ± am. Then taking AB = a ±: m, that is
AB z=i a ±z — > ^nd BC zz a, and defcribing the femicircle, it will be BD =:
\^ aa ± bb = X. In like manner, having given x z=z \^ aa + bb -^ ccy make
hb = am, cc -=. an^ and it will be \/ aa + am — an = x ; and taking AB =
a -\- m — ff, and BC = ^, it will be BD = \/ aa + bb — cc = x.
Quadraticks
conftruftcd
without
transforma-
tioo.
J^^*
93. But however the terms may be, without making any alteration, qua-
dratick radicals may always be conftrucled, either by a right-angled triangle
alone, or by that and a circle together. Let it be a: n: \/aa + bb, and take
AB rz a, and BC zr b perpendicular to AB,
it will be AC = \laa + bb = .V. If :v = ^laay
make AB = ^, and BC =: tf, and it will be
AC =r a/ zaa. \{ x ■=. \^ ^aa, make, as at
firft, AB zz BC = a, and upon the right line
AC raifing the perpendicular CD zz a, it will
be AD =zV^^aa. if at zi -/ 5^*31, make AB
= 2a, BC = a, then AC = V c^aa. \i x zz.
\/aa + bb +"77, make AB rr: ^, BC =: b and
perpendicular to AB, and upon AC raife the
perpendicular CD = c ; then the hypothenufe
AD will be zz ;f = V aa + bb + cc j and fo on
to quantities more compounded, li x ==. \^ aa -^ be, though the term be be not
transformed.
SECT, II.
ANALYTICAL INSTITUTIONS.
«5
Fig. 1 6
A C
AE =
Fig. 17
A
transformed, in the manner fliown above,
taking AB =: a, BC ir h, BD n: c, upon
the diameter CD defcribe the femicircle
CED, then the ordinate BE will be = ^bc\
and drawing the hypothenufe AE, it will be
=^ y = '^ aa -{■ he- Vi X "=. */ aa -^ be + ee^
upon AE draw the perpendicular EM zz ^,
and it will be AM zi at = \^ aa + ^c + ee*
_ Let X — \^aa + be + cc, taking BC zz b-^c,
-^ BD = c, it will be BE r= ^^JT+Tc, and
i^aa + he + cc. If there fhould be more terms, the operations might
increafe, but not the difficulty. Let x zi \/'aa'-bbi
on the diameter AB = <?, let the femicircle ACB
be defcribed, in which infcribe the chord AC
zz h ; then, becaufe of the right angle ACB, it
will be BC = \/ aa-bb» If ^ = V'aa-W 4-/^/7,
produce AC to M, fo that it may be CM — h ;
and drawing BM, it will be zz a: = ^aa-bb-\-hb*
If AT iz ^/ aa — bb — hby in the fcmicircle ACB
infcribe the chord AC =: V bb + hh ; then BC
~Lhb " hb. U X zz '^ aa " be, or a; =z \/aa -. be - ce i taking AB = ^
in the firft cafe, and = ^ + ^ in the fecond,
add direaiy AD =: <:, AH = a, if with the
diameters BD, AH, be defcribed the two
femicircles BCD, AEH j the ordinate AC in
the firft cafe will be zi v/^f, and in the fe-
cond zz V'^c + cey and therefore, taking AE
zz AC, and drawing the chord EH, it will be
y/aa - be in the firft cafe, and zz Vaa-bc^ce
in the fecond. If it were x zz \/aa — be — ee,
make AB = ^, AD z: f, and befides, taking
CF = e perpendicular to AC, it will be AF
= v'^c + te. Wherefore, making AI =s AF,
it will be IH zz ^ zz >/ aa ~ be — <*.
If ;tf = ^a* + b^i that h X =:. s/^a* + b^y transform the fecond term h*
into aamm^ and it will be x i: V^-Za* + «»^ ; and taking the fquare aa out of
K
the.
66
ANALYTICAL INSTITUTIONS.
BOOK I.
Fig* 19
the fecond radical, it will be xzi^ a^aa ■\-mtrf
Make AB z= «, and the normal BC = w,
it will be AC = V aa ■\- mm. Produce CA
to H, To that it may be AH =: AB =: ^.
Upon the diameter HC defcribe the femi-
circle HDC, and from the point A draw AD
perpendicular to the diameter; it will be AD
ZZ 'V a^aa + mm — y/a* + 7* ZZ X,
Cafes mdfc compounded are eafily reducible to thefe here fpecified. I (hall
add nothing about fradtions compounded with rational quantities or radicals,
becaufe they require nothing more than applying, or perhaps extending, the
rules already given.
AfFefted 94. As to the confl:ru6Hon of afFeded quadratick equations, which are the
quadraticks highefl I intend to treat of in this Sedion, I thought their refojution to be
n°V^j"". neceffary, and have given rules by which to obtain the values of the unknown
dependent 9f quantity, and fo to conftrud: them in the manner jufl now taught. Yet this
their folution. previous refohition is not abfoluiely neceffary, and without this they may be
conftrudled after the following manner.
All the infinite number of affeded quadratick equations may be compre-
hended and expreffed by this formula, xx ■±, ax ±i bb zz o, that is, by thefe
four, which arife from the four different combmations of their figns.
i. ii» + ax — hb -=1 o.
2. XX —^ ax -^ bb ZZ o.
3. XX -^ ax + bb ZZ o,
4. XX. — ax 4- bb =z o. '.
It is to be underftood, that the letter a reprefents the whole quantity
which forms the co-efficient of the fecond term ; and b is the fquare-root of
the aggregate of all the known terms. Now to contradl. the two firfl, take
CA zi -^a, AB at right angles to it, and equal
to b. With radius CA let a circle AED be de-
fcribed, and from the point B let the rig it line BD
be drawn, terminating in the periphery, at D, and
paffmg through the centre C Then will BL be the
pofitivc value of the unknown quantity, in the equa-
tion XX + ax-^bb ZZ o, and BDit's negative value:
as on the contrary, in the equation xx — ax—- bb zz o,
BD will be the pofitive value, and BE the negative
value. And in efFed, byrefoiving the two equations,
they
Fig, 20.
SECT, ir.
ANALYTICAL INSTITUTIONS,
67
they are a; = — f^ dt \^faa + bb, and a? = t^ ± S/^\aa + bi. And by the
conftrudtion, it being CA =: CD ri: CE — t^7, and AB = by it will be CB =
x^^aa + bbi and therefore BE = \/%aa + ^3 — i<7, which is the pofitive value
of the unknown quantity in the firft equation; and BD taken negatively,
= — t^ •— V\aa + bbi will be the negative value. Thus BD, taken pofi-
tively, n f ^ + ^-^aa + bbj is the pofitive value of the unknown quantity in
the fecond equation ; and becaufe of CB greater than CE, EB will be negative,
iz f (J — . i^iaa 4- bbi which is the negative value.
The third and fourth formulas are thus conftruded. Taking
CA zz. ia, and AB at right angles equal to ^, as in the fore-
going conRrudion ; and with radius CA defcribing a femicircle
ADH i draw BD parallel to AC. The two right lines BE,
BD, will be the two values, or the two negative roots of the
equation xx + ax "\~ ifl> =: o ; and the two pofitive values in
the equation xx — ax + bb zz o. Now refolving the equa«
tions, the third will give us x zz — ^a + V^aa — bb, and the
fourth X z=:-\a ± y^^aa — bb» Therefore, drawing the right lines
CD, CE, and CI perpendicular to BD, it will be ID =: IE =
V^aa — bb, and therefore BE negative = — t^ + \^^aa — bb,
the negative value of the unknown quantity in the third equa-
tion, becaufe BI is greater than IE. And BD taken negative will be zz -^ ^a
— X^^aa — bbi the Other negative value in the fame third equation. On the
contrary, BD will be pofitive, = -^-^ + ^^aa — bb, and BE pofitive, = f«
— S^^aa - bb, both being the pofitive values of the unknown quantity in the
fourth equation.
Therefore, to conftru<£t any affe(5Ved quadratick equation, it will fuffice to
affume the radius CA equal to half the co-efficient of the fecond term, and the
tangent AB equal to the fquare-root of the laft term ; and the reft as in one or
the other of tlie two figures, according as the laft term fliall be pofitive or
negative. Thus, for example, toconitrudl the equation xx + ax — bx ~- aa
•^ cc zz Of make AC = ^ " , and AB zz ^aa - cc in the firft of the tvyo
figures, if a be greater than c ; and AB = \/cc — aa , in the fecond, if a be
lefs than e. By this example it may be feen how we are to proceed in all other
cafes.
A cafe may happen, that, in the conftru6tion of Fig. 21, the right line BD
fliall not cut, but touch the circle ADH ; or that it may neither cut nor touch
it. It will touch it when it is AC = AB, that is, ^a zz. b, and the two values
of the unknown quantity of the equation, BE, BD, fhall be equal, one pofitive
K 2 and
63
ANALYTICAL INSTITUTIONS.
BOOK I.
and the other negative. It will neither touch it nor cut it when BA is greater than
AC, that is, l> greater than f <3 ; and the unknown quantities will not have any
value at all, but will be impofllble or imaginary. And this agrees perfectly
with the analytical refolLition, becaufe when it is ta — i?, it will be ^aa — lfif:=:Q,
and therefore the two values x z: — -fa + V\aa — bb, and x z=.^a + <J^aa — bb<t
will be ;^ = — f ^, and a zrr f^. And when \a is lefs than by then \/^aa - bb
will be an imaginary quantity, and therefore the two values of the unknown
quantity will be imaginary.
Orotherwife 95. In thefe conftrudions it is necelTary to find the fquare-root of the laft
thus con- term of the equation, which is to fupply us with the tangent AB of the circle.
If therefore this laft term is equal to a redlangle, or if we have a mind to make
it fo, which thing is in our own power, the four formulas aforegoing may be
thus conftruded, after another manner.
ftrudcd.
1. XX ■\- ax — be
2. XX •-~- ax — be
3. XX -{- ax -\- be
4. XX — ax + be
o.
o.
o.
o.
Fig. 22. Let the circle BAD be defcribed with any dia-
meter, provided it be not lefs than either a or b — c;
where I fuppofe b greater than f, or that b is the
greater fide of the reftangle g,iven, and c the lefler
fide. Now, from any point A in the periphery let
the two chords AB nr a and AD n i — ^ be in-
fcribed in the circle, and let this laft be produced
to F, fo as that DF z= c. With centre C of the
fitft circle, and with radius CF, let a fecond circle
FGH be defcribed, which may cut the chords AD,
AB produced, in the points F, E, G, H. This
being done, AG will be the pofitivc value or root,
and aH the negative, in the equation xx + ax
— be •=: o» And on the contrary, AG will be the negative root, and AH the
pofitive, in the equation xx — ax — be = o.
Now, to apprehend the reafon of this, it is neceflary to have recourfe to two
properties of the circle, which are demonftrated by geometricians ; which are,
that the right lines EA, DF, are equal to each other, as alfo the two GA, BH,
are equal, and that the reftangles EA X AF, and GA x AH are alfo equal,
Thefe two theorems being fuppofed, the line BA is to be bifected in M. Then,
by Euclid, ii. 6, the fquare of MG will be equal to the fquare of MA, together
with the redlangle BG x GA, that is HA X AG, that is FA x AE. But
the
SECT. n. ANALYTICAL INSTITUTIONS. 6^
the fquare of MA, by the conftrudion, is equal to ^aay and the reflangle
FA X AE is equal to i>c. Therefore it will be MG = v' ^aa + be, and thence
AG = — t^ + *^\aci + bci the pofitive value. But AH z=.^a + sf ^aa + bcy
whence AH negative zz — \a — s/ \aa 4- be the other value which is negative ;
both exadiy as they arife from the refolution of the firft equation. For the
fame reafon, AG negative will be = f « — s/\aj, + i<r, and AH pofitive = \a
+ ^ \aa + bci which are the values of the unknown quantity in the fecond
equation.
Fig, 23. As to the third and fourth equation, let any circle RAD
be defcribed widi a diameter not lefs than a, ox b -\- c.
From any point of the periphery A let two chords be
infcribed in it, that is AR =: a, and AD zz h -{■ c ; and
making DF = c, with centre C of the firft circle, and
with radius CF, let another circle GHF be defcribed,
which fliall cut the two chords AR, AD, in the points
G, H, F, E. This being done, AG, AH, Hiall be the
two negative values in the third equation, and the two
pofitive in the fourth. For, bifeding RA in M, it will
be, by Euclid, ii. 6, the fquare of MA equal to the
reiflangle HA X AG, that is RG x GA, that is DE x EA, together with
the fquare of MG. Therefore it will be -^aa zz be + MGq, or MG z:
\/iaa - be* And therefore — MA + MG, that is GA negative, will be
r= — f « + */xaa - be. And — MG — MR, that is GR negative, will be
— — ^a — V*:a« - be, both the negative values of the unknown quantity in
the third equation. In like manner, MG + MR, that is j-a + V^^a— *f,
will be GR pofitive ; and MA — MG, that is t« — V^aa - be, will be AG
pofitive, both the pofitive values of the unknown quantity in the fourth equation.
It is plain, both by the conftrudion of Fig. 23, and by the refolution of the
third and fourth equations, that when it is h = -^aa, the circle HGEF will
touch the right line RA, and the two values will be equal. And if be (hall be
greater than -^aa^ it will neither touch it nor cut it, and then the two values
will become imaginary.
Having thus laid down the principal rules, I fhall proceed to fhov/ their ufc
in the foiution of fome particular Problems,
PRO.
70
ANALYTICAL INSTITUTIONS.
BOOK I,
PROBLEM I.
An arithme- 9^. Let there be a certain fum of fliillings, which is to be diftributed among
tical problem. fome poor people; the number of which fhilhngs is fuch, that if 3 were given
to each, there would be 8 wanting for that purpofe ; and if 2 were given, there
would be an overplus of 3 {hillings. It is required to know, what was the
number of the poor people, and how many (hillings there were in all.
Let us fuppofe the number of poor to be x -, then, becaufe the number of
(hillings was fuch, that, giving to each 3, there would be 8 wanting j the
number of (hillings was therefore 2^ — 8. But, giving them 2 fliillings a-piece,
there would be an overplus of 3 ; therefore again the number of (hillings was
2x + 3. ^ow, making thefe two values equal, we (ball have the equation
3^ — . 8 = 2A? + 3, and therefore x zz 11 was the number of the poor. And
becaufe 3^ — 8, or 2x + 3, was the number of the (hillings, if we fubftitute
II inftead of x, the number of (hillings will be 25.
PROBLEM IL
A problem 97. The Velocities of two bodies being given, their diftance, and the differ-
of equable ence of time in which they begin to move in a right line j the point in that line>
motion. gj^^j [jjg ^i^Q 15 required, in which the bodies will meet.
Fig, 24.
A
Let the (irft body be at A, the velocity
of which is fuch, that it would defcribe the
■^ ? 5 fpace c in the time /. Let B be the fecond
body, with fuch a velocity, that it would
defcribe the fpace d in the time g. Let the difference of time in which they
begin to move be b, and let their diftance AB be e. Firft, let them move the
fame way, and let them come together at the point D. Make AD zz x, then
BD 7Z X — <?. To obtain an equation it muft be confidered, that, having
given the difference of time from the beginning of the motion of the body A,
and of the body B, the time muft be found employed by the body A, and alfo
by the body B, and to the lefTer of thefe times, or to that of the body which
moyes laft, muft be added the given difference, and then thefe two portions of
time ought to be made equal. Therefore, by the rule of proportion, we muft
fay, if the body A defcribe the fpace c in the time /, in what time will it
5 defcribe
SECT. II. ANALYTICAL INSTITUTIONS, ^l
defcribe the fpace x ? That is, c ./;: x , ^=^,, which is therefore the fourth
c
term. Likewife, if the body B defcribe the fpace d in the time gy in what
time will it defcribe the fpace x '■^ e} That is, d .g ll x — e , ^^LJZJL^ which
is the fourth term. Therefore the time of the body A is ^^, and the time of
the body B is ^^ J ^^ > and their difference is b. And if the body A began
to move after the body B, it will be ^ + v& = 4!f_Z_£! ; and reducing to a
common denominator, it will be fdx 4- cdb =: cgx — cge, that is, cgx — •//*
= cdb + ceg; and, dividing by eg — fd, it is ^^ — i^ = ;^.
If the body A move before the body B, it will be — zz b -^ ^JLlLiL ;
e a
and reducing to a common denominator, it is dfx = cdb + cgx — ceg^ riiat is^
€gx '" dfx zzz ceg — cdb. And, dividing by eg — <^, it is * = ^^^ Z/w " •
Now, if inftead of x we fubftitute it*s value now found, in the expreflion of
the whole time ^ + ^ in the firft cafe, and in — in the fecond, we (hall
C ' C '
have the time required.
I (hall apply the formula to fome examples. Let the body A have fudh a
velocity, as to move 9 miles in i hour, and the body B to move 15 miles in
a hours; and let them be diflant from each other 18 miles, and let B begin to
move I hour before A. Then it will be ^ = i, /z= i, r zz 9, ^ =: 2,
d zz ic, e zz 18; and therefore x zz ~ !ii = 152, Subftitute this value
J' ' 18 — 15 ^^
fx
inftead of x, and alfo the others, in the expreffiori of the time ~ \- b, and it
will be = 18. Therefore the two moving bodies will be together at the
diftance from the point A of 153 miles, after 18 hours from the beginning of
the motion.
Let the body A have fiich a velocity as to move 4 miles in i hour, and the
body B to move 5 miles in i hour, and let them be diftant 6 miles, and A
begin to move 2 hours before B. Therefore it will be /» = 2, /z: i, r = 4,
^^ = I, ^ z= 5, tf = 6. Taking the formula of the fecond cafe, it will be
IC zz ^^ ~ ^° = 16. And fubftituting this value oi x with the others in the
7^ ANALYTICAL INSTITUTIONS. BOOK I,
expreflion of the time ^^, it will be =: 4. Therefore the two bodies A and B
c
will be together at the diftance of 16 miles from the point A, after 4 hours
from the beginning of the motion.
But if the two bodies move contrary ways, or towards each other, let them
meet, for example, in the point M ; then calling AM z= x, and retaining the
fame denominations as above, BM only will be changed, which will now be
zz e ^- X ; and confequently the time of the body B to defcribe the fpace BM
will be ^SJ^-^. Wherefore, if A begin it's motion after the body B, it will
be ~ + ^ = ^' "j^"" i and if it begin it's motion firft, it will be ~ = /&
+ ' — -1 — j of which equations the firft is fdx + cdh zz cge — cgx, that is a? =:
^^' 7 ^j 3 and the fecond is fdx = cdh + cze — cgx, ov x :=. ^^ — — .
eg •\-Jd J • 6 ib J fd->c eg
Let the body A have fuch a velocit)% as to defcribe 7 miles in two hours,
and the body B 8 miles in 3 hours, and let them be diftant 59 miles, and A
begin to move i hour before B. Therefore it will ht h -zz. i, /— 2, r =: 7,
1^=33, d =: S, e = ^c^ ; and therefore, taking the fecond formula x zz ^-^ — ^,
and fubftitutinff thefe values, it will be x zz — ^ — |-, that is a; = 2 c. There-
to » 21 + 16 ''''
fore the two bodies will meet each other at the diftance of 35 miles from the
point A, after lo hours from the beginning of motion -, as will be feen by
fubftituting thefe values in the expreflion - — , which is the whole time of
motion.
PROBLEM III.
Evpyjxtty a 98. Having given the mafs of the crown of King Hiero, made up of a
famous pro- mixture of gold and lilver, and the fpecifick gravity of gold, of filver, and of
^'^^^oijir- [j^g crown ; it is required to find the quantity of each metal in the crown.
Let the mafs of the crown be reprefented by m, the fpecifick gravity of gold
td filver be as 19 to loj-, and to the fpecifick gravity of the crown as 19 to 17.
Make x the quantity of gold in the crown, and therefore m -^ x will be the
quantity of the filver. The mafs of a body divided by it's denfity or fpecifick
gravity
SECT. rr.
ANALYTICAL INSTITUTIONS.
73
gravity is equal to it's volume •, therefore the volume of the crown will be
-^, that of the sold — , and that of the filver "^ "* '^. But the volume of
17 ' ° 19 ' loj-
the crown is equal to both the volumes of the gold -and filver together which
compofe it. Therefore we fl:iall have the equation -^ r= -^ -f. ^SZJl that is,
i^ ^ 17 19 10^
ig — lof- 17 -- lOf ^
6^ X 19
by reducinj^ it to order, ■- 1 .v — ~ 7 w, and therefore x ~ -;|
^ '^ ' 19 X io| 17 X 10 j- S| X 17
or ;v = -— m. Hence, fuppofing, for example, the mafs of the crown to be
5 pounds, the quantity of the gold in it will be 4-AV pounds, and of the filver
41.4. parts of a pound.
PROBLEM IV.
99. Let there be two weights fo related, that if we take from the firfl 1 pound, An arlthmc-
the remainder fliall be equal to the fecond weight increafed by i pound. And, ''ical problem*
adding i pound to the firft, and taking i pound from the fecond, the fum Ihall
be double to the remainder. The quantity of each weight is required.
Let us call the firfl: weight .v, and the fecond y. Then it will be x
J =
jy + I by the firfl: condition, and
X + I
— y — I by the fecond. By the fird
we obtain this value y ■=. x — 2, which, fubttituted in the fecond, will give
= vY — 3, and therefore x -{• i -zi ix — 6; that is, a; r: 7, and coft-
fequently y ^ $*
PROBLEM V.
100. In a given circle DCM, a line AB being given, ^ geomeiri.
which is intercepted betweien the centre and the linecalprobktr..
MB, drawn from the extremity of the diameter DM
perpendicular to AC : it is required to find a point O
in the tangent MO, from whence the redlangle of
OM into MB may be equal to the redlangle of DM
into AB.
Make AB = h, AM = ^^, MO = a; ; it will be
MB rz '^aa — hh -, and by the condition of the pro-
blem, x^ aa - bb = a^^, that is, it n ■ ^" : ■ '
yaa — bh
L From
74
Analytical iijstitutions.
BdOK Ir
From the point D let there be drawn DO parallel to BM ; then the triangles
JVIBA, DMO, will be fimilar, and therefore it will be MB . BA :: DM . MO,
that is \/ aa - n ,h\\2a, MO =
V aa — l^b
- — ji;.
PROBLEM VI.
Another. 17/. 2.6, loi. A redangle being given, a paral-
lelogram is required, the fides of whkh are
multiples in a given ratio of the fides of thcr
redangle, and it's area fubmultiple.
Let ABCD be the given redan gle^
AB ~ at BC = h, and therefore the area
zr ab. Let the parallelogram required be
BFHG, whofe fide BF Hiould be to AB a*
nio e ; and therefore BF = — . The fide
e
bm
BG (hould be to BC as w to ey and therefore GB zz . Laftly, the area
BFHG (hould be to the given redangle ab, as e to r. Make BL = x, and
Fi£. 2.6,
M
U
A.
\
I
I,
II
^^-^^^'^ ^^^--^
C 3
r
Gr
S
/
XX 9
therefore^ drawing FL perpendicular to BG, it will be FL ~ /-
Wherefore the parallelogram BFHG, that is FL X BG, will be —J'^-^^xx,
e ^ ee
And, fincc this (hould be to the redangle ABCD as e to r, we (hall
have the analogy -^ J ——■ — xx , ah W e , r % whence the equation
—^ flfl»« ^ ^^ __ ^y^ ^^^ taking away the radical, it will be -
•~ XX zz
aac*
, that is XX =
aatin
ee
m'r^
; and extrading the fquare-root, x zz
. / aaan
"~ V ee
mmrr
In the fide BA take BI z: — , and IM = — ; and with centre I, radius
IM, defcribe the femicircle MLP. The ordinate will be BL = V ^^ -^^
' ^ ee mmrr
= «f. Then from the point L raifing the perpendicular LP = BI, and drawing
9 BF,
SECT. n.
ANALYTICAL INSTITUTIONS.
7S
BF, take BG =: — ^ , and compleating the parallelogram BHFG, it will be =
ai
BG X FL zz —^ ; that is, it will be to the redangle BADC iz ab, as e to r.
And the fide BF will be equal to \/BLq + LF^ =
ftruded.
: which was to be con-
The extradion of the fquare-root has introduced an ambiguity of figns, and
therefore two values of the unknown quantity, and confequently two folutions
of the problem. But it is eafy to perceive, that thefe two values are the fame,
and differ from each other only in this, that the fame conftrudion may be made
on the fide of B towards C.
PROBLEM VII.
102. To infcribe a cube in a given fphere.
Fi^. 27. T Let KQEP be a great circle of the fphere, A geometrl-
^ ' ^ A it's centre, AT =: a it's radius, AR half ofcal problem,
the height, or of the fide of the cube to be in-
fcribed, and therefore make AR ~ x. Through
the point R let there be conceived to pafs a plane
perpendicular to AT, the common fedion of
which, with the fphere, fliall be the circle
QNSKFO, and the fquare infcribed in this
circle Ihall be one face, or one plane of the pa-
rallelopiped infcribed in the fphere. But, be-
caufe this parallelopiped ought to be a cube, it
will therefore follow, that GR =: SN zz NO,
or AR zz RI = lO ; and befides, that the
planes which inclofe it fhould be at right angles. In the circle KPEQ*^ the or-
dinate will be KR — QR zz V aa ^ xx ; and taking RI zz RA =z x, it will
be KI = "/ aa — XX + Xt and IQ^= \/ aa — XX — ►v. And in the circle
NKOQ^ the ordinate lO = V^KI x ICi = s/ aa — zxx. Therefore the equation
will be V aa - 2XX = Xy and thence aa = ^xx, or a; zr 4; v^^aa. Now, taking
AU equal to a third part of the radius AB, upon the diameter CU defcribe the
femicircle CRU ; the point R in which it cuts the radius AT (hall be the point
required. And it will be AR zz \/ ^aa, half the fide of the cube, taking it's
pofitive value on the fide of T, and the negative towards Z. Whence taking
AG =: AR, and through the points R, G, the fphere being cut by two planes
L 2 perpendicular -
76 ANALYTICAL INSTITUTIONS, BOOKI^
perpendicular to RG ; and taking RH = RI z: RA, and through the points
J, H, the fphere being cut by two other planes perpendicular to HI, and by
two others through SN, FO^, perpendicular to NO, the cube will be infcribed..
For, becaufe, by the conftrucftion, as it plainly appears, the planes are perpen-
dicular to one another, and it being AR =: RI zz V \aa^ it will be, by the
property of the circle KQEP, the ordinate RQ^zz- \^ \-aa, and therefore ]Q^
.-=r '\/\aa — '^\aa^ and lO zz v/KI x IQ^— s/'^aa \ and confequendy all the-
lides are equal, as was to be demonftrated.
From the conftrudion of this problem arifes a pretty fimple fynthctical de-
monftration. Since AU is a third part of the radius AC, the rectangle CAU,
that is the fquare of AR, will be a third part of the fquare of the radius, and
therefore AR zz RI. If from the centre A of the fphere be drawn a right line-
AI to the point I, the fquare of AI will be double the fquare of AR,that is, two.
third parts of the fquare of the radius. And if from the faid centre A a radius
AO be fuppofed to be drawn, the fquare of lO will be equal to the fquare o£
AO, lefTened by the fquare of AI ; that is, equal to the Iquare of the radius,,
leifened by two third parts of the fame fquare, and therefore equal to one third
part of the fquare of the radius, and confcquently lO is equal to AR, &c.
PROBLEM Vlir.
Another,
producing
an identical
uquation.
Fig. 28.
103. Two concentric circles AGO, BDH, being
given, from the point O to draw a chord in iucii
manner, that it may be CM zz DC.
Let QC be the chord required^ and let F be the
centre. Make FH — a, FO zz b^ and letting
fall the perpendicular ME to AO, let FE zz .y.
Then EM zi s/T^ZTT^, EO zz i^ — x, and
therefore GM zz \/ aa — %hx + bh* From the point C draw CA to the ex-
tremity of the radius FA. 1 hen the two triangles OEM, OCA, will be fimilar,
and therefore OM . OE :: OA . OG. That is, \/ aa ^ ihx ^ bb . b — x :',
2b. OC zz
2hl> — zbx
'^ aa — zbx •\- hb
But^ by Euclid^ iii. :^6y it is DO X OM
BOxOH ; and therefore DO . BO :; OH . OMj that is DO =
a -^-b X b—a'
And confequently CD = CO ^ DO zz ^±zj!l+J^ _ -/ JTrTJI^T^*
'V aa —-zbx -\-bb
But, by the condition of. the problem, it. ought to. be. OM = CD. Therefore
it
SECT, n^
ANALYTICAL INSTITUTIONS,
77
k will be \/l?l> — zbx + aa ~ \/aa — zbx + bb, which is an identical equation.
Whence we gather, that, however we may draw the chord OC from the point O,
it will always be OM =i CD. And this may alfo be known, by drawing from
t.l>e centre F the perpendicular FL to any chord whatever OC. For F being
the centre of both the circles, the right line FL will bifedl both DM and CO ;
and therefore, if from the equals LC, LO, we take the equals LD, LM^ there
will remain equals CD, MO.
PROBLEM IX.
Fig,.igy ^^ 104. The indefinite right line NZ being
propofed, and three points N, A, K, being
given in it, a fourth point M is required, fuch
tliat NM may be a third proportional to NK,
AM.
Becaufe the three points N, A, K, are
given, make NA zz a, NK =: ^, AM — Xy
and therefore MN = ^ + at. Then, by
the condition of the problem, we (hall have
l> . X :: X . a + X ', and, reducing this ana-
"" logy to an equation, it will be xx zz ai^ + h',.
or XX — ^A? zz alf, which is an affeded quadratick. Wherefore, if we add to
each fide the fquare of half the co-efEcient of the fecond term, that is -^i^i?, it
will be XX, -^ ^x -^ \hb 1= ah ■\- \hb \ and extracting the fquare-roor, it is
, , , , . h ±. \/i,ab + bb
A^ — t^ = ± V ab + ^J}hy. that •" " —
/
^
^ «
^-
I
A K M Z
V
i^/
"■■■'•••-. ) \
V
#
/ '■■•• Jt?
—y' --"JK
A jreometri-
cal, or rather-
arithmetical, -
probk'ni.
is X =.
On the right line NZ produced both ways indefinitely, take AR, AQv equal
to each other, and each equal to NK zz /-j and RF four times NA, or RFzz 4^.
Then it will be AF zz A^a ■\- b. With the diameter FQ^let a femicircie FHQ^
be defcribed ; at the point A the ordinate will be AH = \^ i^ab + bb. Then
adding diredly AO = NK zz b, and bifeding OH m S, it will be OS =.
h + 'J^ab + bb _ ^^ ^^^^ ^^j^.^g ^j^ _ Q3^ ^^^^ p^j^^ required will be M>.
as to the pofitive root. For, defcribing the redangks SN, AU, MO, and-
Jf -L W A(xl) -X. bb
drawing the diagonals AI, AE j becaufe it is OS zz , it will be.
AS zz "^"^ "^ II— , and the reaangle OS X SA will be equal to ah^ that is,.
equal to the redangle OA x AN. Therefore the. fides of thefe re^angles will,
be
78
ANALYTICAL INSTITUTIONS,
SOOK I.
be to one another in a reciprocal ratio, that is, OA . OS :: SA . AN, or
EM . MA :: IN . N A. Wherefore the two lines I A, AE, will be diredly to
each other, and confequently the triangles lUE, AOE, will be fimilar, and
therefore it wiil be AO . OE :: lU . UE; but AO ~ NK, OE =z AM, lU
- OS - AM, UE = NM. Wherefore NK . AM : *. AM . NM.
The foregoing condrucflion belongs only to the pofitive value of the unknown
quantity, that radical being taken which is affeded by the affirmative fign. But,
in a like manner, that will be conftruded in which the fign is negative. For
the other femicircle F/??Q^ being defcribed, and drawing the ordinate A-^, it will
be O^ = ^ — \^^alf + bby a negative quantity ; and bifeding Oh in S, it will
be Os zz ' = *". So that a; is a negative quantity, and therefore,
taking Km zz Os from A towards F, m will be the other point which folves the
problem. For, becaufe it is ks z: kh
sh
— — ^ - ^g^ + ^^
it is there-
fore Os y, sk zz ab — OA X AN j fo that, making the redangle Nj, and
drawing the diagonal Az, becaufe Aj X ^O — OA x AN, and AN =: i/, it
will be ks .si \\ AO . Oe, and therefore Os zz Oe. But Os zz km, therefore
\Je = iSIw. But, by the fimilar triangles AO^, iUe, we (hall have AO , Oe '.",
iV . TJf, and it is AO = NK, /U = Oj = O^ zz km. Therefore it will be
NK , km :: km . wN.
Fig. 30.
Without refolving the equation xx — hx — ah zz o, the problem may be
conftruded independently, by the help of § 94, in the following manner.
Take RO z= NK = /^, and diredly to it OD =
"Nk = a. Then with the diameter RD let the
femicircle RMD be defcribed ; the ordinate will
be OM = V ab. With the diameter OR let
another circle ARPO be defcribed, and from
the point M through the centre H let the right
line MN be drawn. And taking AN" = «,
NK = by AM will be the pofitive value of the
unknown quantity. And taking the part km
= P/» from A towards N, km will be the nega-
tive value. 1 omit the conftrudion of the fame
equation by means of § 95, becaufe it is evident
enough of itfelf.
PRO-
SECT. U»
ANALYTICAL INSTITUTIONS.
79
PROBLEM X.
^ig' 3^*
105. The diameter AE of the circle AFE being A geometric
given, and the two portions CB, CD, from the ^^^ P'^°^^^'"*
centre C, and raifing the perpendiculars DF, BH ;
in BH produced, fuch a point G is required, that^
drawing the right line GF to the point F, the
re6tangle GF x FD may be equal to the redlangle
AC X BD.
Draw FH parallel to AE, and make the radius-
CA = r, CB = a, CD = b ; it will be DF =
\/rr - hb = BH, and make HG = x. Therefore
HF = CB + CD = ^ + ^j and GF = s/ aa + 2ab + ^^ + xx. Then, by the
condition of the problem, we fliall have »/ aa + zab ^ bb ■>(- xx X \/ rr — bb ~
ar 4- Ir, and, to take away the afymmetry, it will be ^V* -j- 2^^r* + ^V* ==
^V, and, by reducing.
«V» + zahr'^ + ^V* + r\v* — a^h" — 2ab^ — h^
^V — d'b'- — 2ab^ — 3* z: o. That is, a:* =
ab + bb
'3^ + 2ab^ -f M
r* - P'
and.
extrading the fquare-root, it is ;? = ± ■ . . Therefore x, the quantity
fought, is a fourth proportional to FD, DC, and FH. Now, becaufe the
angles in D and H are right, if we make the angles GFH, ^FH, each equal to
the angle CFD, the triangles GFH, ^FH, CFD, will be fimilar, and the
points G, gy (that is G in refpeft to the pofitive value, and g in refpedt of the
negative value,) will fatisfy the queftion. For it will be FG (or F^) . FH : :
FC . FD. But FH = BD, FC =z AC ; fo that it will be GF (^F) . BD :::
AC . FD. And therefore GF (gF) X FD = BD x AC.
It is eafy to perceive, that, in refpeft of the pofitive value, it is enough to
draw the tangent FG at the point F, becaufe the angles GFC, HFD, are right
a9gles. And taking away the common HFC, the angles GFH and CFD will
be equal.
PK.O-
8o
AK A L'Y T IC At I NST I TU T I *J S.
BOOK J.
P ^R O B L E M XL
A pjeometrl- Fig, 2Z,
eel probleno.
106. From the extremities of the given
line AB, to draw two right lines AC, BC,
in fuch a manner, that they may make the
angle ACB equal to the given angle GDP;
and that the fum of the Iquares of AC and
BC may be to the triangle ABC, in the
given ratio of 4^ to a.
Let AB be bifedcd in E, and letting
fiill the perpendicular CH, make EH — jt,
HC zz y. Now, becaufe the problem is
determinate, and here are taken two un-
known quantities, it will be neceflary to
'find two equations. Make EA = ^, then it will be AH — a — x, HB =
a + X ; therefore the fquare of AC will be aa — 2ax -{- xx + yy, and the
fquare of CB will be aa + lax + xx + yy, and the triangle ACB rr: ay j but,
by the fecond condition of the problem, the fum of thefe fquares fhould be to
the triangle ABC in the given ratio of 4^ to a ; therefore we (hall have
2aa 4- 2XX + ayy . ay '.I \d . a, and thence the equation aa 4- xx •\- yy — idy,
Befides, the angle ACB ought to be equal to the given angle GDP, and there-
fore, PD being produced, if the angle GDP be obtufe, and taking GD at
pleafure, draw GF perpendicular to PF ; then the angle GDF will be known,
the angle GDP being given. And, becaufe alfo DG is known, which was
taken at pleafure, the two lines will be given, DF which make = by and GF,
which make = c. Then, from the point A draw AI perpendicular to BC pro-
duced, the two triangles GDF, ACI, will be fimilar. Now, becaufe of the
fimilar triangles BCH, BAI, we fhall have AI = " ^^
Vaa 4- 2ax -^^ xx -^ jy
aa '- XX ^ yy
laa •'f '2.ax
BI =
V aa •\- 2ax + xx 4 yy
, and therefore CI
Vaa 4- 2ax 4- xx +yy
And now, be-
caufe it muft be CI . AI :: DF . FG, we (hall have
nay
aa — XX — yy
'v aa 4- "i-ax -'t- kx -^ yy
— - cxx — cyy.
w aa 4- lax -{• xx ■\- yy
\\ h , c \ and thence the fecond equation lahy — aac
To eliminate one of the two unknown quantities j from the two equations
(by § 82.) may be deduced the value of xx, that is, from the firft xx = idy
8 -^yy
SECT, n, ANALYTICAL INSTITUTIONS. Si
^r- i^y — aa^ cind from the fecond, xx = aa — yy ^. Whence the equa-
tion 2dy — yy — aa ~ aa — yy j- . That is, dy ^ aa — ^, or (mak-
ing — =/) ^ = -^f> which is a vakie of j exprefled by known quantities
only. This fubftituted infteadof^ in the equation xx — 2dy — yy —. aa, we
(hall have at laft xx - -T-r7 — =t=~^ — ^^j or xx = — ===::i , and thence
^ = ± 7 -^^ "^"^ , a value exprefled by given quantities only.
a +/
Draw AK indefinitely, making the angle KAB equal to the given angle
GDP ; and from the point E let fall the indefinite perpendicular EM, and from
the point A the right line AL perpendicular to AK. Then making DR per-
pendicular to PD, the angle RDG will be equal to the angle DGF. In like
manner, the angle LAE will be equal to the fame DGF, and befides, the
angles at E and F are right ones. Therefore the triangles LAE, GDF, will be
Similar, and thence EL = — = fj and AL = s/aa ■\-ff' In EL produced
take LM = ^, and with centre L, radius LM, let a circle be defcribed, which
(hall cut AK in K. And, becaufe the angle KAL is a right one, the ordinate
will be AK = \/ dd - ff - aa^ Whence, making E;? = AK, and drawing MA,
and «H parallel to it from the point », it will be ME . EA : : «E . EH ;
that is, ^ 4-/ . a\\ y/ dd-ff - aa . ' ' dlf~'"^ = EH = ;^. This
being done, with centre L, and radius LA, let a circle OCQ^be defcribed, and
at the point H railing the perpendicular CH, draw CA, CB, and ACB fhali be
the triangle required. For, by Euclid^ iii. 32, the angle ACB is equal to the
angle KAE, that is, by the conftrudion, to the angle GDP j and, by the pro-
perty of the circle, PC = \/op x PQ. = J, . '^'* J and therefore HC =;
•^r—f And, by making the calculation, we fhall find, that the fum of the
fquares of AC and CB is to the triangle ACB precifely in the ratio of 4^ to a.
The ambiguous fign of the final equation gives us two equal values of x^
one pofitive, and the other negative. If, therefore, EH taken towards A be
confidered as pofitive, then E/& taken towards B, and equal to EH, will be the
negative value; which will require the fame conftru6lion.
It is evident, that the problem will be impoflible as often as dd is lefs than
//-{■ aa^ that is, LM lefs than hJ^; for then the radical will become impofTible,
or only imaginary,
M PRO.
Ba
ANALYTICAL INSTITUTIONS,
BOOK 1»
PROBLEM XII.
Another. Fig. 33.
107. The femicircle BED being giveD>
and a point A being given in the diameter
produced ; from that point to draw a fe-
cant AE in fuch manner, that the inter-
cepted part GE may be equal to the
radius CB.
Make CB = f, AB = h, AD = «, and
AG = x. Therefore, by the condition of
the problem, it will be AE = c -{- x^
Now, by Euclidy iii. 36, the redangle EAG is equal to the redangle DAB,
and therefore we fliall have the analogy AE . AD : : AB . AG. That is,
c -{• X . a \\ b , X. Whence the equation xx -^^ ex = ab; which is an affefted
quadratick, and, being refolved as ufual, will give us x — ■+: V \cc +11 — 1^»
On the right line DA produced, taking AR = AB = b, let the femicircle
ROD be defcribed on the diameter RD ; and drawing the ordinate AO, it will
be — V ab. Draw OM i=i\c perpendicular to AO, and it will be AM n
\/ Ice + ah. Then with centre M, and radius MO, let a femicircle QOP be
defcribed, and it will be AQ^zz \/ \c^ 4- ah — t^, the pofitive value of a?; and
AP ~ \/ \cc + ah + \c. Wherefore AP, taken negatively, will be the nega-
tive value. Then,, if with centre A, and radius AQ>^ an arch were defcribed,
it would cut the femicircle BED in G the point required. And if, on the other
fide, the femicircle RGH be defcribed on the diameter RH -=. BD, an arch on
the fame centre, defcribed with radius AP, will cut it in the point required ^,
which belongs to the negative value. For it being EA x AG = DA X AB,
ah
that is EA X s/ \cc + ah
:C ZI
it will be EA —
And
therefore EG n
ah
Vice + ah w if
'^icc + ab — ic
— \/u7T^b + t^ J that is, reducing to a
— ffr + cv-^cc + ah
And adually making the
common denominator, EG zz
'/j;cc + ab —
^ivifion, it will be at laft EG n c, as it ought to be.
The fame calculus will ferve for the conftru^ion of the negative value, only
making ufe of the redangle HAR inftead of DAB.
Alfo,
SECT, n,
ANAI^YTICAI, INSTITUTIONS.
83
Alfo, the folution of the problem may thus be demonftrated fynthetically.
Becaufe it is OA^ = RA.D, and EAG — DAB, and, by conftru(f\ion,
AR = AB, ACtzz AG, QP = EC, <^0 = MC^ it will be AO^ + OM^
=z AM^ =: EAG + QMq i that is, by Euclid, ii. 4, AQ^ -i- 2AQ!yi -f-
QMy zz EAG 4- QM^. And, taking away the common Qjyr(7, it will be
AQq + 2AQM =z EAG; and, by the third of the fame book, AQ^- + 2AQM.
= EGA + GA^. But AG = AQ^; therefore it will be 2AQ^''I zz EGA,
that is, AQ^. AG ;: EG . 2QM. And therefore EG zi 2QM = BC,
CL E. p.
PROBLEM XIII.
108. Two arches of a circle being given, and their tangents, to find theAtngono-
tangent of the fum of thofe arches. metrical pro*
blem, with
Let the two given arches be AH, HD, and the a general
tangents AI zz a, HK = ^, the radius CA iz r, Solution,
the tangent of the fum of the given arches AB:i:a;,
It will be CB = */rr + xx, CI = \^rr -j- aa, CK
= Vrr + bb. And> letting fall DE perpendicular
to CA, and DF perpendicular to CH ; becaufe of
fimilar triangles CBA, CDE, it will be CE —
rr T-vT-i rx
'^ rr-\-xx
triang
have EO =
, DE =
= ; and alfo, becaufe the
'^ rr + XX
triangles CAI, CEO, DFO, are fimilar, we fhall
a. ^,cO = ^:^^;^, and DO
"^ rrArxx ' \/rr-\-xx
= ^ri-^ . Wherefore we fhall have the equation ED zz EO + OD, thac
IS,
+
lwrr-\-aa rx
'^rr-\-xx */rr + bb
= , Of
rx — ar hVrr-\-aa
i and, fquaring
'^rr-^-xx' '^rr + xx "^rr-^-bb
this to free it from the radicals, it will be ^^^^ - ^'^rrx + ^arr __ birr + a alb
rr ■\- XX rr + bb '
Then, reducing to a common denominator, and taking away fuch terms as
deftroy one another, it will be r^xx — iar*x — lahbrrx + aar'^ zz aabbxx -i-bbr*;
thar ic 1.V ^'"'^ + ^abbrr hbr* — aar* , • , • rr- o t
mat IS, XX -___.;; - ,4 _ aabb ' "^^'^^ '^ ^" ^^edcd quadratick.
Therefore, adding to each member the fquare of half the co-efficient of the
M2 fecond
84 ANALYTICAL INSTITtTTIONS. -ftOOK I,
fecond term, that is the Tquare of ;^ ,, ■■» it will become ^a: — — —-^ — jr—x
4. — — zr •— -— 4 ., ; tnen extract-
^ T^—r^bi^ r^-a-b- r^-aH-y
ing the root, and reducing the homogeneum comparationis to a common denomi-
nator, it will ht x -. rrr- — ±: V —^ -t ^ . But the quan-
tity under the 'vinculum is a fquare, and it's root is \^ _ ^,^ , or otherwife"
M + <7fl ;t Therefore, in the firft place, taking the pofitive root, it will
\^Q X "==■ — - — ^ __ i ; and, taking the negative root, it will be
_ ^r^ + al^r^ - aair- ~ hr'^ ^ ^^^ .^ ^j^^ ^^^^ ^^^ ^^^ ^^^ numerator and
r* — aabb
the denominator are divifible by rr + «<^, and the quotient is — ZTT" » ^"*^*
in the fecond cafe, the numerator and the denominator are divifible by rr -- ahy
and the quotient is- ''^^"" ^^ . Therefore the two values of the unknown quan-
tity are x zi ^^ ^ ^ . , and x = — — ^-y— . The firfl: of thefe will ferve
for the tangent of the fum of the given arches, and the fecond for the tangent
of their difference, as will eafily be feen by folving the problem in this cafe.
This value will be pofitive or negative, according as the arch, or it's tangent ay
will be greater or lefs than the tangent b.
This foundation being laid, it will not be difficult to go on to the general
folution of the problem ; thit is, as many fucceffive arches as you pleafe, with
their tangents being given, to find the tangent of the fum of all thofe arches ;
which may be done in the following manner.
Firft, let there be three arches given, and let their tangents be a, h, c. By
the foregoing folution, -^ \_ab ^^^^ ^^ ^^^ tangent of the fum of two of
thofe arches, the tangents of which are aj h. Let this tangent be called z, and
therefore it will be z z= ^^ ^ ^ ^b ' ^^^' ^^ ^^^^ ^^^^ folution, it will be
r^ ^ '^ "^ '^ , the tangent of the fum of the two arches, whofe tangents are z, c;
rr — zc '^
and z is the tangent of the fum of the two arches, whofe tangents are a, h,
J Therefore
SECT. II. ANALYTICAL INSTITUTIONS. 85
Therefore ^ — ^ — - will be the tangent of the fum of the three arches, whofe
rr — zc °
tangents are a, b^ r. And in this expreffion, inftead of z fubHituting it's value
rr X a-\- ^^^ ^^^ \i2iVt the tangent of the fum of the three arches exprefled
rr -^ab
1 , • 1 1 ,.,■,, ^rrXa + b-{-c — ale r, 1
by the given tangents only ^, b^ c, which will be ^^ _ ^^ _ ^^ _ ^^ • ^7 ^he
fame way of arguing, we fhall have the tangent of the fum of four arches,
their given tangents being a, b, c, f, which will be
rr Into arr H- brr + err + frr - aba - abf - acf - hcf ^^^^^ ^^^ ^^^^^^^ ^^ ^^^ ^^^
rr X rr — ab — ac — af — be -\- bf — cj + abcf
of five, their given tangents being a, by c, f, g, vvill be found to be
r* X a + b + c+f+ g — r^ X n b£-\ -abf-\-acf-^abg + hcf+arg-{-bc^ + bff-\-af^-\-cfg'+ahcfg
rr X rr — ab — ac — af—ag — bc—bf—bg—cf—cg—fg + abcf-i-abcg + ab/g + acfg-{-bcfg
And thus for as many more arches as you pleafe. From hence may be
derived a general rule, to form the fracflion which fhall exprefs the tangent of
the fum of as many given arches as you pleafe ; which will be this.
To form the numerator of the fradlion there mufl be taken the fum of all
the poffible produdts of an odd number of fadors, which can be made with all
the given tangents. For example, if the number of tangents be feven, take
the fum of all thefe tangents ; then the fum of all the threes that can be made,
then the fum of all the fives, and laflly, the producl of all the feven. Thefe
fums are to be multiplied by fuch a power of the radius, as each has occafioii
for, that they may be of a dimenfion greater, by unity, than the number of the
given tangents. And to thefe fums muft be prefixed the figns + and — alter-
nately ; that is, to the fum of all, the fign + ;. to the fum of all the threes^
the fign — , and fo on; and thus the numerator will be completed.
To form the denominator muft be taken the fquare of the radius, then the
fum of all the produdts of an even number of factors, which can be made by
the given tangents, that is of all the twos, of all the fours-, 8tc. This fquare of
the radius, and the fum of all the twos, of all the fours, of all the fixes, &c.
muft be multiplied into fuch a power of the radius, as each has occalion for,
that they may be of a dimenfion equal to the numb.er of the given tangents.
To the fquare of the radius is to be prefixed the fign 4-, to all the twos the
fign — , to the fours the fign +, and fo on alternately. And thus the deno-
minator will be completed.
Now the rule for knowing what muft be the number of all the twos pofTible,
of all the threes, &c. in a given number of quantities, will be this following.
Write
B6 analytical institutions, book r.
Write down the number of quantities given, and thence continue the de-
creating feries of natural numbers. Under thefe numbers write down in order
an increafing feries of natural numbers, beginning from unity. Afterwards find
the product of fo many terms of the upper feries, as is the index of the combi-
nation that is to be made. Alfo, there muft be made the produft of as many
terms of the feries below ; and one product being divided by the other, the
quotient will be the number required. So to know how many twos, threes, &c.
can be made of 5 quantities, for example, write down the numbers thus :
5j 4j 3j 2, I,
I, 2, 3, 4, 5.
The produd: of the two firft numbers of the upper feries is 20, which,
divided by the producSt of the two firft numbers of the lower feries, will give 10
for the quotient. And therefore the twos will be 10. The produdt of the
three firll is 60, which, divided by 6, the produd of the three firft of the lower
feries, will give the quotient 10 ; and therefore the threes will be 10, &c.
From the foiution of this problem we obtain, by way of corollary, the fo-
lution of another which is more fimple; and that is, the tangent of an arch
being given, to find the tangent of any multiple of that arch. For, in this
cafe,, it will be fufficient to make all the given tangents equal to one another,
and equal to the tangent of the given arch. For example, make the tangent
of the given arch =: «, and let it be required to find the tangent of the double
arch, the triple, &c. In the formula which we have already found for the
tangent of the fum of two given arches, inftead of the letter ^ we muft every
where put a, and we Ihall have a formula or expreffion for the double arch
, In the formula for the tangent of the fum of three eiven arches,
rr — aa o d *
inftead of b and c we muft put «, and we fhall have the expreffion of the
triple arch " ^ . In like manner, that for the quadruple arch will be
' 4 ^^A T Tl'^ a. ' That for the quintuple arch will be ^"T "" 'T/: — r- • And
fo of all Others fuccefllvely.
"Whence we may form the following progreffion, or general canon, for the
tangent of any multiple arch, according to any whole number whatever denoted
by n.
« — I n.n—l.n — 2n—3 «.«— I .« — 2.n — 5.n — 4 «— < «.« — I .« — 2.« — 3 . «— 4.H — J .a — 6 « — 7 -
nr a r a^-\-- r a'' • ;• a-'
I. 2. 3 I. 2. 3. 4. 5. I. 2. 3. 4. §. 6. 7
&C.
r r -^ tf* + r ^ a* — —^ r ' a^
I. 2. I. 2. 3. 4. 1. 2. 3. 4, 5. 6
The
SECT. II« ANALYTICAL INSTITUTIONS, 87
The tangent being found of any multiple arch, the inverfe problem mil' be
cafily refolved. That is, the tangent of an arch being given, to find the tangent
of any fubmultiple arch, according to any whole number whatever. That is
to fay, to divide an arch or angle into as many equal parts as- we pleafe.
Wherefore let the tangent of the given arch be b, and n the number according
to which we would have the fubmultiple arch ; we muli take the tangent found
for the multiple arch by the number n, inftead of a we muft put x, and thus x
will reprefent the tangent of the fubmultiple arch.' This tangent of the multiple
arch is therefore equal to the given tangent ^, whence we (liall have an equation
to determine the unknown quantity x.
Therefore the tangent ^ being given, and the radius r, the equation for the
tangent of the fubtriple arch will be x' — ^bxx — ^rrx + brr zz. o. That
for the fubquintuple arch vvill be x^ — ^i>x'^ — lorrx^ + lobrrxx + S''**
— br^ zz o. And fo of the reft.
PROBLEM XIV.
^ig* 34.
C JB Q
109. To find a triangle ALO, the fides ofAgeometrl-
which AO, LO, AL, and the perpendicular LI, cal problem.
are in continued geometrical proportion.
Take one fide at pleafure, or AL, = a, and
make OL zi x. It will be, by the conditions of
of the problem, AO ir
XX
a
and LI z=
Therefore AI = ^ aa — — , and lO - V xx — -^. Therefore AI -f lO
=: AO, that is, a/ aa —
/^
+ ,- fl* XX f^ XX / a*
XX a a XX
f. a* 1 1 r • X* 2xx y a* ,
r: A/ a j and, by iquaring, — V^xx — h C'
XX ^ {id d XX
^- — -, that IS \- XX
zz aa
2 XX f
aa zr — v xx
a
— . Now, by fquaring again.
/^aaxx. And laftly.
it will be -^ -f ^ 4- X* — zx'^ — laaxx + «* z= ^
«* aa aa
by reducing to a common denominator, and ordering the equation, it will be
^' — 2^V — <3*a;* + 2^V + rt' m: o. This equation has the appearance of
one of the eighth degree, but it may be obferved to be a fquare, and therefore,
txtracling it's root, it will be found to be .v* — aaxx — «'^ zz o. This is an
afieded
8S ANALYTICAL INSTITUTIONS.
BOOK U
affedled quadratlck ; therefore, tranfpofing — a*, and adding ^a^ to both fides,
and extra6ling the root by the common rule for affedled quadraticks, it will
be XX — i^a = + t-^5^^ ^hat is, xx — \aa ± iV5^*, and finally, x =
2
Therefore the unknown quantity will have four val.ues ; but it may be ob-
ferved, that the quantity \/ ^af^ is greater than aa^ and therefore, if we take the
radical \f ^a^ negative, that is — y/^a'^ftiien the quantity under the common
radical vinculum will be negative ; whence the value oi x will be imaginary, and
therefore two values will be imaginary, that is j^ — ± y/^fSl — 2--. And two
will be real, that is .;«7 = ± v'ff ifL ^ both equal, but one pofitive and
the other negative.
On the indefinite line AQ^take AL = a, LC ~ ^1/5, and CB = f^.
Then on the diameter AB defcribe the femicircle AFB, and ered: the perpen-
dicular CF. By the property of the circle, it will be CF r= y/ zzx,
Bifed AC in H, and with centre A, radius AH zz — = a, defcribe
the arch HO. From the point L draw LO = CF, and terminated at the
arch HO. And if AO be drawn, and the perpendicular LI, then ALO will be
the triangle required. For, becaufe it is AL = «, LO iz a; = V <^^^v_ ^
AO = AH = — z= ^-^^-^ai it will be AO . LO :: LO . LA. But the
a 2
two fquares of AL and LO taken together, that is aa 4- ^^ ff^^ are equal
to the fquare of AO, that is "^"^ '^^'^ . Wherefore the angle ALO is a
4
right angle, and thence it will be AO , LO : : AL . LL But, becaufe it is
alfo AO . LO :: LO . LA, it will be likewife LO . LA :: LA . LL The
other negative value, which is equal to the pofitive, would ferve for the
conftrudion that may be made under the line AB.
PRO.
SECT. II.
ANALYTICAL INSTITUTIONS.
89
PROBLEM XV.
no. To divide a given angle into three equal parts.
The Problem propofed contains three cafes; one is when the given angle is Thetri-
a right angle ; another when it is obtufe ; and the third when it is acute. feaion of
an angle.
In the firft, let the given angle MAB be a right
angle, which is fuppofed to be divided into three
equal parts by the right lines AC, AD. Make
AB r: j, and at B raife the perpendicular BC,
which produced (hall meet the line AD in D j
and from the point D let DM be drawn parallel
to AB. Then making BC = x, it will be AC z:
>/ aa + XX' But, becaufe the angle CAD muft
be equal to the angle DAM, and becaufe of the
parallels AM, BD, the angle DAM is equal to
the angle ADC ; the angles CDA, CAD, will be
equal. Wherefore CD ■=. CA zz: \/aa + xxt.
whence BD zz x -\- \/ aa + xx. But befides, the
two angles BAC, CAD, or CDA, ought alfo to be equal, and therefore in the
two triangles BDA, CAB, the angle CAB will be equal to the angle BDA,
and the right angle at B is common. Therefore alfo the third BCA z= BAD,
and confequently the triangles are fimilar. Whence we fliall have AB . BC ::
BD . AB i that is, a . x :: x + \/aa + xx . «; and thence the equation
aa ^=i XX -^ x\/aa + xx ; and tranfpofing the term xx, and fquaring, it will be
aaxx -i- X* =z a* '-' %aaxx + a;*, and finally, ^^^^axx zz a*, or a; zz ± \/-^aa.
Produce AB to S, fo that it may be BS = ^AB = ^a. On the diameter
AS let the femicircle ACS be defcribed; the ordinate BC will be = \/±aa iz Xa.
Then draw AC to the point C, and take CD zz AC, drawing AD. The
given angle wijl be then divided into three equal parts. For, whereas it is
BC = \/-^aa, it will be AC = \/*aa = CD, and AD = V'ABy + BDq ==
Vaa 4- ^aa + 2a\/^ z: la. Therefore AD . AB :: 2^ . ^ :: 2 . i, and
DC . CB :: \/:*.aa . ^^aa :: 2 . i ; that is, in the very fame ratio as AD to
AB. Wherefore, by Euclid, vi. 3, the angle BAC = CAD ,* and, becaufe of
CD zr CA, it will be alfo the angle CAD = CDA = DAM. The negative
value, which is equal to the pofitive, would ferve for the divifion of the angle
mAB.
N Let
90
ANALYTICAL INSTITUTIONS.
BOOK I,
-^D
/\
Let the angle BAM be obtufe, and draw BD
parallel to AM, and making the reft as above,
draw AR perpendicular to BD. Since the angle
ABD is known, as being the fupplement of the
given angle MAB, and the angle R is right, and
the line AB is given ; the line BR will alfo be
known, which make zi h. Whence AR n:
s/aa-^ hb, CR = .V — b, AC = CD =
sf aa — zbx ^ xx-, and BD "Zl X •\- s/ aa — ibx -{■ xX'
Then, becuufe of fimilar triangles ABC, ABD,
it will be AB . BC : : BD . BA ; that is, a . x
l\ X -{• V aa - 2bx' + XX * a; ov aa zz XX +
x\,^aa — 2bx + xx. Then taking away the afymmetry, it is 2ix^ — 3<?^Ar.v
^ a* zz o, which is a folid equation, or of the third degree, which at prefent
1 (hall leave unrefolved.
B
Laftly, let the angle BAM be acute ; the perpeh-
dicular from the point A to DB produced will fall
under the point B in R, and therefore it will be
RC zz b + Xy and AC zr \/aa + ibx + xx. Where-
fore, repeating the fame argumentation as in the fore-
going cafe, we (hall have the equation ikx^ + ^aaxx
— ^4 — 0, which differs from the foregoing only in
the (igns.
SECT. III.
The ConJiruSlion of Loci, or Geometrical Places, not exceeding the Jecond Degree,
What arc
variable
quantities ;
and what is
the law by
which they
vary.
III. What are Indeterminate Problems, and how they require two unknown
quantities, has been already explained at § 84. Now, becaufe the value of one
of the unknown quantities may be varied an infinite number of ways, fo, in like
manner, the value of the other may be as often varied ; whence they are called
the Vdr'iable ^antitid of the equation or problem, and their relation, or law
which they obltrve in their variations, is expreffed by an equation. Thus the
equation
SECT. III.
ANALYTICAL INSTITUTIONS,
9'
equation b>: — ay informs us, that, varying x as you pleafe, y mufl alfo be
varied^ but with this condition, that .v mud always have to y the conftant ratio
of a to b. Thus the equation ab z: xy exprefles fuch a law, that the produ(5\:
of the two unknown quantities mull always be conftant, and equal to the pro-
dud of a into b. The equation ax "r^ yy implies, that the fquare of ^ mufb
always be equal to the redangle of x into a conilant line a ; and fo of all other
equations.
1X2. One of the two unknown quantities, fuppofe x for example, mud have General pre-
it's origin from a fixed point, and muft be taken upon an indefinite right line. *^^P^^ ^°J^.^^^
Then, if a determinate value be affigned to this, from the extremity is to be^p^^o^ wkh
raifed another right line in the given angle of the problem, which line is to befome ex-
taken of fuch a length as the other unknown line y ought to have, by the na- amples.
ture of the equation, relatively to the affigned or aiiumed value of x. And
this ought to be repeated for every different value that x can alfume. The line
which Ihall pafs through the extremities of all the ys is called the Locus of the
equation. The unknown line, which is taken from the fixed point on the
indefinite right line, is called the Ahjcifs ; and the other, at the given angle to
it, is called the Ordinate: and both indifferently are caJied the Coordinates of
the equation.
Fig, 38. Hy Now, for example, as to the equation
hx :=. ay \ upon the indefinite line AM take
AB = a, and in any angle draw BC :=: b.
Here, if we take ;f =: AD, the fourth pro-
portional will be parallel to BC, that is DEzijy,
And taking x =■ AF, then it will be FG zzjy.
Alfo, taking x = AK, it will be KH 1= y,.
And thus for infinite others. And the line in
which all thefe infinite points are found, C, E,
G, H, &c, which are determined in this manner, will be the locus of the
equation bx = ay, and which will be a right line.
^'fS' 39-
In the fame m.anner, as to the equation
ax zz yy, if we take .v iz AB, and BC = \/aXy
that is, a mean proportional between AB and
the given line «, it will be BC =: y. And
taking x z=. AD, and DE a mean proportional
between AD and a, it will be DE =: y. Tak-
ing ^v n AG, and GF a mean proportional
between AG and a, it will be GF = y. And
fo of all others. Now the points C, E, F, and
infinite others determined in the fame manner,
will form the line ACEF, which is the locus of the equation ax ^=^ yy> And
the fame is to be underftood of all other equations.
N 2 113. Froitt
92 ANALYTICAL INSTITUTIOMS. BOOK I.
Different " 113. From the feveral different laws exprefTed by the given equations, or
equations from the different relations that the two variables or unknown quantities may
rtquire dif- have to each Other, other /<?a or lines will arife, which will differ both in kind
and vice ' ^^^ ^^ degree. So it is eafy to perceive, that the kcus of the equation bx — ay
verfd. will be a right line, as obferved before ; for y x.o x having a conftant ratio,
becaufe it is j rz -^ ; any line ED (Fig. 38.) will be to AD, as any other FG
to AF ; therefore the triangles AED, AGF, will be fimilar. This may be
verified alfo by any other point H, &c. So that it mull neceffarily follow, that
thefe points will all be in the fame right line. But the equation ax zz yy re-
quires, not that the lines BC, DE, &c. (Fig. 39.) but that their fquares, may
have a conftant ratio to the correfponding lines AB, AD, &c. Whence it is,
that the points C, E, F, &c. will not be in one right line, but in a certain
curve line, called- a Parabola. Thus a curve of a different kind from this
would be the locus of the equation xy ■=. ab \ and a curve of a different kind
and degree would be the locus of this other equation a^ -^ x^ zz j'. And the
like of infinite others.
When the 114. As often as the equation (hall not contain, in any term, either the
ItK-'us w'lW be fquare, or fome higher power, of one of the unknown quantities, or the pro-
a right hne. ^^^ ^^ ^1^^ fame, the locus will always be a right line.
When the 115. And when, in the equation, there is found the fquare of one, or of the
^ecus^^fco- other, or of both the variable quantities, or their redangle, either this or that
nic e ion. ^^ j^ ^^^ j^^pp^j^ . ^^^ ^q ^q^^ Q^^lj include a greater power than the fquare of
thofe variable quantities, or a produdl above the redlangle ; that is, in no term
the variable quantities, either alone or multiplied together, exceed the fecond
dimenfion ; the locus will always be one of th'e Conic Sedions oi Apollonius,
Thefe affertions cannot be better demonllrated than by acftually conftrudling all
the feveral equations of this nature.
Ze>« or curves 11 6. Equations which include the unknown quantities of one dimenfion
diftinguifhed only, that is, the loci to a right line, are called Loci or Lines of the Firft Order.
into orders. Thofe which, either alone or multiplied together, include them of two dimen-
fiops, that is, loci to the conic fedlions, are called Loci or Lines of the Second
Order, and therefore Curves of the Firft Kind. Thofe equations in which the
variables afcend to three dmienfions, are called Loci or Lines of the Third
Order, and therefore Curves of the Second Kind. And fo on fucceffively.
Ththcito 117. Now, as to the loci to a right line, they are all comprehended under
a right Hne
conftruacd, jhefg (Jx equations following :y=-^jj'=:— ■T-j.)'=-t- +r, vzz
in fix cafes. ^ to^^^* t, ' ^
SECT. in.
ANA?.YTICAt INSTlTUriOt?t,
93
ax ax J
r- - ^» JV = -T- - ^» and ^^ =
t(X
4- c. For, by multiplication and
divifion, we may always reduce y to be free from fradions and co-efficients.
By -T- is to be underftood the aggregate of all the known quantities which
multiply .V, and by c the aggregate of all the quantities which form the given
or conftant term.
F'tg, 40.
To conftru(5t the two firft, upon AD
produced both ways indefinitely, take AB
iz: AF = ^ on each fide, and draw BC zr a^
making the angle ABC fuch as the two
variables of the problem ought to make.
Through the points A, C, draw an indefi-
nite right line HE ; this will be the locus of
the two equations y ~ ~, and yzz 7- .
For, taking any line AD = at, and drawing
DE parallel to BC, it will be DErz ~ =j.
And taking AF = — .v, and drawing FH parallel to BC, it will be FH =
ax
The third and fourth are thus conftrufled. Take AN z: AM = f, and
parallel to BC j and draw NK, MG, indefinitely, and parallel to HE. NK will
be the locus of the equation y = -7 — V c ; and MG the locus of the equation
y •=■ — -y c. For, taking AD = x^ it will be DE = ■—. But it is
EK = AN = r, making DK parallel to BC. Then DK = -^ + c z= >
And taking AF = »— a,', and drawing FG parallel to BC, it will be FG =:
T- — f = >
As to the fifth, conftrud the fame triangle
ABC, and produce the lines AE, AD, inde-
finitely ; draw AM = r, and parallel to BC.
Then from the point M draw the indefinite
line MK parallel to AE, which will meet the
right line. AD in Q^ Then will QK be the
locus of the equation y = ~ — c. For,
taking any line AD = x^ and drawing DE
parallel
94 ANALYTICAL INSTITUTIONS. BOOK I.
parallel to BC, it will be DE = -^ . But KE = AM = <: ; therefore DK =
.1 c •=. y. The portion QM will ferve when -^ is lefs than r, that is,
when X is taken lefs than AQ»^ or lefs than — ; for, in this cafe, y will be
negative, and therefore ought to be taken below AD, that is, the contrary way
fromDK.
For the laft formula, make AB zz by
BC — a, and the angle ABC equal to the
fupplement of the angle of the variables.
Make AM — c, parallel to BC, and draw
MQK parallel to AC, cutting AB produced
in Q. Then will MQK be the locus of the
equation y "^ c 7- . For, taking any
how AD = .V, and drawing DE parallel to
BC, it will be DE = -^ . But, producing
ED to K, it will be EK = AM = c, and therefore DK z: f — ~ == ;r.
Now, if ^ be taken greater than AQ>^ for inftance zi AI, it will be IT iz ~,
and therefore r — -^ is a negative quantity =: _y =: IP ^ taken direflly con-
trary to DK, and the indefinite line MR is the locus of the propofed equation
in both cafes.
The locus 118. It may fometimes happen, that, in the folution of a problem the locus
when one of of which is a right line, either one or the other of the two variables will difap-
the variables pe^r, and will not enter into the equation. In fuch cafes, the locus will be to
the perpendicular, or to a parallel to the given right line upon which the ab-
fcifles are taken, according as either the ordinate or abfcifs vaniQies. Here is
an example or two of this.
Fig. 43
The right line AB being given, let it be propofed to
find the locus of the points M out of this, fuch that,
drawing the right lines MA, MB, to the extremities of
AB, it may always be MA iz MB. Taking any line
AH = X, draw HM — y, and make AB zz a. It will
be HB = a ^ X, AM z= \/ xx + yy^ and BM =
\/^aa — aax -^ xx '\' yy J and ihence the equation ^/xx +yy
SECT. III.
ANALYTICAL INSTITUTIONS.
95
= \/aa - 2ax + xx + jy, and fquaring, xx + yy r=z aa — 2ax + a'A? +• yy,
that is, X '=. ^a\ where y difappears, and x remains determined. This fhows
us, that, raking x z=. AH, which is half AB, and from the point H raifing
an indefinite perpendicular, every one of it's points will fatisfy the queftion,
and therefore this will be the locus required.
Fig. 44.
C E
Let the parallels CG, AP, be given in po-
j^ Q. fition, and between them let it be required to
' find the locus of all the points M fuch, that,
drawing MP perpendicular to AP, and MG
making the angle MGC equal to a given angle
' AEC i it may always be MP to MG in the
conftant ratio of a to h. Make the diftance
AC -=r c, AP ~ X, PM zi J/, and producing
A^ T PM to F, it will be FM zz r — jy. Now,
becaufe the angle AEC is given, and ACE is
a right angle, and the fide AC is given, the fide AE will alfo be known, which
may be called /. Now, becaufe of the fimilar triangles ACE, FMG, it will be
AC . AE ;: MF . MG; that is, f . / :: c — ^ . MG = ^^l^l^. Butbefides,
it ought to be PM . MG :: a , l>. Then it will be y . ^^""-^ :: a , l>, and
therefore ky = acf — cify, or y zz , ^^ . So that here is an equation, in
which the unknown quantity x does not enter at all. Therefore, taking x as
you pleafe, y will always be conftant, and equal to T~r^i ^^'^ therefore,
drawing the indefinite line BM parallel to AP, and as far diftant from it as
the quantity ^^ ■ , this line will be the locus required.
119. Having thus explained the conftruftion of the Loci to a Right Line, I The lod to a
come now to the conftrudlion of Equations of the Second Degree, or of ti-e^^*"^^ ^°^'
Loci to the Conic Sections. And here 1 muft fuppofe the learner to be fo well
inftrufled in the chief geometrical properties of thefc fedions of the cone, as
to form from thence the firft and more fimple equations of thefe curves ; to
which fimple equations the more compounded ones may be reduced and re-
ferred, by the methods now to be explained.
A^id, in the firft place, it muft be known, that in the circle any ordinate is
a mean proportional between the legments of the diameter ; that is, it*s fquare
is equal to the redangle of the faid legments. Therefore, in the circle MKCN,
if
96
ANALYTICAL INSTITUTIONS.
BOOK I.
^^S45* ^^^ — XT"^""^^ ^f y°^ make the radius AC =: «, and from the
centre A any abfcifs whatever AB = x^ and the
perpendicular ordinate BD zijy, it will be MB
— « -f- «•, BC zz. a '— x^ and iherefore MB X
EC ^=:aa -^ xx; then it will be yyzzaa — xx^ an
equation to the circle, in rerpe(ft of the quadrant
KC. But, becaufe the fame property may be
verified alfo, taking BE for the ordinate, that
is the negative ordinate — jy, and as well the
fquare of — j? as of jy is yy -, therefore the
fame equation belongs alfo to the quadrant
CN. And now, if we take the abfcifTes nega-
tive, as AH ■=. — X, and the ordinates HFir^r,
HG — — y, their fquare yy will, in both cafes, be equal to the reiftangle
MH X HC. But when it is AH = — x, it will be CH i= CA + AH =
^ — ^ ; and MH = AM — AH :z. a + x hy the rules of Addition and Sub-
tradion. And therefore the redtangle MH x HC will be (till aa — xx. So
that yy — aa — xx is the moft fimple equation that belongs to the whole circle
with radius ^, taking the abfcifles from the centre.
If the abfcifles (hould be taken, not from the centre A, but from M the
extremity of the diameter, making iny one of them MH or MB equal to x,
it will be HC or BC "zz 2a — x, and the reftangle of the fegmcnts will be
equal to 2ax — xx. But the fquare of the ordinate, as well pofitive as negative,
is yy, fo that it will be yy = lax — ; xx', the moft fimple equation of the fame
circle, taking the abfcifles not from the centre, but from the extremity of the
diameter.
By the quantity or magnitude a, which denotes the radius, is meant any
given quantity whatever, whether fimple or compound, integer or fradion, ra-
tional or furd j fo that yy -zz aa — hb — xx will be a ciycle with radius =
s/ aa — bb J yy ^=z — XX wiU be a circle with radius = V^', yy = a'/ab
— XX will be a circle with radius = y/a^ ab» Thus yy = lax — bx ^^ xx
will be a circle with diameter = 2a — h^ or with radius = "^ " 5 yy =:
aax + a X — ^^ ^.jj ^^ ^ circle with diameter = ^'^ , ^ ; yy = x^/ab — xx
will be a circle with diameter = x^ab. And fo of others.
Here it is plain, that, in the equation yy zz aa — bb -^ xx, and in all others
like it, if the quantity b fliould be greater than a ; then aa — bb being a
negative quantity, the circle would become imaginary. For then the ordinate
5 :^-
SECT. III.
ANALYTICAL INSTITUTIONS.
97
y = \/aa -^ bb -- xx being equal to the fquare-root of a negative quantity, it
would be therefore imaginary.
For the fame reafon, in the equation yy zz itix — xx, the abfcifs x cannot be
taken negative ; for, taking x negative, the term 2ax would be negative, and
therefore the equation j)^ = — zax •— xx, that is ^ i= v/_ zax — xxt would
be an imaginary quantity. «
120. The primary property of the Apollonian Parabola is this, that the fquare Thefimpleft
of any ordinate whatever is equal to the reflangle of the parameter into the ^o<^^ to the
abfcifs i taken on the axis if the angle of the co-ordinates be a right angle, or P^^^M* *^°^'
on a diameter if that angle be oblique. Then,
making the parameter zr a, any abfcifs AB — x,
the correfponding pofitive ordinate BC — jr, and
the negative BD = — y\ then yy will be the -
fquare as well of BC as of BD, and ax will be
the redangle of the parameter into AB. Where-
fore yy =z ax h the moft fimple equation which
belongs to the parabola with the parameter a.
And here it is plain, that the abfcifs x cannot be
taken negative, becaufe of the avoiding imagi-
nary quantities. And here alfo, by the quan-
tity tf, which exprefles the parameter, is to be
underftood any given quantity, into which the
abfcifs X is multiplied ; fo that f— = — f = yy
Fig. 46.
K
will be a parabola, the parameter of which is =r —
And xs/ ab zz yy
will be a parabola, the parameter of which is V«^, And the like of all
others.
Fig, 47«
\
D
P
c>
y
-h
J
A
L.
B
If the parabola fhould be differently
, / placed, as in Fig. 47, and on the fame line
^ AB, from the given point A, we fhould
take the abfcifles, or x-, the equation would
be XX zz ay, in which we may take the
abfcifs either pofitive or negative, but the
ordinates muft always be pofitive.
O
121. Let
98
ANALYTICAL INSTITUTIONS.
BOOK r.
The fimpleft pig^ .g.
loci to the
hyperbola
conftrutled.
A
121. Let the oppofite hyperbolas be re-
ferred to their axis, or to a dian:eter, ac-
il ■/[ cording as the angle of the co-ordinates is
either right or obhque ; and let CB be the
axis, or the tranfverfe diameter, and HE the
conjugate. By the known property of the
hyperbola, taking D any point whalever,
and drawing DM parallel to HE, the red-
Tx \i angle CD X DB muft be to the fquare of
^* DM, as the fquare of CB is to the fquare of
HE. Then, making CB = ^a, HE i= 2h,
and from-the centre A taking any line AD — x, DM pofitive zz y, DM ne-
gative = — J, it will be CD z=^ a + x, BY} =i x — a, and therefore, by the
^aa . /^bhy that is, xx — aa =: ^~- . And,
and the ordinates as before, it will be Bi iz
faid property, xx -^ aa * yy
taking Ad negative n: ■
— X ■\- a, Qd zz — X
— a, and
the reftangle
Biaf X ^C = at;; — aa,
^^ zz XX — aa\ the moft fimple
bo *-
Whence, in the fame manner, we fhall have
equation expreffing the two entire oppofite hyperbolas referred to their axes or
diameters, taking the abfciffes from the centre. And, if we fhall take the
abfciffes from the vertex C, we iliall have the analogy (by the faid property)
\aa . /i^bb ; that is, the equation — lax + xx zzz
aayy
~lb~
xy, X - 2a . yy
** ______
And laftly, taking the abfciffes from the vertex B, we (hall have a* X aa + ■*
yy
^aa . ^bb ; and therefore the equation lax + xx ■=. -jp .
It is alfo a primary property of the oppofite hyperbolas, that the fame red-
angle CD X DB, taking the abfciffes pofitive, and Bd x dC, taking the
abfciffes negative, is to the fquare of the ordinate, whether pofitive or negative,
as the axis or tranfverfe diameter is to the parameter. Making, therefore, tlxe
parameter zz p, and other things as before, it will be xx — aa ,yy W za ..p ;
that is, -^ zz XX — aa i^ the moft fimple equation expreffing the two oppofite
hyperbolas as referred to a parameter, and taking the abfciffes from the centre.
Now, taking the abfcifs from the vertex C, the equation will be — ^ = xx
— 2ax ; and laftly, taking the abfcifs from the vertex B, the equation will be
2ax -i- XX zz ^^ .
P
If
SECT. III.
ANALYTICAI. INSTITUTIONS.
99
If the hyperbolas be equilateral, becaufe, in this cafe, the two axes or dia-
meters are equal to each other, and equal to the parameter, each equation will
become yy zz xx — aa, taking the abfcifs from the centre ; or yy ~ %ax + xx^
taking the abfcifs from the vertex B j or j^y = — ^ax + xxy taking the abfcifs
from the vertex C. By the quantity aa is to be underftood any plane however
complicated, as alfo by the quantity bh. And by 2^, as alfo by /», is under-
ftood any line whatever. So that, in the equation t/h^^ "" ^'^ "" ^^'-fft
we (hall have s/aa-^ff for the femiaxis, or tranfverfe femidiameter, and 2^/ aa-{-ff
will be the whole axis or diameter. As alfo, \/b^ab'\s the femiaxis or femi-
diameter conjugate, and ^\/h^ab is the whole. In the equation ^~- = xx
^, it will be V — , the femiaxis or tranfverfe femidiameter, and b the
conjugate. In the equation ax — hx ■=. ■ f-^ ■ , it will be b the femiaxis or
tranfverfe femidiameter, and c-^-m the parameter. In the equation ^^^ '^~ =
XX — aa + bb, it will be 2\^aa — tb the axis or tranfverfe diameter, and.
a — b the parameter. And fo on.
Fig. 49.
If the oppofite hyperbolas (hall be differ-
ently fituated, as in Fig- 49, and upon the
fame diameter CB equal to 2a, produced, if
you would have the x*s pofitive, and negative
from the centre A, (it being HE zz zb,) the
equation would be yy "^hh '=. _^ .
^
P
/
711
E
M
3
C
A. B
T)
11
/T^l
^'^^
M
X
P
\
Bg* 50.
122. In the hyperbola between the The fimp'cft
afymptotes, the redtangle of any line AB ^^'^^' o^ t|^e
taken on the afymptote ^B, into the ordi- between It's
nate EC parallel to the afymptote MN, or afymptotes
Kd X ^C, is always conflant, that is, equal conftrudledt
to a known redtangle. Therefore, making
AB = Xy BC = y, and the known red-
angle zz ab^ it will be xy zz ab ; and,
taking Ad negative rz — ;f, and dC ne-
gative = — J, the redangle Kd x dC fliall
O 2 be
100
ANALYTICAL INSTITUTIONS.
BOOK I.
be alfo xy ; and therefore xy zz. ah is the moft fimple equation belonging to
the oppofite hyperbolas between the afymptotes. It is plain, that the equation
— a;)' = ab-, or xy =t: — ab^ will ferve for the oppofite hyperbolas in the angles
BAM, MN, one of the co-ordinates being always pofitive, and the other ne-
gative, and therefore the produ6l is negative.
The fimpleft Fig* 5 1
loci to the
cllipfis con-
llruded.
123. In the ellipfis CEBH, taking from
the centre A any line AD upon the axis or
tranfverfe diameter CB, and drawing DM
parallel to the axis or conjugate diameter
EH ; by the known property of the ellipfis,
the reflangle CD x DB muft be to the
fquare of DM, as the fquare of the axis or
tranfverfe diameter CB is to the fquare of
the conjugate HE. Therefore, making CB
zz ia, HE zr 2^, and from the centre A
taking any line AD =z x, and making DM pofitive =: y, DM negative = — ^;
it will be CD zz. a •\' x^ DB •=. a — x, and therefore aa — xx ,yy ;; /i^aa , ^bb-.
aayy
that is, -^ —
bo
XX*
And taking Ad negative zz — x, and the ordinates
as before, it will be Bi = BA + A^ = ^ — x, dC zz AC — Ad — a + x,
and therefore the redangle B^ X dC ihall be alfo zz aa — xx» Whence, in
the fame manner, we (hall \\2LWt aa — xx ■=: ^^, the moft fimple equation to
the ellipfis, taking the abfciffes from the centre. And if we fhould take the
abfcifles from the vertex C, we Ihould have the analogy lax — xx .yy :;
j^aa . ^h ; and therefore the equation '^'^
hb
2 ax
XX.
It is alfo a known property of the ellipfis, that the fame redangles are to the
fquares of the correfpondent ordinates, as the axis or tranfverfe diameter is to
the parameter. Therefore, calling this parameter p, and every thing conti-
nuing as before, it will be aa — xx ,yy :: la , p. Therefore it is — ^ m aa
— XX, the moft fimple equation of the ellipfis referred to it's parameter, taking
the abfciffes from the centre. And, taking the abfciffes from the vertex C, the
equation of the ellipfis referred to it's parameter will be
P
•— J^QtvQ "^"^ vs,yQ%
If the two axes fhall be equal to each other, in which cafe they are alfo equal
to the parameter, both of the equations will become yy zz: aa — xx, taking the
abfciffes from the centre ; and lax '. — xx zz yy, taking 'the abfciffes from the
point C. But, if we confine it to an axis in which the angle of the co-ordi-
nates is a right angle, the ellipfis will degenerate into a circle with radius = a,
7 The
SECT. Ill, ANALYTICAL INSTITUTIONS. lOI
The obfervation made in the hyperbola, concerning the given quantifies aa^
hh, 2<7, />, in refpe<5l to the diameters and parameter, is to be underftood equally
of the ellipfis, to fave needlefs repetitions.
124. Now, in equations belonging to the hyperbola and the ellipfis, as re- in thefe ^«
ferred to the axis or diameters, taking the abfcifs from the centre ; as the diameters
may be found,
aayy aayy if not given.
-ff- zz XX '•^ aa. -77- :=. aa —• xXf ^
Ob bb
the fquare-root of the conftant term, or of aa, will always be the tranfverfe
femiaxis or femidiameter. And if the co-efficient of the fquare of the ordinate
be the fame conftant term divided by any given quantity, the root of this divifor
is always the conjugate (emiaxis or femidiameter, that is, the root of bb» But
if this co-efficient be not fuch, or do not contain the conftant term after this
manner, then the femiaxis or conjugate femidiameter will be different. Thus,
for example, in the equation •'-^ = xx — auj the fsmiaxis, or half the tranf-
verfe diameter, is indeed always a, but h is not the conjugate. To find this-we
muft make an analogy : As the numerator of the co-efficient of the fquare of the
ordinate is to it*s denominator, fo is the conftant term to a fourth, the root of
which will be the femiaxis or femidiameter required. Then, in equations to
the ellipfis or hyperbola referred to the axis or diameter, taking the abfcifs
from the vertex, as in —^ = lax — • xx, ^~- := xx -^ lax^ -—- zizxx +2ax,
the tranfverfe femiaxis or femidiameter fhall be half of that quantity, which
multiplies the unknown quantity in it's firft dimenfion, and the conjugate as
before. Obferving, that when the co-efficient of the fquare of the ordinate is
not the fquare of the axis or tranfverfe diameter thus found, the analogy for the
femiaxis or conjugate femidiameter will be thus: As the numerator of the co-ef-
ficient of the fquare of the ordinate is to the denominator, fo the fquare of half ^
the quantity that multiplies the unknown quantity of the firft dimenfion, is to a
fourth ; and the fquare-root of this fourth proportional fhall be the conjugate
femiaxis or femidiameter.
Therefore, in the equation to the hyperbola "^ = xx — aa, the tranfverfe
femiaxis or femidiameter will be = J, and the conjugate = -j- . And fince,
by the property of the curve, it ought to be : As the redangle of the fum into
the difference, (of the tranfverfe femiaxis or femidiameter and the abfcifs,) is
to the fquare of the ordinate, fo is the fquare of the axis or tranfverfe diameter
to the fquare of the conjugate; it will be xx — aa .yy :: ^aa , -7^ t or
^^ X //=■ XX — aa^ that is, -^^ zz xx -^ aa^ which is the propofed equation.
'^'^ Thus,
102 ' ANALYTICAL INSTITUTIONS. BOOK I.
Thus, in the equation ^-^ z= xx — aa, the tranfverfe femiaxis or femi-
diameter — ^, and the conjugate r: V^. In the equation x)i — tax ~ -^,
the tranfverfe femiaxis or femidiameter ~ a, and the conjugate =: -j-y/cm. In
the equation ''■^-^ — yy = xx — hh, the tranfverfe femiaxis or femidiameter
will be z: b. and the coniueate iz \/ — ~ , &c.
To find the 125. If the equations be referred to parameters, as -^ = aa — xx^ or
loci when re- P
ferred to a f^ = ^;j; — /?<? taking the abfciflcs from the centre : or ^^ = lax — xx. or
parameter. / •' o P
^^ r= xax + ^^, or -^ zz XX — lax, taking; the abfciffes from the vertex i
p f p i ^
in the firft, the tranfverfe femiaxis or femidiameter vi'ill always be the root of the
conftant term ; and in the fecond, the half of the co-efficient of the unknown
quantity of the firft dimenfion ; and the parameter will always be the quantity
of the denominator of the co-efficient of the fquare of the ordinate, when the
numerator of the fame co-efficient in the firft is double to the root of the con-
ftant term ; and in the fecond, is equal to the quantity which multiplies the
unknown quantity of the firfl dimenfion. But when the faid denominator has
not the afore-mentioned conditions, the parameter fliall be the fourth propor-
tional to the numerator, the denominator, and the axis or tranfverfe diameter.
Therefore, in the equation to the ellipfis ^aa — xx zz: — , the axis or tranf-
verfe diameter fhall be z= 2a, and the parameter = ^ . And, fince it ought
to be, by the property of the ellipfis, as the redangle of the fum into the dif-
ference of the femiaxis or tranfverfe femidiameter and the abfcifs, is to the
fquare of the ordinate, fo the axis or tranfverfe diameter is to the parameter 5
it will be aa -^ XX .yy :: 2a • -j-, that is, ~ = j« — xx, which is the
equation propofed. In the equation xx — aa z: — , which is to the hyper-
bola, the axis or tranfverfe diameter = 2a » the parameter zz — . In the
b — c
equation to the hyperbola 2ax + xx =z yy, the axis or tranfverfe diameter
will be 2a, and the parameter -r-^ . In the equation to the ellipfis aa — bb
— XX
SECT. III. ANALYTICAL INSTITUTIONS, IO3
,^ XX :i: — , the axis or tranrverfe diameter will be = z\/aa - bb^ and the
c
parameter = '^^'~ ; fuppoHng a to be greater than b, for otherwife the
curve would be imaginary.
126, Thefe things being premifed, and well iinderftood, the conftruflion of the ^« to
more complicate equations, or of all other Loa to the conic fed ions, will be the conic
very eafy ; and that by reducins; fuch complicate equations to the fimple primary ^^^'""^ ^if-
•'.•'. . -1 • 1 01 1 J r • • r r 1 • r rs- , ■ triDuted into
equations here exhibited. So that, the delcription ot inch a conic leaion being three fpecies*
fuppofed, we may proceed to the conltruftion of the propofed equation.
Now, to proceed with the greater perfpicuity, I fliall diftribute all equations
to the conic fedions into three fpecies or claffes, I mean all complicate ones.
Thofe of the firft clafs fhall be all fuch as contain the fquare of only one of the
unknown quantities, and the redangle of the other unknown quantity into a
conftant quantity. As, for example, ax ■±_ ab ■=■ yy. And moreover, all thofe
Ihall be faid to be of the firO: fpecies, which contain rectangles of the unknown
quantities one among another, and with conftant quantities, but have not the
fquare of either of the unknown quantities. h% xy -\- ax ■=: aa — ay ,• the
figns being of any kind, which is alio to be underftood of the figns of the other
two fpecies.
Of the fecond fpecies I call thofe, in which there are the fquares of one or
both the unknown quantities, and alfo their redangles into conftant quantities,
but not their reclangle into each other ; as ^a; 4- lax zz ay •{■ by^ or xx — ibx
"zz yy -\- ay — ax^
Thofe are of the third fpecies, in which are contained redangles of the two
unknown quantities into each other, and other terms of what kind foever ; fuch
as XX + ixy + 2yy = aa — xx -^ bx,
127. To diftinguiQi and conftrud equations of the firft fpecies, there isLodoFtUe
occafion to make ufe of one fubftitution, which is, to put the unknown quantity firft fpecies
which has no fquare, plus or minus (according to the figns), a conftant quantity, conftru<^ed,
equal to fome new unknown quantity ; and thus to reduce the equation, (re-^^J,^ i^^JJ'
peating this fubftitution if there be occafion,) to a more fimple expreflion, fo
that the locus of the faid equation may be eafily known and conftruded ; as
may be feen in the following Examples*
EX-
104
ANALYTICAL INSTITUTIONS.
BOOK I.
EXAMPLE I.
Let the equation be ax + ah zz yy, and let the angle be given, which the
co-ordinates make with each other. Becaufe ax + ab is tlie fame as « X x+b,
make x + b — ^zi then, by fubftitution, it will be az zz. yy, which is the
Apollonian parabola.
Fig. 52
On the indefinite line AB as a diameter,
with a parameter = ^, let the parabola CAC
be defcribed, whofe co-ordinates AB, BC,
contain the given angle ; then let AD zr b.
Taking any line AB z= 2, it will be BC = y.
But, becaufe, by the fubftitution, we have
X zz z — ^, DB will be x. Therefore the
origin of the abfcifs x will be the point D,
taking the pofitive towards M, the negative
towards A, and the correfponding politive
and negative ordinates will be y.
If the propofed equation had been put ax — ab = yy, we (hould have made
the fubftitution x — ^ = 2, and therefore x zz z -\- b. In which cafe, taking
AE = ^ in the diameter produced, and doing the reft as before, the point E
would then have been the origin of the abfcifs x.
EXAMPLE n.
Let the equation be xy -\- ax -^z aa '^ ay. Make y + a=: z, and, inftead
of y, fubftituting this value z — a, we fliall have zx + az zz 2aa ; and
making another fubftitution oi x -^ a =: p, it will be pz = laa, the Jpoilonian
hyperbola between the afymptotes.
F^i' 53
Let the indefinite right lines MM, FF,
comprehend the given angle of the co-or-
dinates, and between the afymptotes MM,
FF, let the two oppofite hyperbolas be de-
fcribed, belonging to the conftant redangle
2aa. Taking any line AC = p, and the
ordinate CE parallel to AM, it will be = z.
But, by the fubftitution, it is x = p — a ;
therefore, making ABz:^, it will be BCzr;;.
And,
Sect. iTi, analytical institutions. 105
And, becaufe we have alfo, by the other fubflitution, jy r= 2: — a, makin*
AN = z7, and drawing NFI parallel to FF, it will be DKz=zy. Therefore,
drawing BQ_ parallel to AN, Q^will be the beginning of the abicifs x. Thus,
to any abfcifs QD — x will correfpond the ordinate DE ■= y, pofitive between
the points Q__and P, and neo;ative beyond the point P, as HI. Bur, when p
is taken lefs than ^, that is, AC lefs than AB, then, as it is kV = /> — a, x wiR
be negative, that is, towards N ; and to it will correfpond the pofitive ordi-
nates 7. Now, if we take p negative, and equal to AU for example, a? will
be negative, and equal to QO, and y negative =: OE. If the equation were
xy -{• ax zz aa -}- ay, or eh", xy 4- ax =: — aa — ay, or this, xy — ax zz an
— ay, or this, xy — ax zz — aa -{- ay ; the two firft would be diviiible by
y -f- a, and we fiiould have x zz ^ a. The two others would be divifible by
y — a, and we (hould have x zz ±: a. Therefore they would not be loci, but
equations of determinate problems. But if it were xy — ax zz aa -\- ay, the
firft fubll:itution would be jy — ^ — z, whence the equation z.v — a% zz. laa ;
and confequently the fecond fubftitution would be .v — a zz p ; whence finally
tiie equation zp zz laa ; and therefore, in this cafe, to the co-ordinates p, z,
muft be added the quantity a, in order to have x and y. And therefore,
taking from A towards U the line AR zz a, and drawing RG parallel to
MN and equal to a, then, through the point G drawing GT parallel to
FF, G fhali be the origin of the abfcilTes .v, and the correfpond ing ordinates
ihall be y.
If the equation were xy -^^ ax = — aa + ay^ the fubftitutions would be
^ •\' a zz Zy and x — a zz p, which would give us the equation pz =■ — 2aa.
Let the fame hyperbolas be defcribed, but in the other two angles, becaufe
the conftant redangle laa is negative ; and let them be />, ie. Producing GR
to L, this will be the origin of x both affirmative and negative. And upon
the right line LQ>^ produced both ways, the ordinates y will infill, that is,
negative from N towards H, and pofitive from N to the point / ; and again
negative beyond the point /.
If it were xy ^^ ax zz m^ aa -^ ay, the fubftitutions would be y —^ a zz z,
and X -^ a zz p. Therefore, the fame hyperbolas ie being defcribed, and QB
being produced to q, this will be the origin of the abfciffes x, and the ordinates
J will infift upon TT.
-If, in the equations, the term xy fhould be negative, it may be made pofitive
by tranfpofing the terms.
The diverfity of fubftitutions, and of the pofition of the co-ordinates, which
arifes from the different combinations of the figns in the propofed equations,
and whatever elfe has been confidered here, is to be fupplied in what follows,
where, for brevity-fake, I fhall omit it.
Hitherto I have fuppofed, that the conftant quantities of the equation are
, Itichj as may make room for the aforefaid fubftitutions. If they Ihould not be
P fuch»
io6
ANALYTICAL INSTITUTIONS.
BOOK I,
fiich, as, for example, if the equation were aa — hx zz yy, we mufl: make
aa ~ hc^ and then we fhali have he — hx zz yy^ and the fubftitution to be
made would be that of <: — - a? equal to a new unknown quantity. Thus, if it
were ^ — + ex zr yy^ we mud make bh :zz ef, whence the equation — + ex
= yy. And then we muft put -^ — h ^ equal to fome new unknown quantity.
If it were
anx — hhx •\- m^
7T1
— yy, we might make aa — bh zz ee, and
And the like of others.
CC2C "I" CC t
m^ zz ecf. and then it would be —~ ~ yy
•' a ■\- b "^"^
Loclolxkis: 128. To reduce and con{lru<^ equations of the fecond fpecies ; let all the
fecondfpecies terms which contain the fame unknown quantity be put in order on one fide of
conftrudted. ^^^ ^^^^ ^f eqyjjji^y^ j^u^j on the other fide all the other terms in order iikewife ;
and in the jfirit member of the equation let the fquare of the unknown quantity-
be pofitive, and free from co- efficients and fradions. To the fame firft mem-
ber, (and to the fecond alfo, to preferve the equality,) muft be added the fquare
of half the co- efficient of the fecond term, if it be neceffary, fo as the firft
member may be a fquare. Then put the root of that fquare equal to a new
unknown quantity j which operation muft be performed in the fecond member
alfo, if it require it. This will give us an equation reduced to the fimpleft
terms, or to an equation of the firft fpecies.
EXAMPLE III.
Let the equation be xx + xax zz ay + by. Add the fquare aa on each fide,
and it will be xx + lax -\- aa zz. aa -{- ay + by. And now, making x -\' a
— z, we (ball have zz zz aa -^ ay + by, which is now reduced to the firft
fpecies. Then, making a + b zz e, and aa zz ef, it will be cf + cy zz zz i.
and putting f + y zz p, it will be zz z= ep, an equation to the Apollonian
parabola.
With parameter c zz a + h, on the di-
ameter AB, and with the co-ordinates in a
given angle, let the parabola CAC be de-
fcribed. Then, taking any abfcifs AB zip,
and BC fhall be z, either pofitive or nega-
tive. And, becaufe y'Tzp-^f-zip ^ ^
taking AD z: -^ , it will be DB = y.
And, becaufe of the fubftitution a; + ^=2;^
from
Fig' 54
O^
SECT. III.
ANALYTICAL INSTITUTIONS.
107
from the point D draw DH =z a parallel to BC, which will be terminated by
the parabola in H, (as will eafily be leen by fubftituting, inftead of p in the
reduced equation zz zz cp, the value of AD z: — -^ zz ^: for it will become
ZZ r= aa, and therefore DH zz z z::^ a^) and drawing through the point H the
line OE parallel to the diameter, it will be HE = DB zz p --■ — y, and
confequently EC =2 — « = ^ pofitive, and negative alfo when the abfciflfes
are pofitive. And to the negative abfcifl'es, that is, taking them from H
towards O, both the negative ordinates will correfpond.
EXAMPLE IV.
Let the equation be xx + 2.bx z=.yy ^ ay. Let there be added the fquare
of half the co-efficient of the fecond term, that is hb -, then it will be xx + ihx
■^r bb z^ yy — ciy -^ bb. And making .v + ^ ~ 2, we (hall have zz =: yy
— ay -\- bb, that is, zz — bb zz. yy — ay. And adding the Iquare of \a, it
will be zz — bb + \aa •=■ yy — ay + \aa. Then make y — f <? = />, and
h will be zz — bb + \aa zz pp. And fuppofing bb greater than ^aa, and
making bb — ^aa =s mtrit it will be zz — mtn = />/>, an equilateral hyperbola
with the femidiameters zz w, and taking the abfcifles from the centre.
^^i' 55' In the indefinite line BD I take BG
zz 2m zz 1\/ bb — iaa, and divide it
equally in A. With centre A, the tranf-
verfe diameter s= 2 AG, equal to the
conjugate, and with the co-ordinates in a
given angle, defcribe the two oppofite
and equilateral hyperbolas. Taking any
abfcifs pofitive and negative AD = 2;,
the correfponding ordinates DH will be p,
pofitive and negative. And becaufe, by
the fubftitution, it is x zz z — b, taking
AE = b, it will be ED zz x. But, by
the other fubftitution, it being yzzp+j;a,
from the point E drawing EO zz fa, parallel to the ordinate, which will
terminate at the curve in the point O j and through that point O draw the
indefinite line KK parallel to the diameter BG, it will be KH z=: p + ^a :;: q.
Therefore the point O will be the origin of the abfcifs x on the right line KK,
to which, taken politively, will correfpond the two ordinates y, one pofitive and
the other negative. And taking it negative, but not greater than EG, two
... - P 2 pofitive
I 11^ A G/ P.
JI\
!08 ANALYTICAL INSTITUTIONS. 'boOKI,
pofitive ordinates will correfpond to it ; but taking it neg-itive and greater
than EG, but lefs than EB, the ordinates y will be imaginary ; and taking
it negative greater than EB, and Jefs than I{I, making BI ~ GE, the two
©rdinates will be pcfitive ; and laftl)'', one of the ordinares will be pofitive,
and the other negative, when the abfcifTes, being negative, fliall be greater
than EI.
Here it fliould be obferved, that the root of the fquare yy — • ay 4- ^cta is
not only ^ — -\a, but alfo \a — jy, and therefore the fubftitutions fliould be
two, that is, both y — \a ~ />, and \a — y ~ ^. " Yet, notwithftanding, in
the prefent example, and in others that follow, I only make ufe of the firft.
For, conlidering, in thefe conftruftions, the new unknown quantity p is to be
iindcrftood both as pofitive and negative, herein will be comprehended thofe
determinations alfo, which the other fubftitution would fiipply, and whicli
therefore would be fuperfiuous here.
If the quantity hh, which I have fuppofed greater than ^aa, fliould, on the
contrary, be lefs, the locus would be to the fame hyperbolas, only by changing
the places of the coordinates and of the conftant quantities. 7 hat is, the
final equation would be zz z=. pp — mjUj the conftrucflion of which is here
omitted, becaufe it is not different from the foregoing, only that the femidia-"
meters here are each equal to s/ Xaa — hi — m. Now, if it were bb =
the locus would degenerate into a right line, as is plain.
aa.
Zcijof the 129. To diftinguifh and conftrud equations of the third fpecies, it is ne?
^'""■^ ^^'^'P ceflary that, putting the fquare of one of the unknown quantities made pofitive,
cou rii e , ^^^ £^^^ from fradtions and co-efficients, together with the redangle of the fame,
on one fide of the mark of equality, and on the other fide all the remaining
terms ; adding to the firft member (and confequently to the fecond alfo) fuch a
fradion of the other unknown quantity, that the firft member may be a fquare 5
then putting it's root equal to a new unknown quantity, and making the fub-
flitution ; by means of which an equation may be had, reduced to a more
iimple exprefl[ion, or to one of the two fpecies before- mentioned.
Thus, in this equation, for ex^mple,^ zz — -^-^ = ay^ adding -^ to both
members, the firft mqmber, will be a fquare, the root of which is ^ — ^ — ,
which is to be put equal to a new unknown quantity p -, and, making the
fubftitution, the equation, will be />/> =: — ^ 4- ay^ which is now reduced tQ
tliq fecond fpecies,
1301 But
SECT. III. ANALYTICAL INSTITUTIONS. IO9
130. But It may be obferved, that fonietimes the new unknown quanliry to Complicate
be introd'jced Ihould be affeded by fome conftant co-efficient, otherwife the ^'^'■* "^ ^"^
conflruiftions v.-ould be much incumbered. For example, in the equation dtfceTto^'
icx ± ^-^ 4- -^ — ±Lfy ± h^, the firfl member of which, without any fj-bfti^u/on.
, with ex-
addition, is already a fquare, whofe root is a? + -^ \ if the term hx were not^"^P^'^s»
there, or being there, if we would eliminate x out of the equation, we might
do it, by putting, inftead of x^ it*s value obtained by the fubftitution, fo that
it may be exprelfed by the new unknown quantity, and by y with conftant
quantities; therefore the fubftitution of ;c + — ~ ^ fhould be made.
But if the term fy were not there, or being there, if we would eliminate 7,
w.e muft make a fubftitution of x + -^ — — ^ . And thus,. refpe6lively, if the
equation were yy ± ^-^ H ^ = ±.^6' ± ^^i the term fy not being there,., .
or elfe to be eliminated, a fubftitution muft be made of j it -^ zr 2; 3 or the
term hx not being there, or being to be ehminated, a fubftitution of j + -^
ZL — is to be made..
a-
In general, the redlangle of conftant quantities into that unknown quantity
by which the. equation is ordered, not being in the equation; or being there,
if we would eliminate that unknown quantity, we muft put the root of the firft
member equal to a new unknown quantity. But if the redangle of conftant
quantities into the other unknown quantity, by which the equation is not
ordered, be not in the equation, or if, being there, we would eliminate that
unknown quantity, we muft put the root of the firfh member equal to a new
unknown quantity, multiplied into half the conftant co-efficient of the (econd;
term of the firft member.
EXAMPLE V.
Let the equation be yy 4- ^-^ + — ^ =: ex. Make y -{ ^ zz a, and the
ihxy
a ' aa
equation will be zz = cx^ which is to the Apollonian parabola. If the angle of
the co-ordinates x^y, of the propofed equation be not given, but. left at pleafurc,
8 ^he
110
ANALYTICAL INSTITUTIONS.
BOOK I,
the conftruftion of the locus would be mani-
feft. For, on the indefinite right line AB
defcribing the ifofceles triangle ACD, with
the bafe CD zz b, and the fides AC ^ AD
= a ; and on the diameter AB, with a pa-
rameter =: Cf and with ordinates parallel to
DC defcribing the parabola of the reduced
equation zz r=L cx\ taking any abfcifs at
pleafure AB = x^ it would be BM = z.
But, by the fimilar triangles ADC, ABE,
we fliall have EB = — , and, by the fub-
ftitution, it is y zz z ^ — EM. and
alfo AE zz AB rn x. Therefore, upon the indefinite line AE taking any
abfcifs AE = x, the correfponding ordinate EM, pofitive or negative, will be
the y of the propofed equation. But, becaufe the angle of the co-ordinates
X and y is fuppofed to be given, the conftruftion aforegoing will not obtain,
but we may proceed thus. On the indefinite line AB let a triangle ACP be
defcribed, having the angle ACP equal to the fupplement of the given angle,
which the co-ordinates of the propofed equation ought to make j and let AC=<?,
CP ~ b. Produce AC indefinitely, and, taking any line AE = Xy make KK
parallel to PC, and it will be EH =z —. Whence, if HK zz z, it will be EK=^;
and then AE, EK, are the co-ordinates of the propofed equation, and in the angle
given. But HK cannot be yet the z of the reduced equation ex zz zz, fince the
abfcifles AH are not yet equal to the x's^ nor yet the lines AE. Obferve, there-
AP X *■ fv
fore, that AH will be := , that is, zz -^^ , (making AP = /, becaufe,
in the triangle ACP, having given the fides AC, CP, and the angle ACP, the
line AP will alfo be given ;) whence the curve thus defcribed, calling AE zz x,
and HK =: z, will give us the equation — zz zzj which would be exadlly
our equation reduced, if, inftead of the parameter c, we had defcribed the
curve with the parameter
/
Therefore, to conftrud the propofed locus, on
the indefinite line AB defcribe the triangle ACP, the fides of which are AC=^,
CP = b, and the angle ACP equal to the fupplement of that angle which the
co-ordinates of the propofed equation ought to make. Then with diameter AB,
parameter = -— , equal to the fourth proportional of AP, of AC, and of the
parameter of the reduced equation, (which is general, whenever the locus is to
o the
SECT. III.
ANALYTICAL INSTITUTIONS.
Ill
the parabola,) and with ordinates parallel to PC, the Apollonian parabola muft be
defcribed. Then taking, on the indefinite line AE, any abfcifs AE n x, EK
pofitive and negative will be zz jy, and the curve will be the locus of the
equation propofed. For it will be HKj- equal to the rectangle of the parameter
into AH, or yy + — ^ 4 —
^f
= cx.
The fame artifice may be made ufe of in other equations, to the hyperbola
and to the ellipfis, in regard to their diameters and parameters, with this differ-
ence only, that in thefe the tranfverfe diameter, or conjugate, according as this
or that ought to be changed, (and it will always be that to vvhich the triangle
ACP belongs,) will be the fourth proportional of AC, AP, and the tranfverfe
or conjugate diameter of the equation reduced. But as to the parameter, when
the equation is given by that, the tranfverfe diameter being varied in the
manner aforegoing, it will be the fourth proportional of AP, AC, and the pa-
rameter of the reduced equation. But if the triangle ACP do not belong tO'
the tranfverfe diameter, but to the conjugate, (the equation being given by the
parameter,) it will be the third proportional of the parameter of the reduced
equation, and of AP; as will eafily be known by the examples.
EXAMPLE VI.
Let the equation given ht yy ■\- ^-^ H ^ zz bx — cc "- icy. Making a-
fubftitution of y + — =2 z, it will be zz zz bx — cc — 2cz H ^, that is
zz + 1CZ -{■ cc ■=. hx +
ihcx
And making again another fubftitution of
z -\r c z=: q, it will be finally qq = ii Xj an equation to the Apollonian
^'i' SI-
parabola. Now, to conflrudt it relatively to
our CO ordinates x, y ; ox\ the indefinite right
line BH let the triangle BCD be conflruded
with it*s fides BD — a, DC = b, and with
an angle BDC equal to the fupplement of
that angle, which ought to be made by the
co-ordinates x, jy, of the equation propofed.
Then let BD, BC, be produced indefinitely,
and from the point B draw BA parallel to
DC, and equal to c. Then from vertex A
to the diameter.AE parallel to BC, and with
the ordinates EP parallel to CD, let the
parabola
112
ANALYTICAL INSTITUTIONS,
lOOK I.
parabola PAP be defcribed, with the parameter = ^ — 7~~' (meaning
by / the known line BC,) and on the indefinite line BF taking any abfcifs
BF = X, it will be BH = AE = ^ , and EP = q, and therefore HP = ^
— c =z Zf and FH = -^ . Then FP = 2; ~ ^ y, pofitive and nega-
tive when ;; is greater than BO ; and both ordinates negative, when it is x lefs
than BO.
In the equation propofed, if the reftangle 2cy (hall be affeded by the affirmative
fign, then the Tecond fubftitution fliould be 2 — ^ = ^, and the parameter of the
parabola equal to i— — if. Then doing the
fame things as before, inflead of drawing BH
above the diameter AE,itfhould bedrawn below
it, and the triangle BDC Ihould be made above
it, as is fliown by Fig. 5 8. Moreover, if the term
2-^ be negative, the firft fubftitution (hould
hx
hey =r z, and thence y
hx
a
Therefore, in this fuppofition, as well in regard
to Fig. 57 as Fig. 58, the triangle BDC (hould
be conltraded below BH, fuppofe as B^C.
Wherefore, taking any hne B/ = a; above B^
produced, it will be /P = jv ; obferving that, in this cafe, the angle B^C
ihould not be made equal to the fupplement, but to the angle itfelf, which is
10 be made by the co-ordinates of the equation.
EXAMPLE VIL
Let the equation be xx 4- ^-^ + -^ = ex ■\- ch. Making; the fubfti-
tution of A? + -^ = ~ , it will be -^ ^ ex ■\- ch\ and making; x ^h =p^
a. a. ' /in o • if
it will be 22; = -^^ , an equation to the Jpllonian parabola. On the inde-
finite
"SECT. HI.
ANALYTICAL IJISTITUTIONS.
II
finite line AC defcribe the triangle APQ^
with the fides AP zz b^ PQ^rr «, and the
angle APQ^ equal to the fupplement of the
angle which fliould be made by the co-ordi-
nates of the propofed equation ; and call the
known line AQ^= /, as ufual. Let AP,
AQ»^be produced indefinitely, take AH — hy
and draw the line HB parallel to PQ^ Fromi
the point B let the indefinite line BD be
drawn parallel to AP ; and with vertex A,
to the diameter AC, with the parameter ~
-j^ J and widi the ordinate CM parallel to
PQ»_ let the parabola MAM be defcribed. Taking any line AE =^, it will
be CM — z\ then HE or BD = .v, and DC zz ~, becaufe of the fimilar
triangles APQ>^BDC. Then is DM = 2 — -^ zz y pofitive and negative,
and the lines BD, DM, are the co-ordinates of the propofed equation.
If the equation had been given atat -}- ^-^ H ^ =. ex — ch^ making the
Fig, S9^
■ — M
5-^
/'
A^^'
'm
Z'-'
U-^
^^^
V
Q iB
C
9/h
^-^oa
lb%K
fame firft fubflitution as in the foregoing equation, we fhould have =i ex
— ch ; and, putting x — h = />, it is zz zz. ^^ , which is the fame as the
firfl:, nor is there any other difference, but only in the firfl: cafe there is a' zr
p — h, and here \t\s x =: p + h. That is, in the prefent cafe the vertex of
the parabola mufl: be at B, and the origin of the abfcifs x muft be in (he point
A, taken on the indefinite line AE.
EXAMPLE VJir.
Let the equation be xx -f ^^l j^^J^L^ ch — ex. Make the fubftitution
of X + — zz -^ , and the equation will be -
z: ch
ex; and purtinp^
h — X zz p, it will be zz z= ^^ , an equation to the parabola.
o.
On
114
ANALYTICAL INSTITUTIONS.
Book i.
Fig. 60. On the indefinite line AH let the triangle
APQ^be defcribed towards H, with the fides
AP — b, PQ^n a, and the angle APQ^equal
10 the fupplement of the angle which the co-
ordinates of the propofed equation ought to
contain. Make the known line AQ^ zz /.
Produce AP, and take AE zr h, and draw EH
parallel to FQ^ With vertex H, on the dia-
meter }^h^ with the ordinates CD parallel to
PQ»_ and with the parameter = ~, let there
be defcribed the Apollonian parabola. Taking
any line EB = py it will be AB zz. b — p =■ x.
Then is BD =2; — ~ = y pofitive and negative,
taking x between the points A and O; and both the ordinates y negative, taking
X beyond the point O, The right line AE being produced indefinitely on the
oppofite fide to the point E, and taking any line ^b — p pofitive and greater
than AE, it will be A^ = ^ — p = a?, a negative quantity ; whence in this cafe
the negative xh will be from A towards e, and the pofitive from A towards E ;
and to the fame negative x will correfpond two ordinates ^D, bT>, equal to /,
one pofitive and the other negative.
If in thefe two laft examples, as in the others which will follow, the reftangle
of the two ordinates be affedted by the fign — , it is done upon the fame confi-
deration as is mentioned at the end of the 6th Example ; which it may fuffice
to have mentioned once for all.
BC = -~ , CD = z.
EXAMPLE IX.
Let the equation be yy
2hxy
+
bbxx
aa
■=. XX -^ aa. Make the fubftitution
bx
oi y Z5 z, and the equation will be zz "=. xx -^ aa, which is to the
Fig, 61. hyperbola. On the indefinite line EE
defcribe the triangle ACH, and make
AC = ^, CH = ^, and the angle ACH
equal to the given angle of the co-ordi-
nates of the equation propofed. Let
AC be produced indefinitely both ways
from the point A. With centre A, and
tranfverfe femidiameter AH =/, with
the conjugate =: a, let there be defcribed
the oppofite hyperbolas with the ordi-
nates
SECT. III.
,ANALYTICAL INSTITITTIONS.
11^
nates parallel to CH. Taking any line AB =r a? pofitlve, it will be BE zr -^.
But ED 5S z: Then is BD =: 2; + -^ z=: y pofitive. And taking in the
hyperbola the ordinate z negative, that is z= EM, then will y be equal to the
difference between EB and EM, that is, equal to BM ; and therefore negative
when X is greater than AO. Then to any pofitive abfcifs greater than AO will
correfpond two ordinates, one pofitive and the other negative^ and both the ordi-
Tiates will be pofitive when x is lefs than AO. But when x is taken negative, that
is on the fide of the point Qj^then it rauft be obferved that QE will be negative;
for the analogy will be, AC (a) . CH {b) :: AQ^(- x) . QE =: — -^.
Therefore, if QE = —, taking z pofitive = ED, it will be 2; + — =
QP = y pofitive ; and taking z negative, it will be — z — =z QM zi y
negative.
EXAMPLE X.
Let the equation he yy ^ ^ + ^^ = bb. Adding — , it will be yy -
2h^
, hl)xx J ,
H = bb
aa
gxx
a
■\ —I and making; the fubftitution of y — — =2;,
aa ° a
it will be 2:2: zz ^ - ^"*
au
— f- bh And putting bh — ag zz mm, it will be
22; zz
mmxx
aa
ntmxx
''F'tgvdl.
+ bb, that is, zz --^ bb = , an equation to the hyperbola.
On the indefinite right line DD let the
triangle ABC be defcribed, with the fides
AB ^ ay BC = b, and the angle ABC
equal to that which is to be contained by
the co-ordinates of the propofed equation;
and make the known line zz f, 1 hrough
the point A draw the indefinite line PP
parallel to BC, and with centre A, tranf^
verfe diameter QC^= 2b, conjugate =
-^ taken in the right line EE, at the ver-
tices Q»^ Q»^ let there be defcribed the two
oppofite hyperbolas HQH. Then taking
0^2 any
ri6' ANALYTICAL I N S T I T U T I O N S, BOOK f^
any line AD = x, and drawing DH parallel to BC, it will be EH zz z = AP,
and DE =
hx.
Then DH
■i -■ zz y, and the lines AD, DH, QinJl
be the co-ordinates of the propofed equation.
EXAMPLE Xr.
Let.tliC equation be yy +
2l'xy hhxx ihxx
a aa a .
+, hb,. Making the fubfti-^-
tution of J -{• — m z, the equation will be 22 zz — ^ + bb, that.isj za.—
bb -
zh,
, which is to the hyperbola^
Fig, 63.
On the indefinite line AD let the triangle
AEP be defcribed, and make AE n a,
EP zz b, and the angle AEP the fupple-
ment of the angle, which is to be con-
tained by the co-ordinates of the pro-
pofed equation. The right line AE
being produced indefinitely both ways,
and calling, as ufual, the known line
AP —/; with centre A, tranfverfe femi-
diametcr AI z: b parallel to PE, and
with parameter = — , defcribe the op-
pofite hyperbolas IC, ic, then taking, any
bx
line AB z= x, it will be BD zr — , and
CD = FA =: 2. Then BG zr z — ~ = ;>. Taking z negative = DG,
it. will be B(G = — z. -\ — — zz — y, and therefore to the fame pofitive x will
belong two ordinates y, one pofitive, the other negative, taking x between the
points A, H. Then taking x between the points H, L, both the ordinates y
will be negative ; and again, one pofitive, the other negative, taking x greater
than AL,
Thea.
5S?CT,-III,
ANALYTICAL INSTITUTIONS.
117
Then taking A^ ~ — x, it will be {i>J) zz 1 , and as it is {dg) iz z,
bx
it will be (i^g) = 2 — = J J and taking 2; negative — {dc), it will be ('3<:)
= — z -\ -111 — y. Therefore to the fame Ah :z: x negative will corre-
a "^
fpond two ordinates _y, one of which is pofitive, the other negative, taking a?
Icfs than ^h \ both the ordinates will be pofitive between the points h and /;
and again, one ordinate will be pofitive, and the other negative, taking a: greater
than A/. And therefore the hyj^erbolas thus defcribed will be the locus of the
propofed equation.
EXAMPLE XII.
Fig. 64.
Ket the equation be yy — .^-^ A — zz.cc — xv + ihx — hh, Makino-
the fubftitution -of y ^ zz z, it will be %z = cc — xx + %hx — hh. And
making another fubftitution of x — h ~ />, it will be finally "zz zz. cc — pp,
which is an equation to an ellipfis, and not to a circle, though it may have the
appearance of -fuch. The realbn of which is, becaufe the co-ordinates f, z, do
not form a right angle, yet however are in an angle to each other, one of them
being AC, the other BT, as may be feen in
the following conftrudion. On the indefi-
finite line EB let a triangle EDF be de-
fcribed, with the fides ED zz ^, DF ::z ^,
and the angle EDF equal to the angle which
is made by the co-ordinates of the propofed
equation ; and making the known line
EF zz /. Produce indefinitely the lines
ED, EF, and taking EP = h, draw the
indefinite line PA parallel to DF, and from
the point A the line AG parallel to EP.
With centre A, tranfverfe diameter MN = — , wiih conjugate diameter RR
equal to 2c and parallel to DF, let the ellipfis MRNR be defcribed; then
taking any line AC =/», it will be EQ^zi x, and therefore BQ^= — . But
BT = 2;j then QT = 2 + ~ =z jy ; then will EQ^ QT, be the co-6rdinates
of the loms required.
E.X-.
iiSl
ANALYTICAL INSTITUTIONS.
BOOK I>
EXAMPLE XIII.
hxy
a
Let the equation be yy -\ — ^ ■{■ xx ■\- cy -{- Ix — ag "=. o. Adding on
bhxx
both fides the fquare — ^, it will be yy -\ ^ + -^
bx
\aa
XX
Ix
— cy + ^^. And making the fubftitution oi y -^ = z, it vviJl be zz =
Ihxx — \aaxx , hex — lalx
^aa
za
Let \aa be greater than hh^ and make — Hif^ rz ^ , and ^^ . ~ /^ ;
4flfl
2a
then adding \cc on each fide, it will be zz •\- cz '\- \cc iz — ^^ + i&;i; -]- ^^
+ -^TTf. And making the fubftitution of z + t^ z= />, it will be /jp iz
+ ^cc. That IS, ^ + -^icc + ag X —■ = xx ;
n
nnhb
nhx
m
nh
and laftly, addine; ^^^^ to both fides, and makins; the fubftitution of x — —
iz q. and of -^cc -\- ag X ~ + ^^ = ee, we (hall have .^CcL ::z ee -— aa^
which is an equation to the ellipfis.
Fig, 65, Upon the indefinite right line AC de-
fcribe the triangle ASF, and make AS =
2^, SF = <^, and the angle ASF equal ta
the fupplement of the angle made by the
co-ordinates of the given equation, and
let the known line AF be called /. On
AS indefinitely produced take AR =
— , and draw the indefinite line RQ^pa-
rallel to FS, and from the point Q^draw
the indefinite line QO parallel to AS, and
make QM = \c.
Then through the point M draw HV parallel to AQ»^ and with centre M,
tranfverfe diameter HV = — , and parameter == ^^, let the ellipfis HNVK
be
SiCT. Ill* ANALYTICAL INSTITUTIONS. Up
be defcribed. And taking any line RD = q, it will be PN — pi and there-
fore AD =: .r, DC zz — , CN =;: 2: ; then DN = 2; — — zz y.
Here it is to be obferved, that if the angle of the co-ordinates fliould be
fuch, as that the angle AFS becomes a right angle, and confequently the angle
MPN is fo too ; then it would be Aaa — hb z=z ff whence — z: ^^^-^^ — zr
~ , and therefore the parameter would be ~^ z: — , that is, equal to the
^aa *■ fn a ^
tranfverfe diameter. Then the angle MPN being alfo right, the ellipiis would
degenerate into a circle with the diameter = ~ •
a
131. As to equations of the hyperbola between the afymptotes, which may General coiv
be required to be conflrudled, they may all be underftood to be comprehended fh"^^°to°thc
in the four examples following. hyperbola
between it's
f \ ^xx , afymptotes j
(I.) ^ •{• xy -=: ab ± mx ± ny. vvith ex-
amples.
(2.) — ^T- '^ xy -zz ab ±: mx + ny ,
iZ') ^-r — xy z=. ab ■±, mx ± ny,
(4.) — ^ '•^ xy = ab ±; mx •±^ ny.
EXAMPLE XIV.
Firft, let the equation ht ^ + xy zzz ab + mx -{- ny, in which I take all
the terms pofitive of the homogeneum comparationis. Making a fubftitution of
^ -\- y zz z, we fhall have zx zz mx -{- nz —'^ + ab ; and, making an-
other fubftitution of 2; — w -f- -^ zz ^, it will hepx zznp + mn -{■ ab -^^^
Ag^in, make a third fubftitution of a? — « = ^, and, finally, it will be pq zz
tf^ + »/» — ^. Suppofmg now that ab + w« — ^ is a pofitive quantity;
9 ®^i
I20
ANALYTICAL INSTITUTIONS.
BOOK 1.
on the indefinite line NN, at the point
A taken at pleafure, defcribe the tri-
angle ABC, the fides of which are
AB =: /?, BC — g, and the angle ABC
•equal to the fupplement of the angle
which the co-ordinates of the equation
propofed ought to make, and nlake
the known line AC zz/. At the point
A raife AD parallel to EC, and equal
.to m — -J- t as in Fig. 66, when
m — ~^- is a pofitive quantity; and
let fall A D, as in Fig. Sy, when
«p- . . . -
m 1- IS a negative quantity, be-
caufe of the fubftitution made of z-^ m
•j- ■— = p. Through D draw the
indefinite line PP parallel to AC, and
on AB produced take AE z: ?;, and
through E draw TT parallel to EC.
Between the afymptotes PP, TT, de-
fcribe the two oppofite hyperbolas RR of
the conftant redangle =z ^^ + »?»—. ^
/
X -y- J that is, a fourth proportional to AB, AC, and the conftant redangle
of the equation reduced. Taking any line EQ^zi q, it will be PM ~ ■—,
and PQ^zi p, and therefore AQ^= q + n := x. But PN = AD zz /» — ^,
therefore NR zz p -{- m
laftly, QR = 2;
«<§• _
-|- iz z ; and becaufe QN z= -^ , it will be,
~- zz y, and the two lines AQ^ QR, will be the co-or-
dinates of the propofed equation. Taking x pofitive, when it is lefs than AE,
V will be negative : when it is greater than AE, and lefs than AO, y will be
pofitive, and when it is greater than AO, y will be negative. Taking x neo-a-
tive, then it will be QN — — ^, a negative quantity ; then j iz 2; — ^
will be = NR -f NQj apd therefore, vvhen x negative is lefs than AO, y will
be negative ; and when it is greater than AO, ;' will be pofitive.
8 But
5^ECT. III. ANALYTICAL INSTITUTIONS. 121
But if the fecond term of the homogeneutn comparationis flioiild be negative,
that is, if the equation were ^ -\- xy -zz ab — mx H- ny ; then the fecond fub.
ftitution would hQ z z=l p — vi — —-, and the equation reduced pq = ab
— mn — ^ • Suppofing then that ab —- mn — ~ were a pofitive quantity,
defcribe, as in Fig. Cy, the hyperbolas RR, but with the conftant redangle
nng
ab — mn — '—■ X -j- > and taking AD iz »; -r -y- , this would be in the
fame manner the Iscus of the propofed equation.
Fig, 68.
If the equation propofed had the laft
term afFeded by the negative fign, that
is, if it were ^~ + xy — ab ■±_ mx — »y,
the third fubftitution to be made would
hQ X + n = q, whereas before it was
X -— n zz. qt and therefore the pofition of
the point A, the origin of Xy would be
changed. Then, in Fig. 68, if the va-
lue of AD be pofitive, and in Fig. Sg^ if
it be negative, the fide BA of the ufual
triangle being produced to E, fo that
AE = n ; between the afymptotes TT,
PP, let the hyperbolas be defcribed of
the conftant redangle belonging to them,
that is, when in the equation the term mx
is affeded by the pofitiye fign, then th-e
conftant redangle =. ab — mn — ^ X
— , and when, on the contrary, it is
affeded by the negative fign, the conftant
redangle will be ^ ab ;+ mn — '-^^j'l
and taking, in the firft cafe, AD — jw 4-
^ , and in the fecond, AD zz -—---mf
h '^
the locus of the propofed equation will be after the fame manner.
Hitherto I have fuppofed, that the conftant redangle of the reduced equation
is a pofitive quantity ; but when it happens to be negative, the conftrudion
R would
1211 A^ALYTIfcAt INSTITUTIONS. BOOKX.
would not be different, only obferve to defcribe the hyperbolas In the other two
angles, relatively to the confiant redtangle, which the reduced equation will
fupply ; taking the line AD pofitive or negative, according to it's value which
the fame equation will give, and the point A either to the right or left of the
afymptote TT, according as the lad term of the homogeneum Ihall be pofitive or
negative, as is clear by Fig. 66, 67, 68, 6^.
The conftant term ab has hitherto been taken for pofitive, but if it were
negative it could make no other aheration, but to make negative the conftanc
reftangle of the reduced equations, which cafe has already been conftruded.
Wherefore the firll of the four equations propofed has now been conftrufted ia
general.
As to the fecond equation of thofe exhibited above, which is — ^^ +
ab ± mx ± ny ; the firft fubftitution to be made is j — ^ = 2, that is,
_. — 9
xy
jK := z -h -^ , and let all the reft be done as before.
Therefore, to obtain the ordinate y, it will be ncceffary to join ~ to z,
whence in each cafe of Fig. 66, 67, 68, 6g, the triangle ABC muft be de-
fcribed under the line NN, as is feen at A^C, with the fides Kb = by bC — g,
and with the angle AbC equal to the angle which ought to be contained by the
co-ordinates of the equation propofed i whence, Ab being produced both ways,
and taking any line A^ zz x, the correfponding line qK will be the ordinate
jf required.
The two laft equations of the four were thefe, but with their figns changed.
-^ ^ + ^ = — ^^ =F w^ T »>
^ ^ xy zz — ab ^i^ mx ^ ny»
But this has been already conftruded in the conftrudion of the firft, and the
other is already conftrufted in the conftrudtion of the fecond ; fo that the four
equations at firft propofed are now conftruded in general, as was required to be
done.
PRO-
SECT. III.
ANALYTICAL INSTITUTIONS-,
1^3
PROBLEM I.
132. The indefinite right line AB is given a geometri-
in pofiiion, and the point F is given out of cal problem,
it ; it is required to find the locus of all the conftrufted
points M, fuch that, drawing trom each of boial"^ ^^'^"
them two right lines, one perpendicular to
AB, the other to the point F, thefe two
lines may always be equal to each other.
Let M be one of the points required, ,
and let the right lines be drawn, MF to the
given point F, and MN perpendicular to
BA. Thefe therefore ought to be equal to
each other by the condition of the Problem;
and therefore, drawing FG perpendicular to AB, and calling it z: a, let MP
be drawn perpendicular to it, and make GP zz x, PM z=. y, it will be PF zz
X '-' a, and therefore FM = \/aa - 2ax + xx + yy. But FM = MN zr GP^
then X = \/aa - 2ax + XX + yy, that is, -va; zz xx ~ 2^*-. 4- aa ■{■ yy, or
^ax — aa ■=-yy. And making the fubftitution of ,;i? — \a z: z, it will be
zax -=■ yy, an equation to the common parabola.
Take GL equal to half GF, and with vertex L, and parameter =r 2^, de-
fcribe the parabola LM. This Ihali be the locus required, in which taking any
line LP = 2, it will be PM zzjy. But GL = \a ; therefore GP = s + ^i
= AT, and therefore GP, PM, will be the co-ordinates of the equation propofed.
It is known from the property of the parabola, that AB is the dire5lriXy and
F the focus of the curve.
PROBLEM n.
Bg. 71
133. The indefinite right line PAPAnotlier,
being given in pofition, and two fixed conftruftcd
points A, D, one in the fame line, and ^^^ ^^f ^J"
*, 1 r • II • • , pcrbola be-
the other out 01 it j the locus is required tween the
of all the points M, fuch that, drawing afymptoies.
the lines MA to the given point A, and
DME from the given point D through
the point M, it may always be AM equal
to the portion ME, comprehended be-
tween the point M, and the point E, in
which the fame line DME meets the
given line PAP.
R 2 From
124
ANALYTICAL INSTITUTIONS,
BOOK 1,
From the given point D, and from the point M, which is fuppofed to be
one of thofe required, draw the lines DB, MP, perpendicular to the given line
PAP. Then the lines AB, BD, will be known, and therefore make AB =z 2^,
BD = ^h, AP = .V, PM - y. Let the right lines AM, DME, be drawn.
Now, by the condition of the Problem, AM =: ME, and it will be alfo PE =
AP =: X, And becaufe of fimilar triangles EBD, EPM, it will be EB . BD :;
EP . PM. And, fubftituiing the analytical values, 2.x -— ia . zb ','. x .y.
Whence the equation xy — ay = bx. Make the fubftitution of a? — a zz. Zy
it will be zy z=z bz -{- ah, or zy — bz -ziz ab. Make another fubftitution of
y — b =z pj and it will be at laft fz rrz ah, an equation to the hyperbola be-
tween the afymptotes.
On the line PAP given in pofition, from the given point A take AL ir a,
and raife LC r: b perpendicular to it. Then through the point C drawing'
the right line RF parallel to PP, between the afymptotes RF, HG, draw the
two oppofite hyperbolas DM, AM, with the redangle ab, which fhall pafs
through the points D, A. Taking any line CK zi 2, it will be KM =: ^.
But AL iz ^, LC = ^ ; therefore AP =1 ^ + 2; ==: ;f, and VM — p + b — y^
(hall be the co-ordinates of the Problemj and the hyperbolas (liall be the locus
required.
PROBLEM III.
A problem Bg. 72,
with three
cafes, con-
ftrufted by
the parabola,
fllipCs, and
hyperbola.
134. Two circles EGF, BNO, being
given, and alfo their centres C, A ; if, from
any point G of the periphery of the circle
EGF, be drawn a tangent GNO, which meets
the other circle BNO in the points N, O ;
and from thefe two points, if we draw two
tangents NM, OM, the locus of all the
points M is required, in which the faid
tangents meet one another.
From the point M, which is one of thofe to be found, let be drawn MP
perpendicular to CA, and from the centre A draw the right line AM. Becaufe
the triangles ANM, AOM, are equal, for the angles at N, O, are right ones,
and the fides AN, NM, are equal to the fides AO, OM, it will be alfo the
angle NMA — OMA ; whence in the triangles NMQ^OMQ*^ becaufe the fide
MQ^is common, and MO ~ MN, it will be QN := QO, and AM perpen-
dicular to NO. From the centre C to the point of contad draw the right line
CO, which will be parallel to AM, it being alfo perpendicular to NO. Make
AB = <7, CE = b, CA = f, AP = x, PM ;= ^, and therefore AM = V'^^^y
. 1 In
SECT. in.
ANALYTICAL INSTITUTION S.
125
In the fimilar triangles AOM, AQO, it will be AM . OA :: QA . AQj and
fubftltuting the analytical values, we (hall find AQ^zz . Draw CH
*<f XX -^yy
perpendicular to MA, produced if need be ; it will be HQ^:;;:: CG, and there-
fore HA = ^ —
"^xx +yy
. But the triangles CAH, AMP, will be fmiilar^
therefore PA . AM :: AFI . AC i that is, x . \/xx + yy :: l> — '^'* . c;
V XX +j'y
and multiplying extremes and means, ex zz h\/ xx + yy — aa^ or ex -\- aa ■=.
hs/ XX +yy. Then fquaring, ccxx -f aaaccx -^ a* — bbxx + bhyy^ that is,
yy +
hb — cc
~~bb
XX —
hb
bb
o.
In this equation there are three cafes that ought to be diftinguiflied ; that is,
when h — d when b is greater than c^ and when c is greater than b,
Firil", let b zn c, then the equation will be yy -^ = o, or yy zz
-^ + -|j- . And finding a redlangle ^bf zz aa, put it inftead of aa in the
laft term of the fecond member, and it will be yy = ^ — ^ i and making
the fubflitution of ;<■+/= 2,, it will be at laft yy zz -j- , an equation to the
/VV 72, ;iVI, Jpollonian parabola. On the right line CA,
towards C take AI =
2b
/, and wuh vertex.
Q I, axis IL, parameter ~ , let the parabola IM
./ be defcribed. This will be the locus required ^
in which, taking any line IP = z, it will be
VM- y; but AI =/, then AP = 2: — / = x,
and the lines AP, PM, will be the co-ordinates
of the Problem.
Secondly, let h be greater than f, which will make the term
hb
bb-
xx to be
pofitive. If we write the equation thus,
thus, XX
hb -
bb
2aacx a*
-jT-ib'-yy'^ ^^
Ti^c ~ TTITo ■" 'ib^c ' ^"'^ adding to both members the
iquare 7==^ , it will be xx — r^ • +
*T-7?1'
a^bb hhyy ,
lb '- cc ' M^^* ~~ WTT^^ "" bb^c'* ^^"^
9
making
m6
ANALYTICAL INSTITUTIONS.
BOOK I.
making the fubflitution o( x —
bb — cc
zz z, it will be finally
ibjy
bb -
a'^bh
,, -—->.; — zz, which is an equation to the ellipfis.
Fig, 74. ^
From the point A towards Y take the
portion Al = ,,^^ ■ , and with centre I,
laab
tranfverfe axis ZY = , , __ ^. , and conju-
gate RT z=
2aa
., let the ellipfis RZTY
'\/ bb —cc
be defcribed, which will be the locus re-
quired. ]n this, taking any Jine IP —
— z, (that is, on the negative fide,)
and it will be PM =: j. But AI —
; therefore AP = z + rr- = x^ and therefore the lines AP, PM,
bb — fic' '" "*" " ' bb—tc
will be the co-ordinates of the Problem.
Laftly, let c be greater than by then the quantity
hi —cc
and therefore the equation is ^^r^ — xx + ^^^ =: yy ~, or xx -|-
U
M
XX will be negative,
if
cc — bb
•~ cc — bb ' "^^^ ^^^ fquare
Cf — ItS)
- on both fides, and the equation will be
zaacx
— ^by
2*bb
""^ ■*" ^rrir + T^W- =■ T^^TTb + r^^' -^^^ making the fubftitution
«*^^ _ bbj/y
of Z = X -{■
, it will be at laft zz —
cc — bb
an hyperbola, when referred to it*s axis.
T^W^ = ^TTm' ^" equation to
On the right line CA, towards the point
C take the portion AI = ■ ''^ ■ , and with
■centre I, tranfverfe axis ZY =
2 a ah
cc — bb
y and
2aa
;, defcribe the oppofite
conjugate zz
Vcc — bb
hyperbolas YM, ZK ; thefe (hall be the locus
required. In which, taking any line IP Z12;,
it will be PM =y. But AI =
- bb
; then
AP
liCT. Ill,
ANALYTICAt INSTITUTIONS.
127
AP = 2
= X, And therefore the lines AP, PM, will be the co-
cc — bb
ordinates of the Problem.
Jn this Problem it is always fuppofed, that the circle EFG is greater than the
circle BNO, or that b is greater than a'y but if it ihould be either b — ay or bLff\
the locus of the points required In the firft cafe would always be a parabola,
in the fecond an ellipfis, and in the third two oppofite. hyperbolas ; fo that
it would be needlefs to diftinguifti thele cafes, which make no variation in
the loci.
PROBLEM I\r.
Fig. 76.
135. Two right lines AC, CB, {V\g. K keus to
76, 77.) are given in pofition on the right t'^'^ 9°"'*^
line AB, which cut one another in C ;• the a°"L a
locus IS required or all the points M, iuch
that, drawing through them a perpendi-
cular PMN to AB, which cuts the line AC
in the point Q»_ and the line BC in the
point N, the fquare of PM may be eqvia:!
to the rectangle PQ^X PN.
Let the right line CD be drawn pa-
rallel to PM ; this will fall either between
the points A, B, as in Fig. 76, or on one
fide of them, as in Fig. 77.
Firft, let it fall between the points A, B,
and make AB = a, AP = «, PQ^= ;f,
PM =_y, PN ■=. z. By the condition of
the Problem, it will be zx =yy. But the
ratio of AP to PQ^is given, which there-
fore may be put as «; to n. Alfo, the ratio
of BP to PN is given, which may be as ^ y
to c. Then it will be PO = a; = -^ , and PN = 2 = 2^1^ . Thefe va^
^^ m b
lues therefore being fubftituted in the equation zx ^==- yy, it will be jjy zr
^—^ — X ~ , or -^ •=. au — «», an equation to an ellipfis with tranfverfe
axis AB. = a, conjugate a\/j- . Such an ellipfis AMB being defcribed,
the uppcx half AMCB will be the locus required.
Now
12S
ANALYTICAL INSTITUTIONS.
BOOK 'I,
Now let the point D (Fig. 77.) fall on one fide of the points A, B, and
make, as above, AB = ^, AP = w, PM =;', PQ^= .r, PN n 2; ; it will be
BP — « — a^ and therefore PN ~
But, by the condition of the
Problem, zx "=■ yy, and x = — , as before. Therefore, making a fubftitii-
tion of the values of z and x. it, will be yy = . X -^» or -^ = uu
* m en
— aUj an equation to the hyperbola.
At the vertex B, with the tranfverfe axis zz ^, and the conjugate axis =:
as/^ i defcribe the hyperbola BCM ; this will be the locus required.
Fig, 78. If the right line AC (hould not fall upon
AB, but Ihould be parallel to it, as it would
be in the pofition aQ^ AB, the right line PQ^
would be given j therefore, making PQzi m,
AB = ^, BP =: «, PN = 2, PM zi y, and
fuppoiing BP . PN :: m » Hi the equation
xz z: yy would become yy zz un. Where-
fore, with vertex B, axis AB, parameter iz »,
defcribe the Apollonian parabola BMC, and
% this would be the locus required in this cafe.
PROBLEM V.
Another. Fig. 79.
136. Let there be a curve AM, the equa-
tion of which is given, and let it*s axis be the
right line AT, out of which let there be a
fixed point F, from whence let be drawn the
right line FM, which cuts the curve in the
point M, and the axis in the point P. Now
the right line FM, moving about the point F,
caufes the whole plane AMP to move parallel
to itfelf upon the line ET, the point P being
fixed in refped: of the point A, but moveable
upon the axis TA, that is, AP being a given
line. In the mean while, the point M will
defcribe a curve CMD. It is required to
know what kind of curve this is, -
Let
SECT. Ill,
ANALYTICAl. INSTITUTIONS.
129
Let the curve be now arrived at the point a of the right line ET ; it will be,
by the conftru<^ion of the Problem, P/> = Ka, and therefore AP = a^> Make
AP "=:■ a, FT = h ; and from the point M letting fall the perpendicular MQ^
,to ET, make TQ^— a;, QM = y, KQjn t, Becaufe of the fimilar triangles
FOM, PMC^ it will be FO . OM :: QM . PQ»_ that is, ^ + j . x ;: y . PQ^
:=. -—- . But PO = <z — / i therefore t^^ z= <«--/, or xy :=i ab — bi +
ay — ty.
Now, in this canonical equation, if we fubftitute the value of / given by y^
and by the known quantities of the equation of the curve AM, we fliall have
the required equation of the curve CMD.
Firft, let AM be a right line. The ratio
of / to ^ will be given, which let be that
of m to n I then / — — . And, fubftitut-
ing this value of / in the canonical equation,
it will be — zz ab — xy
n
locus to the hyperbola between the afym-
ptotes.
To conftru(51: it in the given figure, on FO
take any portion TH, and in a right angle
draw HG fuch, that it may be FH . HG ::
n , m ; draw TG, and upon TA taking the
portion TV = ^" " "^ j from the point V draw VS parallel to TG j and be-
tween the afymptotes VS, VE, defcribe the hyperbola CMD with the conftant
redangle zi ~; (making the known line TG — g.) Then taking any ab-
fcifs TQ^= x^ the correfponding ordinate will be QM zz y, and the hyperbola
hmy
hmy
will be the locus of the equation -^ rz ab — xy =^ + <?y.
n
Fig, 81,
c
--
"^*-->^
/
'\ — -^
-^
T
/^
Q IK
E
r
*rai><<>»
•
/
In the fecond place, let AM be a circle
defcribed with centre P, radius AP zi a.
By the property of the circle, it will be
AQ_^=: t •=! a — s/ aa — yy J and inftead
of / fubftituting this value in the general
equation, it will be xy z: h~^y X ^~aa^yy^
an equation to the conchoid of Nicomedes,
And the curve CMD, which is defcribed
by the interfedion M of the right line
FM with the fuperior arch of the circle
AM, will be the upper conchoid, ET
S will
130
ANALYTICAL INSTITUTIONS.
BOOK I.
will be the afymptote, F the pole. And the curve which is generated by the
interfedion N of the right line FM with the circle under ET, will be the lower
conchoid. This appears evidently from the nature of the conchoid, and from
the condition of the Problem. For the two lines PM, PN, intercepted between
the afymptote and the curve, will always be equal to the radius of the circle AP.
In the third place, let the curve AM be an
Jfollonian parabola, with a parameter AP zz a.
On this hypothefis, it will be / n: ~j and this
value of / being fubfticuted in the canonical
equation, it will be xy '— ay -\- -^ zz. ah -^
~, that is, y'^ + mxy + lyy —■ amy — ahm
— o. This is an equation to two parabolical
conchoids, one of which is defcribed by the
interfe(5tion of the line FM with the fuperior
part of the parabola ; the other by the inter-
feftioh with the inferior parr. And the right line ET will in this cafe be the
afymptote of the curve.
B
'g, 82.
c — -^
"^
Ja-k
.... ,
T
y^^ J'
/
^
--^
F
Z)
-
PROBLEM VI.
Another. Fig. 83.
137. Two equal circles being given,
cutting each other in two points A, N,
and their centres D, B, being given j it
is required to find the locus of all the
points M fuch, that their diflances from
the faid circles may always be equal ta
one another.
Let M be one of the points required ;
then drawing from the centres D, B,
through this point the right lines DM,
BO, then MS, MO, will be the diftances
from the given circles, which ought to be
equal by the condition of the Problem.
T4ierefore make DS =r BO = a, DB — b, and the perpendicular MP being
let fall upon DB produced, make DP = ^, PM =jy; it will be DM =
\/ XX + yyi and SM iz \/^x -\' yy — ^. But BP zr x — ^, therefore BM zz
V'icliriri^-Mr+^j and thence OM zz a — \/ xx - zbx + ^^ + j_y. But it
ought
SECT. HI.
ANALYTICAL INSTITUTIONS.
131
ought to be SM =: MO ; whence we (hall have the equation ^ sx + ^^ — a
zz a — \/xx — 2lx + Ifb •{• yy- By the methods already taught this will be
reduced to xx — l>x -i- ^bb zz aa -^^ — --^,. > and making the fubftitution of
— ^b =. z. it will be zz zz aa —
a,aayy
or
■^'=.aa — 22, which is
an equation to an ellipfis.
Let the right line DB be bifetfted in the point C, and with centre C, tranf-
verfe axis FE = 2^, and conjugate AN -=. \/^aa — bb, let the ellipfis FAEN
be defcribed, which will be the locus required. For, taking any line CP n z,
it will be PM '=^ yi but CD = t^, therefore DP = 2; + f^ iz /v, and there-
fore the lines DP, PM, are the co-ordinates of the Problem propofed.
It would be needlefs to diftinguifli the cafes, in which a is greater, equal to,
or lefs than b, becaufe the Problem will ftill be of the fame nature, b being
always lefs than la, as plainly appears.
It follows from this conftruftion, that the points D, B, will be the foci oi the
ellipfis, and that it's conjugate axis will be terminated at the points, in which
tlie two circles cut each other. And firft, becaufe DS n BO, and SM = MO,
it will be DS + SM + MB, that is, DM + MB = 2DS ; but 2DS = FE,
therefore, by the known property of the ellipfis, the points D, B, will be it's
foci. This fuppofed, by another property of the ellipfis relating to the focit
conceiving the lines BA, BN, to be drawn, it will be BN — BA zz. CE. But
this is verified in the points, in which the two given circles will cut each other ;
for D, B, are their centres, and CE, by conftruflion, is equal to the femidia-
meter of the fame circles. Therefore the ellipfis will pafs through the faid
points of interfedion of the given circles. ci^ e. p.
PROBLEM VII.
Fig, 84.
138. The right line AB being given, to Another,
find the locus o{ fuch points D, that, in the
produced line DA, taking AC half of AD,
and drawing to the point B the right line
CB, this may be tqual to CD.
Let D be one of the points required, from
whence let fall DP perpendicular to AB.
Make AB zz «, AP = a;, PD =: >• ; it will
be AD ~ \/ XX + yyy and, by the condition
of the Problem, AC = ^'v^xxTJy'" where-
S 2 fore
132 ANALtTICAL INSTITUTIONS. BOOK I.
fore CD r= CB = i-x^TTTyy- F""©"! the point C draw CQ^ perpendicular
to BA produced. Now, becaufe of the fimilar triangles AQC, APD, and
AD = 2AC, it will be AP = 2AQ^ and PD = 2QC ; whence CQ^zz iy,
and AQ^z: -^x. Therefore BQ^= a + ^x. Now CBq = CQ^ + BQj zz aa
-\- ax -h -^xx -f. lyy. But CB^ zz CDq = |- X xx + yy ; whence we (hall have
the equation ^xx + ^yy — aa + ax + ^xx + ^yy, which is reduced to
^x — j;ax = j:aa — yy. Now, adding to both members the fquare tV^^j and -
making the fubftitution oi x — -^a = z, it will be finally zz = -^aa -^ yy, an
equation to the circle.
Therefore, taking BM = i.^, and with centre M, and radius BM, defcribe
the circle NDB, this will be the locus required ; in which, taking any line
MP =2, it will be PD = J ; but AM z= \a, therefore AP =: z -f -^^ =: ^,
and the lines AP, PD, will be the co-ordinates of the propofed Problem.
If we would have alfo the locus of the points C, this would be another Pro-
blem of a like nature, which might be refolved in the following manner.
Make AQ_= />, QC = y, which is perpendicular to BN ; it will be
AP = 2/), PD = 2^; but AM iz -^«, and MB = ia. Then NA =: f^,
and therefore NP x PB = \aa ■\- ap — 4/>/'. But, by the property of the
circle, NP x PB •= PD^ and — 4^^. Then it will be 4^^ =r: \aa + ap
— 4/)/^. Whence \aa — jj "^ fV — ^^V* Add to both fides the fquare -i^aa^
and making the fubftitution of ^ — \a — /, it will be qq^ '=■ -^-^aa — tu
"Whence, with diameter MN =: ^a defcribing the femicircle NCM, this will
be the locus of all the points C ; in which, taking from the centre S any line
SQ^=: /, it will be QC zz q. But AS z: \a by the conftrudlion. Then AQjz:
/ 4- -f ^ — ^, and the lines AQ»^QC, will be the cO-ordinates of the Problem.
THefe two Problems may be demonftrated conjunftly in form of a theorem,
after the following manner.
In the given line AB is taken MB equal to \ of AB, and with centre M,
radius MB, a circle NDB is defcribcd ; and alfo with diameter MN the circle
NCM J through the point A drawing any how the right line CD terminated at
the periphery of each circle, and from the point C the right line CB to the
extremity of the diameter, it will always be DA the double of AC, and CD
equal to CB.
Let S be the centre of the circle NCM, and let the right lines SC, DL, be
drawn through the centres S, M. Becaufe SM is half of MB, then will SM
be 4 of AB. But AM is -^ of it ; therefore SA will be f of AB, and therefore
t of AM. But SC is alfo half of DM, and the angle SAC is equal to the
angle DAM j therefore it is eafy to perceive, that the triangle SAC is fimilar
to the triangle DAM, and that therefore AC is half of AD, which was the
firft thing.
But
SECT. III.
ANALYTICAI. I KSTITUT IONS*
«33
But if the triangles SAC, ADM, be fimilar, then the angle SCA will be
equal to the angle ADM; whence the right lines SC, D-L, will be parallel, and
confequently the triangles BLM, BCS, are fimilar, and therefore ML will be'
the fourth proportional to BS, SC, and MB. But BS — -lAB, SC :=i 4AB,
MB z= 4.AB. Therefore ML = f AB = AM. But MD = MB, and the
angle AMD = LMB. Therefore the triangles AMD, BML, are equal, and
the angle ADM = MBL. But ajfo the angle MDB =: MBD, fo that the angle
CDB — CBD, and therefore the fide CB = CD j which was the fecond thing.
PROBLEM Vlir.
139. The two fides AC, CB, of the Another,
norma ACB being given, the locus is
required of all the points, through which
the extremity B of the fide CB will pafs,
whilft the norma moves in fuch manner,
that it's point A fhall always be upon the
line DM, and the point C upon the line
DP, which is fuppofed perpendicular to
DM.
From the point B let fall BP perpen-
dicular to DP, and make DP = x,
PB -y, AC = ^, CB zz 3 ; it will be
CP — ^^^hh -yyy DC ^1 X V hb -yy>
But the angles DCA, BCP, taken together, are equal to a right angle, as alfo
the angles BCP, CBP ; and therefore the angles DCA, CBP, will be equal to
each other. Then the triangles ADC, BCP, will be fimilar, and it will be
AC . CD :: BC . BP, that is, a , x ^ x/TT^Jy :: b ,yy and thence ay zz hx
— h\/ bb —yy ', and, by fquaring and ordering, the equation will be xx —
saxy
+
aayy
-hb
yy.
Make the fubflitution of ;; ~~ =: z, and we
b ' bb
fliall have the equation zz -^ bb -^ yy^ which is to the cllipfis.
On the indefinite line DM defcribe the triangle DEH with it's fides T>Y. — b,
EH = aj and with the right angle DEH, becaufe the co-ordinates of the
Problem rnake a right angle ; and let the known line DH = /, With tranf-
verfe femidiameter DH :=/, and with the conjugate femidiameter DQ^iz b
and parallel to EH, defcribe the ellipfis HBQj it fhall be the locus required.
For, taking any line DF = PB = y, it will be GB = 2?, FG = -^ j there-
fore
134
ANALYTICAL INSTITUTIONS.
BOOK I.
fore FB =z 2 + -l^ = ;^ = DP. And therefore the lines DP, PB, are the
CO brdinates of the Problem.
PROBLEM IX.
Another. Fig. 86.
31
C E
140. The angle BAP being given, and the
point P being alfo given; it is required to find the
JojCuj of all fuch points D, that, drawing the two
right lines, BD parallel to AP, and DP to the
given point P, the lines BD, DP, may always be
to each other in the given raiio of ^ to e.
Drawing DC parallel to AB, make AP = ^,
AC — x\ CD z=. y, CP zz a — x. Becaufe d:ie
angle BAP or DCE is given, drawing DE per-
pendicular to AP, the ratio of CD to- CE is
given, which may be CD : CE ;: d . If ; then
__ h
by
-4- , or elfe n .v H — 7-
d a
CE = -^, AE zz ^ + -^, EP = ^ ~ A-
a a
- .7, PD = -^. Then it will be CD^ — CE^
_, r^r* ¥-»!-< 1 • eexx 2ahv ihxv
r: DPq — rh.q, that is, yy =. — aa — xx + zax + —-— j^ , or
zbxy , bbxx ee + bb — dd . . 2aby , , ,. ,
-+- -77- — -n XX + 2ax — aa + —j- , by addmg the
yy + ^
hbxx
dd dd "^ ' """ ' d
on both fides. But here it may be obferved, that the quantity
fquare ,,
'■ ad
ee -^ bb ^ dd may either be equal to, greater, or lefs than, nothing ; and, firft,
let it be equal to nothing, in which cafe the equation will become yy -f i^ +
ibxx 2aby , * 1 1 • 1 r . n* • r bx
-yj- — ~T' ">■ "^^^ " ^^' -^^^ making the lubftitution of ^^^ + — zi a, it will be
2abz 2aibx „,, , ,. aabb
zz J- ~ 2ax — -—7- — aa, 1 hen addmg
dd
on both fides, it will
, 2<z/« , aabb za^bx . aalb — aadd ^^ , . , /- .
be ZZ — — : — h -77- = 25^ 1-j — I . Now, making the fub-
dd
ab
dd
ftitution of 2 -7- 1= p, it will be pp
dd
ladJx — Zabbx + aabb — aadd
.__ __ ^
or pp
•=. X — \ay, ~— ; and making x — -la — q, it will become at laft
dd
pp =
2add — 2ahb
Id
y, an equation to the Apollonian parabola.
Let
SECT, III.
ANALYTICAL INSTITUTIONS.
135
Fig.z^, ^B.
Let BAP be the given angle; the given
line, AP — a. On AP, produced indefi-
nitely, let there be defcribed the triangle
AMN with the angle AMN — BAP;
and let AM.MN :: ^. ^. Produce AN
indefinitely^, and in AB take AH zz -^,
a
and draw HE indefinitely, and parallel to
AN. Biled AP in O, and draw OCL
parallel to AB. With vertex Q>^ on
the diameter QE, with parameter zz
'—'—jr—'i (making / zr AN,) and
with the ordinates parallel to AB, defcribe the parabola QD. Take any line
QE ~ .V, it will be ED ~ y, and this parabola will be the Locus required.
In the fecond place, let ee ■\- bb — di be greater than nothing, or a pofitive
quantity. Affuniing therefore the equation, and making ee + bb — dd zz: hh^
it will be j?y + ^-p +
bbxx
"IT
hhxx
Id
— aa -{- zax +
Zahy
And making the
lame fubftitution of y + -^ zz 2, it will be zz ^ =r -^ — aa 4- <iax
a it da
— —Tj- ; and adding -jj- , and making the fubftitution of z — — = p, it
will be ddfp zi hhxx + laddx — 7.abb>i — aadd + aabb ; that is, xx +
zad'^x — 2ab*x ___ ddp/> aadd — aabb
make
add — abb
ZZ my then xx + 2mx zz
i>h hb ' hh ' bh
'-j~ + am i and adding mm to each fide, it will be xx + imx + mm — ^^^^
+ am + mm, and making *• + w z: ^, it will be finally qq =z ^
mm, that is, qq ^ am -^ jnm = -.^, an equation to an hyperbola.
hb
+ am -^
Fig. 88.
Let BAP be a given angle; the given
line, AP=:^, Upon AP, produced inde-
finitely, let the triangle AMN be de-
fcribed with the angle AMN equal to.
BAP ; and let it be AM . MN :: d . b.
Produce AN indefinitely, and in AB
take AH zz —J- > and through the point
H draw the indefinite line OE parallel
to AN. Then make AK zz m, and
draw KO parallel to AH. With centre
13^
ANALYTICAL INSTITUTIONS.
BOOK I.
O, tranfverfe femidlameter 00-'^'^'"\^ """
^"^ a
h^ am -f- mm
, and conjugate femidiameter =
, parallel to AH, (by / is denoted the known line AN,) let the
hyperbola QD be defcribed. Then taking any line AV == x, it will be
VD zz 7, and this hyperbola will be the locus required.
LaClly, let ee + bb — dd be lefs than nothing, that is negative. Make
then ee + bb — dd :=. — hh, and making y H — ~ =r z, the equation will be
zz
2ahz
hhxx , 7.ahhx , , , . aahh . , ^ ,
—• aa •\- lax -— —^rr ; and adding —jj- on both fides,
it will be zz —
dd
2al)z
dd
aahh __^ hhxx laddx — 2ahbx
■*' IT " Id "^ Td "*"
dd
aahh — aadd
Id
and
making the fubftitution of 2 — -^ n p, it will be ddpp — — hhxx + 2addx
77, 7 7 7 » 1 • . zahhx •— 2addx aahh — aadd ddbb
— 2abbx + aabb — aaddy that is, xx + t-. = tt jj-.
Make
add
— = m, and we fliall have xx — 2mx = -^ am — ^^
hh
, and
adding mm on both fides, ;if;v — 2mx + ww = mm '- am *^ hT'^ laftly, mak-
ing the fubftitution of a* — m = j, it is -j^ = mm — qq -^ am, an equation
to an ellipfis.
Fig. Sg,
Let BAP be the given angle, and
the given line AP z=. a. On AP, in-
definitely produced, defcribe the tri-
angle AMN with the angle AMN equal
to BAP. Make AM . MN :: d . by
and produce AN indefinitely, and in
AB take AH r: ~, and through the
point H draw the indefinite line HE
parallel to AN. On A P produced take
AK zz m, which in this cafe is always
greater than AP— ^, and draw KO pa-
rallel to AB. With centre O, tranfverfe
femidiameter OQ^zi -^ — f^^Uff (mak-
ing AN =/,) with conjugate femidiameter s=s ^J^ — , and parallel to AH,
defcribe the ellipfis QD. Then taking any line AV zz x, it will be VD = y ;
and this fliall be the iocMS required,
5 HI. I
SECT. III.
ANALYTICAL INSTITUTIONS.
137
141. I faid above that AK = m was greater than AP zi a; m relation to A method to
which I think it neceflary to explain how we may know which of two com- ^^^^'^'?^"^
plicate quantities is the greater. Let there be made between them a comparifon mmority In -
of majority or minority, as you pleafe, and then proceed as in an equation, by complicate
tranfpofing, dividing, &c. and making other operations, till you arrive at a quantities,
known confequence ; which, if it be true either abfolutely or hypothetically,
the comparifon that was made will be abfolutely or hypothetically true; but if
falfe, this will likewife be falfe. So, if we defire to know whether m, that is,
add — all ^^ o;reater than a, or not, make the comparifon or fuppofition
. ^^^ " ■ > ^, and reducins; to a common denominator, it will be add — abb
Jd-bb-ee ^ ^
> add — abb — aee, and expunging the terms that deftroy each other, it will
be o > — ^^^; which is very true, for nothing is greater than a negative
quantity. Therefore it was true that
add — abb ^ ,
-r, — -, was greater than a»
dd — bh — ee °
Thus, to know if aa + 2ab be greater than hb, fuppofe aa + 2ah > bb,
and add to each fide the fquare bb. It will be aa + zab -{- bb > ibb, and
extrafting the root, it is ^ + ^ > \/2bb, or a > \/ ibb — b. But, becaufe
the quantities a, b, are given, it may always be known whether a be greater
than \/zbb — b, or not. And if it fhould be fo, then alfo aa + lab would
be greater than bb. The manner is the fame in cafes more compounded, and
therefore I ihali infift on it no longer.
PROBLEM X.
Fig* 9°'
142. Two right lines VB, VE, being given A geometrl-
in pofition, and alfo the point P, about which cal problem,
as a pole the right line PE revolves ; to find the
locus of all the points D, fuch that it may always
be CD to DE in a given ratio.
Draw VP, and parallel to it the right lines
AD, BE, and let the ratio of CD to DE, or
rather of CD to EC, be as </ to ^ ; and the
angles EVB, EBV", being given, let it be EB
to BV as e to h»
T Make
138 ANALYTICAL INSTITUTIONS. BOOK I»
Make YP =: a, YA zz x, AD =z y -, it will be EB = -^ , and therefore
VB = i . Becaiife of the fimilar triangles CVP, CDA, it will be DA . PV
a
:: CA . CV •, and, by compounding, DA + PV . PV :: CA + CV . CV ;
that is, a + y . a :: X . CV, and therefore CV zr —^ . Again, becaufe of
fimilar triangles PVC, EEC, it will be PV . VC :: EB . EC, that is, ^ . -^
: : -^ . BC J whence BC rz /^ , and therefore the equation EC + C V
T>Tr 1 ^ • '•^V + ^^J^ -^y exy adx
= ^^' '^^' ''' -tr+7^ = -3- ' °' yy - -r = — - ''y-
To conflrud this, make y — -^ z: -^ , and, by fubffiitution, it will be
^ = ^ ^;, --f^ + f^, that is, ;^y + ^ - f^ = - f^.
Ji>
e e se e
1 ^ , oh adh , . •,, , ^ aadh aaddh
gain, make 2 + — — — = /> ; then it will be ty = • — — t-
— ~ . And making a third fubftitution of ^ + ~ iz: gr, it will be /)^ =
~ 5 an hyperbola between the afymptotes, the conftant redangle of
which is pofitive, becaufe e will always be greater than d.
Let PV be produced indefinitely, and take VQ^= — . From the point Q^
draw the indefinite line QS parallel to VB, and, taking any point M in the
right line PH, draw MN parallel to VB» Then, becaufe of fimilar triangles
VMN, EBV, it will be VM . MN :: e , h. Make VI zz f!ijZ_f£i , and
through the point I drawing the indefinite right line RIK parallel to VE, be-
tween the afymptotes RS, RK, defcribe the hyperbola OVD with the conftant
redangle =: ^ — " "^^ X — , (making the known line VN =/,) which
will necefiarily pafs through the point V. Taking any line VH = y, it will be
HD = ;f, that is, VA zz x, AD zrjy, and the curve thus conftruded is the
locMs of the points D»
A fpecimen 143. We may obferve here, that the equations expreffing the properties of
ofthcde- the curves deferibed in thefe Examples, or Problems, ought to be the fame
T"h?^'°" with the equations propofed to be conftruded, when the operations are truly
examples, performed -, and therefore may ferve as a demonftration of the method itfelf.
This
SECT, in;
ANALYTICAL INSTITUTIONS.
39
This I have purpofely omitted to do, to avoid being too prolix. However,
to give a fliort fpecimen of it, I (hall take the conftrudions of Example XIII.
and of Problem VIII,
Fig. 65,
And, firfl:, for the example. Having
made AD zz x, and it being AS = 2^,
AF = /, it will be AC = ^ , and
hn
therefore AR 1= — ; it will be AQ =:
-^, and thence QC = ^ - ^^ =
MP. Therefore, the femidiameter being
-^ , we (hall have HP ~ 'l^^
HM
« ^/
-. J^, and PV = -^ - ^ + -i^.
\am la 2a ^am
Ix
Thus, becaufe DN =;., CD = ^, CP = QM = f^, it will be PN zz;.
J. il + f f. But, by the property of the ellip(is, it muft be HP X PV
2a
• PN7 :: HV . parameter = ^^ . Thence we (hall have the equation ^^*J~JJ'"'
j^a
^ffbnx_ __ ffhhnn^ '^^^^ ^ =:.,€+ ^-^ + ^ + cy + ^-^ + yy. And,
. ^ , - _ . . , , ccmm + /iagmn + nnhh • mi i t
inftead of ee, reftormg it's value ^^ , it will be "^cc + ag ^
^mm
^ hx •=. ice -{- — ■\ ^ + 9' + — + JiT. And laftly, reftoring the
values of — — = — t:^ y ^^^ ^ = ^ "" ^"^ , we (hall have ag — x)i — Ix
4<wt 2
zz cy + — -{- yy, which is the very equation propofed to be conflruded.
J!tg, 90. In the conftrudion of the laft Problem it was
^ — ^-^ — X -^ the conftant redan gle of
the hyperbola, and VI = ae ^ a ^ ^^^ ^^
rallel to the afymptote RS. Alfo, it will be
RI = -^ . But, becaufe of limilar triangles
ee ^ D
VMN, VHG, it is VM . VN :: VH . GV, and
T 2 therefore
Ujpi ANALYTICAL INSTITUTIONS; BOOK I.
therefore GV = -^ =r IK. Then RK = ^ + -^ • But HG = il,
e ee e e
GK z= VI rz ffiri^. Whence HK = ^^JL±Jl±z^ ^ Bat HD = VA
ee ee
rz x ; then it will be KD = ^ — fli-ILf — r_ffl ^ and therefore, by the pro-
perty of the curve, the redlangle RK x KD ought to be equal to the conftant
redangle, or ^M±J^ x ^i2Jt^-ZJ^±ZJlfL =: -J!±zL^i± ^ / ^ ^hat is,
'-' ee ee e^ e
exy , adx • i •
yy ■— -\r ay — -r- = o, as it ought to be.
If the fame care and induftry were ufed in every Example and Problem, it
would fufficiently prove the method of folution to be juft.
SECT. IV.
Of Solid Problems and their Equations,
What are the 144, Any one of thofe quantities is called the Root of an Equation, which,
roots of equa- being fubftituted in the equation inftead of that root or letter, according to
which the equation is ordered, (or inftead of that letter which reprefents the
unknown quantity,) (hall make all the terms of the equation to vaniQi or
become nothing. Or, which is the fame thing, the root of an equation is each
of the feveral values of the unknown quantity, or of that letter which performs
the office of an unknown quantity in the equation.
Therefore the roots of the equation xx ~-' ax -{• Ifx — a^ :=: o will be two,
one of which is a, the other — b -, for each of thefe, being fubftituted inftead
of X, will make the terms of the equation to vanifh ; or, becaufe either a or
— if are the values of the letter x in the propofed equation. The roots of the
equation x* '^ x^ — 19^:* + ^gx — 30 = o will be i, 2, 3, or — 5 j be-
caufe any one of thefe numbers, being fubftituted inftead of x, will make all
the terms to vanifh, and therefore any one of them is the root, or value of the
unknown quantity x. The roots of the equation x"^ — bbxx — aabb — ^'* = o
will be + -s/— tf<7, — v'— aa, -j- \/aa + bbp — ^/aa + bb I and fo of
all others.
145. Again,
SECT. IV. ANALYTICAL INSTITUTIONS. I4I
145. Again, in another fen fe, thofe equations are ufed to be taken for the Or otHerwIfe,
roots of an equation, which are formed by fubtrafting, one by one, the poficive the feveral
values from the unknown quantity, or by adding the negative value, and ^"''^"."'^ °^ ^"
making them equal to nothing. Therefore, in this fenfe, the roots of the^^"^^'°"*
equation xx — ax -{- bx — ab r=z o will be .v — ^ =r o, and x + b zz o,
Thofe of the equation x"^ — x^ — 19.V* -f» 49;^ — 30 zz o will be a* — i =0,
X — 2 1= o, X — 3 ~ o, and a? + 5 =r o. And fo of others. And, in this
fenfe, it is faid, that every equation is the produd of all it*s roots, becaufe,
being continually multiplied into one another, they will exa£lly produce the
giveri equation, or that of which they are the roots. Hence it is, that the roots
of an equation will be fo many, including alfo the imaginary roots, as is the
degree to which the equation arifes. And therefore a quadratick equation will
have two roots, a cubick equation three roots, a biquadratick four roots ; and
fo on,
\i X -{- a — o be multiplied into a; + ^ = o, there will arife the quadratick
equation (I.) xx -^ ax + ab = o.
-i- bx
And if this again be multiplied into x — c zz o, there will arife the cubick
equation (II.) x^ + ax"" + abx abc = o.
-f- bx* — acx
— ex* — bcx
And if this again be multiplied into x + d zz o, k will produce the biqua-
dratick equation (III.) x*" + ax^ ~|- abx^ ' — abcx — abed =. o,
-f- bx^ — acx'^ + abdx
— cx^ + adx* — acdx
H- dx^ ^ — bcx* — bcdx
^ bdx"-
— cdx*" y
Thus, if ^ + \^ab = o be multiplied into x — \^ab = o, the product
will be XX — ab zz o ; and if this be multiplied into ^ + r = o, it will make
x^ + fAf* — abx — abc zz oj and again, if this be multiplied into x -\- c zz q,
it will make x* + 2cx^ — .abx'- — zabcx — abcc z; o.
+' ccx*
If X + a/ — ^7^ :z o be multiplied into x -^ \/ — ab =. o, and then into
X -\- a :zi o, k will produce the cubick equation x^ + ax"" + abx + aab zz o.
146. Therefore, if we had the means of knowing what were the values of Equations
all, or of any of the unknown quantities of an equation, wc might always J?^!^^,^ ^^ ^^'
divide it by fo many fimple equations as are thofe known values, by adding the yif^^^ i/their
negattve values, to the unknown quantity, and fubtradting the politive. Whence roots were
the firft equatipn before will be divifible by x + a, and' by x + b. The ^^owxu
fecond.
142
ANALYTICAL INSTITUTIONS,
BOOK Ii
fecond, by ;c + ^, x -\- l>, and x — c. The third, by x + a, x + If,
X — c, X -\- i^. By this, compound equations will be reduced to fo many
fimple equations as is the number of the roots, if all be known ; or may be
deprefled by fo many degrees as is the number of the known roots, if they be
not all known. So that, for inftance, an equation of the fifth degree may be
reduced to one of the fourth, if one of it's roots be known ; or to the third,
if two roots be known ; and fo on.
Hence is
known the
nature or
formation of
the feveral
co-efficients.
147. From the method by which equations are produced, (which equations
are always underftood to be reduced to nothing, and in which the greateft term
in refpect of the unknown quantity, or in reiped of that letter by which the
terms are ordered, muft be pofitive and free from a co-efficient,) it is eafy to
perceive that the co-efficient of the unknown letter, or that by which the
equation is ordered, in the fecond term is the fum of all the roots of the equa-
tion affeded with contrary figns ; the co-efficient of the third term is the fum
of all the produds of all the pairs of roots which can be formed j the co-effi-
cient of the fourth term is the fum of all the produds of all the ternaries or
threes ; and fo on to the lad or conftant term, which is the produd of all the
roots multiplied continually into one another.
When the
fecond term
will be want-
ing.
148. Hence it may be inferred, that the fum of the pofitive roots muft
neceffarily be equal to the fum of all the negative roots, in all fuch equations
in which the fecond term is wanting : and that the fum of the pofitive roots is
greater than the fum of the negative, when the fecond term is afFeded with a
negative iign ; and contrarily, when it is affeded with a pofitive fign.
How the ab- 149. When any term is wanting in an equation, it is ufual to fupply it's
fence of a place by an afierifm * ; as in ^* * -|- aaxx — Px -+■ ^* = o, the fecond
denoted!^ ^^^^ ^^ wanting. In x* — ax^ * — Px + a*, the third term is wanting; and
fo in others.
Surd roots
and imagi-
nary roots
always pro-
(;eed by pairs.
150. Jf an equation have no term affeded by an imaginary quantity, either
it's roots fliall be all real, or, if it have any imaginary roots, they fhall always
be even in number, and equal two by two ; only with this difference, that one
mull be affirmative and the other negative. For, becaufe the fecond term is
the fum of all the roots, if this be prefent in the equation, when the imaginary-
roots do not deftroy one another, two by two, with contrary figns, fome ima-
ginary root muft neceffarily be in the co-efficient, which is contrary to the
luppofition. Now, if the fecond term be wanting, it muft needs follow, that
the fum of the pofitive roots is equal to the fum of the negative, and confe-
qucntly the fum of the pofitive imaginary roots muft be equal to the fum of
the negative imaginary roots, otherwife they cannot deftroy one another i{» the
manner aforefaid. Wherefore equations, whofe degree is an odd number, will
neceffarilv have one real root at leaft ; and thofe of an even degree may have
all
SECT. IV, ANALYTICAL INSTITUTIONS. I43
all their roots imaginary or impoffible. For the fame reafon, we may make
like conclufions about furd roots. That is to fay, if the equation have no furd
or irrational terms in it, it's roots will either be all rational, or the irrational
roots will be in even numbers, and will be equal two by two, but with con-
trary figns.
151, There are equations which have all their roots pofitive, others have all Affeaions of
their roots negative, others have both pofitive and negative. So fome have all J,'^n-'"°^?f^°r
their roots imaginary, others have all peal, and laflly, others have both real and ' ''^°"^
imaginary. Various rules are given by writers of Algebra, to determine in any
given equation the number of pofitive and negative roots, alfo of real and
imaginary roots. But, becaufe thefe rules and their demonftrations are very
perplexed and prolix, and of but little ufe, I (hall here omit them, thinking it
lufficient to take notice, firft, that if all the roots be negative, all the terms of
the equation will be pofitive. For, in this cafe, fince all the terms of the
fimple equations are pofitive, that is, of all the roots taken in the fecond fenfe,
§ 145, from whence the propofed equation is fuppofed to be produced, all the
produds will alfo be pofitive. Secondly, that if all the roots be pofitive, the
terms of the equation will be pofitive and negative alternately. For the firft
term will always be pofitive by fuppofition. The fecond term will be negative,
becaufe it contains the fum of all the roots, which being pofitive, will be nega-
tive in the fimple equations. The third term, containing the ternaries or pro-
duds of all the pairs, will be pofitive. And fo on. And therefore an equation
compofed of pofitive and negative figns alternately, will have all it's roots
pofitive.
Whence, if the terms of an equation fhall not have all their figns poficive,
or (hall not have them pofitive and negative alternately, there will be both
pofitive and negative roots. It fliall alfo be another fure proof, that the
equation contains both pofitive and negative roots, if there be any term
wanting ; for no term can be abfent, but that the produdls of which it is
formed muft deftroy one another by contrary figns ; that is, there mu(l be both
affirmative and negative roots. This obfervation will aflTift us in it*s proper
place, among the many divifors of the lall term of an equation, to felec't thole
only by which the divifion is to be attempted. Becaufe, if the equation (hall
have only pofitive roots, it would be of no ufe to try the divifion by pofitive
divifors; and if it fhall have only negative roots, it would be needlefs to try
by negative divifors. And the trials muft be made by each of them, when
there are both pofitive and negative roots.
But all this muft be underftood of fuch equations in which all the roots are
real; for where there are imaginary roots the rule does not obtain. For ex-
ample, let the equation be x^ + i'x'' 4- aax + aal> z: o, in which all the
terms are pofitive, and yet the roots are one pofitive and two negative, that is,
» = — ^, a real root, and x zz ^ -y/ — aa, two imaginary or impofl[ible
roots, one pofitive, the other negative,
152. Equations
144 ANALYTICAL INSTITUTIONS. BOOK I.
Affeaionfe of 152. Equations of the third and fourth degree, in which the fecond term is
the roots of ^yanting, if the third term be affedled with the pofitive fign, will certainly have
the thiVd^or Jniaginary roots ; for, if all the roots were real, the third term could not but
fourth de- be affeded with the negative fign ; the reafon of which is, that in cubick equa-
grec. tions, when the fecond term is wanting, the fum of the pofitive roots is equal to the
fum of the negative, and therefore either one pofitive is equal to two negative,
or two pofitive roots arc equal to the one negative. Let the three roots, for
inilaiice, be reprefented by a^ b, and — r, or elfe by — a^ — ^, and + c ;
then the co-efficient of the third term will be ab — ac — be. But, on fuppo-
fition that the fecond term is wanting, it will be ^z + ^ = ^. Therefore ac
will be greater than ab^ and confequently ah — ac --^ be will be a negative
quantity.
Now, in equations of the fourth degree, there may be either three pofitive
roots and one negative, as + <7, + i', + c, and — d\ or there may be three
negatives and one affirmative, as — a^ —^ b, — c, and -j- ^; or there may be
two negatives and two affirmative, as — ^, — b, + c, and + d. In the firft
and fecond cafe, the co-efficient of the third term will be ab -{' ac -\- be — ad
— bd — ed. But, by fuppofition, it ought to be a -\- b + c zz d, (o that
ad will be greater than ab, cd than acy bd than be j and therefore ad -\- hd -{- cd
will be greater than ab -{• ac + be, and confequently the third term will be
negative. In the third cafe, the co-efficient of the third term will be ab — ac
— be — ad — bd + cd, and it ought io ht a ■\- b = c ■\- d. Here, if we
make m ■:== a + b ■=. c -\- d, it will be mm =: a + b X c + d ^=. ac + ad
+ be -^ bd, and mm iz a + b* =z aa ■{■ 2.ab + bb^ and alfo mm zr. c -{• d^ = "
cc + 2ed + dd. Therefore it is ab = "" " ^^ ^ ^ and cd = ffJliilL^f,
22'
and ab + cd zz mm — ^ . Therefore mm is greater than
ab -\- cd, and ac + ad -{■ be -{• bd will be greater than ab + cd. Whence
the co-efficient of the third term will be negative.
The pofitive ^53* ^^ ^^ always in our power, in any equation, to make all the pofitive
roots may be roots to become negative, and the negative to become pofitive. Nothing more
made to be- Js required to perform this, than to change all the figns which are in even
come negja- pig^es, that is, in the fecond, the fourth, the fixth, &c. ; the reafon of which
c'lS^. " is, that the fecond term being the fum of all the roots, in this therefore are
the negative with a pofitive fign, and the pofitive with a negative fign, as has
been plainly feen at ^ 145. In forming equations, compounded of the produfts
of fimple equations, by changing the figns they alfo will be changed. The
other even terms in order are formed from the produds of an odd number of
roots, that is, the fourth from three, the fixth from five, &c. Wherefore, if
they have the pofitive fign, they will be formed from the product of all the
negative roots, or from an even number of pofitive roots, and an odd number
7 of
SECT. IV, ANALYTICAL INSTITUTIONS. I45
ibf negative roots. And if they have a negative fign, they will be formed from
the product of all the pofuive roots, or an even number of negative roots, and
an odd number of pofitive roots. Therefore, by changing the figns of all the
even terms, the pofitive roots will become negative, and on the contrary.
As to the odd terms in order, they being formed of even produds of roots,
if they have the pofitive fign, they will be formed either of an even number
of negative roots alone, or of an even number of pofitive roots alone, or of an
even number of pofitive, or an even number of negative together. VVherefore,
by changing thefe reciprocally, the figns of the terms them(elves will not be
changed. Now, if they have a negative fign, they wiil be tormed of the pro-
dud: of an odd number of pofitive roots, into an odd number of nejative.
Wherefore, by thefe alfo reciprocally, the fign of the terms themfelves will noc
be changed, and therefore they muft be left as they are.
The equation x^ -f- ax"" + al?x — ak -=z o has three roots. Two arc
+ /^x* — acx
— ex* — IfCX
negative, viz, — a, ^ ^, or otherwife, x + a^:zo, x-\'lf:zzo, and one is
pofitive, viz. + c, or otherwife, a; — <r r= o. By changing the figns of thofe
terms which in the order of the equation are even, it will become
x^ — rx* + ai>x + ah = o, and the pofitive roots will be x ^ a zz o,
— ^a:^ — acx
+ ex* — hx
X — ^ = o, and the negative root will be a? + f = o. It is of no moment
whether or no any term be wanting, becaufe in this cafe the afterifm fupplies
the vacancy, and then the fame rule obtains. Thus, in the equation
x^ • — 28.V + 48 — 0, the affirmative roots of which arc x — 2 iz o,
a; •— 4 = o, and the negative root is a; + 6 — o. By changing the figns of the
even terms in order, it will be x' * — - 28a: — 48 zz o, the negative roots of
which are ;c+2Zzo, a:4-4 = o, and the affirmative root is a? — 6 ~ o,
154. Any equation being given, by means of congruous fubftitutions it is The roots of
eafy to increafe or diminifh all it's roots, though yet unknown, by any given an-equatlon
quantity ; that is, it may be transformed into another equation, the roots of ™^^r^f '"*
which (hail be the fame as thofe of the propofed equation, but increafed or dimlniftied
diminiOied by fome given quantity. Let the unknown quantity of the equation at pleafure.
be put equal to a new unknown quantity, adding or fubtrading the given quan-
tity ; adding, if we would have it increafed, or fubtrafting, if we would have
it diminidied. Then, in the propofed equation, inftead of the unknown
quantity and it's powers, their values muft be fubftituted, exprefled by the
other unknown quantity and the given conftant quantity ; from whence another
equation will arife, the roots of which will be fuch as are required. Let the
equation be x* + 4^^ — 19X* — io6;v — 120 = o, the roots of which we
U would
14^ ANALYTICAL INSTITUTIONS. BOOK I.
would have increafed by the number 3. Make ^ + 3 =: ^, whence .v z::^ — 3,
X* = y* — 6y 4- 9, x^ — y^ — 9/ -+- zjy — 27, and x^ — y* — 12^ 4.
54/ — loSy 4- 81 ; therefore, in the propofed equation, fubftituting thefe va--
lues inftead of .v and it*s powers, it will be transformed into this other equation,
/ — 12/ + 54/ — io8y 4- 81 \ ,
+ 4)'' — 3^/ + i^Sj — »o8 I
— 19/ 4-114)' — 171 )* = Oi that is,/ -- 8j^ — >* 4- Sy = o;
— 106^ + 318 I
— 120 y
and dividing by y, it is y — 8/ — ^4- 8 = o, in which it is plain, that the
roots will be greater than the roots of the propofed equation by the number 3 ;
becaufe it was made y zz x -^ 3, and therefore the root^ will be equal to every
value of X increafed by 3. And here it may be obferved, that, in thus in-
creafing ihe roots, the pofitive are increafed by fuch a quantity, but the negative
are diminifhed by the fame quantity ; for, by adding a pofitive to a negative,
if the negative be greater than the pofitive, it will become lefs in it's kind than
at firft ; if they be equal, it becomes nothing, if it be lefs, it makes it pofitive.
Whence, in the propofed equation x* + 4.x^ — 19X* — 106* — 120 z: o,
the roots of which (though they cannot be found by the methods hitherto
taught,) are 4-5, — 2> — 4> — 3, that is, x — ^ = 0, x-{-2 = o,
;e4-4 — o>^ + 3~o» one of which is affirmative, the other negative ; as
I delired to increafe them by the number 3, in the transformed equation
yi — Sy* — y + 8 zz o, they ought to be 4- 8, 4- i, — i, that is, y — 8
= 0, ^ — i— Oijy4- i=zo, and are really fuch. And that which fliould
correfpond to the fourth is =z o, becaufe — 34-3=0. And, for this
reafon, the reduced equation is only of three dimenfions, though the propofed
equation is of four.
On the contrary, when the roots of an equation are to be diminifhed by a
given quantity, for the fame reafon the negative roots are increafed in their
kind by the fame quantity, but the pofitive may become nothing, if the given
quantity be equal to them, and negative if greater. In the fame equation
X* 4- 4X^ — igx* — io6a; — 120 = o, if I (hould defire to diminifti the
roots by the number 3, I muft make x — 3 :=: y, and therefore a- zz ^y + 3,
;v* =:/ 4- 6y 4- 9, x^ = / 4- 9/ 4- 27^ 4- 27, x* =r y* -\- 12/ 4-54/
4- 108^ 4-81, And therefore, making the fubftitutions, the equation will be
y* 4- 12/ 4- 54y 4- io8j' 4- 81 "J
— 120 J
wmm. j^ — 420 = o. And, becaufe the roots of the propofed equation are
+ 5, — 2, — 3, — 4, that is, X — s — o, X -{- 2 = o, X + 3 = o,
* 4- 4 =: o ; thofe of the transformed equation ought to be 4- 2> — 59 — ^*
— 7>
4- 4>' 4- 36/ 4- 108^^ 4- 108
— 19/ — ii4y — 171 ^ zz o. That is, ;'* 4- i6y* + 71/
— io6y — 318
#ECT. JV, ANALYTICAL INSTITUTIONS^ 147
— 7, that is, ^^ •— 2 =: c, ;' -f 5 = q, j -i- 6 = o, ^ + 7 = o. as they
really are.
Let the eqnatTon be x^ + ex* — hhx — bhc zz o, and we defire to increafc
the roots by a given quantity a. Make x -{- a zz y, and therefore x zz y — a,
X* :z: y* — 2ay + aa, x^ :=■ y^ — 3^y + J^i?)' — ^'. Wherefore, making
the fubllitutions, the equation will be y — 3<?y + 3^*^ — • a}
4- 0'* — 2acy 4- «V
— /'^y + ^^^
> = o.
The roots of this are greater than thole of the propofed equation by the
quantity a. And, in fad, the roots of the propofed equation are x — ^ z: o,
x-^-b — o^x + c:=:-o\ but the roots of this are ^ — ^ -f. « = o, ^ H- ^
— a = o, and y •{• c — a zz o,
155. In like manner, if an equation be given, we may transform it into Or the roou
another, the roots of which are the fame as' thofe of the propofed equation, "?Y ^*^ '""^''
but multiplied or divided by a given quantity, fuppofe /•, making a fubfti- d\^|'^e/at
tutionof/v z=. y, {x being the unknown quantity of the given equation,) ifpkafure.
we would have it multiplied ; or of -^ = y^ if we would have it divided.
Thus, alfo, we may make x zz- ~^, if we defire that the roots of the tranf-
formed equation fhould have to thofe of the propofed equation the ratio of /
to g. And we may make \/fx zz y, if we would have them to be mean
proportionals between the quantity /, and the roots of the propofed equation.
In like manner, we may make x zz — , if we defire they may be recipro-
cals, &c.
156. The reafon of thefe rules is evident. For, afluming the firft cafe, orxhereafon
that of incr^afvng the roots, if we make the fubftitution oi x + a zz y^ the of thcfe ope^
values of J extra<iled from the transformed equation will be equal to x + a, '^^'°"**
or equal to the values of x in .the propofed equation increafed by the quan-
tity a. And by a like analogy in the other cafes.
157. Many are the ufcs that may be piade of thefe fubftitutions ; one ofAndthdi'
which may be, that not having as yet a method of knowing what are the roQts^fes.
of the propofed equation, by transforming it after fome one of the afore-
mentioned manners, we may difcover the roots of the transformed equation;
which being increafed, diminiflied, multiplied, divided, &c. by the conftant
quantity, according as the. fubftitution is made, we fhall alfp know the roots of
the propofed equation.
U 2 158. Another
148 ANAtVTICAL INStlTl/TIONS. BOOK f.
Equations 1^8. Another ufe may be, to fcee equations, whenever we plcafe, from
maybefrfeed fradions, and very often from furds. As to fradions, we muft make the un-
jrom fractions , .• r .u .• 1 r 1 • i- ■ 1 1
or furds. known quantity ot the equation equal to lome new unknown quantity, divided
by the lead quantity that is divifible by every one of the denominators of the
terms of the equation ; which fliall be the produd of the fame, in cafe that
thofe denominators are prime to each other. Then making the fubftitutlons,
and reducing the terms to a common denominator, we fliali have another
equation which will be free from fraflions, the roots of which will be thofe of
the propofed equation, multiplied into the quantity by which the new unknown
quantity was at firft divided. Let the equation be y^ + iay* — -^«^_y + aa^
— o ; if we make y zz -;-2;, jy* z= ^-V^;*, y^ = ttx-s-^', then, by fubftitution, the
equation will become — - -J- r — g — — rz + ^^^ — o. And, reducing to a
common denominator, it will be z^ -f- az^ — izai?z + 2i6aaif =: o. The
roots of this equation divided by 6 will be the roots of the equation propofed.
Let the equation be x^ ^ 4. _ffi ^ ^ — o. Make x :=. ~ , and,
fubflituting in the equation, it will be transformed into this, -^— -, — , f ^ „ 4-
-7 — r H 7- = o. Then reducing; to a common denominator, it will be
7? — acd-x^ -f- aH'^cd'^z + a^Pc^d^ zz o, Wherefore, if the value of z were
known, the value of a; would be known alfo. In like manner, to free equations
from furds, we may often proceed thus. Make the unknown quantity equal to
a new unknown quantity divided by the radical, and fubftitute this in the
8
equation. Let the equation be x^ — 'v/3 x ^* + ^r^ j-=o. Make
X = —J-, and therefore x"" z=. — , x^ zz -^—-i and, making the fubflitutions,
V3 3 ^^^ %
it will be ^ - ^ + ^ - ^ = o. Now, multiplying by ^^^^
it will be z^ — 32* 4- ^z — |- z= o. Laflly, by freeing this from fradions
after the foregoing manner, that is, making z =: ^y, or rather, z zz -^y, which
in this cafe will be more compendious, the equation will be y^ — c^'' + 26y
— 24 = o. And becaufe, by the firfl fubftitution, it is ;f = — , and, by
the fecond, z = ^y^ it will be * = -=y- ; or the value of x will be equal to the
value of y divided by 3 V 3.
Let the equation be x^ — x^^nn + px^^^n -^ qx + ^ =0. Make x =
4r 9 and therefore xx z: tr— » a:' =: — , at* =: -^t- » and making the fub-
ftitutions.
SECTi IV. -ANALYTICAL INSTITUTIONS. I49
/»• • • Ml t V* y'^l/nn pyy\/n (ly , *" a 1 1 •
flitutions. It Will be ^7^ <-^ h -jt 77- + 77- = o. And muki-
plying by n^fn^ it will be j* — ny'^ + npf- — nqy 4- r« n o. If we would
obferve the law of homogeneity, equations may be delivered from radicals :
but then fradions would thence arife, which mull be reduced as above.
159. Becaufe, by taking away radicals by means of the foregoing fubfti- Conditions
tutions, nothing elfe is done than multiplymg the roots of the equation by that fo'" expung-
radical, it is eafy to perceive, that if the radical be quadratick, for example v^«/"^ radicals,
it is neceffary, in order to expunge it out of the equation, that the fecond term
of the equation propofed fliall contain s/ n. For, as that term is the aggregate
of all the roots of the equation, it muft be multiplied by »/ n. It will be ne-
ceffary that the third term Qiould not contain »/ n, becaufe, as it is the aggre-
gate of the pairs of the roots of the equation, it muft be multiplitd by the
fquare of s/ n. Thus it will be neceffary that the fourth fliould contain s/riy ^^^
becaufe, as it is the aggregate of all the ternaries, or produAs of three roots, !& "
it muft confequently be multiplied by nV n. It will alfo be neceffary that the
fifth term fliould not contain the radical ; and fo on alternately. For the fame
reafon, if the radical to be taken away were ^n^ ii will be neceffary, that in the
fecond term of the propoled equation there Qiould be found '^nn, in the third
^n, in the fourth none at all, in the fifth ^nrij in the fixth ^«, in the feventh
none at all, &c. And the like is to be concluded of other radicals.
160. By means of thefe fubftitutions we may alfo take away the fecond term Thus the fc-
from any equation. And that will be done by putting the unknown quantity cond term of
equal to a new unknown quantity, adding or fubtradlmg the co efficient of the^" equation
fecond term divided by the index of the degree of the equation given : that is, ^JJy,
adding, if the fecond term have the negative fign, and fubtrading, if that fign
be pofitive. Let the equation hz x^ -^ ax — i^h zz o ; put a: = 2: — ^a,
and, by fubftitution, it will become 2* — az
•+• az — -^aa S- zz o. That is.
^ bb }
zz « — \;aa — bb '=2 o, or zz = ^aa + bb. Hence it may be feen, how all
afFeded quadratick equations may be refolved more expeditioully in this man-
ner, than by that before taught at § 74. Then, only fubtrafting ^a from the
value of z fo four^sd, we (hall have the value of x.
Let the equation be x^ + bx* — abx — a' zz o. Make x zz z — f ^5.
and, by fubftitution, it will be 2* # — "^bbz ■{- -^^P "j
«— abz + -^al^b > rz o.
— a^ }
Whence, taking -^b from the value of 2, we fhall have the value^ of x,.
Let
, 1^ AKAtYTTCAX- INSTITUTIONS, SOOK f.
Let the equation be a:* — aax^ 4- zaaxx — 2a^x + ^* = o. Make
— • ccxx
K zz z 4- ~ , or j; = a; + t<?. Tlien, by fubftitutlon, it will be
4
- ccz^ ^ accz ^ iaacc j = °- ^^^^ ^^^. t^' to the value of z,
and we (hall have the value of at.
Or the third i6i. And tlius we may take away the third term from any equation, pro-
term may be ceeding after the following manner,
taken away.
Let the equation be x* — -i^ax^ 4- i^aax^ — ^a^x — 2«* = o. Make
X -=. y -- hy where ^ is a general quantity, to be determined as occafion may
require. Now, making the fublliturions, it will be
O
/ — 4/^j' 4- dhhy* — ^h^y + b* ^
•*- 2>^y 4- ^ahy^ — ^ab*y 4- 2;^P l
4- 3^^J* — 6^% + 3«*^" ^ = o. Now, in this equation, that
•—' Kja rjy t ja rj~ ^
- 2fl* J
the third term may be nothing, it is neceflary that dhhy"^ 4- ^ahy- 4- ^aay*-
= o, that is, h"^ 4- 4^/6 4- ^^^ n o ; and therefore /-»=: — -^-^ ± ^<7. Hence
we are informed, that the fubftitution to be made inftead of ^ ~ i;, is either
y 4- \a, or y -{-a; for, indeed, either the one or the other takes away the
third term, making the equation y* — ay^ ♦ — *-^a^y — .^^a* zz o, or, fe-
condly, y* + ay* » — ^a^y — 6a* = o.
By this artifice it may be known, that, to take away the fecond term, we
muft make fuch fubftitutions as have been (hown at § 160.
X)rthelaft , 162. Now if an equation, in which the fecond term is wanting, is to be
but one, if transformed into another, in which the laft term but one Qiall be abfenr, it will
^he fecond be ^^ fufficient to fubftitute any given quantity, divided by a new unknown quan-
wanting* . • n j r 1 1 '• r V • -r 1 • ** .
tity, mltead 01 the unknown quantity or the .equation. Let the equation be
^ » 4- aax* — r^te 4- 4^ zr'O, and m^ke x zz — . 'By fubftitution, it will
be --J- * 4- -^ •" — 4--«* := o, And reducing this to a common denomi-
nator, and dividing by,/jr*, it will he y* ^^.ay^ & aay* * 4- «j*=^.q. ,I|i,the
fubftitution of x zz — ;, inftead of the given quantity <?, if we had taken any
other, we fliould have arrived at the fame conclufion, but then ttbe transformed
equation would have involved fraftions.*
163. If,
SECT, IV. ANALYTICAL INSTITUTIONS If!
163. If, In the propofed equation, not the fecond term, but the third, or Or any other
founh, 6:c. (hould be wanting, by the fame method we might make that term °" ^5?"^'"
to vanifh, which is equally diftant from the laft term, as the abfent term is *^°°°*^'°"*
diftant from the firft.
164. And on the contrary, if one or more terms be wanting in an equation, Or an equa-
we may always make it compleat, by taking a new unknown quantity, plus or^'°" may be
»«/«^^j fome- known quantity, and making it equal to the unknown quantity of raSd higher.
the equation, and then the transformed equation will have all it's terms com-
pleat. Moreover, if we would have the transformed equation to be of a fuperior
degree, let every term of the propofed equation be multiplied by fuch a power
of the unknown quantity, by which we would have the degree to be increafed,
and then the fubftitution may be made. Thus, the equation x* — a* zz: o
being given, if we would have it to be changed into another which is complear,
and of the fixth degree, let it be made x^ — aV zz o, and then making the
fubftitution oi x zz z •±, a^ (where by a is underftood any known quantity,)
and we (hall have the equation required. The calculation, for brevity, is
omitted.
165. When equations are reduced to fuch a form, as that they have their Problems are
greateft term pofitive, and without a co-efficient except unity; that they may °^^^" '"^^""'^
be free from fradions and furds, and compared to nothing, in order to j^dge^" °^^
whether the problem propofed be of that degree as is (hown by the equation, divifion.
we muft examine whether it have a divifor of one, of two, or more dimenfions,
by which, being divided, it may be reduced to a lower degree. For the pro-
blem is properly of that degree to which the equation may be reduced, and
not of the degree of the firft equation. If a cubick equation have a divifor of
one dimenfion, by being divided by that, it may be reduced to two dimenfions;
and the two roots of this, (which will be had by the rules delivered at § 73, 74,)
and the divifor, will be the three roots of the propofed equation. Whence the
problem, which has brought us to fuch an equation, is not really cubical but
plane, and may be conftruded by ruler and compafl'es only, that is, by right
lines and circles. If an equation of the fourth degree have two divifors of one
dimenfion, and if it be divided by them, it will be reduced to two dimenfions j
the roots of which, together with the two divifors, will be the four roots of the
propofed equation, and therefore the problem will be plane. After the fame
manner, if it have one divifor of two dimenfions, another of two dimenfions
will be the quotient, the roots of which, together with the roots of the divifor,
will be the four roots of the propofed equation, and therefore the problem is
plane. Further, if it have one divi(or only of one dimenfion, the reduced
equation will be of three, and the problem will be folid indeed, but of the
third degree only, and not of the fourth as it feemed to be. If an equation of
the fifth degree ftjall have three divifors of one dimenfion, or one of one and
one of two, (which is the fame cafe as if it had two of two dimenfions, becaufe
9 then
I^a A'NALYTICAt INSTITUTIONS. BOOK I.
then it will neccfllirily have one of one dimenfion alfo,) it will be reduced to
two dimenfions, and therefore the five roots may be had, and the problem will
be plane. If it have only one of one dimenfion, it will be reduced to the fourth
degree, and the problem will be of the fame degree. If it Ihall have two of
one dimenfion, or one of two, it will be reduced to the third degree, and the
problem will be of the fame. And the like of others. The manner of finding
divifors of one dimenfion has been taught before, at § 56,
And romc- 1 66. But befides, as equations may have divifors of two or more dimen-
titnes by fions, whether rational or irrational, we may operate with them in like
compound jxianner, and, by a like way of reasoning, we mud attempt the divifion of the
propofed equation ; but, firfl, having tried the divifion, by divifors of one
dimenfion, which ought always to precede, whatever the equation may be.
How cqua- 167. The manner of finding thefe divifors for equations of the fourth degree
i''"\"d^^-^ may be this following. For thofe of the third degree are either irreducible, or
may often be ^^V be reduced by a rational and linear divifor, being free from radicals, as is
reduced by here fuppofed.
two quadra-
tick divifors. Admitting, then, that the equation of the fourth degree is not reducible by
a diviTor of one dimenfion only ; let the fecond term be taken away (§ 160.),
and, for example's fake, let there be produced this equation, x* * — i jaax*
— 2oa^x — 6a* zz o. Let this be fuppofed equal to the produd of thefe two
equations of the fecond degree, x^ + yx -{- z z:z Oj and x"" — yx -1- « zz o,
in which jy, z, u, are general quantities, which are to be determined afterwards
as occalion may require; and z and u may have any fign. The produ6l of
thefe two equations will be x* • + zx^ — yzx + «2 = o. Now let this
equation be compared, term by term, with the equation propofed, and, from
the comparifon of the third terms in each, we fhall have z zi — 17^?' -|-y — «,
From the comparifon of the fourth terms, it will be a 1= Il-i-1 + z ; and,
inftead of z, putting it*s value already found, that we may have u exprefTed
by y only, and known quantities, it will be k zz — — ^aa + -fyy.
And, putting this value of u in the equation 2 =r — lyaa -i- yy — u, we Ihall
have 2 = — ^aa + fyy + - — • From the comparifon of the lafl terms,
we fliall have uz :=. — 6a*, and, inflead of z and », putting their values
fxprefTed by y only, and known quantities, it will be ~-a* — ^aayy ^ 3££fL
■i- -^y* ~ -^ 6a* i or, reducing to a common denominator,
/
SECT. IV. ANALYTICAL INSTITUTIONS. I53
y — 54'2!>'* + 289^^)'* — 400^2*^ — o. This transformed equation may be
confidered as of the third degree, becaufe it involves neidier _y^ nor y\ nor _y.
In tliis equation, let the divifois of the lad term be found, and, becaufe it may
be confidered as of the third degree, though it is really of the fixth, try if it be
divifible by yy ± thefe divifors, among which we are to choofe thofe only of
two dimenfions, as is plain. And it will be found divifible by yy — iGaa = o,
whence it will be yy — i6aa, and jy = Hh 4^. This value of jy being fubfti-
luted in the equations u — :^aa + "^yy, and z zz — ^aa -f-
2y Zy
\yyy we (hall have « ~ — 3^^, z = laa. Therefore the two fubfidiary equa-
tions ^* + j)w + 2 rrr o, and x^ — yx -^ u zz o, mud be .v* + A^ax + 2aa
zz o, and x* 4- ^^x — ^aa ■=. o, into which the equation x* * — ^7^*'^*
— 2oa^x — 6a* zz o may be refolved, by dividing by either of them.
But the roots of thefe are (§ 74.) x zz — 2<2 ± s/ laa for the fird, and
X zz ^a ±_ \/ jaa for the fecond ; which are therefore the roots of the given
equation, being all four real, one pofitive and three negative.
If the transformed equation diould not have any divifor, it would be to no
purpofe to feek another in this cafe ; for neither would the propofed equation
admit of any.
Although in the value of jy we have ^ = + 4^, yet I have made ufe of the
pofitive fign only, becaufe it is indifferent whether wc take the pofitive or the
negative root, the refult being the fame in both cafes. For, if we put y ■=
— 4^, it will be u zz laa, z zz — 3<z<2, and the two equations will be the
fame as before, that is, x' — /i^ax — '^aa z=. o, and x^ + ^ax -f- zaa zz o.
Let the equation be x* — 2ax^ + 2tfV — la^x 4- a* z=: o. Taking away
— ccx'^
the fecond term, by the fubditution oi x zi z 4- ^a^ it will be changed into
z* * 4- i^*2;* — a^z + ^^a* = o. Wherefore, making a comparifon of
— ccz^ — ac'z — ^a^c^
this with the equation 2* • + uz^ — pyz -i- pu zz: o, which is the produ(^
— y'^z^ 4- ^yz
+ pz\
of the two equations z^ + y'z -{- p zz o, and z^ — yz -^ u — o ; from the
comparifon of the third terms, as'ufual, we fliall have^ =^yy — a 4- ^aa — cc.
From the comparifon of the fourth terms, we fliall have u zz p " " ; or,
indead of />, putting it's value, u zz \yy -j- ^aa — \cc — ^ '^; and there-
ay
fore p z=. -\yy 4- \aa — \cc 4- "- —, Ladlyv fronvthe comparifon of tl>e
X lad
'54 ANALYTICAL INSTITUTIONS. BOOK I.
laft terms, we fliall have pu = ^^a*' — '^aacc; or, fubftituting the values of p
and «, it will be y + aay* — a*y^ — a^ ')
— accy* 4- cy — 2a\'' I = o.
— a'c* J
Now the divifors of the laft term, meaning thofe of two dimenfions, are aa
and aa -|- cc, and the divifion will fucceed by yy — aa — cc — o. Therefore ^
it will be yy =: aa + cc, and ^ = ± \/aa + cc Whence u zz :^aa —
a' + ac*- fl' + ace . . .
— / > P = T^^ H / y ^^d the two equations z + yz + «> iz o.
zV aa-^cc 2V aa +CC
and %■' — yz + « rz o, will be zz + zx/^a + t-c + -^^<3 -j rr=_ =r o,
iy aa + <^<^
and zz — 2V«^ + 6r + -|^z« .. = o, or zz + zv^aa + rr + i^^ -
7.y aa + cc
+ \a\/7a~\~c'c = o, and zz — Z\^aa + t<^ + :!-«« — t^-Z^TT^ ~ O.
Thefe two equations, being refolved, will give us four values of z j z =
— ■~:'/aa + cc ± v^— iaa + -^cc — -^ax^aa + cc from the firft equation, and
z n \>^ aa + cc ± V — t^^ ■\- \cc -^ \a»/ aa + <•<: from the fecond equation.
And, becaufe thefe are tire divifors of the equation
%^» ^2^ I 2 3 > =1 o, the fame roots fliall alfo belong to
this equation. And now, making the fubftitution oi x ^ \a -J^- z, we (hall
have a; =z -^^ — f v^^^ _}_ ^ ^ _l_ v/ - ^aa -t- -^ff — -\a>s/ aa + c<r, and at = f <3
+ ^\/aa 4-fc ± v/— f<za + ^<-f 4- t^V'a^ + CO which are the four roots or
values of the propofed equation.
This reduc- 168. But a general formula or canon may be formed, as vvell for the tranf-
tion may be formed equation as for the two fubfidiafy equations, which are affiimed in order
performed by jQ obtain the divifors ; to which formulas any equation whatever of the fourth
canon. degree, in which the fecond term is wanting, or taken away, may be uni-
' verlally applied. Therefore let there be this general equation x* * -±2 px'^ ± qx
±: r zz o ; and taking the two fubfidiary equations, x'^ + yx -{• z zz o, and
x^ —yx + « — o, and finding their produdV, x* * -{- zx~ — yzx -{- uz zz c,
— yy^"^ + ^yx
let it be compared, term by term, with the equation propofed. Now, from
the comparifon of the third terms, we fliall have z zi ± p -{- yy — «. From
the comparifon of the fourth, u zz z ± -^; and, inllead of z, it's value being
c fubftituted.
i;
SECT. IV, ANALYTICAL INSTITUTIONS. I55
fubftituted, it will be u — ± ip -i- t.7 ± -fj ; where it is + py if in the
propofed equation the third term be pofuive, and — p, if negative. And thus
alfo for q, if the fourth term be pofitive, and — q, if negitive. And this being
put inflead of u in the firft comparifon, we ihall have z — ± j-p + \yy
qp _i_ . that is, + p, if the third term of the propofed equation be pofitive,
and p^ if negative. And, on the contrary, — q, if the fourth term be
pofuive, and + j, if negative. From the comparifon of the laft terms, we
(liall find zu = ± r, that is, ± t/> + t;:y ± y- into ± \p -f t):)' 4= ^
— + r; and, by adual multiphcation, and reducing to a common denominator,
it will be y^ ± ipy^ + p'^y^ — qq :^ o, the transformed equation, which may
be called cubick ; in which it will be -{- ap, if the third term of the propofed
equation be pofuive, and — p, if negative. And it will be — 4^, if the laft
term of the propofed equation be pofuive, but + 4''> if negative. In the two
fubfidiary equations, inllead of z and u, if we put their values found before,
they will be xx + yx ±: --p -\- \yy =F — = o, and ;v.v — yx ± t/> -f- tjVJK
± -^ := o. Wherefore, if the transformed equation fhall be divifible by
yy -^i z. divifor of two dimenfions of the laft term, we fhould have the value
of ^, which, being fubftituted in thefe two laft equations, will fupply us with
divifors of the propofed equation. And if the transformed equation be not
divifible, neither will the propofed be fo.
Let the given equation be x* * — /\a^x^ — 8^^v + -i^ia* — o. Comparing
this with the canonical equation, it will ht p "=. ^.aa, q = 8a^, r ■=. 35^*; and
therefore the transformed equation will be y^ — ^d'y* -f i6«^'' — 64a* — o,
— 140^'^*
that is, y^ — Za^y* — 124^*^ — 64^*' = o. And the two fubfidiary equations
will be x^ + yx — zaa + iyy + — = o> and .r* — yx < — ^aa -f ^^yy
— — = o. Now, finding the divifors of the laft terra, becaufe the tranf-
formed equation is divifible by yy — i6aa = o, we fliall Irave yy n: i6aay
and thence _y — 4 j ; which values, being fubftituted in the two fubfidiary
equations, will give x"^ -\- /\.ax + yaa zz o, and x"^ — ^ax -f ^aa — o, which
are the divifors of the given equation ; the four roots of which are Af = — la
H; \/ — ^aa, and x z:=. 2a ±: %/ — aa^ all imaginary.
X 2 169. Some-
J 5^ ANALYTICAL INSTITUTIONS.
BOOK T.
Sometfmesa 169. Sometimes it will be fufficient only to take away the fecond term of the
rrll^bc re- ^^"''^^'0"' J" or^er to reduce it to a plane, and io to fpare any fuither operation.
duced to a Thus, for example, it will be in the equation
quadrauck. ^^.4 ^ ^^.cx^ — ^^^^r — ^^^^^^ — ^^^^ __ ^ . vvhich, becaufe it is not reducible
+ ccx"
by any divifor of the lafl term, if we take away the fecond term by making
X "izy — \Ci will be changed into this, jy* * — 2.7V ^ + 7^^ * 1 __
an affefled quadratick equation, the roots of which, being diminillied by the
quantity \c, by the fubftitution oi x z=: y — \c^ will be the fame as of the
. propofed equation.
Sometimes 1 70. This method requires, that the fecond term fbould be taken away
higher equa- from the equation, nor can it be expended be\ond equations of the fourth
refolved by degree. But here is another method, vvhich does not oblige us to take away
this method, any term, and which may be applied, not only to equations of the fourth
degree, but to thofe of the fifth or fixth, and fometimes to thofe of ftill higher
degrees. Let the given equation be x^ + ax^ + ^^V — a^hx — a^b — o ;
— abx^
and let there be taken two fubfidiary equations of the fecond degree, at* ■\- yx
H- « =r o, and ;v* + ja: + 2 rr o, in which the indeterminates, ^, ?/, j, 2,
are to be determined afterwards as occafion may require. The product of ihefe
will be x*" 4- jat' + ux"^ + usx -j- %« = o, which is to be compared, term
-f- sx^ + syx^ + 2yx
+ zx"
by term, with the propofed equation. From the comparifon of the fecond
terms, we (hall have 5 ■=^ a — y \ from the comparifon of the laft terms,
« ~ — — ; and from the comparifon of the fourth, yz -^ su r=z — a'h :
and, inftead of j and z, fubftituting their values, that we may have an equation
exprelTed by y and u only, and known quantities, it will be ^ = "^^ "^^ " .
And, becaufe we have found z« =: — a^b^ froin the comparifon of the laft
terms, therefore u ought to be a divifor of — a^b. Whence let the divifors of
— a^b of two dimenfions be found, (for thofe of one or three dimenfions will
not ferve to be fubfidiary equations of the fecond degree,) which are ± ab,
■±, aa. Let us begin by taking, inftead of «, one of thefe divifors, for ex-
ample ab^ which, being fubftituted in the equation y •=. '^"" . ^\^ , gives y zr
*[ , Therefore, putting thefe values ofj and a in the fubfidiary equation
SECT. IV. ANALYTICAL INSTITUTIONS. 157
x'^ + yx -\- u "==. o, it will be x"^ + —377 ■\- ah — o. And by this, if we try
the divifion of the propofed equation, and if it fhould fucceed, then x"^ + ^—--^
-\- ab zn o would be one divifor, and the quotient would be the other. Bur,
becaufe the divifion does not fucceed, we muft mai<e another trial, by taking,
inftead of «, the other divifor — ab of the laft term, and it will be v n: o ,*
and therefore the fubfidiary equation x"^ -\- yx -{■ u -zz o will become x'^ — ab
~ o, by which the propofed equation being divided, it will fucceed by giving
the quotient x'' + ax + aa =: o. So that the divifors of the propofed equation
are xx — ab zn o, and xx -[- ax + aa iz o.
Alfo, inftead of a, taking the divifor aa of the laft term, by which we fhall
find y zz a, and the fublidiary equation will he xx + ^.v -\- aa zz o. The
divifion by this will fucceed, giving the quotient xx — ab zz o \ that is, the
very fame divifors as before.
When all the divifors of the laft term are put in the place of //, and if the-
operation will not fucceed by any, it may then be concluded, that the equation
propofed cannot be deprefled, at leaft by this method, and that the Problem
remains of fuch a degree as the equation indicates.
But, without trying the divifion, taking, inftead of «, every one of the
divifors of two dimenfions of the laft term, and the correfpondent values of
J, J, z, we may fubftitute them in their ftead in the fubfidiary formulas,
XX -{- jy;f + « = o, and xx ■\- sx ->r "z zz o. And if the produdl of thefe will
give the propofed equation, they will be the divifors required. Thus, taking,
inftead of «, the divifor — aby we fhall have _y zz o, and therefore s zz a^
z zz aay and the two fubfidiary equations will be xx — ab zz o, and xx + ax
-\- aa zz o, the produd of which will give us the propofed equation.
Let the equation x* — lax^ 4- 2aax'^ — 7.a^x + <»* be given, and let it be
— ccx*
compared with the produ6t of the two fubfidiary equations
x^ 4- yx^ + wt* 4- sux -f- 2« zz o. From the comparifon of the fecond
4- ix^ 4- iyx* 4- zyx
4- zx^
terms, we Qiall have j zz — 2^ — y. From the comparifon of the laft terms,
2; zz — . We muft take the comparifon of the third, and not of the fourth,
in order to have the value of ^ exprefl^ed by c, (which letter muft neceffarily be
in the divifor, which could not be had from the comparifon of the fourth,) it
will be then « 4- jry 4- z zz 2.aa — cc. And fubftituting the values of j and 2,
it will be yy 4- 2^y zz -^ 2aa + cc -\- w, in which fubftituting, inftead
u
of u, one of the divifors ± aa of the laft term, fuppofe 4- aa^ and refolving
the
158 ANALYTICAL INSTITUTIONS. BOOK I.
the equation, it will be ^ rr — a ■+: \/ aa +cr. And putting, in the equation
' xx 4- r;^ -f- « — o, the values of « and y, {taking for the fign of the radical
cj4iantity either flui or minus as we pleafe, becauie it will be all the fame at lafl:,)
we /hall have xx — ax + as/ aa 4- cc ■\- aa — o^ by which the divifion
of the propofed equation will fucceed, making the quotient ax — ex •—
x\/ aa + cc ■\- aa zz o\ and confequently the four roots of the propofed
equation will be x ~. \a — ^s/ aa +rf ±- v/ — \aa 4- ^cc — \a'^ aa^cc^ and
X — l.a ■\- l\/aa +71- ± a/ — -laa + ~cc + -^a y^ ^^^r+~cc.
Let the equation be x* + 2hx^ + bbx'^ — a^b — o, and let it be compared
with the produd; of the tuo fubfidiary formulas as before. From the com-
parifon of the fecond terms, we Ihall have s — ib ~- y. From the comparifon
3/
of the laft, z = — ; — . From the cornparifon of the fourth, zy -j- j« zz o ;
and fubflituting the values of s and z, it will be + ibu — uy =: o,
that is, V r=z — -^ — . But, taking; every one of the rational divifors, -h aa,
•+;^ ab, of the laft term, and fubflituting in the place of «, and doing the reft
as ufual, the operation does not fucceed. 1 herefore we muft try by means of
the irrational divifors ± a^ab of the laft term j and therefore putting, inflead
of u, the irrational divifor a\^ ab, it will be _y ~ ^. Wherefore the fubfidiary
equation xx ■\- yx + u zz o will become xx -^ bx -\- as/ ab zz o, by which
the propofed equation being divided, there will arife the quotient xx + ba
-— aV ab iz. o.
Exemplified 171. Asto equations of the fifth degree, it is manifefb, that if they be not
in equations divifible by a linear divifor, as already fuppofed, they cannot be divided but by
rf the fifth ^^^ ^f ^j-^g fecond degree, and one of the third. Therefore for fuch equations
<J«^gre€. ^^^ ^^ taken two fubfidiary equations, one of the third degree, and another
of the fecond, and the product of thefe muft be compared, term with term,
with the propofed equation, in like manner as before.
Let this therefore be the given equation, x^ — 4^^* 4- 6aax^ — 8«V 4- c^a*x
— ^* = o. And let us take the two fubfidiary equations xx + yx + u =: o,
and x^ 4 ix* 4- j.v 4- 2 iz o. Of thefe the produdt is
x^ 4- yx* 4- ux^ 4- fax* 4- j«^ 4- 2« zz o j which "is to be compared with
4- tx* 4- iyx^ + syx* + zyx
4- sx^ 4- zx'-
the propofed equation. Now, from the comparifon of the fecond terms, we
ihall have / = — 4^ — jy. From the. comparifon of the laft terms, z zz
—. ^ . From the comparifon of the fifth, s = ^^ "^^ \ or the value of z
being
SECT. IV, ANALYTICAL INSTITUTIONS. 1^9
being fubftituted, s = - — • H ~ . From the comparifon of the third, we
fliall have finally « + ty -{- s zz. 6aa ; and, inftead of t and s, putting their
values, in order to obtain an equation exprefled by y, «, and known quan-
tities only, it will be yv + 4«?y — — — — 6aa + a + — . And be-
caufe, from the comparifon of the laft terms, we have z = — — , therefore u
will be a divifor of — a^. So that, finding all the divifors of two dimenfions
of — /2% they are to be fubfiituted, one by one, in the foregoing equation, in
order to have the value of ^, which is then to be put inftead of ^ in the fubfi-
diary equation xx -\- yx + « zz o, as alfo the value of u. And if the divifion
of the given equation fliall fucceed by this, we Qiall have our defire. Now the
divifors of two dimenfions of the laft term are Hh aa. Let us take + aa,
which being fubftituted inftead of u in the equation aforegoing, we ftiall have
yy + ^ay — o, that is, j =: o, and jy =z —3a. If, in the fubfidiary equation
ata: 4- JV.V + « ~ o, we put the divifor + aa inftead of u, and befides, if we
put o, which is one of the values found, inft,ead of ^, it will become xx + aa
rz o, by which the divilion of the propofcd equation does not fucceed.
Therefore, inftead of _y, we may put it's other value — 3<?, and we (hall have
XX — ;^ax -i- aa zz o^ by which the divifion fucceeds, and gives x^ — ax*"
+ zaax — a^ zr o in the quotient. If the operation had not fucceeded by
means of the divifor -t- aa, we muft have tried the divifor — aa ; and if
neither by this we had obtained our defire, we muft have concluded the equa-
tion to be irreducible, at leaft by this method.
Let the equation be x^ + ax* * -4- ^V — aabbx — a^b ~ o, which Is to
— a*i?x*
be compared, term by term, with the produd of the two ufual fubfidiary
equations J and from the comparifon of the fecond terms, we (hall find f — a —y.
From the comparifon of the laft terms, z zz — - — . From the comparifon of
the fifth, su -{- zy zz — aalb. Now, inftead of z, fubftituting it's value, it
will be J = 1 — . F'rom the comparifon of the third, we (hall
have « -f (y 4- J — o, in which, inftead of t and s, putting their values, it
will be yy — ay ^ n "" ~" "'^ . The divifors of two dimenfions of a*b
are + aa^ and ± ab. We muft try the operation by means of the divifor
— ab. And therefore, inftead of », putting it's value — ab in the laft equa-
tion, it will hz yy — ay — ^ = o. Thence ^ z= o, and y = 1-i-^. In
the fubfidiary equation xx -{■ yx -j- « = o, inftead of j' let it's value ^'^ "^/^ .
be
l60 ANALYTICAL INSTITUTIONS.
BOOK I.
be fubflituted, and — al^ inftead of u, and it will be xx -\- ax + -^ x — ai
=: p, by which the divifion does not fucceed. Therefore take the other vali>e
of ^, which is o, and the fubfidiary equation will be xx — ab =. o, by which
ihe divifion of the propofed equation will fucceed, and the quotient will be
A"' -{- ax'^ 4- abx -1- a^ =z o.
We were at liberty to make a comparifon between the fourth terms ; but, for
greater fimplicity, 1 made choice of the third terms.
Ecjiiations of 1^2. Equations of the fixth degree, fuppofed not to be reducible by any
tie "'t oe- jjj^eg^^ divifor, cannot be otherwife reducible but either by three divifors of two
"Tee reioived. . ^ . , ■'
" dmienfions, or by one of two dimeniions and one of four, or by two of three
dimenfions. But it will be fufficient to examine the two cafes, in which they
are reducible by two of three dimenfions, or by one of two and one of four.
For as much as reducing them by one of two, the reduced equation will be of
four dimenhons, which may afterwards be reduced by two divifors of two di-
.menfions, if the propofed equation be reducible by three of two dimenfions.
Let the equation given be this: a;* — iT,ax^ + ^^^aax* — jia^x^ + 57^?*;^'
•— i6r<\v + 2«* ~ o, which is required to be reduced by one of two dimen-
fions, and one of four. Let therefore be taken the two fubfidiary equations
XX -I- j'X + « = o, and x* + px^ -{- tx* + sx + z := o, of which the
produd; is x^ + /'a* + A* + sx^ + zx"" + zyx + 2a = o.
+ jA.' -f- pyx"^ + iyx^ + syx^ + sux
4- ux* + pux^ + tux^
Now, from the comparifon of the fecond terms, we fhall have p zz — 13^
— y. From the comparifon of the laft terms, z z=. — -. From the compa-
rifon of the third, t -^ -py -\- u zz 4saa ; and by fubftituting the value of p,
it will be / zz. 4^aa -f 1 7,^y -\- yy — u. From the comparifon of the fixth,
' zy -{• su ■= — \6a^ j and puttmg here the value of z, it will be J zz —
— i-^ . From the comparifon of the fifth, z + 47 + /« — ^ya* ; and fub-
^flituting the values of z, s, and /, that we may have an equation exprefTed by
u and y alone, and by the known quantities of the propofed equation, it will
be at laft ~ — ^* — ^^ + 45«'« + ' 3^J« + «/ — «* = 57^"- That
js, yy 4 — ^ — I ^--1 — ; y- i— !2 — q. And, becaufe the
divifors of two dimenfions of the lafl term 2«* are ± aa, and ± laa, we mufl
make a trial, by putting in this laft equation, inftead of «, the divifor + aa,
and it will be yy + ^ny + iiaa zz o, which, being refolved, will give
y =2
SECT. IV, ANALYTICAL INSTITUTIONS. l6l
y '=• "" 3^ ± \ - 3^aa ^ Whence the fubfidiary formula xx -f-^.v + k zi o
will be XX — — ^^x •\-aazzo. But by this, even though we Ihould take
the alternative of the figns of the radical, the propofcd equation is not divifiblci
nor will it fucceed if we fhould take the divifor — aa ; therefore we mud take
4- 2<3^, and we (hall have yy + nay + 2o^a == o, that is, ^ ::z — 6^ ± 4^,
or _y r= — 10^, and y :=. — la. Take y z=. — loa, and fubflitute it in the
fubfidiary formula xx -\- yx -\- u z=. o, and — loa inflead of jy, and + 2aa
inftead of «, and it will be xx — loax + laa zz o. But by this the divifion
of the propofed equation does not fucceed. Therefore take the other value
of ^, or — 2aj and the formula will be xx — 2ax + laa zz o, by which the
divifion fucceeds, making in the quotient x* — ii^.v' + 2iaax'^ — 'ja^x
+ <?* zz o.
Here it may not be amlfs to obferve, that, inftead of the comparifon of the
fifth terms, if I had made a comparifon of the fourth, I (hould have fallen
upon the cubick equation 2jy' + 26^_)'* + Ziaay + 74^' zz o. But the com-
parifon of the fifth terms has brought me to a quadratick equation only.
Hence it may be feen, that the choice of the comparifon of fome terms rather
than of others may be of good advantage. Yet, however, this cubick equation
might have been of ufe ; for, finding it*s roots, which are y ■\- ia zz. o, and
J H ± \/ ^*jaa = o, one of thefe, _y z: — 2^, would have given me the
fame equation xx — ^ax + ^aa =: o, by which the propofed equation may be
divided.
Let x^ + '3,ax^ + A^aax'' + (id^x^ + 6^V + "^a^x + la^ =■ o, be the given
equation of the fixth degree, not reducible by a divifor of two dimenfions.
Let us therefore attempt the redudlion by two equations of three dimenfions,
and let us take thefe two fubfidiary equations, x^ + yx"^ -{- px -{- u zz o, and
^ + /.v^ + sx + z =z Oy oi which this is the produd ;
A* + yx^ H- px^ + ux^ + tux^ + sux + z« =: o.
4- tx^ 4- tyx"^ + ptx^ + psx'^ + pzx
-f- sx^ ~f- syx"^ -i- zyx^
4- zx^
Now, from the comparifon of the fecond terms, we (hall have / zi 3^ — jy.
From the comparifon of the lad terms, z zz. — , From the comparifon of the
fixth, su -\- pz zz 3^* ; and fubflituting the value of 2, it will be j iz ^ —
-^ . From the comparifon of the third, p 4- /y 4- j = /^aai and fubflituting
the values of t and ,, it will be p = -""""'- ^''''' "^ 'f " ^"'"-^ . From the
Y comparifon
l62 ANALYTICAL INSTITUTIONS. BOOK I.
compnrifon of the fourth, « + p/ 4- jjy + z — 6^' ; and, indead of /, j, 2,
fubflitiinng their values, that we may have another value of />, expreffed by //, j,
and the known quantities ot the equation, it will be ;6 z= — —t .
Now, between thefe two values of p let an equation be made, to obtain the
value of y expreffed by u only, and the given quantities of the equation. This
Will be -^ ^— ^ zi — -—^ . 'Then, re-
uu — za° ^auu — u74y — 2a y
ducing to a common denominator, and ordering the equation by y^ it will be
— 6au^y'^ — 6a^uuy -f- p^V
-f I'^a^u^y -— I'la'^u
+ 4^"
r
= o.
u^ -j- 2a u
And, becaufe it is uz zz 20^, we (hall have u a dlvifor of 2^*. But the di-
vifors of three dimenfions of 2a^ are ± a^, and ± 2^'. Whence, taking one
of thefe inftead of «, fuppofe + a\ and fubftitutmg it in the laft equation, we
Ihall have _>'^ — 4<3)'* + s^ay — 2a^ zz o. From hence muft be extracted the
values of jy, one of which is y zr 2a, which, being fubftituted in one of the
values of /> inftead ofjy, and putting inftead of u the divifor a^, it will be
p ■=. aa. Wherefore, fubflituting thefe values of _y, p, and Uy in the fubfidiary
formula x^ + yx^ + px -i- u "^ o, it will become x^ + 2ax^ + aax -^ a^ zz o,
by which the propofed equation being divided, will give the quotient x^ + ^a?*
-j- aax + 2a^ zz o. If the divifion had not fucceeded by taking y = 2^, I
muft have taken y zz a. And if I had not attained my purpofe by this, I muft
have made trials with every one of the other divifors, repeating the fame ope-
rations. And if it had fucceeded by none of thefe, the propofed equation
could not have been deprefled, at leaft not by this method, but would have
remained of the fixth degree.
Let x^ + ax^ + ciax^ + g^^jv' + a^x"^ + a^x + 2a* z: o be the equation,
which is to be compared with the produdl of the two fubfidiary equations, as
in the foregoing example. From the comparifon of the fecond terms, we (hall
have t -zz a — v. From the comparifon of the laft terms, z zi — . From
the comparifon of the fixth, su -{- pz zi a^ ; and, inftead of 2, putting it's
value, it will be j zz -^ ^ . From the comparifon of the third, p + ty
+ J = tf« ; and putting the values of / and s, k'ls p zz ^^"---"-y+Jj-yy-^'-^
uu — 2a°
From the comparifon of the fourth, u + pi i- sy •{• z zz ^a^ i and fubllituting
the
SBCT. IV, ANALYTICAL INSTITUTIONS. 1 63
the values of z, s, /, in order to have another vakie of />, exprefTed only by
u, y, and known quantities, it will be /> ~ 3^ nu - a^xiy ^ ia u^ n ^ jyj^j^^ ^^
auu — uuy — ^a y
equation between thefe two values of />, that we may have the value of ^ given
by u only and known quantities -, and when all the neceflary operations are
performed, it will be
^' — lau^f" 4- iaaii}y + ^a'^li^
— 2a' uy^ — lev' uuy + ^,V
+ la^uy — (iC'^u
— - «* \ — o.
The divifors of three dimenfions of ia^ are +; «^ and ±i 2aK Inftead off/,
take the divifor + a^, to be fubftituted in this laft equation, which then will be
reduced to y^ — -tay^ + ^aay = o. And dividing by j, it will be _y 1:1 o,
and J* — ^^ay + iaa iz o j that is, j = ^^ — ^ " , Of thefe three va-
lues of ^ take the firft, or ^ n o, and fubftitute this inftead of y in one of the
two values of />, and a^ inftead of «, and it will be /> zz o. Then the fubfi-
diary equation x^ -f yx^ + px + u zn o will become .v' + ^' n o i by which
the propofed equation being divided, will give x^ + ^a'- + a^x -{- la^ z= o
for the quotient.
In fuch equations as thefe, if it were known at firft that they are divifible by
a divifor, in which fome term is wanting, much labour might be fpared, by
taking one of the two fubfidiary equations without that term. But, becaufe this
4s not known, we may firft try the operation with one of thofe fubfidiary equa-
tions, which wants either one or more terms. Neverthelefs, becaufe the labour
would be loft, if the propofed equation be not reducible by this means, and there
will be need at laft, notwithftanding this compendium, to have recourfe to
compleat fubfidiary equations, it will be better at once to ufe this general
method, becaufe it gives the divifors in both cafes.
Without repeating the operations at every example, I might have formed a
general canon, to which every particular equation might be referred, after the
lame manner as that at § 168. But befides, as this may create fome confufion,
it fcems to me that adtual operations made on purpofe afford more light, and
have a better effedl j therefore I have rather chofe to confine myfelf to them.
173. After the fame analogy, we may apply this method to equations- of a Applied to
fuperior order, but the calculation increafes beyond meafure. For, if we are ^vs^er equa-
Y 2 to^>°"*'
164
ANALYTICAL INSTITUTIONS.
BOOK I.
to reduce an equation of the eighth degree, for example, by means of tv\o
equations of the fourth, in which no term is wanting, each of the two fubfi-
diary equations muft have four indeterminates, or general co-efficients. Whence,
if we confider one of thefe equations, fuch as this, x* + yx^ + px"^ ^ qx + u
=z o, and take for u one of thedivifors of the laft term of the propofed equa-
tion, there will remain three indeterminates, j, p, q, to be determined by the
ufual comparifons, in which there will occur folid equations, whofe roots are
to be extracted, in order that the operation may proceed.
PROBLEM T.
metical pro
Applied to 174. To find four numbers, which exceed one another by unity, and their
thefoIutJon product is lOO.
or an anth- *^
Make the firft number equal to x, the fecond will be a* + i, the third
.V + 2, and the fourth a? + 3. Therefore their produdt will be x^ + 6x'
+ 11 A* + 6a; zz LOG, or ^* + 6a*' + i\x^ -{- 6x — 100 zz o. Now, be-
caufe this equation is not divifibie by any divifor of the laft term, we muft
make the fecond term to vanifli by the fubftitution of a; =: z — -f, and there
will arife the equation z"^ • — \z* * — '-^ = o, which is an affeded
quadratick, the roots of which are zz zz \ ^^ Vioi, and therefore z "Z:,
± y/l ± a/ioi. Whence we fhall have .v n — ■ 4 + y/^. ± y^ioi. There-
fore, of the four values of a, two are real, that is, ;; zz — |. it v/| + a/ioj,
and the other two are imaginary, if we take one of the real roots, — -I- +
a/-| + v/ioi, for the firft number of the four that are required, then — f -h
\/i + Vioi will be the fecond, ^ + v/| + -v/ioi will be the third, and ^ 4-
\/|. + vioi will be the fourth : the produd: of which numbers will be found
to be 100. If we fhould take the other real value of .v, that is, — -I- —
y/ ^ + ^loii for the firft number, then — t — • \/^ + ^loi would be the
fecond, f — s/^lTvToi would be the third, and -I- — v/s- -j- ^loi would
be the fourth i the produd of which numbers would alfo be found to
be 100,
PRO-
SECT. IV.
ANALYTICAL INSTITUTIONS.
165
Fig. 91 i
PROBLEM 11.
175. In the right-angled triangle AECAgeometri-
the leller fide AB is given, and, letting fail^al problem,
the perpendicular BD to the bafe AC, the
difference of the fegments AD, DC, of the
fame bafe AC is given alfo ; it is required
to find FC, the difference of the fides AB,
BC.
With centre B, diftance BA, let the circle
AEFG be defcribed, and make AB 1= ^,
CE = ^, the given difference of the feg-
ments AD, DC ; and make FC, the dif-
ference required, zz x. It will be GC —
2a + X, and, by the property of the circle,
GC X CF z= AC X CE, that is, lax -{■ xx — AC x i^, and therefore
AC = !fl+±!. But, becaufe the angle ABC is a right angle, we lliall
have the equation ^^^ — ir~~^~~ ~ ^^^ "^ ^^^ "*" '^^' °^' ^^ redudion,
«* + 4ax^ + 4aaxx — zai'l'X — 2aal>l^ iz o. Nov^f this is not divifible by
— i?l^xx
any divifor of the lad term, and therefore we muft take away the fecond term by
the fubilitution of jc zz z — a ; whence we Qiall have the affeded quadratick
2.* — 2aazz + a*
}=o.
the roots of which are 22; =
y. 2aa -\- hb ± y^sTabT+l?-
± ^/ .
taa ■\- hh ± \/%aabb + h^
and thence z r:
So that ;v — — <2 + v/
2aa + bb ± y/'iaabb + b*
which are the four roots, and all real, when a is greater than b. The root
X "=■ — a ■\- V aa -{- \hb + b\/ iaa-\-\bh, which is pofitive, is adapted to the
propofed Problem. The negative root x ■=. — a •\- \/ aa + '^bb ^b\^2aa-\-\hb
is adapted to the cafe, when the fide BC is lefs than the fide AB ; the other two
roots ferve for the angle ABG.
PRO-
i66
ANALYTICAL INSTITUTIONS.
BOOK. I.
PROBLEM HI.
Another
geometrical
problem.
176. Having given the fqiiare AD, in
the fide AC produced, to find fuch a point
E, that, drawing the right hne EB to the
angle B, the intercepted Hne EF may be
equal to a given right line c.
Make BD = a, DF — x; it will be
CF = a — X. And, drawing BFE, make
FE = c. Now, by fimilar triangles, ECF,
BDF, it will be CF {a — x) . FE [c) ::
FD (x) . FB =
But, becaufe of the
right angle at D, it will be alfo FB zz
v/«« + AA- ; whence we fliall have the equation \/aa + xx = -^ ; and, by
Iquaring, ^^ __ ^^^ . ^.^ = aa -{• xx ; and, reducing to a common denominator,
and then ordering the equation, it is x* — 2ax^ + laaxx — 2a^x + a* = o,
— ccxx
the roots of which may be feen, at § 167, 170, to be x •= ia — i\/aa +
± ^^ircc —
aa
+ tv/
aa + cc
a^/aa .+ cc-i and x z=. \
The two laft roots are always real and pofitive ; the latter of which, being
lefs than ar, determines the point F, through which the line BE being drawn,
EF will be equal to the given line r, and refolves the Problem propofed. The
ciher of the two, which is greater than a, determines the point /, to which
drawing the right line B/, gives us alfo ef equal to the given line, and ferves as
if the Problem had been propofed by the angle AC/.
The two firft roots are imaginary whenever cc is lefs than Zaa, and the
Problem will be impoffible. But, fuppofing cc not lefs than ^aa, the two
roots are real and negative. Taking, therefore, DG ~ f^ — \\/ aa + cc
— ^\^ aa + ^^ ~~
+ y/^'^tc —
\/ \cc ^ -iaa
aa
— ^a\/ aa 4- cci and D^ = \a
a\/aa^^T, and through the point B drawing the right
lines GM, gniy they will both be equal to the given line r, and would ferve
were the Problem propofed for the angle ACD«
177. Very
SECT. IV,
ANALYTICAL INSTITUTIONS.
167
i^6^- 93-
177. Very often, when the Problem is not Veally folld, but plane, it may How lil^Ker
appear as an equation of three dimenfions, by making ufe of fume certain line '^'l"i^^i'^"''^
for the unknown quantity; but, by ufinj fome other line for the unknovvn ^jnieg tn.
quantity, it may put on the form of an equation of two dimenfions only. I Qiall avoided.
take an example of this in the foregoing Problem, in which, making DF rr: x,
there has been found an equation of the fourth degree, by which means we
have been obhged to take the trouble of reducing it. But, fuppofing E to be
the point required, draw ER perpendicular to
BF, which may meet BD produced in R, and
EL perpendicular to BR. Then makeDRrr^-,
and, as before, BD n a, FE zz c, and BF —y,
another unknown quantity to be elitninated
afterwards; it will be BR zz a -{- x^ BE r:
c -{- y. Now, becaufe of fimilar triangles,
BDF, ELR, it wiil be ER — y, becaufe of
EL — CD =: BD. And, becaufe of finiilar
triangles, BRE, ERL, it will be BR . BE :: ER . EL. Therefore it will be
^~ -i- X . c -{- y :: y . a ; whence cy -\- yy =z aa -j- ax. Bur, becaufe of the
right angle BER, the fquare of BR is equal to the fum of the fquares of
BE and ER ; that is, aa + 2ax -^ xx zz lyy + icy + cc. Therefore, in-
flead of cy + yy, putting it's value aa -\- ax, the equation will be aa + 2ax
+ XX zz. laa + zax + cc, that is, x zz =i= >/ aa + a-.
Again, after another manner. Bifecfl FE in H, and inaking CD = a, let
the given line be ^c, to which FE ought to be equal. And making BH — .v,
it will be BF = ;tf — c, and BE =: ;v + ^. But BE^ — AB^ = AE^ ;
therefore it will be AE = \/l^x'^r-J^'Vcc~^^^a. Now, becaufe of the fimilar
triangles, BDF, BEA, it will be BF {x — ^j . BD {a) \\ BE (at + c) . AE r:
\/ XX + zcx + cc — aa • Whence ax -\- ac zz x — c X s^xx + 2,ax + cc
aa \
and, by fquaring and ordering the equation, it will be finally
X* — laaxx — %aacc "1 o-ni j-i • ri-i
— ^ccxx 4- f* > == o, an anected quadratick equation, or which
the four roots are * — dt V aa -^ cc ziz a-^TT'^r^c.
Fig. 91.
After the fame manner in Prob. II. § 175,
if, inltead of making FC :=: x, I had deno-
minated BC "zz X \ by purfuing the fame
argumentation, I (hould have found the
equation x* — lanxx + ^* 1
— bbxx — aabl? J ••
an affeded quadratick, of which the roots
are ^ = ± \/aa + "^bb =1= by/zaa-^^bb, which
agree with thofe before found.
Again,
1 58
ANALYTICAL INSTITUTIONS.
BOOK r.
Again, in a fimpler manner. Make AE r: x, and, arguing as before, we
fliould have the equation xx -f kx zr 2aa, and therefore x zi — ^l> dz'^2aa-\-^l>b^
And, bccaufe we fliould find the e^preffion — a + \//,6 4- zLv + xx -^ aa for FC,
inftead of x putting the value now found, we (liould have what is required, or
the (ame value for FC as before.
Or otherwife, 1 78. Another artifice may be tried for fuch like Problems, when they bring
by finding ^,3 ^q ^ (oWd equation, and yet are not fuch in their own nature. This is,
the'fame'^^ ° retaining the fame line for the unknown quantity, by which the firlt equation is
quantity. found ; then, by means of another property, to find a fecond equation, and to
equal one to the other. From their.com parifon, a third equation will arife of
an inferior degree. See an example of this in the following Problem,
PROBLEM.
This exem-
plified in a
geometrical Fiz, 04,
problem. ^ ^^
179. In a given circle, to infcribe a regular heptagon.
Let the given circle be ABFGCDE, with
centre H, radius HA :=: r, and let the fide
of the heptagon be AB zz BF zz FG, &c.
iz X. Let AB be bifefted in I ^ it will be
AI rr -^^ = IB. And drawing IC, which
will neceffarily pafs through the centre H, it
will be HI zz ^rr - %xx> CI z: r + y/rr-^xx,
CB zz V 2.rr + 2r'/rr — j^xx. Let there be
drawn CE and HD ; the triangles CDK,
HI A, will be fimilar, becaufe of the two right
angles CKD, HIA, and of the angles DCK,
AHI, the firft of which, becaufe it infifts on
the arch DE, will be double to the angle
ACI, which infifts on the half of DE, and therefore is equal to the angle AHI
the double of the fame angle ACI. Hence we (hall have, by the fimilitude of
thefe triangles, CK =
and HK = /
V rr — \xx —
2r
CE =
V
4rrxx
rr
\rrxx — AT* 2rr — XX
4rr
2r
But the triangles CEN, CHK,
are alfo fimilar, the two angles at K, N, being right ones, and the two angles
KCH, CEN, are equal, becaufe they infift on two equal fegments. Therefore
8 it
■SECT. IV, ANALYTICAL INSTITUTIONS. 1 69
v/:
It Will be CN — ——' , and CB =
2rr
— XX
X
V ^rrxx
—
X*
X
ri '
277- —
■ XX
\^^rrxx
-
X''
r3
•-a:'*;
-^nd thence the equation s/ rr + r'/4rr — aa- = -
Therefore, Squaring, it will be ^rr + r\/ ^rr — xx — 4> "^"^^-^ +-^ y^ ^rrxx-
and fquaring again, and ordering, we (hall have x^* — i6r*.v" + i04r'*ji:''*
— 352?-^^^ H- 66orV — 6j2r'°x* + 336r'V — 63r'-^ zz o. But this equa-
tion is divifible by x'' — ^r'^ zz o. When the divifion is performed, we ihall
have A^" — i3rV° + 65rV — i57rV + iSq^V — i05r'V- -{- 2ir'"z:o,
which is not divilible by any divifor of two dimenlions; wherefore the Probletn
feems to be of twelve dinienfions. Therefore I refolve this Problem in another
manner, retaining the fame unknown quantity x ■=: AB =: BF n &c. Becaufe,
in the triangles HCD, CDL, the angle CDH is common, and the angle at the
circumference DCL, which infifts upon the arch CD, the half of DA, thefs
triangles will be fimilar, and therefore we fliall have DL=— , and LH — r — — .
But the angle DLC =:: DCH i= EDH ; wherefore the angle HLM, which is
equal to the angle at the vertex DLC, will be equal to the angle EDH; whence
the two right lines LM, DE, will be parallel, and the triangles HLM, HDE,
will be fimilar, and therefore it will be LM = ^^^~^ . But CL zz CD zz x,
rr ^
(the triangle LDC being fimilar to the ifofceles triangle HDC,) and CLrz MA,
becaufe the angles HLC, HMA, are equal, and therefore the triangles HLC,
HMA, are equal and fimilar. Therefore CA zi 2X + ^-^^LZJL, And, becaufe
CA — CB, the equation will be ^^^^ "" " = ^2rr + r/4rr - xx. And, by
fquaring, gr*x^ — 6r^x* + x^ zz 2r* -f rW ^rr - xx. And, by fquaring
again, and ordering the terms, the equation will be «'° — iirrx^ 4- 54rV
— ii2rV + i05rV — ^^r^° = o.
And thus I am arrived at another equation, which, becaufe it is of an inferior
-degree to the firft, muft be multiplied by fuch a power of the unknown quan-
tity, as is neceffary to bring it to the fame degree, fo that it may be compared
with that. Therefore, multiplying it by .vat, it will be a?" — i2r*x^° + 54? V
— ii2rV + io5rV — 35r'V zz ^" — i3r^v'° -f 65rV — ly^rV 4-
i89rV — 105?^^ + 2ir". Now, fubtrading the firft from the fecond, it
will be ^" — iirV + 45r*;c* — 84rV + yor^x"" -^ 2ir'° =z o. Which,
becaufe it is of the tenth degree, being compared with the fecond equation
found above, and fubtraded from the fame, will be x^ — 9^^ 4. 28rV
— 35^^^* + 14^' = o, which may be divided by xx — 2rr; and making this
divifion, we (hall have at laft this equation of the fixth degree, x^ — yr'^x*
rt i^^x^ — yr^ zz o.
Z ^ I have
I/O ANALYTICAL INSTITUTIONS. BOOK I.
I have proceeded in this way, to fliow the ufe of the method. For otherwnfe,
I n-.ight have gone more diredly to the fame equation, by comparing together
the two vakies of tlie fquares of CA, found in the two different folutions of the
Problem \ that is, ^— i of the firft, and ^^--
of the fecond. For, making an equation between thefe two vakies, and takinor
away the terms that deftroy one another, it will be x^ — /rV + \\r*ic*'
— 5rV^ — o. And, dividing by a?*, it will be x^ — 7rV* -f i4rV — c^r^
= o, as before. We might alfo, after a more compendious manner, have
divided the equation firft found by x^ — 6rV -h 9rV — Cyr^ ~ o, and the
fecond by .v* — 5?'V + 5^* = o ; and in each cafe we fliould find the equation
tK^ — 7rV + 14^ V — 7r* zz o.
Yet the propofed Problem is not of the fixth degree, thonglx it may feem to be
fuch, notwithftanding all this care we take to deprefs it. To make this appear,
we will retain the fame compofition of the figure, and make HI — x. Then it
will be AI = \/rr - XX 1= IB, CI = r -f iV, CB ZZ \/rr + zrx + ar;f + rr - ^x
= \/ 2rr + %rx* Then, by purfuing the fame way of arguing as before, we
Hiall have CK z= idlL^L^ ^ HK = ^ !liLi!:!£ +J£! ^ rr - ..x ^^
r rr r
2CK = -^ s/77':rrx, CN = I2:f_zii' x v";^^^;-, cb = 2CN =
rrx — 1 X ^ ^^ ^ ^^ g^^ ^^ h2L\t before found CB = \/ 2rr + irx* There^
' . - .,11 • %rrx — i6a'* , > . .
fore the equation will be s/ 2rr + zrx :=■ 5 x v rr -- xx»
Now I (hall feek another equation after a different manner, but fhall retain
the fame unknown quantity HI zi x. By the fame reafoning as above, it will
be DL = l!L=L±fl , LH ZZ r - flZLlff - l^^jiin, lM = a^^^^^
f r r rr
X 4;rA: — 3rr, CA ZZ 4v/rr - ^^^ + is/ — - — X "^^^T^^^^r ', that is, by re-
duftion, CA zi ^ffL:zJILy/j;—r^ z= CB. Whence the equation V 7.rr + 2rx
-- 8x.y - arr ^^^ ^ ^^ , ^^^ laftly, by equalling the homogeneum comparationis
rr
- , . ••Ml ^rrx — i6;i;3 %xx — 2rr . , . ,
of each equation, it will be 5 v rr — ;rx — — jij: v rr — xxi wnicn,
being reduced, will be 8*' + /^rax — js^rx — r' = o, an equation only of the
third degree.
180. When
SECT. IV. ANALYTICAL INSTITUTIONS. I7I
180. When the methods above- defcribed have been put in praflice, if the Solid Pro- •
equations cannot be deprefled, but ftill remain above the fecond degree, we may^i^'^f"^
proceed two ways in the folution of Problems, which arife to three or more Cardan's
dimenfions. The way of leafl: general iife belongs only to equations of the rule?, or by
third or fourth degree, and confills in refolving them by unravelling the ana- <^°'^^'""*^^°"*
lytical values of the unknown quantity, which therefore will prefent ihemfelves
under the form of cubick roots ; which method is called Cardan's Rule. The
The fecond way is more general, and of much more extenfive ufe, and confifts
in finding the geometrical values of the unknown quantity, by means of the
interfeflions of certain curve-lines, which are purpofely introduced into the
equation ; that fo the propofed Problem may be conftrudled.
181. But, to begin with the analytical folution. I fuppofe the equations to How by the
be without the fecond terms, becaufe they may always be reduced to fuch, if four cafes of
they are not fuch already. And all equations of the third degree, wanting the ^'*''^*" ^"^"^^^
fecond terms, are comprehended under diefe four canonical formulae.
I. x^ -— px — q zz: o, \\. x^ -{• px — q zz o.
III. x^ — px + q zz O. IV. x^ •\- px ■{■ q zz o.
Make x zz y + Zy then px :r, py + pz, and x^ zz y^ + 3^*2; -f- 3^-2* -|- 2'.
And, fubftituting thefe values in the firft equation, it will be y^ -\- ^y^z
+ 3y^^ +2,' — py — pz — q zz o. Of this we may form two equations,
which are 3^*2 + ^yz^ =z py -^ pz, and y^ + z' zz q. Dividing the firil
by ^ + z, we (hall have 3J2 zz />, or ^ = -^ . This, fubftituted in the fe-
cond, will give -^ -{- 2' zr ^, or 2" ^ qz^ zz — -^\p'^. Whence, by the
rule for affeded quadralicks, 2* — qz^ + ^qq =t "^qq — ■y'^tP^^ and 2^ = f ^
+ V^i?!?-Vr/'- Laftly, it will be 2 zz V \q + v/iyy - Vy/^. In the ex-
tradion of the fquare-root, I have taken only the pofitive fign, becaufe the
negative would bring no variation, and gives at laft for the value of x the fame
quantity as the pofitive, as may be feen from the calculation. And it is to be
iinderfl:ood in like manner in the other canonical equations. Now, becaufe
y + 2' = q, it will be therefore j^ — 5' — f^ -— N/iy^-^v/^, and thence
y = \^iq — ^"kll - -rV/'' But it was at firft x zz y -{- z ; therefore x zz
V t^ — V^qq^^^p^ + \^iq + V iqq _ ^,pi. Hcncc it is feen, that the
alternative of the figns, which was omitted, makes no variation.
182. The fecond equation a;' 4- /^a? — ^ — o, making the fame fubftitutions, By the fecond
will be/ + 3/2 + 32;^ -{- z^ -\' py + pz ^ q zz o. From hence let ^^^^ ^^^ *^^
the two equations be formed, 3^*2 + 3^2' ^ — py — Z^, and j^ + 2* = q,
Z 2 From
^73* ANALYTICAL INSTITUTIONS. »00K 1*
From the firfi:, we have qyx iz — _p, or v =: ^ ,, which, f;'.b{titiited in die
fecond, gives — -^ -f z^ =z f, or 2.* — qz^ = V_p'. And therefore 2' —
iq + VJ?f+l^, and 2: = V-^^ + '/JfT+T^- I^i^^ JV^ + a* = f, there,
fore ;' - '^W^'VW+7^> and a; = V^^^^^WTT^ + v^i2 + v'i'i/TT?^.
Tlie third 183. The third equation x^ — px -^ q ::: o, making the fubftitutions, will
^^ *-'• be y^ + ^y^z + 372^ + 2;* — /vy — pz + q = o. Let the two equations be
formed, 3^^^ + j^a" =z py + pZy and jy' + 2^ z= — q. From the firfl, we
have 3^2; 1= />, or jy z= — , which, fubftkuted in the fecond, gives -^ — + 2.^
= — ^, or s' + qz' = — -.\p' ; and therefore 2' zz - t^ + -/I^TZ^^,
. and thence 2; = '^ — i^ + \^ Xqq-^^p^- But _>^ + 2;' = — q -, whence
y = n/— ^^q — > 7I7^ ., _.^^3, and lallly, x - %/— j^q — ^^^^--V/^ +
The fourth 184. The fourth equation a'' + /)a; + ^ rr o, making the fubftitutions, will
aadJaft cafe, be ^^ + 3^2 + 3J2;'' +2;^ + /^^ + pz + ^ = o. Forming the two equations^
^y^z 4- 3J2:* zz — ^^ — P^t and jy^ + 2;^ ~ — ^, from the firft we (liall have
• ^yz — — pi OX y zz — . This, fubdituted in the fecond, gives ^
+ z^ — ^ ^^ or %* + ^'2;^ = -tV/^'j a^<^ therefore 2;' iz — -J^ + ^i^y + ^V*
and thence z = \/ — f^ + ^iq^^TT^' But y + 2;' z: — y ; whence
y - V— t?— i/^^^ 4- ^p3, and laftiy/ ^ = v^— - 1? — "^^n^P^ +
Otlierex- 185. The fame roots or formulas may be had, by putting x zz iz -^^ ~ 9
preffions of 3*
the fame ^j^^j- jg j, _^ jf in the equation it be — px^ and — ^, if it be 4- tx in
roots. ' * 32 ' * * 3a ' ■'^
the equation. Whence x^ zz z^ -^^pz •\- — ± -'^» Make therefore the
fubftitutions in the firft canonical equation, and it will be z^ + -^-r — ^ = o,
or 2;^ — qz^ =s — -V^^ and z^ zz. \q + V^qq - ^^pi^ and then 2; =:
\/\q + >/ i,/y - J^f. Therefore, becaufe it was made x zz. z -\- — , it will
be ;f z: V\q + '^iii-i^p^ + - 3 /
3»
8 . To
SECT. IV. ANALYTICAL INSTITUTIONS. I75
To reduce this to the fame expreflion found in the firft manner, it will
be fufficient to rauhiply the nuinerator and denominator of the fecond term
of the homogeneum comparationis by v i^' — ^ -^'j — iV^> ^"<^ it will be
^ — ""'^^ ; .^^^ '^^ , that Is, V^t? — ^ i'jq — aV^S ^^^ therefore x will be
the fame as before. And the like may be obferved in the other cafes.
186. It is evident that the values of the unknown quantity x, found by theTodiafn-
firft fubftitution of a: =jv + 2, require the extraclion of two different cubick guii^ when
thefc roots
roots, whereas the fecond, by the fubftitution of .v — 2: + — , require the ^""^ """^^^ ^".<^
32 when imagi-
cxtracflion of one only ; and that the value by the fecond and fourth canonical "^""y*
equation will always appear under a real form, becaufe the quantities under the
quadratick radical are wholly politive. But that of the firlt and third will be
under a real form, if -^qi] be greater than yt/*' » ^'^^ under an imaginary form,
when -^qg is lefs than -^-V^^ And this is called the Irreducible Cafe ; but, not-
withftanding this, it does not follow, but that all it's roots are real. For all the
three values in the firft and third equation are real, when ^gq is lefs than ytP^*
But when ^qq is greater than -V/^'> in the firft and third equation, and, in
general, in the fecond and fourth, the roots or values alone thus found are real,
and the other two are imaginary.
As to the fecond and fourth equation, this has been already demonftrated at
§ 152, when they have the third term pofitive. Then, as to the firft and »
third, when the third ttrm is negative, each of thefc will have three real roots,
which are a, — ^, — r, or — a, -f ^, + ^ ; and, becaufe the fecond term
is wanting, as is here fuppofed, it will be <: = ^ + f, and the equation there-
fore, which arifes from (uch roots, will be of this form,
x^ — l/lfx ±: be X b -^ c = o.
— bcx
— ccx
When h, c, are real quantities, then J"ir7* will be a pofitive quantity ; and
therefore, if we put lb — 2bc + cc zz D, it will be alfo bb + be -•{• cc = D
+ ol,c, and '±+JL±]f}l = ^VD3 ^ ^d^^^ + D^^,, 4. p,^. But befides,
it will be bb + ibc -{• cc z=. flTc]^ zz D + /\.bc, and therefore ^bbcc X b + c *
= j^Dbbcc + PcK And ^V^^ + "^D^bc + Bbbcc + Pc^ is greater than
^Dbbcc, and therefore it will alfo be greater than ^bbcc X b + c*, and therefore
tV X bb + bc + c7\^ will be greater than ""^bbcc x b + c] ^ That is, the cube of.
the third part of the co-efficient of the third term, taken pofitively, is greater
than the fquare of half the laft term j that is, -^f is greater than ■'^qq. There-
fore,
174 ANALYTICAL INSTlTtTTIONS. BOOKl.
fore, if all the roots be real, the third term will always be negative, and befides,
-sVP^ will be greater than ^qq. When it happens to be oiherwife, two of the
roots will be imaginary.
After the foregoing manner, having found one value for each equation, we
fliall have the other two roots by dividing the propofed equation by this value ;
for the quotient will be an equation of the fecond degree, which may always be
eafily rcfolved.
Acompen- 187. But, if it (hall be thought convenient, the trouble of this divifion may
^h"*" '\ k ^^^^ ^^ fpared by confidering, that as unity itfelf has three cubick roots, which
rodts of "" are I, — f + iv/— 3, and — f — iv^— 3 ; fo it may be underftood of
unity. ^jiy other quantity ; of f ^ -f- \/J^Y+~^^ for example, which, being multi-
plied into unity, it's three cubick roots will be i X "^^q 4- s/ ;^^ + ^LpJ,
— t + t^^- 3 i"to v^i? + ^iqf+T^'' and — t - iV- 3 in^o
"^t? + ^ ill + ^''
Whence the three cubick roots of the firft equation x^ — px — q :=z o, by
ordering them in a due manner, will be as follows: x zz \/fq + V ^^^ _ ^\p3 _{-
7
And, in fad, if we find the produd of thefe three roots into each other,
making, for brevity-fake, V^q + ^ \n-^^p^ - »^ ^"^ \/^fy — y/^q^^^pi
= », the product of the laft, x + ^ ^ m + ^ ~ '^ ~ ^ « into the fecond,
X + — f^ H n will be xx + mx ■\- nx -^ mm ^~ mn •{' nn,
which, multiplied into the firft, x — m — k, will give x^ — ynnx — m^ — «'j
and, reftoring the values of m and ;/, it wiil be finally x^ — px — q zz o,
which is the equation propofed. Nor will it be otherwife in the other equations.
Example of 1 88. The foregoing general formulae being thus found, to apply them to the
this reduc- particular ufe of any given equations, it will be fufficient to compare the pro-
pofed equation to that of the four canonical equations which correfponds to it,
thence to obtain the values of q and p j which, being fubftituted in the formula,
will give the roots require^.
Let
lion.
SECT. IV. ANALYTICAL INSTITUTIOMS. I75
Let the eqnaiion be x^ -\- ^acx — 9^^ rz o. The correfpon ling one of the
four canonical equations will be the fecond, x^ •\- fx — q zz o \ {o that it will
be f — ^aa^ q ~ ga\ Then, making the fubftitution of thefe values inftead
of p and q, in the general expreffion of the root of this fecond equation, we
(hall have x = Vf^' 4- \^\'a'> + 7\^^ + Via' — y'V^+^V^j or, laHl)',
X = yT^' + v/^^V?^ + \^-S'->/¥5T^'- T'le other two roots will be
^ ^ ::li^3^y7^r^,7II^ + ^H^^^^^>^~ v/^.% and .V =
"I - a/-^^^ 3 , ^i2il^ 4. - ' + v/- 3 ^1^3 — y/LLifZ; the nrodu.a
of which roots will reftore the propofed equation,
189. But, without having recourfe to the general formulae, particular equa- Examples
tions may be folved independently of them, by making ufe of the given rule, without the
Thus, for the equation x^ + 2aax — ga^ = o, making x z= y -^ z, it will be^°'^™"^'
2aax zz laay + zaaz, and .v' ~ j' + 3/2 + ^yz* + z^ ; and, fubftituting
thefe values in the propofed equation, it will be changed into this other,
y + 327* 4- 32*7 + z' + 2aay + laaz — ga^ z= o. Of this equation may
be made thefe two, 32^* + ^zzy =: — laay — zaaz, and y^ + z^ in 9^7'.
From the firfl, by dividing by y -\- z, we have jzy n — 2aa, or jy — ;
p 6
which, fubftituted in the fecond, gives + 2' = ga^, or z^ — ga'z^
= ^V«^ And therefore z^ — \a? + \/^/a<*+^«S and z = ^\a^ + /Y^^S^*
But it \% y^ + 2' 1= 9^^ therefore y := \a^ — \/-V"^* + ^VS ^"<^ JV =
-^f tf' — v^i^-ia64_8^^6. But it is ^ -{- 2; r= ^, therefore x zz '^t«M-~V^^M^5
4- ^\a^ — ^/ ^^a^ + ^»ya'*, the fame as above.
Let the equation be z^ + 3^2' — 5^^z + 2^' = o. Let the fecond term
be taken away, by making a = a: — a^ and there arifes x^ — ^d'x + ga^ — o.
By comparing this with the third canonical equation, we fhall have p z= 8^^,
^ =1 9^'; whence, fubftituting thefe valties in the general formula for the root,
it will be a: = ^ — \a^ + \/-V-a« - Vt^ + ^ 1^'' — "^^^-a^ - V/^, that
is, ^ = ^ - f «^ + i/44>' + ^- ftf^ — -/t4I-^'. The like for the other
two roots. And, becaufe it was made % •— x — a, by fubtrading the quan-
tity a from each of the three roots, we ihall have the roots of the propofed
equation.
Let the equation be x^ — gd'x + 2^' = o. This will correfpond to the
third of the four canonical equations, and therefore it will be /> n: 9^% q = 2^';
therefore, making a fubftitution of thefe values, inflead of ^ and q in the general
exprefliort.
^y6 ANALYTICAL INSTITUTIONS. LOOK I,
exprefTion of the root of Ithat third equation, it will be x =. 4^ -^ a^ -^ /-— '-°-^'*
+ ^ — a^ -^ y — W^* J which expreffion is imaginary, notwithftanding all
the three roots are real i as the irreducible cafe requires.
Reduftion of '1 90. In equations of the fourth degree, we may proceed after this manner.
equations of iJLet i\^q canonical equation be a:* * + px"^ + gx — r 1= o, in which the fecond
flctrce! term is wanting; and if it had not been abfent, it might have been taken away.
Let this be transformed into a cubick. equation, after the manner explained at
§ 167, by means of the two fubfidiary formulas, x^ + yx -\- z = o, and
x^ — yx -}- zf zz o; and it will be transformed into y^ + :2py* 4- ppy^ -. ^j~o.
And the two fubfidiary equations, by putting, inftead of tt and z, their values
found from the comparifon, of the terms, will become x"" + yx + ip + ^yy
— ~ = o, and x'^ — yx + t/' + ^yy + -^ = o* Now, as it is fuppofed
that this equation has no divifor of two dimenfions, the fecond term muft be taken
from it by th.e fubftitution of jyj =z / — 4^, and then we Ihall have this new
equation, /' — :i^fpt — ^^p' = 0.
4- 4r/ — 4pr
Let this be compared with the firft or fecond of the four canonical equations
of ^ 181, according as ^r is lefler or greater than ^pp, that we may have it's
cube-root, which, for brevity-fake, we may call h. Whence it will be i zz h ;
and, becaufe it was made yy "^ t — \p^ it will \it yy zz. h — -|A and therefore
y — \f h — 1/>, which, for brevity, may be called g. In the two fubfidiary for-
mulae put g indead of jy, and gg inltead of yy^ and they will be xx + gx
+ \gZ + i/' — -^ = o* ^"^ XX —gx -V -\gg -^ ip + -|- =: o J the roots
of which are A? = — -j^ ± / -^ r- ~p — igg of the firft, and x zn j^g
r— -^ \p — \gg of the fecond. And, reftoring the value of ^ =
^^75, they will be a; = — iv//. -|/> ± J ^^1-—. — # — ^^, and
X — ^is/ h -i-h -+- / — " ^ — \P — ^h. the four roots of the propofed
equation .x'* * + px"^ •\' qx — r = o.
Let the equation be «* ♦ — Z(>aax^ 4- 600^'^ — 851^* iz o. This being
compared with the foregoing canonical equation, we fhall have /> n — 86^«,
q zz 600a', r — 851^*. Therefore the transformed cubical equation will be
y — . lyzaay^ ■{■ loSoo^y — 360000^* =: o. Now, becaufe this is divifible
by
SECT. IV,
ANALYTICAL INSTITUTIONS.
^77
by y'^ — 100^* =: o, without refolving it by the rules of cubick equations, as ~
we know already the root to h^ yy zz looaa, and y ~ ica ; fubftitutc thefe
values inO.ead of jy and jyy, as alfo the values of p, q, in the two fubfidiary
equations, they will be x^ + loax — 2-^aa =r o, and x^ — lO:.?^; -f ^yaa ~ o,
and their roots are jc zz — 5^ ± \/^8t?a, and ^ ~ 5^ 4: V — 12^7^7, which'
are therefore the four roots of the propoied equation. This example is infeited
only to (how the ufe of the method ; for the given equation may be reduced to
two of two dimenfions, after the way already explained in iCs place.
191. This method of refolving equations can be of ufe only in arithmetical How equa-
queftions, and not in geo netrical : becaufe, in this way, we have the value of ^'""s may be
the unknown quantity exprelfed by a cube-root, which it is fuppofed cannot be '^^^"'V^.^.i^'iJ*"
. aftually extrafted ; for, oiherwife, the equation would have a divifor, and would ^combtnLiJii
not be of the degree it feems to be. Now, to find this cube-root geometrically of /Wi..
cannot be done oiherwife than by the interfedion of curve-lines; which is the
fecond manner, and the general one which I have mentioned before, at § 180.
This method confifts in introducing a new unknown quantity into the equa-
tion, by which we (hall have two equations, each of which contains both the
unknown quantities, and both of them together all the known quantities of the
propofed equation. Thefe two equations are two loci geometricii which are
therefore to be conftrucfVed ; the interfedions of which determine the geometrical
values, or the roots of the equation propofed. And the reafon of this is mani-
feft. For, as from the combination of two places, or from two indeterminate
equations, by putting in 'one of thefe, inftead of one of the two unknown
quantities, it's value given by the other equation, there arifes a determinate
equation, which determinate equation may be refolved into two indeterminates.
Let there be given the two equations ax = zz, and xx — 522 + 2az
-f- ^aa = o. If from the firft^ for exatnple, we derive the value of ^ ~ ~ ,
' . . . ^
and fubftitute it in the fecond, there will arife the determinate equation
2* — 5^^222 -f 2a^z -I- 3«* — o, of the fourth degree. Then, taking the
locus to the parabola ax = 22, if we make the fubftitution of the value of 22
in the equation 2** — 5^7^22 + 2a^z + 3^*
iz o, there will arife the fecond locus aaxx
— 5^<22* -i- ia}z -f ^a^ =: o, or x^ — 52*
4- 2^2 -1- Q^aa z= o. To conftrud this
fecond locus, with centre A (Fig, (^c^.") and
tranfverfe axis CB ~ \a^ and with the pa-
rameter n 8«, let there be defcribed the
two oppofite hyperbolas BN, CP, which
fhall be the locus of the equation x^ — c^z^
-f 2^2 4- 3^2* zz o, taking the abfcifs z
from the point D, which is dilfant from the
centre A by the quantity \a towards the
vertex C.
A a Rightly
I>A. 3
lyS ANALYTICAL INSTITUTIONS. BOOK 1,
Rightly to combine this with the firH: locus ax — zz, it is necefTary that the
origin and the axis, of the unknown quantity x^ may he in common to both
the loci. And therefore at the vertex D, with the parameter =: a, upon the
axis DO, parallel to the conjugate axis of the oppofite hyperbolas, the parabola
of the firH: equation ax :=. zz fliould be defcribed. This will meet the two
hyperbolas in the four points M, N, R, P, from which drawing the perpen-
diculars MI, NO, RV, PS, to the axis DO, they will be the. four values of 2;,
that is, the four roots of the equation z* • — 5^-32* -f- la^z + 3*2* =: o.
The two IM, ON, will be pofitive, and the other two VR, SP, will be nega-
tive. For, as z of the determinate equation, (that is to fay, every one of the
roots of the fame,) ought to be common to both the loci, this can happen only
in the points M, N, R, P, in which thefe two loci interfed: each other. There-
fore the right Imes MI, NO, RV, SP, which exprefs z, will be the four roots
of the determinate equation propofed.
When two of 192. Hence it is plain, that the nearer the points M, N, approach to each
the roots will other, fo much the lefs will be the difference of the ordinates IM, ON. So.
^f?."^'^'^" that when one point falls on another, (in which cafe the two curves will no
when ima- longer cut but touch each other,) the two ordmatcs become equal, or the
ginary. equation will have two equal roots. Alfo, if the curves cut each other at the
vertex, in which place the ordinate is nothing, the equation will have one of
it's roots equal to nothing. And laftly, if the two curves neither cut nor touch
in any point, the roots of the propofed equation will be imaginary or im-
pofTible.
The loci 193^ Now, in the introdu<5lion of the new unknown quantity, it fhould be
fhoiild be endeavoured, that it may be done in fuch a manner, as that the two loci may be
fu'^^l ^thT ^^^ fimpleft pofTible, it: -efped of the degree of the propofed equation. That
fimpkft con- Js to fay, if the equation be ot the third or fourth degree, the two loci fhould
ftrudion. be of the fecond, that is, conic fedions. And it might be convenient, as any
one would think, that one of them Ihould always be a circle, as being the
fimpleft curve. But it ought to be confidered, that, by determining one of the
loci to be a circle, the equation to the other Iccus in many cafes may become
perplexed ; and therefore in fuch cafes I fhould prefer any other locus before the
circle, if it would afford a greater fimplicity. If the eqviaiion be of the fifth or
fixth degree, the two loci may be one of the fecond, and the other of the third.
If.it be of the feventh or eighth, they fliould be one of the fecond, and one of
the fourth ; or two of the third, firft reducing that of the eighth to the ninth.
-And fo on, obferving the lame analogy.
Taking, therefore, this equation of the fourth degree, x* + 2l>x^ + acx*
— fiV-jf — a^f =r o, aflume the equation (I.) xx + ^a; ~ ay, and, by fquaring,
it will be x4 -f il^x^ + b'^x^ — a^-^ and therefore x^ + 2bx^ zz d'y'^ — ^V.
Jn the propofed equation let this value be fubftituted inftead of x'^ -{- zbx^, and
.there will arife this other equation, i^^-) jy "— —5- H — cix — af zi o,
.Nowj
,5ECT. IV. ANALYTICAL INSTITUTIONS. I 79
Now, putting the value of xx obtained from the firfc equation, that is, ay — - I?x,
in the fccond term of this, and letting the third term alone, there will arife
(III.) yy ~y + ~x + — ;;' — iix — af z=l c. Or, fubftituting the
value of XX in the third term of the flime equation, letting the fecond terra
alone, there will arife (IV.) yy xx -\- cy x — dx — af z=. c.
And in this, putting the value of xx^ it will be (V.) yy -{• cy y -x
— dx — af — o. Laflly, if from this be fubtraded the firft made equal to
nothing, or xx + c^x — ay — o, and then adding it to the fame, there will
arife from the firft operation (VI.) yy + cy y + ay — xx — l>x ~x
+ —^x — dx ^ — af zn o-j and from the fecond, {VU.) yy + cy y
1 he b^ J .
m^ ay -^ XX + ex x -| ~ x — ax — aj zz o,
194. It is plain, that the firfl: equation is a locus to t\\t Apollonian parabola. To dlftmgmfh:
To diftinguifh the reft, we mufl make ufe of the redudions explained at^^'^^^^^"*
§ 127, 128, by which we (hall find, that the fecond will be a locus to the
parabola, when it is ac — bb ; to the ellipfis, when ac is greater than bb ; and,
finally, to the hyperbola, when ac is lefs than bb. The third will be to an
ellipfis, which will degenerate into a circle,, when it is c = <?, and the co-ordi-
nates are at right angles. The fourth will be to an hyperbola, which befides
will be equilateral, if it is b -=z a. The fifth will be to a parabola. The fixch
will be to the equilateral hyperbola. The feventh will be to the circle, when
the angle of the co-ordinates is a right angle.
From hence we may make choice of fuch a combination of the two loci, for
the conftrudion of the propofed Problem, as (hall be thought mod con-
venient.
195. If the fecond term of the propofed equation had been negative, we Cautions to
fhould have made xx — bx "=1 ay -, and the equations thence arifing would have^^ obferved*
been the fame as before, only changing the fign of thofe terms, in which the
letter b is of odd dimenfions. And if the propofed equation had at firft been
without the fecond term, I Ihould have taken xx ~ ay. Therefore, expunging
the terms in which b is found in the other equations, they would have been
fuch as this cafe requires.
196. In the propofed equations, the fecond term beirvg 4: ihx^i we fliould Conftruaion
take the locus to the parabola xx ± bx z=. ay, rather than xx zz. ay ; becaufe °^ ^ ''"^"^^
thus the other loci which arife have not the redangle xy^ and , therefore areexarap"e,'bya
conftruded with the more eale. parabola and
Aa2 EX-*"'^'='"^-
i8d
ANALYTICAL INSTITUTIONS.
BOOK I.
EXAMPLE I.
Fig, g6.
JLet the equation be of the third degree, x^ — aax + 2d^ rr o. Let it be
multiplied by ;v zr o, to reduce it to the fourth degree ; whence it will be
X* -^ aax^ -f- 2a*x z= o; which is required to be conflru6led by means of a
parabola and a circle. As the fecond term is wanting, make xx zz ay, a locus
to the parabola. Then fubftituting, inftead of x* and a:*, their values aayy
and ay, it will be yy — ay 4- 2ax zr o ; to which adding the firft equation
XX — ay zz. o, we (hall have the equation yy — zay + 2ax -\- xx zz o, which
is a locus to the circle.
"With radius BD z= s/ laa let the circle
ADME be defcribed, and make BC ■=. a,
and alfo the ordinate CA =z CB ~ a. From
the point A drawing the indefinite line AP
parallel to ED, and on it taking the abfcifles
AP zr j>', and making the ordinate PM = x^
this will be the locus of the equation yy — lay
+ lax -^^ XX zz o. Upon the axis AP, on
which ape taken the ys, with vertex A let
the Apollonian parabola MAM of the equa-
tion XX zz ay be defcribed, which (hall cut
the circle in two points A, Mi from whence
the ordinates being drawn, they fl)all be the
the real roots of the equation .sf* * ~ aax*,
4- za^x zz o, and two will be imaginary.
But at the point A the ordinate is nothing, and therefore one of the roots
will be a: zz o, as it ought to be;, it being now introduced by multiplying the
propofed equation by ;tf zz o. Therefore PM will be the real negative root of
the equation x^ — aax + ia^ = o, and the other two will be imaginary. If I
had multiplied the propofed equation by x equal to fome quantity, the circle
would have cut the parabola in two points out of the vertex, one of which
would have given me the introduced root, and the other that of the propofed
equation.
Now, to (how that PM is one of the roots of the equation x'^ — mvf^
+ 7.a^x zz o, it may be confidered, that, from the nature of the circle, it is
EO X OD =: OMo. But OM =— ;^-~^, EOz=jk4- \/2^^ — ^, and
OD -=. a — y ■\- y/ 2aa, Therefore xx + lax + aa zi aa i- aay — yy.
But, by the equation of the parabola AM, it is xx zz ay, and therefore — =^>
Then
SECT, IV.
ANALYTICAL INSTITUTIONS.
I8l
Then fubftituting thefe values of y and yy, and reducing the .equation to no-
thing, it will be X* — aax"" + 2a^x zz o, which is the very equation of the
fourth degree, vvhofe roots we were to extrafl.
197. If we would conftrud the equation x* — aax^ + 2a^x ~ o by means— By two
of two parabolas, it would be convenient to make ufe of the equation found P^''^^°^^®'
above, yy — ay -^ lax =r o ; and the locus of this, together with the parabola
of the equation xx zz ay^ might determine the roots required.
Fig. 97.
Therefore, with parameter — ia^ let
there be defcribed the parabola MCA, in ^
which make CD — \a. And letting fall
DA = ^a^ which will meet the parabola
in the point A, and through that point
drawing the indefinite line AP parallel to
the axis CD ; and taking the abfcifs x
from the point A, pofitive towards B and
negative towards P, and the ordinates
PM zr jy, this will be the locus of the
equation yy — ay + 2ax ~ o. Then
with vertex A, to the axis AQ»^ let the
'Other parabola MAS of the equation xx r: ay be defcribed j this will cut the
'firft in the points A, M. And letting fall the perpendicular MP, it will give
the negative root AP of the propofed equation. And becaufe at the point A
the perpendicular is nothing, therefore there is no other root; jufl as it ought
to be, the propofed equation being multiplied hy x "z:: o.
For, in the parabola MCA, it being CN :=z — x -\- ^ay and NM =:^ — f^?,
it will be, by the property of this parabola, ^aa — lax 3: j>7 — ay -f ^aa-y
and fubftituting the values of jy and yy, which are given by the firft equation
to the parabola MAS, that is, xx rz ay, and ordering the equation, we fliall
have at laft x^ * — aax'^ + la^x = o, which is the equation of the fourth
degree, of which the roots were required.
198. Now, if I had intended to have made ufe of the parabola, and of the — By a para-
equilateral hyperbola, it would have fufficed, from the fame equation yy — ay^o\a and an
H- 2ax iz o, to have fubtraded the firft equation xx — ay zz o, and the equation ^^"''^'^^^1
yy + lax — xk zi o would have arifen from thence, which is a locus to the equi-
lateral hyperbola ; which, being conilrucled, would have given me the roots re-
quired, by means of it's interfe6tions with the parabola of the equation xx == ay,
199. Finally, if I had defired to folve the Problem by the circle and the ^By a circle
hyperbola, I fhould have conftruded the third equation yy — zay + %ax •^ ^A:'and hyper«
rr o, a locus to the circle, and the fourth equation yy + 2ax "—• xx — o, 2ilocus^'^^^'
to the hyperbola, as is feen before j the interfedions of which loci would have
.given me the roots required.
8 200. But,
18*2
ANALYTICAL INSTITUTIONS.
BOOK X>
Thefe equa-
tions con-
ftr lifted by
various loci^
with ex-
amples.
20O. But, without multiplying by .a? the equation propofed, x^ — aax + 2a^
=r o, we might have conftruded it after the following manner, when we do not
choofe to introduce one locus rather than another. Make therefore xx — ay^ andj
inRead of xx, put it's value ay in the equation, and there will arife the equation
ay — ax -\- zaa zz. o,. a l^cus to thehyperbola between it's afympcotes.
Fig,^Z,
Therefore let the two indefinite right
lines SR, QT, cut each other at right
angles, and let thefe be the afymptotes of
the two hyperbolas MM, mm, having the
eonftant redlangle — ^aa-, taking the ab-
fciffes from the point A, diftant from the
point B by the quantity a. At the vertex
A, to the axis AR, with the parameter
r= a, let the parabola of the firft equation.
XX zz ay be dcfcribed ; it will cut the
hyperbola MM in the point M. Then
drawing the ordinate PM, it will be the
real and negative root of the propofed
equation.
For, by the property of the hyperbola MM, it will be BP x PM = — laa,
that is, xy — ax :=z — laa. And, by the property of the parabola AM, we
fliall have y zz
XX
a
Therefore, inftead of y, fubftituting it's value, and
the equation pro-
ordering the equation, it will be x^ — aax ■+• 2^' = o,
pofed.
In general, all equations of the third degree may always be conftrudled after
this manner, without being reduced to the fourth: by a parabola, and an hyper-
bola between the afymptotes.
EXAMPLE II.
Let there be given the equation of the fourth degree, z'* * — 5^*^* + 2^'2
H- 3^* = o, which is to be conftrufted by means of a parabola and a circle.
Take the equation ax — zz, fquare it, and in the equation propofed, inftead
of 2,* and 2% fubftitute their values, and there will arife a fecond equation,
XX — ^ax + laz + ^^a zz o, from whence fubtrading and then adding the
firft equation zz —^ ax zz o, we (hall have, in the firft cafe, a third equation,
XX — Ji^ax + 2az + 2>^a -f- 2:2 =r o ; and in the fecond cafe, a fourth equa-
tion, XX — tax + 2.az -f« ^^a + zz zz o; which is a locus to the circle, and
therefore I ftiall make ufe of it to conftrucft the propofed equation of the fourth
degree.
With
SECT. IV,
ANALYTICAL INSTITUTIONS.
iS?
^^S' 99' With radius z= s/'jaa let there be de-
fcribed a circle BMF, and from the centre
C towards B taking the line CL = 3^?,
and from the point L make LA zz a, per-
pendicular to the diameter, from the point
A draw the indefinite right line AP parallel to
the diameter BF; it will be AP zz x^ and the
correfponding ordinates in the circle PM = 2.
And therefore A will be the vertex, and AP
the axis of the parabola of the equation axzz zz*
Whence, with the vertex A, axis AP, and pa-
rameter — a, defcribing the parabola AM, it will meet the circle in four
points M, from whence drawing the perpendiculars PM to the axis AP, they
will be the roots of the propofed equation, two being pofitive and two ne-
gative.
For, producing PM to D, if there be occaflon, it will be, by the nature of
the circle, BD X DF = DM^. But DM = 2: + ^, BD = ;^ — 3^? + y/yaa^
and DF zz — a? + 3<2 + s/'jaa. Therefore zz -|- ilcz -\- aa ■=. — xx -^
6ax — laa ; but, by the nature of the parabola AM, it is ax = zz, and
XX zz
■ . Therefore, making a fubftitution of thefe values, and ordering the
equation, and bringing the terms all to one fide, it will be 2* — ^aaz^ -f la^z
-J- 3^* = o, which is the equation propofed.
EXAMPLE IIL
Let there be given an equation of the third degree, x^ — ^aax + ^a* zz o,
^nd let it be multiplied by a: -f- 2a, that it may be reduced to one of the fourth
degree; it will be x* + 2ax^ — ^aax* — a^x + 10a* zz o.
Take the equation to a parabola xx -h ax = ay, which, by fquaring, will
become x'^ + 'i-ax^ + aax^ =: aayy. Let the value of it's two firft terms,
x^ + lax^i that is, aayy — aaxx, be fubftituted in the equation, and there will
arife (IL) yy — 4^a: — ax -\- loaa = o. And in this, inftead of xx, fubfti-
tuting it's value ay — ax, there arifes (IM.) yy — 4ay + ^ax + loaa ~ o ;
from thence fubtrading the firft, xx -{- ax — ay :=z Oy and alfo adding it, there
will arife thefe two equations, (IV.) yy — o^ay + 2ax + loaa — xx :=. o in
the firft cafe, and (V.) yy — c^ay + /i^ax + loaa + ^a; = o in the iecond
icafe. 'I (hall make ufe of the firft locus, and alfo of the laft.
For
i84
ANALYTICAL INSTITUTIONS.
BOOK I,
Fig, loa. For the conftruAlon of the laft, let
the circle OSN be defcribed, with radius
OP cz \a ; and, producing it to F, that
it may be OF n la, and at the point F
erefting the perpendicular FC zz FO zz
2.a, draw the indefinite right line CQ^ pa-
rallel to FP. Taking any line whatever,
CQ^ zz jy, the correl ponding negative
ordinates, QS, QN, v,'ill reprefent x^ and
the circle will be the locus of the fifth
equation. Now take in FC the line
CB =: \a, and from the point B draw the
perpendicular BA = -^a. Then with
vertex A, and with parameter zz <3, let
the parabola NAM be defcribed, which fiiall be the locus of the firft equation,
faking the abfcitfes y on the right line CQ^ From the points O, N, in which
the parabola cuts the circle, railing the perpendiculars OH, NQj^ thele will be'
the two real negative roots of the equation, x"^ + lax^ — 3a V — a^x -|- 10^*
= o, of the fourth degree which was propofed.
And becaufe OH, taken negative, is equal to 2^, which is the root intro-
duced by the multiplication of the given equation into x + 2<2, NL will be
the real negative root of the propofed equation x^ — "^aax + 5«* z: o, the
other two roots being iinaginary.
For, by the property of the circle OSL, it will be OG x GL =: GN^.
But OG :=. y — 7.a, GL. = 3^ — j', and GN zz — 2.a — x. Therefore,'
making the fubftitutions, it will be xx + ^ax + loaa -\- yy — ^ay zz o. But,
sex *T" fix
from the equation to the parabola NAM, it will be ^ zz , and yy zz
X + 2ax + aaxx ^ ^^^ fubftituting thefe values of y and yy in the equation to
the circle,, it will be at laft x^ + lax^ — ^aaxx — a^x + lo/?* =s o, as it
ought to be.
EXAMPLE IV.
Let the equation be a® — A^aax^ — %a^x^ -{- 8^V + 32^® = o ; and becaufe
it is divifible by a?* — ^x -f- 4^?^;, and the quotient x'^ + d^ax^ + 8^3:^ + %a^x
-}- Sis'* z: o is an equation of the fourth degree, which we thus conftrud:;
take the equation xx + lax zz ay, of which finding the fquare ;v* + 4^x'
4- ^c^x\ = <?y, and, inftead of x^ + ^ax^y fubftitute it's value aayy — £^aaxx
in
"SECT. IV. A-NALYTICAL INSTITt/TIONS. 1 85
in the equation, and there will arife (II.) yy + ^xx -j- ^ax + %aa n o ; in
which, if we put the value of .r^, or ay — lax^ there will arife (III.) yy -f 6,ay
^ %aa =r o, from which, if we fubtrad the firft, there arifes (IV.) yy -f- t^ay
_^ %aa — XX ' — lax = o ; and laftly, if we add the hrft to the third, it will
be (V.) yy + '^ay + %aa + ^^ + 2^^ =z o.
The fecond locus is imaginary. The third is a determinate equation, but it's
roots are imaginary. The fifth locus is alfo imaginar3% But the fourth locus is
real, and is to an equilateral hyperbola.
To the axis DC =i v/i laa^ wirh centre
A, let there be defcribed the hyperbolas
CR, DG. Take AB = a, and let the
indefinite perpendicular BM be raifed, in
which take BM = \a\ and from the
point M let there be drawn MQ^ parallel
to the axis DC. Taking the x\ from the
point M upon MQ»_ the correfponding
QR or MT will be the y\, and the curve
is the locus of the fourth equation. Pro-
ducing QM to N, and making MNT -zz a,
and drawing NA to the centre of the hy-
perbola, take NO = a, and with vertex
O, parameter n: Cy to the axis OS let the
parabola OM be defcribed, which will pafs through the point M. Then taking
the ys on MT, and the correfponding ordinates TL. zz Xy this will be the locus
of the firft equation xx + 2ax zz ay. But now, as thefe two loci can never
interfeft each other, as is evident, all the four roots of the equation a?"* + 4ax^
-4- 8«V + Sa^x -\- Sa* m o will be imaginary. Whence the propofed equa-
tion X* — 4^V — Sa^x^ + Sa*x* -f- 32^^ iz o is found to have only two real
roots, which are equal to each other, being each equal to 2a,
201. But if, befides, we fhould be willing to conftrud equations of the third —By given
'and fourth degree, not only by the help of conical loci, which are to be thus'^'^^' °'j,^".':'^
found, but ot fuch of them as may be given, or fimilar to given, loci ; which jQ-y^^^
may be of ufe when a conic fection is given in the ftate of a Problem : It may
be done after the following manner, fuppofing, however, that the equations of
the third degree are reduced to the fourth, and that thefe are freed from their
fecond term, if they have any.
Yet I muff here obferve, that though, for the mod part, it may be better to
436 determined to this conical locus which already enters into the Problem; yet
we fliould always have it in view, that the iife of this given locus ought not 10
fuperfede a greater fimplicity of conftrudion. For, in this cafe, without any
•regard to the given locus, it may be better to introduce two new loci.
£b
Bdng
i86
ANALYTICAL INSTITUTIONS,
BOOK U
Being willing tlien to make ufe of given lod^ or fuch as are finillar to thofe
that are given, the artifice confills in introducing two indererniinate or general
quantities into the equation, and to determine them after\vards as occafion may
require. Therefore let the equation be z* -\- ab%^ — aacz 4- <=^''^ ^ o. Make
2 = —:- , in order to introduce the firfl indeterminate /. Making the fubfti-
tutions, it will be a;* +
a a a *
Let us take the firft locus
(I.) x^ ^^fy 11 o; and, fubf^iruting the values of x^ and ^*, there will arife
— a; + — =: o. To this let be added
H .. /^ „ , //^
the fecond locus (II.) j* •-{ — —y
the firft, and we (hall have (III.) x" — fy ■\- yy ■\- -^y — ^^x -\- :^ - o:
Now, to introduce a fecond indeterminate ^, let the firft: locus be multiplied
by — , and we ftiall have ^^ ^ ^ ^ — o; which, added to the fecond, will
■'a a
give (IV.) / + ^y — A;^+^ + ^^-— . iLy - o; and, being fub-
¥
fc
ffi , gf
traded, will give (V.) y + -^v-.2L;^+^^+ IJLy ^A^x" - o.
The firft locus and the fecond are to a parabola ; the third to the circle, when
the co-ordinates are at right angles ^ the fourth to the eliipfis ; and the fifth to
the hyperbola.
Now, let it be required, for example, to conftru6l the equation by means of
a given circle and a given hyperbola. Let us therefore aflume the third and
fifth /<?<:/ ; and as to the third, with radius CG = -^\/ cc - i^ad + bb - 2ab + aa,
let the circle EMG be defcribed, and,
fc
taking CD = ^^— , from the point D let
fall the perpendicular DA =
Fi^. 102.
2a
(fuppofing a to be greater than l> ; for it
muft be rarfed the contrary way, when if
is greater than <«,) then from the point A,
on the right fine AP parallel to DG,
taking the abfcifles AP n x, the corre-
fponding PM will be the y, and the
circle EMG will be the locus of the equai-
tion x^ — fy +y H — ~y ^x +
ffd
:^^ z: O.
a
As
SECT. IV,
ANALYTICAL INSTITUTIOKS.
187
As to the fifth locus ; to conftru6t it and combine it with the circle, through
the point A, the origin of at, produce the right line DA to H, fo that it may
be AH = ^-^-^^ \ and through the points A, H, draw AP, HK, parallel to
2d
- /•
DG. On HK, towards the point L, fet off the portion HI z= -^ , and with
f
centre I, tranfverfe axis LK zz — \/ aacc 4- ^a^gd — ai?'^g — ag^ — zabg"^, (fuppofing
(c + 4^0- to be greater than ^ '^' ^ ,) let the hyperbola KM bedefcribed.
with parameter KO ~ -^—y/aace + ^aagd — abbg — agi — 2abgg \ in which, if it
be AP zr x^ PM = jy, it will be the locus of the fifth equation. From the
points M, in which it cuts the circle, drawing to AP the perpendiculars MP,
the lines AP, AP, will be the roots of the equation a^ ■\- -^x^ — ^—x 4-
— = o. And, becaufe it was made 2 ■=. -*r , and x is given, and alfo z,
they will be the roots of the firft propofed equation.
Fig, 103.
But, if we had fuppofed cc -f- ^gd
to be lefs than -^ LMjLi! ^ the locus
a
of the fifth equation would be the hyper-
bola MM, half the tranfverfe axis of
which z: . / y ^^S + ^hg +g^- ^^-^ - 4^^^
2a ^ g ' '
the conjugate femiaxis IK =
%g a '
parameter KO of the conjugate axis z:
/ .y ^h + ^^'gg + g'^ — ace - ^ngd
This fuppofed, to fatisfy the firft condition, that it fhall be a given circle,
let it*s radius be zz r, and then it would be r zr -^\/ic - ±ad + bi> - zab 4- aay
from which equation the value of the affumed indeterminate may be derived,
or/ =:
lar
Vcc — ^d -i-ib — zab + aa
be that, the radius of which is = r.
. And then the circle defcribed, EGM, wili
Bb2
To
l83 A N A L T T I C A t INSTITUTIONS. BOOK I.
To fatisfy the fecond condition, that the hyperbola may be given -alfo, let zt
be the given tranlverfe axis, and p the given parameter. Then ic will be
2/
- Z /.. -i- .„^ _ % t."i+ -'^■^ _^ /• fif_
But it is alfo p zz ^y^f + A^g ~ %+^J.+ ^^^g . therefore, inftead of /,
putting it's vakie now found, it will be ^ zi ~ , from whence we have the
value of ^ z: -^ . And putting this inftead of g in the value of/, it will be
/= — — — — — '■ • ' Wherefore the tranfverfe diameter
^ ^/cc + 8aJ>tJ — 2bl>J>i lalfp
and the parameter of the hyperbola (^efcribed (Fig. 102,) fl:iall be truly the
given lines 2/ and p ; and thus as to the firft cafe.
Then, as to the fecond, which is when cc + 4dg is lefs than — — ^-^,
let the conjugate axis of the given hyperbola be LK = 2«, and it's parameter
= g; then it will be 2u zz lJl!l+_l^L±l! ^ cc ^ 4^^, and q =
^^t!l±^^iL±Jl ^cc -- ^dg. Whence it will be found, by operating a3
before, / = — ^ ^" ^ ^ r==., and ^ = ^ . And the
/ zhbuq -\- zbaqq + — Accuu — Baduq
hyperbola will have for it's conjugate axis LK = 2U, and for it's parameter ta
the faid axis KO = q. And thus the Problem will be conftruded by means of
a given circle and a given hyperbola.
Now% if the hyperbola fhall not be given, but ought to be fimilar to one
given ; that is, if the axis be to it's parameter in a given ratio, or as m to n ;
becaufe it has been feen above, that the ratio of the axis to the parameter is that
of a to g, it will be fufficient to make the analogy, a .g :: m ,n, and thence
to have the value of ? = — . .
By making ufe of the fame method, we may conftrud equations by means
of any other given loci, or which are fimilar to thofe given. As, for example,
by means of the aforefaid given circle, and of a given ellipfis, or like to a given
one, by taking the fourth equation before, inftead of the fifth.
EX.
SECT, rv.
ANALVTICAt INSTITUTIONS,
182
EXAMPLE V.
Let the equation be x* — ax^ — a/ix'' — a^x — atj'* ~ o, which it is
required to conftrudl by means of a parabola whofe parameter zr a, and of an.
ellipfis fimilar to one given, whofe tranfverfq axis is to the parameter in the
given ratio of ^ to ^.
Let the fecond term be taken away by the fubditution of .r =z z -f ^a, and
the transformed equation will be 2* * — -^^aaz^ — "T^^^ — iH^* — o»
I put 2 =: -—3 to introduce the firft indeterminate /, and it will be y — V^J^**
— V/!)' — iHf* = o- Now, taking for the firft locus yy — fq to the para-
bola, and making a fubftitution of the values of y* and jy*, we Ihall have the
fecond locus alfo to the parabola, qq — ^^-fq — ^fy — -Irsff — o. Now,
becaufe the given parabola has it's parameter — a, we may here make ufe of
the firft locusy by taking f = a, and therefore it will be yy zz aq. And fubfti-
tuting the value of/ iii the fecond, (for, the ellipfis not being given, the firft
indeterminate /, in refped of this, is ftill arbitrary,) it will be qq — 'tH
Now let the firft locus be multiplied by -^ , in order to introduce the fecond
indeterminate g, and it will be ^^LZ-fi^ — o, which, being added to the fe-
cond, will give the third locus, qq —
which is to an ellipfis.
For the conftrudion of this third Iccus,
v-^? — ^^y
5 9 5^* ^.ilLZJi^ - o
Fig, 104.
we ftiould have the ellipfis MSQ_ to de-
fcribe, with the tranfverfe axis SQ^=:
^^21^£i±lT^f£+l^ll±l^ ^ and with
256^
parameter rr — y- 7 .
But, becaufe in this the ratio of the axis
to the parameter is that of g to tf, which,
by the given condition, ought to be that
of b to d, it will be ^ z: -^ . And
therefore, inftead of g, fubftituting it's
value.
190 ANALYTICAL INSTITUTIONS.' BOOK I.
value, the elllpfis MSQ^ mud be defcrlbed with the tranfverfe axis =:
-gj V ■ ^ ^^^ , and with parameter r= -
Now, from the rentre C taking CA rz —-—rr— 3 and from the point A
letting fall the perpendicular AB ~ -^ , if from the point B be drawn BR parallel
to the axis SQ»^ taking any line BR. = ^, it will be RM — ^, and the ellipfis
will be the /o£Us of the third equation gq — ^^aq — '-^ay — ^l-^a 4- ?lLllJi3
= o«
With vertex B, axis BR, and parameter = a, let the parabola MBM of the
equation yy — aq be defcribed \ it will cut the elliplis in two points M, M.
From which points drawing RM, RM, perpendicular to th« right line BR,
they uiil be the two real roots of the propofed equation.
For, by the property of the ellipfis, it will be SP x PQ^to PM^, fo is
ibd
the tranfverfe axis to the parameter. But CP zz q — ' "^ ^ , and there-
fore SF =z -TjV ~ r — + ? — ri »
\od b ^ ltd
, -n^ I , '7\f>a}hd* + i-iBa^db^ + 6Aa^b^ + i6oa*^3 Wad -^ Sab
and VQ^ zz -— >/ ^ ^ — i , — ~ a A rz — .
^^ l6d b •* ltd
And befides, PM =^ — "^H" • Therefore we (hall have the analogy,
Tstbd?- ^ "^ 8^ Tstd"- •
^^ — "^ir "*" ?!^t •• ~j" • "T" •• ^ • ^- A."^ therefore the equation
^^^^, ^^1^ + -^-^^-^a — — = ^^7 — ■^~-^' ^^^' ^y ^^^^ equation to the
parabola, it is ^ zr aq. Therefore, fubftituting, inftead of q and qq, their
values — and ~- , and ordering the equation, dividing by d and multiplying
the terms by aa^ it will be y* — i!|^ — i^i! — d2i^ — q. But, by making
thefubftitution of 2 = —-^ (or making ^ = z, for « 1= /,) it will be
2* — Vtf'z* '^—■a^z — -iri^* = o> which is the reduced equation; to the
fjoots of which adding ^a, they will be the roots of the equation propofed.
Ir was indeed unnecelTary to take all tJiis trouble about an Example, which,
by nature, is not fohd but plane ; for the propofed equation is divifible by
X 4- a, and by ;v — 2a, But, however, it will ferve to ftiow the ufe of this method.
202. Equa-
SECT. IV,
ANALYTICAL INSTITUTIONS.
191
202. Equations of the fifth and fixth degree are conftruded by means of two Equatums
iQciy one of the third degree, and the other a conic feclion. ^"^" fifu/and
fixth degree.
EXAMPLE VI.
Let the equation be **.+ aax'^ — (3* = o. I take the ylpoUonian parabola
XX =z ay, and making the fubftitutions, there arifes the fecond locus xyy + axy
— a^ zn o.
Hitherto I have not mentioned the conflriiflion of loci above the Conic
Seftions, having referved the treating on thefe for the following Se(5bion j for
thus order neceffarily required. At prefent, therefore, let there be fuppofe ^
and alfo let there be defcribed, a curve
with three branches MCH, FE, PNO,
whofe equation is xyy -J- axy — a' = o,
in which AB reprefents the .v's, and BC
the y's. With vertex A, axis AL, and
parameter =: tf, let the Apollonian parabola
RAC be defcribed. It will meet the
branch MCH in the point C ; and there-
fore, letting fall the perpendicular CB, it
will be AB — Xy the real and pofitive
root of the propofed equation, and the
other four will be imaginary. If we defire
to conftrudl: the fame equation by means
of an hyperbola between it's afymptotes,
and alfo by a locus of the third degree, make xy zz aa, and, by fubftituting,
it will be x^ + aax — ayy zz o.
FJg, 106,
To axis AB, with abfcifs AB z: x, and
ordinate BC =^, let the curve CAN be
defcribed, which is the locus of the equation
a' -)- aax — ayy = o. And between the
afymptotes AB, AG, let the hyperbola
MCH of the equation xy = aa be de-
fcribed, taking the x's on the fame axis
AB ; this will cut the firft curve in the
point C, from whence letting fall the per-
pendicular CB, it. will be AB = x^ the
root of the equation propofed.
Now
192
ANALYTICAL INSTITUTIONS.
BOOK I*
Now I multiply tlie fame equation by x = o, in order to reduce it to the
fixih degree, and I ihall have x^ + aax* — a^v = 0. I take the fame locus to
the parabola .v-v =: ay, and, making the fubftiiution, there arifes the fecond
/octfs y^ + ^y* — ^^'V zz o, which is the
curve NBAM, taking the abfciffe AP =:y,
and the ordinates PM — x.
With vertex A, to the axis AP, with
parameter zr a, the Apollonian parabola AM
of the eqijation xx r: ay being defcribed, it
will cut the faid curve in the vertex A,
which gives us one of the roots ;^ = o, the
fame that was introduced into the equation.
Befides, it will cut it in the point M, and
letting fall the perpendicular MP, it will be
another root of the equation.
If we dcfire to make ufe of the firfl cubic parabola x^ = aay, make the
fubftitution in the equation x^ 4- a'^x* — a^x — o, and there arifes the fecond
locus, yy -f- xy — ax ■= o, to the Aplhnian hyperbola.
Fig, 108.
On the indefinite h'ne AP let the tri-
angle ACP be defcribed, being right-
angled at C, (fuppofing, if you pleafe,
the angle of the co-ordinates of the equa-
tion yy + xy •— ax ^= o to be right,) and
let it be AC . CP :: 2 . i. At the centre
A, with the tranfverfe femidiameter AF =
2^
a\^ S, with the parameter = -^,'let the
Jpollcnian hyperbola FM be defcribed ;
then from the point F drawing the inde-
finite line FQ^ parallel to AC, and taking
any line FQ^zz x, and QM parallel to CP and equal to y, this fhall be the locus
of the equation yy -\- xy — ax z= c. To the axis FL parallel to PC, let there
be defcribed the cubical parabola NFM of the equation x^ zz. aay. This will
cut the hyperbola in the vertex F, which gives us the root a: zz o. And from
the point M letting fall the perpendicular MQ^ upon FQt^ this will determine
the other root FQ^of the equation x^ + aax* — a^x.
If Our equation had had the fecond term, and if we had defired to make ufe
of the cubic parabola, a fecond Iccus ot the third degree would have been
derived. Therefore we ought to make the fecond term to vaniQi, or make ufe
of another locus.
EJC-
lECT. IV.
A,KALYTICAL INSTITUTIONS.
193
EXAMPLE VII.
Let the equation of the fixth degree be this, x^ + ax^ + ^^''^ — ^2*110.
I take the locus to the Apollonian parabola xx ~ ay. Making the fubftiiutions,
the fecond locus will be y^ + Ary* + a^v
— ^»' zi o, which is the curve CBM, taking
the abfcifles AP ~ j, and the ordinates
PM = X.
At the vertex A, with parameter zz a, to
the axis AP, let the parabola MAM of the
equation xx = ay be defcribed. This will
cut the faid curve in two points M, M, from
whence drawing to the axis the perpendiculars
MP, MP, they will be the two roots of the
propofed equation, of which one will be po-
fitive, the other negative, and the four others
will be imaginary.
403. Equations of the feventh degree are conftrufled by means of two loci —of
of the third, or elfe by one of the fecond and one of the fourth. But, becaufe, ^'t^f^^J'
by multiplying them by the unknown quantity, they are reduced to the eighth ^^f^^^
degree, and thofe of the eighth are conltrudted in like manner by a locus of the
fecond, and another of the fourth, I (hall content myfelf with giving an Ex-
ample of thofe of the eighth degree.
EXAMPLE VIII.
Fig* no.
Let the equation of the eighth degree
be x"^ + a}^ + a^x^ — a* z=. o. Taking
the equation to the Apollonian parabola
XX iz ay, and making the fubftitutions,
there arifes the fecond locus x^ -f- xy^
+ axy"^ — fl* zz o, which is the curve
GBFCMN, taking the abfcifles AP n^,
and the ordinates PM z: x. With vertex
A, parameter zi a^ and axis AP, let the
parabola of Jpollonius, MAN, be defcrib-
ed, belonging to the equation xx zi ay,
C c This
1:94-.
ANALYTICAL INSTITUTIONS.
BOOK r*
This will meet the aforefaid curve in the points M, N, from which drawing the
pcrpendiciilars MP, NQ>_ to the axis, they will be the two pcaJ roots, one-
poluivc,, the other negative, of the propofed equation, and the others are
imaginary.
—orof higher 204.. Here it may be obferved, that equations of the ninth decree, (;is well as
degrees. thoie of the eiglith, reduced to the nin-th by multiplying them by the unknown
quantity,) may always be contlruded by means of two loci of the third degree^
making the fecond term to vanidT, if it have one.
Thus, in general, equations of the tenth degree may be conftrufVed by means
of a locus of the third degree, and oae of the fourth. And, in like manner,
thofe of eleven and twelve degrees, obferving to reduce thofe of eleven 10
twelve, by multiplying them by the unknown quantity, and by making the
fecond term of an equation of the twelfth degree to. vanilh, if it have any.
And the like is to be underftood of equations of higher degrees.
All equations 205. Another manner of conflrucling equations of any degree may be, by
may be con- means of a iccuj of the fame degree as the equation propofed, and a right line j
fe„f:"hV ^fier the following manner.
fame degree. ^^^ jj. ^^ ^^ equation of the fifth degree, .v' — kx* + acx^ — aadx"" + a^cx
— a"^/ ~ o. Let the laft term a*f be tranfpofed, and taking one of the linear
divifors, /, of the laft term, make it equal to z, for example, and divide the
equation by a* ; then we iliall have 2 z: ■ .
jFVg-. III. On the indefinite line BQ^ defcribe
the curve BMDRNLFC of this laft
equation, taking the x.*s from the fixed
point B. The ordinates PM, SR, &c.
will be equal to z ; and therefore, from
the point B draw the right line BA =1/,
parallel to the ordinates PM, SR, and
through the point A draw the indefinife
right line KC both ways, and parallel to-
BQ^ From the points in which it cuts
the curve, let fall the perpendiculars
MP, RS, CQj they will determine the
abfcifles BP, BS, BQ>_ which are the roots
of the equation propofed. Thofe from A towards Q^are pofitive, and thofe
the contrary way are negative.
If the right line AC (hall touch the curve in any point, the correfponding
abfcifs X (hall denote two equal roots j and if it meet it in no point, all the roots
win be imaginary,
8 H
W 1>
5ECT. IV.
ANALYTICAL INSTITUTIONS.
*95
If the laCl term had had it's fign pofuive, we muft have made x ■=: — f^
and therefore muft have taken BA nr — /, that is, below the point B, or on
the negative fide,
206. Tins method may be of ufe to verify con (Iruifl: ions, which have been Ufe of thfs
made by the combination of two curves, by confronting with each other the™
number of the roots, whether real or imaginary, pofitive or negative, which are
found by each method.
PROBLEM I.
207. Between two given quantities, to find as many mean geometrical A Problem
proportionals as fhall be required. to exemplify
* ^ this method.
Let the two given quantities be a and ^, and let x be the firft of the
mean proportionals j they will form this geometrical progreffion following :
>»i^ v3 ^c^ ic^
a, X, — , — r > — r > — r » ^c* Now, if we would have two mean pro-
' a a* a* a"* , ' '^
portionals, the fourth term of the progreffion muft be <^, and therefore we
fliould have this equation — j- n ^, or x^ = a^lf. To cenftrud this by the
help of a parabola and a circle, I reduce it to the fourth degree, by multiplying
it by A" zi o, and then it will be x* — a^i'X =. o. Taking the locus to the
parabola xx — ay, and making the fubftitutions, there arifes the fccond locus
yy — l^x zn o, which is alfo to the parabola ; from, which fubtrai^ling the firft,
there arifes a third, yy — bx ^ xx -^ ay zn o, which is to the hyperbola; or,
adding the firft and fecond together, there arifes, laftly, yy — l^x -{- xx — ay
zz o, a locus to the circle, fuppofing the co-ordin?,t.es to contain a right angle.
Fi^. I J 2,
With radius C(jf'±:«f v'^^z + M let the circle
DMA be defcribed ; iind taking CB r:: ia, leu
fall -the- perpendicular BA =: j;l', which wUl
meet the circle in the point A ; fiom whence
drawing AQ^ parallel to the diameter OG, and
taking any portion AQj^y, it will be QjVI — .v,
and this circle will be the Iccus of the equation
yy — Ifx •}- XX — ay zz o. With vertex A,
axis AQ»_ and parameter =: a, let the parabola
XX = ay be defcribed, which will meet the
circle in the point M ; from whence letting fall
the perpendicular MQ> it will be the root of the
propofed equation. For the vertex of the pa-
C c 2 labola.
19^
ANALYTICAL INSTITUTIONS.
BOOK I.
rabola, being in the periphery of the circle, will give the other root x :=: Oy
which was introduced, and the other two are imaginary.
Taking the firft and fecond equation, the Problem will be conflruded by
means of two Apollonian parabolas. Taking the firft and third, it will be con-
ftruded by means of the parabola, and the hyperbola referred to it's diameters.
fame 2o8. Without multiplying the equation x' — aah zi o, it may be con-
rwtfe ftruded by a parabola and an hyperbola between it's afymptoies ; for, taking
.ruaed. jj^g y^^^j ^^ _ ^^^ ^jj^ making the fubftitutions, there arifes xy = ah.
Fig, 113. Q
Between the afymptotes NN, QC^ let
there be defcribed the hyperbola MM wich
the conftant redangle ah, and let AP be
the /s, and PM the at's. To the axis AP,
with the vertex A, the parameter — a, let
the parabola AM be defcribed ; from the
point M, in which it cuts the hyperbola,
drawing the ordinate MP, it (hall be the
root of the propofed equation.
The firft of the two mean proportionals
being thus found, we have alfo the fecond,
being equal to the abfcifs AP — j = — .
A fimpler ^°9* "^^ ^^^ three mean proportionals, the Problem becomes plane ; for,
cafe of the having found, geometrically, that in the middle, which let be m for example,
fame Pro- the mean between a and m will be the firft of the three, and the mean between
^^'='"- m and b will be the third.
Carried
higher.
210. Let it be required to find four mean proportionals; then b ought to be
the fixth term of the progrcffion, and therefore we fliall have the equation
*' iz a*bj or ** — a*b zz o.
Fig, 114.
mj
KV
^
Jt^^.,-^
A
/
^
"-^^
a
K
?
a
I take the locus to the Apollonian parabola
XX zz. ay, and making the fubftitution, there
arifes the fecond locus xyy — aab zz o,
which is an hyperboloid of the third degree.
Therefore, between the afymptotes Qg^RR,
let there be defcribed the hyperboloid MN,
mn, of the equation xyy zz. aab, making the
abfcifs AP iz j, and the ordinate PM — x.
Now, to the diameter AQ>^ vertex A, de-
fcribingthe parabola of the equation xxzzay;
and from the point M, in which it meets
the
SECT. IV.
ANALYTICAL INSTITUTIONS.
197
lig. II
the hyperboloid, drawing the ordinate MP, it (hall he the root 6f the equation
x^ — a*b z= o, and the firft of the mean proportionals required j by means of
which the others may be found alfo.
211. Alfo, the Problem may be conftrinfted by means of the JpollornanC^mQ.Tu&ed
hyperbola between it's afy^mptotes, and the fecond cubical parabola. ocherwife.
Make therefore aa z: xy, iht locus^ 10 the aforefaid hyperbola; and, inftead
of tf*, fubflituting it's value x>yy, there arifes the locus x^ — byy^ which is the
fecond cubical parabola.
Wirh the axis AQ^let there be defcribed
the fecond cubic parabola RAN, in which
AQ^ gives the .»'s, and QN theys. And be-
tween the afymptotes S F, MQ»^ let there be
defcribed the hyperbola NN. And from the
point N, in which it meets the parabola, let
the ordinate NQ^be drawn. Then will AQ^
be the root of the propofed equation, that is, '
the firll of the four mean proportionals. a ^
212. To find five mean proportionals the Problem Is only cubical. For, Extended t»
having found the middle term geometrically, which, for example,' let be ;»; higher cafes.
to have the two means between a and m, is a cubical or folid Problem, as has
been feei\ juft now.
It may be eafily perceived with a little attention, that the Problem for finding
fix mean proportionals may be conftruded, either with a locus of the fecond,
and one of the fourth degree, or with two of the third degree. But to find
feven fuch, having found the middle one, the Problem will be reduced to the
finding of three. And in the fame way of reafoning, we may go on to greater
numbers.
PROBLEM II.
213. In the circle ABCD, having two chord s The /«>« ex-
given, BA, DC, which proceed from the extremities^f^phfied hy
of the diameter BD, and the third chord AC being ^j'^^^'^"" ^'^'*^
given alfo ; to find the diameter BD.
Draw the chord BC, and make AB zz ^, AC ~ by
DC = Cf and the diameter BD zz a:; and let fall the
perpendicular BM upon the chord AC, Becaufe the
angle in the femicircle BCD is a right one, it will be
BC zi ^xx — cc 9 and becaufe the angles BAC, BDC,
infift
198
ANALYTICAL INSTITUTIONS.
BOOK I.
infift on the finie arch EC, and a,Ko the angles M, BCD, are right angles, the
two triangles BCD, BAM, will be fimilar. Wherefore it will be AM =: — ,
Bur, by £//r//V, ii. 13, it is BC'7 = AB^ -f AC^ — 2CA X AM ; therefore
the equation will be xx — cc "zz aa ■{■ bb "— , that is, a^ — ccx — aax
— Ibx + labc z: o.
I multiply it by x, to reduce it to the fourth degree, and thus conftrudl it,
by means of the parabola and the circle. It is then at* — c^x^ — a^x*" — b'^x*
H- zdbcx — o. Taking therefore the locus to the parabola, the parameter of
which is the lead of the three chords, which let be c for inftance ; that is,
taking xx =: cy^ make the fubllitutions, and the fecond locus will arife yy —
— — — '- H — o, which is alfo to the parabola. To this add the
firft equation xx — cy — o, and we ihall have finally a locus to a circle, taking
lUe co-ordmatcs at right angles, that js, yy -I — y -\ x -{• xx zz: o.
J^ig, 11 J..
Tj r • L r A r' _— yOabb -^ccmm
1 hereiore, with radius AC =: \/ ,
cc
(tor brevity-fake writing m tor ,)
draw the circle AMBP, and taking CD zz tn,
from the point D raife the perpendicular DE z:
\-r\ J
^ '^ — , which will terminate in the periphery of
the circle at the point E; and drawing the in-
definite line ECLparallel to the diameter AB,
upon this line take any how EL = jy, the cor-
rcfponding ordinate will be L,M =: a*, and this
^circle is the Iccns of the equation. With vertex E, axis EQ»^ and parameter
— f, let the parabola of the equation ax ~ cy be defcribed. " This will cut the
circle at the vertex in the point E, which will give the introduced root x = o*
It will cut.it befides in the three points M, N, P, from whence, to the right
line EQ^ letting fall the perpendiculais ML, NR, PQ»_ they fhall be the ihree^
roots ot the equation propofed, two politiye and one negative. The firft po-
litive root ML cannot ferve for this Problem ; for, fuppofing j = r, it will be
in the parabola.MV— ^j and in the circle, x — — }- ^ hbb + aa + cc,
, * ■, zz:\^I\ •■:.:.'. . c ^ cc
But this value-o^ A-,^eJat?vely to the circle, is greater than r, if the two chords
rt, i*, be not fqual to eaoh other; and it is equal to c, if the two chords be
equal. Wherefore the point in the parabola which correfponds to the abfcifs r,
either, frills in M, or falls within the.cucle. Therefore ML is either lefs than f,
i on.
SECT. IV.
'/Analytical institution
199
or, ar moft, is equal to ir, and therefore mu^ needs be Icfs than either of t!ie
chords a, b, and confequentiy car>not be the diameter of the circle.
The fecond pofitive root RN will fupjily us uitii the diameter required.
The negative root Q^P fupplies- us with a di;imerer for another cafe ; that is,
when the two chords which terminate at the diameter are drawn from the fame
fide, as in Fip:. 118. Fo^r, doing the f\me things as
above, draw hkewilb the chord AD..Tlie angle DAS'
being right, t'le two angles DaC, MAB, vvill be eq')>:d
to a right angle.. But alio, tlie two angles MAB, M jA,
are equal to. a right angle j therefore MBA — DAC —
C3D, as infilling on the fame arch DC. Hence the two
triangles CBD, MBA, are fimilar, and therefore MA —
— ; but, by Euclid^ il. 12, ic v/iil be CB^.. ~ CAq. -j-
118.
BAy + 2CA X AM J whence the equation xx
2aic , ■ -
, that IS, .v^ — ccx
hh -{' aa +
hhx
cc — •
nax — icilK
=: o i the condrutflion of which is the fame as the pre-
ceding, except that now, the lalt term being negative, we muft draw Di^
(Fig. 117.) the negative way, becaufe the axis of the parabola v/ill be below
the diameter of the circle ; and the two pofitive roots in the firft cafe are
negative in this, and the negative becomes pofitive.
And becaufe the fecond term is wanting in both the equations, it proceeds
from thence, that the two pofitive roots in tlie firft cafe are ec[ual to the nega-
tive, and the pofitive in the fecond is equal to the two negative. Hence we
learn that the firft of the three roots, which gave us no folution of the Problem,
yet however belonged to it, as being the difference of the two diameters.
PROBLEM III.
fig* 1*9
214. The redangle ACDB being given, Another
in the fide AC produced to find the point g^"'"^*'"'*^-!
E, fo that, drawing the right line BE^'""^^""^-
from, the angle B, the intercepted line EF
may be equal to a given right line c.
When a fquare is given inftead of. the
redangle ABDC, the Problem is plane,
and has been already folved in Sed. IV.
\ 176. But, fuppofing ABDC to be a
redlangle.
200
ANALYTICAL INSTITUTIONS.
BOOK I.
re(f\angle, it changes the nature of the Problem, and makes it folid. Therefore,
making AB = ^, BD i= b^ DF = x^ and repeating the argumentation in the
place above cued, we (hall have an equation of the fourth degree, which is this;
x^ — rax^ + aax^ — zabhx + aalh r: o.
€CX^
To conflruft t|jis by an hyperbola between the afymptotes, combined with a
circle, I put ab :=z zat, and making the fubftitutions, there arifes the fecond
/ocus XX — 2ax -\- aa -\~ bb — cc — 2bz + 22 zz o, which is to the circle.
Between the afymptotes BA, BD, let the hyperbola OM be defcribed, of the
equation zx :ii ab, which fliall pafs through the point C. Taking any abfcifs
BP, BN — r.y the ordinate will be PO, NM =z a-. With centre C, radius
equal to the given line f, let the circle OMV be defcribed. This fhall be the
locus of the equation xx — 2ax -\- aa ■\- bb — cc '•^ 7.bz + zz ~ o.
From the points O, M, in which this cuts the hyperbola, let fall the perpen-
diculars OP, MN ; they fliall be the two pofitive roots of the equation. The
lefler will ferve for the Problem in the cafe propofed, of the angle BAG. The
greater for the angle AC/. And if the given line c be fuch, that the circle
cannot reach to cut the oppofite hyperbola mo, the other two roots will be ima-
ginary. But if it (hall cut it, they will be real and negative, and will ferve for
the angle ACD.
PROBLEM IV.
A Problem
for angular
fc£lIons»
JO'
circle, HL zz \/rr —
HLC, CDE, we (hall have DE = ^^
Vrr — yy
215. To divide a given angle FCB, or arch
FAB, into three equal parts.
Let H, I, be the points of divi(ion required ;
then the chords FH, HI, IB, ought to be equal:
and the arch FAB being given, it's chord FB
will alfo be given, which let be equal to if.
Then, drawing the radius CA zz. r perpendicular
to FB, which will bife6t it in D, it will alfo
bifed the chord HI, and CD will be known,
which make — a. Drawing the radius CK
perpendicular to CA, and from the point H
drawing HL perpendicular to CK, make CL
iz ^, and it will be, by the property of the
And drawing the radius CH, by the fimilar triangles
But, becaufe the angle FHC
ought
SECT. IV,
ANALYTICAL INST ITU T I O N S.
20t
ought to be equal to the angle CHI, by the conditions of the Problem, and
CHI = CED by the parallels FB, HI, and CED r: FEH ; then FHC =::
FEH, and therefore FE = FH. But FH z: HI = 2j, therefore FE zz ^y\
And the whole line FD zz. ^y -\-
a)>
\^rr —
yy
But FD -=. /; therefore zjy -{-
ay
=:/; and taking away the afymmetry, it will be jy* — fy^ ■\- \ffyy
*/rr — yy
+ j^aayy — rryy + frry — -^//rr = o ; or, becaufe rr — ff + aa, it is
_y* fy^ — ^rry^ -f- frry — \ff^^ •= o, an equation of the fourth degree,
which may be conftrudled after the manner already explained, making ufe of
fuch conical /o<:/ as fliall be moft agreeable. But this equation is divifible by
y — /, and the quotient is the equation ^' — ^rry + \frr = o, which I (hall
conftrud by a parabola, and an hyperbola between the afymptotes. Make
therefore yy — rz, and making the fubftitutions, it will be zy — \ry -f -\fr
•=. o, an equation to the hyperbola.
Fig, 121.
Make AR zz f r, and AB =r 4/.
Producing AR, AB, each way indefi-
nitely, between them, as afymptotes, let
the hyperbola TP//) be defcribed, which
(hall pafs through the point O. Then,
taking RC z= \r, and from the point
C drawing the indefinite line CI parallel
to AL, take any line whatever, CI zr y,
and it will be IP =r z, and the hyper-
bola will be the locus of the equation
zy — ]^yy + \fr zz o. With vertex
C, diameter CM, and parameter zz: r,
let the parabola NCH be defcribed ; it
will cut the hyperbola in three points
T, P, N, from whence drawing the lines
TS, PQ»_ NM, parallel to AL, thefe fhall be the three roots of the equation.
It is plain that the parabola will cut the hyperbola TP in the points T, P,
becaufe, it being CR =r ^r, putting this value inftead of z in the equation to
the parabola, yy — rz, it will give us ^^ z: fr. But fr is always greater than
i/, and therefore the ordinate in the parabola, which correfponds to the point
R, will always be greater than RO; and therefore the parabola will pafs within
the hyperbola.
Now, becaufe the circle is given in the Problem, it will be much more
convenient to make ufe of this for the conftrudlion, by introducing it, firft, to
be added to the final equation, and that by putting the line HL (Fig. 120.) or
V rr — yy — Z»
Then it will be DE = -^ , and DF = 2y +
Dd
and
therefore
202
ANALYTICAL INSTITUTIONS.
BOOK I,
therefore the equation is ly -f- -^ — /, that is, zyz + c.j -=2 fzy a locus to the
hyperbola between the afymptotes.
Fig, 122,
I4H_A,_T
Bifefting DF in P, through the point P
draw the indefinite hne PN parallel to AC,
and taking -Qp ~ \a, thrciigh the point O
draw th.e indefinite line VA parallel to KC.
Between the afymptotes PN, VA, defcribe
tliC hyperbola whofe redangle is -^af^ which
(hall pafs through the point C ; and taking
the ^'s on the line CQ»^ pofitive towards the
point K, the correfponding ordinatcs (hall'
be 2;, and the hyperbola be the locus of the
equation 2zy 4- ay — fz "zz o.
This will cut the circle in four points H,
R, M, S, from which drawing perpendi-
culars HX, RG, MY, ST, to AC, thefe
fhall be the roots of the equation, three, HX, RG, MY, politive, and one,
ST, negative.
It Is plain that the root HX, or CL, ferves for the divifion of the given
arch FAB ; and the root YM ferves for the divifion of FMB, the remainder
to the whole circle. For, if I had propofed to divide the arch FMB, I (liould
have had the fame equation, and therefore the fame locus.
The root RG ferves to no purpofe, but, however, it informs us, that it is
equal to /, or that by which the equation is divifible, which refults from the
two loci rr — yy zn zz, and 2zy -\- ay — fz =10; that is, the folid equation
found before.
Now, to demonftrate ir, taking Ooo zz -rci zn OQ>^ the correfponding ordinate
of the circle will be GR ~ /. But coG = PD zz ff; therefore wR zz ff.
But the confiant redangle of the hyperbola is j^af; therefore the hyperbola will
cut the circle in the point R, and therefore it will be the root which correfponds
to this point.
The other root TS ferves for the divifion of the whole circle into three
equal parts, which may be demonftrated in this manner.
Becaufe FD =: RG, the arches FK, KR, will be equal ; and therefore, RG,
being produced to Z, the arches FAB, RMZ, will be equal. Therefore FR,
or BZ, will be half the difference of the two arches FAB, FMB. But if we
iiiould folve the Problem relatively to the arch BZ, we (hould find the fame
hyperbola HCS, and ZS would be a third part of the arch BZ, that is, a third'
part of half the difference of the arches FAB, FMB ; and therefore BS is a
third part of the faid difference. But HB is two-thirds of, FAB, and therefore
one»lhird.
SECT. IV,
ANALYTICAL INSTITUTIONS.
21^:
one- third of the faai of the two arches FAB, RMZ. Therefore the fura of
HB and BS, that is, the arch HS, will be a third part of the whole circle.
Qj E. D.
ai5. This Problem has been refolved before, at § iio, but after another Other cafesof
manner. There it is feen, that, in the cafe wherein theeiven anale is a iio;ht^''"^/'"S^^'j'^
, iTAii -III 1 T! I r r ir contlrucled.
anole, the Problem will be plane. In the other two caies, or an obtule or
acute angle, we arrived at thefe tvvo cubic equations, zi^x^ — ;^aax^ -j- a* zz Oy
and 2^a' + ^aax^ — a"^ ~ o.
But if it be confidered, that in the firft equation, which Ccrvcs for tlie obtufe
angle, taking x negative, it will be changed into the fecond, which ferves for
the acute angle ; it v/ili be fufficient to conftrud: the equation for the fii ft cafe,
becaufe the negative root of this will give the folution for the other cafe.
Therefore I multiply the firft equation by x zz: o, in order to reduce it to
the fourth degree, and I divide it by zi^ j then it will become x* —
2b~
+
Zb
zz o.
I take the equation to the parabola xx — '— ^ zz ay^ and fquaring it, it will
— aayy. Then, inftead of the two firft terms, fubfti-
be x^
^aax
3 ^ ga
i6bb
aax
tuting their value,- it will be yy — ' ^yl ' "^ l? — - °* Here, inftead of xxy I
fubftitute it's value ay + ^^ , and I fliall have the equation yy — ^^~ —
2 7(3'**
circle, taking the co-ordinates at right angles.
- H ^ = o J to which adding the firft, xx — ^ ay zz Oy it will be
ay zr o, an equation to the
Fig. 125.
With radius CG = y/ mm + ««, (making,
for brevity, ^—j^^- = im^ and -L^^~
= 2«,) let the circle MNH be defcribed,
and taking CD — i», from the point D
draw DA perpendicular to CD, and equal
to n. This will meet the periphery of th^
circle in the point A. Through this point
A draw AK parallel to RG ; and, taking
any line at pleafure, AK = y, the corre^
fponding ordinate will be KH rz .v, and the
circle will be the locus of the equation.
D d 2 On
204
ANALYTICAL INSTITUTIONS.
BOOK I.
On the right line AD take Al zz ~- , and through the point I drawing LO
parallel to AK, let there be taken a portion of it, IL i= jr^, and with vertex
L, axis LO, and parameter zr a, let there be defcribed the Jpollcnian parabola
ALH. From the point A taking the abfcils y on the axis AK, the corre-
fponding ordinates will be KH — x, and the parabola will be the locus of the
equation xx — -^—r- = ^y
this will meet the circle in four points. A, M,
H, N. The point A will give the introduced root r: o. The three perpen-
diculars, (^M, PN, KH, to AK, will give the three roots of the equation.
The firfl pofitive root, QM, will ferve for the obtufe angle. The fccond, PN,
which is negative, will lerve for the acute angle. The third, KH, will ferve for
the divilion, into three equal parts, of that angle which is the difference be-
tween the given angle and a right angle.
Now, to fhow that this is true, let
the given angle be MAB. Let AH be
perpendicular to AB ; and let us divide
the angle MAH into three equal parts,
which is the difference between the
given angle MAB, and the right angle
HAB. Suppofe it fo divided by the
right lines AC, AD, and repeating the
reafoning of § tig, it will be AC n CD, and the triangle ACH will be fimilar
to the triangle DAH, and therefore we (ball have the analogy, CH HA ::
HA . DH. Naming the quantities, therefore, as in § no, AB = a, BR z: /^
and BC = X, it will be RC = ;^ — • /^, BH = -^, CH =r ;c — ~ , AR =
zbx, DH ■=! X
aa
T
\/ aa — bb, HA — -7— >/ aa — bb 3 AC — \/ aa + xx
+ \/ aa ■\- XX — 2hx. Therefore, fubftituting thefe analytical values in the
foregoing proportion, it will h^ x ^ ~ . —s/ aa - bb :: -7- \^ aa - bb . x
aa
+ V <ra -f- XX — zbx'
Whence the equation ~ X, aa —' bb zz x — -^
X X
—r- + "^ aa ^ xx — zbx ', which, being reduced, and finally divided
by aa — bb, will be found to be ibx^ — "^aaxx -{• a* z=z o, the very equation
which was to be conftruded.
Befides the angles lefs than two right ones, which infifl on arches lefs than a
femicircle, and which the architects call Entrant yfm^les, there are alfo angles
which are greater than two right ones, and which infill on arches greater than a
femicircle, and are called Salient Angles, The inclination of the two lines
AB,
S ECT. IV,
ANALYTICAL INSTITUTIONS,
205
Fig. 124. AB, AM, which point towards C, may be
confidered as pofitive, and that negative which
poincs towards D. As long as the inclination
of the two lines AB, AM, (hall be pofitive,
and fhall point towards C, fo long the angle
MAB (hail be entrant, or lefs than two right
angles, and (liall infill; upon an arch, BCM,
lefs than a femicircle. If the two lines A2B,
A2M, fhall make a right line 2B2M, the
inclination will be none at all. But if the
inclination (hall become negative, the two
lines A3B, A3M, winding to^vards D, then
the angle 3MA3B will be changed into a
falient angle, greater than two right ones, and
will infifl: upon an arch, 3MC3B, greater than a femicircle. Therefore the
trifecVion of any given angle may alfo include that of a falient angle.
Now it is to be confidered, that, as the line AB (Fig. 123.) infifts upon the
line MAE, whilft it forms the angle MAB, three other angles will confequently
arife, that is, the entrant BAE, which, united to the given and alio entrant
angle MAB, makes up the two right angles ; and the falient angles MAB,
BAE, which, united to the correfponding entrant angles, complete the four -
right angles.
Wherefore the three roots of our equation, ibx^ — 3^^.v' ■\- a* zz o, ferve
for the trifedion of all the fore-mentioned angles. By means of the leafl:
pofitive root, the obtufe angle MAB is divided into three equal parts j and, by
means of the negative, the acute angle BAE, as has been feen. Befides, it has
been (hown, that the greater pofitive root ferves for the angle MAH ; and this
ferves alfo to trifed both the falient angles MAB, BAE. For, indeed, the
falient angle BAE is equal to three right angles, together with the angle MAH.
The third parr, therefore, of the falient angle BAE mud be equal to one right
angle, together with the third part of the angle MAH ; and fuch is the angle
CAB. In like manner, the falient angle MAB is equivalent to three right
angles, taking away the angle MAH, or ;&AE ; and confequently fAB will be
it's third part, as being equal to the right angle ^AB, taking away the angle
hACf a third part of the angle hKE,
217. Now, to divide the given angle into three equal parts, if I had made The fame
ufe of Prob. Xlll. § 108, I fliould have come to the equation x^ — 3^x*conftiuaed
— yrx + Irr ~ o ; and, multiplying by ;c =; o, it is x"^ — ;^bx^ — ^rr^^ another way.
4- brrx zi o. Wherefore, affuming the locus to the parabola xx — ^bx =: by^
and doing the reft as ufual, we fliall have another locus to the circle, taking the
co-ordmates at nght angles. This will ht yy ^ ^
-f x;f zi o.
Ub
Ub
Thefe
2o6
ANALYTICAL INSTITUTIONS.
BQOK I.
Thefe two loci being defcrihed and combined, will give the fame conftru6tion
as in Fig. 125, differing only in the known quantities. For, in this cafe, the
radius of the circle will be CG zz \/^mm + nn ; (niaking, for brevity-fake,
^,— — = 2m, and ^-^—^- = 2»,) and it will be CD = ?», DA r= »,
8bb
Uo
AI =: ^by and IL zz -^^h.
ThisProbiem 2 1 8. From the fame Problem we have a general method for dividing any
raifed higher.^iyen arch Of angle into as many equal parts as we pleafe. Thus, to divide it
into five equal parts, we fhall have this equation, \ '—, ; ~ h. that is,
^* — 5^A* — lornv' 4- lobr^x^ + ^r^x — br^ zz o.
To conftrudt this, I take a locus to the j^pollonian parabola xx zi ry, and,
making the fubllitutions, there arifes a fecond of the third degree, xyy — ^byy
— lorxy 4- lo^ry + S^^^
'Fig* 126.
brr = O, that is, X ZZ shy - ^0b ry_+Jrr ^
yy — lory + 5rr
Therefore, having defcribed the locus
of the equation, which (hall be the curve
with three branches, Fig. 126, that is,
HT between the alymptotes RK, BC ;
GMQ^between the afymptotes DI, KR;
and fnil^ between the afymptotes DF,
DI } in which, on the axis AV, are the
^'s, and the correfponding ordinates are
the at's. With vertex A, parameter zr r,
and axis AV, if the parabola of the
equation xx — ry be defcribed, it will
meet the curve in five points, O, M,
T, /, Q>^ which will determine the five
roots, <7r, inn, TV, Sr, and PQj three
pofitive, and two negative, of the equa-
tion propofed.
— raiTedftlU 219. So, to divide an arch or angle given into any greater odd number of
higher, equal parts, other curv-es may be found, relative to the degree of the equation*
SECT.
SECT,. V> ANALYTICAI. INSTITUTIONS, ZO/
SECT. V.
Of the Co7ifiru5iion of Loci which exceed the Second Degree.
220. The Geometrical Loci may be conftru<5led after two different manners ;Hi'glier/o«
rhat is to fay, by defcnbing curves exprelTing equations which exceed the fecond conftruaed
degree; if we may call ihat d'efcribing, in each manner, which is rather tracing^"^° w;:ys.,
them out, fo as to give fome notion of fuch curves.
The firfl; manner of tracing them is, by finding an infinite number of points.
The fecond is, by means of other curves of an inferior degree, which are
already defcribed. Thus, a locus or equation of the third degree may be con-
ftrucled by means of a right line and a conic fe<5tion i a locus or equation of the
fourth, by means of two conic fedlions; a locus or equation of the fifth, by-
means of a conic feftion and. a locus of the third degree. And fo on, as far as
you pleafe.
221. Now, as to the firft manner, by an infinite number of points; firfl, —firfl, by
the equation muft be reduced in fuch manner, that one of the two unknown finti'^r^f^
quantities, which (hall feem fitteft for the purpofe, muft be freed from fradiions'" ,"'^^x,.
or co-efficients, muft be of one dimenfion only, and placed alone on one fiJiepoi,it<j,
of the (ign of equality ;. which may always be done by the methods explained
in Se6f. 11. Then, in refpedl of fuch unknown quantity, (the other being"
confidered as conftant,) the equation muft be af it's own nature plane, that is,
muft not exceed the fecond degree. As, for example, the equation xyy + zaay
= a^, that is, yy H z: xXj which, managed- by the rules for aflected qua-^
draticks, will give y z= — , — -I__^ ,
Equations being given or reduced in this manner, the way of conftrufting
the locus^ or curve exprefi^ed by it,, confifts in giving an arbitrary value to that
unknown quantity which is included in the homogeneum comparationis ; taking it
from a fixed point on a right line, which ferves as an axis or diameter, according
as the angle of the co-ordinates is to be a right or an oblique angle. As in the
— act i V iV* -'f a'*' - r
equation y = — • ^ if vve fhould give to * a value at pleafure, by
8. thajti
2o3
ANALYTICAL INSTITUTIONS.
BOOK I,
that means we flionUI have a congruous value o( y alfo. Then, from the ex-
tremity of the allumed value of .v having drawn the value of j, in the given
angle of the co-ordinates, this would fupply us with a point in the curve to be
defcribed. Another value that we may give to the fame- unknown quantity x
will fupply us with another jy, and that with another point in the curve ; and
thus, one after another, by affigning different values to x, we fhall have fo
many /s, or fo many points of the curve. Now, the greater the number be
of thele points, fo much the more exad: will be the defcription of the curve,
and then only we can have it perfedly exad, when we take an infinite number
of fuch points, at due diftances.
Theordlnates 222. For the fake of greater fimplicity, I (liall at prefent fuppofe, that thefe
to be at right curves are referred to their axis, or that the angle of the co-ordinates is a right
^hf T^ *° "^^^^ angle; for, in cafe the angle be oblique, no alteration will thence follow.
An Example
of defcribiug
the curve by
points.
223. For the more eafy underftanding the application of this method, I (liall
take a fimple example of a curve already known, that is, of the equilateral
hyperbola yy iz. xx "• aa, or ^ ~ ± \^ xx — aw
Fig. 12 J'. Let A be a fixed point, or the beginning
of the ^*s, to be taken on the indefinite line
AE. Firft, then, I examine what ordinate
correfponds to the point A, that is, what will
y be when a: = o. Therefore, fubflituting o
inflead of x in the given equation, it will be
found jy — Hh V'o — aa, or J is imaginary and
impoffible. Therefore, to the point A there
belongs no point of the curve. By making
* n o, if ^y had not come out imaginary, but
only o, the curve would have begun at the
point A. It may be obferved, that as often as
X is lefs than a, the radical ^xx — aa will always be negative, and therefore y
an imaginary quantity. Therefore, making AB — a^ to every x lefs than AB
an imaginary y will always correfpond, fo that there will be no point in the
curve. I take at = ^ = AB, then jy =: ± V aa - aa :=. o ; and therefore B
will be a point in the curve, or rather, the curve will have it's origin in the
point B. I take x zz 2a ^ AC, and it will bey zz ±: \/^a - aa = ± '^3^%
pofitive and negative. Therefore make CD pofitive, and Cd negative, each
equal to V ^^a, and D and d will be two points in the curve. 1 take x zz ^a
= AE, and it will he y zz ±, \^Saa. Making therefore EM pofitive, and
Hm negative, — V^aay and M, m, will be two points in the curve. And
thus going on continually, by giving other values to x, we fhall have the con-
gruous values of y. And it is eafy to perceive, that, as the x*s increafe, fo the
7 quantities
SECT. IV. ANALYTICAL INSTITUTIONS. 20(J
qnantiiies \/xx - aa v/ill perpetually increafe, that is, the values of y, bot!\
affirmative and negative. Thus, the curve will always proceed, enlarging and
lengthening itfelf both above and below the axis j and, laftly, taking x infinite,
becaufe, to fubtract a finite quantity from one that is infinite, is the fame tiling
as to fubtrad nothing-, therefore \/ xx - aa will become \/xv, or a;, and we fhall
have J' 3: ± ^, and y pofitive and negative will be infinite, and tlierefore the
curve will go on ad infinitum ^
224. And becaufe, in the equation y = ± y'.v.v — aa, the unknown quantity In even
«• is raifed to an even power, that is, to the iquare; if we take x negative, the P*^^^^'"^» ^'"-' '
equation itfelf receives no alteration. Hence it is, that, if we aflign negative ^'Jl^ "^ .^.^^^
values to x^ or if we take it on the fide of A towards F, the fame curve would biguous.
be defcribed as before, but in a contrary pofition with it's vertex H, it being
AH — AB. And to no abfcifs .v, pofitive or negative, taken between B and
H, any real ordinate, pofitive or negative, will correfpond j that is, there will
be no point of the curve.
225. Now it is plainly feen, that the given curve cuts the axis in no point To find where
out of the vertices B, \\ ; for, as x increafes, y always increafes. Neverthelefs,tl'e curve cuts
it very often happens, that, befides the vertex, they cut it in other points, in^^'^^^^s.
which cafe y muft neceffarily become nothing. Therefore, to have thefe points,
in the g'ven equation we muft fuppofe y zz. o, and find the values of .v on this
fuppofition, which will give us the points required. Wherefore, in the equation
yy ss. XX — aa^ fuppofing jy = o, it will be xx zz aa^ that is, .v zz ± ^.
Therefore, in the points B, H, only, the curve will cut the axis, and not in
any other,
226. If, between the points B, C, other values of .v be taken, we fhall alfoThe more
have the correfponding values of ;>', that is, other points of the curve between points we
B and D, as alfo, between B and d\ fo that the more points we have, the more^^^^* ^^^
exa<5t will be the defcription of the part BD, or BJ; but we can never have it
perf£<fl, unlefs the number of thofe points were infinite. And the fame may be
iaid of any other portion.
' 227. Now it is plain, that if either of the two indefinite quantities be madeTo find when
infinite, and the other be neither infinite nor imaginary, but be either finite or^ curve can
equal to nothing, the firfl indeterminate will be an afymptote to the curve, '^^^^^"'^^y"^'
which will correfpond to fome determinate point of the value of the fecond.^
Therefore, to inquire if a curve have afymptoies, and where they are, it will
fufiice to make y infinite, and to fee what value for .v will then relult from the
equation. Then, to make x infinite, and fee what value for y will thence refult.
In the equation jy zz ± ^xx - aa, making y infinite, it will be \/xx — aa ^ 00,
and therefore xx — aa ~ 00, or xx = 00, and therefore x is infinite ; for the
root of an infinite fquare muft alfo be infinite. So that y cannot be infinite
E e except
2IO
ANALYTICAL INSTITUTIONS,
BOOK T,
except when x is infinite alfo; and therefore the axis of the jy*s cannot be an
afymptote. Making x infinite, \^xx — aa vvill be the fame ; for a finite quan-
tity, added to or taken from an infinite quantity, can make no alteration \ ic
will be now y "=■ ^i x, or, if x be infinite, y will be {o alfo, and it's axis
cannot be an afymptote.
—found by 228. But it is not fo in the equation ay ■\- xy zz. hb^ which we otherwife
changing the j^^qw to belong to the hyperbola between it's afymptores. For, taking y infi-
nite, the two terms ay + xy will be infinite, and, in refpcd: of them, the term
bb will be nothing, and the equation will become ay 4. at); zz o, and, dividing
by ^, it is AT — — a 'y fo that, taking x zz. — tf, the ordinate, which in this
point is infinite, vvill be an afymptote to' the curve. Then, taking x infinite,
becaufe the two redlangles ay^ xy^' having the fame altitude ^, are to each other
as their bafes a, x, the fecondmull be infinitely greater than the firll, or ay will
be nothing in refped of xy. Therefore, expunging ay out of the equation^
there will remain xy = bb, or y =. — . But x is infinite by fuppofition,
therefore y = — r: o. So that when y — o, then x will be infinite, and
therefore is an afymptote to the curve.
Cautions to 229. But here it mufi; be obferved, that this way of arguing takes place only
be obferved in the cafe of afymptotes parallel to the co-ordinates, and not otherwife. For
m finding ^^^ truth is, the hyperbola yy zn xx — aa has indeed it's afymptotes, but which
are not parallel 10 either of the co-ordinates ; therefore, in this cafe, the prefent
way of arguing cannot be applied, but there is need of a further artifice;
which, as it depends on the Method of Infinitefimals, I fhall referve for
another place.
afymptotes.
To find If the 230. It remains to inquire, whether the faid curve y zz ±: \/ xx ^ aa be
curve be con- ^oncave or convex towards it's axis j for which purpofe, we mufi: take from it's
caveorconvex
towards .it's
origin any abfcifs AE of a determinate value, and, by means o^ the given
AXIS.
eauation, we mufi: find the value of the correfponding ordinate EM. Then,
taking another abfcifs AC of a determinate value lefs than the former, we miift
find the value of the correfponding ordinate CD ; and drawing the right line
BM, which (hall cut CD (produced, if occafion) in I ; and the lines AE, AC,
being known, or BE, BC, and the ordinate EM, by the (imilar triangles BEM,
BCI, we fhall find the value of CI; and if this be lefs than CD, the curve will
be concave towards the axis AE, as is plain ; but if it be greater, the curve
will be convex. In the given Example, I take x zz. AE zz 3^ ; then y zz -^^a".
Again, I take x — AC ~ 2a; then y — CD =: V ^aa. Now, becaufe
BE = 2a, BC zz a, it will be CI — =: V 2aa, that is, CI lefs than
CD, and therefore the curve is concave towards the axis AE.
231. But
SECT. IV,
ANALYTICAL INSTITUTIONS,.
aij
231. But thefe conclufions are valid only in fiich curves, which have no Further to
point of contrary flexure, or of regreflion. But, becaufe thefe have their J*^*^*^™'"^*^*^
particular methods, of which, at prefent, this is not a place to treat, we cannot ^,°™gg°^^j}^
as yet form a jiift and complete idea of fuch curves. examples.
EXAMPLE II.
Fig. 128^.. li ^7 -y^ ^^^ ^^^^ equation be y^ =: aax, or y "iz
'^aax. Drawing the two indefinite lines
BH, DC, making a given angle BAG equal
to that of the co-ordinates ; in AC, from
the point A let the .v's be taken, and the
^'s upon AB, or a line parallel to AB.
Firft, I inquire if the curve paffes through
the point A or not, that is, what will y be
when X zz. o^ But, making a; = o, we find
y rz '^aa X o, that is, y zz. o. Therefore
the curve pafles through the point A. I in-
quire further, if the curve cuts the axis AC
in another point, that is to fay, what is .v when y = o, and I find ^aax 13 o,
that is, A? zz o. Therefore the curve cuts the axis in no other point but A.
Make 5c zz AM — ^a, and the given equation will be jy zz ^^a}. Therefore,
drawing MP zz ^^a^, and parallel to AB, then P will be a point in the curve.
I make x zz AC zz ^, and it will be jy = -^tf' =: a-y then drawing CN = a,
and parallel to AB, N will be another point in the curve. And doing this
fuccefiively, we may find as many points as we pleafe, through which the curve
of this equation will pafs. Laftly, make x infinite, or a; zz 00, and it will be
y zz '^aa X 00, that is, y is infinite, and therefore our curve pafles on to infinity.
And becaufe, taking ;f zz o, it is alfo y zz o, and taking a; = 00, it is alio
y zz CO, the curve will have no afymptotes that are parallel to the co-ordinates.
Let the line AN be drawn beneath, which cuts in I the line MP, produced
if neceflary. Now, fince AM = ia, AC =z a = CN, it will be MI = -fa.
But MP = ^ia\ therefore MI will be lefs than MP, and therefore the curve
is concave to the axis AC.
Now, if we take the abfcifs negative, becaufe in the given equation
jv' zz aax, the exponent of x is odd, when x is taken negative it's fign fliould
be changed, and the equation will then be ^ zz 4^ — aax -, here it is evident,
that, taking the values of x the negative way, that is, from A towards D, but
equal to thofe already taken the pofitive waj^, it will give as many negative
values of _y, equal to the pofitive. Whence the branch AE will be juft the
fame as the branch AN, but contrarily polited.
E e 2 EX-
212
ANALYTICAL INSTITUTIONS,
SOOK. !•
EXAMPLE III.
Let the equation be a^ — zyjy z: o, that
IS,
y
= ± v/
and let us take the
z's from the point A on the axis AC. Firft,
I inquire if the curve pafles through the
point A; making therefore z n o, the
equation will be j zz ± -/ — , that \Syy zi
± CO. Therefore BD, being infinite on
both fides of A, will be an afymptote to the
curve. Next, I inquire if in no point the
curve cuts the axis; and therefore put jizo,
and the equation will be ± -/— = o, or — = o, or 2 i= — , that is,
z = 00. Therefore AC will be another afymptote. Taking z ■=, a zz AE,
it will be^ = ± >/-f_ = H-^. Making therefore EF pofitive and EG nega-
tive, and each zn a, the pioints F, G, will be in the curve. Taking 2; zr 2^
•=. AH, it will be ^ =: ± s/ ~ — + >^ \aa. Therefore, making HI pofitive,
and HK negative, each equal to s/^aa^ the points I, K, will be in the curve.
Taking new values of z ahvays greater and greater continually, there will refuic
new values of y always lefs and lefs, fo that the two branches, FI, GK, of the
curve being in every thing equal and fimilar, will ftretch out on each fide,
approaching to the afymptotes BD, AC, yet without ever touching them, but
at an infinite diftance from the point A.
As to the negative abfcifs 2; ; becaufe the exponent of 2; is an odd number,
if it be taken negative it will be convenient to change the fign of the term
— 2xy, and then the equation will be a^ •\- zyy z=. o \ that is,^ = ± ^-.^,
That is to fay, the ordinate y is imaginary, and therefore on the negative part
of the abfcifs there will be no curve.
To examine whether the curve be concave or convex towards it's axis AC,
I take AC ■=: ^a; then it will be CM = V-^aa-, and drawing FM, which cuts
HI (produced, if occafion) in O, and MN parallel to AC, it will be JSIF =
a — \^^aa, PI z: \/j;aa — \^-^aa. Then making the analogy, MN . NF ::
MP , PO, that is, 2a , a -^ V'^aa :: a , PO ;. it will be PO = f-H-i^;
and.
SECt. IV.
ANALYTICAL INSTITUTIONS.'
21^
and therefore, if PO be greater than PI, the curve will be convex towards the
axis AC. This is to be examined thus. If it be
a — v^T''^
> V^aa — V-^aa,
then multiplyjng by 2, it will be ^ — \^^aa > 2\/j:aa — 2\/-^aat and
a 4~ \/^aa > 2\^i:aa, and fquaring, aa + iu\/-^aa + ^aa > 2aa, and mul-
tiplying by 3, 2^a + 6a\/^aa 4- aa > baa, and reducing the terms, ta\/^^aa
> 2<2i?, and dividing by 2a, ^x^-jaa > a, and, laftly, fquaring, -^aa > <7^, or
3 > I. Now, as this is a true refult, fo it is alfo true PO is greater than PI,
and confequently the curve is convex towards tue axis AT.
EXAMPLE IV.
Fig. 130.
Let the equation of the curve be ^y =r
_i_ y 4<?-y + d* — 2x^ ± a\f aP- + Sfljy Q„
2
the indefinite right line AB, taking the x's
from the fixed point A, and the j's on
AD, which makes the angle DAB of the
co-ordinates \ if it be put ^ =r o, it will
be J = ± y/ — = — ^—i that is, y =:
± V — , and y 1= -I- 4/ — ; or y =:
± ^7, and y •=. o. Therefore, making
AE pofitive and negative iz: a, the points
E, A, E, will be in the curve. To
find where the curve cuts the axis AB, I put jy = o, and therefore
- o. inen, Iquanng and tranfpofing.
± s/
/^ax + aa -^ ixx zz ± a\/aa + Sax, and fquaring again, i6aaxx + Sa^x
+ a* -i^ 4A* — i6ax^ — 4aaxx zz. a^ -\- ^a^x ; then, reducing and dividing by
4^jf, it is 3^^ — ^ax -\- xx zn o, and refolving the equation, x zz + a -{• 2a,
that is, X 1:1 a, and x zz ^a. Therefore, taking x zz AF zz a, and x = AB
zz 3^, the curve will cut the axis in the points F, B. Make x zz ^a — AH,
it will be y zz ± s/^ z^^vA^a^ therefore the four values of 7 are real, be-
caufe 2a^^aa is lefs than taa ; which roots are, y/^-fflif.^ ^Sf^z^^J^-
4 • 4-
-s/
^aa — 2aiy^aa
J and — v/
^aa + ia^ ^aa
The two pofitive roots are
Eelatively
Zl^ ANALYTICAL INSTITUTIONS. BOOK 1.
relatively equal to the two negative ; therefore, taking HI =: Hi zz
^/l --^—-^ and HG = H^ — s/ l --^—y the four points, I, G,
^, /, will be in the curve.
Examples to 232. As often as the quantity under the common radical vinculum is a
wh^en"thTor- "^g^^ive quantity, (for that under the fecond vinculum, or \/^rT"8^, cannot
dinatesarc b^ negative, the abfcifs being pofuive, as I now fuppofe it,) the ordinate y will
real. be imaginary. Now, therefore, that there may be an ordinate, it will be
cr ^L ^ '^ u / ^ax + aa ■— 2VX + avaa + Sax
neceflary that it be v^ • =— —
> o.
In the firft place, I take the fign pofuive of the fecond radical, in which cafe
the whole quantity will be certainly pofitive, if.it be 4.ax -\- aa -^ 2xx > o,
that is, 2XX — 4ax < aa, and therefore xx — 2ax < faa, and xx — 2ax
-^ aa < ^aa^ and extrading the root, x — a < x/^aa, or a — ;^ < \/\aa.
From the firll root, in which x is fuppofed to be greater than a, I infer that it
muft be X < a -\- s/ \aa» From the fecond, in which it is fuppofed that x<a,
I conclude that it mull be a? > ^ — s/^aa. But, as « — V ^aa is always a
negative quantity, it will be always x > a — \^\aa^ when x is taken lefs
than a. Therefore, taking x lefs than ^, the quantity ^ax -i- aa — ixx will be
pofitive, fo that much more the quantity 4ax + «* — 2^* + ^\/^^^~+~8^ 'will
be pofitive. And therefore, in general, taking a? lefs than AF, or a, it will be
, yA^x 4- a* — 2x* + aA/a* + Sax 1 J- -n 11
y — ±: V , a real ordinate. But, even though
, • . • • Afix + aa — 2XX ^- a'v aa 4- Sax
j^ax -\- aa — zxx were a negative quantity, yet \/- -I — -
may be a pofitive quantity; that is, whenever it is y^- ■ — — >o,
it will be, by fquaring and tranfpofing, a\/aa + Sax > 2xx — aa ^^ ^ax, and
by fquaring again, a* + ^a^x > 4x'^ — i6ax^ + i6^V — 4^V + 8a^x + a'*,
that is, 4.^* — i6<?a;' + iiaaxx < o, and dividing by ^xx, it is xx — ^ax
4- ^aa < o, and therefore xx — ^ax ■+ 4aa < aa, and extrafting the roof,
X — 2a < ay as alfo 2a — x < a. From the firft root, which fuppofes x to be
greater than 2a, arifes x < o^a. Therefore, taking x greater than 2a, but lels
than AB, or 3<f, the radical will be pofitive, and therefore the ordinate y will be
real. From the fecond root, which fuppofes x lefs than aa, I obtain x > a ;
and therefore, whenever a; is greater than a, and lefs than 2^, the radical will be
pofitive, and therefore y real. But we have feen by the firft, that, taking x lefs
than a, the ordinate y is real ; therefore, in general, the ordinate y will be real,
if we take x lefs than AB, or ^a.
Taking
SECT. IV. ANALYTICAL INSTITUTIONS. 215
Taking the fi^n negative of the fecond radical, it would be
\/- — ~ > o, and Iquanng, /^aDc + aa — 2xx >
a\^aa + Sax, and fquaring again and reducing, and dividing by ^xx, it will be
XX — ^ax > — '^aa^ and thence alfo xx — 4<2a' + /i^aa > aa^ and extrading
the root, x — la > a, as alfo, ia — .v > a. From the firlt root we obtain
.V > o^a. But we have feen, that x > 2<i gives the value of jy imaginary, when
the fecond radical has a pofitive fign, and therefore much more when it has a
negative fign. Wherefore, omitting this root, I make u{q of the other,
2tf — a: > ^7, which gives me x < a. Therefore, taking x Itfs than AF, or ^,
the quantity under the common radical vinculum will be pofitive, as well if we
take the fign of the fecond radical pofitive as negative, and therefore betweea
A and F thei"e will correfpond four real ordinates, that is, two pofitive and two
negative, which are relatively equal to the pofitive. But when x is greater than
AF, or ^, the negative fign of the fecond radical gives an imaginary ordinate,
and the pofitive fign gives it real ; becaufe it is x lefs than AB, or 3^, and
therefore between F and B will correfpond to the fame abfcifs only two real
ordinates, one pofitive, the other negative and equal to the pofitive; and beyond
the point B they will be only imaginary.
Now let the abfcifles be taken negative, that is, from A towards K. Tn this
cafe, changing in the equation the figns of all the terms in which the exponent
/-•ill . / aa — 2XX — a.ax + a Vaa — Sax , ■,
of X IS odd, then y :=: ± v^ . I put a; = o, and
it will be jy 1= ± V^ — — , that is, jv =: ± a, and y = o. Therefore the
points E, A, E, will be in the curve, as in the firft cafe. To fee if the curve
cuts the axis, put y = o; then ^ -^ - ^-^ - ^-^ ± -"^ -- - ^^- - ^^ ^nd
fquaring, and tranfpofing, aa — 2xx — /\.ax =1 a\/aa — 8a*-, and fquaring
again, and reducing, and dividing by ^^xx, it will be xx -\- ^ax + 3^<? in: o;
and refolving, x zz — 2a ± a.
Therefore the curve will cut the axis when it is a- — o, a divifion being juft
now made by ^xx -, when it Va x — — 3^2, and when it is x =. — a; that is,
by being a negative quantity, on the fide oppofite to this, towards which we
now take x j and therefore only in A, F, B, as has been already feen. Now I
put a: :z: 00, to fee if the curve goes on to infinity, or to the afymptote AK,
and it is j? =: ± v/— 2 x oo» ± ^— Sa x 00, that is, y is imaginary. I inquire
then what are the limits of the real ordinates. It is certain that then x is greater
than f^j the fecond radical will be a negative quantity, and therefore the
ordinate y imaginary; fo that x rnuft not be taken greater than -Ja ; but in this
hypothefis, becaufe the whole quantity under the common radical is pofitive,
taking the pofitive fign of the fecond radical, it will be enough that aa —■ 2xx
7 -^^ax
2l6
ANALYTICAL INSTITUTIONS.
BOOK I.
— /\.ax be pofitlve, that is, aa — 2xx — agx > o, and therefore xx + iax<, faa,
or .V < \/ -l^aa — a. But when x is not ^i eater than {a, an J alio < */ j-aa — a,
making then x not greater than ja, the oidinate will be real. Taking the nega-
- /- , r 1 !• 1 • -111 yfia — zx-x — a.ax — a^'aa — Sax
tive fign or the Iccond radical, it will be v/ >o,
that is, fqtiaring and tranfpofing, aa — zxx — /^ax > a\/aa — 8ax, and fquaring
again and reducing, x + la > a. But x -^ la h always greater than a, and
therefore, fuppofing x to be taken not greater than ^a, the ordinaies will always
be real. I take x = i^?, and it will bey— ± ^-^^ ; and therefore, making KM
pofuive, and KN negative and equal to ^-—^j the points M, N, will be in the
curve. I take x = 75" > ^'^ ^"^ be jk = ± -^^ jg , that is, the
four values are real, two pofitive, which are relatively equal to the two negative.
And, becaufe the fourth proportional of f^, — ^^, and t'^^, or ^^^/"* ^ is
lefs than —^ — -!-g ^-^— , but greater than — ^ , the curve
will have two branches above AK, one concave, and the other convex, and
fllfo two below, hke and equal to thofe above, as in Fig. 130.
EXAMPLE V.
Let it be the curve of this equation y z:
bbxx — x^ H-' 2ax^ — aax , ^
; here, tor one
± V
X — 2a
cafe, let a be greater than b, and let the ;^'s
be taken from the point A, upon the inde-
finite line AM, and the y*s upon AD in a
given angle, or parallel to a given line;
Making A^ = o, it will be jy =1 o, and there-
fore the point A is in the curve,. Making
7ZZ0, it will be V
hhx — x'^ -i- zaxx —aax
= o.
that is, hbx — x^ + <iaxx — aax = o, and
dividing by x, it is bb — xx 4- 2<2a; — aa
n o, and therefore xx — 2ax -^ aa zz. bb, and extracting the root, x — a ■=:
+ b ; therefore the values of x will ht x ':=^ a -{■ b, x "=. a — />, and x — o,
becaufe the equation was divided by x, "Whence, making AB n; BM = a,
BN
•SECT. IV, ANALYTICAL INSTITUTIONS.
217
BN z= BC = ^, the curve vvill cut the axis in the point A, as has been already
feen, and alio in the points N, C. Making x zz AM zz 2a, y will be pofuive
and negative infinite, and therefore there vvill be an afymptote at M. Put
;^ zi CO, it will be ^ =z ± \/ — xx, that is, imaginary. Therefc-ire the curve
is not continued to infinity. Now, that the ordinate y may be real, it follows
that the quantity under the vinculum mud be pofitive j it is therefore neceflary
that, the numerator of the fraftion being pofitive, the denominator muft be fj
alfo; and the one being negative, the other mu(l be the fame. But, that the
numerator may be pofuive, it muft be bbx — ^' + zaxx — aax > o, or, di-
viding by X and tranfpofing, xx — 2.ax < bb — aa. Therefore xx — lax
-{• aa < bb, and extracting the root, x — a <. b, taking x greater than a j and
a — X < b, taking x lefs than a. From the firft root, x — a < b^ we have
X < a + b. From the fecond, a — x < b, we have x > a — b. Therefore,
taking x greater than «, it muft bt x < a -{• b \ and taking x lefs than a^ it
muft be .V > ^ — b, fo that the numerator may be pofitive. Now, that the
denominator may be pofitive, it muft be a: > 2rf ; and, as it cannot be greater
ihan 2a, and at the fame time lefs than a + b, and than a, the numerator and
denominator cannot be both pofitive; and therefore between the points N and
C there will be no real ordinates. If we take x > a -^ h, the numerator will
be negative ; as alfo, if we take x < a — ^. And if we take x < ia, the
denominator will alfo be negative. Therefore, between A and N, and between
C and M, there will be real ordinates, and the curve vvill be nearly as ia
Fig. 131.
Take x negative ; changing therefore the figns of thofe terms, in which the
-exponent of x is an odd number, the equation will be j zi + ^ — ^
, . ,hbx — «' — 2axx — aax rr^, j • -n . ■
that IS, _y — ± V*^ T • -^ "^ denominator will always be pofi-
tive ; bur, that the numerator may be pofitive, it will be neceflary that b^x — x*
— 7ax^ — a^x > o ; and, dividing by x and tranfpofing, xx + lax •\- aa < bb,
that is, X H- a,< b, and therefore x < b — a. But, if we fuppofe b < «,
then b — a will be a negative quantity, and therefore it can never b^ x <b — a,
that is, the numerator can never be pofitive. So that the ordinates jy will always
be imaginary, and there can be no part of the curve on the fide of the negative
abfcifles.
Ff EX-
2l8
ANALYTICAL IKSTITUTIONS.
BOOK S.
EXAMPLE VL
Let the equation bejy' —lay'^ ^aay + za^
= axy, that is, x - y2^^_-y' - ^^y + ^^'
ay
From the fixed point A, upon the indefi-
nite line AQ»^ I take the jy's, and on the
indefinite line AM, or it's parallel, in the
given angle of the co-ordinates, I take the
A,'s. Putting J' r= o, it will bt x — ~,
o '
that is, X =00; fo that the curve will ap-
proach to the afymptote AM. To fee if
the curve cuts the axis, and where, I put
X zi Q, and therefore jy^ — lay"^ — - aay
+ 2«^ zz o ; and, refolving this cubic equation, we fliall have three values of jy,
that i^i y ■=. a, y zz la^ and y z= — a. Therefore, making AB — AD =
BC =: a, the curve will cut the axis in the points B, C, on the fide of the
pofitives, and in the point D on the negative fide.
To determine 233. If the equation j' — 2jy* — aay + 2a^ = o had been irreducible,
thefamewhen fQ t[,^|. ^^^ could not have had the analytical values of j, we muft have con-
are frredii°"^^^^'-^^^ this equation, and by that means have found the values of y geometri-
cibJe. cally, that is, exprefled by lines, which would have given us the points required.
And this is to be underftood of any other fuch cafe. Thus, I put y ~ 4^,
and it will be .v = — -j-V^, that is, the ordinate is negative, and therefore the
curve pafles below the axis AQ^at B, and returns above at C. I put y = 02,
— — — 00, and therefore the curve goes on to infinity. It is
it will be X
plain that the infinite branch BE will be convex towards the axis AM, the
branch BC will be concave to the axis AC^ and CF convex, when the curve
ihall have no contrary flexures. Let us now take the abfcifles y negative from
— >3 _ 2ay^ + aay + 2a^
————— ,
A towards D. Then the equation will be ^ =
- «^
or
__ ^3 + 2 ay* — aay — 2a^
X — ■ ■
ay
laa
I take y zz Oy then it will be;^=: z: — 00;
therefore MA, produced infinitely on the fide of the negatives, will be alfo
an afymptote to the curve. I take y zz \a, it will be .v = — ^a ; I take
y zi a, then it will be x zi o, and the curve will pafs through D. I take
J =: 00, it will be x zz
^ yy --
a
CO, and the curve above AD will go on ad infi^
nitum.
SECT. IV, ANALYTICAL INSTITUTIONS. 2T9
nitum. I take y "=. ^a :=: AK, then x =: ^a r: KP. I take y z=. ^a r=z AN*,
then it will be .v zz 6^ = NR. Now, becaufe, drawing the light line DP, it
will be NT = ^a, and ^a > ta ; therefore NT > NR, and the curve in R.
is convex to the axis AK, that is, concave to the axis AM. But, if it go on
towards the afymptote AV, below AK, it mud therefore neceflarily be convex
to it, and therefore will have a contrary flexure ; but, to determine this does
not belong to this place.
234. But, if the propofed equation of the curve to be conftrndted fliall It may tic
involve both the indeterminates raifed to a power higher than the fecond, fo ^°"^ ^>' ^^'^^
that it cannot generally be reduced in fuch manner, as that it may have one of ^"^ P°'° *'
the two indeterminates alone, on one fide of the equation, of one power only;
ttien, indeed, the trouble of the operation may increafe, but not the difficulty
of the method. For, fixing a known value upon one of the indeterminates,
for example .v, we fhall have a folid equation, given by y and conftant quantities,
which is to be refolved or conflrudted ; from whence we fliall have the values
of ^, which will determine fo many points of the curve. Then, fixing upon
another value for a:, we fhall have another folid equation to be refolved or
conftruded, which will furnilh us with other points of ihe curve; and thus
working from one to another fucceirively, we may find as many points as we
pleafe of the curve to be defcribed.
23^. But, on this and fuch other occafions, as it is required to refolvc and An objtaion
conftru(fl folid equations, as in the fixth Example, it may feem as if we fdl"'^^'^'-*^'^'
into what logicians call Circulus Vitiofus, becaufe, in treating of Solid Problems,
I have fuppofed the deftription of curves which are fuperior to conic fedions.
But, upon further refltdtion, the matter will be found to be much otherwife.
For, if the curve to be defcribed be of three or four dimenfions, the folid
equation to be conftruded will be of the third or fourth order at mod, and be
performed by means of the conic fedions. Therefore, without any circulus
'uttioj'us, any curve of three or four dimenfions may be defcribed. If the equa-
tion of the curve to be defcribed fliall be of five dimenfions, the folid equation
to be conftruded will be, at moft, of five ; and this is done by means of a
curve of three, and one of two dimenfions. And fo, in like manner, of the
higher orders ; whence it plainly appears^ that there can be no objedion of our
falling into fuch a fallacy.
Ffs VRO-
220
ANALYTICAL IKSTITUTlONg.
BOOK I,
PROBLEM I.
Example, for ^^g' ^33*
determining
the forms of
the loci from
the equation.
236. Having given the femicircle AEB, it is
required to find the locus of the points M fuch,
that, if through every one of them a right line
be drawn from the extremity of the diameter A,
which thall cut the periphery in D, and if the
lines MP, DO, he let fall perpendicular to the
diameter, the intercepted lines from the centre
JB CP, CO, may be always equal.
Let M be one of thofe points, and make
AB zi ^, AP = X, PM —yv and, becaufe it
mud be CP zi CO, it will be OB = AP = .v,
and OD — \/ ax — xx. And, becaufe of fimilar
triangles APM, AOD, it will be x . y :: a — ;;
• ^/ ax — xxi and therefore y =1
'V ax —
, that is, y —
x^x
*y ax — XX
Diccles,
, , or;' =
" ~ " W a — X
, the equation of the curve to be defcribed, which is the Cijfoid of
To defcribe it upon the given figure by various points, it may be obferved
^bat the right line AB is the axis of the ^*s, and A is the given point from
whence they take their oiigin. And, becaufe the ^''s are perpendicular to this
axis, from the point A drawing the tangent AQ^ this will be the axis to which-
the ordinates y are to be referred. Theie things being premiled, if we make,
firfi, ,r 1= o, to fee if the curve cuts the axis AQj and, becaufe we find alfo
y iz o, therefore A will be a point in the curve to be defcribed. Make jy ~ o,
to fee if the curve cuts the axis in any other point. But, becaufe we find
Af = o, the curve will not meet the two axes in any other point but A.
Make x z= ^a^ it will be y n —^ ; make .v = ^a, it will be y =:f^7,-and'
therefore, from the centre drawing CE perpendicular to the diameter AB; the
curve will pafs through the point E. Make x ~ ^a, then y — — ^ ; and,
laftly, making x — a, we (hall find y zz — — 00 , and therefore the tangent
BR to the circle will be the afymptote to the curve. Taking x greater than a^
the quantity under the radical fign in the denominator will be negative, and the
curve imaginary. "Which being alfo imaginary, if we take x negative, it will
be
S5CT. IV.
ANALVTICAt INSTITUTIONS.
221
be wholly comprehended between the two tangents AQj^ BR, produced in iyifi^
iiitum. And, becaufe it approaches to the afyniptote BR, having no contrary
flexure, it will necellarily be wholly convex to the axis AB, and will appear as
i^i Fig. 133.
PROBLEM 11.
Fig, 134.
237. The angle ABC being a right Another ex*
angle, and the point A in the fide AB ^"^Pj^ ^"'' tj?«
being given, the lotus is required of iJl mepurpoe.
the points M, fuch that, drawing through
every one of them the right lines AE,
terminated at the fide BC in the point E,
it may be always. EM zz EB.
Let any right line AE be drawn, and
let M be one of the points requiretl ;-
from the point M let fall IVIP perpendi-
cular to AB, and make AP ~ x, PM z=j,
and AB = a. It will be PB z= « — x,
atid AM =1 \/xx 4- _>:>•. Now, becaufe of
fimilar triangles APM, ABE, it will be x . y :: a , BE, and therefore BE —
EM zi -^. But it is alfo AP . PB :: AM . ME ; that is, x , a — x ::
x/xx + yy * -^ ' Therefore ay ::z a ^ x X y/ ^x -^ yy^ and fquaring, aayy zz
aa:<v — lax' -j- •*'* ^__
aaxx "-^ 2ax^ + ^ + ^^yy — aaxyy + xxyy, or zz yy.
And, laflly, fince the root of aaxx — 2ax^ + x* is as well ax — xr as
XX
— ax. it win be V =
, and y =
XX — ax
w zax—i
'^f 2ax—xx
'vzax — XX
, the equation to the curve which is required.
that is,
y =
The ordinates y will therefore be pofitive and negative, and equal to each
other ; and the pofitive and negative will correfpond to the fame abfcifs ; and
therefore the curve will be both above and below the axis AB, and will be
altogether fimilar and equal.
From the point A. drawing AR perpendicular to AB, which (hall be the axis
to which the ordinates y are referred, as AB is the axis of the abfcifs x \ firft, L
make
3a2a
ANALYTICAL INSTITUTIONS.
BOOK f.
make .v = o, to fee if the curve. pa0es through A ; and, becaufe I find alfo
y — o, the point A will be the vertex of the curve. Now make ^ z= o, it
will be ax — xx = o, and therefore x in o, and x zz a. Hence I find that
the curve will pais through the point B alfo. Make x — ^a, and it will be
J- y — . — —
3\/5*
Make x = ia, and it will be ± ^ —
Make x rz 4«,
it will be -4- v z:
^ . Make x zz 2a, and it will be •+- y = — zz 00 : and
therefore, taking AD = 2a, and drawing the indefinite right line SQ. parallel
to PM, it will be an afymptore to the curve. If x be greater than 2a, the
quantity under the radical vinculum will be negative, and therefoie the ordinate
y will be imaginary, fo ihat there is no part of the curve beyond the poijit D.
It is plain that, between the points A and B, the curve will be concave towards
the axis AB. And becaufe, beyond the point B, it applies itfclf to it's afymptote
6Q»^ it will be convex to the axis BD between B and D, provided it has no
contrary flexure.
Taking a; negative, the quantity under the vinculum will be always negative,
and therefore the ordinate y is imaginary; fo that, on the negative part of the
abfcifs, there will be no curve j whence it will be nearly as in Fig. 134.
Another ex-
ample of che
curve cnlled
the Witch,
PROBLEM III.
238. The femicircle ADC, on the
diameter AC, being given ; out of it a
point M is required, fuch that, drawing
MB perpendicular to the diameter AC,
which fliall cut the circle in D, it may be
AB . BD :: AC . BM. And, becaufe
there will be an infinite number of points
that will fatisfy the Problem, the locus of
thofe points is required.
Let M be one fuch point, and making AC zz ^, AB n x, and BM = y,
by the property of the circle, it will be BD zz \/ax - xx ; and, by the con-
dition of the Problem, it is AB . BD :: AC . BM i that is, x . \/ ax - xx ::
a ,y, and therefore y zz
aw ax — XX
or ^ =
aSr a —
*Jx
will be the equation of
the curve to be defcribed, which is vulgarly called the Witch,
8
Becaufe
SECT. IV.
ANALYTICAL INSTITUTIONS.
a^3
Becaufe AB = .v, BM zr _y, the axis of the x*s will be AC ; and AQ»^ pa-
rallel to BM, will be the axis of the ordinates y. Firft, make x zz o^ k will
be ^ n oo, and therefore AQ__is the afymplote of the curve. Make _)? in: o, it
will be a\/a ~ x ~ o, and therefore x = a. So that, when it is ^ = a^ the-
curve will cut the axis AC, and confeqiiently will pafs through the point C,,
which will be it's vertex. Make x = AR n: fa, it will be y :z: a. Make
X = AP ~ l^, it will be _y =: ^v/y. Make x =z AF ^ ^a, it will be jy zi:
^-/^ z= fa. Putting x greater than a, the quantity under the vinculum vvill
be negative, and the curve imaginary. To fee whether the curve be concave ^
or convex towards the axis AC, make this proportion. As CP = fa (which
correfponds to x zz 4<?,) is to jy zi as/f, Co is CF zr fa, (which correfponds
to a: z: ^a,) to a fourth, which will be ^ x tV't. But x ~ ±a gives y =z
a\/f, and a x x^^t '^ ^^^^ ^\\2iXi w/f. Therefore the curve will be concave
towards the axis AC. But, becaufe of the afymptote AQ^ it ought alfj to be
convex ; therefore it will be partly concave and partly convex, and therefore it.
will have a contrary flexure, which will be found by the method to be given in,
it's proper place. And taking x negative, becaufe the quantity under the
vinculum will be negative in the denominator, jy will be imaginary. Wherefore
the curve will be as may be feen in Fig. 135, obferving that this curve has a
branch fimilar and equal to the branch CLM, on the other fide of AC, corre-
fponding to y negative.
PROBLEM IV.
Fig, 12,6.
239.The indefinite right line NN being Another ex-
given in pofition, and a point P out of ^'^P^^' ^^'"S
I r t • A/T • • 1 r I the Concboid
the lame, the pome M is required, ^^^^ oi jsficcmales.
that, drawing from it to the point P the
right line MP, the line NM, intercepted
between the indefinite line NN and the
point M, may be equal to a given right
line. And, becaul'e there are infinite
points that fatisfy this demand, the locus -
of thefe points is required.
From the point P draw the right line PA perpendicular to NN, and the
right line PM to any point M, which is one of thofe required ; and drawing
the right line ME parallel to NN, make PS = ^, SE = Xy EM = y, and let
SA zn a he the given line, to which the right line NM is to be equal by the
condition of the Problem. From the point N draw the right line NO poj'pen-
dicular to EM, and it will be MO z= \/aa - kx* And, becaufe of the
fimilai".-
*-^4 ANALYTICAL INSTITUTIONS. BOOK t,
fimilar triangles PEM, NOM, it will be PE . EM :: NO . OM, that is
h ■\- X , y W X , \/aa - .XX i and therefore i + x X \^aa — xx — xy, and
fquaring, xxyy == aaxx — a* + 2aabx — 2bx^ + aabb — blxx ; and laftly,
y _ _f_ ^ ^ the equation or the curve to
be di-^fcribed, which is the Comhoid of Nicomedes,
Tliree different cafes may be diftingnilhed in this Problem. That is, it may
be b ~ a; it may be b lefs than a; and laftly, it may be b greater than a.
Firft, let it be b — a and the equation will be changed into this following :
y — ± .
X
Since it is SE rr-y, and EM z: ^, the axis will be NN, to which the ys
are referred, and PA that of the at's, the origin of which is at S. Firfl, I
make x n o, to fee if the curve paffes through the point Si and becau(e there
arifes _>• ~ ± — , that is, y pofitive and negative is infinite, NN will be the
afymptote of the curve. I make y rz o, to fee where the curve cuts the axis
TA, and it will be — at'* + za}x — lax^ + <a* — o. Now, this equation
being refolved by ihe rules before laughr, it's roots will determine the points
in which thecurv-e-meets the aforefaid axis PA. Now the roots of this equa-
tion are four, that is, ^ zr ^j pofitive, and three negative roots equal to it, or
:4? — — a. Therefore the cirrve will meet the axis in two points, diftant from
the point S by the quantity a. But, becaufe, at prefent, we are concerned only
vwith the pofitive a;'s, it will be fufficient to conlider the pofitive root ; and
. therefore the curve will pafs through the point A, it being SA ~ a^ as is fup-
pofed. Make x ~ \a, it will be j = ± — ii!!f . Make x = 4^, then y =s
^ A — 2££ ^ ^^^ ^ ^^ greater than a^ and the quantity under the vinculum
will be negative, the firfl term, on this fuppofition, being greater than tlie
fourth, and the third than the fecond. Wherefore, taking x greater than a,
the curve will be imaginary. It remains to examine whether the curve be
iiivvays convex towards the axis PA ; for it muft be fo in part, becaufe of the
aiymptote NN. Make then this proportion : As AE zz f^, (which corre-
fponds \.o X zz \a;) is to ^ zz i— I^, fo is AI zz \a to a fourth, which will
be ■2=— ^, But AI zz \a correfponds to a- zz -|^, and therefore to y =
t — 1_ . Now -—~ — IS greater than ■- , and therefore the curve will be
partly concave towards the axis 1*A. Confequently it will have a contrary
flexure^
SECT. V. ANALYTICAL INSTITUTIONS, • 22^
flexure, as (hall be feen in it's due place. And, becaufe two equal values of j,
^ne pofitive, the other negative, correfpond to the fame value of .v, the curve
will have another branch on the negative fide of y, fimiiar and equal to that on
the pofitive fide; and it will appear as is defcribed in Fig. 136.
To defcribe the curve on the negative part of x, it will be neceffary to
change the figns of thofe terms in which the indeterminate is raifcd to an odd
r 1 , • -111 I \/_ jt-+ — 2ii^x -f- zax^ -\- a*
power; fo that the equation will then be jy r: ± .
aa
Now, firft, let it be « z= o, then y = ± — , and therefore NN is ftill the
afymptote to the curve on the negative part. Make y zz o, and it will be
— X* — 2^^V + zax^ + ^* r: o, whence we obtain four roots, as above :
three are equal and pofitive, ^ iz rt, and one negative, x = — a. The nega-
tive root, which was pofitive in the foregoing cafe, is already fixed in the
fuperior conchoid. Then the three equal values fignify, that, in the pole, which
is diflant from the beginning of the x*s by the quantity a, the curve will have
a regreffion, of which we lliall treat in the Method for Contrary Flexures.
Make x = \a, then y zn ± ^5^. Make x zz 4^, then v = ± ^^ . If
we take x greater than a^ the curve will be imaginary ; becaufe, as the quantity
under the vinculum is the produft of xx — lax + aa (a quantity always po-
fitive,) into aa — xx, which, in this fuppofition, is negative, the whole
quantity under the radical will be negative, and therefore the ordinate y is
imaginary. Now, make this proportion : As PR ~ ^a (making SR z= |a,)
is to ^^ , fo is PQ^zz ^^a (making SQ^zz \a,) to a fourth, which will be
^ . But >' zz -^p" correfponds to SQ^zz ^a, or to PQ^zi ■^^, and ^
is lefs than — 1^^ fo that the curve will be always convex towards the axis NN,
fuppofing it not to have a contrary flexure ; and it will have two equal and
flmilar branches; for two equal values of j correfpond to the fame x, one of
which is pofitive, the other negative. So that the curve will appear as defcribed
in Fig. 136.
240. Now let h be lefs than a-, the equation therefore will be y = Another cafe
, */aaxx - X* + 2aabx - zbx^ + anbo - bbxx , - , . .,, , of the fame.
± . Make .v zz o, it will be v =
•*■ "^
±: ~ = ± cc. Therefore, in this cafe alfo, NN (Fig. 137,) will be the
G g afymptote
£l6
ANALYTICAL INSTlTUTIOITSv
^OOK R.
Fig, 137.
of the curve. Make ^ z= o, then <zV
= o ; the four roots of which (that is,.
A" ^^ ± <», and two equal to each other,.
X "iz — ^,) will determine the points in
which the curve cuts the axis PA. Bur,,
at prefent, it will be enough to confider
the pofitive value x — a ; and,- becaule
SA :=z a, A will be the vertex of the
curve. Make x :=i -^a zz SE^ then it will;
, , V-iaa 4- I2ab + \2blf i- a /r
be y zr ± -^ — zz EM-
Make x zz ^a zz SI, then it will be y zz.
yUoa a + go-^T +Tg _. jj^^ ^^^^ ^j^g proportion, AE =z \a to EM =
tij %aa + Mab + \zbh
r • Kr ' r .. c t 1*1 Ml 1 'y%aa-\-X2ab-\-\2bb
i fo IS AI ZZ -^^, to a fourth^ which will be -^^ f
ifl order to fee if the curve be concave or convex to the axis SA. But, taking.
AI zz ^^a, we have SI :r: -ay to which correfponds IK =_y =
*^loaa 4- 60^-^ + 4 ^ 33 '
J V • r 1 ^ L TTT V 5d'a4-i2rt^4-i2/JiJ , (. , TTr- v'20jta4-6ofl34-4s3^^
and It IS found to be IV zz —^ — lefs than IK, or ^ -^ -
Therefore the curve will be concave towards the axis SA. But, as it applies
itfelf continually to the afymptote NN, it- will be alfo convex, and therefore it
will have a contrary flexure.
It is plain, that, taking the abfcifs beyond the point A, that is, x greater
than a^ there will be no curve ; for the fecond term of the radical will be
greater than the firft, the fourth greater than the third, and the lixth greater
than the fifth ; and therefore the quantity under the vinculum will be negative,,
that is, y will be imaginary.
And, becaufe to the fame abfcifs x two equal ordinates y eorrefpond, one of
which is pofitive, the other negative, the curve on the fide of the negative-
ordinates will alfo be the fame, and nearly as in- Fig, 13.7.
To defcribe the curve on the fide of the abfcifs x negative, in the equation
I change the fign in thofe terms wherein the power of x is odd, and. it is y zz
"^ aaxx — A* — 2aabx 4- 2bx^ 4- aabb — bbxx iv/r t i , ab ■
+ > — . Make ^ z: o, then y zz ± — ,
'that is, infinite, and therefore NN (hall be an afymptote. I make y = g,
and it will be aaxx — x* — 2.aabx. + ihx^ + aakh — bbxic zz o ; the four
roots
SECT. V,
ANALYTICAL INSTITUTIONS.
£27
roots of this equation, which are thefe two, x "=1 -^^ a, and two equal ones,
X z=: bj will determine the points where the curve cuts the axis AP. The ne- .
gative root av=: — <j gives me the point A, the pofitive root x ^ a the point
in, and the two equal roots x zz b give the point P, fo that there will be a
node in the curve. Taking PR ~ SR zz ^h zz x, it will be jk — ± — '^aa^ —
= RT. Taking PQ^zi ^^b, that is, SQ^= x = 4^, it will be j =
v:::^rr-^ - QH. I make the analogy, PR (t^) . RT {/.^^)::?Q^
rh -
qaa
{\h) . QP = (^
,W \aa — bb
J, lil UlUtl WJ itc VVIJ
3
convex towards the axis PS. But QO (
), in order to fee whether the curve be concave or
"^ s.aa — bb
) is greater than QH
(•— ^^^r— ~ ) ; fo that the curve is convex towards the axis PS. And this
follows alfo from it's approaching to it as an afytnptote.
Taking the abfcifs beyond the point m, that is, x greater thc^n a, there will
be no curve, becaufe the radical aforegoing is the fame as ^ aa — xx x
V^;e* - zbx + i*. Bur, fuppofing X greater than a, the quantity aa — xx will
be negative, and xx — 2/\v 4- hb is pofitive; therefore the produdl is nega-
tive, and the ordinate y is imaginary. Taking the abfcifs beyond the point P,
that is, X greater than b, but lefs than a, it will be aa — ^.v, a pofitive quan-
tity, as alfo, XX — ^hx -f- bb ; therefore the produd is pofitive, and the ordinate
y i& real ; fo that between P and m the curve will correfpond, and will form a
foliate Yxmy, having a node at P; and the curve will have the appearance nearly
as in Fig. 137.
241. Laftly, let b be greater than a j the equation will be the fame as in the A third cafe
former cafe, and, taking the abfcifs x pofitive, the curve will be alfo fimilar. °^ ^^^^ ^^"^*^«
Then taking x negative, and fuppofing
y zz o, the four roots of the equation,
that is, a: zi ± a, and the two equal roots
X "zz b, will give, indeed, the fame points,
A, w, P, in the axis PA : but the point m
will be above the point P. And, afTuming
the abfcifs greater than Sw, that is, x greater
than a, the quantity aa — xx wjll be nega-
tive ; and becaufe xx — ^bx + bb is po-
fitive, their produdl will be negative, and
therefore the ordinate y will be imaginary.
Therefore the curve will not have the
G g 2 foliate
Fig, 138.
azS
ANALYTICAL INSTITUTIONS.
BOOK f.
The method
improved of
defcribing
curves by
points.
foliate of the former cafe, but will have it's vertex in m. And, becaufe the
curve is firft concave, and then convex towards it's axis PS, as is eafily feen,
and approaches to the afymptote NN, it will be nearly as m Fig. 138.
242. This method of defcribing curves by an infinite number of points,
may perhaps be reduced to a greater perfedlion, by making ufc alfo of geome-
trical conftru6lions. I (hall give fome Examples of it, wliich may ferve to put
the matter in a proper light.
EXAMPLE I.
Let us conftru6t, by various points, the curve
of Prob. I. § 236, which is the CtffbU of £)/-
odes, the equation of which was found to be
= . With radius AC iz: f « let the
circle AEB<? be defcribed ; and, taking at plea-
fure AP = Af, I obferve that the correfponding
ordinate P/ is n \/ ax — xx* Through the
point / 1 draw the diameter /CD> and joining
the points A, D, with the line AD, the point m,
in which it cuts the upper ordinate PF, conti-
nued if need be, will be in the cijfoid. For, the
angle in the femicircle/AD being a right angle,
as alfo the angle APM of the co-ordinates, the
triangles A/P, APM, will be fimilar, and therefore we Ihall have the analogy
/P . AP :: AP . PM ; that is, V ax - xx . x ;: x .y. Whence it is jv =:
VflJC _
Q. £• I.
After another manner. Becaufe the triangles PC/", CDO, are fimifar, the
angles P, O, being right, and the angles at the vertex PC/, DCO, are equal*
and alfo C/ =: CD, it will be alfo CP = CO, a property of this curve^
EX-
SECT. V,
ANALYTICAL INSTITUTIONS.
229
Fig, 134.
EXAMPLE II.
Let the curve be that of Prob. II.
§ 237, the equation of which h ±: y —
J ^-Z^^-^ . With radius AB = ^ let
V zax — XX
the circle AFD be drawn. Taking any
line AP — x, from the point P draw
the ordinate PF zz */ zax — xx ; and
drawing the radius BF, let AHE be
drawn perpendicular to ir. This will
cut the ordinate PF, continued if need
be, in the point M, which will be in the
curve AMB required. For, the triangles
AMP, FMH, being fimilar, and likewifc
the triangles FMH, FBP, the triangle
AMP will be fimilar to the triangle BFP, and therefore we (hall have PF . PB
:: AP . PM, that is, V zax - xx
propofed equation y ^^ ^
ax — XX
lax — XX
:: X . y,
Qi E. I.
Whence we have the
After another manner. Becaufe the triangle AMP is fimilar to the triangle
AHB; and it has been feen above, that the triangle AMP is alfo fimilar to the
triangle FPB. But the fide AB = BF -, therefore it will be alfo BH z=: BP.
Let the right line Ml be drawn parallel to AB, and then the triangles BHE,
MIE, will be fimilar. But they will be alio equilateral to each other, it being
BH = BP =: MI. Therefore it will be EB = EM, which is the fundamental
property of the curve propofed.
EXAMPLE III.
F^S' nS-
Let the curve to be defcribed be that of
Prob. III. § 238, called the mtch, the
equation of which is y iz , the
diameter of the circle, being AC =z a.
Take any line at pleafure, AB = x, and
draw
23<5
ANALYTICAL INSTITUTIONS,
3iOOK X.
draw the indefinite lines BM, CE, perpendicular to AC. Then through the
point D, in which BM cuts the circle, let AD be drawn, which, produced,
{hall cut CE in E. Through the point E draw a parallel to AC ; it (hall meet
BM in the point M, which will belong to the curve. For, by the property of
the circle, it is BD = V ok - xx, and, by fimilar triangles ABD, ACE, it is
AB . BD :: AC . CE. That is, ti . V ax - ^x
tlie equation to the given curve.
a , CE —
\'^ ax —
= y*
EXAMPLE IV,
Let the Conchoid of Nicomedes of Prob.
IV. § 239, be to be defcribed by va-
rious points. It's equation is i; jy z:
-=^ 1 . Make SA z: S<7 n a,
SP = h. With radius SA = e, let there
be defcribed the circle ABC<7, and taking
at pleafure two abfcifles SE, S^, equal to
each other, which may be called x pofitive
and negative, draw the ordinates EB, ^C,
each of which (hall be — V aa ^ xx^ and
let them be produced indefinitely beyond the points B, C. Through the points
S, B, let the right line SB be drawn, and through the point P a parallel to ir,
PM. The two points M, w, in which PM cuts the two right lines EB, ^C,
£hall belong to the curve required} that is to fay, the point M to the fuperior
branch, and m to the inferior branch of the conchoid.
And as to the point M ; becaufe the two triangles SEB, PEM, are fimilari
it will be SE . EB :: PE . EM ; that is, x . V aa - xx :: I' + x ,y. And
V aa — XX . r c\ r \
, in relpect or the
b •{■ X X
confequently the equation will be jy ~
upper branch of the conchoid.
Then, as to the point m ; drawing the line SC, the triangle S^C will be
fimilar and equal to the mangle SEB. For the triangle Pm is fimilar to the
triangle SEB ; therefore alfo it will be fimilar to S^C, and therefore we fliali
V aa — XX :: b — X .y,
which is the very fame
Through
em
S^ . ^C ; that is, — X
b — X X
•^aa —
have the analogy P^
Whence we have the equation y —
as (hould belong to the lower branch of the curve.
SECT. V.
A,N ALYTICAL INSTITUTIONS.
2J1
. Through the point S drawing the indefinite line SN parallel to the ordinates
EM, etJiy fioni the conftruitiou above vve iball eafily obtain the principal pro-
perty of the conchoid', which is, that, from the point or pole P, it we dravv PM.
cuitiiTg the curve in the points M, w, and the line SN in tiie point N, the
intercepted lines /«N, NM, between the curve and the indefinite line SN, vvill
always be of a conftant leng'h, and equal to SA rz SB = <«. For, by the
conftrudlion, SBMN will be a parallelogram, and therefoic NM ir SB, B-it,^
^rawing NO parallel to S^, the triangles SEE, otNO, will be (idiiiar ; and
befioes, NO n:; S*? z: SE. Therefore it will be wN zi SB, and coniequentiy
«wN ~ NM. (i. E. D.
243. The conftru<fVions oi the three firfi: Examples come out pretty fimple, — Improved
there being nothing required to be done, but to draw a circle with a given |?y'.'^^ '^°"'*^
diameter, and fome right lines. On other occafions the Conic Sections muil: be
admitted, which are fomecimes to be defcribed with variable diameters, para-
meters, and re(5tangles. But thefe may be taken as cotiLlant, in determining
one or more points of the carve.
Fig, 140. To give an example of it. Let iis con-
ftrud, by points, the curve belonging to thia
equation x^^ax — xx = yy» Draw the circle
AHB^, whofe diameter is AB = 2a. Take
at pleafure AD iz KB = ;^; it will be DE =
KG rz */ lax — xx> With parameter DE, to
the axis AB, dclcribe the Apallonian parabola
GFAj^, and DF, D/, will give the pofitive
and negative values of jy, making x zz AD.
And KG, K^, the polltive and negative va-
lues of y^ making x "=. AK. Wherefore
the four points F, /, G, g, will be in the
curve required. By a like method, and by
varying the value of^, we may determine
other points of the curve^
244. A fecond manner of conftrufting curves beyond the fecond degree, —By para-
will be that mentioned at §. 220, by means of other lines of a lower degree. J^?^^^ °^ .
And, to begin with parabolas of any degree^ it may be firft obferved, that thegJ.fg\Y '^^
Jpllonian parabola is the only one of it's kind, and is expreffed by the equa-
tion ax — yy. The cubic parabolas are two, that is, aax z= j% and axx ■=. y^.
Thole of the fourth degree are three, a^x = j^ aaxx — y^, and ax^ ~ y*.
And, in general, thofe of the degree expreffed by « are in number n — 1^
and are ^;i? "" = > ,
a —2 n
aax "^ y J
a'x
,«-3 _
fucceflively, till the exponent of x is unity.
= J , arx
«-4
«
and fo ort
245. All:
2^3
ANALYTICAL INSTITUTIONS.
BOOK I.
The firfl cu-
bical parabola
conftrufted.
245. All thofe which have ;^, with unity, for it's exponent, are called firtl
parabolas. Thus, aax = j', a^x — j*, a^^^x^y", are all firfl: parabolas.
To conftrud any parabola of any degree whatever, the beginning muft be
from the firft cubic parabola aax =: y^, ^
It is plain that this muft have two branches,
one pofitive, the other negative ; for, taking
X pofitive, y will alfo be pofitive, that is,
y zz. ^ aax, and this will be it's pofitive
branch. But, taking .v negative, y will alfo
be negative, or y zz ^ — aax, (which is no
imaginary quantity,) and this will be the
negative branch. It is evident that thefe
two brandies go on ad infinitum^ and are
concave to the axis AB.
To proceed to the conftrudion. Make
yy zz az', and, fubflituting in the equation
aax zi y^ this value oi yy, the equation to
the cubic parabola will be changed into
this, ax = zy, which may be rcfolved into the following analogy, a . z '.'. y , x.
This fuppofed, let the parabola of the equation yy zz az be defciibed to the
axis AB, and let it be DAE. Make AB = z, BE = j^, BD zz — ^,
AC = a. Draw CB, and through the point A draw the line KAF parallel to
CB ; and making AG = BE, draw GE. It will be CA . AB :: AG . GP\
that is, a , z ',*, y » x. Whence taking AB at pleafure, the correfponding lines
BE, or AG, and GF, will be the co-ordinates of our cubic parabola, and F will
be a point of it. For, in the analogy a .z :: y , x, refloring the value of z.
or
yy
it will be a . -^ l". y , te, or the equation y^ r: aax.
Now, becaufe, when x is taken negative, y will be negative alfo, the analogy
a . z '.'. y . X will be changed into this following, a , z :: — y . — x i whence,
taking AV = BD, it will beCA. AB :: AV. VK; that \s, a , z :: -^ y . — x,
^ and, the point K will be in the cubical parabola. The branch AMF will be
\pofitive, and ANK the negative branch.
llie firA: pa- 246. Let it be propofed to conftrud the firfl: parabola of the fourth degree
rabola of the ^3y = j*. This will have alfo two branches, one above the axis, the other
i^'^^j^^ below it, becaufe to x pofitive correfponds both y and — y, for the index of
the power of y is an even number. Thefe two branches will be concave
towards the axis, and will proceed in injinitum. To go on to the conftrudion.
1 make y r: aaz, and, inflead of j', fubflituting this value in the equation
propofed, we Hiall have zy zi ax, ox a , z ll y , x.
To
SECT. V.
^ig' 142.
ANALYTICAL INSTITUTIONS.
23i
To axis KC let the parabola of the
■equation y'^ zr aaz be ^fcribed, which,
becaufe it is the firft cubic, we know
already how to conllruft ; and let this
be QAD. It will be AC ~ GD = z,
AK — — z, and CD = AG = y,
KQ^r: — y. Take AB ::= a, and draw
the right lines BC, BK, and through the
point A draw AF parallel to BC, and AP
parallel to KB. This fuppofed, it will be
BA . AC :: AG . GF, that is, a . z ::
y . x; and the point F will be in the
curve-line propofed to be conftrucled.
For, it being a . z ll y , Xj and z =
^^ , it will hQ a . -^ :: y . X ; that is, a^x zz y*.
But, becaufe when x is pofitive we may take y negative, which in this cafe
will be KQ^ and AK will be — z, we (liould have alfo BA . AK :: KQ^
(zr AR) . RP ; or a , — z :: — y , x» Therefore the point P will alio be ia
the curve ah' = j*.
247. Let it be propofed to con(lru(^ the firil: parabola of the fifth degree, The firl^ pa-
^*x —J'*. This will alfo have two branches, one pofitive, the other negative, rabolaofthc
For, taking .r pofitive, 7 will be pofitive, that is, y = l/a^x. But, taking -^ ^^^Jjif^^'^X
negative, y will be negative, that is, y ':=: \/ — a^x, Thefe two branches go*^°
on infinitely, and are concave to the axis AB. To proceed to the conftrudion.
Make y* zz d^z, and fubftituting this value in the propofed equation, it will be
ax ~ yzy 01 a , z :: y . X,
To the axis AB (Fig. 141.) defcribe the parabola of the equation y* = d'z,
and lee it be DAE. It being AB = z, it will be BE = y, and BD = — j>'.
Make AC = a, and draw CB, and KAF parallel -to it. Then draw the right '
line EFG, and the parallel DVK. This fuppofed, it will be CA . AB ::
AG . GF, or a . z '.'. y , x ; and the point F will be in the curve to be
conftrudted. For, it being a , z \\ y . x, as alfo, a^z n: jy% it will be
i;4
y . Xi or y^ zn a*x, the equation to the curve propofed.
Now, becaufe, x being negative, y will alfo be negative, the analogy
^ . z ','. y . X will be changed into this, a . z ','. — y . — x. Wherefore, taking
AV zz D^^ it will be CA . AB' :: AV . VK, or a , z :: — jy . — a;.
Whence the point K will be in ihe curve propofed to be conllruded. The
branch AMF will bepofiuve, and ANK will be the negative branch.
Hh
248. And,
234
ANALYTICAL INSTITUTIONS.
BOOK I.
Thcfiiftpa- 248. And, in general, let it be propofed to conftrud the parabola vvhofe
rabi)la of any
degree con. equation is a^'^x ~y": Makej""*^ = a^^Zj and fiibftituting tiiis value
in the propofed equation, we (hall ftill have zy = ax. Whence it may be
perceived, that we may always conftrud: any firft parabola by means of a.
triangle, and of the firlt parabola of the next inferior degree.
Conftruaion 249. Now it will bc cafy to go on to the conftrudion of the other fucceedlng
^^p^j^^^"^^ ^""; parabolas, or thofe of the fecond, third, fourth, &c. of any degree ; for thele
^las. ^^^'^^"alfo may be conftruded by the conflrudion of their firfl parabolas.
Let it be propofed to conftrufl the fecond cubic parabola, whofe equation is.
axx zzy. I make y^ zz aaz, and, by fubftituting, inftead of j', it*s value ii>
the propofed equation, it will be xx zz az,
^^S' HJ* I 'Ry^^:^''^ '^^ ^^ ^^^'s ^ ^^^ there be defer! bed the
Apollonian parabola AC, whofe equation is xxzzaz;
then to the fame axis defcribe the firfl cubic para-
bola of the equation jy' — aaz ; and it being
— AB zi 2, it will be BE = y. But, in the Jpol-
lonian parabola AC, becaufe AB zi z, it will be
EC = ^v. Therefore we fhall always have the
two co-ordinates x, y^ of the fecoiKl cubic para-
bola.
Let it be propofed to conftrufV the third parabola of the fourdi degree, whofe
equation is ax^ = y^, I make a^z rz y\ and, by fubftitution, it will be
X' = aaz. Let this firft cubical parabola x^ zz aaz be conftrucled, and to the
fame axis let there alfo be conftrufted the firft of the fourth degree, y* = a^z.
The two ordinates of thefe curves, correfponding to the fame abfcifs Zy will
give the co-ordinates a*, j, of the propofed curve.
In the conftrudlion of all others, of any fuperior degree, we may proceed ia
the fame method ; thefe examples are fufficicnt, the thing itfelf being very
plain.
Squaring the 250. It only remains to be obferved, that the fecond parabola of the fourth
ducesTredu- ^^^S'^^^' ^^^^ = ^*> Js no Other than the Apollonian parabola, but redoubled the
plication of contrary way. For, firft, if it be aaxx = j^ it will be alio, by extrafting the
die curve, fourth xoot, ^aaxx =z V'a^ ~ Hhjy. But \/ax =: ± jy, or ax :=^ yy, is no
other than the equation to the y/^(?//£>«//?» • parabola. Our curve is therefore a-
common parabola, but redoubled ; becaufe the term aaxx is alike generated,
as well from 4- ^a; x 4- ^^t as from — ax x — ax -, which may be equally.
verified, becaufe y/aaxx zz \/+ ax )^ + ax = t/ -ax x -<?x = \^ax z=z ±y»
Wherefore,,
EECT. V.
Fig. 144.
ANALYTICAL INSTITUTIONS.
^3S
Wherefore, to negative x will correfpond
real y, and the branch MAN on the negative
fide will he perfedly like the branch BAG oa
the pcHtive lide^ having refped to both the
expreffions y/^aaxx zz \/ax = ± 7» But
the Jfollcnian parabola has no branch on the
negative fide ; for, putting x negative, it will
be y/ — ^^ = ± jy ; fo that the curve will
be imaginary.
If we raife the equation ax zz yy to the third power, the curve correfponding
to the equation a^x^ ■=■ y^ will be no other than the Apollonian parabola only.
Raifing the equation ax zz yy to the fourth power, the curve correfponding to
the formula a^x^ = y^ becomes the common parabola redoubled the contrary
way. And, in general, if the power to which the formula ax zz yy is raifed
(hall be even, the Apollonian parabola redoubled will exhibit the curve ; if the
power be odd, the common parabola will be fufficient.
The fame do<ftrine may be applied to all fiiil parabolas and hyperbolas,
whofe canonical equations 3iVt a ~ x ^zi y , taking for n any integer number,
affirmative or negative. This being raifed to an even power, the proper curve
of the new equation will be the parabola or hyperbola a ~ x = y redoubled
the contrary way. If the power be odd, the reduplication vaniflies, and there
will remain the fimple genuine curve of the equation a^^ x = y*^.
251. From the conftrudion of parabolas of any degree, we may go on toConaruai'on
the conftrudion of hyperbolas alio of any degree. of hyperbo-
loids.
The hyperboloids of the third degree are two ; that is, «' — xxy, and
fl3 — xyy. Let it be propofed to conftrud the hyperboloid of the equation
a^ = xxy. This curve will have two branches which approach to afymptotes ;
both of them will have their ordinates pofitive, but the abfcifles in one will be
pofitive, in the other negative.
To conftrudt it, make xx zz az, and, by fubftitution, it will be aa zz zy.
Between the afymptotes AM, AG, (Fig. 145.) defcribe the hyperbola FQ_of
the equation aa zz zy. Then taking AG = 2, it will be GF zz y ; then from
the point G, at half a right angle, let be drawn GB, and it will be AB z:
AG = z. To the axis AB let there be defcribed the parabola CAE of the
H h 2 equatioa
—■of higher
Rypcrboloids,
ANALYTICAL INSTITUTIONS.
BOOK I.
equation az =z .vjf, and drawing the ordi-
nates BC, BE, and the indefinite lines CK,
EP, parallel to BA, it will be AH — BE
n X. And drawing FK parallel to GD, ic
will be HP = GF = ;f. In the fame
manner, it will be AD zz BC zz — ^,
DK zz y ; and the points P, K, will be ia
the curve propofed*
I forbear to give the conflrudion of the
equation a^ zz xyy, becanfe it is the very fame
curve, only the co-ordinates have changed-
their places.
252. Let there be propofed an hyperboloid of the fourth degree, and let it*s
equation be «* zz ii^y. This curve will have two branches, which apply to^
afymptotes, in one of which x will be pofitive, and y pofitive, and in the other
X will be negative, and y negative.
Fig, 146.
zz aaz.
and, by fubftitution.
Put x^
we (hall have zjy zz aa. Between the
afymptotes MP, TG, produced indefi-
nitel-y, let the hyperbola of the equation
%y zz aa, or ER, KO, be defcribed.
Then it will be AF zi 2, FE = jy,
AM = — 2, MK =z — J'. From the
point F, at half a right angle, draw FG,
to which let MT be parallel, and it will
be AG = AF zz z, and AT zz AM
=z — 2i. To the axis TG let be de-
fcribed the cubic parabola SAH of the
equation x^ =: aaz, and it will be AI zz
GH = X, and AP zi TS zz — x.
Whence, drawing the right lines EC, KV, parallel to AI, it will be IC ^ y^,
and PV =z — y, and the points C, V, will be in the propofed curve.
Here, alfo, I omit the conftruftion of the equation ^2* =: xy^^ becaufe, only-
changing the places of the co-ordinates., it is the fame as before. Alfo, I omit
the conftruction of the equation «* z: xxyy^ becaufe it is reduced to the /Ipd-
Ionian hyperbola.
Other hyper
boloiMs con-
ILruded.
253. Let the hyperboloid of the fifth degree be propofed, and, firft, let the
equation be a^ =: x^y. This will have two branches, which approach to
afymptotes j in one of which, talking .v pofitive, y will alfo be pofitive. In the
others/taking x negative, yet, however, y will be pofitive.
Make-
S£CT. V.
ANALYTICAL INSTITUTIONS.
237'
Fig. 147
Make a* ~ a^z ; then, fiibftituting, it will be aa — ?;_^y. B:twcen' the
afymprotes AG, AM, (Fig. 145.) defcribe the /Jpollontan hyperbola FQ_of the
equation aa zz zy -, then, taking AG ::=: z, it will be GF — y. From the point
G, at half a right angle, draw the right line GB, and it will be A8 r= AG — 2;.
To the axis AB defcribe the parabola CAE of the equation x* — a^Zf and, it
will be BE iz AH — x, BC zz AD zz -^ x i and, drawing FK parallel to
GD, and CK, EP, perpendicular to the fame, it will be HP z: DK zl GF
=: jK» and the points P, K, will be in the curve propofed.
Let a^ zz x^y^ be another equation of the hyperboloid of the fame degree;
this will have two branches, becaufe to the fame pofilive x will correfpond two.
ordinates y^ one pofitive, the other negative...
Make x'^ zz aaz ; then, fubflituting, it
will be a^ zz zyy. Between the afymptotes^
DM, CN, let there be defcribed the hyper-
boloid RG, FV, of the equation a^ = zy~,
and making AH. — y, AP zz — _y, it will .
be HI =: z = PK zz AB. To the axis
PH let there be defcribed the cubic para-
bola AS of the equation x^ zz aaz, and'
from the point B draw BQ^at half a right
angle, and raife the perpendicular QS ;
then it will be AQ^ = z, QS =: x.
Through the point S draw* the right line
OT parallel to the afymptote NC, which
may meet the produced lines HI, PK, in the points T, O. Then, it being
AH —y, it will be HT zz ;^, AP = — y, ?0 zz x i^ and the points O, T,
will be in the curve propofed.
The conftruftions of the other two equations, a^ = x^^, and a^ :z: xy*, will
be after the fame manner, only making the co-ordinates to change places.
And by the fame artifice may all the hyperboloids of any degree be eafily
conftruded.
254. It may be obferved, that all the firfl: parabolas, which are defcribed Obfervat ion
about one and the fame axis, will cut one another in the fame point. For, "" ^jj^ j^'""'^
taking for every one of them the fame abfcifs x = a, they will all have the j^^jj^^^
fame correfponding ordinate y z= a; which could not be, except they all cut
in the fame point.
ins
255. Alfo, the parabolas of higher dimenfions (meaning higher than the firft,)— -of In'ghcr
tend firft to arrive at the point of fedion, above thofe of an inferior degree, P^^^jj"'^_*
approaching nearer to. the tangent of the vertex, and after the fedlion theybyiyj^tj^/
approach to the axis, thefe more than thofe. For, in the Apollonian parabola,
it being y = s/ax, in the firft cubic, y zz ^.aax, in the firtl of the fourth
decree, .
.253 ANALYTICAL INSTITUTIONS. ^OOK I.
degree, y = ^/ah; and fo on ; if we take x lefs than a, then \/ax will be lefs
than 4^aax, and this will be lefs than ^a'x, and fo on. But, on the contrary,
taking x greater than a, it will be V ax greater than '^aax.y this greater than
iya^x ; and fo on.
After the fame manner, and for a like tieafon, the hyperboloids (meaning alfo
the firft,) all cut one another at the vertex, and thofe of higher dimenfions tend
after the point of fedion between thofe of lower dimenfions and the afymptote
in which the x's arc taken. And on the part of the afymptote, parallel to y,
the inferior tend within, between thofe of higher dimenfions and the afymptote.
Curves of- 2^6, There remains now to conftrucl fiich equations as have feveral terms,
^'^^'^a^^ ^*^'^T^ in which I fliall diftinguiili three cafes. Thofe of the firft cafe I call fuch, which
du"ided into' ^^^^'^ ^^^^ term Only, in which the indeterminate y is found, and that of one
•three cafes, dimenfion alone. Of the fecond cafe are thofe, which have one term only ia
which y is found, but that raifed to any power, Thofe are of the third cafe
which have many terms in which y is found i and that raifed to any power.
CASE I. EXAMPLE I.
An example
of the firft
cafe.
Fig. 14&.
257. Let it be propofed to conftrud the
curve of this equation a* — if* — a^y»
Make y :=z f — q, by which the given
equation may be refolved into thefe two.
a* =z flV, X
* = a'q.
To the axis AB let
the parabola MAC of the equation x* = a'q
be defcribed ; and it being AD :z: q, it
will be DH = x, DF z: — ;f . But, by
the equation «* = ^V, it is / iz ^ ; and
therefore, taking AB = a zz i, it will be
t -^ q = y. Whence, taking at pleafure
any abfcifs BS =: DH zr Xy and BO zz
DF r: — x, the lines SH, OF, parallel to BA, will be the correfponding
ordinates of the curve propofed, which is one portion of the fame parabola of
the fourth degree.
EX-
SECT. V,
ANALYTICAL INSTITUTIONS.
23^9
EXAMPLE U.
258. Let it be propofed to conftrua: the curve of ihe equation x* + ax^ — c^y. Another
By the rules already known, we may perceive this curve to have three branches, '■'^'""p'^'^*
two infinite and pofuive, and one negative, together with a maximum^ which at
prefent we can take no notice of ^ and the axis will be cut in two points.
Make jy zi 2 4- /, whence we may have
Fig, 149.
two equations,
X* ~ a'
and x^ rr aat\.
Fig. 150.
To the axis AB let the parabola MAD of
the equation .v* z= a^z be defcribed ; and
it being AF = z, it will be FD = AE
n X, Through the fame point A let the
cubic parabola CAP of the equation x^ zzaat
be defcribed, and PE — / will correfpond
to the fame x. Whence, it being AE r: x,
it will be PE + ED zz 2 + / zz ^, mak-
ing PD parallel to AF. Whence it may
be feen, that, taking x pofitive, the ordi-
nate y increafes in injinitum.
Then, taking x negative, /will be nega-
tive, and confequently y "zz z — /. Let
AG :=. X negative, it will be GM zz z,
GT iz /, whence y — MT negative ; and
among all the values of MT, there will be
a greateft. Taking x "zz — <?, it will be
GM z= GT, whence y :=: o. Taking .v
negative, and greater than a, it will be
GM — GT, a pofitive quantity ; whence
y will be pofitive, and will increafe ad infi-
nitum. The curve will be nearly of the
form of Fig. 130, taking x from the point A,.
EXAMPLE III,.
259. Let it be propofed to conftrud the curve of the equation a?* -f ^A'AYiothei- ex-
— aax"^ z= a^y» This curve will have four branches, two pofitive and infinite, ample to .the
two negative and finite. It will cut the axis in two points, and will touch it in^"^^^ ^^^'■-
S one..
-a^o
ATSr A L y T I C A L INSTITUTIONS.
BOOK r.
• one. If wil4 have two negative maxima^ &c. as will be known by the rules to
bo delivered in their due place.
Put y Tz % — ^, and make the two
•equations x* + ax^ = a^z, and — xx ^z
] — aq. The curve of the equation .v* +
^x^ zr a}z we know already how to con-
ftru<5l by help of this method, and let it be
CBADG (Fig. 151.); in which, taking
AK zz X pofitive, it will be KG zz 2.
Taking x negative zz AP, it will be z
negative =: PB ; taking x negative and
greater than AO, it will be z pofitive.
To the axis AN let the parabola TAH of
the equation xx = aq be defcribed. Ic
being then AK = x pofitive, it will be
KH zz <j', and GH zz 2: — q iz. y, which
will increa'fe in infinitum as x increafes in infinitum. In the point F it will be
z zz q, and y zz o. Between the points F and A, q will be greater than 2;
whence z — q will be a negative quantity, and y negative, and there will be a
negative maximum. In the point A, it will be 2; ~ o, j' zi o, j zz o. Taking
X negative equal to AP, it will be z = BP, and negative; whence y is always
negative. Between the points A and O there will be a maximum BQj whence
' there will be a greateft q negative. Taking x negative and greater than AO,
z will be pofitive, but lefs than q ; whence y is negative. Taking x negative
and equal to AM, it v/ill be 2; zz ^, and y z=. o. Taking x negative and
greater than AM, it will be always z greater than q ; whence it will be always
y pofitive in infinitum.
If the equation fiiould more abound in terms, the fame artifice might be
•iifed ; and, though the conftrud:ion in this cafe might become more com-
pounded and perplexed, yet, however, the method would Hill obtain.
We might conftrud the laft equation in a different manner, by making
y zz z + / — q^ and thence deriving three equations, x"^ zz a^z^ x^ = aaty
— XX zn — aq^ and, by means of thefe three auxiliary curves, we might pro-
ceed to the conltruftion of the principal curve j but I omit this for brevity.
■The-co-or'di- 260, Perhaps, in tliefe conftruclions, and in the few that follow, it may feem
nates may neceflary that the angle of the co-ordinates fhould be a right angle, it being
always fuppofed to be fuch. But it will appear, after a little refledion, that
this angle may be as we pleafe ; efpecially if we give a little attention to the
angle of the co-ordinates of the fubfidiary curves introduced, relatively to the
angle of the co-.ordinates of the curve of the given equation.
make any
anglf.
CASE
SECT. V, ANALYTICAL INSTITUTIONS. I4I
CASE 11. EXAMPLE IV.
261. Let it be propofed to conftrud this equation, .v" ± a x" ' ± a"x^ '^^j cafe df cm-ve»
A t n r % t t—l ^ r ^ n- • i* i • n i r t . COnftrUtlcd.
&c. =1 y . Make y zz a z, and lubllituting this value inltead or y , trie
equation will be a; + a x ~ ±: a x ~ , &c.. zz a^ z. By the method of
the firil cafe, this curve may be confl:ru6led ; then defcribe the parabola of the
equation y = a" z, and we (hall have the relation between x and y in the
propofed equation.
CASE III. EXAMPLE V.
262. Let it be propofed to conftruft the equation x ±: ax ±1 bx , &c. = The third
cafe con-
y^ 4- /, &c. Make y^ ±. y^, &c. = z; then, by fubftitution, the equation ft»""^ed, with
•^ "^ * a general
will be x"" ± ax"" ± hx\ &c. zz z. By the method of the firfl cafe, each of ^''^™P^^'
thefe two auxihary curves may be conftrufted to the fame axis, in which z is
to be taken ; and we Ihail have the relation of the two co-ordinates x and y of
the curve propofed.
263. Hitherto I have confidered only thofe equations which have their To feparate
indeterminates feparate; fo that, when the indeterminates are involved with ^^5 '"'^etc''-
€ach other, the rules hitherto given cannot take place. mmates when
' D t involved.
In thefe cafes there is need, either by the common diviflon, or by the ex-
tradtion of roots, or by a congruous fubftitution, or by other expedients, to
contrive a feparation of the faid indeterminates. As, if we had the equation
fi'y + ax*y = ^V + x^, dividing by «' + ax^y it would be y = ^t^^ * .
And, if the equation were aaxy + xxyy =:;?* + <?*, making the fubftitution of
2 = ~ , we fhould have the equation a^z -j- aazz — «*-]- ^^ in which the
indeterminates or unknown quantities are feparate.
The propofed equations being thus prepared, we may proceed to their
conftru^ion in the following manner.
li EX-
242^
ANALYTICAL INSTITUTION Si
BOOK I.
EXAMPLE. VL-
Example of 264. Let the equation to be conftruaed be y = ^!fl±fl. Make «V + a;*"
the conftruc- c^ + ax^
tjon of thefc = a^p. Make alfo a^ + ax" = «V j and fubftituting thefe values in the equa.
tion propofed, it will be ^ = -^, that is, / .j) \\a .y.
The propofed curve will have two branches, which ^rach o\it ad infinituw.
Pofitive y will correfpond to x either pofitive or negative.
-%• '52. To axis HD let the curve LAC of the
equation aaxx H- at* = a^p be defcribed ;
and, taking AD= ;f, it will be DC z=:p
= AB. Take A¥z=za — AM, then witli>
vertex F, to the axis HD, let the curve
PFE of the equation a^ + ax^ z=: a^t be
defcribed ; and, taking AD = x^ it will
be DE == t. Whence, it being DC =^^
and DE = /, draw EG parallel to AD,
and from the point G draw GH at half
a right angle, and it will be AH = /.
From the point C draw CB parallel to DA,
and draw the line BH, to which let MK be
parallel. It being AD = *, it will be
AK =;^ J for, becaufe of fimilar triangles AMK, AHB, it will be AH . AB
::: AM . AK ; that is, t , p \\ a , AK = -^ zzz.y. Whence, drawing KQ^pa-
rallel to the axis, the lines AD, DQ*^ will be the two co-ordinates of the curve
propofed. To obtain the other branch of our curve, it will fuffice to take x on
the negative fide, and to repeat the fame conftrudion on the contrary part..
\t
"R
Q
\
/
/
/y
K
/
H\Iyr
J4
^
y^
\
F
G
E
\
EXAMPLE VIL
Another 7<7^«j 265. Now let it be propofed to conftruft the other equation a^xy -|- x^y* =^
conftrufted. ^ _j. ^^ which, being managed by the rules for aifeded quadratick equations,
may have the indeterminates feparated. Or, by the fubftitution of 2 = — ,
it will be reduced \q a^x. + aazz = *^ + a\ This, equation, may be con-
ftruded
SECT. V,
ANALYTlCAt INSTITUTIONS.
243
flru6ted by the method of the third cafe, and we fliall have the two co-ordinates
K and z. Then make the analogy, x . z :\ a , y, which will be the ordinate
required. If one fubftitution be not enough, to free the indeterminates from
being involved together, we muft try more than oncj and when none will
fucceed, the equations elude this method, and we muft have recourfe to other
artifices.
265, A convenient fubftitution may alfo be of ufe in other cafes, in which An obferva-
the indeterminates are already feparate ; and may often fuggeft a conftrudion ^^o''*
which is more eafy and elegant. Wherefore it may not be amifs to try feveral
ways, that we may choofe that which will prove to beft advantage.
EXAMPLE VIII.
267. Let the equation be y^ — /^ay^ + ^aayy = la^x. Make la^x r=z 2*, Condufioa
and therefore it will be y^ — 4^' 4* ^<''^yy = 2% that is, yy — lay = zz, or of the cx-
lay —yy^ zz. ^"^P^'^^'
Therefore I conftruft this locus, which in the firft cafe will be, by two oppofite
equilateral hyperbolas, with tranfverfe axis equal to 2a ; and in the fecond cafe,
by a circle with diameter = la : and, in general, by this and that together.
With tranfverfe dikmeter AB = 2a,
(Fig. 153.) let there be defcribed the
two equilateral hyperbolas AMH, BMH,
and the circle AMB. Then with vertex
A, let the parabola of the equation za^x
zi z* be defcribed, and raifing the inde-
finite perpendicular AQ»_and taking any
line AD = zj then drawing MM parallel
to AB, it will be DS = A-, and DM =^,
pofitive in the circle and in the hyperbola
from A towards B, and negative in the
hyperbola on the oppofite part ; and
the curve will be nearly as KAGBF
(Fig. 154.) ; in which the two branches,
BF pofitive and AK negative, will go on
ad infinitum ; and there will be no branch
under the axis AB, becaufe it can never
be X negative.
l\z
SECT.
ft44
ANALYTICAL INSTITUTIONS.
BOOK I.
S E C T. VI.
Of the Method De Maximis et Minimis, of the Tangents of Curves , of Contrary
Flexure and Regreffwn ; making ufe only of the Common Algebra.
To find the 268. Althoogh the Calculus of Infinitefimals be the (impleft and the fhorted'
maxima and method, and alio the mod univerfal, for managing fuch fpeeulations j yet I wa*
minima oi willing, before I finilhed this Tradt of Analyticks, or of what is called the
comparifon ^ ^^rteftau or Common Algebray to fliow very briefly, and by way of introdudlion,
•withanequa- how the folution of fuch qucftions maybe performed, in geometrical curves^
tion of two or fuch as are expreffed by finite algebraical equations, without the affiftance of
equal roots, jhg Differential Calculusy or what is alfo called The Method of Fluxions,
And to begin by the Maxima and Minimal
that is to fay, to find in geometrical curves the
greateft or the leaft ordinates. Let the curve be
AGB (Fig. 155, 156.), and taking any ordinate
DM, draw MF parallel to the axis of the abfcifles
AB, the two ordinates DM, EF, will be equal,
to which two different abfcifles AD, AE, will
correfpond. But the more the ordinates DM,
EF, IhaM move approaching nearer to each other,
the difference of the abfcifles AD, AE, fliall be
fo much the lefs; till at laft the two ordinates
DM, EF, coinciding with the greateft ordinate
CG, or the two LM, NF, with the leaft IG, the
abfciflTes AD, AE, or HL, HN, (hall become
equal in refped of the axis HK. Therefore,
when the ordinate is the greateft or the leaft, the
equation of the curve, difpofed according to the
letter which exprefles the abfcifs, ought to have
two equal roots. To determine which, there is
to be formed an equation of two equal roots, for
example, xx — 2.ex + ^<? = o, which is the
product oi X — e into X'^e-, and let. the curve
Q whofe
F/f. 156
H 1
L 1
[ N K
i :i
^^
a
\
> i
:> J
i £>
SECT. VI, ANALYTICAL INSTITUTIONS* J^j
whofe greateft or lead ordinates are required, be the ellipfis xx — lax + ^^
= o, for example, the abfcifles being taken from the vertex. Let this equation
be compared, term by term, with the equation formed from two equal roots, in
the following manner : xx — aax -j — ^ zi o. •
P
XX — lex + ee -=: o.
From the comparifon of the fecond terms, we find a -= e ; but e is the root
of the equation xx — 2ex + ee := o, and therefore e =: x^ and aifo a =: x ;
and becaufe x is already determined, the comparifon of the laft terms will be
fuperfluous. Wherefore, taking x := a, the correfponding ordinate in the
ellipfis will be the greateft, as is already known, it being then half the conju-
gate axis.
But if the equation of the curve had been of the third, fourth, or higher
degree, that we might make the comparifon, it would be neceffary that the
equation of two equal roots, xx — lex + ^«? = o, fliould be reduced to the
fame degree as is the equation propofed, by multiplying it by fo many roots,
whatever they may be, as there may be occafion for. Let the curve belong to
this equation of the third degree, x^ * — axy 4-^ = 0, (the afteriik * is put
in the place of the fecond term which is wanting, and which fhould always be
done, as often as any term is abfent,) of which we require the greateft ordmate.
Therefore I multiply the equation xx — lex + ee = o by .r — /== o, and
Compare the produd with the equation propofed, x^ m — axy + y^ =: o.
x^ — 2.ex^ + eex — ^^= o.
— fx^ -f- zefx
From the comparifon of the fecond terms, I find — 2^ — / = o, and
therefore / = — ^e. From the comparifon of the third, I find 2ef + ee =
— ay, and fubftituting the value of /, it is — 3^^ = — ay. But e => Xy
therefore jr = ^. Inftead oi y, if we fubftitute this value in the equation
of the curve, it will give us ;f = -i-i^ , to which correfponds the greateft or-
dinate y, which will be tJL^^ or ^^4^ .
3 3
269. But, without comparing the given equation with another, which con- To find the
tains two equal roots, to fatisfy the condition of the Problem, it will be fame by mul-
fufficient to multiply it, term by term, by any arithmetical progreffion. For, g'lf g^/ithm^-
if the equation has two equal roots, as it ought to have in the cafe of a maximum tical pro-
or minimum, one of thofe roots will alfo, of neceffity, be included in the product greffion.
of that equation multiplied by the arithmetical progreffion. Whence, by thus
multiplying the equation, the condition will be included, under which the value
146 ANALYTICAL INSTITUTIONS. BOOK I.
of the abfcifs will be found^ to which the greateft or lead ordinate correfponds.
Now, to demonftrate this, let the equation of the two equal roots be in general
this, XX — 2bx + hh = o, which let be multiplied by the arithmetical pro-
greflion a, a -{- b, a •{• ib^ and the product will be axx — zahx -^^ahh — o.
— ibhx + ibhh
In this fubftitute the quantity h inftead of Xy and all the terms will deftroy one
another. Or elfe, dividing it by x — /&, the divifion will fucceed. Therefore
X — h will be one root of that produd, as it is of xx — ^hx + hb = o.
The fame willx)btain if the arithmetical progreffion be decrealing, as a^ a —~ b^
a — lb, a — Q^by &c.
Now, becaufe the equation of the two equal roots is general, and the arith-
metical progreffion a, a -^r b^ a -\- ib^ &c. is general alfo, it will always be
true, that when an equation of two equal roots is multiplied, term by term, by
any arithmetical progreffion, the produ6t will be divifible by one of thofe roots.
For the fame reafon, if an equation (hall have three equal roots, and be multi-
plied by an arithmetical progreffion, the produdt will have two of thofe equal
roots. And if this produdl be multiplied again by an arithmetical progreffion,
the new product will have one of thofe roots. And fo we may go on to
fuperior equations.
I refume the equation to the ellipfis xx — lax + -^ = o, which I multiply
by the progreffion 2, 1,0.
XX — 2ax H — =^ = o.
P
2, I, o.
The produd is ixx — ^ lax — o^ which gives .« = j, as is found above, I
multiply the fame equation by another arithmetical progreffion, 3, 2, 1,
XX — %ax ■{ ^ = o
P
3> 2> i»
The produ6l is 3^^ — i^x + ^^ = o, in which, inftead of yy^ I fubftitute
it's value, ^ax - xx X —• , given from the equation of the curve, and find
AT = «, as before.
I take the fecond equation above, x\ * — «xy + y = o, and multiply it by
the progreffion 3, 2, i, o,
«' * — axy -f- j»' = o,
Zi 2, I, o.
The product is 3*' •— axy = o, or 3*' = ay, as before.
270. By
SECT, vi;
ANALYTICAL INSTITUTIONS,
247
270. By a' like method may be found the tangents and perpendiculars 10 Tangents
curves in any given points. ' ^/}^ perpen-
The queftion is reduced to this; to find a circle that fliall touch the curve °"" *
in this point. For, in this cafe, the tangent of the circle in this point, as alfo
the perpendicular or radius, will be in common to the curve alfo in the fame
point.
Let the curve be ACM, of which we
defiie the tangent at the point L \ and let
the circle be GMH, which cuts it in the
two points M, C. Drawing the two ordi-
nares CE, MP, and the right line MCT, .
through the points M, C, it will cut the
curve alfo in the points M, C. But the
nearer thefe points fhall approach to each
other, the lefs always will be the dif-
ference of the ordinates CE, MP, and
alfo of the abfcifles AE, AP ; fo that when the two points coincide, for ex-
ample at L, they will make the values equal of thefe ordinates, or of thefe
abfciffes; and then the circle will touch the curve in the point L. (Except
when the curve and the circle are of equal curvature ; for, in this cafe, the
circle will both cut and touch the curve in the fame point, as will be feen in the
Differential Calculus.) The right line MT (hall be a tangent both to the curve
and the circle in the fame point L; as alfo, FL will be a common perpen- ^
dicular.
Fig, is^*
Therefore, in the curve ALM, make
AQ^= X, QL = y, and from the given point
L drawing the right line LN, which we fup-
pofe to be perpendicular to the curve, and
confequently to the tangent at L ; make
LN = J, AN = Uy and it will be QN = u
— X, Then the right-angled triangle QLN
will give the canonical equation ss = uu —- lux
^- XX + yyj from which we are to have the value of y, or of .v, and to fubfti-
tute it in the equation of the given curve ; by means of which we muft have
the value of s, or of «, confidering x or y as given, becaiife we affume the
point L aS' given.
Let the curve ALM, for example, be the Jpollcnian p^irahola. of the equation
ax = yy, Inftead of yy, make a fubllitution of it's value given by the canonical
equation, and we. fhall have ax = ss — uu +• 2ux — xx ; which being ordered
according to the letter x^ will be xx — 2ux + «« = o. This equation, there-
-f- ax — ss
fore, ought to have two equal roots when the right line LN = s is perpendicular
to
243 ANALYTICAL INSTITUTIONS. BOOK I.
to the parabola in the point L, that is, in the cafe of a tangent. Therefore, the
value of the indeterminate AN := u being found, on the hypothefis of two
equal roots, we (hall have the point N, from whence drawing NL to the
given point L, and LT perpendicular to NL, that fliall be the tangent re-
quired.
Now, to determine the unknown quantity u on the fuppofition of two equal
roots; I compare the equation, term by term, with one of two equal roots,
that is, with xx — zex -j- (?^ = o, after the following manner :
XX 2UX -{■ UU \
+ ax — ss J
XX — 2ex •{• ee =1 o.
Now, from the comparifon of the fecond terms, we (hall have — 2« + ^
= — 2^, or « = j,a + s* But e = x, by the equation xx — 2ex •{- ee = o.
Therefore u = j;a •{■• x. Wherefore, from the point Q»^ taking QN = ia,
NL will be the perpendicular, and LT, perpendicular to it, will be the tangent
to the curve in the point L.
Inftead of comparing the faid equation with one of two equal roots, it may
be multiplied by this arithmetical progreffion 3, 2, i, thus :
^X — 2UX + ^« 1 _
+ ax — ss J
3» 2, I,
The prod^ia is s^^at j ^ux + uu ^ ^ g^^ ,, = ««-, t,ux + xx
+ zax — ss J
+ yy; and, by the parabola, it is yy = ax; whence ss = uu •-■ 2ux -\- xx -{- ax,
Subftituting, therefore, this value inftead of ss, it will be zxx — 2ux •{• ax = o.
That is, u = ^a + jc, as before-
We might have had our defire more compendioufly, by multiplying the
equation by this arithmetical progreffion, 2, i, o.
Example. ^7'" Let the curve be the fecond cubical parabola x* = ayy. Making the
fubftitution of the value of yy, derived from the canonical equation, there arifes
the equation x^ + ax^ — laux + auu — o, which, becaufe it is of the third
ass
degree, muft be compared with the produd of the equation xx — 2ex + ee = o
into X — /= o; thus, ^' + ax* — laux + auu "> _ ^
-^ ass ^
)c^ .. 2ex* + eex — ^^ = o.
— yit* + lefx
By
SECT. VI.
ANALYTICAL INSTItVTlONS,
249
By comparing the fecond terms, we have — ^e — /=s a, that is, /= — a
— 2^. From the comparifon of the third, it is ee -f. 2ef — — 2au ; and
putting the value of/ now found, it is « = — ^, that is, u = 3j)f_+_2ff^
becaufe e = x.
Now I (hall multiply the equation by the arithmetical progreflion 3, 2, i, o,
x^ + ax"^ — 2aux + auu 1
3, 2, I, o.
The produd is 3^' + 2ax^ — 2aux = o, and therefore, in like maaner,
^xx + 2ax
U =
za
^ 272. Concerning the choice of a proper arithmetical progreffion, it may be How to
be obferved, that, generally, that will be the mod convenient, which forms the choofe a
exponents, beginning with the greateft index of that letter according to which P'^°g'"^"^°"'
the equation is ordered.
273. Another m.anner of folving this Problem may be this, which is fome- ThisProblem
thing different, but perhaps more fimple, and which will be of ufe in contrary ^"'/^^ ^"'
n ^ 1 rr ^ other way.
flexures and regreliions.
Fig' 159. ^ -Let the curve AEMD be cut by the
right line HED in the points E, D; and
make the abfciffes AB or AC = x, the
ordinates BE or CD = y. It is plain that
the right line HD going on to be the tangent
FM of the curve in the point M, the two
points E, D, will coincide in M, and con-
fequently will make the two lines AB, AC,
equal to each other, as alfo the two lines
BE, CD. Draw AN parallel to the ordinates, and make AF = «, AN = s.
By the fimilar triangles FAN, FKM, it will he u , s :: u + x .j ; that is,
y — , and X = "^ ~ "^ . In the equation of the given curve, fubftitute
thefe values inflead of y or x, and another equation will arife from hence, which
will have two equal roots, fince AF, AN, are fuch, as that the right Une FNM
touches the curve. Therefore, making a comparifon with another of two equal
roots, or multiplying it by an arithmetical progreffion, we fhall have the value
of AF or AN required ; and one being given, the o'.her will alfo be given. I
forbear Examples, becaufe the manner of operation is the fame as that ulcd before.
274. As ihe nsituvQ o{ maxima and minima, and likewife of tangents, necef- ^'^'"tsorcon-
(arily requires equations of two equal roots, fo, in contrary flexures and re- anj'^re^reffioo
Kk grelTions.whatT^and^''"
how found.
Fir
250
ANALYTICAL INSTlTlfflONS,
BOOK I.
gredions, three equal roots are required. By contrary flexure is meant that
point, in which from concave the curve becomes convex, or the contrary; and
by regreflTion is meant that point in which the curve turns diredly back, again,
whether concave or convex.
Fig* 1 60.
G AH
Let the curve be ACFM, which has a
contrary flexure in the point F, and let be
drawn the right line GCM, which touches
it in the point C, and cuts it in the poinC
M ; from which draw the ordinates CH,
MP. It is eafy to perceive, that the more
the point C of the tangent fliuU approach
to the point F of contrary flexure, fb much
the more alfo the point M fliall approach
to the point F ; fo that when the point C falls in with F, the point M will alfo
fall in with it ; and confcquently AH, AP, will become equal, as alfo CH, MP,
and the right line GCM will both touch and cut the curve in the point F. But
the nature of the tangent already requires two equal roots, and now they are
joined by a third ; fo that the property of contrary flexure is fuch, that three
equal roots are correfponding to it.
From the point A drawing AN parallel to the ordinates, and making AN = /,
AT = u, and drawing TNF; becaufe of fimilar triangles TAN, TVF, it will
be y = "^ + ^^ ^nd X = ^y " ^^ making VA = x. and VF = y. Wherefore,
fubftituting thefe values oi x or j in the equation of the given curve, the
equation that arifes ought to have three equal roots, when AT or AN are fuch
that TNF, drawn from the point T through the point N, may meet the curve
in F, the point of contrary flexure required.
In like manner we may reafon about the
curve ACM, which has a regreffion in the
point C. For the tangent TC of the curve
in the point C, will alfo cut it in the fatn«
poii^Vt 'i an<^ thence the three equal roots will
arife after the fame manner.
Let AFS be the curve of the equation
^yy — ^yy — ^^^ ~ ^-^ '^"^ ^vhich are AQ_= x,
and QF = y\ and let the point F of contrary
flexure be required. Make AT = a, AV =/,
and QF parallel to the ordinates. Now, in-
ftead of ►¥, fubftituting it's value '^^ ~ "\ in the
equation of the curve, it will be
A
^
\^
Q
^^
<F
-'-—-§_
B
SECT. VI. ANALYTICAI, INSTITlfTJOWS* %$t
y^ ^ + aaxy — aas
— syy
}"■
This equation ought to have three equal roots, and therefore we muft com-
pare it with an equation of three equal roots ; or eife multiply it by two
arithmetical progreffions.
Let us multiply it, therefore, by the progredion i, o, — i, — 2, and the
produd will be y * — aay + la^s = o. Multiply it again by the progrefBon
3, 2, I, o, which will give us 3^' — aay = o, and therefore yy = ^^aa. This
value, being fubftituted in the equation of the given curve, will laftly produce
X = j^a,
275. The manner is the fame for finding the regreffions of curves, and thisTo Jlfltn-
meihod is applicable to both. So that, to diilinguilli them, there is no other S"'*^^ ^o"-
way, but to find, by means of a conflrudiion, the figure and proceeding off^ol^ recn-ef- '
the curve. fions, and
maxima from
The fame ambiguity arifes in quefhions de maximis et minimis, _ whicn only can*"'"""<^'
be removed by acquiring fome knowledge of the difpofition of the curve. By
the fame condition of three equal roots we may find the Radii of Curvature j
but as 1 fhall further treat of fuch things in the following Volume, not to be
too tedious, 1 fhall here put an end to this.
END OF THE FIRST VOLUME,
ANALYTICAL INSTITUTIONS.
ANALYTICAL INSTITUTIONS,
IN FOUR BOOKS:
ORIGINALLY WRITTEN IN ITALIAN,
BY
DONNA MARIA GAETANA AGNES I,
PROFESSOR OF THE MATHEMATICKS AND PHILOSOPHY IN
THE UNIVERSITY OF BOLOGNA.
TRANSLATED INTO ENGLISH
BY THE LATE
REr. JOHN COLSON, M.A.F.R,S.
AND LUCASIAN PROFESSOR OF THE MATHEMATICKS IN THE UNWERSITY OF CAMBRIDGE.
NOW FIRST PRINTED, FROM THE TRANSLATOR'S MANUSCRIPT,
UNDER THE INSPECTION OF THE
REV. JOHN HELLINS, B.D.F.R.S.
AND VICAR OF POTTER 's-PURY, IN NORTHAMPTONSHIRli:.
VOLUME THE SECOND,
CONTAINING THE LAST THREE BOOKS.
With
AN ADDITION BY THE TRANSLATOR.
LONDON:
Printed by Taylor and Wilhs, Cbancery-Iane ',
AND SOLD BY F. WINGRAVE, IN THE STRAND ; F. AND C. RIVINGTON, IN
ST. Paul's church-yard ; and by the booksellers
OF OXFORD AND CAMBRIDGE.
1801.
ANALYTICAL INSTITUTIONS,
B O O K II.
THE ANALYSIS OF QUANTITIES INFINITELY SMALL,
THE Analyfis of infinitely fmall Quantities, which Is otherwife called the introduftion.
Differential Calculus, or the Method of Fluxions ^ is that which is converfant
about the differences of variable quantities, of whatever order thofe differences
may be. This Calculus contains the methods of finding the Tangents of Curve-
Lines, of the Maxima and Minima of Quantities, of Points of Contrary Flexure,
and of the Regreffion of Curves, of the Radii of Curvature, &c. ; and therefore
we (hall divide it into feveral Sections, as the nature of the feveral fubjetls
may require.
SECT. I.
Of the Notion or Notation of Differentials of feveral Orders j and the Method of
calculating with the fame.
I. By the name of Variable ^antities we underftand fuch, as are capable of v triable
continual increafe or decreafc, while others continue the fame. They are to be quantities,
conceived as Hawing ^antities, or as generated (as ic were) by a continual *^^^'' ^
moiion.
Vol. il. B For
ANALYTICAL INSTItUTIONS.
SOOK IT.
F^.u
For indance, in Fig. i, let there be a right
line ABC, which is conceived as generated by the
motion of the point A, and is produced in infi^
nitum. Upon this, at any incHnation, let another
right line BD infift, and let it be conceived that,
whiift the point B moves from B to C, carrying
with it the line BD from the place BD to CE,
always remaining parallel to itfelf, the point D
(hall defcribe the line FE in fuch a manner, as
to pafs through all the points of the curve ADE.
It is plain that the abfciffes AB, AC, as alfo the
ordinates BD, CE, and likewife the arches AD, AE, will be quantities conti-
nually increafing and decreafing, and therefore are called Variable ^antiiies^
or Fluents^ or hlowing ^antities.
Conilant 2. Conftant ^tantiiies are fuch, which neither increafe nor diminifli, but are
quantities. Conceived as invariable and determinate, while others vary. Such are the para-
^iiat, meters, diatneters, axes, &c. of curve-lines.
Conftant quantities are reprefented by the firft letters of the alphabet, a, h,
f, di &c. and variable quantities by the laft letters, z, jy, x, v, &c. juft as is
ufually done in the common Algebra, in refpeft to known and unknown
quantities.
A fluxion or 3. Any infinitely little portion of a variable quantity is called it*s Difference
difference,, or Fluxion ; when it is fo fmall, as that it has to the variable itfelf a lefs pro-
portion than any that can be affigned ; and by which the fame variable being
either increafed or diminifhed, it may ftill be conceived the fame as at firft.
what.
Fig, 2.
S" B T ^
Let AM (Fig. 2, 3.) be a curve whofe
axis or diameter is AP ; and if, in AP pro-
duced, we take an infinitely little portion P/>,
it will be the difference or fluxion of the
abfcifs AP, and therefore the two lines AP,
Af, may ftill be cogfidered as equal, there
being no affignable proportion between the
finite quantity AP, and the infinitely little
portion Pp. From the points P, pj if we
raife the two parallel ordinates PM, pm^ \i\
any angle, and draw the chord mM pro-
duced to B, and the right line MR parallel
to AP ; then, becaufe the two triangles
BPM, MRw, arc fimilar, it will be BP . PM
:: MR . Rw. But the two quantities BP,
PM, are finite, and MR is infinitely little 5
then,.
SECT. 1.
ANALYTICAL INSTITUTIONS,
then alfo Rm will be infinitely little, and is therefore the fluxion of the ordinate
PM. For the fame reafon, the chord Mm will be infinitely little ; but (as will
be (hown afterwards,) the chord Mm does not differ from it's little arch, and
they may be taken indifferently for each other; therefore the arch Mm will be
an infinitely little quantity, and confequently will be the fluxion or difference of
the arch of the curve A M, Hence it may be plainly feen, that the fpace VMmp
likewife, contained by the two ordinates PM, pm, by the infinitefimal P/), and
by the infinitely little arch Mw, will be the fluxion of the area AMP, compre-
hended between the two co-ordinates AP, PM, and the curve AM. And
drawing the two chords AM, Aw, the mixtilinear triangle Mhm will be the
fluxion of the fegment AMS, comprehended by the chord AM, and by the
curve ASM.
4. The mark or charadleriftic by which Fluxions are ufed to be expreffed, is by How fluxions
putting a point over the quantity of which it is the fluxion. Thus, if the abfcifs are reprefcnt-
AP = Xi then will it be Pp or MR = x. And, in like manner, if the ordi- *^'^» ^"4 '''^*^
• arc tncir Ic*
nate PM izjv, then it will be Rw zzy. ^"^^ m^xzX Qx>l<^r^
making the arch of the curve ASM "=. s, the
fpace Al-*MS = /, the fegment AMS = «, it
will be Mm zr i, PMmp — i, AMm =. ii. And
all thefe are called Firji Fluxions^ or Differences
of the Jirji Order. And it may be obferved,
that the foregoing fluxions are written with tlie
affirmative fign ■+• if their flowing quantities '
increafe, and with the negative fign — - if they
decreafe. Thus, in the curve Nfc-C, (Fig. 4.)
becaufe AB = Xy BF zz x, BC =^, it will be
DC = — j), the negative fluxion of j.
That thefe differential quantities are real things, and not merely creatures of
the imagination, (befides what is manifeft concerning them, from the methods
of the Ancients, of polygons infcribed and circumfcribed,) may be clearly
perceived from only confidering that the ordinate MN (Fig. 4.) moves conti-
nually approaching tov^rds BC, and finally coincides with it. But it is plain,
that, before thefe two lines coincide, they will have a diflance between them,
or a difference, which is altogether inaftignable, that is, lefs than any given
quantity whatever. In fuch a pofition let the lines BC, FE, be (uppofed to be,
and then BF, CD, will be quantities lefs than any that can be given, and
therefore will be inajjignable^ or differentials^ or infiniujimalsy or, finally, fluxions.
Thus, by I he common Geometry alone, we are affured that not only thefe
infinitely little quantities, but infinite others of inferior orders, really enter the
compofition of geometrical extenfion. If incommenfurable quanrities exift in
Geometry, which are infinites in their kind, as is well known to Gcometrici-ins
B 2 and
ANALYTICAL INSTlTUTlOSTSr
BOOK rr.
and Analyfl:?, then infinicefiraal magnitudes of various orders mud ncceCirily
be admitted.
Fig. 5.
i\
B
A
E GK C
For ihe fake of an example, let AB be the
fide of a fquare, and AC it's diagonal or dia-
meter ; which two hnes (by ihe lall propofition
of the tenih Book of Euclid,) are incommen-
furable to each other. Now it may be proved
that this afymmetry of their's does not proceed
from any httle finite line CE, how fmall foever
it may be taken, but from another which is
infinitely lefs than it, and therefore of the infinitefimal order.
Let it be fuppofed then, if pofTible, that it is the finite line CE which is the
caufe of the afymmetry or incommenfurability between the two lines AB, AC ;
confequently the remaining line AE will be commenfurable to the fide AB,
Let the right line F be their common meafure, which can never be equal to
EC, for then the diameter and fide would be commenfurable. It muft therefore
be either greater or lefs than it.
In the firfl: cafe, let F be fubtraded from CE as often as can be cbne, and
let the remainder be CG. Now, becaufe F meafures AB, AE, and alio EG>
the two right lines AB, AG^ will have to each other a rational proportion; and
therefore it was not the magnitude CE that made the lines AB, AC, incom-
menfurable, but fome quantity lefs than it, fuppofe GC, which therefore is
finite, the finite line F being once or oftener fubtrafted from the finite line CE,
Let F be biieded^ and each part bifeded again, and fo on, till there arife an
aliquot part of F which is lefs than CG, and which being taken from CG, there
will remain CH. But this, by the fame way of argumentation, is not the
quantity that caiifes the incommenfurability of the lines AB, AC. And as the
lame way of reafoning obtains in all other finite magnitudes, we may thence
fairly conclude that the incommenfurability proceeds from an inaffignable
quantity, or which is lefs than any that can be given. The fame may be alfo
proved in the other cafe, or when the common meafure F is greater than CE.
From hence I (hall proceed, further, to take notice, that the fquares upon the
right lines AB, AC, which are to each other as one t5 two, notwithftanding
that their fides are irrational, are neverthelefs commenfurable to each other ;
and that this proceeds from an infinitely little
quantity of the fccond order. The two fquares
AB, AC, being propofed, (Fig. 6.) let the two
quantities ED, FI, equal and infinitefimal, be
thofe which render the fides AD, AG, AI,
AH, incommenfurable ; and the conftrudion
being completed as in the figure, it is known
that the two redangles DK, IK, arc incom-
menfurable
SECT. I.
ANALYTICAL INSTITUTIONS.
menfiirable to the fquare AB. But the whole fquare AC is to the other AB in
a rational proportion : therefore the fquare AC is made fo by the infiniiefimal
fquare KC, a quantity of the fecond order, by which it exceeds the faid incom-
menfurable gnomon.
It may be obferved, that cubes upon the lines AT, AH, are incommenfurable,
although their bafes are rational ; and it may be eafily proved, that they are
made fuch by means of an inaffignable magnitude of the third order, and we
may go on in like manner as far as we pleafe.
5. After the fame manner that firft diiferences or fluxions have no afTignableHowlifgher
proportion to finite quantities ; fo differences or fluxions of the fecond order orclers of
have no aifignable proportion to firft differences, and are infinitely lefs than "'"°r"n,^JJ
ihey: fo that two infinitely little quantities of the firft order, which differ from
each other only by a quantity of the fecond order, may be aflumed as equal to
each other. The fame is to be underflood of third differences or fluxions in
refpeft of the fecond ; and fo on to higher orders.
Second fluxions are ufed to be reprefented by two points over the letter, third
fluxions by three points, and fo on. So that the fluxion of x, or the fecond
fluxion of a:, is written thus, x ; where it may be obferved, that x and x* are
not the fame, the firft fignifying (as faid before,) the fecond fluxion of x, and
the other fignifying the fquare of x. The third fluxion of x will be x\ and fo
on. Thus, y will be the fluxion oi y, or the fecond fluxion of jx; and fo of
others.
But, to give a juft idea of fecond, third, &c. fluxions, the following The-
orems will be convenient.
THEOREM I.
6. Let there be any curve MBC, and BC an Infinltefimali
infinitely little portion of it of the firft order. P''?^^^ ^^
From the points B, C, let the right lines BA, ^*'^*
CA, be drawn perpendicular to the curve, and
meeting in A. I fay, the lines BA, CA, may-
be alTumed as equal to each other.
Let the tangents BD, CD, be drawn, and
the chord BC. If the two lines BA, CA, be
unequal, let one of them, as CA, be the
greater, and to this let the perpendicular BH
ba
6 ANALYTICAL INSTITUTIONS. BOOK II.
be drawn. The difference between the lines BA, CA, will be lefs than the
intercepted line CH, which is lefs than the chord CB, becaufe of the right
angle at H. Bat the chord BC is an infinitefimal of the firft order, the arch
being fuppofed an infinitefimal ; therefore the difference between BA and CA,
at leafi:, will not be greater than an infinitefimal of the firft order, and therefore
thofe lines BA and CA may be alfumed as equal.
Coroll. I. Therefore the triangle BAC will be equicriiral, and thence the
angles at the bafe ABC, ACB, will be equal ; and being fubtraded from the
right angles ABD, ACD, will leave the two angles BCD, DBC, equal to each
other, and confequently the two tangents BD, CD, will be equal.
Coroll. II. The right line DA being drawn, the two triangles ADB, ADC,
will be equal and fimilar ; and that line will bifed: the angles BaC, BDC.
And, becaufe the two triangles AEB, AEC, are fimilar and equal, the fame
line AD will be perpendicular to BC, and will divide it into equal parts in E.
Coroll. III. And the two triangles DAC, EDC, being fimilar, the angle DCE
will be equal to the angle DAC ; and the two angles DCE, DBE, being taken
together, will be equal to the angle BAC.
Ccroll. IV. From hence it follows, that any infinitefimal arch BC, of any
curve whatever, will have the fame affedlions and properties as the arch of a
circle, defcribed on the centre A, with the radius AB or AC.
Coroll. V. The two triangles AEB, BED, being fimilar, we (hall have
AE . EB :: EB . ED. But AE is a finite line, and EB an infinitefimal of the
firft order ; therefore ED will be an infinitefimal of the fecond order, and it's
value will be zz -j-^ . But the redangle of twice AE into EI is equal to the
fquare of EB, from the property of the circle. Therefore EB^ =z 2AE X
EI =z AE X ED, and confequently 2AE . AE :: ED . El. But the firft
term of the analogy is double to the fecond, therefore the third is double to the
fourth. Confequently the two lines El, DI, of the fecond order will be equal.
CorolL VI. And therefore the difference between the femichord BE, and the
tangent BD, is an infinitefimal of the third degree; for as much as from the
centre B, and witli the diftance BE, drawing the arch of a circle EL, a mag-
nitude of the fecond clafs, which coincides with it's fine; the two triangles BDE,
EDL, will be fimilar, which, befides the right angles at E and L, have a
common angle in D. Thence it will be BD . DE :: DE . DL. But BD is a
6rft fluxion, DE is a fecond fluxion by the foregoing corollary, and therefore
PL wiil b^ a third fluxion. Wherefore the arch of the curve BI being greater
3 than
SECT. I.
ANALYXrCAL INSTlTUTIOliS,
than the femichord BE, and leCs than the tangent BD, it cannot differ from,
either of them but by a magnitude of the third order.
THEOREM ir.
Fig.S.
Fig. 9.
H
M
7. Let there be any curve whatever, DAE
(Fig. 8, 9.), in whofe axis are taken two
equal infinitefimal portions of the firft order
HI, IM ; let parallel ordinates HA, IB, ME,
be drawn, which in the given curve fhall cue
off the little arches AB, BE, which are like-
wife infinitefimals of the firft order. Let
there be drawn the chord ABC, which (hall
meet the ordinate produced, ME, in the point
C. I fay, that the intercepted line CE, be-
tween the curve and the chord AB produced,
fhall be an infinitefimal of the fecond order.^
E^
C^
/
c
.^
B
A
^
F
D_^^
Let the chord AE be drawn. If the right
line IM were a finite and aflignable quantity,
then the triangle. ACE would alfo be finite.
But ME continually approaching, [from a
finite diftance,] to the ordinate HA, [while IB
remains fixed,] fo that IM may alfo become a
fluxion, or may be an infinitefimal of the firft
order J the angle ACE always continuing
the fame, the angle AEC increafes, making
the angle CAE always lefs and lefs, till at laft
it becomes lefs than any given angle, that is, an infinitefimal. In this cafe, as
the fine of an infinitely little angle of the firft order, having a finite and aflign-
able radius, is an infinitefimal quantity of the firft order; fo the fine of an
infinitefimal angle, CAE, of the firft order, with a radius AE or AC, which is
an infinitefimal quantity of the firft order, fhall be an infinitefimal quantity of
the fecond order. But in triangles the fides are proportional to the fines of the
oppofite angles, and therefore the right line CE fhall be an infinitefimal of the
fecond order.
Wherefore, calling DH — *', HA =jy, HI = IM =r ;c; then FB = GC
~ j, and EC zz: — y\ the negative fign being prefixed, becaufe jj/ does not
increafe but diminifh (Fig. 8.). And thus, on the contrary, it will have the
pofitive fign if j/ increafe, that is, if the curve be convex in this point to the
axis DM (Fig. 9.).
CorolU.
s
ANALYTICAL INSTITUTIONS.
BOOK II.
Corcll. If from the point E the normal ES be drawn to BC, then alfo
ES, CS, will be the fluxions of the fecond order; for each of them is lefs
than EC.
THEOREM III.
8. If in the circle be taken an arch which is an infinitefimal of the firfl:
order, I fay, that it's verfed fine (hall be an infinitefimal of the fecond order;
and the difference between the right fine and the tangent (hall be an infinitefimal
of the third order.
Fig» lo. Let the arch DC be an infinitefimal of the
firft order, DB it's right fine, CE the tangenr,
and let DF be drawn parallel to AC. From
the nature of the circle, it is GB . BD ;:
BD . BC. But GB is a finite quantity, and
BD an infinitefimal of the firfl order. There-
fore, as GB is infinitely greater than BD, fo
BD will be infinitely greater than B^^. There-
fore BC or DF will be an infinitefimal of the
fecond order. By the fimilitude of the triangles
ABD, DEF, it will be AB . BD :: DF . FE.
But AB, a finite quantity, is infinitely greater
than BD, an infinitefimal of the firfb order,
and therefore DF, an infinitefimal of the fecond order, will be infinitely
greater than FE, which is therefore a third fluxion, or an infinitefimal of the
third order.
9. Coroll, I. And whereas the tangent is always greater than it*s arch, the
^rch greater than it's chord, and the chord greater than the right fine, the
tangent and the right fine may be afTumed as equal, they not differing but by
^n infinitefimal of the third order. Alfo, ihefe following may be affumcd as
.equal, the tangent, the arch, the chord, and the right fine.
10. Corcll. II. If we conceive the radius of the circle AN to be an infini-
.tefimal of the firft order, the arch NO and it's right fine OM will be infinite-
iimals of the fecond ; and therefore the verfed fine MN will be an infinitefimal
oi the third order.
II. Coroll.
SECT. I.
ANALYTICAL INSTITU-TIONS.
9-
(i-i
Fig. I2<
N
Fig. II.
I r. CorolL III. In the axis DM (Fig. 1 1, 12.)
let there be twofirft differences HI, IM, equal
to each other, to which correfpond the two
infinitefimal arches AB, BE, of the curve
DABE ; and let be drawn the two chords BE,
AB, of which this is produced till ic meets in
C the ordinate ME, produced alfo if neceffary.
Let ES be drawn perpendicular to BC, and from
centre B, with radius BE, let the arch EO be
drawn. By the' corollary of Theor. II. CS is
an infinitefimal of the fecond degree, and, by
the foregoing, OS is an infinitefimal of the
third degree. Then CO is alfo an infini-
tefimal of the fecond degree, becaufe an
infinitefimal of the third degree being-
added to, or fubtrafled from, an infini-
tefimal of the fecond degree, makes no
alteration in it. Now, becaufe HI zz IM,
or AF :r: BG, and, becaufe of equal and
fimilar triangles AFB, BGC, it will be
alfo AB z: BC. But the arches may be
afliimed equal to their chords ; then CO
will be the difference of the two arches
AB, BE ; and therefore, if the arch
DA — s, it will be AB zz BC = i, and
CO zr — ^ y with a negative fign, becaufe AB decreafes when BE is lefs than
AB, as in Fig. 11. And, on the contrary, with a pofitive fign, as in Fig. 12.
^
SCHOLIUM.
12. In determining the fecond differences (or fluxions) of the ordinate, and
of the arch of the curve, I have fuppofed, both in Theor. II. and in this lad
corollary, that the firft differences HI, IM, are equal ; that is to fay, that the
firft difference of the abfcifs does not alter, but remains confiant, in which cafe
the fecond difference of the abfcifs is none at all. So that, calling the abfcifs .v,
it's firft difference will be x^ and it's fecond x zz o.
Wherefore we may further make thefe two other conclusions, one of which is,
that if the firff: difference of the ordinate be conftant, thofe of the abfcifs and
of the curve will be variable. The other is, that if the firft difference of the
•curve be conftant, thofe of the abfcifs and ordinate will be variable.
Vol. II. C No\r,
fO
ANALYTICAL INSTITUTIOKS.
BOOK ir.
Now, thefe things being premifed, we
may cafily proceed to thefe two other hypo-
thefes. Suppofing what has been already
advanced, let BF (Fig. 13, 14.) be equal
to EG ; that is, let the fluxion of the ordi-
nate be conftant ; and let EP be drawn pa-
rallel to BG, and PT perpendicular to it.
Then will BF = PT, and therefore AF =:
BT, AB = BP, and GP or EP will be
the difference between H! and IM. And
with centre B, diftance BE, defcribing the
arch EO, PO will be the difference betweea
the arch AB and the arch BE, becaufe the
chords may be affumed inftead of the infi-
nitefimal arches. But, becaufe of the fimilar
triangles BTP, CEP, we Oiall have PT.TB
:: CE . EP, PT . PB :: CE . CP; and
PT, TB, BP, are firfl fluxions, and CE is
a fecond fluxion j therefore EP, CP, and
much more OP, will be fecond fluxions.
Whence, if DH =: x, DA zz s, it will be
TG
zz
PE =
. X,
PO
,
•^7
in
Fig.
1^7
and
PE
zz
— ^,
PO
ZZ
—
Sy
in
Fig.
i4i
and
y =
0.
i M ''
. Let the firft differential of the curve be conftant, that is, AB = BE. From
the point O let fall ON parallel to TP. Becaufe, by fuppofition, it is AB zi
BE zz BO, it will be alfo AF zz BN. Then VE or NG will be the difference
between HI and IM. But it will be alfo FB = NO ; then VO will be the
difference between BF and EG. But it is plain that, EC being a fluxion of
the fecond order, EV and VO will be fo too. Then, if it be DH zz x,
HA zz y, it will be NG zz x, OV = — j', in Fig. 13, and NG = — ic,
OV ~ j', in Fig. 14, and s = o.
The fuppofition of a conflant firfl fluxion makes calculations more fhort and
eaf}', as will be feen in applying it to ufe. However, on many occafions, for
the fake of greater univerfality, we fhall proceed from firfl to fecond differ-
ences, without making the fuppofition of any conftant firft fluxion, which it will
he always eafy to determine.
Let HI, IM, (Fig. 15, 16.) be firfl fluxions of the abfcifs DH, though
not precifely equal to each other, and let their difference be ML, a fecond
fluxion. Let the reft be as above, and draw the ordinate LN, and E? parallel
toBG. Therefore, LM being the difference of HI and-IM, it will be HI zz IL;
that is, AF = BR 5 and therefore the triangles ABF, BRN, will be fimilar and
equal.
Analytical institutions.
tf
equal. Confequently BF =: NR, and N/,
will be the difference between BF and EG ;
that is, the difference of BF, or the fecond
difference of AH. In like manner, it will
be AB =: BN, and therefore NO will be
the difference between the arch AB and the
arch BE; and therefore the difference of the
arch AB, or the fecond difference of the
arch DA. Wherefore it is plain that N/,
NO, are fluxions of the fecond degree.
The fame things will obtain, if, inftead of
fuppofing IM greater than HI by a fecond,
differential, we fliould fuppofe it lefs,
13. It is to be obferved here, that the
foregoing determinations do not require any
reftridions concerning the angles of the co-
ordinates, though the figures may feem to
infinuate that they are at right angles ; for
the conclufions will be all the fame, what-
ever the angles may be.
LEMMA.
14. Right-lined angles are to one another in a ratio compounded of the dired
ratio of their arches, and the inverfe ratio of their radii.
J^ig- 17-
-.1)
Let there be two angles EAB, FAC (Fig. 17.).
Producing AE to D, from the fimilitude of the
fedors ABE, ACD, it will be AB . BE ::
AC . CD ; therefore CD = ^^ "" ^^
AB
But the
angle EAB, or DAC, is to the angle FAC, as CD
to CF J therefore the angle EAB will be to the
, r-^r^ BE X AC ^ ^rr .1 ^ ' „ BE
angle FAC, as — rs — to C^; that is, as
CF
AC •
AB
AB
to
C2
THE-
xa
ANALYTICAL INSTITUTIONS,
BOOK II.
T H E O R E M IV.
F^^' iS. I ^, Taking the archCF,an infinltefimal of the firtl
degree, in any curve whatever ACF, and drawing
CJ, FI, perpendicular to the curve; with centre 1,
and radius IF, if we defcribe the circular arch FS,
I fay that it will fall all within the curve ACF, to-
wards C, and the intercepted line CS will be aa
infinitelimal quantity of the third degree.
Upon the curve AQR a thread may be con-
ceived to be ftretched, lo as that, being fixed in
any point below, as in R, and taken by it*s tn(\ in
the point A, it may continually recede from the
curve, but in fuch a m inner as to be aUvays equally
ftretched, and wiih it*s point A to deicribe the
curve ACF. Now, the thread being in the pofition CQ»^ it will be a tangent to
the curve AQR in the point Qj and in the polition FR, which I fuppofe to be
infinitely near to CQ»_ ic will be a tangent in R; then producing CQ^it will
meet FR in I. Now, fince, by the generation of the curve ACF, the right
line QC is equal to the curve QA, and the right line RF to the curve RQA,
and the two infinitely little tangents QI, RI, are together greater than the
element QR ; therefore CI, IR, taken together, will be greater than the curve
RQA, or than the right line FR. Then, taking awav the common IR, IC will
be greater than IF, and therefore the circular arch FS, defcribed with centre I
and radius IF, will fall within the curve. But, by Theor. I. and III., the two
tangents QI, RI, do not exceed the arch QR but by a third fluxion. There-
fore the curve AQ^^ together with the right lines QJ,, IR, exceed the curve
AQR, or the right line FR, by the fame quantity. Then taking away the
common IR, AQ»_ together with QI, that is, IC, will be greater than IF by an
infinltefimal of the third order.
16, Coroll. Therefore we may conceive the circular arch FS as coinciding
with the arch of the curve FC ; and one may be taken for the other indiffer-
ently. And the tangent RF will be perpendicular to the curve ACF in the
point F, and QC in the point C.
The curve AQR is called the Evolute^ the curve ACF is the Involute, or
curve generated by the evolute ; thai is, produced by the unwinding of the
firing or thread AQR; and the circle FS, defcribed with centre I and radius IF,
is the Ofculating or equicurved circle ; alfo, IF is called the Radius of Curvature
©f the curve ACF in the point F,
THE-
SECT. I, ANJ\LYTICAL INSTTTUTIONS, ^^
THEOREM V.
17. If In the curve DABE (Fig. 11, 12.), at the points A, B, E, infinitely
near, (that is, the arches AB, BE, being infinitefimals of the firft order,) be
drawn the perpendiculars QA, QB, and NE, which meet BQ^in the point N ;
I fay, that the angles AQB, BNE, may be affumed as equal.
For, by the foregoing Lemma, the angle AQB is to the angle BINTE, as
A R FR
—- is to rr^ , that is, as AB x BN is to EB x AQ^ But the redangle
EH X AQ^is not lefs than the redVangle AB x BN, but only by the.redangle-
BE X QNj and by the redangle of BN into the difference of the arches AB,
BE. And, as QN, BE, are infiniiefimal quantities of the firiT: degree, their,
red:angle will be an infinitefimal of the fecond degree j as alfo, the difference of
the arches AB, BE, being an infinitefimal of the fecond degree, the rectangle
of thefe into BN will be an infinitefimal of the fecond degree. Therefore the
two redangles AB X BN and EB into AQ do not differ from each other, but
by two infinitefimal rediangles of the fecond degree, and therefore may be
affumed as equal, and confequently the angles AQB, BNE..
18. CorolL I. If PBR be drawn a tangent at the point B, it will bife<3: the
angle CBE, made by the two chords ABC and BE. For, by Theor. I.
Coroll. III. the angle BQA being double to the angle PBA, to which the angle
CBR is equal; thence the angle BNE (hall be double to the angle CBR. But,,
by the fame Corollary, the angle BNE is double to the angle RBE. Therefore,
the angles CBR, RBE, are equal.
19. Coroll. II. Therefore the angle CBE will be equal to the angle BNE,
and thence the fedor BNE will be fimilar to the fedor EBO.
THEOREM VI,
20. If in two circles, the diameters of which exceed each other by a firft:
infinitefimal, be taken two right fines equal to each other, and infinitefimals of
the firft degree, the difference of their verfed figns fhall be an infinitefimal of
the third degree.
8 Let
ANALYTICAL INSTITUTIONS.
BOOK II.
Fig. 19. Let the two circles be ABC, PFH, and
let the equal right fines be BE, FG, infini-
tefimals of the firft degree, and the verfed
fines EC, GH. Let the chords AB, BC,
be drawn. The fine BE, and therefore the
arch BC, being a firft fluxion, the angle
BME will be an infinitefimal of the firft
order, and therefore alfo the angle BAC,
which is the half of it, and the angle EBC,
which is equal to this. Therefore, fince the angle EBC, and the fides EB, BC,
are iirft infinitefimals, the verfed fine EC will be a fecond infinitefimal.
The fame obtains of the verfed fine GH. But the verfed fine EC (by the
property of the circle,) is found to be -j-g, and the verfed fine GH =: -^
=: ^. Therefore we (hall have this analogy, EC . GH :: PG . AE. But
PG, a finite quantity, exceeds AE, a finite quantity, by an infinitefimal quan-
tity in refped of itfelf, that is, of the firft order, by hypothefis. Therefore
EC, an infinitefimal quantity of the fecond order, will exceed GH, an infini-
tefimal of the fecond order, by an infinitefimal quantity in refpe(5t of itfelf, that
is. of the third order.
THEOREM VIJ.
I'lg, 20.
difference of the fecond order of the ordinate AB,
21. Let the curve BEG (Fig. 20, 21.) be
referred to a focus, that is, fuch, that all the
ordinates proceed from a given point, which is
called the Focus, and let this point be A. From
hence let be drawn three ordinates, which are
infinitely near, AB, AE, AG, which contain
the two infinitely little arches of the firft degree,
BE, EG; and draw the chord BE, which, pro-
duced, meets the ordinate AG (produced if need
be,) in the point L. With centre A let the
arches BC, EF, be defcribed, and let BM, EN,
be their right fines. Laftly, make the angle
NEP equal to the angle MBE. I fay, that the
intercepted line GP (hall be the infinitely little
Let
SECT, I»
ANALYTICAL INSTITUTIONS.
'^
Fig. 21. .Q ' Cry -^^^ ^^^ chord EG be drawn. Since tlic
angles MBE, NEP, are equal by conftrudion^
and the angles at M and N are right ones,
the triangles EBM, PEN, will be finiilar ;
then taking the fine BM for conftanr, that
is, fiippofing it equal to EN, the fore(aid
triangles will alfo be equal. Therefore ii will
be ME = NP. But, fuppofing BM = £N,
by the foregoinar Thcoreai the difference of
the verfed fines MC, NF, is infinirefunal in
refpeCt of them. Therefore, alio, CE, FP,
will be equal, and thence GP will be the
difference between CE and FG. But the
right lines VX^ QG, being drawn perpendicular to the curve in the points
E, G, the angle LEG will be equal to the angle EQG, by Theor. V. Coroll. U.
[which is true whether the curve be referred to an axis, or to a focus,~\ And
the angle EQG is infinitely little. Therefore, alfo, the angle LEG will be
infinitely little. And, becaufe the right lines EG, EL, are infinicefimals of the
firft order, GL will be an infinitefimal of the fecond order; and much more GP,,
refped being had to Fig. 20.
By Theor. III. Coroll. I. the line BM is equal to the arch BC. Then, in-
ftead of the fine, taking the arch for conftant, and making it = x, AB =_)>,
CE :rrj)/, it will be GP zz — y. And with centre E, and diftance EG, de-
fcribing the arch GV, it will be VP — — }", if BE = i.
22. Corolh The angle LF'P will be equal to the angle EAG. For the angle
EPA, by conftruftion, is equal to the angle BEA ; but the external angle
EPA is equal to the two internal angles L and LEP ; and the other, BEA, is
equal to the two, L and EAG. Then, taking away the common L, there will
remain the two equal angles LEP, EAG. Wherefore this will be true, whether
the curve be concave towards the point A, (Fig. 20.) or whether it be convex,
(Fig. 21.) as it is eafy to perceive. In the fame Fig. zr, the angle LEP will
be an infinitelimal, and therefore LP is an infinitefimal of the I'econd order.
But it has been feen, that GL is alfo an infinitefimal of the fecond order.
Therefore the whole, GP, will be fo alfo, which will be zz. y ; and with centre
E, diftance EG, the arch GV being defcribed, it will be PV n >'. ^
If we fuppofe y to be conftant, with centre A, and diftance AG, let the arch.
GT be defcribed, and from the point T let the riglit line TOA be drawn..
Becaufe FG — EC, by hypothefis, the triangle TLO will be fimilar and equal
to the triangle EBC ; and therefore BC ~ EO, and BE z=l ET. Then
OF = X, and TV = y, in Fig. 20. But OF = — x, and TV =: — }"; in
Fig, 21.
Taking
^i6
ANALYTICAL INSTITUTIONS.
BOOK IT,
Fig, 22.
Taking s for conftant, and drawing the
right line VRA, k will be EG = EV =
■BE; and therefore the triangles EBC, EVR,
are equal and fimilar j thence is BC ~ ER,
and CE z= RV. Whence RF = 3c, VI iz
— y, in Fig. 20. But RF — — x\ and
VI =
y^
m
Fig.
21.
If no firft fluxion be taken for conftant,
let EG be greater than BC, (Fig. 22, 23.)
by the fecond fluxion RF j let the right line
ART be drawn ;* with centre A, diftance
AG, draw the arch GT ; and with centre E,
diftance EG, draw the arch GV. There-
fore, fince BC = ER, it will be alfo CE
=: RI, and BE = EI. Therefore TI will
be the difference between CR and FG, and
VI the difference between BE and EG.
SCHOLIUM J.
23. It may not be befide our purpofe to obviate a difficulty, which feems
likely to arife. And this is, that in the foregoing Theorem the lines CE, FP,
are affumed as equal, in virtue of Theor. VI.; which Theorem fuppofes as
equal the fines BM, EN. Whence it may feem, that the determinations con-
cerning fecond differentials can only take place in the cafe when we make a
fuppofition of a conftant fluxion BC, and in no other. But, to remove this
difficulty, it will be fufficient to confider, that, though BC be fuppofcd variable,
the difference will be an infinitefimal of the fecond degree, which does not
iiiader the equality of the firft fluxions BC, EF, nor of the fmes BM, EN.
SCHOLIUM II.
24. In the foregoing Theorems are contained the principles, by which infi-
uitefinials of any order may be managed, and which prepare the way to make a
right
SECT. r. ANALYTICAL INSTITUTIONS. I7
right life of the Method of Fkixions, whether dired or inverfe; and befides,
to apply the fynthefis of the ancients to infinitely little magnitudes of all
degrees; and to make ufe of the flri(S:eft Geometry, which proceeds vvith a
particular iimplicity and elegance.
Now, to avoid paralogifms, into which it is but too eafy to fall, it will be
needful to rellccl, that infinitely little lines of any order, (agreeably to what .
obtains likewife in thofe that are finite,) have two important circumftances to
be confidered, which are their magnitude and their px>fition. And as to their
magnitude, I think they cannot be rejected except by thofe, who fancy fuch
infinitefimal quantities to be mere nullities.
Now, although quantities, by diminifliing ad infinitum, may pafs from one
order to another, the proportions in every order continue the fame. And, be-
caufe of three lines of any the fame order a triangle may be formed, it may
be confidered, that if, by leiTening proportionally the fides, fo as to pafs
from one degree to another, the angles are not thereby changed, the fides mufl
always preferve the fame ratio to one another; that is, infinitefimals with the
finite, and infinitefimals of the fecond order with thofe of the firfl, and with
finite ; and fo on.
But if two magnitudes, of any order whatever, fliall differ by a" magnitude
which in refpeft of them fhall be inaffignable, then with the utmofl fecurity,
and without any danger of error, one of them may be taken for the other ;
nor need it be apprehended that fuch a comparifon will introduce the leaft
error.
Therefore it is necefTary to be much upon our guard, when the pofition of
lines and angles is concerned ; for, to confound them, when they ought to be
nicely diftinguifhed, mufl needs lead us into unavoidable paralogifms.
25. The principal foundations of this calculus being thus laid, I fliall pafs
on to the methods or rules of finding the fluxions or differences of analytical
formulas or expreflions. And, firfl, let us take the differences of various
quantities added together, or fubrraded from one another ; for example, of
a-\-x + z-\-y--u. As the fluxion of x is ^c", of z is z, &c ; and as the
conflant quantity a has no fluxion ; then, conceiving every variable to be in-
creafed by it's fluxion, according to it's fign, the formula propofed will be
changed into this other, a •\- x •{- x -\- z -i- z +^-hJ^ — « — i^ i from
which fubtradting the firft, the remainder will be x + 2; + j/ — «, which is
exadly that quantity by which the propofed quantity is increafed, that is to fay,
it*s difference or fluxion.
Hence we derive this general rule, that, to find the fluxion of any aggregate
of analytical quantities of one dimenfion, it will be fufficient to take the fluxion
of every one of the variable quantities vvith it's fign, and the aggregate of thefe
fluxions fliall be the fluxion of the quantity propofed. So, the fluxion of
Vol.11. . D ^ —
l8 ANALYTICAL INSTITUTIONS. BOOK II.
b — 5 — 2 will be — i — ir. The fluxion of aa — 4^2 + ly will be
— - £^hz + hy. And fo of others.
26. But if the quantity propofed to be differenced fliall be the produdl of
feveral variables, as xy ; becaufe x becomes x + x^ and y becomes y -V y, and
xy becomes xy -Ar yx •\- xy -^ xy, which is the produft oi x + x into y + y ;
from this product fubtrading, therefore, the propofed quantity xy^ there will
remain yx-+ xy + ^j- But ^ is a quantity infinitely lefs than either of the
other two, which are the redangle of a finite quantity into an infinitefimal.
But xy is the redangle of two infinitefimals, and therefore is infinitely lefs, and
muft be fuppofed entirely to vanifh. The fluxion, therefore, of xy will be
xy + yx.
Let us difference xyz by this rule. The product o^ x -\- x into y -{■ y into
z -^ z \% xyz + yzx + xzy + xyz + zxy + yxz -f xyz + xyz ; which, fub-
•trading the quantity propofed, will give the remainder ^x.v + xzy + xyz + zxy
+ yxz + xyz + xyz. But the flrfl, fecond, and third terms are each the
produft of two finite quantities and one infinitefimal ; the fourth, fifth, and
lixih are the products of one finite quantity and two infinitefimals, and therefore
every one of ihefe is infinitely lefs than any one of thofe, and therefore will
vanilh : and much more the lafl, which is the product of three infinitefimals.
Therefore let all thefe terms vanifli, beginning at the fourth, and ih^nyzx -^xzy
-f- xyz will be the fluxion of xyz.
Hence arifes this rule, that, to take the fluxions of the produd of feveral
quantities multiplied together, we muft take the fum of the products of the
fluxion of every one of thofe quantities into the products of the others. Thus,
the fluxion of bxzt will be bxzt + bxtz + bt-zx -f xzt x o ; becaufe the
fluxion of the conftant quantity b is nothing. That is, the fluxion of 2>xzf will
be bxz} 4- bxlz -J- b/zx. The fluxion o{ a +x X ^— jy will be x X b —y —jf
' X a + x, that is, bx — yx ^ ay — xy.
27. Let the formula to be difterenced be a fradtion, fuppofe -^. If we put
— c: z, it will be then x — zy. And therefore their differences will alfo be
y
equal, that is, x z:z zy + zy. Wherefore z zz l-H-^. But z =: — ; there-
fore, fubftitutine this value inflead ot z, it will be ^ := — — . -^ = .
' & ' y yy yy
But if z — -^i then z will be the differential of ~ , and therefore the dif-
y y
ferential of — will be ^ - ^ " ^-^ .
y yy
Now
SECT, I. ANALYTICAL INSTITUTIONS. I9
Now the rule will be, that the differential of a fradion will be another
fradion, the numerator of which will be the produ<ft of the difference of the
numerator into the denominator, fubrradling the product of the difference of
the denominator into the numerator of the propofed fraflion ; and the denomi-
nator muft be the fquare of the denominator of the fame propofed fradion.
Therefore the difference or fluxion of — will be — — ^ . The fluxion of
X XX
f will be ^'^ — ax — XX ^^^^ .^ ax_ ^ ^^^^ fluxion of —— win be
X XX XX —y
£-^+^^ ^^^^ .^^ _^^^ ^^^ ^^^^^^ ^^ J5^ will be
^ [ ' b-y
3xy + syk X a- X + X X ^xy , . ^axy + gayx - ^x\y
—^ ~;a -— J tnat is, ==:::rj .
a — x^ a — x\
28. Now let us find the fluxions of powers, and, firft, of perfed and
pofitive powers, that is, whofe exponents are pofitive integer numbers ; for
example, of at*. But xx is the product of x into x, and therefore, by the rule
of produdls, it*s fluxion will be xx + xx, that is, zxx. To find the fluxion
of x^. Now this is the product of .v into x into .r, and therefore the fluxion
will be XXX + xxx + xxx, that is, ■^xxx. And, as we may proceed in the fame
manner in infinitum, the fluxion of ►v , m being any pofitive integer, will be
mx X,
If the exponent be negative, fuppofe ax^ , or --^, the fluxion, by the rule
of fradlions, will be the produft of the fluxion of the numerator into the deno-
minator, fubtrading the produ(5t of the fluxion of the denominator into the
numerator, the whole being divided by the fquare of the denominator. But
the fluxion of the denominator is 2xx ; fo that the fluxion of ax" or — will
XX
be — ^~-i that is, — ~. The fluxion of x"^, or —> will be — ^^ ,
X* ' x^ ' *' ' x^
• ■— OT J. Tft—J.
or ~ , And, in general, the fluxion of ^ , , or -^—3 will be — flif
that is, —
hx * hbx
. —m — l
maxx
b
Let it be an imperfed power, and, firfl, let it be pofitive ; that is, let the
exponent be an affirmative fraflion, as y/x\ or x n , where — flands for any
pofitive fraction. Make x n zz z, and, railing each part to the power », it
D 2 will
20 ANALYTICAL INSTITUTIONS. BOOK II.
m — i
will be x^' zz 2", of which taking the fluxions, we (hall have mxx
nzz" ', whence z — "^ . But, becaufe x"^ — z, and thence z zz
»2
. «— I
mxx
x*"" n y which being fubflituted, it will be 2; zz , that is, z —
m
m
nx n
m
m . I
' OCX n ■
n
If the exponent were negative, as — ^ , that is, ^ " « , or elfe —^ , the
fluxion, by the rule of fradionsj would be
X n
m
m • ^
" i\i\ 7ft
or — — XX
m . » X
%/n n
xlT
Therefore the general rule is, that the fluxion of any power whatever,
whether perfeft or imperfed, pofitive or negative, will be the produ6t of the
exponent of the power into the quantity raifed to a power lefs by an unit than
the given power, and this multiplied into the fluxion of the quantity.
i- . . i— I . . - .
Let it be required to find the fluxion of x^ j it will be l-x* x, that is, ^x^'x,
or elfe ^x^^x,
5 4 — 1 . •*• • • /
Let be given x'^ ; it's fluxion will be {x^ Xy that is, ix'^x, or |x</ ;v.
Let be given —j that is, x~"^ i the fluxion will be — ^xx^''"^, or
2^x ^, or, laftly, —
s •
The fluxion of ax + xxV willbe 2 X ax-\-xx X ax-^zxx, that is, 2aaxx
+ 6ax^x + 4X^x,
The fluxion of xy + ^.v^* will be 3 X xy + ax\* x xy-^yx-\-ax, that is,
^j^y'^y 4- Sax^yy + ^a^x^y + ^^x^x + ^ay'^x'^x + i^a^yx^x + jaV^.
The fluxion of ' ■-, or ax ^yy)"^^ will be — 2 X ofT^^^"^ X
ax — jjyj
ax-^2yy, or .^ — .
ax '—yj\3
The
SECT. I. ANALYTICAL INSTITUTIONS. 21
The fluxion of \^ax-^xxj or ax-^xx)^y will be f X ax--xx\ * Xax^Izxx^
that IS, -^ TT «
The fluxion of v/a;;^ + ^^y, or ata* -f- x^^"-, will be -^ x a-a; + xy)"* x
2 XX + xy + yx
- . . — , . 2XX -r •*/ T yj
2XX'{-xy + yx, that is, =|=^-j-
"^ ax ^*' + a;)'1i
The fluxion of 4^ax^xx, or ax^xx) 3 , will be ^ X ax — xxi ^ x ax —2xx,
. . ax — 2XX
that IS,
3 X ax — xx^j
The fluxion of 3. ^ . , or =L=-^, or ay + ^^1~% will be — -5- x
. —4 — : : :- ay + xy + yx
^y + ;0m ^xayi-xy -^yx, or = -- — si •
The fluxion of ^^-^7+7, or «-a; X tfT^ ^j is -- x x "^T^'^ + f
X ^-.;tf X a^x\ ^ X ^, or — x^a+;f + -—===7^.
3 X tf+vlT
The fluxion \^ ax -\- xx -{■ -C/^TT^^, or a;? + xv + a^-x^^V will be t X
tf;i: + s^f^c- + ^ X — 4^^' X a* - AT*] ^'^ X ^x + ^* -fa*- A'^li]"*, or
ax + zxx — . . 4-/==:rr
^4 _ ^4\4: a + aj? X ;f '^ fl'* — x^Y — xx^
2 X ax -{- XX + a* — x^ * 2 '^ ax ^xx + V a'' — ** x ^fl*— ^c*)'
The fluxion of , or ^^+xv X ax + .vxl""*, will be 2x;v x
__x _________ — _ 3.
ax + xx\ — ^ X ax^^xx x ax-^-xx^ * X «« + a;a-', that is,
z'xx X a^ + XX — f X air + zxx x «<* + ** 3tfjrjf* + I'xx^ — a^;;: _ laaxx
The fluxion of ^±^ will be 3^V^ + 2. ^.^. , 3 ,^3^ „ .^.^j + :>^^i
aV ay — ^_y 2« X 'VflA- + XX X V^flv — xy\ '
29. After the fame manner as the fluxions of finite quantities are found, (o
are found the fluxions of infinitefimal quantities of the firft order, and the
fluxions of infinitefimal quantities of the fecond order, and lo on fucceflively,
making ufe of the fame rules which have now been explained.
Here
22 ANALYTICAL INSTITUTIONS. BOOK II,
Here it mnft be confidered, whether any firft fluxion be affumed as confhanr,
and which it is ; for then it's fluxion will be nothing, and fo ought to be
omitted in taking the fluxion.
Let the formula yx — xy be propofed, to find it's difference or fluxion.
Let no fluxion at prefent be fuppofed to be conftant, and it's fluxion will be
xy -\- yx — xy — xy, that is, yx — xy. Now let the fluxion x be aflurned
as conftant ; then the difference will be xy — xy — .vy, or — xy. Let the
fluxion y be conftant, then the difference will be xy + yx — xy, that is, yx.
Let the quantity be ^, in which no flift fluxion is taken for confl:ant.
The fluxion will be iiliL^Zi^ ^^ ^ ^ iL ^ yJL . Here, taking x for
conftant, it will be x — ~ , Taking j/ for conftant, it vvill be ;^ + ^.
Let the formula be ^—^, — —, and let z be conftant. The fluxion will be
z
yy^^^+yy + ;, X .";i:JJi , that is, i±+yL+J^_+JE , Taking y for con-
ftant. It will be rr-^-^ , that is,
zz
xJcyi + y^i+yz^x^ySriz-y^i Taking X for conftant, it will be
xzvxx +yy
yzyy .. _____
yzVxx+yy + / .. . . . "yZVxx+yy •••• . •5- . • •••
— — ^^ , that 13, —^ 7 ± — = •
^2; zzVxx-i-yy
And, laftly, if no fluxion be conftant, the differential will be
yz\/xx +yj' + yz X .. — yzs/'xx +jy
v ^x 4- yy , .
_^ ^ . . ^ ^-^'^ , that is,
zz '
xxyx -^y^z+yzxx-{- yzyy — y xx'z —yyy'^
ZZ v/ XX + _yj
Now in this, if we expunge all the terms in which z is found, that is, if we
affume the hypothefis of z being conftant, this expreffion will be changed into
the firft. And if we cancel thofe in which y is found, it will be changed into
the fecond. And, by expunging thofe in which x is found, it will become the
third, as is manifefl.
Let
SECT. I, ANALYTICAL INSTITtrTIONS. ^
XX "4" vv
Let be given ■ , and let x be conftant. Then the fluxion will be
\r v<V> ^L V4I
'^XX +
yy
f .11. jf^ + Vy — ____
A-x + ry + ^7 X V ;«-x + _y^ X .v^ + jyJ
— r* '■ , or
^y.^^y^^^^^y.y^^.y,y ^ Taking > foF conftaiit, it wiU bc
■ M - XX -^^ yy X x'x + J'i
.V* + Ar;e + >* X V ^.v + >7 ' -. "
, that is.
XX -^ yy
:,^ii Ar x^}- ■\. y-^-x- ^Jxk ^ 2xyiy ^ ^^^ j^^j^^ ^^j^j^^ ^^.^j^^^. ^^ ^^^^ fluxionS for
xx+yy\^'i
.. *. XX 4- jpv __-___—_
i» -^ x'x +>■* + j5' ^ V^ x:i -\- yy — . " X ;r*' + x^
conftant, it will be l^V^y ^-^^—^ ,
K^x + xS' * + «*yv + y*jf* + y*.vjc + y^y — ixyxy
IS, ■• ^— -= — ,3
Let it be required to find the fluxion of this differential formula of the fecond
degree, "^ +^'' ^ ."^"^ ±^1 , or of this.'^S^, taking x for conftant. The
fluxion will be — 3_LJ r^ . The hypothefis of y
being conftant, cannot take place in this formula, becaufe here is already-
found j". Taking neither of the fluxions as conftant, the differential will be
3 X XX Jr yy X x^ •\- p^i X — xy + x^ + x^ X .v' + jy
-I
x'-y
In a like method we muft proceed in all other cafes, ftill more com-
pounded.
SECT.
14
ANALYTICAL INSTITUTIONS,
BOOK I,
SECT. II.
^he Method of tangents.
Fig. 24
JB C N
Fig. 25.
30. Let the right line TDG (Fig. 24, 25.)
be a tangent to the curve ADF in any point
D, and the ordinate BD be perpendicular to
the axis AB in the point B, to which let CF
be infinitely near, which produced (if need
be,) (hall meet the tangent in the point G,
and let DE be drawn parallel to the axis-AB.
By what has been already demonflrated in
the foregoing Theorems, and their Corolla-
ries, GF will be an infinitefimal in refpeft
of EF, and alfo the difference between DF
and DG will be an infinitefimal in refpedt
of the little arch DF. Therefore we may
affume as equal the two lines EF, EG, as
alfo the two, DF, DG ; and therefore, if
AB =: ^, BD =^, it will be EF = EG
=:j)/, DF = DG = \^ITTJJ' But the
fimilar triangles GED, DBT, give us this
analogy, GE . ED :: DB . BT ; that is, in
analytical terms, y . x i: y . BT, and there-
fore BT zz -^ ; and this will be a general
formula for the fubtangent of any curve.
Wherefore, in the cafe of any given curve, in order to have the fubtangent,
nothing elfe is required to be done, but to find the fluxion of the equation, and
to fubftitute the value of ^ or y in the general formula — , by which the dif-
ferentials will vanifh, and we (hail have the value of the fubtangent expreffed
in finite terms. This will belong to the curve in any point whatever; and if
we would have it at a determinate point, inftead of the unknown quantities we
are to fubftitute fuch as (hall belong to the given points,
31. Becaufe
SECT. II. ANALYTICAL INSTITUTIONS. * %$
31. Becaufe we may affume EF ~ EG, and DF — DG, it will follow,
that we may confider the point G as coinciding with F, that is, that the tangent
DG, the arch DF, and it's chord, are all confounded together, or that curves
may be confidered as polygons of an infinite number of infinitely little fides.
This conclufion obtains only when we confine ourfelves to firft fluxions ; but
when we are to proceed to fecond fluxions, the point G mufl: not then be con-
founded with the point F, for GF will then be a fecond fluxion. Now, whereas,
in the Method of Tangents, there is no occafion for fecond fluxions, it may be
fafely fuppofed that the tangent coincides with the little arch and it's chord.
32. The fame triangle GDE will fupply formulas for the other lines, which
are analogous to the fubtangent.
Becaufe the triangles GED, DBT, are fimilar, it will be GE . GD ::
DB . DT ; that is, y . V^JT+TJ :: y , DT, and therefore DT 1= ^^^^+'7 i
which is a general formula for the tangent.
Let DN be perpendicular to the curve in the point D. The triangles GDE,
DBN, will be fimilar, whence it will be DE . EG :: DB . BN ; that is,
X .y \\ y . BN, and therefore BN = ^ , a general formula for the fub-
normal.
It will be alfo DE . DG :: DB . DN, or x . y/kTTTy •• ^ • ^^ ; therefore
DN = — =^ , a general formula for the normal.
From the point B draw BM perpendicular to DN, and BH perpendicular to
DT. The triangle GDE will be fimilar to DBM, whence GD . GE ::
D3 . BM, or \^xx + jy 'jf :iy » BM = ■ ;^^ — — , a general formula for
the line BM.
The fame triangle GDE will alfo be fimilar to DBH i whence it will be
GD . DE :: DB . BH, or ^/^^+7J .x::y,BH = -7-=^==, a generalfor-
V XX + Jy
mula for the line BH.
2^. The fimilitude of the two triangles GED, DBT, will alfo be a means of
difcovering the angle, which the tangent makes with the axis at any point of the
curve at pleafure. For, becaufe the angle DTB is known, therefore the ratio
of the right fine DB to the fine of the complement BT will be known alfo ;
that is, the ratio of GE to ED, or that of _y to x.
Vol. II. E Therefore,
26
ANALYTICAL INSTITUTIONS.
BOOK II.
Therefore, the equation of the curve being given, if it's fluxions be found
and refolved into an analogy, of which two terms are y and x, we may have
the ratio of the fines of the angle DTB, and confequentiy the angle will be
known.
Ftg
34. By the fame way of argumentation,
the fame formulas may be derived for fwch
curves as are referred to 2, focus, (Fig. 26, 27.)
if we only confider, that, drawing from the
focus B the right line BT perpendicular to
the ordinate BD, meeting the tangent in T ;.
the triangles DTB, DGE, will be fimilar,
becaufe the angles TBD, DEG, are right
angles, and the angle TDB is not greater
than the angle DGE, except by an infinitely
little angle DBG, which is plainly feen by
drawing GQ^ perpendicular to TB. There-
fore the two angles TDB, DGE, may be
aflumed as equal, and confequentiy the two,
BTD, GDE ; therefore the two triangles.
DTB, GDE, are fimilar. But GF is an
infinitefimal in refpeft of EF j therefore^
&c.
EXAMPLE I.
Fig. 24.
S C N
35. Let the curve ADF be the Apollonian
parabola, whofe equation is a% n yy. Tak-
ing the fluxions, it will be ax = iyy, or
Wherefore, fubftituting this va-
X
— ^yy
lue infl:ead of Xy in the general formula for
the fubtangent —-^ we fliall have — , or
2A'^ putting, inllead of yy^ it's value axy
givea
SECT. II. ANALYTICAL INSTITUTIONS. 27
given by the equation of the curve. Therefore the fubtangent in the parabola
is doable to the abfcifs ; fo that, taking AT = AB, and from the point T
drawing the right line TD to the point D, it Qiall be a tangent to the curve at
the point D. Inftead of the value of x, given from the equation of the curve,
if we fubftitute the value of y, or — , in the general formula ^ , it will be
alfo -^ , as before ; which may fuffice to obferve in this Example.
In the fame parabola, if we require the fubnormal BN ; the general formula
of the fubnormal is ~ . But, by the equation of the curve, it is ^ zz ^ ;
X a
.fo that, making the fubftitution, the fubnormal in the parabola will be zz \a^
that is, half of the parameter; and therefore, niaking BN — \a, and from
the point N drawing the right line ND to the point D, this (hall be perpen-
dicular to the curve in D.
If we feek the tangent DT, the general formula of which is - — ~ — ^ , by
the equation of the curve we have x zz ~. Then, fubftituting this value
Itiftead of x in the formula, we (hall have ^^^. — ^^ = — '/^j.j; + aa =
ay a ^^^
^/ i^xx + ax, (putting, inftead of jj, it's value ax from the given equation,)
which will be the tangent required.
Jf we would have the normal DN, fubftituting the value of ^c* zz -^ in the
aa
general formula -.-i-^ , it will be - — ^^^L^- ^- = ^^^^--^ — —
t> X ' 2yy z
4g-* + ^^ ^ putting, inftead Qi yy, it's value from the given equation.
If we would have the right line BM ; fubftituting the value of ;ic' = ^
yy . . ^yy ^y
in the general formula ,/ . . . . . , it will be ./ . . , — r-r zz ./ . =
° V XX +yy V ^yyyy + ^oyy V ^yy + aa
y/ /^ax + aa
If we would have the right line BH ; fubftituting the value of x in the
general formxila -7====, it will be ^^-^^^ =: ^^ zz
y^x'x-^-yy \^ Ayyjy "^ o^jy V^yy-i-aa Vi^ax^aa
E 2 Having
28 ANALYTICAL INSTITUTIONS.
BOOK ir.
Having found the fubtangent, there is no need of any formulas for finding the
other lines, though here, by way of exercife, I have made ufe of them. For,
when BT is known, the triangle TDB, right-angled at B, will furnifii us with
the tangent TD, and the iimilar triangles TBD, DBN, DMB, DHB, with all
the other lines. So that, in the following examples, I fliall apply the metliQd
to finding the fubtangents only.
If we would have the angle which is made by the tangent of the parabola
with it's axis ; taking the fiuxional equation ax zz 2jrK, and refolvino it into
an analogy, it will h^ y . x W a . ly. That is, that the right fine BD is to the
fine of the complement BT, as the parameter is to the double of the ordinate;
whence is determined the point D. And if we v/ould determine the tangent to
any certain point, for example, to the point D, to which correfponds the abfcifs
AB zz sc =z ^a-, from the equation of the curve finding the ordinate jv, corre-
fponding lo x ~ ^^, which, in this cafe, \s y zz \a, we fiiall have the analogy,
y , X \: a . ai that is, the angle DTB will be half a right angle, when it is
y =: ia, or x — \a.
At the vertex A it is y = o, and therefore the analogy for the angle of the
tangent at the vertex will ht y , x :: ^ .o; that is, the ratio of j/ to x is
infinite, which is as much as to fay, that the fine of the complement will b«
nothing at all, or that, at the vertex, the tangent is perpendicular to the axis.
EXAMPLE II.
36. Let the equation h^ x — y , which is a general equation to all parabolas
<>f any degree whatever; where m flands for any pofitive number, integer, or
fradion, and unity fupplies any dimenfions that are wanting. By taking the
w—
fluxions, it will be ^ — myy * ; and, fubfl:ituting this value inftead of x in
the general formula -^ , the fubtangent will be my"* = mx. Let «« =r 3, that
is, let it be the firffc cubic parabola ^ zzj'j it's fubtangent will be o^x. Let
ro = I, that is, let it be the fecond cubic parabola xx = j' ; the fubtangent
will be \Xi &c.
The fluxional equation of the curve x zi myy *" gives this an-alogy, y . x
\\ 1 , my . But, putting jy = o, if w be greater than unity, the analogy
will be j^ . ^ :: I . o ; or the ratio oi y \o x will be infinite, and therefore the
tangent at the vertex is perpendicular to the axis. And if m be lefs than unity,
the
SECT. II.
ANALYTICAL INSTITUTIONS.
19
the analogy will he y . x :: i . -7:1^; that is, making y—o,y,x.,i, — ,
which is as much as to fay, that the ratio ofj/ to ^ is infinitely little, and
therefore, at the vertex, the tangent is parallel to the axis.
EXAMPLE III.
37. Let the curve be DCE, of which we de-
fire the fiibtangent, the equation of which is
xy zz aay being the hyperbola between it's
afymptotes. By taking the fluxions, we fhall
have xy + yx :z: o, or X zz — . Where-
fore, fubftituting this value of x in the formula
of the fubtangent ~ , the fubtangent will be
— X with a negative value, which is as much
as to fay, that the fubtangent BT muft be taken on the contrary part of the
abfcifs.
Therefore, taking BT =z BA, and drawing the right line TC to the point
C, it (hall be a tangent to the curve at the point C.
Now, becaufe in the curve DCE, as the axis increafes, the ordinate y will
decreafe, in taking the fluxrion we might have put y negative; but becaufe,
for the fame reafon, we ought to have taken the fame y negative alfo in the
general formula, I have omitted to do it in both places, becaufe it comes to the
fame thing, without incumbering ourfelves with changing figns ; and what is
now mentioned may be underftood on other like occaiions.
Let X iz — be a general equation to all hyperbolas ad infinitum, between
their afymptotes, where m flands for any pofitive number, integer, or fradion.
By taking the fluxions, we fliall have x zz.
myy
tn — l
ffZ + I
And,
fubflituting this value in the general formula ™, the fubtangent will be
•— — , or — ntXj by the equation of the curve.
EX.
SO-
ANALYTICAL INSTITUTIONS,
BOOK II.
EXAMPLE IV.
38. Let the curve ADF (Fig. 24.) be a circle whofe diameter is 2^, ABz:;^,
BD zz y ; the equation will be xax — .v.v = yy^ whofe fluxion is zax — 2xx
= 2.yy, and therefore x ~ — ^^ . Then, fubflituting this value in the formula
, the fubtangent will be ^3^,, that is, -fl£lff , by putting, inftead oi yy^
it*s value from the given equation. Therefore the fubtangent in the circle will
be a fourth proportional to ^ — a*, la -^ ,v, and x.
Fig, 29.
But if the circle fliall be denoted by
this equation, aa — xx zz yy^ in which
the abfcifs AB — x \% taken from the
centre -, by taking the fluxions, we fliall
have XX zz — yy, and therefore x •=.
yy
Wherefore, fubflituting this value
in the formula, the fubtangent will be =
-. ^ , that is, a third proportional to AB and BD, but negative ; that is to
fay, it muft be taken from B towards T.
EXAMPLE V.
39. Let the curve ADF (Fig. 24.) be an ellipfis, with this equation ax — xx
~ J taking the abfcifs from the vertex A. The fluxional equation will be
. Now, fubflituting this value
ax — 2XX =. f^, ^nd therefore x zz — ^"^^
in the general formuU ^ , then — ^===. will be the fubtangent ; or elfe,
y b X a — 2x
2a» —2XX
a
'■ — — , inflead of ~ , putting it*s value ax — xx from the given equation.
Making x = ia, half the tranfverfe axis, in the value of the fubtangent, it
will be —f that is, infinite. Therefore the tangent will be parallel to the
tranfverfe
SECT. II. ANALYTICAL INSTITUTIONS. 3I
tranfverfe axis in that point, in which the conjugate axis meets the curve. And
this we fhall find to be true alfo, if we inquire what is that angle, which the
tangent itfelf makes with the (ame axis.
Let the equation, in general, to ellipfes of any degree be this ; — 7 — =
x" X a — ^'"j where m and n reprefent any pofuive numbers, whether integers
or fradions. The fluxion of this will be —7— X ayy = ?nxx x
. m+a— I
a-^x) — 'fjxx X a — x^ ; and therefore .v n ■
bmx xa — A'l —bnx Xa—x\
And, fubftituting this value in the general formula, it will be
m^n^ay _ ^ ^^ .^^^^^ ^^ ay_ j^ .^,^ ^^^^^
tn—\ \n » m _ ^n — i ' b ' ^ "
bmx X a—x\ — bnx X rt — aI
from the given equation, the lubtangent will be
« — I ' ^ « '" w 1 \ « — I
mx X « — ;cl — n.v X <»— ;«?i
n — l
And, dividing the numerator and denominator by ;? X ^' — ^1 i it will
be, finally, — .
ma — ntx — nx
Make »; 1= i, » =: i, that is, let it be the ellipfis of Jpollonius ; then the
fubtangent will be ^^ "" , as before. Make ;» z: 3, n — 2 ; then the
equation is ^ zz x^ x a — x)"", and the fubtangent will be ^-—-^-t^* And
fo of others.
If the equation were -^^-r — :=: x*" X a + ?i", it would exprefs all hyperbolas
of any degree, when referred to their axis ; taking, in the fame manner, the
beginning of the axis from the vertex A. Then, by a like operation, we
ftiould find the fubtaneent to be ; — ^ — ^, which differs from the fore*
" ma -\- mx -\- nx '
going only in it's figns ; as alfo, the equation, from whence it is derived, differs
only in it's figns.
Make m r= i, « =z i, which is the JpcUonian hyperbola. The fubtangent
ill be — — . Make w zz 3, « = 2, then the equa
X a 4- xy I and the fubtangent will be • ^^'^ '^ ^^'^ &c,
3^ + 5**
will be — — . Make m zz ^, « = 2, then the equation will be ^= **
40. From
3a
ANALYTICAL INSTITUTIONS.
BOOK II.
Asymptotes. ^ 40. From this method of tangents may be further derived a way of difcover-
ing whether curves propofed have afymptotes, and the manner of drawing
them, when they are inclined to the axis. For, as to the more fimple cafes, ia
which they are either perpendicular or parallel to the axes, fufEcient has been
faid in the firft Part, Sed. V.
EXAMPLE I.
Fig' SO
41, Let the curve be ADE, with the equa-
m4-n
tion -~ — zz X X ^ ^--'^l , as above, thefub-
tangent of which is TB "=.
T AJtJ
Then the intercepted line AT zz
— Xf that is,
m -\- n X ax -f xx
ma •+ *nx -^ nx
m + n y. ax-\-xx
ma + mx + nx:
ma + tnx + nx
It is plain that the tangent TD will become an afymptote, when touching the
curve at an infinite diftance ; that is, when the abfcifs AB =: x becomes infinite,
the intercepted line AT fhall remain finite. Now, putting x infinite in the
exprefTion of AT, the firfl term ma of the denominator is infinitely lefs than the
others, and therefore vanifhes. Whence, in this cafe, it will be
mx + «;e '
or
^— — , which is a finite quantity. Wherefore the curve has an afymptote.
which will begin from the point M, making AM zz
na
m -f- n
Now, to draw it.
let AH be raifed perpendicular to AB, and let it be, for example, MHP,
This being fuppofed, if we take x infinite, it will be x ,y :: MA . AH, and,
m-\-n
in the fuppofition of x being infinite, the equation of the curve ^^ — = sT
X a H- ;c]", [a being nothing in refped of ^, will be changed into this other,
^^ — = 3i*^ , Or, extracting the root, and, for convenience, making
?« 4, « r= /, it will be y^a — x^/ b-, and, taking the fluxions, ys/ a zz x^b\
fo that X ,y \\ y/a , ^b. Whence MA . AH :: s^a . </^. And, becaufe
MA
*ECT, II. ANALYTICAL I K S T I T U T I N S. 33
MA = -^, it will be ^ . AH :: ^a . ^by or AH = ^ X ^/4•• If>
therefore, we take AM iz — , and raifing the perpendicular AH = — x
^ — , and drawing the indefinite right line MHP ; this will be the afyniptote
of the curve ADE.
Make ;?z = i, » =: i, that is, let the curve be the Apollonian hyperbola,
whofe equation is —■ ::= ax + xx ; it will be / =z 2, and therefore AM r± ^a,
AH =1 — X \^ — ~ ^'Z ab. That is, AM is half the tranfverfe axis, and
z a
AH half the conjugate, juft as it fliould be from the Conic Sedions.
EXAMPLE II.
42. Let ADE (Fig. 30.) be a curve whofe equation is jv' — .v^ ,= axy ;
Tnaking AB z: x^ BD zz y. By taking the fluxions, we (hall have 3jy^ — 3^^
r= axy + ayx^ and therefore ^ zz 3.y — ^^y ^^^ ^-p __ 21. — x zz
p — 3^ — ^^y ^ Qj.^ inftead of 3^ — ^x'y putting it's value ^axy from the
equation of the curve, it will be AT zi — —■ — • And, making a: infinite,
~ T ^xx •{•ay " '
that is, in cafe of an afymptotc, in which AT becomes AM, the term ay is
nothing in refped of 'ixx, fo that it will be AM z: ^ = ~.
But, becaufe, in the propofed equation, the indeterminates cannot be fepa-
rated, nor, confequently, can the value of AM be determined ; if we put
AM zz -^ :::: t, (which expedient may alfo be ufed in other like cafes,) it
3*
will be ^ =r - — i which value being fubflituted in the propofed equation, it
will be ^-~ x^ zz Qtx'^. or ^-^ — x zz: rit. But, as x is infinite, the laft
term will be nothing in comparifon of the others, fo that it will be -^ — x
0,
zz o, or t zz -^a. Taking, therefore, AM ^ ^a, the afymptote muft be
Vol. IL F drawn*
^
ANALYTICAL INSTITUTIONS,
BOOK ir.
drawn from the point M, Moreover, it muft be MA . AH :: x .j/y and the
propofed equation y — x^ zz axy^ or y^ zz x^ + axy, will be reduced to-
x^ •= y\ ov X zz y^ when x is infinite, and therefore .v — y. Therefore,
making MA i= AH, if from the point M, through the point H, a right line
be drawn, it will be an afymptote to the curve.
I add further, that the lifft AT mufl: neceffarily approach to a certain limir,
beyond which it cannot pafs, and that the aforefaid limit is then an infinitefimal,.
or nothing. Here follows a plain Example of this.
Fig^Z^
Let BCF be an equilateral hyperbola ;
and making AB ~ a, AD = .v, DC — v,
we fhall have the equation aa + xx zz yy,
the fluxion of which is xx zz yjf. Thence
the fubtanffent will be ED =: ^ = —
__^ aa + XX
aa + XX
, and eonfequently ED — AD
— ;.• = ~ = AE.
Putting X ~ o, AE will become, infinite, and the tangent at the point B will
be parallel to the axis AD. And, making x rr w , it will be AE n o.
Wherefore the point E defcribes the whole line AE infinitely produced, and
finiOies it's courfe at it's origin A, beyond which it palTes not, though the
curve turns it's convexity towards the axis. Therefore the afymptote AG
procee Is trom the point A, and makes half a right angle with the line of the
abfcifles ; foralmuch as, in the equation of the locus aa + .v.v ~ yy, making
« r: 00 , the conftant quantity aa will vanifli, and it becomes xx — yy, or
X =: J.
JF/V. 32
4-3. Hitherto I have fuppofed that the
angle of the co ordinates is a right angle;
but, if it were obtufe or acute, making, ^
before, BC — x^ CE = j, CD m x,
OG = J/, (Fig, 32, 33.) the fubtangenc
will be neither more nor lefs than -r-,
y
for the two triangles GEO, EAC, will be
fiill fmiilar ; but the other formulas will
have need of fome reformation.
In the triangle EGG, the angle at O,
equal to the angle ACE, is fuppofed to
be known ; therefore, from the point G
letting fall GI perpendicular to AD, and
producing
€ECT. II. ANALYTICAL INSTITUTIONS. |5
producing EO to H, if there be occafion, in the triangle GOFi the angla GOH
will be known, and the angle at H be a right angle. Wherefore the angle
OGH is known, and confequently the triangle OGH is given in Jfech, that is,
the ratio of GO to GH is given. Let this be the fame as a to ;;?, and therefore
it will be ^ . w :: y . GH rz ■^~ . Alfo, the ratio of GO to OH will be
given, which may therefore be as a to 7^ ; and confequently a . n :: y . OH ~
— . Then EH z= .v ± -— , (where the fign muft be aftirmative in Fig. 32,
, . . ^. . ,,,, r t^r> aaxx i zanx'y + nn'^v A- mmyv
and negative m tig. 2^.) Wnereiore iLUq ~ ■ — .
But if OG be expreffed by a, GH by m, OH by ji, then it will be aa zr m-dj
+ m, and aajy zz imyy + nnyy^ which, being fubftitut-ed in this value of
EG^, Will make EG^ iz — . -, and EG = j =
^ff — ZL^— ll-, the expreffion of the element or fluxion of the curve,
a •
This being fuppofed, by the fimilitude of the triangles EGO., AEC, it will be
GO.GE::EC.EA, that is, j) . v/ "'" ~ '7 "" "'' ::j.EA; or EA =
X. v/ liii^ijL^ which will be the formula of the tangent.
.y ^
Let TE be perpendicular to the curve, and ES to the diameter AL Then,
by fimilar triangles GOH, ECS, we fliall have ES =: ~ , and CS = — .
And, by the fimilar triangles GEH, EST, we fliall have EH . HG :: ES . ST.
That is, ^"-"^ . ^ :: -^ . ST = — =L=. And therefore CT -
a a a a X ax :^ny
mmyy
ny_ _ mmyv ^ nn_yr ± anyx _ ayy±n;p: ^ ^^^^^^^^ .^ ^^^^ ^^^^^^^^ ^^
■a X ax±7iy '~ '^ a X ax ± nj ^-^^ ± »/
the fubnormal.
In a like manner, the other formulas may be reduced, which it is fufficient
only to take notice of here.
44. But the curves, whofe tangents we defire, may be Trajifcenient or Tangents t<»
Mechanical, that is, are not expreflible by any Algebraical equation, bit may ^'"^"^^'^"'^'^"^
^pend on the redification of other curves, which are not rei^tifiabie. Let the"'""""
F 2 curve
curves.
3«
ANALYTICAL I-NSTITUTiaNS.
BOOK ir-
curve be APB, whofe tangent PTK we
know how to draw, at any given point P.
Then, producing to M the line QP per-
pendicular to AQi^ let the relation of MP
to the arch PA be exprelled by any
equation, to find the tangent MT of the
curve CMA, defcribed from the point M-
Draw c/m infinitely near to QM, and MR.
parallel to PT j and fuppofing the re6lifi-
cation of the arch AP ; make AP =: .r,
PM —y, and it will be Vp zz x, Rm zij)/,
and the two triangles wRM, MPT, will
be fimilar, and therefore mK . RM ::
MP . PT, that iSy-y . x :: y , PT zz -^ , the formula for the fubtangent of
the curve CMA, by taking it on the tangent of the curve APB. From the
given equation of the curve AMC is found the value of x or jy to be fubfti^
tuted in the formula. All the reft is to be done as ufual^
^^i' ^•
x/
K
]\-
A
Q
M^^
p/^
m.^ A
J'
/
EXAMPLE.
^^S* 55^ 45- While the circle DPC revolves
uniformly upon the right line AB, be-
ginning at the point A ; the point C of
it's periphery, which at the beginning
of the motion fell upon A, leaves an
impreffion in the plane of it's motion,
which it continues till the point C ar-
rives again at the right line AB. It will
defcribe a curve ACB, which, from it's
generation, is called a Cycloid, It will be the ordinary cycloid, when the circle
fo moves upon the right line AB, as that it (hall meafure out the whole exadly
by it's periphery, after that the point C Ihall have palfed from A to B, fo that
AB may be equal to the periphery of the fame circle. It will be a prolonged
cycloid when the motion is fuch,, that the right line AB is longer than the
periphery of the circle; and a contraded cycloid when the fame AB is (horter
than the periphery.
From the defcription of this curve it plainly follows, that, drawing from any
point the right Ime MQ^ parallel to AB, the intercepted line MP, between the
curve and the circle CPD, will have to the arch CP the fame ratio as the line
AB has to the whole circle.
N. B. The chord ME is omitted in Fig. 35.
Suppofe
SECT. II, ANALYTICAL INSTITUTIONS. 37
Suppofe the generating circle to be in the two poficions EMF, DPC ; draw
the chords ME, PD. Now, becaufe the arches EM, DP, are equal, the
chords EM, DP, will be equal and parallel, and therefore MP zi ED. B it,
by the nature of the curve, it is AE . EM :: AD . EMF :: AB . EMFE.
And in the fame ratio is alfo ED . MF. And MF zz PC, FD = MP : there-
fore it will be MP . PC :: AD . EMF :: AB . EMFE Therefore, if we call
the light line AB z= a, the periphery of the generating circle EMFE = ^,
and any arch or abfcitfa CP = x, the ordinate PM zi y ; the equation of the
curve of the evcloid will be y zz — ,
■' a
Having therefore the equation of the curve, in order to find the fubtangent,
it*s fluxion will be ^ ~ — ; and, inftead of x, fubftituting this value in the
formula ^, it will be PT =r — = x Therefore, takins;, on the tangent
y a ' o' to
of the circle, PK, (Fig. 34.) which is fuppofed to be drawn, a portion PT
equaJ to the arch of the circle AP, and drawing the right hne TM to the point
M, it Qiall be a tangent to the cycloid in the point M.
Now, befides, if the cycloid be the ordinary one; becaufe, in this cafe, we
fhall have b — a^ and therefore y z= x^ it will be'PM — PT, and the angle
PTM = PMT. But the external angle TPQ^is double to the angle TMP,
and the angles TPA, APQ»_ are equal, by Euclid, iii. 29 and 32, therefore the
angle APQ^will be equal to the angle TMP, and therefore the tangent MT
is parallel to the chord PA. ,
46. Without the affiftance of the tangent of the curve APB, (Fig. 34.) we
may have the fubtangent of the curve AM, taking it in the axis KAB. Make
AQ^zz Xi QP zz J-, the arch AP zz /, QM zz 2, and let the relation of the
arch AP to the ordinate QM be exprelied by any equation whatever. Let qm
be infinitely near to QjVl, and MS parallel to AB. It will be MS zz x,
S?n zz z, and the fimilar triangles wSM, MQN, will give us z . x :: z . QN
=: ~, a formula for the fubtangent.
Inftead of taking for the ordinate QM = Zy if we take PM zz u ; drawing
MR parallel to the little arch P/), it will be wR zz «, RS = po zz y^ and
therefore mS zz u -^ y. And the fimilar triangles ^^SM, MQN, will give us
u ■\- y y. X
4- J/ . .V :: « + V . QN zz — r-^— r — , another formula for the fubtangent.
EX-
38
ANALYTICAL INSTITUTIONS.
BOOK fl.
EXAMPLE I.
-^^- 34.
/£
47. Let the curve APB be a circle
TZ-Kj" wliofe diaineter is zr, and let the ratio of
PM to the arch PA be that of a to b ;
that is, let the curve AMC be a cycloid.
Make AQ^=: a;, QP := jy, QTvI zz z, the
ar<:h AP = s\ then drawing »;j infinitely
near to MQ^ MR, parallel to Vp\ MS,
P^, parallel to AB ; it will be m'^ ^.-s;*
RS = fo =: J/, Vp :=^' i ; and wR, the
-diffeicnce or fluxion of MP, v/ill be
2 — J/. But, becaufe, by the property of
lli the curve, we have MP, to the arch PA,
as ^ to i* ; in the fame ratio, alfo, will be
their differentials /»R, ^P; and therefore it will be 2; — j/ . i \\ a \ b -, that is.
■X
•y -
But i ~ ^yicx +yy, and, by the property of the circle,
y = y/irx — XX. Tlicrefore y zz
•whence i zz
rx — XX
^2rx —x.\
, and yy zz
2rx^
+ A-'
arAT — XX
yzrx — XX
Wherefore, thefe values being fubftituted inftead of s and y in the equation
-r- "=■ z — jfy we fhall have z zz
ar'x -|- hrx — hxx
b'^irx —
tlie differential equation
of the cycloid.
Therefore, the value of i;, given from the equation, being fubflituted in the
formula for the fubtangent ~y we fhall have ON zz ^''•^'^•" ~ ^^'^
^ ^ ar -{- br - bx'
Now, if the cycloid be the ordinary one, it will ht a ~ b, and therefore
QN
zv 2rx — XX
zr — X
- J that is, 2r — ^ . \/zrx-^xx :i z . QN j or zr
:: z . QN. But, by the property of the circle, it \s ir -^ x . y :: y . x.
Therefore it will be ;y . ^ 1 : 2 . QN j that is, QP . QA : : QM . QN. There-
fore MN will be parallel to PA. ^ ^ ^ ^ ^ •
EX^
SECT. II,
ANALYT, ICAL INSTITUTIONS*
39
EX AMPULE ir.
48. Let the curve APB be a parabola, the equation of which is px = ^/,
Make AQ^zn x^ QP z= _y, and let the arch AP — j, PM — u, and the ratio
of MP to the arch PA be that of ^ to b. Therefore it will be wR .Vp'.'.a. b.
That is, u ,s ','. a , b^ and therefore -f- z= u. But, in the parabola, it is
y — \/'pXi and jf ~
_ P^
Therefore } __ 1_4£5 — pp^ And this value being;
Z\/pX o
fi-ibftituted inftead of } in the equation ~ = ii, the equation to the curve
Z^/pX
v:
AMC will be ^ ^ — ^ n: «, Wherefore, taking; the formula of the fub-
2bjypx ' ^
tangent
K + y X X
, which is proper to this cafe, and making the fubftitutions
inftead of u and j, it will be QN
— " +.y ^ ih^px
But jy z= K^pXi by the
a V^px + pp + hp
property of the curve APB, and -^ = u,hy the property of the curve AMC
_^ %asijpx + ibpx
wherefore QN
a \/pp + \px + bp
49- From the different manners by which many curves may be generated,,
arile diiferent formulas of their fubtangents, though the method of finding
them is alike. It will be enough to (how it in one, to give an idea of the
manner, and of the artifi<:e, which is to be ufed on all other occafions. Where-
fore, two curves AQC, BCN, being.
Fig, 36.
given, having a common diameter
TF, whole tangents can be drawn ;
let there be another curve MC fuch,..
as that the relation of the ordinates
PQ^ PM, PN, in refped of any
point at pleafure, M, may be ex-
prefTed by any equation whatever j.
and let the tangent MT be required^
at any point M. Let ^S be drawn
infinitely near to PN, and the lines-
:NS,, MR, Qp, parallel to AB, and
maker
40 ANALYTICAL INSTITUTIONS. BOOK II.
make PE = /, PF = /, known by fuppofition, PQji: x, PM z:^, PN = z.
Becaufe of fimilar triangles QPE, ^OQ^ it will be QO = -^ zi MR = NS ;
and, becaufe of the fimilar triangles /«RM, MPF, it will be PT = ~,
a formula for the fubtangcnt. Now, by differencing the equation of the curve
MC, in order to have the value of ;i', to be fubftituted in this tormala, it will
be given by y and % ; but [he fubrangent itfelf is not to be had in finite terms.
It is to be confidered, then, that the fimilar triangles NS^/, NPF^ will give us
NP . PF :: »S . SN, that is, z . / :: ± i; . SN zz ± — . (That is, z muft
have a poficive fign, if, when x and y increafe, 2; will increafe alfo ; and a
negative fign, if, when x and y increafe, 2; will decreafe.j But it is alfo
SN — ~ i then 4- — ^ zr — , and therefore 2 z: h . Therefore, in-
ftead of 2;, putting this value in the fluxional equa^on of the curve MC, we
fliall have the value of x expreflTed by j, which, being fubltiiuted in the
formula for the fubtangent -^ , will make the fluxions to vanilli, and the fub-
° xy
tangent will be exprefled in finite terms.
EXAMPLE I.
50. Let Dcz. = yy be the equation of the curve MC, the fluxion of which
will be zx + xz zz lyy i and, inftead of z, fubftituting it's value Hh -^, it
will become 2^- ± ^ z= 2yy, and therefore x = ^^f . Wherefore, inftead
* tz ^ sz
of X, fubftituting this value in the formula^for the fubtangent, it will be
PT =: — ^^^ — = — T- J when, inftead of ry, we put it's value xz. Now let
the curve AQC be a parabola whofe parameter is b ; the curve BCN a circle
whofe diameter is AB = 2^. IP, therefore, the point N falls in the periphery
of the firft quadrant beginning at A, in which z is pofitive ; the formula of the
fubtaneent PT will be — f- , and the fubtangent of the circle will be ^"^^'-- ■zif^
{making AP =: ^,) and that of the parabola will be 2^ zz s. Therefore, thefe
values of / and s being put in the expreflion — _^ , we (liail have PT =:
6^7 —All
4^-3f * 7 51. But
iECT. II. ANALYTICAL INSTITUTIONS. 4.1
51. But if the point N falls in the periphery of the other quadrant, :<-. will be
negative, and the formula of the fubtano;ent will be PT z= -^ . In this cafe,-
the fubtangent of the circle is "^^J" ^'^ = /, and that of the parabola continues
to be 2q zr s. Therefore, making the fubftitution of the values of / and s in
the expreffion -^ , we {hall have PT zr ^'^ " "^"^^ -, the fame as before,
52. Let A? be denominated as before, AQ^being a parabola; it will be
PQ^rz X = \>^l>q. And BCN being a circle, it will be PN zr 2 =: \/^ zaq — qq.
Then the equation yy zr zx of the curve MC will be yy zz q\/ lab — t?q.
And thus, the equation being given by the two co-ordinates AP, PM, the
fubtangent PT may be found by the ufual and ordinary formulas ^ . There-
fore, differencing the equation yy = qs/ lab — bq^ it will be jyy zz ^~==l=.
4 V tab — bq
Now, multiplying the numerator and denominator of the formula -^ by. y^ it
will be — ^ , and fubftiiuting the refpedive values inftead oi yy and yy^ it will
be m. = SfLZLlTi ^ p-p ^3 before.
53. Let the equation of the curve MC be more general, thus, x*z ziy"* ,
the fluxion or which is mz xx + ^^ ^^ zz m + n X yy . And,
.«.«? — It n • m~l
inftead of z, putting it s value ± — > it will be — z:
ore X ^ -^ .
mt ziziis X a *•
J/, if we put it's value x z inftead of y
w + « X y)' ; and therefore x rz -^^ , Whence PI =:
mt ziziis X a *•
«; + «
*y ~T~; ^ n m mi + ns
54. If the two curves AC, BCN, become right lines, in the cafe of the
fimple equation xz zz yy of the curve MC, it will be one of the Conic Sedions
o{ Apolloniusy as is to be feen in Se6l. lU. of Vol. I. § 135. It will be an
ellipfis, when the ordinate CD falls between the points A and B : an hyperbola,
when it falls either on one fide or the other: and laftly, a parabola, when the
points A, B, are infinitely diflant one from the other, that is, whtn one of the
Vol. II. G ri^hc
4«
ANALYTICAL INSTITUTIONS.
BOOK II.
right lines AC, BC, is parallel to the diameter. Hence it is manifefl, tliar, in
the fame ciicumftances, the fame curves will be conic fedions, but of a fuperior
degree in injinitum^ whea the equation to the curve MC fliall be this general
jn n
one, X z
viJfn
~ y
Fig^ 37* y^ 55. If the curve AP be given, having
it*s oriffin in A, of which we know how
to draw the tangent ; let there be another
curve CMD fuch, that, from a given point
F drawing the right line FMP any how,
the relation of FM to the portion AP may
be exprefled by any equation : we are to
find the tangent of the curve CMD.
Let PH be a tangent to the curve APB
in the point P, and let FH be drawn per-
pendicular to FP, and Fp infinitely near ;
and with centre F let the infinitely little arches MR, PO, be defcribed ; and
Jet MX be the tangent required of the curve CMD. Make PH zz /, FH zz s,
FM = J, FP = Zf and the arch AP zz x. Becaufe, infl:ead of the infini-
tefimal arches, their right fines may be affumed, the triangle MR»» will be
right-angled at R ; and, becaufe the angle M.mK is not different from the angle
TMF, but only by the infinitefimal angle MF»;, the two triangles MRz«,
TFM, may be confidered as fimilar ; and, for the fame reafon, the two tri-
angles PO/, HFP, are fimilar. Therefore it will be »;R . RM :: MF . FT ;
tlut is, > . MR : : 7 . FT, and pT zi ^^ "" ^
y
So that, to have the value
of FT, it is neceflary to have that of MR firft, which we might have if PO
were known. Now, by the fimilar triangles PFH, P0/>, it will be PH . FH
: : P/» . PO ; that is, -
FMR, it will be FP .
Whence FT = ^,
, s :\ X . OP rr " . And, by the fimilar fedlors FPO,
PO :: FM. MR; that is, z.-^':: y .MR = ^^ .
the formula for the fubtangent. Now if, infl:ead of x.
we fubfiitute it's value, which may be obtained from the fluxional equation of
the curve CMD, we (hall have the fubtangent expreflTed in finite terms.
EX.
SECT, ir.
ANALYTICAL INSTITUTIONS.
43
E X A M P L E I.
Fig, 38. ^6. Let there be a circle ABCD defcribed
with centre H, and radius HA; and whilfl:
the radius HA, widi one end fixed in the
centre, moves uniformly round, and with the
other extremity A defcribes the periphery
ABCD ; let the point H move uniformly upon
the radius HA, fo that when the radius returns
to it's firft fituation HA, the point H, in the
mean time, fhall pafs through the radius, and
fhall then be found at A. The point H will
then defcribe the curve HEf A, which is called
the Spiral of Archimedes. From the generation
of this curve, it is eafy to perceive that any
arch of the circle whatever, as AB, fliall be to the correfponding portion of the
radius HE, as the whole circle is to the whole radius. Therefore, making the
radius zi r, the periphery of the circle zi f, the arch AB zz x-, and the ordi-
nate HE = ^ ; it will ht tt * y '»'. c , r -, and therefore y —
rx
c
an equation
to the fpiral, in which the ordinates proceed from the fixed point H. This
being premifed, if we would find ET, the tangent of the fpiral ; becaufe, in
this cafe, FP (Fig. 37.) is the radius HB of the circle, it will be 2; zz r, and
the two lines, PH the tangent, and FH the fubtangent, (in the fame Fig. 37,)
are in this both perpendicular to the radius HB, (by the nature of the circle,)
and confequently parallel to each other, and alfo equal ; whence it will be
s — ti and therefore the general formula, in this cafe, will be ^ . Then, dif-
ferencing the equation ^ = ~ , it will be j/ = ~ ; and the value of x being
fubflituted in the formula, it will be 4?- = HT. Or elfe, putting, inftead of>,
it's value — ,
it will be — 3: HT. Therefore, with centre II, and radius
HE = y, defcribing the arch EQ>_ and taking HT equal to the arch EQ^ it
fhall be the fubtangent. For, by fimilar fedtors HEQ»_ HBA, it will be
HA.AB :: HQ^. QE, That is, r .;?::;'. QE z: ^.
Gi
If,
4 +
ANALYTICAL I N S T I T U T i o K ^»
ROOK II,
If, inflead of making the equation y '==. -— , it were, in general, y
m
r X
■- ; that is, the periphery to the arch AB, as aay power integral or fraiflionai
c »
of the radius, to a like power of the ordinate : Then taking the fluxion of the
... . mcyy , . fncyy
equdtion, it would give us X =: ^ — ^ and yx — -^^
r. r
rng this in the formula of the fubtangent ^ , it would be
y'" ^LJL- therefore ^ = HT - ;;/ x EQ^
f»cy
. Then fubftituE-
m-\- 1
za+l
ziHT. But
Fig. 37
57. We fhall have the formula of the
fubtangent more fimple, if the equation
of the curve APB were given from the
relation of TM to FP. For the fimilar
triangles ^OP, PFH, will give us PO zi
-^, and the fimilar fedors FPO, FMR,
will give us MR = ^ ^ and laftly, the
limilar triangles MRm, TFM, will giY,e
us FT = ^ .
zzy
EXAMPLE II.
Fig. ^g.
58. Let the curve CMD be above
the line APB, which makes no alte-
ration, and let APB be a right line,
and let FM, FP, always differ from
each other by the fame quantity, that
is, make the conftant line PM zr a.
Then will j — z =z ahe the equation
of the curve, which is the Conchoid of
Nicomedes^ whofe pole is the point
F, and afymptote AB. Taking the
fluxions
SECT. II,
ANALYTICAL IKSTITUTJONS.
fluxions of the equation, k will be y — z, and thence the fubtangent FT
sry
Drawings then, ME parallel tO' PA, and MT parallel to FE, MT will be a
M. For it will be FA zi j, FE = -^ , and FT
tangent to the curve in
■Fig. 40^ X ''--^ 59, Any curve AM being given, to
the axis EAT of which curve wc know
how to draw the tangent MH, at any
point M; and a point F being given out
of the curve, from which let be drawn
the right line FPM ; if we conceive the
right line FPM to revolve about the
immoveable point F, making the plane
PAM to move upon the right line ET,
always parallel to itfelf, the intercepted
line PA always continuing the fame :
Then the point M, which is the common
interfeftion of the two lines FM, AM,
by this motion will defcribea curve CMD, the tangent of which i» required.
Let the plane PAM m.ove, and, in. the firft inftant, let it arrive at an infinitely
near pofition pam, and let SKm be drawn parallel to ET. The fimilar triangles
MRw, MHT, would give the right line HT, which determines the tangent
required, if the fides MR, Km, were known. Therefore, to obtain them, let
us make FP, or Fp zi x\ FM, or ¥7/1 := y, Fp — z, and the known lines
PA = a, HM = /, PH =: s. It is plain, by the confliudion, that it will be
Pp = A« zz Km zz z -y and, by the fimilar triangles tPp, FSm^ it will be
Fp . Vp : I Fm . Sm, That is,
:: y , Sm =
— y
Then SR
__ yz — x^
And, by fimilar triangles MPH, MSR, it will be HP . HM :: RS . RM.
That is, J . / :: ^Izjli . MR = '=^^^-^^. Laftly, by the fimilar triangles
MRw, MHT, it will be MR . Km :: MH . HT. That is, !^^-^^ . z :i
/ . HT = -i^ .
y — X
From the point F draw FE parallel to the tangent MH, and taking
HT =r PE, draw TM, which (ball be a tangent to the curve at the point M.
For, becaufe of fimihr- triangles PMH, PFE, it will be PM . PH M
PF . PE 5 that \%,y-^x.s\\x, -^^ = PE iz HT.
y - fc
60. It
.4'6
ANALYTICAL INSTITUTIONS.
BOOK If.
60. It has been already demonftrated, Vol. I. Sect. III. § 136, that, if the
line AM were a right line, the curve CMD would be an hyperbola, which
would have ET for one of it's two afymptotes. If AM were a circle with
centre P, the curve CMD would be the conchoid of Nico-rnedes^ the pole of
•which is F, and ii'siifymptote ET. And laflly, if AM were a parabola, the
curve CMD would be the companion of the paraboloid of Cartefius, that is,
one of the two parabolical conchoids.
fig. 41.
61. To the diameter AP let there be
any curve AN, whofe tangent we know
how to draw, and a fixed point F olU of
it J and let there be another- curve CMD
fuch, that, drawing, as we pleafe, the
right line FMPN from the point F, the
relation between FN, FP, FM, may be
exprefied by any equation whatever. It
is required to find the tangent MT, at
any given point M.
Througii the point F draw HK perpendicular to FN, which meets the
diameter AP produced in K, and the given tangent NH in H. Let FQ^be
infinitely near FN, and with centre F kt the arches MR, P<7, NQ^ be de-
fcribed. Make FK = s, FH = /, FP - x, FM - y, FN = 2; then it
will be »;R zz. y^ fo ■=. x, Qn zz — z. And, becaufe of like triangles NQ^,
NFH, it will be NQ^= — -^ . Alfo, becaufe of like fedtors FNQ^ FMR,
it will be MR = — — . Laftly, becaufe of like triangles MRm, MFT, it
will be FT — — ^~, the formula reouired for the fubtangent. But here it
might be fufpeded, that, taking the fluxion of the equation of the curve, the
value of j|/ to be fubftituted in the formula will be given by x and z, by which
means the fluxions would not vanifli. Yet, however, the fimilar fedtors FNQ»,
FP^, will give us P^ = —; and the fimilar triangles Vop^ PFK, will give
us the analogy, X . —
and therefore
txz
zz
szzx
s. Whence the equation szzx = — ixx'^
z zz . Therefore, fubftitute the value of y in the~
formula for the fubtangent, which value is to be obtained from the fluxional
equath.n of the cuive, and then this value infl;ead oi z ; by which the fluxions
will vanifli, and we fliall have the fubtangent in finite terms.
If
SECT, ir.
ANi^LYTICAL INSTITUTION 9.
47
If the line AP were a curve inftead of a right line, drawing the tangent PK,
by the fame way of argumentation we (hould find the fame value of the fub-
tangent FT.
EXAMPLE,
Fig> 42,
T F
62. Let the curve AN be a circle
which pafTes through the point F, and is
fo pofited, that, from the point F drawing
the perpendicular FB (produced) to AP,
it may pafs through the centre G of th&
fame circle; and let PN be always equal
to PM : the curve CMD of the foregoing
figure, that is, FMA in this, will be the
cilfoid of Diodes f the equation of which
will be z + ^ rz 2X. Then we (hall
have, by taking the fluxion, z + j z=l iXy
01 y ^=. ^X -' z ; and fubftituting this
value of y in the formula — -^ of the fubtangent, it will be —
we fhall have
yyt^
txx
2%%x — zza
styy
2txx + -f^^:
and laftly, putting, inftead of — z, it's value
zz FT, the fubtangent required.
Here it is plain, that if the point M, at which the tangent is required, fliould
fall upon the point A ; in this cafe, KH being perpendicular to FA, it would
be FN = FP zi FM = FA := FK = FH ; and therefore FT z= 4^
= 4^AF.
63; Perhaps we might find the fubtangent of the cifToid more fpeedily, by
means of the ufual formula, at § 30. For, drawing NE, ML, perpendicular
to FB, and making FB =: 2«, FL zz x, LM n jy ; by the property of the
curve FMA, it will be BE — FL — x ; and, by the property of the circle, it
will be EN rz 's/iax'—xx -, and the fimilar triangles FLM, FEN, will give
FL . LM :: FE .FN, and therefore FL . LM :: EN .EB; that is, x .y\'.
, or jyy = ~ — , the equation of the
\/ 2.ax —- XX » Xi whence y = — —
y 2ax ~-xx
curve FMA. Therefore, by -taking the fluxions, we fhall have lyy :=:
, -^ — ; and taking the ufual formula •— , and making all the neccifary
za '- xY ° - y
8. fubftitutions^
4«
fu;bfiitutions, it will be -^ z=: yy x
■inftead of rv, it's value — ^
2,a
3d — A-
BOOK II.
, by putting.
£4. Let there be two curves ANB, CPD, and
a right line FK, in which are three fixed points
A, C, F. Further, let the curve EMG be fuch,
that, drawing through any of it's points, M, the
right line FMN from the given point F, and from
the point iM the right line MP parallel to FK ; the
relation of the arch AN to the arch CP fliall be
exprefled by any equation at pleafure. It is re-
quired to find the tangent of the curve EG at the
point M,
Let MX be the tangent required, which meets in
T the right line FK> produced if need be, and from
the point T let there be drawn TH parallel to FM,
and through the point M let be drawn MRK parallel to the tangent in P, and
MOH parallel to the tangent in N, and let FmOn be infinitely near to FN.
Make FM = s, FN == /, MK iz a, and the arches AN = y, C? =: x ; and
therefore N« ~ y, Vp — x. By the llmilar triangles FN;/, FMO, it wmU be
FN . N« : : FM . MO ; that is, / . j)) : : j . MO =: -|- . And, by the fimilar
triangles M»;R, MTK, and MOm, MHT, it will be MR . ;«M : : MK . MT,
and M;/2 . MO : : MT . MH ; and it will be alfo MR . MO : : MK . MH.
That is, X .
«.MH =
my
tx
Wherefore, by taking the fluxion of the
given equation, we fliall have the value of j} given by X ; and, by making the
necelTary fubftitutions, we fhall have MH exprefled in finite term?. Taking,
therefore, MH equal to the value now found, and parallel to the tangent in N
of the curve ANB, and drawing HT parallel to MF ; if from the point M be
drawn the right line TM to the point T, it will be a tangent to the curve EMG
in the point M.
N. B. The letter r has been put, by miftake, for the letter /, in Fig, 43.
EX-
SECT. II.
ANALYTICAL INSTITUTIONS.
49
EXAMPLE.
Fig, 44.
6^. Let the curve ANB be a quadrant of
a circle, whofe centre is F ; and let CPD of
Fig. 43 be the radius APF of Fig. 44, which
is perpendicular to the right hne FKTB, and
let the tangent AR be drawn. Let the radius
FA be conceived to revolve equably about
the centre F, and, at the fame time, the tan-
gent AR. to move equably upon AF towards
FB, always parallel to itfelf j fo that, when
the radius FA falls upon FB, the tangent AR
may coincide with FB. By this motion, the
point M, which is the interfedion of the
Radius and the tangent, will defcribe the curve AMG, called the ^ladratrix of
Dinojlratus,
It is plain, from the generation of this curve, that the arch AN will be to
the intercepted line AP, as the quadrantal arch AB is to the radius AF.
Therefore, making AN — y, AP =z x, AB z= ^, AF — r, it will be fy zi axy
ax
and j/ = -^ ; then, fubftituting this value of j)/ in the formula -:-, it will bfc
MH =: ~ ; but, in this cafe, FN is the radius of the circle, and MK zz AF
— AP ; then t z:z r^ u zr r — x i whence MH =
asr — asM
•y
putting, inftead of ax, it's value ry from the given equation. From the point
M raife MH perpendicular to FM, and equal to the arch MQ^defcribed with
centre F, radius FM, and let HT be drawn parallel to FM ; then MT will be
a tangent to the quadratrix in the point M. For, becaule of fimilar fedors
FNB, FMQ^ it will be FN . NB : : FM . MQ^ That h,r ,a ^y W s . MQ^
=: 'l^Ul - MH.
66. Let there be two curves BN, FQ>_
of which it is known how to draw the
tangents, and which have the light line
BA for a common axis, in which are two
fixed points A, E. And let there be
another curve LM, fuch, thar, drawing
the right line AMN through any pf it's
points M, and with centre A and radius
AM defcribing the arch MG ; and from
the point G letting fall GQ^ perpendi-
cular to AG ; the relation ot ihe fpaces
Vol. Jl. H ANB,
50
ANALYTICAL INSTITL'TIOXS.
BOOK Uf
ANB, EFQG, and of the lines AM, AN, QG, may be given by the means
qf any equation. The tangent of the curve LM is required at the point M.
Drawing the right line ATH perpendicular to AMN, let there be another
Awn infiniuly near to AMN, and the arch w^, and ihe perpendicular ^^;
Then, with centre A defcribing the little arch NS, making tlie given fubtan-
gents HA == a, GK zz /^, and make AM ~ y, AN zz z, QG zz ii, and the
fpaces EGQF — s, ANB zz /, it will be R;7i = Gg = j, S;z = z. And,
becaufe of the funilar triangles KGC^QO^, it will be Oq zz — it — —-. And,
by the fimilar triangles I-L\N, NS;?, it will be SN = — . The fpace CQjg
may be taken for the fpace GQO^, becaufe their difference QO^ is an infinitefimal
of the fecond order. Whence it will be GQ^^'^ zz t^y -=z — s. Thus, there-
fore, it will be AN« z= iAN X NS zz \az — — L Wherefore, thefe values
being fubftituted, inftead of ii, s, f, in. the fluxion of the propofed equation,^
we IhaJl have an equation from whence may be deduced the value of z given
byj. Now, becaufe of fimilar feftors ARM, ANS, it will be MR zz
^■, and, by the iimilar triangles ^»RM, MAT, it will be AT = ■^, the
formula for the fubtangent ; in which, inftead of z, if we fubftitute it's value
given by y from, the equation of the curve, the fluxions will difappear, and the>
fubtangent v/ill be given in finite terms.
EXAMPLE.
a
67. Let the fpace EGQF be double to ABN, that is, j zz 2/-; then i =: iti
But i zz — «y, and / zz — ^az \ therefore it will be ny — az^ and z z:;.
Then the fubtangent is AT = ~ — .
Let the curve BN be a circle with centre
A, radius AN zz c\ whence z z; f ; and
let the curve FQ^be an hyperbola with the
equation uy zi ff ; the fubtangent will be
AT zz &i that is, the ratio of AM to AT
cc
will be conftant. The curve LM (Fig. 46.)
will be called, in this cafe, the Logarithmic
Spiral,
7 Here
SECT. II. ANALYTICAL INSTITUTIOiJS, 5I
Here it is manifeH:, that the curve LM will make an infinite nuiTiber of
circnmvolvifions before it arrives at the point A; forafmuch as, when the point
G (Fig. 45.) coincides with A, the fpace s will be infinite, as may be fcen from
ilie Inverle Method of Fluxions. For then, alfo, the Ipace / muil be infinite,
which cannot be but after infinite revolutions of the radius AM.
68. It remains, laftly, to confidcr a particular cafe belonging to Tangent?.
It has been ieen that, the coordinates of any curve being x and y, the general
formula of the fubtangent will be —, or -— , according asjy or x fupplies the
place of the ordinate. Wherefore, the fluxion of the equation of the curve
being taken, if from thence we deduce the value ofx or y, this value, being
lubftitured in the general formula, vvill give us a fraction in finite terms, which
is the expreffion or value of the fubrangent for any point of the propofed curve.
Now, if we defire the. fubtangent for any determinate point of the curve,
nothing elfe is required to be done, but to fubftitute in this fra6lion, inflead of
X and_>', their values which they have at the point given. But it may fometimes
liappen, that, by fubfiituting, infiead of x or jy, a determinate value in the
fradion which expreffes the fubtangent, or otherwife, in the ratio of x to y
deduced from the fluxional equation of the curve, all the terms in the nu-
merator and denominator may vanifh of themfelves, and that there will only
arife -:- — — , and thence, alfo, the fubtangent will be — , from whence,
however, we are not to infer that the fubtangent is nothing in this point.
For an example, let us take the curve belonging to this equation y* — fioy^
— izaxyy + i6aayy -{- /\.Saaxy + ^aaxx — 64^^v, ;z: o, and let y be the
abfcifs, and .v the ordinate. Therefore -1- will be the formula for the fub-
tangent. Therefore, by taking the fluxion of this equation, we (liall have
■^ n , <■ -r- — TT J and the fubtangent will be -<- zr
X yi — tayy — oaxy -\- taay -f izaax ° x
7,axyf — J2aaxy — laaxx 4- \6a^x «- ./. , , , i r t .
', ' 7 —5 : . Now, 11 we would have the lubtanp;ent to that
y — oayy — baxy + ^aay + la^a.v °
point of the curve, which correfponds to the abfcifs y zz. zr, it being alfo in
this cafe x zz la^ by the given equation ; make the fubflituiions in the
fraction which expreffes the ratio of x to y, and we ftiall find it to be
^-,- ~ r-, — r-; r, that IS, — , becaufe all the terms deftroy one
another ; and therefore the fubtangent alfo, at this point, is — , which informs
iU£ of nothing, although one or more fubtangents may belong to that point,
H 2 6g. This
53
ANALYTICAL INSTITUTIONS,
¥00K II.
Fig, 4.7. 69. This cafe will always happen, wlien-
ever ihe curve has icvcral branches which
iiutTfcft one another, and when we would
have a tangent at the point of concourfe.
And, indeed, the curve NO PQ^IR (Fig,
/\'j,) cf the propofed equation lias two \i\q\\
branches, which cur one another in the point
G, to which exadtly correfponds y zz za,
C)T being the axis of the jy's, and it's begin-
ning; atO. Alio, X zz la^ taking the .v's
in the axis OQ^
To give a reafon for this cafe, it is enough to take notice of two things.
The firft is, that, at the point of concouife of the different branches of the
curve, feveral roots of the equation become equal to one another. 7'hus, as to
The propofed equation, in the point G the two values of .v are equal, and alfo,
two are equal of the four values of jy. The fecond is, (what is demonflrated
in Des Carles's Algebra,) that if an equation which contains equal roots be
multiplied, term by term, into any arithmetical progreffion, the produ(ft will btr
equal to nothing, and will contain in it fewer by one of the equal roots. And
if this produd be again multiplied by an arithmetical progreffion, the produd
will, in like manner, be equal to nothing, and will contain dill fewer by one of
the equal roots, than were contained by the firft produd ; that is, fewer by two
of the equal roots, than were contained by the firft; equation. And thus on
fucceffively to that produd, which fliall contain only one of the equal roots.
If, therefore, any equation of a curve, treating x as variable and y as con*
ftani, Ihall be multiplied by an arithmetical progrefHon which terminates in
nothing ; in the cafe of e^ual roots the produd fliall be equal to nothing ; and
it will alfo be fo, if the produd be divided by x, which divifion will fucceed
when the laft term is multiplied by nothing. The fame thing will obtain alfo
by treating V as variable and a\ as conftant, and multiplying the equation by
inch an arithmetical progreffion as has nothing, or o, to put under the laft:
term.
This being fuppofed, it is eafy to perceive that fuch an operation as thi^
performs the very fame thing as taking the fluxion ; that is, if it treats x a^
variable, and multiplies the equation by an arithmetical progreffion, the firft
term of which is the greateft exponent of x, and the laft term is nothing, and
produces a produd multiplied into x. Then, if it treats y as variable, and
multiplies the equation by an arithmetical progreffion, the firft term of which
is the greateft exponent of y, and the laft is nothing, or o, and produces a
produd multiplied into y. But, in the cafe of equal roots of x, and in that of
equal roots of ^, as well the produd multiplied by x, as that by y, are equal
to nothing. So that the ratio -r- zi — • ought to arife, in that point wherein
two branches of the curve interfed each other,
8, That
S E CT. ir. ANALYTICAL INSTITUTIONS. 5^3
That this ma)' be (ccn more fully, I here fct in order the equation of the
propofed curve according to the letter jy, and multiply it by an ariihmetical
progreffion, the lad term of which is o,
y* — Say^ — iiaxy^ + ^Saaxy + ^.aaxx ? «.
The produd will be
4_y* — 2^ay^ — 24.axy* -f. ^laay'' -|- /^Saaxy = o.
That is, dividing by 47,
jy' — Bay"^ — 6axy + Saay + iiaax = o.
Then I fet the fame equation in order according to the letter a;, and multiply
it by the arithmetical progreffion, the laft term of which is o.
Say' y = Q.
i6ay J
4.aax^ + 4.Saayx + y^,
— 64.aaax
— I2ayyx +
2, I, O,
The produ(fl will be
Baax"^ + ^Saayx — 64^'^ ~ 1 2ayyx ~ o*
That Is, dividing by 4.V,
2aax + iiaay — 16a' — '^ayy z= o, "
This being done, I take the fluxion of the propofed equation, which is
4yy — i^ayy — i^axyj — iiay'^x 4- ^2aayy + ^.Saaxy + ^Sa^'yx + 8^^;^;^'
— 64^2'^ =. o; that is, dividing it by 4, and tranfpofing the terms belonging
to Xf
y — 6ay^ — Saxy 4- Sa^y -f I2a^x into j/
= ^ay"^ — i2^^_y + ^aax + 16^' into .v.
Now here the multiplier of J/ is the Erft product into the arithmetical pro-
greffion, and confequently iz o in relation to the point G, in which y has two
equal values. And the multiplier of x is the fecond produd into it's arith-
metical progreffion with it's figns changed, which does not hinder it being ~ o,
in relation to the fame point G, in which x has two equal values. Therefore it
will be J/ X o =z jv X o, that is, -4- = ~ in the point G.
But, if to multiply any equation by an 'arithmetical progreffion, or to find
it's fluxion, (which is the fame thing,) bring it to pafs, that, on the fuppofition
of
54 ^ ANALYTICAL INSTITUTIONS. BOOK If.
of equal roots, ihvit cafe will arife of which we are treating, that is, -4- z: — ;
it alfo brings it to pafs, that, in the equation derived from thence, there will be one
lefs of thofe equal roots. Wherefore, il the equation propofed have two equal roots,
when differenced it will have but one of thofe equal roots. And, if the propofed
equation have three, by dilierencing again that which was differenced before,
(aifuming as conflant the differences or fluxions x, j',) the equation thence
arifing will have only one ; and fo on. Therefore, if we affijme as conftant the
fluxions X, j, as well the terms multiplied into x as thofe multiplied into _)/,
will mutually deftroy each other, in the fuppofition of fuch a determinate value
of X and y ; alfo, the terms muliiplied into x and y will deflroy one another.
By proceeding in this way of operation, equations will be reduced to contain
only one of the number of equal roots which they had at firtl ; and therefore,
finally differencing the lafl, to obtain the ratio of j/ to Xy there can no longer
arrfe the c^afe of -4- z: — ■ .
Therefore I refume the foregoing equation vvhofe fluxion was found to be
jy.3y — (^ay'^y — 6axyy — '^ay'^x + '^aayy + iiaaxy + iiaayx + laaxx
— \(ia^x ~z= o. But, becaufe, by fubftituting, inltead of j, it's value 2<2,
and, inftead of x, it's correfpondent vahje 2^, in order to have the tangent at
the point Gj I find only -4- = — : 1 go on to difference that already differ-
enced, taking always for conflant the fluxions .v, y, and I fliall obtain 3/j)/*
•— iiay^ • — 6axy^ + Saay^ — iiayyx + 2^aayx + 2aax^ = o.
Inflead of j and x, I fubflitute their values 2a, in relation to the point G ;
and 1 find x =: ±j/'/8. Then, in the general formula for the fubtangent
-~, putting the values of at zz 2a, and x = dt>V8, I fhall finally have the
fubtangent iz + -^; or, to fpeak more properly, the two fubtangents corre-
fponding to the point G, one pofitive, the other equal to it, but negative.
If the curve fliall have three equal roots at the point in which the tangent is
required, that is, if the curve fliall have three branches which meet one another
in that point ; becaufe, after the equation has been differenced once, it will ftill
have two equal roots ; it muft be differenced again, that we may have the ratio
of J/ to .x* : It will give us, notwithftanding, by what has been already faid, the
ratio -4- zz - — : and therefore it will be necefTary to take the difference or
fluxion a third time. And, in general, the equation muft be fo often differ-
enced as is the number of equal roots, or the branches of the curve ; and from
the laft difference muft be obtained the ratio of y to x. And fo many will be
ihe tangents as are the branches of the curve, which cut one another in that point.
Let
SECT. II.,
AN ALYTICAL INSTITUTIONS,
55
Let the curve be QfvDHA/^iAI,
\^hore equation is a* — ayxx -\- ^j^no,-
and which has three branches QAD,
lAc/, i=AII, which cut one anothtr in
A. And let AP be the axis belonging
to Xj and AB perpendicul.ir to AP, the
axis belonging to y; and the point A
their common orii^in. B3' differencing
tile equation, it wiil be ^x'X — 2ayxx
— axxy + sh'yj' =^ o ; that is, -.-
r= ^^l — 3 yy ^ gi^jj.^ j^ ^^,g would have
the tangent at the point A, becaufe
there it is ;f n o, y ^ o ; it will be 4- zz ~ . We mud therefore go on to.
•TO "
fecond fluxions, and the equation will be iixxxx — layxx — ^nxxy + 61>yjy
~ o. But from this we fliall only obtain — ^ zz — ,. every term being multi-
plied by .V n o, by fuppofition, or by y zz o. Therefore, differencing for-
the third time, it will be 24^:^^ — Sayx^ + 6/^v' =: o. Here, making a; ~ o,,
the firfl. term vaniQies, and therefore it is ayx'^ z= ^j/', from whence we have
three values of j/ ; that is, j/ = o, and / r= ± /!> which give us three ratios
of X to y ; that is to fay, three tangents at the point A. One of them will be
infinite, which coincides with the axis AP, and ferves for the branch ^AH. ,
The other, taking any line AS, and drawing ST perpendicularly in fuch a.
manner, as that it may be ST. SA :: \/^ . \/^ ; the lines TA will be
tangents in the point A, one of the branch QAD, the other of the branch lA^.
70. The truth of thefe conclufions may alfo be demonflrated after another
manner, and, as they fiy, a pojieriori. The differentials of finite equation?,,
which are found by the foregoing rules of differencing, are not really the com-
plete differentials, the rules giving us only thofe terms which contain the firfh
differences, or of one dimenfion only ; and omitting, for brevity-fake, and for
greater convenience, the differences of other degrees, or of greater dimeniions :
which, by the principles of the calculus, would make thofe terms in which they,
are found to be relatively nothing.
Refuming the equation >"* — ^ny^ — iiaxy^ + /i^^a'yx + 4^V
it's fluxion or difference will be ^y'^y — 24ayy — izayyx — 24^.5^;^ -i- $2ayy
+ /\.^aaxy + 48aayx + Saaxx — C^a^x rz o. But here, if jy be confidered as-
iucreafcd by it's fluxion or difference, and likewife x ; and that in the propofed
equation, inftead of y and it's powers, we Ihould write jy-fy and it's correfponding.
powers 5 and Ihould do the fame by writing x -^ x and it's powers inflead of
thofe.
}=..
•5$ ANALYTICAL INSTITUTIONS. BOOK II.
ihofe of X ; we Qiould then have the terms as they are fet in order in the
following Table.
I. II. III. IV. V.
-J- jv* + 4y[y ^ + ^yyyy + Ayy^ + y'^
— %ay^ — 2.i\ay'^y — T-'^ayyy — %ay'^
— i7axy'^ — i^axyy — izaxyy — iidxy*
4- iGaay'^ — iiayyx — CL^ayxy
-f ^"iaaxy + '^i.a^yy ^- i6aayy r = o.
+ ^aaxx + 4Sa'^yx + 4.Saaxjf
— 64.a^x + ^Sa'^xy + /^aaxx
+ Sa^xx
— 64.a^x
Now the fum of all thefe columns, excepting the firft, which is the propofed
equation itfelf, will be their complete and entire fluxion. But, becaiife the lafl:
or fifth column is infinitely little in refpecl of the fourth, and the fourth in
refpect of the third, and the third in rcfpeft of the fecond ; we affume the
fecond column alone for the fluxion of the propofed equation, which compen-
dium proceeds from the comm.on rule of differencing. But it can be fo only
when the columns after the fecond are abfolutely nothing. If, therefore, a cafe
(liall arife, in which the fecond column is abfolutely nothing, the third may not
be nothing in refpecl of ir, and therefore ought not to be omitted, but will
itfelf be the differential of the firft. And the fame m.iy be faid of the fourth,
when the fecond and third are nothing; and fo of the reft. But this cafe pre-
cifely happens, when we feek the relation of x lo y in the propofed equation,
in that point in which it is _y m la, and x zz: 2a ; becaufe, making the ne-
cefTiry iubflilutions, we find the fecond column itfelf to be nothing ; and
therefore we go on to make ufe of the third. And this is exadly the fame
thing as to ditf}:rence the equation twice, as appears from hence.
71. By the fame principles, and after the fame manner, a like cafe may be
relolved, which arifes in the conftruftion of curves, when the ordinate is ex-
preffed by a fradiion, the denominator and numerator of which become each
equal to nothing, when a determinate value is afligned to the abfcifs.
Now, to remove this difficulty, it is enough to confider the fradlion as if it
exprelled the ordinates of two curves, which meet in fome point of their com-
mon axis. And becaufe, in this point, their ratio cannot be exprefled otherwife
than by — , it is necellary to find what may be their ratio in a point infinitely
near it, that is, when they are increafed by an infinitefimal. That is to fay, we
muft proceed to differencing the numerator, and then the denominator of the
did fradion, and that once, twice, or oftener, till at laft, putting the deter-
minate value of the abfcifs in the fradion, it may no longer be — , for the
fame reafon mentioned before, concernijig the columns of differentials.
Let
SECT. II. ANALYTICAL INSTITUTIONS. 57
T 1 ' • 1 ^ la^x — X* — a& aax ,_, , . ' < ,
Let the equation be ^ z= ^ . Taking x •=: a, and mak-
ing the fubftitiition, it will be ;' zi — , from whence we cannot therefore infer,
that when the abfcifs x = a, the correfponding ordinate will he y •=. o. For,
by differencing the numerator, and then the denominator of the fradion, it
_4- _2.
• II 1 a^x — 2x^x X 2a'x — x*\ ^ — ^a^x x a ^x - rr>t j- • i* 1 i
Will be y = ^3 L:> ^ • Then, dividing both
— ^axxx X a "^x
above and below by x, and making x zz a, it will be j' =: \~a.
Let the equation be ^ zz — — , in which, it we put x — a,
tj 2aa-\-zxx — X — a
it will become J z= -2_. Wherefore, differencing, firft, the numerator, and
then the denominator of the fradion, it will be ;' = 4^'^--^ x 4a^ + 4^ ^^^^ -^^
2x X zaa-^-ixx] * — I
omitting x, which fhould be in both the numerator and the denominator. But
now, in this fradlion, if we put x -=: a, \t will be flill _y = ~ . Therefore,
proceeding to difference this fecond fradion alfo, we (hall have y =
— 5
^37-, omitting the x. And now, making ^ — ^, it will be
^aa X 2aa + 2xx\ *
y ~ 2a,
Vol. II. I SECT.
58
ANALYTICAL INSTITUTIONS.
BOOK ir.
SECT. III.
The Method of the Maxima and Minima of Quantities*
'i' $^'
't CD K
72. In any cnrvne whatever, whofe or-
dinates are parallel, if, the abfcifs BC
(Fig. 49^ 50, 51, 52,) continually in-
creafing, the ordinate CG Increafes alfo to
a certain point E, after which it decreafes,
or is no longer an ordinate of any kind ;
or, on the contrary, the abfcifs increafing,
the ordinate CG goes on continually de-
creafing to a certain point E, after which
it either increafes, or elfe is no more : In
this cafe, the ordinate EF is called a
Maximum or a Minimum^
In the curve GHF, let EF be the
greateft of the ordinates, (Fig. 49.) or the
lead, (Fig. 50.) taking any abfcifs BC,
and drawing the ordinate CG i let GA
be fuppofed to be a tangent at the point
G, and DH to be infinitely near to CG.
Make BC zi x^ CG = y, and drawing
GI parallel to BC, it will be GI = CD
~ Xy and IH = jK. Now, becaufe the
triangles ACG, GHT, are fimilar, in
Fig. 49, it will be AC . CG :: GI . IH.
And, becaufe the triangles ATG, GHI^
are fimilar, in Fig. 50, it will be AT . TG
:: GI , IH. This being fuppofed, let
the ordinate GC, being always parallel to
itfelf,.
K B. The letter A is omitted in Fig. 50.
SECT, in» ANALYTICAL INSTITUTIONS. 5<j
F^^i' 52. ' itfelf, be conceived to approach to the
greaiefl or leaft ordinate EF. It is plain,
that, as CG approaches to EF, the flib-
:S — A , — tansrent AC, or AT, will always become
greater and greater; (o that, when CG falls
upon EF, the tangent will become parallel
to BC, and confequently the fubtangent
will be infinite. Therefore, in this cafe,
we fhall have AC to CG, or AT to TG,
'••♦H an infinite ratio, CG dill remaining a finite
.■'^'
•^Hx
/•
^ ^^ ^ quantity. But, fmce it is always AC to
CG, or AT . TG :: GI . IH, GI to IH
will alfo have an infinite ratio. Therefore it will be as nothing in refpecfl of a\
that is, jj/ — o in the point of the greaieft or leaft ordinate.
Let the curve be GHF, (Fig. 51, 52.) EF the leaft of the ordinates,
(Fig. 51.) or the greateft (Fig. 52.); therefore, taking any abfcifs BC, and
drawing the ordinate CG, the tangent GA, DH infinitely near to CG, and
GI parallel to BC ; and making BC zr x, CG ~ j, it will be GI =2: CD — a',
IH zr y. Now, becaufe of the fiinilar triangles ACG, GUI, it will be
(Fig. 51.) AC . CG :: GI . IH ; and, becaufe of the fimilar triangles ATG,
GIH, it will be (Fig. 52.) AT . TG :: GI . IH. Now, .the- ordinate CG
always remaining parallel to itfelf, and continually approaching towards the
greateft or leaft ordinate, the fubtangent AC or AT will always become lefs
and lefs ; fo that, when CG falls upon EF, the tangent will become perpen-
dicular to BC, and confequently the fubtangent will be nothing. Therefore^
in this cafe, we ftiall have AC to CG, or AT to TG, in the ratio of nothing to
a finite quantity ; and therefore, GI to IH being in the fame ratio, x will, be
nothing in refped: of J/, that is, j/ — 00, in the point of the greateft or leaft
ordinate. Wherefore the general formula for the greateft and leaft ordinate
will be J/ iz o, or elfe j/ zi 00.
73. Therefore, the equation of the curve being given, of which we would
find the greateft or leaft ordinate, we muft difference it to find the value of the
fradion or ratio -4- ; then making the fuppofition of j z: o, or elfe of x — o,
that is, J/ — 00, we fhall have the value of the abfcifs x^ to which belongs the
greateft or leaft, y ; and this value, being fubftituted in the propofed equation,
will give us the greateft or leaft ordinate, as required. Only here we muft
obferve, that, in the cale of the fuppofition of j/ z: co, that is, of x zz o,
X will fupply the place of the ordinate j if in the other fuppofition, it is y that
does the fame. That, if neither the firft fuppofition of j/ z: o, nor the fecond
of J/ zz 00, will fupply us with any real value of j, it is then to be concluded,
that the propofed curve has no greateft or leaft ordinate,
1 2 . 74. This.
6o ANALYTICAL INSTITUTIONS. BOOK II,
74. This meihod will help us to acquire a complete and exaft idea of
curve-lines ; to find in what points the tangents are parallel to their conju-
gate axes, See. Befides which, it may be applied to an infinite number
of qiieftions, which we may want to have refolved, whether geometrical or
phyfical. Such it would be to inquire, among the infinite paralielopipeds of a
given folidity, which is that which has the leaft fuiface : as it would be to in-
quire, among the Infinite different ways along which a moving body may pafs,
to go from one point to another not in the fame vertical line, which is that which
may be defcribed in the O-iorteft time, according to fome given law of motion :
and many others of a like kind. In fuch queftions muft be found an analytical
exprefiion of what we would have to be a maximum, or a minimum^ which may
be put equal to y. Then taking the fluxion, we muft proceed according to the
rules here given.
EXAMPLE 1.
75. Let there be a curve with this equation lax — xx n yy, and let it be
required to know, to what point of the axis, or of the abfcifs x, the greateft
ordinate y correfponds, and what that ordinate is.
The fluxional equation of this will be lax — 2xx — 2yy, that is, -4- n
Making the fuppofition of j/ ~ o, the numerator of the fradlion ought
y
to be nothing, or a — x zz o, whence x zz a» Therefore the greateft ordinate
belongs to that abfcifs which is equal to a. This value being fubftituted inftead
of X in the propofed equation, it will be 2aa --^ aa zz yy, that is, jp z: + a.
Therefore the greateft ordinate, pofitive and negative, will be equal to a.
Making the fuppofition ofj/ zz 00, the denominator of the fradion ought to
be nothing, and therefore it will be y zz o. Wherefore, fubftituting this value
inftead of j in the propofed equation, we (hall have a; =z o, and x zz 2a ;.
which is as much as to fay, that ^ = o will be thp leaft, and x zz 2a the
greateft : Or, more properly, that, when x zz 6, and x zz 2^, then y being
infinite in refped of x, the fubtangent will be nothing, or the tangent will be
parallel 10 the ordinate j.
EX-
SECT. Iir. ANALYTICAL INSTITUTIONS. 6i
EXAMPLE II.
76. Let it be the curve of this equation xx — ax zz yy. By taking the
fluxions, it will be 4- = ^^-^^ , The fuppofition oi y :=. o gives here
X = \a. But this value being fubftituted inftead of .v in the propofcd equation,
y will be found imaginary ; {o that the curve has no ordinate correfponding to
iuch an abfcifs, and therefore much lefs will it have a greateft or a lead. The
fuppolition of J)/ n 00, that is, of .v =: o, will here give j = o: which de-
clares that the tangent will be perpendicular to the axis of the abfcifs x in the
point in which jy zz o ; which correfponds to the two abfcifl'es x = o, and
X zz a. For, inflead oi y, fubflituting o in the propofed equation, it will be
.Y.V '-— ax -=. Qy and therefore x ■=. o, and x iz. a.
EXAMPLE IIL
77. Let the curve belong to this equation laxy •=. a} . -\- axx — hxx^ in
which X is the abfcifs, and y the ordinate. By taking the fluxions, it will be
7.axy + layx zz laxx — ibxi', and therefore -^ — -, The fup-
pofition of J/ zz o gives X zz — ~ ; and this value being fubftituted in the
propoled equation, it will be -~^^ zz a} 4- ^ i -^ , that is, ^^ zz ^ X
a -^ b, and jy 1= ± \/aa w. ab, the greateft or leaft ordinate. And, fince we
have X zz -^, fubftitutino; this in the value of r, it will b^ x — -^ - ^^^ ■ • .
the abfcifs, to which belongs the greateft or leaft ordinate now found. The
fuppofition of J/ zz cx>, or x = o, gives us ax = o, that is, x -zz o. And
making the fubftitution in the propofed equation, it will be ^^ — o ; which
implies that a given finite quantity is as nothing : fo that the curve will .have
no other maxima or minima but thofe found in the firft fuppofition, which, be-
caufe of the ambiguity of the figns, are two, and thofe equal ; one of which is
pofitive, and correfponds to the pofitiveabfcilles, the other negative, and belongs
tg the negative abfcifs.
4 78. This
62
A.^ALYTICAL INSTITUTIONS,
SQOK II,
^8. This method, indeed, gives us the maxima and minima^ but ambiguoufly
^nd indifcriminately ; nor by this can we diftinguifh one from the other. But
they become known when the .progrefs of the curve is known. But, without
fuch knowledge, we may proceed after this manner. Let there be affigned a
value 10 the abfcils in the given -equation, which is either a little greater or a
little lels than that which anfwers to the greatefl or leaft ordinate with which we
are concerned, and the value of the ordinate which arifes from thence will
determine the queftion. For, if it Qiall be greater than that which the method
difcovers, the queftion is about a minimum ; but, being lefs than that, the
queftion is about a maxi'mum. Therefore the curve of this Example will have
two leall ordinates.
EXAMPLE IV.
^ig* 63*
79. Let the curve MADEAN belong
to this equation x^ + y^ =. axy ; make
AB = Xj and BE => By differencing,
we fliall have ~ ii: ~ — ; and there-
X -i^yy — ax
fore, making the fuppofition of j/ z: o,
it will be ^ n: ~ . Then fubftituting
this value in the equation, we fhall find
X = ^a^i. Wherefore, fince y == ^,
it will be ^ = ^a^4. — BE, the greateft
ordinate in the curve, which correfponds to the abfcifs x =■ ^a^i = AB.
The fuppofition of ^ = o will give us x = — , and making the fubftitution
in the given equation, it will be j = ^a4^2, whence x = t^^4, the greateft
AC, to which correfponds y = CD = ^a'^z, which is the tangent in the
point D.
So. But, before wc proceed to more Examples, it will be convenient to
provide for a cafe, which fometimes is wont to happen ; and that is, that as
well the fuppofition of j/ =r o, as that of j/ = 00, will furnilh the fame value
of the ordinate, or of the abfcifs j in which cafe, no maximum or minimum will
be determined, but only a point of interfe^ion or the meeting of two branche*^
of
SECT. irj. ANALYTICAL INSTITUTIONS. 63
of the curve. And the reafon of this is plain ; forafmuch as, -4- being equ:il
to a frac^lion, if from the numerator we derive the fame value of at, for example,
as from the denominator, this value or root being fubftituted, will make each of
them equal to nothing, and therefore in fuch a point of the curve it will be
^ n: — . But it has been already fliown before, at ^ 69, that when
4- = — , It always indicates the meeting of two branches of the curve*.
Therefore, &c.
EXAMPLE V.
81. Let the curve GEM (Fig. 51.) be the cubic parabola with the equation
y — ^ z: 4^a^ — 2aax + axxy BE r= EF =: a, BC — x, CG =jK' Taking
the fluxions, ic will be 4- = .^ » The fuppofition of y zz o
•f J. X a* — 2aax + axx] i
will give US x = a, and the fuppofition of j/ = 00 will give, in like manner,
X zz a. Therefore the curve has a point of interfedion F, which correfponds
to ihe abfcifs x zz a, and to the leaft ordinate y n: a ; which is derived from
the propofed equation, by fubftituting it's value in the place of x.
Let us take the fame equation, but freed from radicals, that is, y — 3<2y*
+ S^'^y — a^ = a^ — laax + axx. By taking the fluxions, it will be -4- r=
— ^^^ '- ^^^ — ^ xhe fuppofition of y = o will give x = a, and putting this
3yy - (>^y + 3^^ if j b ^ t^ b
value in the propofed equation, we have y zz. a. The fuppofition of y = 00
will alfo give y — a, and therefore x ^= a ; and y ■=: a gives us the point F,
which is a point of meeting or contad: of the two branches GF, FM, and, at
the fame time^ the lead ordinate y.
But, if we {hould operate upon the equation y* — a ~ a^ X a ^x)^j whiclt
exprefles the branch GF alone, (the other branch FM would be exprefled by
^ — 2a^
y ^ a ZZ ai X x — a] ^,) we fhould have -7- iz — =r-. • The fuppofition
of j) zz o, informs us of nothing. The fuppofition of j/ — 00 gives us x zz a,.
and therefore y zz a. And the point F, in this cafe, fupplies us with a,
maximum in refpedt of x, and a minimum in refped oi y,.
-8u 8^,L
64
ANALYTICAL INSTITUTIONS.
BOOK II.
82. I faid that the fuppofition of j/ = o, which here gives la^ ~ o, informs
us of nothing, meaning in refped ot finite maxima ; for, taking in the infinite
alfo, it fiipplies us with two of them. If la^ rz o, it will be then x zz o ;
and fubftituting this vahie in the propofed equation, it will be — — '^xx, that
is, ;? iz + -/— ; and therefore x and jy are infinite. The maxima are two, one
belonging to the branch FG, die other to the branch FM ; for, putting a zz o,
the equations exprefs them both.
This cafe will generally arife, as often as the fuppofition of j)/ zz o, or of
J/ zn CO, exhibits a conftant finite expreffion, or a conftant divifor, to be equal
to nothing ; which value, being fiibftituted in the propofed equation, does not
bring us to an imaginary quantity, or to a contradiflion. And the reafon of it
is this, becaufe a finite quantity cannot be taken for nothing, but only in refped
of an infinite quantity.
EXAMPLE VI.
Fig. 54.
83. Let the curve belong to the equa-
tion X* — 2ax^ + aaxx zz y'^. Make
AB = ay AC or AP z: x, CM or PM
= y. Taking the fluxions, it will be
y ___ 4Jif' — 6ax'^ -f- laax
4_y3
7ji-
The fuppofi-
tion of j/ =z o will give us three values
of x^ that is, X zz Oy x zz a, x zz -fa.
The value x zz o, being fubflituted in
the propofed equation, makes j = o.
I ' -.. The value x =■ a, makes y = o. The
value X zz ^a, makes ^ zz ± 4^, The
fuppofition of J/ zi CO gives us jy = o ; fo that y has the fame value in both
the fuppofitions, when ;^ = o and x :zz y. Whence the points A, B, will be
points of meeting of the branches of the curve, and x -zi -^a zz AC will give
the greateft ordinate ^ = ± 4^ =: CM, or Cm. The locus of the foregoing
Example may be called a double locus ^ which arifes from one or other of the
two fimple formulas, [ax — xx — yy, to the circle, and xx — ax zz yy to the
hyperbola,) being raifed to it*s.fquare. Whence it would not be fufficient to
reduce the equation to a firnple circle, or to a fimple hyperbola ; but it will be
necelfary to have a view to the complication of the two loci or curves with each
other.
EX-
SECT* III.
ANALYTICAL INSTITUTIONS.
65
EXAMPLE VII.
Fig' 55
84. Let it be the curve of Fig. 55, the
equation or which is yy — 2a — Z *
Make AP = ;^, PM = jy, AD zi 2a,
The fluxions will be 4-zz 1^-— ^=- ;
X y X 2a — xr
Before
, . v a^ — A.aax + ±axx — at'
that IS, 4- — =-^ - 7.
•*■ a^x X i^x X 2a— A^*
I proceed, I fliall here obferve that both
the numerator and the denominator of
the fraction are divifible by a •-' a \ there-
fore, in the fuppofition of j) iz: o, and in that of j/ 1= 00, we fliall have
tf — .V n o, or a: =: tf. And this, being fubftituted, will give jy = o, and
therefore the curve will have a node in the axis at the point B,. making AB =s j.
Therefore, making the divifion, it will be 4- = ga — lax ^ rpj^^ ^^^^
pofition oi y — o will give x -zi ^^-=^-^ . But the value x =: 3^ + ^v/?
cannot be of ufe, becaufe, being fubftituted in the propofed equation, it makes
the ordinate imaginary; and this, in general, is imaginary, when x is affumed
greater than 2^, as may be plainly feen. Wherefore, fubftituting the other
value, .V
2 ' o ^ — ^ a + a-v/5
Making, then, AP z=
, PM, Pm, will be the greateft ordinates, one pofitive, the other
negative j as above.
The fuppofition of j zz .00 will give x zi o, and a; =z 2a. Thefe values
being fubftituted in the propofed equation, v/e fhall have y = o, and y zz or;
that is, taking ;v ~ o, or in the pomt A, the tangent will be parallel to the
ordinate PM. And taking x :=: la zz AD, the ordinate will be infinite, that
is, will become an alymptote to the curve, in refpeCt of the branches
BH, BL
N. B. By miftake of the Wood Cutter, a Roman M has been put in the lower part of Fig. 5^,
irili:tad of an Italic ;«.
Vol. II.
K
EX-
M
ANAtYTXCAL 1 N 8 T I T U T X N S..
BOOK IX«
EXAMPLE VIIL
Fig. 56.
85. Let the curve be the eonchoid with the equation yy zr
caxx ~ x^ + 2aahx ~ zbx^ - bhxx + aabh ^ ^^j^i^ ^^^ fluxions, it will be 4- =r '
^ X*- bx^ - aah x - aahb j^^ y^j^ j^ ^ ^^^^ j j^^^g already
confidered three cafes of this curve. The firft is, when a •= b. The fecond^.
when b is lefs than a. The third, when b
is greater than a. As to the firft cafe, the
curve will be that of Fig. 56, and the equa-
fl* + ta^x — 2ax^ — X* A/r 1 •
tion yy zi — . Making
GA = GP = ^, GE zz X, EM = y ;,
R and, taking the fluxions, it is -^ =n
— Thelupponrioa
i xxy a^ + aa^iv — 2fljf* — .**
oi y z=i o will give the numerator equal to-
nothing, that is, ;c H- ^ x ^^ + «' =: o ^
and therefore x :=: '- a, which value, fub-
fiituted in the cqiiatibn of the curve, gives y zz o. The fuppofition of j/ = 00-
gives the denominator equal to notliing, that is, xx^x t a' x aa ^ xx = o,,
and therefore x zn o, x zz — a, and x ~ a. But the value x •=: -^ a was
alfo found in the ftippofition of j/ ~ o. Therefore, when it is at z= — a, that
is, taking GP zz a^ the curve will have a point P, where two branches meet
each other.
The value x — a, being fubftituted in the equation,, will give us j' = o ;■
and therefore the fame x will be — a zz GA, to which correfponds y = 0..
The value x zz o, being fubfliituted, will give y zz co. Therefore, through
the point G, where ^ = o, if a line be drawn parallel to the ordinates, it will
iQuch the curve at an infinite diftance, that is, ic will be an afymptote.
Af
SECT. Ill,
F^S' 57'
ANALYTICAL INSTITITTIOMS.
67
As to the other two cafes, Fig. 57, 58.
Let GA = GK = a, GP = ^, and the
reft as above. The fuppolition of j/ = o
will give — X* ^ hx^' — aabx — aabb
= o ; that is, ;v i- ^ x — x^ — aab zr o,
and therefore x m •-• by x -sz ^ — aab.
The fuppofition of jj/ = 00, will give
xxs/a'x'- — ;«* + za'-bx — ibx^ — b'-x* -j- a'b'^
=: o, that is, xx\/ x-\-S^'^ X aa — xx — o^
and thence x :=: o,
X ZZ -" by
O
J)Z/'0
T?
•/K
The value x = —• b^ which is the
fecond cafe, being fubftituted in the
equation, makes j zz o, and is exhi-
bited by both the fuppoficions. Therefore
(Fig. 57.) taking GP on the negative
fide, and equal to — b, the point P (hall
be a meeting or an interfedion of two
branches of the curve. The fame value
A? = — ^, bping fubftituted in the equa-
tion of the curve H- y = \^aa-^xx,
in the third cafe, gives the radical nega-
live, becaufe of b greater than a, and therefore the curve is imaginary, and of
no ufe.
The value x = ^ — aab, fubftituted in the equation of the curve, gives us
,aa — bb X %/ alb + '^ahi^ — aab + labb
y-±s/'
yabb
which is therefore imaginary
when b is greater than «, (Fig. 58.) and therefore, in Uke manner, ferves to no
purpofe in this third cafe. But it gives y real when b is lefs than a ; and there-
fore, (Fig. 57.) making Q\ •=. If — aab^ IN will be the greatefl ordinate,
or jy as above. The value x zz o here gives jv n co, that is, an afymptote.
The value ;^ =: + « gives ^ = o ; that is, the tangent in the points A, K, is
Darallel to the ordinate.
K2
EX^
68
ANALYTICAL INSTITUTIONS.
BOOK II.
EXAMPLE IX.
86. Let AMF be half the contrafted
cycloid. Make AB = 2^, BF r: b,
p AP = .r, PM n z, the femipcriphery
p ANB — Cy the arch AN — j i it will be
j£ PN ~ -^zax—xx^ NM n z — ^lax—'XXi
and, by the property of the curve, it is
ANB . BF :: AN . NM ; that \%,c,b ::
E
z
^ . NM =: -^ . Therefore -^ =
c
Z —
\/2ax — xx. By differencing, it is -^ —
/ ■ ' ■ ■ . Now, drawing mp infinitely near to MP, it will be N» zi
Whence, making the fubftitution in the equation, we (hall
\^ 2ax — .
have -r- =z
ai
X __^ ab -\- ac — ex
cV]
The fuppofition of i; zz o will give here x =
-^: — 1- ^. Therefore, if H be the centre of a circle, taking HE equal to the
fourth proportional of the femiperiphery ANB, of the right line BF, and of
the radius ; the correfponding ordinate will be the greateft, as was required.
The fuppofition of z =: co gives us ^^ = o, and x zz la ; which is as much
as to fay, that, in the points A, F, the tangent will be parallel to the ordinates*
PROBLEM I.
Fig. 60,
87. A redangle ADCB being given, the leaft right
A line QH is required, which can be drawn through the
point C in the angle QAH.
Make AB = «, EC = b, BH = ;tf; it will be
CH rz \/bb + xx ; and, becaufe of the fimilar triangles
HBC, HAQi we Qiall have HB . HC :: HA . HQj
X -]- a
that is, X , ^bb + xx :: x + a , HQ^iz v^bb + xx.
Wherefore,
SECT. Ill, ANALYTICAL INSTITUTIONS. 69
Wherefore, fuppofing HQ^zi y^ as if it were the ordinate of a curve, we fhall
have y zr ^^ — -y/bb + izx., and, by difFerencinff, it will be -4--r: — : — ^
The fuppofition of j i= o will give x — -^^^^ ; and therefore, making
BH =: ^ abbt and drawing HCC^ it will be the leaft line, as required. The
fuppofition of j = CO will give x =. y/ — bb^ and x zz o, which anfwers no
purpofe ; it not being meant that the right line drawn through the point C,
which, in this cafe, would be BC infinitely produced, fliould be a maximum^
for that reafon becaufe infinite. Wherefore, in fuch cafes as thefe, it will be
fufEcient to difference that expreffion, which we would have to be a maximum or
minimum^ and afterwards to fuppofe the numerator equal to nothing, and then
the denominator.
PROBLEM 11.
•pi„ 5 1, 88. The right line AB being divided
'. into three given parts, AC, CF, FB, the
X C E if li point E is required, in which the middle
portion CF is to be divided, fo that the
reftangle AE x EB to the redangle CE x EF, may have the lead poffible
ratio.
Make AC = ^, CF = ^, CB = c, and CE = jf i then AE = ^z + x^
EB = c — ^, EF = b — x; and therefore the ratio will be ^^ ^ ^^ =
ac Ar ex - ax - xx ^j^|^j^ ^^^ be a minimum. The fluxion, therefore, will be
bx — XX
cxx-axx-bxx+2acx-abc ^ ^ ^ ^^^^ making the numerator equal to nothing, we
bx — xx)
Ihall have x = " '" ^ ^""^'^ "" ' - ' " ^'^' '^ ""' . One of the values is pofi-
c — b — a
tive, which gives the point required,* E, from C towards B. The other is
negative, which would give us the point E, from C towards A. Making the
denominator equal to nothing, we iLall have x — q» and x =: b, in which two
cafes the ratio of the redtangles will be a maximum ; for, taking * zr o, the
point E falls in C ; and takiug x — b, the point E falls in F ; and therefore,
in each cafe, the re<5langle CE x EF is nothing.
PRO-
70 ANALYTICAL INSTITlTTIOilS. BOOK XU
PROBLEM III.
89. The given right line AB is to be fo cut in the point C, as that the
product AC^ X CB (hall be the greatefl: of all fuch produds.
Make AB = <3, AC = x, then CB =: a — at. Therefore ACq x CB =:
axx — x^. The differential will be 2.axx — ^^xx, which, compared to nothing,
will give X = 4^, and ^ =r o. Wherefore, taking AC zz x :=: ^a, the pro-
duft will be the greatefl: poffible ; and taking a; = o, the produd: will be a
kind of mimmum, becaufe it will be nothing, the point C falling in A. The
differential not being a fradion, the other ufual fuppofition cannot take place,
of the denominator being made equal to nothing. But if we will confider the
expreffion of the produd axx — x^ as an ordinate of a curve, by the laws of
homogeneity that produd may be divided by a conftant plane, and thus the
differential will be a fra(5lion vvith a confl:ant denominator. But that conftant
quantity can never be nothing, but only relatively in refped of ;v being affumed
infinite ; and furely then the produd muft be a maximum, when it is AC =: x
s=; CO.
I faid that the produd AC^ x CB is a maximum, when it is AC = -1^ ;
which will be plainly fecn by defcribing the curve of the equation ^"^^ '~^- :z.yi
For all the ordinates between A and B are lefs than that which correfponds to
the abfcifs x •=z ^a. The other value, at = o, being fubftituted, it will be
y zz o, from whence it may be concluded, that this value will be of no ufe.
90, In the foregoing Problem, and in all others of a like nature, this method
may be made ufe of to difcover, whether the queftions propofed are concerning
a maximum or a minimum.
PROBLEM IV.
91. Among all the parallelopipeds that are equal to a given cube, and of
which one fide is given ; it is required to find that which has the leaft furface.
Let the given cube be «*, apd the known fide of the parallelopiped = h
Let one of the fides fought be x, and then the third will be ~ , becaufe the
p produ<^
SECT. ni. ANALYTICAL INSTITUTIONS. ft
produ(fl of the three makes the given cube a^. The produ(5ls of the fides,
taken two and two, that is, hx^ — , and —r- , form the three planes which are
half the fuperficies of the parallelopiped, and therefore the fum. of thefe, that
^ l;x + — — \- -f- i n^"ft b^ ^^^ minimum required. Therefore, taking the
JS,
fluxions, we fliall have bx —^t or -^ ^ " " x. The fuppofition of the nu-
werator equar to nothing gives x =: V-t-. Therefore the three fides of the
required parallelopiped will be h, ^-r-"* ^"^ — ~~r > °^ ^"T" • Therefore-
the two fides required will be equal. The fuppofition of the denominator,
being equal to nothing, ferves to no purpofe ; for then x = o^, which contra*-
diets the Problem.
If we would have a parallelopiped with the conditions affigned, but without
afluming any fide as given ; making one fide = x, the two others will be equal,
and each = -/ ~. The fum of the three fides or planes, which is to be a
minimum, will be zxV— — | , which, by differencing, is -^ — ; ot
x^ —
X.
a'
a^xx — c^x ^ — -
t^us, 1 — ^. Here, making the numerator equal to nothing, we
XX J
X
(hall have x :^ a, and, in like manner, the other two fides will be = ^ •
lb that the cube itfelf will be ihe parallelopiped required..
PROBLEM V.
92. Among the infinite cone5 that may be infcriHed in a fphere, to determine
that whole convex fuperficies is the greateit j the bafe being excluded.
Itt
7^
ANALYTICAL INSTITUTION Si
BOOK II.
Fi^. 62. E j^ In the femicircle ABD let there be the
triangles ABC, AEH, and let a femicircle
revolve about it's diameter AD. At the
fame time that it defcribes a fphere, the
triangles will defcribe fo many cones. But,
as it is demonftrated by Archimedes, that the
fuperficies of the inlcribed cones will be to
each other as the rectangles AE x EH,
AB X BC ; the queftion is reduced 10 this,
to determine fuch a point C in the diameter AD, that the produd AB x BC
may be a maximum.
Therefore make AC = x, AD = ^ j by the property of the circle, it will
be CB = s^'^x
XX, AB := V ax, and AB x BC = \/ax x \^ax —
= ^aaxx — axK Therefore, taking the fluxions, we fliall have ifffiz|£^.
And making the numerator equal to nothing, it will be a; == ^a^ and ;; — o.
Making the denominator = o, it will be ^ — 5, and ^ z: o. Taking, there-
fore, AC — jAD, the fuperficies of the cone defcribed by the triangle ABC
will be the greateft, as required. The other two values x :=: o, and x ^=: a,
can be of no ufe in this Problem, as is evident.
PROBLEM VI.
Fig, 63.
93. The angle FDG being given, and
the point A being given in pofition, to
find the leaft right line, which, in the
given angle, can pafs through the point A.
Let CB be the line required, and let
AQ^ be drawn perpendicular to FD, FAP
perpendicular to DG, and CK perpendi-
cular to FP. Becaufe the angle FDG is
given, and the angle FPD is a right one,
the angle AFQ^will be known. But the
point A is alfo given in pofition ; then
the lines QA, QF, FA, QD, will alfo be
QF = a, QA =1 c, QP = ^, and QC = .v.
Therefore it will be FA =z \/aa -^ cc, CA ss >/^^ -f- xx, FD zz b -{■ a, and
FC = « — X, But, becaufe of fmiilar triangles FAQ»^ FDP, it will be
FA
/ 1^ H/
known. Therefore make
?ECT. tl r; ANALYTICAL INSTITUTIONS^ 7^
FA . FQ^:: FD . FP. Wherefore FP n ^!_il!., , and AP - ""^ " '"
"^ aa + cc */ aa + cc
Now, becaiife of fimilar triangles ACK, ABP, it will be AK . CA :: AP . AB.
Therefore AB rr *^ ~ ^^ ^ — - — —, and thence CB = \/<,c + xx 4-
tx + ax ^ VII
— : — s/cc + XX, which is to be a minmum. Therefore, takino; the fluxions.
ec -^ ax ' D »
• II , XX , XX X ab — cc X cc ■\- ax — ax X tf' — cc X cc -i xx . ,
It Will be + — ; . And,
'^CC •\- XX cc ■\- ax]^ X cc + XX i
putting the numerator = o, (firft reducing to a common denominator,) it will
be x^ + -^ -{ — ^ + — — . ifc^ rr o, which is a folid, equation.
a a a ^
To conftrud it, I take the equation to the parabola xx = ay_; making the
fubftitution, it will be xy + ~ -\ — ^ + ^ ~ o, a locus to the
hyperbola between it*s afymptotes.
'This fuppofed, on the right line QD is taken QM — — , and drawing the
right line MN =: — from the point M, and parallel to AQ» NS is drawn
parallel to QD, and between the afymptotes NS, NT, the hyperbola HOV is
defcribed with the conftant redangle ^— ~^LlI-ff- . And, on the right line
<JF, from the point Q^let the x's be taken, and the ^'s perpendicular to them.
Then, with the axis AQj^ vertex Q^ and parameter z=z a, let the parabola QO
of the equation xx zz. ay be defcribed. From the point O, in which the
parabola cuts the hyperbola, let OC be drawn parallel to AQj and from the
point C let the right line CAB be drawn through the point A. This flull be
■the minimum required.
And, indeed, by the conftruclion, it is NS = a? + — , SO = y -f ~ ,
And, by the property of the hyperbola, it ought to be NS x SO, equal to
the conftant redangle. Therefore ^9, + if2 + ffl^* + i^ :z ^±l±^,zffl.
y^ fubftituting this value, we fhall have — + ^^^ +
But CO = ^ = ~ , by the property of the parabola. Therefore, inftead of
:' Zccxx hccx Ice c* *
a aa aa a aa ^
that is. A?' + — ^ 4- iff -J- -f hcc z=z o^ which is the very equation from
-whence the value of x was to be derived. Therefore, &c.
Vol. II, L I have
74
ANALYTICAL INSTITUTIONS.
BOOK ir.
I have here made the fuppofiiion, that the numerator of the fraflion, which
cxprefles the mhiimum, is to be nothing. The other fuppofition, that the deno-
minator muft be nothing, will give cc t- ax\^ x */ cc -\r xx rzr o, that is,
*/ cc + XX =z o, cc •\- ax 7Z. o. But \/^£c -j- xx = o gives us x ■=. a/ — cc,
which is imaginary, and therefore of no ufe. cc -|- tf;c = o gives us ;? = — — ,
Bur, taking Q£ = a: = , and drawing he, the triangle QA^- will be
fimilar to the triangle QFA, or PFD, and therefore the an2;le QfA will be
equal to the angle FDP. Whence ck will be parallel to DP j which is as
much as to fay, that a line drawn from the point Cy and through the point A in
the given angle FDG, will be infinite, which is a kind of maximum.
It may be (hown ftill in a (horter manner, that the right line here fought will
be infinite. For, in the expreffion \^cc + xx +
ah — cc
s/cc -h XX =• CB^
cc -j- ax
inftead of Xy if we fubftitute it*s value — — , the denominator becomes no-
thing, and therefore the line is infinite.
SECT. IV.
0/ Points of Contrary Flexure^ and of Regreffion,
Fig. 64.
94. In Seift. VI. Vol. I. it has been faid already, what are Contrary Flexures
and Regreflions of Curves. Suppofing, therefore, that ta be already known,
let ADEM be a curve whofe ordinates are
parallel, and which in E has a contrary
flexure or regreffion. Taking any abfcifs,
AB = *, and it's ordinate BD =^, and
drawing CF parallel and indefinitely near to
BD ; It is plain, that, aflliming x = BC as
conftant, that, as the abfcifs AB = x conti-
nually increafes, the fluxion GF of the ordi-
9 nate
3ECT. IV.
ANALYTICAL INSTITUTIONS.
7S
nate BD, that is, y, will always become lefs and lefs, till the ordinate becomes
HE, which correfponds to the point of contrary flexure or of regrelTion : after
which point, in both cafes, the fluxion jf will go on continua'lly increafing.
Therefore, in the point of contrary flexure cr rcgreflion, y will be' a minimum.
Whence, by the M-thod of Maxima and Minima^ y - o, ox elfe y z=z oo, will
be the formula of contrary flexure or regreflion.
Fig. 6
If the curve fliall be firfl convex, and
afterwards concave to the axis AH; the
abfcifs increafing continually, the fluxion or
difference of the ordinate will increafe to the
point E of contrary flexure or regrefiion,
after which it will go on decreaiing. There-
fore, in this point, y is a maximum, and, fir
that reafon, we may put y = o, or elfe
y zzz 00.
The fame thing may alfo be inferred from this confideratlon, that, in a curve
firfl; concave towards it's axis, the fecond fluxion of the ordinate j, that is, y,
is negative to the point E of regreflion or contrary flexure, after which it
becomes pofitive. And, in curves that are firfl convex, that fecond fluxion is
pofitive as far as the point E, after which it becomes negative. But no quantity
from pofitive can become negative, or from negative can become pofitive, but
it mufl pafs through eitlier nothing or infinite. Therefore, in the point E of
regreflion or contrary flexure, it ought to be y zz o, or elfe j' = oo.
Let the right line DT (Fig. 64.) be a tangent in the point D to the curve
AEM, which is firft concave towards the axis; and alfo, the right line EP af
the point E. As the abfcifs AB increafes, the line AT, intercepted between
the tangent and the origin of the abfcifs will always increafe fo far till the point
B falls in H, after which, in the cafe of contrary flexure, the abfcifs (till in-
creafing, that intercepted line will decreafe. Therefore, in the point E of
contrary flexure, that intercepted line AP n 4^ — x ought to be a maximum.
Wherefore, by differencing, taking x for conflant, it will be -^ZlSlIl-^—ZJ^ ^
equal to nothing, or to infinite ; that is, by reducing, and dividing by — yx,
and multiplying by yy, it will be, finally, y =: o, or y =r 00. In cafe that
the point E be a pomt of regrefl~ion, if the intercepted line AT increafe, the
abfcifs AB will alfo increafe, till the point T falls in P, and the abfcifs fhall be
AH ; beyond which point T the abfcifs will go on decreafing. Therefore AH
will be a maximum^ and it*s difference will be equal to nothing, or infinite.
Therefore, relatively to fuch a difference, the difference of AP will be infinite,
or nothing. Therefore y = cc, or y zz o, as before.
L2 If
ANALYTICAL INSTITtJTIONS. BOOK 11.
If the curve be firfl: convex to the axis,
the intercepted hne AT will be zz x — ~,
jj C P H
and the difFerence
xyy — x' V •\- yx'v
that Is,
-~ •, and therefore, dividing by yx, and
yy
multiplying by yy^ we (hall have neither
more nor lefs than y zz. o, or elfe j/ ~ oo.
In the curve DEM, the origin of the
abfciffes x being A, and E the point of con-
trary flexnre, the intercepted line AP will be
equal to AH + HP. But, in this cafe, the
fubtansfent HP is negative, that is, — 4- .
jf
Therefore it will be AP — x — ^-. Hence
y
we fee, that in no cafe the intercepted line
AP can be A? + ^ ..
gc^. The formula here found will ferve for curves which have parallel ordi-
nates, or fuch as are referred to an axis or diameter. But it is different in
curves that are referred to a focus.
Let the curve be ADE, (Fig. 67, 6%.)
it's focus Q> ^rom whence the ordinates QD
proceed; and let Q^ be infinitely near to
op. Draw QT perpendicular to QD, and
and Q/ perpendicular to Q^. Draw DT a
tangent to the curve in the point D, and df
a tangent in the point d. Let Q/ (pro--
diiced if need be,) meet DT in the point 0.
Now it is plain, that, as the ordinates in-
creafe, if the curve be concave towards the
focus Q»^ (fig* 67.) Q£ will be greater
than QJ'. But, if the curve be convex
towards the focus Qj, (Fig, 68.) Q/ will be
lefs than QT. Therefore, as the curve
changes from being concave to convex, or
vice verfd^ that is, in the point of contrary
flexure or regrefiion, the line or quantity <?/,
from .
Fig, 68.
SECT. IV.
ANALYTICAL INSTITUTIONS'.
77
from being pofitive, ought to become negative, or the contrary, and there-
fore mud pafs through nothing or infinite.
Wherefore, make QD — y, DM zi x, and with centre Q^let the infini-
tefimal arches DM, TH, be defcribed. The two triangles fi'MD, dC^J , will
be fimilar, as alfo, ^Q^, THo, and therefore it will be ^M . MD :: dQ^ (or
DQ^) . QT. That is, / . x :: y . QJ = ^ , But the two ibdors DQM,
TQH, are alfo fimilar j whence QD . DM :: QT . TH. That is, y . .v ::
And, becaufe of the fimilar triangles dQ^, THo, it will be
J
i^.THr:
dQ^ioT DQ) . Q£ (or QT) :: TH . Ho. That is, y . 4- :: ^ . HO=4i..
But H/ (Fig. 67.) is the difference of QT, that is, H/ zz ^-'^ ,-7'^"'"^ ? takings
for conftant. Therefore fo = /H + H^ ~
xj[y — yxY 4- .T-
.}jy
yj
which mufl be
equal to o, or to co. And therefore, alfo, mukiplying by j/j, and dividing
by X, it will be yy — yy + xx^ equal to nothing, or infinite.
In Fia". 68, the line ot becomes negative, andtherefore iz "" '^"-^ — 15UZJL ^
..... ^^
Therefore, dividing by — x, and multiplying by jy, it will be xx + yy —• yy
equal to o, or to oc.
Fig. 69.
"Wherefore, if any curve be referred to a focus Q,
whofe ordinates are QB ~ y, and the little arches
BC = x^. and fliall have a contrary flexure or re-
grefiTion ; the general formula to determine it will be
yy + XX — yy = o, or = 00.
Here, if we fuppofe y infinite, the two firit terms-
of the formula will be nothing in lefpedt of the third,
and therefore it will be — yy^ equal to nothing, or
infinity ; and dividing by — y^ we (ball have y = 0,.
or y = 00 ; which is the formula of the fiifl cafe of
curves referred to a diameter, as it ought to be. For, fup.poling y infinite,
the ordinates become parallel to one another,
96. The nature of a curve being given by means of an equation, and x
being fuppoled confiant ; by differencing twice, if the curve be algebraical, or
once, if it be a differential of the firft degree, that we may have the value of J/
expreffed by x ; this, compared to o or 00, will give ihofe values of the abfcifs
.V, to which will correfpond that ordinate y, which meeis the curve in the points
of contrary flexure or regrelTion. Wherefore, if thole values be fubditutcd in
the.
7*
ANALYTICAL INSTITUTXOSS.
BOOK !I.
the equation of the curve inftead of x, we dial I have y either real or imaginary.
If y be imaginary, or fhall involve a con trad iiStion, then the curve will have no
fuch points,
97. To diftinguifli the points of contrary flexure from thofe of regreffion,
becaufe this method giv\;3 us each of them indiicriminately, it will be fufficient
to fee the progrefs of the curve, by taking an ordinate very near. And this
will afford light enough to remove any doubt about it.
98. Curves may have another kind of rtgreffion, different from this which
has been confidered^ And that is, when the curve returns backwards towards
ii*s origin, turning it's cavity the fame way as it did before it's regreflion. After
I have firft treated on the Radii of Curvature, I fliall give a genera! formula,
alfo, for regrefTions of this fecond fort, at the end of the following Sedtlon.
EXAMPLE I.
Ffg^S^'
"^
\
r
,--'M
,
/'
\
F \.
-" B
C
:i
> }
4
A
99. Let there be a cubic parabola with
the equation y =: a •}- ^a^ — 2aax + axxf
which, in § 81, has been found to have
a point of inter fetlion. Now, by difFer-
.,, , . — 2aax + 2axx
encmg, it will be y = — ^, ,
and differencing again, taking x conftant.
it will be j/ = —
2axx
The
9 X a^ — 2aax+axx^y
fuppofition of j' ~ o will give us — zaxx
^sz o, which is of no ufe ; making, therefore, the fuppofition of j' = oc, it will
be 9 X a^ — 2.aax -r ^xx\ "^ = o, that is, aa ■— 2ax + xx = Oj and therefore
X :=. a. This value being fubftituted inftead of x in the propofed equation, it
will he y ^ a, and therefore the curve has a contrary flexure, or regreffion,
which correfponds to the abfcifs x == a, to which belongs the ordinate y = a.
And, becaufe we know otherwife, that this is alfo a point of interfedion; it
cannot therefore be a point of contrary flexure, but muft be a regreffion.
In
ANALYTICAL INSTITUTIONS,
79
In the fame cubic parabola, taking the
abfcifs AB = x from the vertex A, and
the ordinate BC =7; the equation is axx=y^,
the fluxion of which is zaxx = SXvy. And
taking the fluxions again, making x con-
ftant, it will be y =
— dyyy + 2axx
m
But, by
the equation, it is ^yy = -^x^aax^ and, by
the firft differencing, y = — t, — • There-
7^x1^ aax
fore, making the fubftitutions, it will be y = -^
— 2axx
aax
The fuppofition of j^ = o has no ufe. The fuppofition of jy = 00 will give
^X'¥aax = o, that is, x = o \ which value, being fubftituted in the equation,
gives ^ = o. Therefore the curve has a regreffion at the vertex A,
EXAMPLE II.
TOO. Let the curve be DFM, commonly
called the Witch, the equation of which is
y = tfv/^-=^, AB = :v, BF = y, AD = ai
by differencing, J/ = — — - — ; and
ixw ax — XX
taking x conftant, and differencing again, it
• 11 I .. Xa^xx — AAaxx'x
will be y = ^ — -- — , , .
4* X ax—xx^i
The fuppofition of y = o will give 3a' — 4<z^v = o, that is, x = |^';
which value, being fubftituted in the equation of the curve, gives y ~ as/ ^»
Whence, taking AB = \a^ the ordinate BF = « V4- will meet the curve in the
point F, which will be a contrary flexure. The fuppofition of y =z 00 gives
us 4* X ax — xx^^ = o, that is, ^ = o, and x = a. The firft value fubfti-
tuted in the equation makes J' = 00, the fecond, ^ = o. But neither the one
nor the other cafe infer a contrary flexure, but only that the afymptote AQ»^ as
alfo the tangent in the point D, is parallel to the ordinaies.
a
EX.
8o
ANALYTICAL INSTITUTIONS.
BOOK ir.
EXAMPLE III.
F/V. 72,
loi. Let AMF (Fig. 72, 73, 74.) be a
"■ cycloid With the, equation z = — — ,
§ 47. By differencing, it will be ii =
arx — arr — Irr . .
^''i' n-
h X ITX—XX^A
The fuppofition of ;S = o will give
^r;^ — hrr — «rr .= o, that \%^ x — r
•\- — , If ^ be greater than h, it will be
a
the protradled cycloid. Whence, taking
CE from the centre, and equal to the
fourth proportional of BF, the femicircle,
and the radius, and drawing the ordinate
ED, (Fig. 73.) it will meet the curve in
the point of contrary flexure D. If a be
lefs than b, (Fig. 74.) the cycloid will be
contradled. But when a < by the line
X = r ■\ ^ will be greater than zr, that
a
is, greater than AB, in which cafe the
ordinates are imaginary ; becaufe there is
no part of the curve under the point F.
Therefore the curve has no point of con-
trary flexure or rcgrefTion. If it be « = b,
it will be the common cycloid, (Fig. 72.)
hr
and therefore ;vr=r + — = 2r = AB,
and y — BF ; which gives no contrary
flexure or regreflion, but only informs u«
that the tangent in F will be parallel to the abfcifs or diameter AB.
Fig. 74.
The fuppofition of .^ = 00 gives us b x 2r^ — xx)'^ = o, that is, ^ = o,
and X = zr. The value x = o, in all the three cafes, gives the tangent in
the point A parallel to the ordinates. The value x = 2r, in the firft and
fecond cafe, gives the tangent in the point F, in the fame manner, parallel to
Xhe ordinates. But, in the third cafe, it gives us a coniradidtion. For, the
equation
3
S£CT. IT.
A S A t Y T 1 C A L
INSTITUTIONS.
8 I
equation being % z: — — , inftead of x fubftitiuing it's value 2r, it will
be 2; =: o. But it cannot be i: ~ o, and at the fame time ;^ — 00 ; therefore
fuch a value ferves to no purpofe in this cafe.
EXAMPLE IV.
102. Let the curve be the conchoid of Nichomedes, confidered above at
§ 85, the equation of which is ;>;;' = ^ f£fij-f££f^
or J = .^^^- X -^^^ - ^^ . Taking the- fluxions, it will be _y = ZJ^lZ^ .
*' xx'^ aa—xx
11- . • u- • n. ^ '• "^^^^ — '''•*•' ~ 'in'hx'^ . .
and'takine them a^am, making A'conltant,j/=- ;= ■%- — x xx,
° ° x^ X aa — .v^|l
As to the thr^e ufual cafes, which this curve may have, I beginwith the firfl:,
when azzb, (Fig. 56.) This fuppofed, it will be y = =^^' " '''''" " ^^'^-^
x^ X aa — xx\'^
The fuppofition of y = o will give ^a> — aax^ — -^^a^xx = o, that is,
K^ -J- 3^;;* — 2^^ — o; and, refolving the equation, it is ;c = \/ '^aa — ^,
X zz. — \/ "^aa — a, and x zz — a. The firft value gives us the abfcifs
GE zz X •=. \/^na — ay to which belongs the ordinate EM zz y zz
t/^aa X V2aj^c'aa - ^aa ^j^j^j^ mcets the curvc in M, the point of contrary
flexure ; the fecond value is of no fervice, becaufe it makes the equation of the
curve imaginary ; the third gives us a rcgreflion in the point P.
As to the other two cafes, the fuppofition of j/ = o gives 2aab — x^
— "^bxx zz o, or x^ -h ^bx^ — laab = o. Now, to have the roots of this
equation, I make xx zz bzj a locus to the Apollonian parabola ; and, making
the fubftitution, there arifes the fecond locus xz ■+■ 3^2: — 2aa zz o, which is
to the hyperbola.
Between the afymptotes AQ»^ AD, take ACizQ.a,
the perpendicular CN = a^ AD ~ 3-^, and taking
the abfcifs x from the point D on the afymptote
AD, let the hyperbola GNF be defcribed, with
the conftant rediangle zi laa; it will pafs through
the point N. Then railing DM perpendicular to
DA, on the axis DM, with the vertex D, and
parameter zr by let the parabola of the equation
.VA." =: bz be defcribed.
Vot.IL M If,
82
ANALYTICAL INSTITUTIONS,
BOOK ir..
Fig, 58.
D- /O
T?
•/K
If, therefore, we affLime b greater than a, becaufe AD = 3^, AC = 2^,.
CD will be greater than b. Now, taking in the parabola the abfcifs 2 n ^
n CN, the ordinate will be at iz V ab. But if a be lefs than b, alfo >/ab will
be lefs than ^, and thence alfo lefs than CD. Therefore the parabola will cut
the hyperbola between N and D, fuppofe in the point I.
Now, if w^ afTume x "=. — ^, it will be in the parabok 2 — -^ , and in.
the hyperbola z zz r ; but ~ is greater than ■ — j ; therefore the
parabola will cut the hyperbola in fuch a point I, as that it will be HI = — x
lefs than a. Therefore this abfcifs will
have in the conchoid a real ordinate,,
which here determines the contrary flexure
in the point N, for example, of the lower
branch KN. The line GM, drawn from
the point G, another interfedion of the.
parabola and hyperbola, will neceflarily
be greater than <2, and therefore to fuch an
abfcifs there can be no correfponding real
ordinate in the conchoid ; fo that this
value is of no ufe. Laftly, the third
value TF will give us an abfcifs, to whicli
an ordinate belongs in the upper branch,,
which meets the curve in the point of contrary flexure M.
Fi^, 76. Let ^ be lefs than ^; then CD will be-
lefs than b ; and in the parabola, taking:
z =■ azz CN, the ordinate will be x'^L^ab^^
that is, greater than by and therefore greater
than CD. Whence the parabola will pafs
between N and C : fo that it will either not
cut the hyperbola, and the two negative-
values of X in the equation a?' + '^bx'^
— laab =: o will be imaginary j or, if it
cut it, they will always be greater than ^,
to which, in the conchoid, (Fig. 57.) ima-
ginary ordinates correfpond, and therefore
are of no fervice. Wherefore the parabola
will certainly cut the hyperbola, on the pofitive fide, in the point F for example,.
Whence TF, which is lefs than a, will be the value ot x, to which the ordinte
correfponds in the branch AM of the conchoid, which it m.ects in M, the point
©f contrary flexure.
I faid that if the parabola cut the hyperbola between N and O,^ the two^
jRegative values of x would be greater than a» For, taking ►v = — « in the
parabola^
gjSCT. IV, ANALYTICAL INSTITUTIONS. $3
it will be z = — , and in the hyperbola z = ^ . But -y is lefs than
■ ^^^ y for is lefs than a. Now, if fo be that .v negative be not greater
2tf — a
than a, the parabola would not cut the hyperbola; fo that it will cut it in a
point in which x Iliall be greater than a. Taking x pofitive equal to rt, it will
be in the parabola z zz — -, and in the hyperbola 2 zr —. . But -,- is
*■ b ■''■ lb + a b
greater than —r^ — ; fo that the parabola will cut the hyperbola in fuch a
point F, that TF will be lefs than a.
The fuppofition oi y "zi 00 gives iis a;' X aa-~xx^^ n o, that is, x =z o^
and X zz ±: a ; which is as much as to lay that the afymptote and tangent in
A are parallel to the ordinates in all the three cafes, as likewife the tangent in
K, in the fecond and third cafe • and in the firlt, that in P there is a point of
interfeftion, (as the regreflions alfo intimate,) becaufe the fame value x :==. — a
has alfo been already fupplied from the fuppofition of j/ n o ; which point of
interfediion has alfo been found before, at § 85.
103. The fame after another manner. I take the fame conchoidal curve,
but with all it's ordinates proceeding from a fixed point, or from the pole P.
Therefore make PM zz y, (Fig. 56, 57, 58.) and draw PF infinitely near to
PM. Then with centre P defcribe the little arches MB, DH ; make MB zzx,
AG zz a, GP zz h, and make PD zz 2, HO zz z. By the property of the
curve, the equation will bt y zz z •±: a\ that is, ^' z: 2; + ^ in refpedl of the
curve above the afymptote GR, and y zz z — a in refped to the curve
below it.
Therefore, finding the fluxions, it will be in both cafes y ■= z. Becaufe of
fimilar triangles PCD, DHO, (for the angles GDP, DOH, do not differ but
by the infinitely little angle DPH, and the angles at H and G are right angles,)
we (hall have PG . GD :: DH . HO ; that is, h . s/zz - bb :: ^.z; and
y
. r z'cVzz — lb -n • • ^ r • %XVX,% — hb .
thereiore z zz ■ . But z zz y, therefore y zz — ■■ — j- ; and
taking the fluxions again, making x conftant and putting z inftead of y, y n
. -^^^ ~ — - Xxz; and then putting the value of Zy we fliall have y zz
bbyyVzz -^ bb ■
"^^ "" — ^^rj— X XX -J and laftly, fubftituting the value of y zz z ztz ^9
bby^
bb y. ^'±. d]^
Mz The
;t Will be y = r— ^ X xx,
^ bb y. T±^3
84
ANALYTICAL INSTITUTIONS.
BOOK II.
The formula of curves referred to a focus has been found to be xx + yy
'•^yy zz. o, or elfe n oo. Therefore, putting the values of jy, of j', and of j'*.
It Will be '——=—- X XX =: o, or elie =z oo. The luppolition or
bb y, % -±0]
the formula being equal to o, will give abb ± 'Tf^bz T 22' — o. In the firil
place, let it be ^ =: b, and let us confider the upper branch ; it will be
z^ — \aaz — 4-tf' :=:. o, and the three values of 2; are ;s = — a^ z ::z
<Lz.^l2^ and z zr t±j£^21 , But it is ^^ = 2 + ^ ; therefore it will be
2 i
jf = o, _y = 3^ + ^^'^^ ^ and y — '^.^-IZ. ff . The third value is of no ufe,
becaufe it gives the ordinate lefs than 2«, where there is no curve. The fe-
cond gives the ordinate jy, which meets the curve in the point of contrary
flexure, for example, at M. The firft is alfo fupplied by coniidering the
lower branch, and determines the point of regreffion P ; and, in refpedl of the
inferior branch, will be z^ — -aaz + -i^' — o. Hence the three values,
z ^. Uy z •=. r ^ ^ '^'^''^ . But, in this cafe, y m z — /?, fo that we fhall
have y :=z o, y '=1
— 3« ± V^aa
The two lad values ferve to no purpofe.
becaufe they give y negative, where there is no curve.
As to the other two cafes, (Fig. 57, 58.) it will be 2' — ^bz =F ^abh rz o.
To obtain the roots of this equation, I put zz = -J<^^, a locus to the Apollonian
parabola ; and making the fubftitution, there arifes a fecond locus which is to
the hyperbola, fz — 2t^z zz ±: ab; that is, the homogeneum comparationis is
pofitive in regard to the upper branch of the curve, and negative in regard to
the lower. Between the afymprotes PQ^
NM, perpendicular in A, are defcribed
the oppofite hyperbolas (Fig, 77.) in
the angles PAN, MAQ»^ if the homo^
geneum be pofitive, and in the angles
PAM, NAQ>, if it be negative. And,
fuppofing b to be greater than a, make
AB = bj BC = <J i the hyperbolas will
pafs through the point C. And taking
AM =;= 3^, from the point M in the
afymptote MN let the p's proceeds
Then at the vertex M, with axis MN,
and parameter 4^, let there be defcribed
the parabola HMD of the equation
zz = ^bp. Then taking p zz MB.
= 2h, the ordinate in the parabola is « = b, greater than a^ that is, than bcy^
the parabola will pafs without the points C, and will cut the hyperbolas DC,
CT,
SECT. IV,
ANALYTICAL INSTITUTIONS.
^S
Fig, 5^'
G
JlUiL
-R
YH.
CT, in the points D, T, 1, from which the right lines DH, TV, lO, being
drawn parallel to the alymptote QP, will be the three roots or values of z in
the equation z^ — -l^ii^z — -la^l> r: o, that is, in relped of the upper branch
of the conchoid, But y zz z i- a, then DH + a Ihall be the ordinate yj
which meets the curve in the point of con-
trary flexure, for example in M, (Fig. 58.)
The other two roots VT, Ol, ferve to no
purpofe ; for, being negative, and a ad-
joined to VT, the difference, or _y, will
be negative ; and a, adjoined to OI, the
difference will be pofitive, but lefs than a;
and, in this cafe, the curve will not corre-
fpond to y negative, or lels than a. As
to the inferior branch of the conchoid,
that is, in the equation z^ — j^bbz + f^M'
= o, the three roots will be OG, VK»
HE ; but if from the firfl, and from the
third, a be fubtradled to have _>', the difference will be negative, that is, y ne-
gative, to which the curve does not correfpond, and therefore they will be of
no life. If a be fubtracted from the fecond, VK, the difference LK will be
the ordinate y, which meets the curve in the point of contrary flexure, tliat is,
in N.
Fig* 57. ' Suppofing if lefs than a, the parabola
will pafs between the points c, C, of the
hyperbolas GrK, ICT ; and ibereforfe the
two negative vahies of z in the equation
z^ — iififz — ialfJl/ •=. o, by adding a, will
^' give y lefs than ^, to which the curve does
*' not correfpond. The third, by adding ay
. will give y, which will meet the curve in
1^ the contrary flexure, as at M, (Fig. 37.)
As to the inferior branch, that is, to the
equation z^ -i- \bhls, -f ^abb — 6, from the
two pofitive roots, tv'hich are lefs than by
fubtradt a ; and aifo, being fubtraded from
the negative root, we (hall always have negative y greater than PK, to which
the curve does not correfpond. Therefore the inferior branch of the conchoid,.
when b is lefs than a, has neither contrary flexure nor regreflion.
The fuppofition of the formula being = oc, gives, in all the three cafes,
2i = + <j, and therefore ^ = o. In Fig. 58, the value y — o ferves to no
purpofe, becaufe there is no curve. In Fig. c^6f 57, it gives the tangent in P^,
which is alfo a point of regreffion in Fig. 56, but not fo in Fig. 57.
/""'^
P^^Tt
/ ^
V
G
B/ /O
/•/H
K
-^
^--^
<i
^N
84
ANALYTtCAI* INSTITUTIONS,
SOOK 1^.
Fig, 7S.
EXAMPLE V.
104. Let the circle A ED be defcrlbed with
centre B, and let AFK be fuch a curve, that,
drawing any radius BFE, the fquare ot FE may
be always equal to the redangle of the corre-
fpondent arch AE, into a given right line b ; and
the contrary flexure of the curve AFK is re-
quired.
Let the arch AE be called z, BA = BE = j^
BF = y, and FG = x. Drawing B(? iniinit ly
near to BF, and with centre B, radius BF, de-
fcribing the little arch FG ; by the nature of the
A ^HB
curve, it will be hz = aa — lay + yy. Then taking the fluxions, it is hz =
— xay + %yy^ whence ^ = ^"^^ 7 ^'^ = Ef. But, becaufe of fimilar fedors
BE^ BFG, it will be BE . BF :: E^ . FG ; that is, a ,y :: ""^-^ "^ "-"^ , xi
Whence x = ^>^ ~ "^^yy ^ ^^d taking the fluxions again, making x conftant.
it will be J\yyy + '^-yyy — "Z-ctyy — 2.ayy = o, whence yy =
_ oyy — yyj'
y - a
In the general fortmala of curves referred to a focus, xx + yy — yy = o ;
fubftitute the values of xx and of yy given by j), and we fliali have
'^^ "^^ -^"^ — ' '■ =^ i which, reduced to a common de-
aabb y — a
nommator, will be ^ -—^ ^^i-J — ^L = o, or = 00.
aahb x y — a
Wherefore, this equation being conftrudled, one of the roots will give the value
of the ordinate y, which meets the curve in the point of contrary flexure.
S EC T.
f
SECT, V»
^NALVTICAi INSTITUTIONS,
S;
SECT. V.
Of Evolutes, and of the Rays of Curvature,
Fig, 7^» 105. Let the curve be BDF, and let
it be involved or wound about by the
thread ABDF ; that is, the thread being
faftened by one of it's ends in the fixed
and immoveable point F, let it be con-
ceived to be ftretched along the curve
BDF, fo that the portion AB may fall
upon the tangent of the curve AR in the
point B. Let the thread move or un-
wind by it's extremity A, continually
evolving the curve, but in fuch a manner
that it may always have the fame degree
of tenfion. By this motion, the point A
will defcribe the curve AHK.
The curve BDF is called the Evolute of the curve AHK, as has been already
faid before, at § 16. And the curve AHK is called the Involute of BDF, or
the curve generated by the evolution of BDF j and the portions AB, HD, KF,
of the thread are called the Rays of tiie Evolute, or Rays of Ofculation,
106. Now, becaufe the length of the thread ABDF always continues the
fame, it follows from thence, that the difference of the rays of ofculatif)n AB,
HD, will be equal to BD, the correfponding portioa of the curve. As alfo,
the other portion DF is equal to the difference of the radii HD, KF, and the
whole curve BDF is equal to the difference of the radii AB, KF. And if the
radius AB (hould be none at ail, that is, if the point A fhould fall in B, the
radius HD would be equal to the portion BD, and the radius FK to the whole
curve BDF.
107, From
8^
ANALYTIC A.L, I K S.T. I X IT T I O N S.
E.OOK II.
107. From the generation of the curve AHK, by the unwinding of the
thread, it may be clearly feen that every radius HD, KF, at it's extremities
D, F, is a tangent to the evolute BDF.
108. Let the arch HK of the curve AHK be an infinitefimal ; therefore,
i^lfo,. th« ajrch: DF of tjjg. evoUjte, wijl be an ifliank-elm^aJ;- ai5d:, a& k ha& been
demonftrated in Coroll. 4. Theor, I. § 6. that any infinitely little arch of a curve
has the fame properties as the arch of a circle : and in Theor. IV. § 15. that
tlie radius HD being produced, fo that it may meet the radius KF in S, the
hnes SH, SK, differ from each other only by an infinitely little quantity of the
third degree ; therefore thofe lines SH>, SK, may be aiiumed as equal : and
therefore they are perpendicular to the curve AHK in the points H, K. But
the two lines HD, HS,. differ from- each other by DS, an infinitcfimal of the
firft order, and HD is finite ; therefore they may be affimied as equal. Where-
fore, to determine any point D in the evolute, that is, to determine the length
of any ray of olculation or of curvafure HD-, it-wili fuflice to have given in
pofition the perpendicular HS of the given curve AHK, (which is done by the
Method of Tangents;) che point S may be determined, in which it is cut by
thp infinitely near perpendicular KS. This may be done in the following
manner.
F/g. So,
109. Firft, let the curve DABE be
referred to it's axis ; let the two infinitely
little arches be AB, BE, the perpendi-
cular BQ>_and the other EQ»_ which meets
it in the point required, Q^ Make, as
ufual, DH = Xf HA =7; draw AF,
BG, parallel to DM, and the chord
PABC which meets ME produced in C,
and draw the other chord EBR. Now,
with centre B, and diftances BE, BP,
the little arches ES, PO, being defcribcd,
it will be AF == Xy FB = j/, AB = i
= \/xx + yy. But, by Coroll. 2.
^heor. V. § 19, the fedors QBE, BES, are fimilar. Therefore we (hall have
QB . BE :: BE . ES, that is, QB . i :: i . ES, (calHng the element of the
curve s,) and therefore QB = ~. Now, becaufe the little arch PO may be
expreffed by it's right fine, (Cor. i. Theor. III. § 9.) the triangles RPOj
-BEG, will be fimilar, and therefore BE . EG :: RP . PO ; that is, i ,y ::
RP . PO = -LirH. But the feftors BPO, BES, are alfo fimilar; and there-
fore it will be BP , PO :: BE . ES 5 that is, ^
J'
s ys
And
SECT. V« ANAtTTICAL INSTITUTIONS. 89
And laftly, QB rz »'^^p i a general formula for the rays of ofculation, or the
radii of curvature, in which nothing elfe remains to be done, but to fubftitute
the value of RP, the fluxion of DP z= ~- — x, according to the different
hypothefis of the firft fluxion which is to be taken for confl:ant.
If no firft fluxion be taken for conftant, it will be RP zi • ^^'^ ."•^''^ , and
therefore QB = ?l±2ll .
If ^i- be afliimed as conftant, it will be RP = — •??^, and therefore QB =
xy
If J/ be aflumed as conftant, it will be RP z: ^, and therefore QB =s
x'x •\' yy\k
yx
If J be aflumed as conftant, that is, */ xx + yy^ it will be xx + yy = o,
and ->; = —; whence RP = ^JLilif-liL, and therefore (^ zz
4r-\/yy'+~xx ; or elfe, fubftituting the value ic-, QB =: "^ -^^xv' , There-
fore, in the expreflion of QB z= lfJl221i., in which, as no fluxion is taken
^^ yx — xy ^
for conftant, it will be fufficient to expunge the tertnyx, in the fuppofition of x
conftant; to expunge the term xy, in the ftippofition of j/ conftant j and to put,
inftead of — y, it*s value -^ , in the fuppofition of s conftant.
110. The curve may be referred to a diameter, or the co-ordinates may be
inclined to each other in an oblique angle. Make the abfcifs DV zr a*,
VK = X, the ordinate VA ~ jy, and the reft as above. Becaufe the angle
DKB is known, the angle BNF will be known alfo. Wherefore, it being
MB zzjf, NF and FB will be given, and therefore AB, or i. But the triangle
RPO is fimilar to the triangle ABF, for the angles at O and F are right ones,
and the angle ORP does not differ from the angle FAB but by an infinitely
little angle RBP. Wherefore there will be given RP, PO, and thence ES,
and finally, QB.
Vol. II. N iii. From
as
X
50 ANALYTICAL XNSTXTUTIOl^S, BOOK II.
Suho/culatrixy III. Froiii the extremity of the radius of curvature BQ^is drawn QT parallel
or Co-iuidius, ^q ^]^q ^xis DM, which meets in T the ordinate Bl produced j the right line BT
^ is called the 6'w^o/6-«/«/n>, or the C<?-r^^ii/^. The radius B(^ being given, the
co-radius BT will, in like manner, be given alfo ; for, by the method of.
tangents, the normal of the curve Bw is given, and therefore BT will be given
by means of the fimilar triangles Bwl, BQT.
, But if we would have an expreffion for the co-radius independently of the
radius, we may make BT = 2. The triangle BT(l_is fimilar to the triangle
BCG, or BAF ; for, the two angles TBG, QBC, being right pnes, take away
the common angle QBG, and there will remain the equal angles TBQ>^ CBG,
and the angles at T and G are right ones. Therefore it will be ^c- . i :; z , BQ
X ^^'
caufe they differ from each other only by an infinitefimal of the third degree ;
therefore the difference of QB fliall be nothing; and, by differencing, without
_ . xz X XX + yy + zx'xx + zxyy — »Jc X xx •{• yy
aifummsf a conllant fluxion, n o.
'-' xxy XX •\- yy
But i^ =: j, becaufe TB and IB have the fame difference. Therefore z =
X y. XX +yy _. grp^ ^ formula for the co-radius, in which no fluxion is yet
yx — xy
affumed as conftant. If x be conftant, the term yx (hall be nothing, and
therefore the formula, on this fuppofition, will be ^ — ^ = BT. If j/ be
conftant, the term — xy will be nothing, and therefore the formula, on this
fuppofition, will be - — ^t— ^ = BT. If the element of the curve be con-
ftant, it will be — y zz -r-, and therefore the formula, on this fuppofition,
will be ■— = BT, the value of j^ being fubftituted : or elfe — 4^ z:BT, the
X J
value of X being fubftituied.
The co«radius being given, by the fimilitude of the triangles B;»I, BQT,
the radius QB will be given in a like manner.
112. If the co-ordinates (hall be at an oblique angle to each other, in the
analogy x . s \\ z . BQ»^ infl-ead of x and i, it will be enough to put the
refpedive values, which in this cafe agree to AF, AB, and to do the reft as
above ; and then you will have the formula of the co-radius BT, in that cafe
when the co-ordinates are at any oblique angle.
113. After
4
^ECT. V.
ANALYTICAL INSTITUTIONS*
§f«
mq X
jyj
113. After feveral other manners the fame formula of the radius of curvature
may be had. As, with centre Q»^ diftance Qw, defcribe the little arch mn,
AiTuming the infinitefimal arch mn by tile tangent at «, the two triangles BCG,
ffifiq, will be fimilar, and therefore BC . BG :: mq . mn ; that is, \/xx -h yj
But mq is the fluxion of Dm, that is, of the
fubnormal I;», with the abfcifs DI or DH j that is, of ;^ -j- ~ . Therefore,
by differencing in the hypothefis, that no fluxion be conftant, it will be mq rr
^'+y^ +ky^-^-, Therdorc mn = iL±2^^M=r-^ . But, becaufe
of fimilar fedors Qmn, QBE, it will be BE — fnn . BE :: Bm ( ^^■^^+-^> ) , qb^
xx+JJ)l
that is, fubflituting their analytical values, QB
JfX — XJ>
Which formula.
being modified according to the fnppofition of fome conftant fluxion, will give
an expreflion for the radius QB, correfponding to that fuppofition.
114. In another manner, thus. Let EM be produced to /, and BG to L.
Becaufe the triangle EGL is fimilar to the triangle BIw, the angles GEL, IBw,
being different from each other only by the infinitefimal angle BQE, it will be
GL = =^ . Therefore BL = "Il-i^Z.. But it has been feen, that mq =
+ yxy -r^xyy - y j^ ^ ^^^ ^^^ fioiilar triangles QBL, Qniq, give BL — mq
. BL :: B;w . BQj^ Therefore, fubftituting the analytical values, we fliall have
BO = ^^+x^l^ ^
yx — xy
Fig, 81
115. Now let us refume the curves
which are referred to a focus. Therefore
let the curve be BEG, the focus A. ' And
taking the two- infinitely little arches BE,
EG, and dravying the ordinates AB, AE,
AG, with centre A let the litde arches
BC, EF, be dcfcribed; and to the chords
GE, EB produced, let Af, AD, be per-r
pendicuiar. Laftly, let the chord DE,
produced, meet the ordinate AG in L,
N a and
92 ANALYTICAt INSTITITTIOJJS. BOOK II.
and wilh centre E let the Utile arch GR be defcribed. Make AB = 7,
CE — J/, BC 3: X, AD zz p. The httle arch DH being defcribed with
centre A, it will be HI n p. But HM is an inftnitefimal quantity of the
fecond degree; Theor. III. ^ 8. Therefore we may take as equal HI, IM,
and thence it will be MI =: />. The triangles EBC, EAD, are fimilar, which
gives ED = ^ = EI, as being different only by an infinitefimal. And,
afluming the little arch GR by it's tangent, the triangles EIM, EGR, will be
fimilar. Hence GR in ^. Now, drawing EQ»_ QG, perpendicular to the
curve in the points E, G, the fedors QEG, EGR, are fimilar; fo that QE zr
4-.. The fimilar triangles EBC, EAD, will give us p = ^ z= ,!'^ , ;
P ^ Vx'x -jryy
and, by differencing, without affuming any conftant fluxion, p ss
yx + xy X xx +yy - S(x ^yy X yx ^ q^. v _. ^^y + y.^y x + xy^ - y'xyy ^ \;Vhcnce
x'x-\-yy\\ XX +yy^k
fubftitucing this value inftead of p in the expreflion of QE, it will be QE =
- — J' X -y^ +yy\ — __ ^ general formula for the radius of curvature of curves
x3 +yyx + xyy — yxy °
referred to a focusj without taking any fluxion as conftant.
If we would have x conftant, taking the value of p in this hypothefis, and
fubftituting ; or, without any thing elfe but expunging the term yyx in the
general formula, it will be QE = • , . '^•'^- • "^"^ •*• •
o ' ^^ X* + xyy — yxy
If we would have y conftant, expunging the term — yxy in the general
formula, it will be QE = ^^""^^. .
' ^N. X* + xyy + jyx
And laftly, taking i for conftant, that is, \/xx + jij, we (hould have x =
— 4^; and, inftead of x, fubftituting this value in the general formula, it
will be QE zi - ^"^ . /' "" V-^ -, or elfe, fubftituting the value of j'', it is QE z=
XX — }y
yy V XX +yy
xy -i-yx
116, If
»ECT. V.
ANALYTICAL INSTITl/tlONS.
93
1 1 6. If, in any of thefe formulas, we flioiild fnppofe y infinire, all thofc
terms would vanilh in which it is not found, and the formnlas will be the fame
as thofe found for curves referred to an axis ; which ought to obtain, becaufc,
if J be infinite, the point A will be at an infinite diftance, and therefore the
ordinates will be parallel.
Fig. 82.
117. After another manner. In the
point E let ER be a tangent to the infi-
nitely little arch EG, and let QE, QG,
be the two radii of curvature, and pro-
duce QG to R. From the fccus A draw
AN perpendicular to QG, and AK per-
pendicular to QE, and make EK := / ;
then is KM zz /. Becaufe the triangle
AKM is fimilar to the triangle QNM,
and this is fimilar to the triangle QER,
it will be QE . ER :: AK . KIVI iz /.
But, becaufe of the fimilar triangles ELC,
or EGC, EAK, it is AK r: 4^ , and ER
may be affumed for EG. Then it will be QE . i ::
yy
and therefore QE
z= ^
i
But EK ■=. t "zi^. Then doino; the reft as before, that is, differ-
encing the value of /, and fubftituting in the exprefTion of QE, we (hall obtain
the fame formulae as above.
118. Making QP perpendicular to EA produced to P, the triangles EAK,
EQP, will be fimilar, and therefore EA . EK :: EQ^. EP. But it has been
(hown, that EQj= ^. Then ;»./:: ^ . EP = ■^. And, inftead of /,
fubftituting it's value 4-, and, inftead of/, the differential ^ "^yr^^-y^y ^
yxss
xss + yy'x — yxy
, a general formula for the co-radius, in which no fluxion
without afl'uming a conftant fluxion, it will be EP zz
y'x^ -Vy'xyy
x^ -|- xyy + yyx — yxj
is made conftant ; from which, being modified, we obtain the other formula,
which correlpond to the fuppofition of a conftant differential. And if in thefe we
(hould fuppofe y to be infinite, that is, if we Qiould cancel the terms in wnich
it is not found, we fhould have the fame formulae which have been found for
curves referred to an axis or diameter,
119. Now,
H
ANALYTICAL INSTITUTIONS,
BOOK II.
119. Now, whatever the curve may be, as we find but one expreffion only
for the radius of curvature, and for the co-radius ; and that as well in curves
referred to an axis, as in thofe referred ro a focus ; it follows from hence, that,
whatever the curve be, it can have but one evolute.
^ig' 83.
120. Therefore, any curve being given, expreffed by any equation whatever^
of which curve the radius of curvature, or the co-radius is required ; it will be
neceflary to difference the equation, in order to
have the values of j/, jfy, and y given by x ; or
thofe of X, kc. given by j 1 and to fubftitute
them in the formulas now found, by which we (hall
have the expreffion in finite terms, and quite free
from diflfereniials, of the radius of curvature, or
the co-radius of the propofed curve.
121. If the value of the radius of curvature, or
of the co-radius, be ,pofitive, they ought to be
taken on that fide of the axis DM, (Fig. 80.) or
o^ the focus, (Fig. 81.) as has been hitherto fup-
pofed, and the curve will be concave to this axis
or focus. But if it fhall be negative, they ought
to be taken on the contrary fide, and, in this cafe,
the curve will be convex. Hence it follows, that,
in the point of contrary flexure or regreffion, if
the curve have any, the co-radius, from pofitive,
will become negative j and two radii of curvature
that are infinitely near, from being convergent will
become divergent. But this cannot be, without
they firft become parallel, that is, the radius of
the evolute muft be infinite in this point ; or elfe
they mufl; coincide one w.itb the other, and thus
make the radius of the evolute nothing. It is
evident, that when the evolute is fuch, as that the
radii go on always incrcafing, as they approach to
the point B (Fig. 83, 84.) of contrary flexure or
regreffion, to pafs from being converging to be-
come diverging, thty muft firft become parallel,
being then AD, FE, the evolute of the curve
-ABF. But if the evolute of the curve ABF,
(Fig. 85, 86.) Iliali be DBE, the thread, un-
winding iilelf from the point B, and proceeding
towards A in refped of the portion BA of the
curve, and going on towards F, in refped of the
poition
SECT. V.
ANALYTICAL INSTITUTIONS.
'95
portipn BF ; bccaiife, as the radius is always lefs, the
nearer it is to the point B, it mud of neceffity become
nothing before it paffes from being pofitivc to become
negative.
EXAMPLE I.
Fig. 87.
122. Let the curve AB be the Apollonian
parabola of the equation ax zz yy^ of
which we would find the radius of curva-
ture at any point B. By taking the
fluxions, it will be ax :^ iyy ; and taking
the fluxions again, making, if you pleafe,
X conftant, it will be lyy + 2yy = o.
aaxx
But V zi — , therefore y zz —
Wherefore, thefe values being fubftituted
in the formula for the co-radius — — ~ ,
it will be
4j' + aay __^
riBE; orelfe, by putting.
inftead of
^ A/ax
y, it*s value given by the equation of the curve, it will be BE —
+ s/ax.
meet
From the point B let the tangent BT be drawn, which meets the axis in T,
d from the point T is drawn TE parallel to the perpendicular BM : this will
BP produced in the point required, E. For, becaufe of the right angle
•:, it will be BP . PT :: PT . PE ; ti^at is, by the property of the para-
^1::^, Therefore BP + PE = BE
bola ^ax , 2X v, 2x
PE = "^^
/i/ax
96 ANALYTICAL INSTITUTIONS, BOOK IT.
zz li— ^ 4- K^ax. Now, BE being determined, draw EQ^parallel to the
axis AP ; the normal BM, produced, will meet EQ^in the point Q»_ which will
be a point in the evolute.
Or elfe, becaufe of the fimilar triangles BPM, BEQ»_ it will be BP , PM '.:
BP^ . EQ^ But, by the property of the parabola, it is PM = -^a. Then
s/ax . itf :: 1^^ + V^^ . EQ^ Whence ECL= ix + i^ zi: PK, and
MK = 2x. Wherefore, taking MK double to AP, or PK = TM, and
drawing KQ^ parallel to PB, it will meet the perpendicular BM produced in the
point Qj^ which will be in the evolute. And, becaufe it is BP . BM II
BE . BQ, and BM zn ^^±^, it will be ^ax . H^J^H^ :: ^JL!^ +
y/ax . BO = 4^^ ->r aa\% ^^ radius of curvature.
laa
,3
Taking the formula -^-^ "^ yy ^ of the radius of curvature, and making the
fubftitutions, it will be OB zz ^JI^ - IflEffl!, as at firft.
^ Zaa 2aa
Proceeding to the fecond fluxions of the equation ax =: yy, without making
any conftant fluxion ; becaufe ax zz 2yy, it will be ax — 2yy + 2yy, or y =
^"^ ~ ^'^ . Wherefore, taking the formula for the radius of curvature - "^..^-^V.. ,
which belongs to this cafe, and making the fubftitution of the value of j', it
will be QB — ^^^^^y^f,'^ y. ■ I and laftly, putting the values of y and y, it
IS QB ~ , as above.
The fame thing will be found in the other fuppofitions ofj/ or i conftant;,
which, confuking brevity, I fliall here omit.
If we would have the radius of curvature at any determinate point of the
curve, it will be fufficient to fubftitute, in the finite expreflion already found
for the radius of curvature for any point, the value of ►v agreeing to that deter-
minate point. Thus, if we would have the radius of curvature in the vertex A,
or in the point N in which the axis AN of the parabola touches the evolute
NQj fmce, at the vertex A, it is ;v ~ o, by expunging the term d^ax in the
expreflion ^^ — ^^ of the radius of curvature, we (hall have AN =: ia i
which
5£Ct. V.
ANALYTICAL INSTITUTIONS.
n
which cannot be otherwife, the radius AN in this cafe being the fame as the
fubnorma], which, in the parabola, is known to be equal to half the parameter.
123. Now it will be eafy to find the equation to the evohite NQ, afrer the
manner of Des Cartes, or the relation of the ordinates NK, KQ>^ m the follow-
ing manner.
Make NK =: «, KQ^ir /. Since KQ^z: PE = ^^', we fhall have the
equation / zz ^^^ . But AK = AP + PK = ja? + ^a, and AN ~ i,a.
Then NK ~ 3^; — u, and x z=: ^u ; therefore, putting, inflead of x, this
value in the equation / =:= ^1^ ^ vve Qiall have / = ±L^^lf!t, and, by fquar-
ing, zyaU = i6«', which is an equation to the fecond cubic parabola, with a
parameter = -^ ; which exprefles the relation of the co-ordinates NK, KQ^
and is the evolute of the propofed Apollcnian parabola.
It is evident that the whole fecond cu-
bical parabola will be the evolute of the
whole Apollonian parabola ; that is, that the
branch NQ^will be the evolute of the upper
part AB, and the branch N^ of the lower
part Ab : and that the two branches N^,
NQ»^ change their convexity, and have a
regreffion at N.
124. It is alfo evident, that If the propofed curves be algebraical, their
cvolutes alfo will be algebraical curves, and that we may always have an equa-
tion in finite terms, expreffing the relation of the co-ordinates; and that, befides,
thofe evolutes will be redifiable, or we may find right lines equal to any portion
of the fame ; for example, to QN. For, if the propofed curve AB be alge-
braical, we may always have the radii of curvature BQ»_ AN, in finite tenps;
and, from BQ^fubcrading AN, the remainder will be the arch NC^
Vol. IL
,0
£X-^
93
ANALYTICAL INSTITUTION*.
BOOK n.
EXAMPLE n.
Fig. 89.
125^ Let the curve MBM be the hyper-
bola between the afymptotes, vvhofe equation,
is aa zz xy. By differencing, it is xy ■+■ yx
= o, and by differencing, again, and taking
X as conftant, it is y = — ^. Subfti-
tuting thefe values ofjzndy in the formula
•^■^ for the co-radius, we (hall have
'J
BE
~ - 2y
a negative value. If^
therefore, it is AP zz Xy PB zr jy, in AB,
produced, taking BN z: ^^BA = \'^ xx~\-yy^
and raifing the perpendicular NE, which may meet the ordinate BP, produced
in E, the co-radius will be BE, as was required. For, becaufe of fimilar
triangles BPA, BNE, it will be BP . BA :: BN . BE, that is, y . \/xx + yy
:: -^s/ XX 4- ^^ . BE zr "^"^ ^^ J and therefore, on the negative fide, it muft
be BE zz "-^ — ^ . Wherefore, drawing EQ^ parallel to AP, and producing
to Q^the perpendicular to the curve FB in the point B, the radius of curvature
will be BQ>_and the point Q^will be in the evolute.
To determine the radius of curvature at the vertex of the hyperbola D,.
make x zz AH r: a. and therefore v zi HD zz a. Then the co-radius ^^ ^^
at the vertex D will be equal to — a, and the radius equal to — A/2aa.
If we do but confider a little the figure of the curve MBM, we (hall find
that the evolute will have two branches, with a point of regreflion at L, in
which the radius DL will revert, and will be the leaft of all the radii BQ^
Wherefore, by differencing the formula of the radius of curvature "^-^ — ~^,
the difference or fluxion will be nothing, or infinite ; that is, fuppofing x to be
conftant, it will be
-3^>-'^^-4-j;- + ^x:^r+^ _ ^^ ^^ ^^ ^^j^ ^j_
xxyy
viding by ^/ xx -Y yjy. and multiplying by xyy^ it will be xxy -jr yjy - 3m
^
= o,
«ECT. V^ ANALYTICAL INSTITUTIONS. ^f
rz o, or 00, But, by the equation of the curve, it is j/ — ~- y j' r:
— —^ , y zz -^ — ■—- . Therefore, making the fubditutions, and fuppoling the
faid quantity to be equal to nothing, we fliall have x — a zz. AH. That is to
fay, the regrelTion will be in the radius of curvature at the vertex D of the
curve. But it has been feen, that that radius is equal to — y/2,aa ; therefore
it will be DL. =: — */ laa — DA.
In the formula of the radius of curvature, fubflituting the values of j/ and y
we (hall have BO = ^ — ^^ =: ^ — ^ — , and therefore, difFerencinsr, that
we may have the leaft radius, that is, the point of regreflion L, it will be
^xx + '^yy X */ XX + yy :=z o; and, inftead of j), putting it's value, it will be
2^xxx — '^yyx x K^xx + ^ n o, that \s, x :=. y :=: a» And fubftituting this
value in the expreflion for the radius of curvature, it will be = — V 2aa zz
DL, as found above.
The radius B(^ may alfo b« conflruifled in another manner. For, becaufc
j' = —• ~ , inftead of :>c' and .v, fubftituting their values by /, it will ht y zz,
— , and therefore the co-radius BE = ^'^ — M^ . And, becaufe of limilar
triangles BPF, BEQ, we ftiall have EQ^zz — J^ -. 21 . Now draw the
tangent BT to the point B, and from the point T the line TS perpendicular to
BT, or parallel to Bq^^ and make BE = fBS, or PK zz fFT. Now, if EQ^
be drawn parallel to AT, or KQ^ perpendicular to it, they will meet the line
BQ^in the point of the evolute (^ For it will be BS =i^.^±ir^ , then BE
= yJ±±2yl', it will be alfo FP + PT =z FT = - 4- - 4-. and there,
fore EO = -.4--4..
If the equation be y^ zi x, which exprefles all parabolas aJ infinitum^ when
m denotes an affirmative number, and confequently the parabola of the firft
example : (and it exprefles all hyperbolas between the afymptotes, when m
ftands for a negative number, and therefore that of the prefent example.) By
taking the fluxions, we (hall have myy^"^ zz i; and taking the fluxions
^gain, fuppofing .?; conftant, it vyill be mm — - /w X yyy^"* + myy"*"^ z: o.
O 2 Now,
100
ANALYTICAL IMSTITUTIONS,
BOOK II,
w — I
Now, dividing by my " , it will he — y zz m — i x =^ . Wherefore,
taking the formula for the co-radius — — ~ , and making the fubftitution of
the value of y, we fnall have BE =: ^ ^"^ ^ -^^ , and therefore EQ. or PK =
m •— \y
+
F/^.87.
From the point T (Fig. Z']^ ^(^^ in
which the tangent B T meets the axis AP,
is drawn, in like manner, 7'S parallel to
BC^ a perpendicular to the curve, which
meets in S the ordinate BP pioduced,
T) C
Then take BE = , on the negative
fide, \i mht z. negative number, as in the
hyperbolas which are convex towards the-
the axis AP, (Fig. 89.) that is, to the
afymptote. But BE mufl be taken on the
pofitive fide, if m be a pofitive number,
and greater than unity, as in the parabolas
(Fig. 87.) that are concave to the axis AP;
and on the negative part, if »;, being po-
fitive, be lefs than unity, in which cafe the
parabolas are convex to the axis A P.
To determine the point in which the
axis of the parabola touches the evolute, I
take the formula of the radius of curva-
ture, which is ^ — 4^, from whence,
' — xy
by
fubftituting the values of ^ = myy
m—x.
and of — y :=-
m — I X yy
y we (hall have
BQ^
mmy
zm
'=^7^-
m — z
It is here underftood, that unity may fupply any.
m X tn — i Xy
powers required by the law of homogeneity. Whence, fuppofing m to be
greater than unity, for that reafon the parabolas will be concave to the axis AP;
if m be lefs than 2, the ^ in the denominator will become a mult4plier m the-
numerator, and therefore, making j =s o, as the prefent cafe requires, it will
be BO = o, that is, the axis will be a tangent to the evolute m A, the vertex
^ of.
SECT. V.
ANALYTICAL INSTITUTIONS!
lOI
of the parabola, as it would be (for inftance) in the fecond cubic parabola
a XX = y^y Fig. 70.
Fig, 90.
Now, if nr be greater than 2, the y of
the denominator would be raifed to a pofi-
tive power, and therefore, making y zz o,
BQ^ would be infinite, that is, the axis of
the parabola will be an afymptote to the
Q. evolute ; as in the firft cubical parabola
AB, (Fig. 90.) whofe axis AP is an afym-
ptote to the evolute LQ.
The evolute CLQ^of the cubical femiparabola ABM of the equation aax~y\-
has a point of regrefiion L, and therefore two branches LC^ LC ; by evolving
the branch LQj^ the portion BA will be generated, and by evolving the branch
LC, the infinite portion BN4 will be produced.
To determine the contrary flexure L, take the value of the radius of curva-
ture, which in this curve is ^^ ^J" ^ > which ought to be a minimum; and therc-
fore, by taking the fluxion, it will be —- ■—■ ^
= o, that is, 45_)'* — «* = o; whence jy zz ^ — . And this value, being
fubftituted inftead of y in the equation aax = y^, we Qiall have x zz ^
91125.
Taking, therefore, AP r: ^ , and drawing the ordinate PB, the point
of regreflion L will be in the perpendicular to the curve at the point B. And,
in the exprefllon of the radius of curvature, putting X/ ~ inflead of y^ we
ihall have the value of BL.
After another manner. By differencing the equation aax zz jy', or ^ =
aTxi, It will be j^ =: ^a'^xx \ y zz — ^aJxxx , y =: -l^a^x^x ^, fuppof-
ing X to be conflant. Whence, taking the formula xxy + yyy — syyy =0, and.
fubftituting thefe values, we fliall have AP = \/— ^ — » as before..
o J ^ qii2C
EX-
roa
ANALVTICAt INSTITUTIONS.
BOOK n.
EXAMPLE III.
Fig, 91,
126. Let the curve ABD be an ellipfis "or
hyperbola, the axis of which is AH = a, the
parameter AF = by AP =: x, PB zz j, and the
abx ^ bxx
equation y -sz */
a <
ahx m tbxx
By differencing, it
a}hhpi
^ z's/ aabxz^haxx* ' 4 x aabx'-S^abx^\:
taking x for conftant. Making the fubftitutions
■^ * of the radius of curva-
in the formula
— xy
ture, it will be BGq^= 4^"*- T '^■"' + '^^.^ a^M' + 4W«^T _ g^^ ^^^ ^^^^^j
will be found to be BG - A'"'" T ^l"" + "■'» T ^aU' +;gBj _ Therefore
the radius will be BQG = ^^^ Z^"^' ; fo that, taking the parameter h for the
firft term, the normal BG for the fecond, and continuing the geometrical pro-
portion, the quadruple of the fourth term will be the radius of curvature B(^
Making a^ =r o in the expreffion for the radius of curvature, it will be
BGQjzz AM = t^. And making x = AO = t^, we (hall have in the eUipfis
BGQ^= DOQ^z: ^— , that is, equal to half the parameter of the conjugate
axis ; and in Q^will be a regreffion; and the evolute of the portion AD = DH
will be MQ^— of the portion DH, will be R(^ Bur, in the hyperbola, the
radius is extended in infinitum.
In the ellipfis, if we make a zz b, the radius of curvature BGQ^will be
= -^a, wherever the point B be fituate. Therefore the radii will all be equal to
one another, and the evolute will become a pointj that is to fay, that the ellipfis,
in this cafe, degenerates into a circle, having the centre for it's evolute.
EX-
IICT. V,
AUALYTICAL I N » T X Ttf T I O H S%
103'
EXAMPLE IV.
Fig» 92. 127. Let the curve ABD be the com-
mon logarithmic curve, the equation of
which is — zi x,
y
By taking the fluxions, making x con-
ftant, it will be y = -^ =: ^-^ , by fub-
•^ a aa ^ ■'
ftituting the value ofj/. Making the ufual
fubftitutions in the formula ^^ .-^-^ of the
H co-radius, we fliall have BE zz ~ "^ " ^y .
y
and becaufe, in the logarithmic, it is
found that the fubnormal PH = — , it will be EQ^zz — j — — . Therefore,
taking PK = TH, and raifing KQ^at right angles, it will meet the normal;
HBQ^in (^ the point of the evolute required.
If we would determine the point of greatefl: curvature in the logarithmic,,
that is, the point where there is the lead radius of curvature; making the fubfti-
tutions in the formula
"+7?)y
— xy
■^^'^ of the radius of curvature, it will be — '^•^^'^ .-
- ^y
— 3ayyy X aa+y^'^ + ay X aa +yy) i .
■ = o, and;
aayy
and taking the fluxions, it will be
therefore PB = y "=. ^\aa.
Or, taking the formula of § 125, xicy -^^ yyy — 3J/J7 = o, and making the
fubftitutions of j/ = ^, jy = -^ , and y =: -^j we fhallcome to the fame
conclufion of PB = ^ = s/iaa..
EX-
104
ANALYTICAL INSTITUTIONS.
BOOK II.
EXAMPLE V.
128. Let ABD be the looarlthmic fpiral, the property of
which is, that, at any point B, drawing the tangent BT, and
from the pole A the ordinate AB, the angle ABT may always
be the fame : therefore, making AM to be infinitely near AB,
the ratio of MR to RB will be conftant. Wherefore, putting
AB z: y, the little arch BR zz Xy the equation will be
ax -zz by I and, by taking the fluxions, and making x .con-
ftant, it will be y "=: o. Therefore, taking the formula of the
co-radius, ^118, -r-; — ^^.. ' ^^'!-^. —, for curves that are
•^ ;f * -f- Ayy •\- yyx — yxy
referred to a focus, which, being managed on the fuppofition
of X being conftant, will be >'-*-^ . /■^•^.. . And in this, ex-
° XX + yy —yy
punging the term yy, becaufe the curve gives us here y rz o,
and making the fubftitution of the value of x or y, or, dividing the numerator
and denominator by xx + jl{y, the co- radius will be BA = y.
Therefore, drawing AC perpendicular to AB, it will meet the perpendicular
BC in C, the point of the evolute required; and, becaufe the fubnormal
AC = 41, it will be BC = >:^tii.
Drawing BT, a tangent to the curve in the point B, the triangles TCB,
CBA, will be fimilar, and therefore the angles TBA, ACB, will be equal. But
the angle TBA is a conftant angle, fo that the angle ACB will be fo too.
Therefore the evolute AC will be the fame logarithmic fpiral ABD, but in an
inverted fituation.
EXAMPLE VL
129. Let ABD (Fig. 93.) be the hyperbolical fpiral, the property of which
is, that the fubtangcnt is a conftant line.
Do the fame things as in the foregoing example, and the equation of the
curve will be y^ = ^, or yk — ay. Then, by differencing, making x con-
ftant.
«ECT. V,
ANALYTICAL INSTITUTIONS.
105
ftant, j' = — . Wherefore, taking the formula of the co-radius, correfpotid-
ingto thehypothefisof ;c conftant, that is, ^-^jt~-,, and, inftead of y,
fubftituting; it's value ^^ , and, inilead of y, it's value ^ given by the equa-
tion, the co-radius will be = ~,
But, becaufe the fubtanoent AT = a, and the fubnormal AC = — , it will
be TC = "^ — — . Therefore the fourth proportional to the fubtangent TA,
and TC, and the ordinate AB, here determines the co-radius. Whence, from
the point C drawing CQ^ parallel to the tangent BT, which cuts in Q^the ordi-
nate BA produced, BO will be the co-radius required.
For the triangles BAT, CAQ^ are fimilar ; fo that we fhall have CA . AQ^
:: TA . AB; and, by permutacion, CA . TA :: /\Q . AB. And, by com-
pounding, TC , AT :: QB . AB ; and, by inverfion, TA . TC :: BA . BQ^
EXAMPLE VII.
B^. 94.
130. Let ADN be a fe(flor of a circle, and
from the centre A drawing any radius ABP,
let it be ND . NP :: Ap"* . ABl^*. The
Q point B (hall be in the curve ABD, which is
one of the fpirals ad infinitum, the equation of
m
a z
which IS y = ""T"* making NPD = ^, NP
= Zf the radius AP =r a, and AB =: y.
Then, by taking the fluxions, it will be
m-
. ««— I a «
myy zz
, ■. Now, drawing the radius
hp infinitely near to AP, and making BR = x ; becaufe of fimilar fedors
AP/>, ABR, it will be 2J = ~ . Wherefore, putting the value, inftead of z^
Vol. n. P Jn
ic6
ANALYTICAL INSTITUTIONS.
fft + i.
BOOK ir.
in the fluxional equation, it will be myy*" =
; and therefore, takings
the fluxions again, making x conftanr, we fliall have mfnyyy"" + my^y zz o,
that is, yy ~ — fjtyy. Wherefore, making a iubftitution of this value,
and of the value of x, iri the formula of the co-radius, it will be BE =
y X mmbby + a ^
— . Make TAG perpendicular to AB, and draw BT a
+ ttt + 1 X a '
tangent to the curve in B, and BC perpendicular to it i it will be AT =
"■'^ , AC = -1—- , and therefore TC = =^^5!— i^ Whence
m+i
mby
m~ 1
mhd
3
the fourth proportional to TA + »* + i X AC, to TC, and to AB, will be
"^ — — ^ = BE. And therefore, drawing EQ^parallel to TC„
it will meet the perpendicular BC in the point Q^ which will be a point in the:
cvolute.
EXAMPLE VIII.
^^g' 95-
A 131. Let the curve ABD be half of
the conkmon cycloid, the equation of
K which \s y ^ Jc>/ — "" "^ ; making AG
^ = ia, AP = X, PB = y.
By differencing, and taking x for
axx
conftant, it will be j' = — . ,
x\ zax — XX
and fubftituting thefe values in the for-
V mula for the radius of curvature -r-^,
^ — ■*j'
it vyill'be BQ^rz zy/ /^aa — 2ux, But
the normal BG = \/^aa — xaa, which is equal to the chord EC. Therefore
the radius of curvature BQ^= 2BG = 2EG.
o ^ Making
3ECT. V.
ANALYTICAL INSTITUTIONS.
}n
Making a? — 0, to have the radius of curvature in the point A, it will be
BQ^m AN zz i\a, and therefore CN zz CA = la,
. Making x = za, the radius of curvature in the point D will be =: o, and
therefore the evolute begins in D, and terminates in N.
Becaufe the tangent of the cycloid in B is parallel to the chord EA, {-§ 47.)
the normal BQ^will be parallel to the chord EC. This fuppofed, complete the
reftangle DC&^S, and vviih the diameter DS zz CN :=: AC defcribe the femi-
circle pis, and draw the chord Dl parallel to BQ»_ or to EC. The angles
CDI, DCE, will be equal, and confequently the arches Dl, CE, and their
chords. Therefore Dl, GQ» are equal and parallel ; and drawing \Qi_ it will
be equal and parallel to DG. But, by the property of the cycloid, DC is equal
to the arch EC, and therefore to the arch Dl. Then the arch Dl = IQ^ and
the femicircle DI3 zz SN. Whence the evolute DQN is the fame cycloid,
DBA, in an inverted fituation.
132. The radius of curvature and it's formula being now fufEciently ex-
plained, it will not be difficult to find the formula for the regreffions of the
fecond fpecies, mentioned before at § 98.
Fig. 96
Let the curve be BAC, with a contrary-
flexure at A, and let it be evolved by the
thread beginning at any point D, different
from the point of contrary flexure A. The
evolution of the portion DC generates the
curve DG, and that of the portion AB ge-
nerates the curve EF ; in fuch manner, that
the evolution of the whole curve BAC will
forai the entire curve FEDG, which has two
regreffions ; one at D of the ufual form,
becaufe the two branches DE, DG, turn
their convexity; the other at E of the
fecond fort, becaufe the two branches ED, EF, are concave towards the fame
parts. Let NM, N«/k, be any two rays infinitely near, of the evolute DA,
and NH, »H, two perpendiculars to the fame; the two infinitefimal fedors
]S;^;M, HN», will be fimilar, and therefore HN . NM :: N» . M»/. But, in
the point of contrary flexure A, the radius HN (^ 121.) ought to be either
infinire, or equal to nothing, and the radius NM, which becomes AB, con-
tinues finiie. Therefore, in the cafe of contrary flexure A, that is, in the point
of regreffion E, of the lecond fort, the ratio of N??, M//?, that is, the ratio of
the dilferential of the radius MN to the element of the curve, ought to be
either infinitely great or infinitely little. But the formula of the radius
V z MN
lOS ANALYTICAL INSTITUTIONS, BOOK II.
MN is ^^ '^■^l'^ , taking x for conftant; the differential of which is
- 3 wV^r+^ -M yj<JgTglj ^^^ j^^ _. ^JFTn* Therefore ^ =i
^^^^ — -^ff. "* ^-^"^^ = o, or 00, the formula for the points of regreffion of the
fecond fort.
This formula is the fame as that already found, § 125 ; but in that place it
ferved for the regrefTions of the firft fort of evolutes, and here for the re-
greffion of the fecond fort of curves, derived from evolutes ; x and y, m both
■cafes, being the co-ordinates of the curves fo produced.
END OF THE SECOND BOOK.
ANA.
ANALYTICAL INSTITUTIONS.
BOOK III.
OF THE INTEGRAL CALCULUS,
THE Integral Calculus, which is alfo ufed to be called the Summatory ixiixoSxadt
Calculus, is the method of reducing a differential or fluxional quantity,
to that quantity of which it is the difference or fluxion. Whence the operations
of the Integral Calculus are juft the contrary to thofc of the Differential ; and
therefore it is alfo called 'ihe Inverje Method of Fluxions, or of Differences,
Thus, for example, the fluxion or differential o{ y is j/, and confequently the
fluent or integral of y is y. Hence it will be a fure proof that any integral is
juft and true, if, being differenced again, it fliall reftore the given fluxion, or
the quantity whofe integral was to be found. Differential formulae have two
different manners, by which their integrals are inveftigated. One is, by the
help of finite Algebraical expreffions, or by being reduced to quadratures
which are granted or fuppofed. In the other, we are allowed the ufe of infinite
feries. In this firft Sedion, I (hall deliver the rules required in the firit manner.
In the fecond Sedtion, I Ihall treat of the fecond manner ; to which I fhall add
a third Sedtion, to fliow the ufe of thefe Rules in the Redification of Curve-
lines, the Quadrature of Curve-fpaces, &c. And laftly, I fliall add a fourth,
which fliall teach the Rules of the Expsnentiai Calculus,
ion*
SECT.
IIO AlIALYTlCAt INSTITUTIONS, BOOK III.
S E C T. I.
^he Rules of Integrations expreJJ'ed by finite Algebraical Formula, or which an
reduced to Juppofed ^ladratures.
1. ^s in fimple quantities raifed to any power, their differential or fluxion is
the produdt of the exponent of the variable into the variable itfelf, raifed to the
fame power lefTened by unity, and multiplied by it's fluxion or difference ; fo
the fluent or integral of the proiu6t of a variable raifed to any power, into the
difference of the lame variable, is the variable raifed to a power the exponent
of which is increafed by unity, divided by the fame exponent fo increaled.
And this obtains, whatever the exponent fliall be of the power of the variable,
whether pofitive or negative, integer x)r fraftion. Thus, for example, the
fluent oi mx ^ x will be ^ , or x". The integral of a; " x will be
« — I + I ^
3 that is, "^ ; and fo of others.
n
2, Any conftant quantities, Ample or complicate, by which the fluxions may
be muhiplied or divided, will make no alteration in the rule; for they remain
in the fluents juft as they were in the fluxions. Therefore the fluent of
''"-^" will be ^"^"■^'
3. Thus, if the differential formula were a fraftion, the denominator of
which were alfo any power of the variable, multiplied (if you pleafe) by any
tn. m.
conftant quantity ; as the formula "^-^ , or ■■ , which will be
aax — box aa •— bo y, k
- m — ft
the fame as ^'^ ^ ,, , and therefore fubjeft to the general rule,
'' 4. But
SECT. I, ANALYTICAL INSTITUTIONS. Ill
4. But here we are to obferve, that, in order to have the integrals complete,
we ought always to add to them, or to fubtradl from them, lome conllant
quantity at pleafure, which, in particular cafes, is afterwards to be derermined
as occalion may require. Of this we Ihall take further notice in it's due place.
Thus, the complete integral of x, for example, will be x ±: a, where a
fignifies fome conflant quantity. That of x*x will be -ja;' ± a^ ; and fo of
others. The reafon of which is, that, as conftant quantities have no differ-
entials, but X may as well be the differential of ;if + ^, or of a; — ^, &c. as
of X ; fo A-, or Jt + <2, or X — ^, &c. may be the integral of x. The fame
obtains in any other formula.
5. The fame rule of integration ferves for complicate differential formulae,
or thole compounded of many terms ; whether they have a denominator,
■whether that be wholly conftant, or contains the variable in it, whether it be
fimple and of one term, or whether it be reducible to fuch.J
For, in thefe cafes, the complicate differential formula may be refolved into
as many fimple ones, as are the terms of the complicate, and then each of thefe
comes under the given rule. Let the formula be — - — -^^^ ; this will
be equivalent to thefe two, ^,, and — rr- > ^^^ therefore the integral of
^ ' aa — bb aa — bb °
thefe two formulas will be the integral of the firfl: \ that is,
/a + I X aa -~ blf
m
A =r^ ±/.
m X aa—bb
Let ic be "^ ^ ~ "^ ^ ; this is the fame as thefe two, , ,
. tf A .V — cxx a ^ c X X* a — c X x*^
or as thefe, -^ — — ~ , and therefore the integral will be
2 X a "'C
— I
± /, that 15, h === i: /•
.— I X a — tr 2a — 2C. a—c X X
w— T
1 rn • Ti. ^
Let It be ; this is equivalent to thefe two, l/x x — aax ^;f,
ftiid therefore the integral will be -~ ■ — ^ ± /,
6. Befides,
112
ANALYTICAL INSTITUTIONS,
BOOK III.
6. Befides, if the complicate differential formula be raifed to any power, the
exponent of which is a pofitive integer, it being actually reduced to the given
power, every term may be integrated by the fame rule.
7. All that I have hitherto faid will obtain, when in the differential formula
there is no terin in which the exponent of the variable is negative unity, iuch as
— , or ax X -y for, according to the rule, the integral would be — — -- — , or
o
, that is, infinite j and which therefore teaches us nothing.
8. In thefe cafcF, therefjre, we are obliged to have recoiirfe to other methods.
There are two of thefe which will affift us. One is, by means of a curve which
is called the Logarithmic Curve, or the Logijlic. The other is, by means of
infinite fenes. As to infinire feries, of which we fhall make very great ufe in
many other cafes alfo, J fliall treat of them hereafter, as may be feen in
the next Section,
Fig. 97. Q^ 9. Now, as to the logarithmic curve, it
is a curve of fuch a projierty, that, in the
axis, taking the abfcifles in arithmetical
progreffion, the correfponding ordinates
will be in geometrical progreffion. There-
fore let the axis AD be divided into equal
parts, AB, BC, CD, DE, &c. At the
points A, B, C, D, &c. erect the perpen-
diculars AE, BF, CG, DH, &c. fuch, that
they may be to each other in geom.etrical
proportion. The points E, F, G, H, &c.
will be in the curve. And again dividing
the lines AB, BC, &c. into eqwai parts, and at the divifions raifing perpen-
diculars in the fame geometrical proportion, we Hiall have other intermediate
points. And laftly, multiplying the divifions in infinitum^ we fljali have infinite
points, or the very curve itfelf.
Therefore, the axis being divided into infinitefimal equal parts, let one of
thefe be CM = ^c-, the ordinate CG :=z y, and MO infinitely near it ; therefore
it will be NO =J)/. Let there be another ordinate DH n: 2, and others as
many as you pleafe, correfponding to the abfciffes that are arithmetically pro-
portionals. Therefore thefe ordinates will have the fame proportion to each
other, and, by confequence, their differentials alfo will be in the fame pro-
portion. So that it will ht y , z W y , z ; ox y , y \\ z . z i whence the ratio
of J/ to J' will be a conflajit ratio. And therefore, afiuming x conftant, it will
CM DF
\>^ y ty 1/ X , a^ ox —
X J which is the equation to the curve.
9
Here
SECT. I.
ANALYTICAL INSTITUTIONS.
113
Here it will be'eafy to perceive that the fubtangent of this curve will always
be conftant ; for, in the general formula of the fubtangent ^^ inftead of jy,
fubHituting it's value given from the equation of the curve, we fliall have
~- ~ ~ = a. Now, as the increafing geometrical progreflion of the ordi-
nates may be continued in infinitum^ the abfcifles alfo increafing arithmetically
in infinitum ; therefore the curve will go on infinitely, always receding further
from the axis. And as the fame progreflion, decreafing, may be al(o continued
in infinitum, the axis ftill increafing the contrary way, the other part of the curve
will go on infinitely, but always approaching towards the axis without ever
touching it, and therefore that axis will be an afymptote to the curve.
. 9. Among many other ways, the logarithmic curve may be conceived to be
defcribed in this manner alfo.
Fig, 98.
Kb
Let the indefinite right line MH be
divided into equal parts MN, NB, BK,
&c. ; and taking Nl at pleafure, at the
point I let the perpendicular lO be eredled
of any magnitude; then draw NO, and
at the point A let the perpendicular AC
be ereded till it meets NO produced to C.
From the point B draw BC, and at the
point E let the perpendicular ED be
erefted, which meets BC produced in D.
From the point K draw KD, and at the
point F let the perpendicular FP be raifed,
which meets KD produced in the point P.
After the fame manner, let the operation be continued in infinitum^ and the
points O, C, D, P, &c. will be in the logarithmic curve. To have the inter-
mediate points between O, C, D, P, &c. let the portions MN, NB, &c. be
bifeded, and the fame operation being repeated, we fhall have other points.
And finally, by multiplying the equal divifions infinitely in the right line MH,
that is, by fuppoling the equal portions MN, NB, &c. to become infinitefimals,
we (hall have an infinite number of points which will mark out the logarithmic
curve, the fubtangent of which (hall be always a conftant line, as appears
from the conflrudion. Making, therefore, Nl ^=. a, and fuppofing the infini-
tefimal conftant portion of the axis to be ;«■ ; make the ordinate GR z=. y,
GH = :>c', TS = J/ ; by the limilar triangles STR, RGA, it will be j/ . ;c ::
y . ai that is, ~ zz x, the equation of the curve.
Vol. II,
Q.
From
114
ANALYTICAL INSTITUTIONS.
BOOK III.
From this conftrudlion we deduce alfo this, which the firft fuppofes ; that is,
the primary property of the logarithmic curve, that the ordinates are in geome-
trical proportion, which correfpond to the abfcifles in arithmetical proportion.
For, fuppofing the equal portions of the axis to be infinitefimals, the little arch
OC, produced, will be the tangent ISIO, the little arch CD, produced, will be
the tangent BC, the little arch BD, produced, the tangent KD ; and lb of all
the oihers. , Therefore the triangles OIN, CAN, will be fimilar, and therefore
it will be OI . CA :: NI . NA. Thus, alfo, by the limilitude of the triangles
CAB, DEB, it will be CA . DE :: BA . BE. But NI = BA, NA i= BE ;
therefore it will be OI . CA :: CA . DE ; and fo fucceflively. Therefore the
ordinates will be in continual geometrical proportion. Hence, alfo, if we con-
ceive the axis to be divided, not into infinitely little parts, but into finite and
equal parts ; becaufe the intermediate proportional ordinates, for example,
between lO and CA, are neither more nor fewer in number than the inter-
mediate between CA and DE, and thus of others; therefore lO, CA, D^,
will be in geometrical proportion, correfponding to the abfciifes in arithmetical
proportion. Therefore, taking any two ordinates at pleafure, and other two
alfo where you pleafe, provided the diftance between the firft and fecond be the
fame as the diftance between the third and fourth, as would be lO, CA, RG,
SH ; then the firft will be to the fecond, as the third to the fourth.
The logarithmic curve cannot be defcribed geometrically, but only organi-
cally, and therefore it is called a mechanical curve ; and the impoffibility of
being geometrically defcribed is the fame as the impoffibility of the quadrature
of the hyperbolical fpace, as will be feen in it's place. Wherefore the integrals
of fuch differential formulae as belong to the logarithmic curve, are alio faid to
depend on the quadrature of the hyperbola.
Hence, in the logarithmic curve, the portions of the axis, or the abfcifTes
taken from fome fixed point, correfpond to the ordinates juft in the fame
manner as, in the trigonometrical tables, the logarithms correfpond to the
natural feries or progreflion of numbers.
Fig, 99.
10. This fuppofed, let DC be the
logarithmic curve, the fubtangent of
which is equal to unity, or, if you pleafe,
is equal 10 the conftant line a 5 and let
the ordinate AD be equal to the fub-
tangent, that is, equal to unity, or to
the conrtant line a, which is in the place
of unity. Taking any abfcifs AB = x,
make BC = y. But the equation of
the curve is ~ zz x. and therefore the
integral
SECT. X« ANALYTICAL INSTITUTIONS. II5
integral or fluent of — will be x. But x r: AB, and AB is the logarithm of
BC, or of ^. Now, to make ufe of the mark / to fignify the integral, fiim,
or fluent, all which mean the fame thing ; and of the mark /, which means the
logarithm, it will be/™ = ly^ in the logarithmic curve, the fubtangent of
which is a. After the fame manner, it will bey*-^ zz /j, in the logarithmic
whofe fubtangent ss i }/— ~ ly, in the logarithmic whofe fubtangent is If i
/ , ^ zz / ^ -j-^, in the logarithmic whofe fubtangent is equal to a. That is,
taking, in the logarithmic, the ordinate BC iz AH = y, if to it we fhall add
HK :z If, and if we draw KG parallel to the afymptote, and draw GE parallel
to AD, it will be GE = ^y 4- ^, and then AE zz I b -t y*
II. From the nature of the logarithmic it is plainty feen, that whenever the
quantity is infinite, of which we would have the logarithm ; which quantity will
be reprefented by an infinite ordinate in the logarithmic ; then the line inter-
cepted in the axis, between that ordinate and the point A, will alfo be infinite,
that is, the logarithm will be infinite. And if it (hall be equal to the firfl:
ordinate AD, that is, to the fubtangent, the logarithm will then be equal to
nothing. And if it fliall be lefs than AD, as if it were CIA., the logarithm will
be UA, and therefore negative. And laftly, if the ordinate were z= o, the
logarithm would be negative and infinite. If the differential formula were
— — , the integral would be — / y. And if it were ^ — , the integral
would be — la-^y. If it were — , the integral would be la — y*
a "^ y
and if it were -^ — , tlie integral would be •— I a —y. Thefe logarithms are
to be underftood in the logarithmic of which the fubtangent is unity. The
reafon of this is, that as the integral of — is I y, fo the differential of ly is
— . And, to fpeak in general, the differential of a logarithmic quantity is
that fradion, the numerator of which is the product of the fubtangent into the
differential of the quantity, and the denominator is the fame quantity. Thus,
the differential of — I a •'^y will be — -^ — . The differential of la — y
will be — -^ — . The differential of — I a — y will be -^ — , fuppofingthe
0^2 fubtangent
ii6
ANALYTICAL INSTITUTIONS.
BOOK III.
fubtangent of the logaathmic zr i : and whenever it is not fo, the numerators
of the differentials mufl be multiplied by the given fubtangent.
12. But, becaufe the logarithmic has no negative ordinates, it would feem
that we cannot find the qi^anrity which correfponds to the expreffion la — y^
that is, what is the logarithm ot a — y^ when a -^ y is a negative quantity,
or when y is greater than a. But, in this cafe, it may be obferved, that
I a — y and I y — a are the fame thing j and that in fuch a fuppolition, Vi/hen
y "^ a is pofitive, it may be the ordinate in the logarithmic; and, indeed, if
we difference the fiift logarithm, we Ihall have ~
a — y
, and if we difference
the fecond, we (hall have — — ; and changing the figns of the numerator and
denominator, it will be —
a — jr
, the fame as the firft.
Fig. 99.
13. Other properties concerning logarithmic quantities may be deduced from
thefe of the logarithmic curve ; and fird, that the multiple or fubmultiple of a
logarithm fhall be the logarithm of the quantity raifed to the power of the given
number. Thus, 2/a; zz h* ; ^Ix zz Ix^ j \lx :=: Ix^ ; ^Ix zz Ix^ ; nix :zz Ix ;
I
— Ix zz Ix n • and the reafon of this is, becaufe, iii the logarithmic curve, if
we take any ordinate whatever, OP ~ y,
(Fig. 3.) whofe logarithm is AO ; if
hO, OS, SV, &c. be equal to each
other, then AO, AS, AV, &c. will be
arithmetical proportionals, and the ordi-
nates AD, OP, ST, VI, &c. will be
geometrical proportionals. Wherefore,
putting AD equal to uniry, OP = j, it
will be ST ~ y\ VI rz /, &c. But
AS, the double of AO, is the logarithm
of J*, or /y* ; and AV, the triple of AO,
is /y'. So that zly — ly^, 3/y zr (y^ &c.
Thus, alfo, making AO = ly, and bi-
M Q, .S
fefting it at M, it will be MN zz^*,
and therefore AM iz t^-O, that is.
■^ly =: /y*. In the fame manner, making QR =7, and dividing AQ^into
three equal parts in M and O, it will be MN =. ^y zzy^. But -AM = ^ly,
and therefore \ly zz ly^ % and, in like manner, of all others.
We
SECT, I. ANALYTICAL INSTITUTIONS* IJ7
We mufl here obferve, that the integral of — ~ is not only — fyt as was
feen before, but may be thus exprefled alfo, / — , or ly" ; for, taking in the
logarithmic any ordinate OP iz jy, and making AH = AO, it will be, by the
nature of the curve, OP . AD :: AD . HA ; that is, y . i :: i . HA z= -i-.
y
But flA is the negative logarithm of OP, that is, of j, and is alfo the logarithm
of HA. Therefore it will be — /y rz /— ~ ly~' ; that is to fay, the nega*
tive logarithm of any quantity whatever will be the fame with the pofitive
logarithm of the fra6tion, of which the fame quantity is the denominator, or of
the fame quantity with a negative exponent. Thus it will be — mly ==
/ — = ly
m
y
14. Moreovef, the fum of two, three, &c. logarithais will be equal to the
logarithm of the produdl of the quantities, of which they are the pofitive loga-
rithms ; and the difference of two, three, &c. logarithms (hall be equal to the
logarithm of the fraction, the numerator of which is the produ6t of the quan-
tities, of which they are the pofitive logarithms, and the denominator is the
producfl of the quantities, of which they are the negative logarithms. For,
becaufe it is OP = ^'j QJ^ = z, it will be AO = ly, AC^= /z. Take
QB = AO, it will be AB zz /y 4- /«. But AB is alfo the logarithm of BC,
and, by the property of the logarithmic, BC is the fourth proportional to AD,
OP, QR, that is, = yz ; therefore it will be AB zz. ly ■\- Iz — Izy, Let
there be another ordinate MN = />, and take BV = AM ; it will be AV =
AM 4- AB = //> -i- /yz ; but AY is the logarithm of VI, and VI — pyz.
Therefore /p + (y + /z = Ipyz,
Now make QR zz z, OP = y, and take QM = AO ; it will be AM =
AQ^— AO zz Iz— ly. But AM is the logarithm of MN, and, by the fame
property of the logarithmic, it is MN = -^ . Therefore AM = /z — ly
y
zz / — . Let there be another ordinate BC = p. and take SA = BM. It
will be 2A = — AB + AM = — Ip ^ I—, But S A is the logarithm of
y
Sn, and SIX = -^ , (becaufe it is the fourth proportional to BC, MN, AD,)
therefore /z — /y — /p r: / ~ .
15. As
Il8 ANALYTICAL INSTITUTIONS. BOOK III.
15. As in other cafes, fo alfo in thefe integrations by means of the loga-
rithms, fome conftant quantity fliould always be added, that is, the logarithm
of an arbitrary conftant quantity, which is to be determined afterwards as
particular cafes may require.
16. But when the differential formula propofed to be integrated are fracflions
with a complicated denominator, fome cafes may be given in which it is eafy to
have their integrals by means of the logarithmic, and this will be as often as the
numerator of the fraftion (hall be the exad differential of the denominator, or
as often as it is proportional to it. And, in this cafe, the integral of the formula
will be the logarithm of the denominator, or it*s multiple, or fubmultiple, or
proportional to that logarithm.
Thus, the integral of — ?^ will be I aa -i- xxx the integral of ^—
will be I aa — .ar-v ; the integral of \ ^ $ ^'^^ ^^ ^ ^^ + x^ ; the integral of
^x
- will be 2I aa + xx, that is, I aa ■\- xx) ; the integral of — —
, , . , ^.„. __ will
sa -^ xx ' °^ aa + XX
be f/ aa + xx^ or laa -^ xx^ * ; the integral of ^^/V^ ^ will be f / a^ + x^, of
B — I.
mx X
l^a^ 4- x^ I and, in general^ the integral of -^ will be ± ~ /«" Hh x'^i
tMt IS, ± ml'm^ '^ , or ± l7±x> "" . Thus the integral of ^^
ax —'2XX
ax— XX
\ax — XX
ax — XX
will be I ax — XX i the integral of will be /\/ax — xx ; and thus of
° ax — XX
all others whatever, taking thefe logarithms from the logarithmic, the fub-
tangent of which is = i.
17. But if the numerator of the fradion be not of the form we have now
confidered, though the denominator may be fuch ; and that no one of its linear
components is imaginary; that is, when all the roots of the produdl from whence
k arifes are real ones ; then we may proceed in the following manner.
18. And, firft, the roots of the denominator are all equal to each other, or
PI.
they are not. If they be all equal, as in the formula ^ ^ , make x ± a
X ± a)"
= 2, and therefore x =z z^ x"" z: z + aT, x ± «!* — z" ; and fubftituting
thefe
SECT. I. ANALYTICAL INSTITUTIONS. Up
thefe values In the formula, it will be — ^ — ■ —, Wherefore, adually raifing
2.
z ^ a to the power m, each term can be integrated, either algebraically, or,
at leafl, tranfcendemally, by means of the logarithmic. Whence, inftead of z,
reftoring it's value given by x, we (hall have the integral of the formula propofed
Let it be, for example, ^ \ , . Put x — a = z, and therefore x zz z,
X — a)
x^ =z z^ -\- laz^ + Q^aaz + tf^, x — ^1' =: a' ; and, making the fubffitutions.
It wiU be "Y^ J and, by mtegration, z + 3/2 — ^--
— — 5 and, inftead of z, reftoring it's value given by x, we (hall have at laft
/ rz a: — <2 + / ;tf — «> ^^^ — =T*5 which integral, being:
differenced again, will reftore the formula propofed to be integrated.
19. Now, if the roots of the denominator (hall not be all equal, but either
all unequal, or mixed of equal and unequal ; then it will be neceifary, firft, to
prepare the formula, by making the teem of the higheft: power of the variable
in the denominator to be politive, if it (hould happen to be negative, and then
to free it from co-efficients, if it have any. Then, if the variable in the
numerator, when there is any, be raifed to a greater or equal power to the
higheft in the denominator, the numerator muft be divided by the denominator
fo long, as that the exponent of the variable in that may be lefs than in this.
Laftly, the roots of the denominator are to be found algebraically. Take this
formula — — for an example. Changing the figns, and dividing by 4,
it will become —2—^, that is, "^ — Again, let the formula
propofed be — ; ^^r—^ — r — r; dividing by 2, it will be — : — iffl— — -
that is, ^'^^ , If the variable (hould be in the numerator, and raifed
X + 2a X X + b
to a higher power than in the denominator, we muft make an adlual divifion,
by which we (hall have both integers and fradions. The integers muft be
treated in the manner before explained ; the fradions in the manner fol-
lowing.
3 20. Let
120 ANALYTICAL INSTITUTIONS. BOOK 111.
20. Let the fradlion be .'' ; I fay, this will be equal to two
fradlions, the numerators of which will be the fame as of the firfl, and the
denominators will be thefe : Of the fiift, it will be the produdt of one of the
roots into the difference of the conftant quantity of the other root, and of the
conftanr quantity of the fame root: Of the fecond, it will be the produ(fl of
the other root into the difference of the conflant quantity of the firft root, and
of the conftant quantity of this fecond root. Thus,
X -^ 2a X X + b
"^^'^ • + ^^ . And if the roots (hall be three, four, &c,
X + 2a X ^ "-ia X + b X aa — b
proceed always in the fame method. And if the fradions found after thfs
manner fliall be reduced to a common denominator, they will reftore the firft
fraction from which they were derived.
Now the integrals of fuch fradlions Co fplit, which will always be in our
power to find, fuppofing the logarithmic curve to be given, will be the integrals
of the formula propofed. Thus, it will he / ^ — — — = —^ — -, X lx+^
^ ^ X + 2a X X + b 2a -b
^.X l^TTTa-, that is, -i^ X /^^, or -±- l^ t±± , [n the
za ~-o 2« — o X -^ 2a 2a ^ b X + 2a
logarithmic whofc fubtangent = a.
Let it be ==^^==: ; this may be fplit into thefe two, ==r-~:
X -^ ia X X — ^a x-^-ia X —ia—\a
+ ^"^^"^ , or — 5fi_ _ -^r-r » and therefore it will be
r ^^'^ = t/ ^ 7x^ > or = ^v/ —T-r > in the logarithmic of which
the fubtangent == tf.
Let it be ■== — — == ; this may be fplit into three,
x + aXx^bxx + c
!'*• a^x j^ a^x
gf + a X — b —a X c -^ a x--bXa-\-bXc-^b x ■\- e X a — c X — b "C
a^x aa
and therefore / — == — -.==: = — == X / a? + « + == — ■-==
X -{■ a X X — b X X + c a + b X a " c a+b X c + &
X Ix ^b ^ ^"^ — ^ xlx + (i in the logarithmic whofe fubtangent
a — c X b-^ e
Let
SECT. I. ANALYTICAL INSTITUTIONS. 121
Let it be ~ ^ ' , that is, := — ^-^ — ; this may be fplit into
thefe three, ■ + == — — h
X + a X — 2a X o — a x — aX2aXo + a x + o X a — ox —a — o'
that is, — " "'^ — '■ + — ; and therefore it will be / " ^ "^ ~ /^v
2 X X -\-a 1 X X — a A'. -^ XT' — aax
— . LI XX — aa^ that is, / — , in the logarithmic of fubtangent zz a,
\/ XX — aa
22. If the denominator of the formula fiiall be mixed of equal and unequal
roots, as, for example, =— ^-^^= , then the formula mud be conQdered
as if it were '==^-^'T==r , and being fplit as ufual, it will be
a^'x
X — b X X ■\- c X — b '>^ X -^ c'
a}'x
4- ■=== — - j and then, multiplying the denominators
— b X c •\- h « + cx— ^ — <•
d^'x
by X — b. the other root of the propofed formula, it will be -
^ ^ X -h\^ X x-^e
= + ' — — ; but the firft term of the homoge'
X — by X c ■{■ b x-{-cXx — bx^b—c
fiettm comparationis has all the roots of it's denominator equal, and the fecond
term confifts of roots all unequal ; fo that, both of them being managed as
before, we may have the integral of ^ ' •= , which will be partly alge-
braical, and partly logarithmical, that is, ^^ x / '^^ v — ^ — ■ ;
b + c\^ X — o X — b X i + c
taking the logarithm from the logarithmic, whofe fubtangent zz a.
If there fliall be a greater number of equal roots, the operation muft be
repeated in the fame manner, as often as fhall be neceffary.
23. That cafe remains to be confidered, in which the fradlions have alfo in
the numerator the variable raifed to any power ; alvvays meaning, as has been
already obferved, that the power of this variable in the numer;itor be lefs than
the greateft which is in the denominator ; and not being fo, it muft be made
fuch by actually dividing.
In thefe cafes the formula muft be treated in the fame manner, as if in the
numerator there were no power of the variable, fplitting it, in the manner before
explained, into fo many parts, as are the roots of the denominator. Then, if
the exponent of the variable in the numerator of the given formula be an odd
Vol. II. R number.
122 ANALYTICAL INSTITUTIONS. BOOK IIf»
number, let ibe figns be changed in the numerators of the fraclions found ;
and if it be an even number, their own figns muft remain to the numerators.
After which, every numerator muft be multiplied by fuch a power of the
conftant quantity ot that root, which is in the denominator, as is the power of
the variable in the numerator of the propofed formula, prefixing fuch a fign to
that conftant, raifed to that power, as it's natural fign requires, which it has in
the denominator.
Let the example be ''^ ^ = . This being confidered as if there were
X + a X X — a
no variable in the numerator, it will be fplit into tbefe two, ^ +
X + a X — za
: ; but, becaufe in the numerator there is the variable raifed to the
X — a X 2a
.power denominated by unity, or the firrt power, the figns are changed in the
numerators, and are multiplied relatively by the conftant of that root which is
in it's denominator, that is, the firft by a, and the fecond by — a, and we (hall
, hlxx — bbx X a hh'x x ~ a , • hb'x
have = — , that is, +
X -\- a X X — a X + a X — 2a x — a X 2a 2 X x + a
Jl.__. J and therefore it will be / .=i=- = ^/y';e -^ a + bly^x — j,
3XX— « X -^ a X X — a
or his/ XX -~ ^^y »" ^hc logarithmic of the fubtangent zz h. Or otherwife, it
will be bbl\/ XX — • aa^ in the logarithmic of the fubtangent r:: i»
But it was needlefs to reduce this formula to two fraclions ; for, as it was
< ^— , the numerator is exadlly half the differential of the denominator, and
therefore, without any other operation, the integral will be bbls/ xx — aa^ (as
is faid at § 17,) in the logarithmic whofe fubtangent is unity.
s^'x
Let it be == , that is, -- — r- — '- 7- 5 ^^^ dividing the nu-
XX - aa X X -V b ' *3 -j- ^^z - aax - aab
pierator by the denominator, we fhall have xx -i- — TTTTZ" — Z — V ' ^"^
dividing again the term — bx^x by the denominator, we (hall have
z:z XX — bx + —^==r — : — . Now the two firft terms
:ix '— aa X X -^-h xx — aa X x -{■ b
2LT^ integers, and the laft has not the variable in the laft term of the nume-
rator, and therefore may be managed j fo that there only remains the term
ftill to be reduced. This being confidered as not having the
»g*^aa X X -k- b
?«CT, I, ANALYTICAL INSTITITTIONS. 122
variable in the numerator, will be T==r i= ^ - +
xx — aa X X + i> x + & X — aa + bb
aa •^- hh X X . aa -^ bb X x , , . . mi i
+ - :: - -j and therefore it will be
* + ^ X — zaZi -\- 2aa X — a X 2ab + 2aa
+ bb X x'^'c a.i + bb X bb'c aa + bb X aav
+ -== : -r
tc + b X XX — aa x + b X — aa + bb x -^ a X — zab + zaa
aa + bb X «*«■ -ttt! i ni x*-x • j ■ aalhx
, v\ hence, laltly, — i(x — bx —
K — a X 2ab + 2a* " xx —aa x x + b x + ^X^x—aa
aa + bb X Ih'x . aa ■\- bb X aax , aa + bb X aax y -r ^
+ ■ ■ + 4. . ■ ; and if we
X -\r b X -' aa ■{■ bb a'+^X— zab + laa x — a X 2ab + zaa
would ftili fplit the term — "^ , in order to have, finally, the integral
X -^ b X XX '—aa
of the propofed formula, it will be * ^ :=: xx ^^ ^x + ^
XX —aa X X + b x-\-bx -—aa+bb
H- ^ ^ + ^ "^ , Then, by integration, we (hall have
X + a X 2aa — 2ab x — a X zab + 2aa
J -"" l— _- = \xx — bx ^^7 X Ix + b + ^^-T X Ix + a -f
-^ XX - aa X x+ b aa-bb laa - zab ^
X Ix^a; taking fuch logarithms in the logarithmic of the fub-
zaa + 2ab
tangent iz i.
Now in this, as well as in all other integrations that can be made, we are to.
conceive a conftant quantity is to be added, though, for the fake of brevity, \
here omit it ; but it will be enough to mention it here.
^4. But differential formulae may have, and often have, fuch denominators,
of which we cannot find the roots algebraically; yet, notwithftanding this, we
may make good ufe of the Rule of Fradions in thefe cafes alfo. For we may
treat the denominator as if it were an equation, and, by means of the inter-
fedions of curves, may be found geometrically, in lines, the values of the
variable, Juft after the fame manner as folid problems are conftruded. And
fuch values or lines may be called A, B, C, &c. wich pofitive or negative figns,
i^ccording as they come out pofitive or negative. Every one of thefe, being
fubtraded from the variable, will form a root of the denominator in fuch
manner, that die propofed differential formula will be converted into one of this
n.
X X
form, ' — -=r — , and with this we may proceed in the fame
X - A xx + Bxx-C,8u:
manner, as the operation has been performed in the cafe of algebraical rootf.
R 2 25. It
124
ANALYTICAL INSTITUTIONS.
BOOK iir»
25. It may be eafily obferved, tliat the rule here produced ferves only in fuch
cafes, when the roots of the denominator are real ; for when it is oiherwife, the
formula being fplit into other fraftions, fo many of thefe vvill be imaginary,
(and confequently the integrals will be imaginary,) as are the imaginary roots
in the denominator of the differential formula propofed.
26. Therefore, when the denominator of the propofed differential formula is
compofed of imaginary roots, either wholly or in part, there is a neccffity of
having recourfe to other means. And, in the firft place, let the given formulc&
have their denominators of two dimenfions only, that is, of two imaginary
roots : and let it be, for example, ^ — .
The integral of this formula, and of all others like it, depends on the redi-
fication or quadrature of the circle j I fay redification or quadrature, becaufe,
one of them being given, the other is reciprocally given alfo.
Fig, 100.
Wherefore let ACG be a quadrant of a circle, the
radius AC =. a, the tangent CD zz x ; it vvill be AB =
^/.
aa + XX
, CB zi ^ —
V
aa + XX
, EB =
^
aa -f XX
EH =
aa + xx\i.
Drawing AK infinitely near to AD, then EO will be
the fluxion or difference of the arch CE. And from the
point O drawing the right line OM parallel to EB, and
EH parallel to AC, then will HE be the differential
of CB, and HO the differential of EB, and therefore
, and HO zz ^''' . , . Thence the little arch EO =1
-v/HEj + OH^, will be zz \/
of the formula
aa + X3c\%
a^xx + a^xxxx
Whence the integral
aa + XX
aa 4- xx\ ^ <^"- + '*'•*'
will be the arch CE of the tangent CD = ^, and of
radius CA = a.
Now I refume the formula
U
Ih'x
nator by aa, it will be
aa + XX
aa'x
- ; multiplying the numerator and denomi-
aa aa + xx
J but the integral of
is the cir-
bbx
aa 4- XX
cular arch, which has for it's tangent x. and it*s radius zz a \ therefore/
' D ' •' aa + XX
z= to the fourth proportional of aa, of hh^ and of the arch of the circle with
radius z: a^ and tangent = ;v.
3
Let
SECT. I, ANALYTICAL INSTITUTIONS. 12
a.itnx
tixx + /mi>
Let the formula be — ^ — r ; as, by multiplying the numerator and deno-
minator by b. it vvill be equivalent to this other, —r- X — — ^-^; it will be
f — ff^l! — . ~ to a fourth proportional to nh^ to am, and to the arch of a
circle, with radius = \/ ah., and tangent r: a^. And fo of all others of a
like kind.
27. On the contrary, therefore, the differential of any arch of a circle is the
produ(5t of the fquare of the radius inio the fluxion of the tangent, divided by
the fum of the fquares of the faid radius, and the fquare of the tangent.
And, as a conftant quantity is always to be joined to other integrals or
fluents, fo alfo to this of the recflification of the circle ; to have the integral
complete, we muft add a conllant arch of the fame circle ; for the difference by
which the arch, thus compofed of a variable and a conftant, can increafe or
diminidi, can never be any other than what belongs to the differential of the
variable arch ; fo that to the fame differential may belong, by way of integral,
the fum of the variable arch, together with any conftant arch of the fame
circle. Let us fuppofe that x is the tangent of an arch of a circle whole radius
is a^ and that b is the tangent of another conftant arch of the fame circle ;
we know that the tangent of the fum of ihefe two arches (Vol, I. § io8.) will
be — _ T^^^ . But the differential of this, multiplied by the fquare of the
radius, and the produd divided by the fquare of the radius, adding the fquare
of the fame tangent, is .found to be — ^ — , which is the differential of the
variable arch.
Let the formula be — ; —-z — — rr* in which .v;*- — 2bx + ^^ is a fquare.
Make x — ^ = z, and, by fubftitution, we fiiall have — ^^ — . Therefore
aa ■{- z%
J ; — = arch of a circle with radius z: a. and tangent — x. But z zz
X — b ; therefore / ■ —-7 — —77- = arch of a circle with radius =z a,
•^ aa -X- XX — zbx + bb '
with tangent zz x ^ b, when x is greater than b. But, taking x lefs than b, the
integral will be minus the arch of the circle, with the fame radius and tangent.
And, indeed, by differencing;, we (hould have — rrr— —i a the fame
■^ D? . aa '\- ib " zbx ■\- XX '
formula as at firfl:.
Let
12$ ANALYTICAL INSTITUTIONS.
BOOK III,
Let this formula be propofed, ■ ^^ "^ — ^-^^ — . Make the fecond term of
the denominator to vanlQi, by putting x — y -{- 2a. Making the fubftitiitions,
rt will be • -^ ^ ^-^ — - , that is, ■— r^-— H
b\y
yy + zaa jy "i" ^'^^ yy + '^^'^
Therefore the integral of the fiift term will be a third proportional to a,
to ^by and to the arch of a circle with radius zz \/zaa, and with tangent zz y\
Of the fecond, it will be I yy + iaa\^, in the logarithmic of fubtangent z: b.
Then, inftead of jy, fubftituting it's value x — 2^, the integral of the formula
— ffJ^ — ^^ — vvill be the third proportional of a, c^, and the arch of the
XX — /{ax + baa ^ ^ ' ''
circle with radius zz »/ laa, with tangent zz a? — ia\ with I xx -^ a^ax -\-6aa\^
&\(o, in the logarithmic of fubtangent r: b-
48. We will proceed now to fuch differential formulae, as contain radical
figns, that is, quantities raifed to a power with a fradion for it's exponent. If
the formula either is, or may be reduced to fuch, that the variable quantity
under the radical does not exceed the firfl: dimenlion ; and out of the radical is
a pofitive power ; then fuch formulae will always be integrable algebraically, and
will obtain their integrations by making ufe of a very fimple fubftitution ; and
that is, by putting the quantity under the vinculum equal to a new variable.
Wherefore let the formula be ax\/ax — aa. Put \/ax — aa zz 2, and
therefore x zz , x — —; and, making the fubflitutions, we (hall have
2ZZZ, and, by integration, -^z^ ; and, inftead of z, reftoring it's value given
by Xf it will be -| x ^x -- a^ i", the integral of the propofed formula.
If the given formula were ^^ , by proceeding after the farne manner
•^ax — aa
we Ihould have 2 x ax — aa"^^ for the integral.
Let it be xx^ya — x ; putting v/a — .v zz z, and therefore x zz a — z\
and ;^ zz — 42'^; and making the fubflitutions, we Qiould have 42;*^ — /{.az*z;
and by integrating, ^z^ ■ — ^az^ j and, inftead of 2, reiloring it's value given
by x^ it will be ± x a — x) ^^ — ~ X a — x]'^' .
If
SECT. I. ANALYTICAL INSTITUTIONS. 12^
If the formula were -.'*^-=, proceeding after the fame manner, we Qiould
W a — X
have the integral ^ x a — x]* — -^ X a — x]^.
Let it be x^x\/a 4- x ; make \/a t x zn z, and therefore x zz z^ '— a,
and X zr 2zi;, and xx zi zz — ^V ; and making the fubdiiutions, we (hall
have zz — ^^* X izzz, that is, 22^2; — a^^^z^z + a^^z'i;; and, by integration,
4-2^ — t'^z* + -^aaz^ ; and, inftead of 2, reftofing it's value given by x^ it will
be, laftly, 4- X ^ 4- x^i — ^^ X i~+n^' + ^^^^ x a + .rl'^^, the integral re-
quired.
If the formula were - , the Integral would be |- x <a + x\^ — — X
Let it be xx\/a + x\^, that is, ^;c' x a + «"5^. Make, as ufual, ia 4- x^»"
= 2, and therefore at = zt — a, x :=i ^z^^^z; and making the fubftitutions,
it will ht z^ — a x |2^2^, that is, 4-2^2; — ^az^z ; and integrating, 4.2T
— 4-^2^ J and, inftead of z, fubftituting it's value, it will be 4- X ^~+~^i
— 4-^ X ^ + ^1^.
If the formula were ■ ::— L , , we (hould have for it's Integral 2A/a + ;v +
2<Z
29. In general, let it ht ax x x a + x] " , and let the exponents /, w, »
m
be pofitive integers j make, as ufual, JhP^ " = 2, and therefore a ^ x '=z
n n It \
m ' n m ■ t tn \t j , . , ^ , «.
^ » ^ "^^ 2;, ^ z= 2 — i»l ; and makmg the fubftitutions, the
formula will be 2 *" — a[ X —az"' z; and adually raifing 2 " « to the
power/, it is plain that every term will be algebraically integrable ; in which
terms, being integrated, inftead of 2, reftore it's value given by x, and we ihall
have the algebraical integral of the propofed formula,
30. If
123 ANALYTICAL INSTITUTIONS, BOOK HI.
30. If the exponent m were negative, fo that the quantity under the vinculum
would pafs into the denominator, in which cafe the exponent m would then
become pofitive ; that is, if the formula were —^^ — ; making: the fame fub-
ftitutions, we (hould have z^" — a X — a'z"^ z; and adually raifing
z — ^ to the power /, every term would then be algebraically integrable^
— I •
excepting fuch cafes in which the power z z fliould infmuate itfelf, and then
we (hould be obliged to have recourfe to the logarithms.
But if the exponent / were negative, the two foregoing formulae would not
then be algebraically integrable, but might be freed from their radicals, and
reduced to the quadrature of the circle and hyperbola, as will be feen in it*s
place.
31. But when the variable under the vinculum is raifed to any power greater
than unity, provided the quantity out of the vinculum is the exad differential,
or any proportional to the differential, of the quantity under the vinculum ;
then, by means of the faid very fimple fubftitution, we might have the integral
of the differential formula, which faid integrals will always be algebraical.
Wherefore let the formula be ixxs/xx + aa; make \/ xx -{- aa nz Zt
whence xx -\- aa zz zz, zxx zi izz ; and making the fubftitutions, we (hall
have 2ZZZ, and integrating, 4.2'; and reftoring the value of z, it will be
v.^
If the formula were ^"^ , we (hould have for the integral 2\/xx + aa.
Let it be 2.ax — ^xx X \/ ax — xx 4- bb, that is, 2 x ^^ — ixx x
\/ ax — XX -\- bb \ put V ax — xx -^ bb zz z, and therefore ax — xx + bb
zz zzy and ax — 2xx = zzz ; and making the fubftitutions, we fhall have
4ZZZ, and integrating, it will be j-z^ ; and, inftead of z, reftoring it's value,
k'ls ^ X ^x — ^^ + bb^'^.
Let the formula be . ^^^" 4 f f_ . j^'g integral vvill be 4 x ax — xx -\-bb\^,
'^ ax — XX + bb
Let It be xxx '- ^axx X V >^ — ^^ > ^^^^ *^» 1 '^ ^^ — ^^
"i^xxx
3
make t/ x^ -^ ax"" = z, and therefore z'* = a;' — ax'', and 2^xx — laxx
SECT. I. ANALYTICAL INSTITUTIONS. 120
= j^z^z; and making the fubftitutions, we fhallhave +2*^;, and by integrating,
r\2' ; and, inftead of 2;, reftoring it's value, it will be ^^ x .v' — ax?j t.
TT
If the formula were 3-^^^-^ ^_ff± ^ ^^^ integral would be ♦ x x^ — axK)^\
V^TZ
Let it be 2xx y/xx + aa] % that is, 2xx x xx + a^i'^ ; put xx + aa\ '^ rr z
and therefore xx + aa — 2*, and 2xx iz: 4-2J* z; and making the fubfti-
tutions, we (hall have -IrZ^z, and by integration, i-z"* 5 and, inftead of 2, re-
floring it's value, ^ x ^'x -ir aa x ^^^^^h^*«
If the formula were - ^ ' — —, the integral would be %^xx + aa»
And, in general, let the formula be />.v x x x* + a'" " ^ in which ^ and
4!W may alfo be fraflions ; put x" -{- a"^ ^ r= z, and therefore z « zzx^ ■{■a'y
u
and wa;*""" x zz — 2 » z\ and making the fubftitutions, we (hall have
^z n z, and by integration, — ~ — X z « ; and, inftead of 2, reftoring
it's value, the integral will be — ^ — X x"* + a" X fe" + a"^ *^
' ° mu + mn
w— I.
If n were negative, or if the formula were — ^ — , in which « is now
W— I
X
n
m , m\ u
X + a
u—n
pofitive, we fliould have the integral — ^ — x x'" +/'''" •
viu — mn
Hence we may form this general rule, that the integral of fuch a formula will
be the quantity under the vinculum, the exponent being increafed by unity, and
dividing it by the exponent fo increafed ; or the integral will be a proportional
to this, according to the proportion which the dilferential quantity out of the
vinculum will have to the precife differential.
\ Vol. II. S 32. But'
I^O ' ANALYTICAL INSTITUTIONS^, BOOK III.
:>
2. But ftlll in a more general manner : Let the formula be px x x
x"* -^ a"' " , fuppofing r to be a pofitive integer. It will be equivalent to thii
Other, px x x x X ^ •{• a ' -^ make, as ufual, z :=. x -{■ a ^ ,
u
and therefore x" -^ d" zz 2 « , and mx^^ x :=: —z » z ; and, becauHi
u u ^ r— I
X*" zz z n — a"j, it will be x ""* zz z » — a"^ , Therefore, making
It
u ■ \^~^
the fubftitutions, we fhall have p X z~ •— a"^ y, -^z n z. Now, fuppol^
' tnn
ing r to be a pofitive integer number, then alfo r — i will be a pofitive integer
u
number ; and aflually raifing 2; " — a* io the power r — i, each term will
be algebraically integrable, in which integral reftoring^ inftead of z, it's value
given by x, we fliall have the integral required.
rm—i.
If n were negative, that is, if the formula were — ^ , in which n is
X + di
now pofitive, making the fubflltutions, it will be ^ x 2
— m\
— z a Zf which is likewife integrable.
In all thefe cafes, if the quantity under the vinculum, Inftead of being
x*^ + a", had been x'" — a", or a" — x"*, we might proceed after the (kme
manner, without hindering the operation*
By this method we may find likewife, that it will be
fax'^^^x X J e + fx"" = -^ X 7+~^».
SECT. I. ANALYTICAL INSTITUTIONS. I3I
2m — 1 • _y w
— = — ^^ -r?— X ^ X e -r fx""! .
, ■37J— I • / , j^ W lOff — 24fAv + ^OJX V / ^ . A.wl*
And fo we might go on as far as we pleafe.
33. Likewife in the cafe, in which the variable out of the vinculum (liall be
in the denominator, the formula will be algebraically integrable by the help of
two fubllitutions, provided the exponent of that variable out of the vinculum
X X x"' + a"^ "
fliall have a certam condition ; thus, let the formula be
mti
rm H 1- I
a* ITn""" int. m m\ u
Then make x — — , X— — ^ x zr , x -\- a \ ■=: ~
+ a y'
y yy m ' • mn
y —
Then making the fubftitutions, the formula will be
n
zm , m m\ u mn
yy ^ Zk ^ y
y " . irn, m ni\ u
—-^n > ihat is, - f^-^y X 1-_±±Z_ .
a u
a formula which has the conditions here required, and which may be integrated
algebraically, by means of the fubftitution mentioned at § 32.
If the formula propofed were — -li=, that is, . ,/ "" ; this having
xWax + xx *v^T^
the conditions required, will be algebraically integrable j which is alfo to be
obferved of others.
S 2 ' 34. But
132' ANALYTICAL INSTITUTIONS. BOOK III.
34. But here it may be obferved, that, in the general formula, it may alfo be
«^ ~ I, in which eafe the power of a;"* + a will be rational, that is, inte-
grable. [qu. integral.]
Alio, in this cafe, fuppofing « to Be a negative number, (for when, ic is
affirmative there will be no difficulty,) we may make ufe of the fame fubfti-
tution, and of the fame method, by which the integrals may be found of fuch
formula, the integrals of which will not always be algebraical. For very often
they will depend in part upon the quadrature of the hyperbola, that is, on tho
logarithmic curve.
Therefore, by a known method, we fliall find that
m — l.
7« — 1 . -v
/■ '*■ I OT , wj — I
X + a \
'2m — I. ■ tM
~ a
/•* X 1 J m . m .
a m X a + X
I
5w— I. m . m VI -I m .
m T m , m 2m
a
x + a ' m X a + X
m— I .
a + a: ' %m X a •{■ X ^
2m— 'I . tn
3w— I. ^m 2m
J ^ . - = — la + ;p ■\ — ' . * &c.
. a + X I m X a +x im X a •\- x '
35. But the manner of proceeding will be very different when the propofed
differential formula containing the radical, are not fuch as that the quantity out
of the vinculum (hall have thofe conditions before mentioned. Thefe formulae
may always be delivered from their radical, provided they contain but one,
which is that of the fquare-root, and that the variable under the fame does not
exceed two dimenfions. Now, for thefe there will be occafion for fome caution
in the choice of fuch fubftitutions as are to be made, that they may be freed
from radical figns. When this is done, we may go on to integrations, either
algebraical, or fuch as depend on the quadrature of the circle or hyperbola, after
»he manner already explained, if they come under the given rules. If not, we
muft have recourfe to other methods, which are to be given hereafter.
8 If:
SECT. I. ANALYTICAL INSTITUTIONS. I33
If the radical of the propofed formula were s/ ax ± xx, or s/xx ± ax \
this radical may be made equal to -^ , meaning by 2 a new variable, and by b
any conilant quantity whatever.
If the radical were ^/ aa ± xx, make it = ,v + :s, or a; — x.
If the radical were \/ aa — aw, or \/fp — xv, put the radical = \/ fp-
-f. ^ , or =1 V//> — -^ . From fuch equations the values of x and S: may
be derived, expreffed by x and conftant quantities j which values are to be
fubRituted in the given formula, and we fliall have other formula free from
radicals, and given by 2. In the integrations of which, if they can be had, the
value of % by x being reftored, we fhall have the integrations of the propofed
form u lie.
36. If the quantity fliould have three terms, that Is, the fquare of the variable
with the redlangle of the fame into a conftant, and befides, a term which is
wholly conftant ; then either the fecond term mull be taken away, after the
ufual manner, as in the common Algebra j or, if the conftant term be pofitive,
as in \/xx -^ ax -^ aa for inftance, however the others may be pofitive or
negative, provided the quantity be not imaginary ; make \/xx -\- ax + aa :=.
xz
/2 + -^ ; and if the conftant term be negative, as, fuppofe \/ xx -\- ax -r'
aa.
it may be made \/xj<: + ax — aa zz x -j- z. ^
From hence it may be feen, that the whole artiftce conlTfts in comparing the
radical quantity to fuch other quantity compofed of the given variable, and of
a new one with conftant quantities, as that an equation may refult from thence,
from whence we may have the value of x and of x, free from radical figns.
Let there be propofed to be integrated the difFerentlal formula x^xy^ax — xx.
■*■» ] ^u r ■*^» I • alb
Put \/ax — XX ^= -r- 9 and therefore a — x zz -— - that Is, x zz rr ,
D bo ^ zz -\- bb
and X = — ===- , x^ = ,, , and \^ax — xx zz -~~ = -~—n .
zz + bb"^ z% + b^^ b zz + bb
za^b'^zzz
\6 f
Make the fubftitutions in the propofed formula, and it will be —
ZZ + bff
formula which, though free from radical figns, yet, as to it's integration, will
not fubmit to the ufual methods..
Let
^34 ANALYTICAL INSTITUTIONS. BOOK III.
aax
x^ ax + xx
Let it b« — J- ' . ■ . Make Vax 4- xx = -r-j and therefore it will be
X = 77 , A' = — " - — ^-.^ , s/ax ■\- xxzz -r- ^ . Making the
fiibRitutions in the propofed formula, it will be — -^ , and by integration,
2az
— -— - j and, inilead of z, reftoring it's value by x, it is /
Let it be — ; put \/ax -^ xx ^ ~- , and making the neceflary fub-
ilitutrons as before, the formula will be — ■ ^^, , that is, ~ j=— -^-^=:=r- ,
zz
But we have already feen how to manage this by the Rule of Fraftions, and it
will have for it*s fluent ^ '^ , + f ^ ^^-^3 i^^ t^^e logarithmic the fubtangent
of which is unity. And,, inftead of z, reftoring it's value by x, it will be
^ , V ax -\- xx — X
/-— ^= =^ ^ ax -)r XX '\' \al 1 — ^ -, in the logarithmic of the
wax + XX *v ax ■\- xx + *
fame fubtangent z= i.
Let it be . Make s/ xx + ax — aa iz x + z, and therefore
V XX -^ ax — aa
.,, , zz + aa . iazk — 2zzz + laaz , ^
It will be x = - — --- , X = .1 , and ^/^xx ^ ax — aa z:
fl — zz a — 221
AT + 2 =: ■ —^- — . Make the fubftitutions, and the propofed formula
•11 u zz -{- aa y. 2z ^1 ^ • 2ZZZ -f taaz j i • ^ • ^ i • ,
Will be . , that is, — , ^ ; and by integration, (which may be
pei-formed by the foregoing rules, it is ^'"' — -i^ + tz + ial a — 22,
4 X a — 2z
in the logarithmic with fubtangent z=z 1. And, inftead of 2;, reftoring it's valufe
by X, it will be, laftly, /—=:£== = [''__ ^^ _^^
^xx H- ax -aa 4^2 + »^v - ^^^^ + ax - aa
+ i\/xx -\- ax ^ aa + ^al a -^ 2x — zVxx + ax -^ aa, in the logarithmic
whofe fubtangent is unity.
9 37- As
SECT. I.
ANALYTICAL INSTITUTIONS.
^35
37. As to feme radical differential formula, the trouble, indeed, would be
rnperfluous to tranfmute them, by means of thefe fubftitutions, into others that
are free from radical figns, in order to prepare them for integration ; and fuch
are all thofe which of their own nature require the quadrature or redification of
the circle. Wherefore let there be a femi-
FtF, loi. i\T circleGMD, (Fig. loi.) iL*s radius AD zza^
"^ AB = Xf whence BF — \^aa — xx ; and
drawing CH infinitely near to BF, it will
be BC = X, EF =z
XX
:. Therefore
ii C
Alfo,
Vaa — XX
therefore /
the expreffion of the infinitefimal redangle
1) BCHE will be x\/ aa — xx, and therefore
JxVaa — XX is equal to the fpace ABFM.
will be the expreffion of the infinitely little arch FH, and
= arch MF. And if the little arch FH be drawn into
*/aa —
half the radius, then
CLdX
" / will be the expreffion of the infinitely little
2Waa~-xx ^ ■'
fedor AFH, and therefore/'
= to the fedor AFM*
a w aa — xx
In the fame circle let it be now DC = x, and CB =z x. It will be CH =
s/iax — xxy EF =
• Wherefore J x\/2ax — xx will be equal to
V 2ax—x*
the fpace HCD. And thus/- "^ r = arch HD, and /- — ^^ = fedor
V 2ax — XX a V i.ax—xx
AHD. In fuch as thefe, therefore, the trouble [of transformation] would beneedlefsj
for, in the firft cafe, we fliould make s/ aa — xx zz a — -^ , and therefore x =:
■iab% • zab^z — 2ahizz
X =
, \/aa
making thefe fubftitutions, it will be — === — 77
^ ^aa - XX ""^ + ^^
— x% alh — flza t.,
^a; = ^ — -r- = TTT- * Now,
b zz + bb '
ax
iabz
i a formula for the
reflification of the circle, the tangent of which is equal to z, as has been feen
already at § 26.
Alfo,
13.6 A.NALYTICA-L I N S.T I T U TI-ON S^ BO(;>KIII,
Alfo, let it be -^-^^4=. = -^^-— rr, a formula which requires the fame
^.- . T ri • '11 u • • zaahz X bb — szl»
reamcation. In like manner, it will be xv aa — xx "n • -r— , a
formula which, though at prefent we cannot manage, yet afterwards we (hall
find to depend. on the fame circle.
x% J ^, r Zahh
In the fecond cafe, I vi\-\x. </ lax — xx — ~, and therefore .v zz — ,
^ *^ za + bb'
X ~ — J^^U!^ , , and ^yiax — XX r= -7- =: — ~t?* Making the fubllitu-
zz + bbT ' ^ b zz + bb ^
tions, it will be '''' = ^^ITJl » ^^^^ redification of the circle.
V2ax^^x ^^ "*"
a /bk
Let it be alfo = — ,, , ,. , the redification of the circle, as
2V2ax —XX ^^ ■!■ ''''
before.
%a^b^z^z
In like manner, it will be x*/2ax —xx iz — — ^, , which includes the
2a 4- bbY
fame circle.
38. If our differential formulas (hall be compofed of two radical quantities,
in this cafe the operation will be double, but ftill it will fucceed as well. For,
in the radical quantities, the fecond term may be wanting, or it may be taken
away, and the formula may be multiplied by an odd power of the variable ;
and that by putting one of the radical quantities equal to a new variable. And
thus the propofed formula will be reduced to another, which will contain one
radical only, and which confequently may be managed in the ufual manner.
Let it be, for example, "^ '^'^ ^^ _f fl , \ puj- ^ ^^ -^ xx iz y, and therefore
^bb-\-xx
XX =:yy — aa, xx = yy. Making the fubftitutions, it will be - ^^ ^ ^ ^^ ~ ^-^ ^
'^yy —aa +bb
that is, •^'^ — ^^-^^-^ , each of which we know how to
"^yy ~~ ^^ ■{- bb vyy •— ea -\- bb
manage.
39. If we confider a little this manner of operation, we may ealily perceive,
that, in thefe radical formulse, it will not fucceed in general, that we Qiall be
able to free them from their radical vinculum, except when it is a fquare-root,
and the invariable under the vinculum does not exceed the fecond dimenfion.
I fay in general ; becaufe, in feveral cafes, it may fucceed, whatever the radical
may
SECT. I. ANALYTICAL INSTITITTIOHS-
may be, and whatever the power of the variable may be, which is under the
vinculum. And certainly it will, in all cafes, be comprehended in the two
^^-r
following formuIsE, the firfl: of which is this, -^ — Z__Z — 1 , in which «,
tm4-l
y
«, /, are pofitive integers, and may alfo be nothing; and this obtains, bf
making y + T ^" =2;, whence / = 2" — T, y = "H £; and making
my
the fubftitutions, it will be "''"'" 'f "" "" — ^, that is, Ullll-UtZl, But
-" my
t+lXm ~~n I«^/+' J L • • , . '11 t
y = z — if ^ ; and when f is an mteger, the power / + i will be
an integer, fo that the propofed formula will be free from radicals.
If / were negative, the formula would be the cafe confidered above at § 32,
which has an algebraical integration.
In other cafes, the integral will depend on the quadrature of the circle, and
of the hyperbola, as will be feen in it's place.
±'
The fecond formula is ^ j/ X y"* -h b^ ^ , which, when is a whole
number, may always be freed from it's radical figns, either in the whole, or, at
ieaft, from radicals of the complicate quantity., which will be fufficient. Where-
fore, make y"* -{- b^ ^ n z, and then it will ht y^ z^ z -^ b^, y zz
^_, iL JL_x
p \ t p t . t ,m \tn
~ — -^--z z y. z — b
2; X z
z — ^ ' , y = , and y =
m
Z
* ^b'
i and making the fubftitutions, we fhall have the formula
±1 t jTn] m m
f±l f X 2 -' X 2 ' — r ' "^ " .But when "-±i is an integer,
I + «
m
Vol. II. T . only
the power ^-^^ — i will always be an integer, [or o,] fo that the formula will have
13^ ANALYTICAL INSTITUTIONS. BOOK III,
only radical figns of the complicate quantities. And therefore, when -— ^ — j
is a pofitive integer number, the integration, at moft, will depend on the quadra-
ture of the hyperbola, or on the logarithmic, and may be had by the given rules.
And when — i is a negative integer, the integration will depend on the
quadrature of the circle, and of the hyperbola, and may be had by the rules
which will be given in due place.
40. Now let us go on to fuch formulse, which being fra(51:ions free from
radicals, the variable is raifcd to any power in the denominator, which I will
fuppofe to be compofed of imaginary roots, becaufe in thefe only there is any
difficulty. I fay, that as often as the denominator is reducible to real com-
ponents, in which the variable does not exceed the fecond dimenfion, the
formula may always be fplit into fo many fradtions, as are the forementioned
real components, each of which will be integrable, fuppofing the quadrature of
the circle and hyperbola ; and confequently the propofed formula will always
be reducible to the faid quadratures. To do this, let there be propofe.d this
formula, ^^"^ . Take a fiditious equation,
A* -i- ax + 6i X Jr4^ -{■ ex + l>c
aax
Axx + Bx . Cxx 4- "Dx . 1 • 1 r 1
— — = — ; TTT H : — -TT—i *" which formuk
xx + ax + bb X XX + ex + be ^'-^ + ax + bb xx + ex -k- be
the capitals A, B, C, D, are conftant arbitrary quantities, which are to be deter»
mined by the procefs.
ahx
Thus, if the formula were ■ ' ■ ■■■ ■ — , we Ihould make
XX + ax ■\- bb X XX ^ aa X a: ± c
, Kxx + Bi , Q.tix + T>'x . Hv A J ^1 ^ —J !^
*t equal to -— rr + 7^^ r— . And thus we may proceed m
-• XX -^c ax '\- bb XX ^ aa x ±c "' ■^
iChe fame order, if the components in the denominator were more in number.
When this is done, the terms of this equation are to be reduced to a common
denominator, and laftly, by tranfpofition, the equation muft be made equal to
nothing. Then, by comparing the firft terms to nothing, the value of the
aflumed quantity A may be found. And fo, by comparing the fecond, third,
fourth, &c. terms in the fame manner, the values of the other capitals B, G,
D, &c. may be found, exprefled by the given quantities of the propofed
formula; which values, being fubflituied in the places of the affumed capitals
A, B, C, D, &c. in the equation, will fupply us with fo many fradions as are
.equivalent to the propofed formula ; and which, being reduced to a common
.djenominator, will exa<5tly reftorc the formula at firfb propofed.
Q Of
SECT. I. ANALYTICAL INSTITtTTIONS. 139
Of this we will take an example. Let it be propofed to find the integral of
this formula ■ ■ ' — '■ — — — ■ . Therefore I affbme this fididous equation
XX + 2a.v -^ aa X xx -{r aa
aa'x A.vi + fti- , Cxx + D
_ + - — ; • Then I reduce the
equation to a common denominator, and, by tranfpofing the term aax^ I reduce
it to o, and find it to be
+ Q>:}x + D.v*i + 2'
+ ^Qax^x — Qi
har.xx 4* ^^ax \
\Daxx — IDaax > n o.
Qaaxx — aax j
Wherefore, from the comparifon of the lirft terms with o, we fliall have
A 4- C rr o, or A iz — C, From the fecond, B + E) -f iQa zz o, that
•is, putting — A inftead of C, B — zA^? — D. From the third, Aa^ 4- ^Da
2D
— Ctf* — o, that is, C zz <« H . From the lafl, '^aa — 'Daa — aa
a
= o, that is, putting, inftead of B, ii*s value given by D and Ay it will
be D z= A/» — t> and therefore it will be C =s i— fJl— j but C = — A,
and therefore A = — ,Dz= — 5^, B = |, C= j whence we (hall
, , n oa'v XX + 'iax xx + ax
have at laft — = — - ^ — ~ .
XX 4- zax — aa X XX + aa /^a X xx + zax — aa 4a X ** + aa
But, by making the fecond term of the denominator to vanifh, where there
is occaliou, the homcgeneum comparationis is integrable by the quadrature of the
circle and hyperbola ; the integral of which, by the given rules, will be found
XohQ -^l\/ XX -^ iax — aa •{ —=. I ^/ x -\- a -- ^ laa -^ — - — l\/'x-{-a-\-^2aa
,_ J^li^xx + aay fubtrading, beiides, from thefe logarithms the fourth
proportional of 4JJ, of unity, and of the arch of the circle, the radius of which
is a, and the tangent ir x. Therefore the integration of this formula depends
on no higher quadratures than thofe of the circle and hyperl^la.
41. If, befides^ the fradion Qiall be ipultiplied into any power of the variable,
aax X
which power is pofitive ; as if the formula were _ =-= * ; make
XX + zax — aa X xx + aa
Ax X + Bx^x . Cx "* X + Dx X J 1 1 1 r ^1 • 1
it ecual to j and let the values or the capitals
^ XX 4- ^^-v — aa XX + aa ' *
A, B, C, &c. be found in the fame manner as above, or you may work as if the
iaid power were not there ; and the refulting fradions may be multiplied by the
T 2 faid
.A^'
14^ ANALYTICAL INSTITt'TION*. BOOK III,
faid power, and we (hall have, in like manner, fo many fradions, which will
not require any higher quadratures than thole of the circle and hyperbola, and
which may be managed by the rules already given.
42i And if the power of the variable (hall be negative, that is, if it (liall be
pofitive in the denominator, all the denominators of the refulting fradions maf
be multiplied by this power, and they will acquire the form following.
— «■
X X
As, for example, — :==:=. . This being refolved as if
XX •\- ax ■\- bb X XX -^aa X « J: c
x~" were abfent, and then multiplying every term by x" , it will be
+ 4-
XX -^ ax •\- hh X xx:haa y. X ±. c xx •\- ax •\- bb y, x xx±.aa xa" ;v ± <• x ;«'
underflanding now by the capitals fnch values, as, being found by the foregoing
method, fliall make the fum of thefe fraftions equal to the propofed formula.
The laft fraflion will have no occafion for any particular artifice, becaufe it's
integration is known by the common rales.
As to the firft, to clear up the example, let it be A = aa, and B := ahb,
whence it will be thus exprelTed, _a__y — _^ which is to be made equal to
XX -k- ax -^bb y, X
— rr H • -^"d t"us we mult eo on till
XX -^ ax -V oh n o
' ' ,- X
the laft term becomes conftant, that is, the laft power of the variable x muft
'have it's index ir o. When thefe fradions are reduced to a common denomi-
nator, and all made =: o, we fhall have the values of the capitals, as was
done before. The fame thing muft be done in regard to the other fradion
XX ■\- X ^ ^^^ thus, finally, the integral will be found of the propofed
~r n
XX "X. aa y X
formula,
"Wherefore generally, fuppofing only the quadratures of the circle and hyper-
bola, we rpay always have the integral of the foregoing formula, if the com-
ponents of the denominator be real, provided in them the unknown quantity
do not exceed the fecond dimenfion.
43. But if the denominator of the propofed formula, or fradion, may not be
refolvable into it's real components, in which the variable does not exceed two
dimenfions, nor can be reduced to fnch by the common rules of Algebra ; yet
^"itm^y always bt reduced to (iich by a little further ^rtifiGe, as often as it is a
- ■ ^ convertible
4
SECT. I. ANALYTICAL INSTITUTIONS. I4I
convertible formula, or the prod ud of feveral convertible terms. I fliall call A convertible
that a convertible formula, in which the variable has the greateft exponent of^'^JJ'""^^'
it's dimenfions an even affirmative number; as, fuppofe n were fuch, then the^ ^ *
laft term would be a"^, and the terms equidiftant from that in the middle mufl
have the fame co-efficient, and be affected by the fame fign, fupplying the di-
menfions by that conftant quantity, of which the laft term is formed. Such
would be the formula x^ + ^% or this, x* + ^^^ + ccxx + aabx + «*, or this
other, x^ — bx^ + /&V — a*bx + a^. Now, if it were x^ + bx* + a*x
4- a^by it would be written in this equivalent form, x* -^ a* X x -\- b, m which
a:* + a* is a convertible formula, and x + b \s linear, which does not increafe
the difficulty. The fame thing is to be underftood of infinite others.
44. Therefore now let us have x — a" to be refolved into it*s real com-
ponents, in which x may not exceed two dimenfions, and which (hall not have
fradions for their exponents ; and, in the firft place, let m be an even affirm-
ative whole number. In this cafe, it will be divifible into x'^'" -f- a^ and x*
— a^ , without any fradlions in the exponents, becaufe of m being an even
whole number. The firft divifor may be refolved by the rules which will be
foon given for the binomial x + <» • The fecond, x^"* — c^' , if ^m fliall
be an even number, may be again refolved into x"^"* + a^„ and x^"* — a^ ,
without a fradion in the exponents. But, if ^;« fhall be an odd number, it
will be refolved by the rules tliat will be prefcribed for the binomial x^ -^ a ^
when,.w is an odd number.
In the fecond place, let it be ;f ■\- a , and let w be an even affirmative
whole nvjmber, in which cafe the formula is convertible. Let us fuppofe
x" + a" ~ o^ and then let there. be formed, a convertible formula, in which the
greateft exponent of x may be m — 2, and which may have all it's terms, and
the laft term may be ^ "" , and the co-efficient of the fecond term may be ^,
for example, that of the third cCy that of the fourth ^'% and fo on ; ..and let this
be compared to o, whence refults an equation. Let this equation be multiplied
by XX -f- fx + aa ', the produ'ft will be another convertible equation, in which
the greattft exponent of x will be r: m. Let this equation be compared, term
bv term, with the fidirjous equation x ^•\- .a zz, o, in which the co-efficie.iis
of the mtermediate terms are =.0; and, j^y the comparifon of the kcord
terms having the va!ue of the afllimed quantity ^, from the comparifon of the
third terms the value of fr, from that otthe.foiirth terms the value oi d\ ord
fo on to ti.e middle term, taking this m aUo j now, fioni thai ol the middle the
ether
I4i ANALYTICAL INSTITUTIONS. BOOK lit.
Other equations will become the fame, becaufe of their being convertible equa-
tions which are compared. From this laft term will be found the value of/
exprefled by an equation, which will have Im for the number of it's dimenfions,
of which all the roots will be real, and will give us the values of/; which being
fubflituted in the trinomial xx + fa + aa^ will give us fo many trinomials,
the produfts of which will reflore the propofed binomial x^ + a"'.
Let the example be x* + a*. I ,take a convertible equation of the fecond
degree, xx + bx -{- aa zzz o, which I multiply by xx + fa -{■ aa = p, from
whence I have another convertible equation,
jtf* + hx^ + 2aax^ + aafa + a* \ ^
+ fa^ + hfa* -\- aat^ J "~
I compare this with the ^iiditious equation x* + a^ = o, and from the comp3«
rifon of the fecond terms I find ^ -}- / z= o, or ^ ~ — /. From the compa-
rifon of the middle terms I find ,2aa -f- ^/ ~ o> and, inltead of h, fubltitutmg
it*s value — /, it will be ff — 2aa == o, or /=: ±: \/iaa..
Let it be x^ 4- ^^. I take the .convertible .equation ** + hx^ + fV + a^hx
+ >»* = o, which I multiply by x'^ + fa ^ aa :;:^ o, and the refuliing equation i&
i(^ -^ bx^ + ccx^ 4- ^aabx"^ f O^x'^ + a^fa + a^ 1
+ fa^ + hfa'' + fccx^ 4- a'bfa'- + a^bx | = o.
4- <j V 4- d^c^x"- J
I compare this with the fiditions equation x^ •\- a^ •=. o, and from the com pa-
rifon of the fecond terms 1 find b + / = o ; from the comparifon of the third
terms I find cc •\- bf -^ aa zz. o^ that is, fubftituting the value of b, cc — ff
■\- aa zz. o\ from the comparifon of the middle terms I find laab 4- fee xz. Op
that is, inftead of b and cc, fubftituting their values, fa — ^^af = o. ^
Now, by uiStually performing thefe operations, we (hall find that
If m =z 4, it will he ff ^-^ 2aa = o.
If »i = 6, then fa — ^aaf = o.
If « =: 8, then /♦ — 4aaf* 4- 2a* = o.
If » = 10, then fa — s^^P + S^V = o-
•If «» z= 12, then/** — 6aafa + ga*fa — 2a'' =s o.
If /» = 14, then/* — 7««/' + H«y — 7^7 = ©•
And fo wc might proceed to the other even values of m*
Inftead
SECT, i;- AKALYTICAL INSTITlfTlONS, I4I
Inftead o£ x* + a% Jet it be x* + 2i^x^ + laahx + a*, which is alio a
^convertible formula, f multiply the convertible equation ^-^^ -^ bx -\- aa z=, o
hy XX + fx + aa zz o, and I fball have, as above,
M* + bx^ + laax* + aa/x + ^* "1 _
+ ^' + b/x"- + aabx J — o.
I compare this with the fiftitioas equation x* + zh^ + 2aal>x + a* z: o,
and from the comparifon of the fecond terms I find b + / zz ih, that is,
V •=. zh — /i from the comparifon of the middle terms T find laa -\- bf zz. Oy
and, inftead of b, fubftituting it's value, we fliall have laa -j- 2bf -^ Jf =z o,
that is, /f — ilf — 2aa zz o.
Let it be x^ + ^V 4- a^, I take the convertible equation x* + l»i^ +
cix^ -^ aahx -j- ^i* = o, which I muhiply by xx -{- fx •\' aa, and I (hall
have this produd,
x^ + hx^ + ccx^ + 2aahx^ + <2 V -f- a^fx + ^'^
+ >* + hf)c*-\- ccfx^ 4- «*^/a;^ + a^bx \ = O.
4r rt«x* -1- /2W
} =
This being compared with the equation x^ + a^x^ + <?* = o, I find, from
the comparifon of the fecond terms, b +/ = o i from the comparifon of the
third terms, cc + bf ■{• aa zz o -, and, inftead of b, putting it*s value, it will
be cc — ff-^- aa z=z o ; from the comparifon of the middle terms, 2aab + ccf
zz a^ i and, inftead of b and ccy. pmtting their values, it will be /' — Sf^^f
«— ^j' zi o. And fo for as many others as you pleafe.,-
Now let us have x* + zbx^ + laabx + a* to refolve into it*s real compo-
nents, in which jf has no fradlion for it*s exponent, and does not exceed the
fecond dimenfioa. The equation which (hould give us the values of /is there-
fore // — ibf =r 2aa, from which we obtain both the real values of/, that is,
f — If -{- s/iaa-\-bb^ and / z= b — »/ 2^a -\-bb» Wherefore, fubftituting
each of thefe values inftead of/, in the trinomial xx -\- fx + aa^ we (hall find
^t ^ + 2bx^ + 2aabx + a^ is the produft of the two real components
XX '{' hx ■{' xs/iaa ■\- tif ^ cm, and xx •{' bx — xsf xaa -{- bb. -j- aa.
Thus, if it were ^ + aax* + a*x^ + a* = q. The equation which gives
the values of / being /^ — 2aaf zz o, from thence we Qiall have the value*
of / all real, that is, / = o, / ~ A/iaa, and / = — V 2aa ; fo that
** + aax* + «V + a^ is the produft of the three real components xx + aa,
itx + xV<2aa •\- aa, and xx — xs/iaa + aa.
Let us have x^^ + a^\ The equation which ought to give the values of •/
is/* — 5«^. + 5«y = o. From whence we derive the values of /all real,
ibai
144 ANALYTICAL INSTITUTIONS. BOOK lit.
thatis,/=o./=V^4^,/= -,,v/^4^,/='»v/^-=^. and
/ zr — a»/ '^ " ^^ . Wherefore, fubftituting every one of thefe values inftead
of /in the trinomial !kx H-/v + aa, we fliall find that a?" + «'° is the pro-
duct of ihefe five real components, xx + <i^i xx + <7.vv^ ? -f ^^,
ATA? — axx^ ^"^ ^^ -{- aa. xx + ax>/'^-^~^ + rt<?, and xx — ax'/ ^"'' ^ +^tf.
2 i 2
Whence it is to be concluded, that the integral of any differential formula,
whofe numerator is ^c" multiplied into any conftant quantity, and the denominator
is of a like nature with thefe here confidered, will not depend on quadratures
higher than thofe of the circle and hyperbola, and may be had from the rules
here given. ^
45. Now let x"* ±: a be given to refolve as above, and let m be any affirm-
ative integer, but odd.
The formula may be divided by ;; ± ^, and the quotient (which in the firfl
cafe will be ^v**"" — ax " + a^x^"^ — a^x^''^ &c. to the laft term, which
will be + «*""" ; and, in the fecond cafe, it will he x"*" + ax'""* 4.
a^x""^ -f a^iT"^^ &c. to the laft term, which will be + a"'' ,) maybe fup-
pofed = o ; and let this fiditious equation, which is a convertible one, be
compared, term by term, with the producft of a convertible equation, in which
the number of dimenfions of the variable ;v is m — 3, into the trinomial
xx ■\- fx + aa-y and, from the comparifon of the fecond terms, we fhall have
the value of the alTumed quantity, for example b ; from the third the value
of cc^ from the fourth the value of ^% &c. ; and laftly, from the comparifon
of the middle terms, we may derive the values of/, cxprefled by an equation
of which the number of dimenfions will be ^ "" . All the roots of which will
be real, and will determine the values of/ all real; which, being fubftitutcd
in the trinomial xx •{■ fx •\- aa^ will fupply us with fo many trinomials, which,
multiplied together, and alfo by ^ ±: ^, will reftore the propofed formula
m m
K ■±,a m
By this method we may find the following equations, which will ferve for the
refolution of the binomial x + a\ when tn is an odd, integer, and pofitive
number.
If
S£Ct. I. ■ A N A L Y T I C A tINSTITUTlONJ, I4J
If m — 3, it will be / + « n o.
If »2 zz 5, then ^ -H ^/ — ^-^ ~ o.
If w z: 7, then /' + aff — laaf — a^ ~ o.
If ;;; — 9, then /* + ^/' — S^^f/ — ^^Y + ^* — o.
If ;;« = II, then /^ + ^/"^ — ^q/^ — 3^'/^ + ^a*/ -i- a^ = o.
If w z= 13, then f + af\^ s^'P — 4^7' + ^-'^y* + 3^'/— «' - o.
And thus we might proceed to find the other values of/, if /;/ be an odd
number.
If the propofcd formula were ii — « , and m were an odd integer affirm-
ative number, dividing by .v — rf as before, the fame equations would be had,
only changing the figns in the fecond, fourth, and fixth term, and in all others
in even, places.
46. If, indead of a* ± ^ , fuppofing »? to be any odd affirmadve integer,
the formula were any other, but fuch, as that, dividing by ;; + fome conftant
quantity, that which refults fliould be a convertible formula; as a;* + >^'^"'* — ^^'><^^
-»— aahx^ 4- a'^x -\- al^h, which, being divided by ^ + ^, will give a* — aax^
•4- a'^ ', this laft being managed as ufual, and the values of/ found and fubdi-
tuted in the trinomial xx + /c + aa^ we fliould have fo many trinomials,
^hich being multiplied together, and alfo by x + hj would reftore the pro-
pofed formula.
Let it be required, for example, to refolve .v^ + a^ into it's real components,
in which x may have no fradional exponents, and may not exceed the fecond
dimenfion. The equation which is to give the values of/ (according to what,
goes before) will be /^ + af ^^ aa zn. o, from whence we derive thefe values
q{ f^ f — " '^ - '^*^^ . Thefe being fubftituted, inflead of/, in the trinomial
i(x + fx + aa, we fliall have the two real trinomials xx — \ax + ^ax^ ^ -{- aa,
and XX — -^ax — ^ax\^ ^ + aa^ the produd of which, together with x + ^,
will reftore the formula propofed.
Let it be required to refolve into real components the formtila x^ + bx*
— aax^ — aahx'' -f- a\v 4- a*b, which> being divided by ^ + ^, will give
^* — aax'^ 4- a^. The equation that gives us / will be //^ = o^aa, and the
values of/ will be / zz ± K^'^aa. Thefe being fubftituted inftead of /in the
trinomial xx -h fx 4- aa, we fhall have thefe two real trinomials xx + x\/ ^aa
4- aa, and xx — xV ^aa 4- aa ; the product of which, together with a: 4- ^,
will reftore the formula- propofed.
Vol. II. U - 47« From
14^ ANALYTICAL INSTITUTIONS, BOOK Til,
47. From hence I conclude, that the integral of any diflfcrcntial fonnula
whatever, the numerator of which is x into any conftant quantity, and the
denominator of a nature like to thefe here coniidered, will not depend on qua-
dratures higher than thole of the circle and hyperbola, and which iTiay be
obtained by the rules here given,
48. But, beeaufe in higher dimenfions the value of/ cannot be obtained by
ai^lual feparation, from the equations b^efore cited ; in fuch cafes it will be
enough to have recourfe to the geometrical conftrudion of the fame equations.
ThiiSj to find the components of x'^ + a\ and thence the integral of the
formula -j— — - , the denominator being divided by ,v + a, the quotient will
be x^ — ax^ 4- artx^ — a^x^ + a*x^ — a^x -j- a^ . The values of/ for the
refolution of this formula muft be furnifhed by the equation /^ 4- af' — ^aaf
— <-<' =r: o. Wherefore, by the ufual methods of Algebra, by means of the
inrerfedions of two curves, or by any other way, having found the values of/
affirmative and negative, which are to be all real; for example, let one be A,
another — B, the other — C ; the quantity x'' 4- a' will be the produdl of
A' 4- ^ into XX 4- A.v -|- aa into xx — ^x •\- aa into xx — Qx 4- aa ; and
the quantities A, B, C, will be real and given. Then we may proceed 10 the
integration of the formula -j^ — - , by the quadrature only of the circle and
hyperbola*
49. By the fame artifice by which we find the equations for the refolution of
the binomial a;"* 4h cT^ we may find them for the refolution of the trinomial
X. '" H- 2aax 4- aa, fuppofing 2m to be an even affirmative integral number.
And thus, in general, as oftetv as it is propofed to refolve a formula which is
eonvertible, or js the produd of a convertible into a linear quantity, and which
has not a fradion in the exponents ; they may always be reduced by the method
here explained.
The cafe of the produd of a convertible formula into a linear, we (hall have
when m is an odd number, and otherwife. Let this be an example, x^ -f- b^x^
— a^x* — a'^h^f that is, 'a:^ + h^ x .v* — a^, or a^ + b^ X xx '\' aa >^
XX — aa. Wherefore, the divifor x* 4- b^ being refolved into it*s real com-
ponents of two diaiienfions, which may be, for example, xx + kx -\- bb^ and
x-.v + Ba: 4- bby it will be x'' + b'' X ** -- ^^ = xx + A^ + <^^ X
w 4- B^f 4- bb X XX 4- aa x a.v — aa. And if it had been x" 4- b^ x
x" + <5*, then, by the refolution of x^ + a"^ into xx 4- Cv + aa^ and
XX 4- Da^ + aa^ it would be x^ + ^* X x" + a^ = xx ■\' hx -^ bb x
XX 4- B.v 4- bb X XX + Qx 4- aa X xx -V Dx -^ aa.
50. To
SECT. I. AN-ALYTICAL INSTITUTIONS. I47
50. To have the integrai of the formula -^— — ^ , in which m denotes any
;jffirmative integer number, let A, B, C, &c. reprefent the feveral values of/
^ith their figns, which ferve for rhe refolution of the denominator .v"' ^4^ a*".
And it muft be obferved, that of thefe values one may fomctiaies be zz o,
which will obtain as often as w is a term in this feries 4, 8, 12, 16, &:c. it
being x"* — a in the given formula. And as often as m is a. term in this feries
2, 6, 10, 14, 18, &c. when it is x"" + a , This being fuppofed, the in-
A , ^ : — : : B
tcgral required will be + — i k/' xx + Ax -^ aa ±: — l\/xx -^ Bx -\- aa
± — l\^xx ■+■ Cx + aa, &c. taking thefe logarithms from the logarithmic
curve, the fubtangent of which h zz a ; adding to, or fubtratfling from this
aggregate of logarithmic terms, (according as the fign of the term a" in the
denominator (hall be + or — ,) twice the fum of lb many arches of a circle,
as are the values A, B, C, &c. of which arches thefe are the radii in order,
>/aa — tAA, A^aa — :J:BB, \/aa — ^CC, &c. and the tangents are in the
fame order, x + ^A, x + fB, x + 7C, &c. Such will be the integral of
ma X
the formula • — , if m fliall be an even affirmative number. But in the
X -\- a
fame formula, if »; (hall be an odd affirmative number, it will be neceffaryto
add to the whole the logarithm of .v + a, becaufe the denominator has alfo the
711'
ma X
real root * + ^. And if the formula fliould be '■ — , m being an odd af-
X —a
iirmative number ; inftead of the logarithm o{ x -\- a, that of a; — a muft be
m .
added. And laftiy, the formula being — "^-^ — , and m beins aa even affirm-
X — a
ative number, it will be neceflary to add the logarithm of ,v — a, and to
itibtrad that of a; ■\- a i ilill taking thefe logarithms from the logarithmic with
fubtangent — a,
ti. But if in the propofed formula — the number m fliould be a nc-
*^ * m J, m
X ± a
gative nwmber, that is, if it were —;—- — -^i it would be expre(re<i thus,
U 2
I4S ANALYTICAL INSTITUTIONS, BOOK HI.
, which, reduced to a common denominator, is equivalent to this.
m m
X a
a X X
m _, m
a ± X
; and dividing the numerator by the denominator till the greateft power
of the variable is lefs in this than in that, we (hall have at laft ± a"x
zm
a X
i '- — , in which m will be a pofitive number. And what has been faid
X ± a
before will alfo take place, in the formula - — ^^^-5 when m is an integer nega-
tive number.
m , m
52. Moreover, if the fraction be fuppofcd to be multiplied by x".
X + a
n being an integer number either affirmative or negative, the denominator
being refolved into it's real components, in which x does not exceed the fecond
dimenfion; this will be the cafe already confidered by me at. ^ 41, 42, and is
therefore reducible to the quadrature of the circle and hyperbola.
e^^* But when « is negative, it may be reduced more expeditioufly thus.
Firft, let n be lefs than m. The formula === may be thus expreffed
X x ** X «•
m-'ft. I ..
by equivalents, — ^ — =~=. And llkewife, the formula
m n
a X
Kit ^, m , m w 'w w . "
a X X -\- a X ~- a X X
m—n.
X X
by ^ j ■■ — . Secondly, let n be greater than m. The formula
^ ^ a X X — a
^ mav be expreffed by the equivalent feries — - — — . +
x"' + a" xx" " '-^ ^ *
f «. 1 &c. till we come to that term, in which the exponent
aw n — zm Am n — irn
oP X or X ^
of ;^ is but jufl greater than ;» J ± == . Here the fign mufl. be
X + fl X a X
+ or — 5 according as the alternate change of the figns fliall require ; and r is
the fame exponent of the quantity a^ as in the antecedent term, and / is the
remainder of the divifion made of the number n by the number w, taken as
often as it can be done.
Now
SECT. I. ANALYTICAL INSTITUTIONS, l^g
Now if it were L^ , fuppofing « to be greater than m ; all the
m m ^ n
X —a X X
terms of the feries ou^ht to be afFeded by the negative fign, and the term out of
the feries, that is, ~ , ought always to have the affirmative fien
m m _ r t °
X —a X a X
prefixed. Thus, if the formula were , it would be equivalent to
x^'-i- a^ X x^
X
-r-r — ==: . But we know that — === — • is equal to 2 —
4- — . Therefore it will be '^ r: -^^ — j-
a^ X x^ + tf 5 x^ ■{• a3 X x^ '»'•*■* o^* a^ X Ar' + as'
all which are quantities that may be managed by the given rules.
54. But if m (hall be a fraclion either affirmative or negative, let / be the
numerator of the fradion which is equal to w, and reduced to the fimpleft
terms, and let p be the denominator of the fame : fo that the given formula
may be thus expreffed, — j— ^ — j- . Put x zz j^, and a zz i/^ and the for-
X / ±a p
p—i.
mula will be converted into this, ~ — ^, which has no fraftions for it's expo-
y ±b
nents, and m'ay therefore be refolved by the given, rules.
Let the formula be, for example, — y~ — r 5 rnake x zz yy, a zz hh, and it
will be X = 2>j i and making the fubftitutions, the formula will be changed
into -~rTr> which has no fractions for it's exponents.
yi ±1,3
ft-
55. Now if the given formula be , in which m and n are broken
m jL. m
X 2: a
numbers ; making r the numerator of the fradion ;/, and p the denominator of
the fame ; and thus making / the numerator of the fradion m, and q it's deno-
minator, (fuppofing thefe fractions to be reduced to their fmalleft terms,) the
formula will be , in which r, />, q, t, will be integer numbers,
X ^ ±1 a (J
pofitive or negative.
Now
I^O ANALYTICAL INSTITUTIONS. BOOK III.
Now let It be made x = y^^, and a = ^^^'; the formula will be converted
into this, — ■-—-* which has no fradions in it's exponents. Let it be,
3...
for example, the formula ^ '" ^ ; make x z: >'% a :=i h''' i it w ill be .v =
loy^y x'^ z: y^\ x^ ^^ y i ^^ making the fubftitutions, the formula will be
dianged into -rrr^t which has no fradlional exponents.
n-
X X
<6. Laftly, if the formula fhall be , , the exponents », w, «, being
X 2: a
pofitive integers, we may always have it's integral, fuppofing only the quadra-
tures of the circle and hyperbola. And the integral will be compofed of
algebraical quantities, and of one fiuential quantity ; which will be done in the
following manner.
^uppofe the formula /;
X X
m ,
X ±a
£f + ^^ ±^^ ^ as far as to a conftant
^«-i
term, or to that term in which the exponent of ^ is o, and let this be K ; then
muft be added A/ — ^-^ — -, that is^ it muft be made/;
11'
X X
+ A/-
X ziz a X :±tf
x'x
g^n+«/«-2m + i ^ ^^n + um^zm ^ p^« + ««-am-l g,^ ^ g^ «
m . m
A" — <* ' X s^ a
Difference the equation, make it — o, and fet the terms in order. From
making the firft terms — o we (hall find the value of the aflumed quantity B.
Making t-he kcond terms r= o, we fhall have the value of C. And fo, one by
one, the values of the others ; which values being fubftituted inftead of the
n
X X
capitals, ^s the fluent of will depend only on the quadratures of the
X dL a
circle and hyperbola, and the other terms in the homogeneum comparationis are
purely algebiaical, fo thepropofed formujia will require no higher quadratures..
57. Sometimes k may happen, that fome one o( the co-efficients B, C, D,
he, may come out arbitrary, or tp be determined at pleafure; but ijc will hs^
^ -only
SECT, I. ANALYTICAL INSTITUTION*, I5I
only when n is greater than m — i. And it may alfo be obferved^ that as often
as it h m r= n ^ li the co-efficient A will be found = o, and confequently
the integral of the propofed formula will be algebraical.
5S. But if, in the propofed differential formula, the exponent « fliould be a
negative integer, fo that it might be reduced to . ; in which it is
now pofitive ; the integral would be
^ ±Si. -_=+Ji- ^ifSuhJE + A/ -^ — - Which
ft'-l m . m »^^ n m , m
X X X dz a X X X d: a
co-efficients B, C, D, &c. will be determined in the fame manner as before.
As, for example, _ ' ' ■ ; in. which cafe we have « =: i, w i= q, « zz 2,
Wherefore it will be/=^ = B.» + C^ + K a/ --^ . And taking
the fluxions, - zz : . • — -j
Then reducing to a common denominator,, fetiing the equation in order, and
making it equal to o, it will be
— 3Ba:*jv — 3Cx^A' + Aa^xx
+ Aa'^jc" — XX
> = o.
Now making the firft, fecond, third, &c. terms rr o fucceffively, we fliall
find A — B = a, or B- =z A i C =1 o, K = o ; zBa^ + Aa^ — i zi o,. or
Aa^ zz 1 — iBa^ ; and putting A inftead of B, it will be A z= — = B.
Whence^ Mly, it is /=^ = -1= + ~ X/-ii-. But/-^
zz — i\/xx — ax + aa — Ix + a; together with — multiplied into the
arch of a circle with radius = \^iaa, and tangent = x — j;a. So that it will
^^ /- Vr^i^x ~ ^""4= + -^ X ^\/xx — ax + aa — -^ X Ix -^ a
+ — 7: X arch of a circle with radius V-^aa, and tangent zz x —-' iai taking
the logarithms from the logarithmic with fubtangent zz a,
59. But
J5J ANALYTICAL INSTITUTIONS. BOOK It I.
59. But if the exponent 711 be negative, the formula muft be changed into
another that is equivalent to if, in which the exponent is pofitive ; according to
the manner fhown at §51 of this Book.
60. And if both m and n (liould be fra(5lions, the fubflitutions mud be
made according to § ^^ of this Book.
61. Again, if the exponent ti \vt\Q not an integer, but a fraction either
affirmative or negative, it will fuffice that the formula be one of ihofe cafes
confidered at § 39. Forafmuch as it may be tranfmuted into another form,
which is capable of being managed by the given rules,
n-
Thus the formula -. ' — ^ , the exponehts 7?, m, u, being pofitive or nega-
tive integers, or elfe rational fractions of any kind, with the figns -4- and — •
at pleafure ; it will be integrable, or, at leaft, may be reduced to known qua-
' dratures, as often as the faid exponents Ihall have fuch a relation to one another,
that one of thefe two quantities compofed of them, that is, u — i
— — , or — — I +— , Qiali be equal to any inteeer number. If this
integer number fhall be pofitive, the formula will admit of an algebraical inte-
gration, except the cafes in which the power x" x Qiall intrude, which obliges
us to recur to the logarithms. If this integer number lliall be negative, the
formula will be red.uced to the quadrature of the circle, or of the hyperbola.
To obtain our purpofe as to the firft cafe, in which u i ij
m
eq[ual to an mteger, make a ■\- a ■=z zx \ then .v zz — — -, x zz ^,
z — i\m
n ,, «+l „ «+l^
X — ■ — • , X = — - ; and therefore a- x = — v
n_ ' i + n m ^
2; — i] m ' z — i^ m
m , m m a ,& , »i , m\u a z
2: — il '" . But X ■\- a zz zx zz — '— , and x + a ' r:
2-1 z - i)u
Therefore, making the necefTary fubftitutions in the propofed formula, it will be
X 2; — 1\ m ■"^+", which is plainly feen to be alo-e-
3 zz ' —
braically integrable, (except the excepted cafe,) when ^i j^ u is equal
3 to
SECT. I. ANALYTICAL INSTITUTIONS. I^j
to a pofitive integer number. And that if " ^ "' ^ ^ i _j. « is an - integer
number, but negative, by what is advanced in the foregoing articles, the inte-
gration of this formula will depend on no higher quadratures than thofe of the
circle and hyperbola.
I come now to the fe'cond cafe, when i A ■ is equal to an inteo;er
number. Make x -{• a =: z, and then it will hQ x zz z -m a ^ x "zz
I a I +«
m\m n rri\ m n + l >n) m a. %
, X = z " a ^ =z ~
z -^ a , X = z -^ a ^ „ =z z — a , x x zz — x
tt+i
a^^ '" , But x* -\- a^ zz z, and x" + ^'^ = z -, therefore, mak-
ing the fubftitutions in the propofed formula, it will become — x
^ I ^±1-1
^ — , or elfe -^ — ^ x 2 — «"*' '^ * which is algebraically inte-
a -I- 1 .
grable, (excepting in the cafe excepted,) when — i is equal to a pofitive
integer, or a negative ; for then the integration will depend on the known
quadratures of the circle and hyperbola, as appears by the foregoing articles,
62. Now if the denominator of the propofed fraftion, raifed to any integral
power, (liould not be a binomial, as has been confidered hitherto, but (hould
be any multinomial whatever ; provided it be reducible into it*s real compo-
nents, in which the variable does not exceed the fecond dimenfion ; either by
means of convertible equations, or fome other manner j the formula may always
be reduced to known quadratures.
Let it be, for example, - =- ====r » railing actually the powers
of the denominator, make a fiditious equation thus :
x_ _ A.r^.^' + ^x^'x + C-y^ + D-y , F.y'.y + Gx^ + H.y
xx-\-bx-\-aa>?- X T+7)^ ~ K^-^2hx^->g- laax'' + bbs-^ + 2 aabx + a* x^ + yx"^ + 2>^cx + c' '
Here are fo many terms taken in general, as are the components of the denomi-
nator ; and in thefe terms fo many capitals, as is the higheft power of the
variable in it's refpedlive denominator, multiplying alfothe tirft capital in each
term by the higheft power, lefiened by unity, of the variable in it's denomi-
nator, the fecond capital by the fame power diminilhed by 2, and fo on to the
Vol. II. X laft
154 ANALYTICAL* IKSTITUTIONS. BOOK III.
laft conftant quantity. Thefe aflumed conftant quantities are to be determined
in the ufual manner, and the firft term will furnilh fo many fradions divided by
xx+l>x-i-aci\'^ J in which denominator making the middle term to vanidi, the
m •
fradions will be a particular cafe of the general canon ■^. . And the
« j_ n *
X ± a
fecond term will give us fo many fradlions divided by x +^% which may be
reduced to the ufual rule of denominators compounded of equal roots.
63. Moreover, if the numerator of the propofed formula be multiplied by a
pofitive or negative power of the variable ; having found the values of the
capitals, and operating as if the fradion had not been multiplied by any fuch
power ; the refulting terms may be multiplied by the faid power, and the reft
may be done as ufual.
64. I fliall finidi this Sedion by fulfilling my promife made to the reader,
concerning the Method of Multinomials, of Sig. Count James Riccati^ which
is as follows.
By the name of Differential Multinomials I call fuch fradions, as have for
their numerators the fluxion x, and for denominators an aggregate of powers,
the exponents of which conflitute an arithmetical progrefTion, which proceeds
till it terminates in nothing. And till this condition is fulfilled, the abfent
terms mufl be fupplied, and their co-efficients made equal to nothing. Suppofe
we had this exprefTion — ; . At firfl view it might feem to be a trino-
x^ •\' x^ -{■ a
mial, but is really a quadrinomial, and is thus to be compleated :
Si I •
In any multinomial expreffed by a fradion, the denominator of which is
raifed to the power p, being a pofitive integral number, there is a method
which would be general, if it were not frequently made ufelefs by the inter-
vention of imaginary quantities.. But there are fome particular artifices, which
often come opportunely to our affiftance,
1 begin with the trinomial .^ =z y, becaufe to fuch an expreflion
X + ax •{■ 0'
as this every trinomial may eafily be reduced. Make x'" :=z z -^ A^ where z
is a new variable afTumed, and A is a conf^ant to be afterwards determined.
The necelTary computations being made, to arrive at the fubflitutions we Ihall
kave as foUows,
2mi
SECT. I. ANALYTICAL INSTITITTIONS,
X = 22; + 2A2 -f- AA, and confcquently
ax -zz az -\- ah.
b - h
^
Z'" + ax" +*h^ = zs: 4- 2A + « x 2 + AA + ah + ^/.
, It ought to be contrived in fuch manner, that the quantities A A -^ ah + h
may difappear, by putting them — o, and in cafes in which A is no imaginary
quantity, this redudion fucceeds very well. It is therefore ^v** iz z + A ;
m
I
and taking the fluxions, mx x ^z z, and * zz z -i- A) "^ • Then k zz
m — i
mx
X r+Ai
' In proceeding to the neceflary fubftitutions, in.our principal formula, inftead
of X and it's powers, are to be fubftituted the aflumed variable 2;, with it*s
i"un(flions ; and we Ihall find
l» X » + a) »* X 5S« + 2A + «X!8^
and freeing it from the quantity 2, which multiplies the binomial 2 -f 2A + <«
— / •
under the vinculum, it will be -~,^ "= ~ — ; •
m X a + Al « xa+zA + d/
The mod fimple cafe is, when the exponent p is equal to unity, the other being
when m is any number, integer or fradion, affirmative or negative; and, for brevity,
making ih -^ a - g, the general expreflion, [when /> = !,] will become this
particular onp, -^;3^ •^j:r7 — ^J/*
g X «T^ "* + K X x+A) "»
1 make a firft divifion by dividing the numerator of the fradion by it's
— I.
denominator, and the firft quotient will be ^- -'^,. i ^"^ making the
g X » + Al «
multiplication and the fubtradion, according to the ufual method, the re-
mainder will be L to be divided by the denohiinator ,- and therefore
m — i
m — i /w — I m-'i ^
£ X » + A:i"ir+«xM^"^ ^X« + A) m gS^^+M m +^X«H-A1 m
X2 The
156 ANALYTICAL INSTITUTIONS. BOOK III.
The firfl term of the fecond member is already reduced to known quadra-
tures, and the other term may eafily be reduced, by making z -|- A z: «, and
performing the necelTary fubftitutions. For then we fliall have
— OT+I
gg X » + A\ m + ^2 X a + A'i m
To purfue our inquiry, let the exponent ^ be equal to any pofnive and
integer number; to obtain our defire it will be fufficient fomething to produce
the operation, Refuming, then, the general formula
=z y. And, for example-fake, making p = 2,
m — l
m X z + A) m x a + gf
this will be reduced to the following,
«-2^
. — my,
m—i m—i m—l -'
ig X z+ATI m + zgz X «4-Al m + zz X a + A) m
Then, as before, 1 divide the numerator of this fraflion by it*s denominator,
— 2.
and the firfl: quotient will be — - — ^ ; and, after the neceflary opera-
gg
X z + Al m
2Z Z »
tions, we Ihall have the remainder — -^ — , to be again divided by
the whole denominator. Then I make a fecond divifion with the fradion
. Here, after the necef-
771 — 1 • m—i
g^ X » + A^ m -h Zgg X z + A) m + gzz X z + A) m
fary operations, we fliall have the remainder H + i^ , to be divided by the
whole denominator. Whence there will 'arife the following equation,
— 2 . _3. — I .
% Z Z Z 2Z Z . 'IZ
— + J
">— I m — i m~ I m~i
a+A^ m X «n)* gg X z+A"^ m ^3 ^ « + A] m g*^z-\-A m x z±g*
. zzz
+ ; •
w — I
^3 X » + Al~^ X 7^^^
The
SECT. I, ANALYTICAL INSTITUTIONS. 1^7
The two firft terms of the homo^eneum comparationis are two binomials, and
the other two may eafily be reduced to the form of binomials, by making
z + A zi z/, or z + ^ — ^» In cafes more, compounded, in which are made
^ z: 3, or 4, or 5, &c. the tedioufnefs of calculation ^ili indeed increafe,
but the method will ftiil be the fame.
This method may be extended to all multinomials in infinitum^ fuppofing p to
be a pofitive integer ; for, if it were a,.negative integer, the matter becomes fo
eafy that there is no need to mention it. To apply the method, nothing elfe
is required but to repeat the fubftitutions x zz. -z + A, z = a + B, &c.
always making thofe terms to vaniQi, in which only conftant quantities are
found ; by which means quadrinomials (for inftance) may be reduced to tri-
nomials, and thefe to binomials. It will alfo be needful, from time to time,
to make ufe of a partial divifion, that we may not be interrupted by negative
exponents, which will often intrude in the numerator of the fraction. After all,
the manner of operation will be better perceived by examples than by precepts.
Let us take the quadrinomial — y. The conftant
quantities a, ht may be zz o. I fuppofe x = 2 + A j then we (hall have
x'^'" -I- ax""" -f bx"" + f = 2' + 3A2* + 3AA2; + A'
4- az" + za^z + <?A*
H- bz + Kb -^ c.
I make A' + ^A* + A^ + £ zz o, and thus I determine the value of the
aflumed conftant quantity A. Then repeating the operations as in the trinomial,
I find . The letters g, h, denote conftant quantities.
z + A m X za +,?» 4- S\P
which are fubftituted in the place of others more compounded. And, fup.
pofing j!) to be a pofitive integer, I raife the trinomial zz ■{■ gz ■\' b to the
power p.
After this, 1 make ufe of as. many divifions as are neceflary, to make the
exponent of the variable in the numerator to be negative ; and in the deno-
minator, that no other quantity ftiall enter but the binomial z -^ a\ "* . And
I fet afide fuch fraftions, as, negleding the co-efficients, Qiall be analogous to
this, — — -^3-j ; ftippofing n to be any pofitive integer. The other terms are
2 + Al » reprefented
15S ' ANALYTICAL INSTITUTIONS* BOOK III,
reprefentcd by the general formula . Then I repeat
the operation, making 2: = « + B, making the lad term to vanidi as ufiial,
and raifing the binomial u + B to any power » + i, and fubftituting, inftead
of z and it's powers, their vaUies expreffed by the new variable u\ all the parts
will appear under the afpeft expreiled by the following formula.
tt + A + B") m X u + ^P
When f is greater than »., fo that the exponent n — /> is negative, then
th€ divifions muft be put in pradice, and the formula thence arifing will be
— «
»
^ ; then n —p, being pofitive, we fliall have
« + A + bY^ u + A+B)~^ X 7+lP
And laftly, making « + il s= w, and, as well » as /> being integer numbers,
the binomials that will arife from the forementioned operations will always be
reducible to more limple quadrat^ures.
It is true, that, upon the account of imaginary quantities, this method re-
mains limited ; but very often the roots, either in the whole or in part, are
real ; and befides that, in many particular cafes, thefe imaginary quantities
may be eliminated. Nor ought we to defptfe the much we may have, becaufe
we cannot obtain all.
, Let us take, for example, the trinomial ■ --^ - . Make x^ ziz + A,
X -{■ 2a/x + z\P
then X -j- is/x -f 2 = zz + 2A2; + 22 + A A + 2A 4- 2. By making
A A -f- 2 A -1-2 = 0, we find A zz \/— i — 1. Now here we have a
magnitude naade up of real and imaginary quantities j therefore, proceeding
— /• J—A'
according to the method, we fliall have • — -^ ■ = ^ -
z-fAl ~' X z+2A+a)/ » + 2^-iy
+ - ^:^--. Now, that the imaginary quantities may be avoided, let us
change our manner, and in the magnitude 22+2 A + 2 X z -{■ AA H- 2 A + 2,
let us bring it about, that the middle term 2A2 + 22; may be deftroyed, by
putting it zi o ; whence it is A = — i, and A A + 2A 4- 2 =1.* So that
the formula will be as follows.
And now, in the two binomials of the homogeneum comparationiSy which are
equivalent to the two others already confidered, we (hall meet with no difficulty.
4 SECT.
SECT, n,' ANALYTICAL INSTITUTIONS, 1^9
SECT. II.
Of the^ Rules of Integration, having reeourfe to Infinite Strtes,
65. Now, to proceed to the other rrranner of Integration, or of finding
fluents, which was mentioned at the beginning, that is, by means of infinite
feries j it is neceflary to premife thefe Rules following.
' Rule I. To reduce a fraction to an infinite feries.
Divide the numerator by the denominator, according to the ordinary method
of divifion, and let the remainder be again divided, and thus from term to term
in infinitum ; and you will have a feries confiding of an infinite number of
terms, which is equal to the propofed fradlion.- Therefore it mufl be obferved,
to make that term the firfl which is the greatefl, and that as well in the nume-
rator as in the denominator of the fraftion propofed. Wherefore, by operating,
after this manner, we (hall have as follows :
/ =/__A4.-^_>l4.>l &c.
m •\- n m m* m^ ra'* m* '
m — n m m* m^ m* fa^ ~
m* ± «* ~ .«»"+■ ^ "T" ;;,* "^ ■;;;5" "•" m»° ' ^^'
Here the figns of the feries mull be alternately •{■ and — , when tli€ fecond
term of the denominator is pofitive ; and all the figns mufl be pofitive when it
has a negative fign.
In like manner, it will be
i»» ± «« — ;n» *"^ ^ + "^ ■+• ^ "^ «* ' ^^•
~-^» =Z I — . ^' -f- ;f* — / + X', aCC.
i 3
7 zi ^x-^ — IX -\- 7;^ * — 13;? + 34^"*" i &c.
1 + ** — 3*'
j»
»»♦ ' «iJ "~" «• ' ml '
^r«^' m^ !«♦ /»» — ot'
Let
l6o ANALYTICAL INSTITUTION Si BOOK III,
Let there be a fradion, of which the numerator and denominator are each
an infinite feries -, for example, this following :
The quotient will be
_ x^V _ ^i^^»^/ -. ^^^a*h\x' ) &c.
66, Rule II. To reduce a complicate radical quantity into an infinite feries.
Take, for example, \/aa ± xx ; let the fquare-root of the firft term be
cxtraded, and then let the operation be profecuted in infinitum, in the ufual
manner of the extraction of the fquare-root, and we (hall have
,8
111 %
^/ax ±^xx — a'-x'- ± -^ 7 ± — j ^~i» ^^'
2«* 8a^ 16^ 128a*
It may here be obferved, that in each of thefe two feriesj if the numerator
and denominator of each term be multiplied by 3, beginning at the fourth,
the numerical co-efficients of the numerators will be in order, 3^ "^ X Sj
3X5X7, &c. arifing from the continual muldplication of the odd numbers.
Then in the denominators, beginning at the fecond, they will be 2, 2 x 4,
2X4x6, 2X4X6>^8, &c. arifing from the continual multiplication
of the even numbers.
67. Rule III. All this may be done more generally by the help of the
following canon :
m m
FTTQi '^ zz P~ + -^ ACL+ '—' BQ^+ ^" CCL+ ^«DQ>_&c.
In which P 4- PQ^is the given quantity, — is the numeral exponent, P
reprefents the firft term, Q^is the quotient of all the other terms divided by the
firft, and every one.of the capitals A,,B, C, D, &c. fignify the preceding terms
3 refpedively
SECT. II. ANALYTICAL INSTITUTIONS. l5l
refpedlively ; fo that by A is underdood P " , by B is meant — AQj^ by C,
ftt — n
Zn
BQ^j and fo on.
Let the formula s/ aa -\- xx be propofed to be reduced into a feries ; thea
it will be P = aa^ CI— — , m •=! i. n zn z ; therefore
5*'
Let it be {/a^ -^ a^x — x^, that is, a^ -f a*x — ^*^t ; it will be P zz ^%
O zz "^ 7 "^ , »? z: I, « zz c ; therefore «* + «V — .v^Vr ~ /? -f- "" ^ ■
— ^——. , &C.
Let it be , ^ ■ = ^ X f — aay\''^ ; it will be P = y', O = ^ il,
m = — I, K zz 3 ; therefore
Let It be ^ . ^^ , which would be exprefled thus, ^ x a -^ x\ ^, and
the reft would be done as before.
Let it be ^ x a -{- x\ "^ j then P = «, Q^=: -^,f«=— 3, «=!.
therefore b X ^ + xy^ = — j ^- H 7- — -— ^ > &c.
68. Let US have a complicate quantity to raife to a given power, or let <3: +Af
(for example) be raifed to the power m. Then P zz a, Q^zz ~ , 7n zz w,
» = I ; therefore
\W
m~—l — — ^; — 2 , — — m — 'i .
»j , ma X . m y. m~\ a *■* , »»X?m — iXnz — 2^2 •'xJ
a ^ x\ = t? + -4 ,&c,
J ' 1x2 1x2x3
Let us have an infinite feries to raife to a given power. For example, let
^ + ^y + by^ + cy^ 4- dy^, &c. be raifed to the power m. Then will P z:/,
Q^zz ay + by^ + cy^ + ^% kc, m — m, n = i ; wherefore
Vol. n. Y y ^
ife ANALYTICAL INSTITUTIONS. BOOK III.
■S.OT m , may
y + af -^ by^ + cf + dy\ &c.^ i= j +
I
m X OT-ifly" + ^ fK X m-i X W7-2 rt'j''"'*'^
+ — +
1X2 1X2x3
mby m X tn — I aby ' -^
-j_ , '+ —
+
I X 1
OT X ;y; — I X m — i X OT--3a'*j" ''^'^
I X 2 X 3x4
W X OT— I X W — 2 rt^^J/"* 4-
T
I X 2 X I
+
m X m—iacy^^^
1 X I
+
?n X m-\l^y^^^
1X3
+
«4« + 4
69, This being now fuppofed, let the differential formula —— be propofed
to be integrated. The fradlion being: reduced to a feries, and every
numerator being multiplied by ^, we (hall have — — =: — • '■ 1 r-
— —^ — 1 ^ , &c. And by mtegration,
. hh ix hx^ bx"^ hx* hx^ „
•^ a -^ X ~fl~ "~ 2aa ' 3^* 4a* 5^* ' *
70. Let the formula be — . Making x — b -\- Zy where b denotes any
• •*
conftant quantity at pleafure, and z a new variable j it will be — = ^Fa'
The fraflion y-^ being reduced to a feries, and multiplied by Zy it
will be
S.ECT. II. ANALYTICAL INSTITUTIONS, l6|.
az az <7zz a-xP-'z a%^z aK*z „ * i i •
m = T" "" >■ + 7i ^ + -^> &c. And by integration,
f. az az az* az'i at* rt-s o i •
.ax a y. X — b a X x — ^ * a y. x — b:^ ax x — bY o^
71. Let the formula be _ J ; this, reduced to a feries, is
\Jx-^A^ Va-tx
-r - —8- + IT - — 71 > &c. And by integration, / = -^
— ^-— J + ^^-^ ^— n J &c. And the fame may be done by any other
propofed formula.
72. If the feries thus found, which exprefs the fluents of propofed differential
formulae, and which are compofed of an infinite number of terms, fhall be
infinite in value ; the fluents or integrals of the propofed fluxions will be infi-
nite. And if thefe feries fliall be finite in value, and alfo fiimmable, that is to
fay, if we know how to find the values of thefe feries, though compofed of
terms infinite in number, and which very often may be done ; we fliall have
them in a fi,nite quantity, and therefore the algebraical integral of the propofed
differential formulas. But, if the feries fliall be finite in value, and yet not
fummable, the more terms fliall be taken of the feries, fo much the nearer we
fhall approach to the true value of the formula ; but we cannot arrive at the
€xad value, except we could take in the whole feries,
73. In order to know what feries are infinite in value, what are of a finite
value, and which are fummable ; the treatife of Mr. James Bernoulli de Seriehus
infinUis^ may be confulted, and other authors who have written exprefsly on this,
fubjedt,
74. But whenever the differential formula fliall be compofed of two terms
only, we may, in general, and with expedition, make ufe of the following
canon ; in which the exponents w, n, t, may be integers or fradlions, affirm-
ative or negative •, and which may be continued to as many terms as we pleafe;
for from thefe four terms fet down, the law of continuation is fufBcienily
manifefl.
Y2 /gr
l54 ANALYTICAL INST ITITT IONS. BOOK III,
t
fay y Xo + cf's ~ i? + cyn ^ into -^ TT~7~ ^ W^ +
t + vt>K+n t + mn + 2n ac* t+zn t + mn + n ,. t-{-mn-\'Zn t-^Tnn-\-%n
The manner of finding this canon is this. Take the fiditious eqtiatlon
/^^ _y X ^ + O'"^ = ^ i- fj'") into A> + B)' +0' + Dy ^"^
4- E/'*''^", &c. ; in which the affumed quantities A, B, C, D, E, &c. are
arbitrary and conftant, to be determined afterwards as occafion may require.
Then, by taking the fluxions of this fiditious equation, we (hall have
ay "" J/ X ^ + cyA^ = m+i X ncy^~^y x b + cy""^ into hy + By -f.
/+2« 5,^ . -irr~z^.^^^ :..^ .A/ '-» , irrz, ^ -dvJ+^-i
Cy^"", &c. + ^ + ry-r"'^' into /Ay/ * + / + « X Byy^'^ + /+2« x
Cyy , &:c. Then dividing all by b + cyA , and fetting the terms in
order, it will be
ayy"^ - thhjy'^ +7+^ X %/"*"""' + 7+^ X^Cy/"^'^"", &c.
+ /f Ay^ ^ . -|- / 4- « X <:Byj ^ , &c.
/ + «— I , ; — ^^ Tj- ? + 2a — I
+ w + 1 XncAyy "^ + »^+i x«fBj/y , &c.
Here the term ayy "" might be tranfpofed to the other fide of the equation
by which the whole will be equal to nothing, and therefore the co-efficients of
each term will be equal to nothing, by which we (hould have as many equations
as there are arbitrary quantities A, B, C, D, &c. by which they will be deter-
mined. Or, making the firft terms on each fide equal, it will be tl>A — ^, or
A — -^, Then / + » x ^B -j- tcA + »a + i X ncA zz o, and fubftituting
the value of A, it is /^B + »^B + ~ + ^ + ^ = o, or B
b ^ tb tb t + »
X -jT' Again, /+a«x^C +/ + «X^B + »?+i x »^B = o, or
Q _ / + «x-cB + ^ + ix -ncB ^^j fubftituting the value of B, it will be-
b X t + 2a
SECT. TI. ANALYTICAL INSTITUTIONS. 165
C zi ■- — ~ . And thus from one to another, till we
/ + « X / + 2n X /i'^
have the values of as many as we pleafe of the feveral affiimed conftants ; and
thefe values, fubilitutcd in the fiditious equation, will fupply us with the
aforefaid canon.
If the exponents nty n, /, of the propofed formula fliall be fuch, .that the
canon or infinite feries will break off, or that any term (hall become — o,
(in which cafe all the others that follow will alfo be = o,) the feries becomes
finite and terminated, or we (hall have the algebraical integral of the propofed
differential formula. But it is necefTary that the feries (hould firft break off in
the numerator, or that the numerator fhould become equal to nothing before
the denominator. For, if the denominator be equal to nothing firft, that term
and all that follow after will be equal to infinite. Now, that the feries fhould
break off in the numerator, it is neceffary that — m fhould be equal to
feme integer affirmative number.
But if the exponents /, w, n, of the propofed formula fliould be fuch, that
the feries never breaks off; then the cxpreffion of the formula fhould be
changed into another equivalent to it. Thus, for example, the formula
ayy"^ X 'b~+~cy\^ fhould be changed into this other, ayy ~^+^'* -^ 4>'"'" + ^"*>
which is equivalent to the fiift, and it fhould be tried whether or not this will
anfwer our expedation. If not, the formula will not be algebraically inte-
grate, at Icafl not by this canon. If the formula were ayy^ X b — 9")'^,
then all the terms of the canon would be pofitive.
Let it be ^ — iJLif , that is, a^xx"* X b ■^x\ ^\ it will be / — i zz — |,
»=z I, OTizif, c iz i; whence the quantity t -^ mn -\- ^n will be equal to
nothing, and confequently the fourth term — o, and the others of the feries
that follow. Therefore we fhall have ftl-H — ^-JUlf. — Ja^xxT'' X b +"^»-
_1
7^
— ^ J X b H-A-l*..
Let
t66
Let it be -- — — - — -
yyViia +jy
and therefore the fecond term of the feries will be := o. Hence
ANALYTICAL INSTITUTIONS. BOOK III.
; then / = — i, » =z 2, m zz — f , f =: i, ^ := «<f j
/
ay
— '^y
—i
yy'^aa -if.yy
X aa -jr yy
— ^ L V aa + yv
ay
SECT. III.
^e Rules of the foregoing Se^ions applied to the ReSfifcation of Curve-lines^
the ^adrature of Curvilinear Spaces, the Complanation of Curve
Superficies, and the Cuhature of their Solids,
Fig, 102,
y^» To (liow the ufe of the foregoing Rules of the Integral Calculus, by
applying it to the quadrature of fpaces, to the reftification of curves, to the
complanation or quadrature, of fuperficies, and to the cubature of folids; let
there be any curve ADH referred to an axis
AB, with the ordinates parallel to each other,
and at right angles to the axis. Draw CH
parallel to the ordinate BD, and infinitely
near to it, and alfo DE parallel to BC ; the
mixtilinear figure BDHC will be the fluxion,
the differential, or the element of the fpace
ABDj and becaufe the fpace DEH is nothing
in refped of the rectangle BDEC, we may
take that redangle for the element of the faid
fpace ABD. Therefore the fum of alPthefe infinitefimal reflangles BDEC will
be the fpace comprehended by the curve AD, and by the co-ordinates AB and
BD. Wherefore, making AB = ^, BD = y, it will be BC = x, EH nj/,
and the redangle BDEC zzyx will be. the formula for fucli fpaces. There-
fore, in this formula, inftead of ^, if we fubflitute it's value given by x, and by
the conflant quantities of the equation of the curve ; or, inftead of x, it's value
4 given
SECT. iir.
ANALYTICAL INSTITUTIONS.
167
given by y and j/, and the conftants, and then integrate the formula, this
integral will be the required (pace ABD.
Other expreffions or formulaE may be had for the elements of fpaces, by
means of fe»ftors, or of trapezia, which, on certain occafions, are fomqtimes-
more convenient than redangles; we ihali hereafter fee the ufe and manner of
them in fome examples.
76. For, if the curve be referred to a focus^ that is, to a fixed point,
fuppofe to M, fcom whence all the ordinates proceed ; drawing MH infinitely
near to the ordinate MD, the infinitefimal fpace MHD will be the element of
the fpace AMD. Then with centre M and radius MD, drawing the infinitely
little arch DK, the little fpace DKH will be nothing in refpedt of the fpace
MDK J and alfo, becaufe the little arch DK may be alTumed for the tangent
in D, or in K, it thence follows that the fpace MDK ihall be the element of
the fpace AMD.
Wherefore, calling MD izjy, KD — 2;, it will be \yx for the general
fbrrnula of the fpaces, in curves referred to a focus. And in this formula,
inftead of jy, or of ^, if the refpedive values be fubftituted from the equation-
of the curve, the integral will be the fpace required AMD,
Fig* 103. H^--'^ 77* Bi^t if the curve *fhall be referred to^
a diameter, fo that the ordinates iliall not be
at right angles to their abfciflesj drawing.
HG perpendicular to AG, the produft of
HG, or of FG into BC. will be the little
parallelogram BCED, and confequently the
element of the area ABD. Therefore the
angle Y)%Qi being given, and confequently
the ratio of the whole fine to the right fine,
which, for example, rriay be that oi m to « j
making, as ufual, AB = a", BD = j, then will HG or FG be = — , and the
parallelogram BCED will be — , a. general formula for this fpace.
78. It is plain, that the fum of all the infinitefimal portions DH of the-
^ /curve will form the curve itfelf, and therefore that DH will be it's element-
Making, therefore, AB = .v, (Fig. 102.) BD zi y, and thence BC — x,,
EH iz y ; in fuch curves as are referred to an axis, that is, with the:
co-ordinates at right angles, it will be DH == \^xx -^ yj, a general for*;
mula for the redifi^cation of thefe curves..
7a- -^
j68
ANALYTICAL INSTITUTIONS.
BOOK III.
79. As to fuch curves as are referred to a focus, making alfo MD zz y,
KD zz i,', we lliall have, in like manner, \/yy + zz for a general formula.
Fig. 103.
80. But as to the curves with the co-ordi-
nates at oblique angles, the given angle
being HCG, the ratio of the whole fine to
the fine of the complement is given, which
fuppofe is that of m to e ; whence it will be
CG = — , and EF — — , and therefore
DH zi ,J XX + yy -|-
2exy
81. Now in each of thefe formulse, inflead of j), or x^ or z, fubftituting
their refpedive values given by. the other variable, and their differentials from
the equation of the curve, and then making the integrations, we (hall have the
length of the curve required.
Fig, 102.
82. Let the plane AHC be conceived to
move about the right line AC, the curve AH
will defcribe a fuperficies, while the plane
AHC defcribes a folid. But the infinitelimal
portion DH will defcribe an infinitefimal
zone, which will be the element of the fu-
perficies defcribed by the curve AH. And
the infinitefimal f)lane DBCH will defcribe a
folid alfo infinitefimal, which will be the
element of the folid defcribed by the plane
AHC. Now, as to curves referred to an axis with the ordinates at right
angles; let the ratio of the radius to the circumference of a circle be that of r
io c i the circumference defcribed with radius BD = y will be - — , and
therefore -^s/xx + yy will be the expreflion of the infinitefimal zone, and
confequently the general formula for the fuperficies,
83. Alfo, -^ will be the area of the circle defcribed with radius BD zz y,
and therefore -^^ will be the expreflTion of the infinitely little cylinder defcribed
by the reflangle ECED. Now this does not differ from the folid generated by
the plane BCHD, but by an infinitefimal quantity of the fecond order;
therefore the general formula for thefe folids will be —■ .
84. But
SECT. HI. ANALYTICAL lifSTITUTIONS. , 169
84. But as to the cafe of Fig. 103 ; that If, when tbe co-ordinates make a
given oblique angle to each other; the melius of the ciicle, on \^hich the
iittle zone and the little cylinder infift, it is not CH =r^, but indeed GH zz
— ; as likewife the element DH, which forms the zone, is not \/xx + J17,
but s/ XX -\- yy -^ —1; and the height of the little cylinder is not BG r: x,
but FD z: ^c- + — . Therefore the formula for the fuperficies, in this cafe.
will be ^*y XX + yy 4- ^^^ .
85. The product of the circle with radius GH into the height FD, that is,
^^^^ X •'^ + ~9 is the element of the folid generated by the plane AGH.
Therefore, from this fubtradling the element of the folid generated by the
triangle HCG, that is, ^-^^^^ X -^ , what remains will be the element of the
folid generated by the plane ABD, and therefore will be ^I^HJl^ the general for-
zrmm
cnnyyx
zrmm
mula for thefe folids.
86. As to the curves referred to a focus, becaufe of the variable angle DMB,
(Fig. 102.) and confequently becaufe we cannot have the value of BD or GH,
the radius of the circle, which muft neceflarily enter the formula of the quadra*
ture of the fuperficies, and the cubature of the folid ; it will be neceffary, from
the equation referred to the focus, to derive the equation of the fame curve
referred to an axis, and then we are to proceed in the manner before fpecified ;
obferving that, in the cubature, it will be necefTary to fubtradt from th€ integral
the cone generated by the triangle MHC, to have the folid generated by the-
plane AMD.
"Sj. From the differential equation of a curve to the focus, to obtain the
equation of the fame curve to an axis, the manner is this following.
Let the curve ADH (Fig. 102.) be confidered, at the fame time, both as
related to the focus M, and alfo to the axis AMB. It is certain that the fquare
of HD, the element of the curve, is equal as well to the two (quares DK, KH,
as to the two others DE, EH ; and moreover, that the fquare of MD is equal
to the two fquares MB, BD. Making MB = ;;, BD zz ;', MD z= z, and
the little arch DK =: ?>, we Ihall have zz + m z^ xx + yy, and xx 4- yy
zz zz.
• ToL. II, Z Now
lyO ANAtYTICAl, INSTITUTION 5# BOOK III.
Now the equation of the curve to the focus is expreffed, in general, by the
formula pz =. u, in which /) is a known function or power of z ; and it will be
zz -{• ppzz z= XX ■\- yy» And putting, inftead ofj/, it's value arifing from
the equation xx -^ yy "zz zz, that is, y — ■• ~ , wefhallfind zz+ppzziz
'XX + ^^ ~ '^'^^ , which may be reduced to this following, ppzz x zz -^ xx
= zzxx — 2XZXZ + xxzz ; and ex trading the fquare-root, it will be pz zz
!tX ■— XZ
\ %Z —XX
It is neceffary to clear again the foregoing equation, by freeing it from a
mixture of unknown quantities, by making at ~ — ^, and therefore x zz
f ^, 2* ^>- . By the help of this affumed fubfidiary equation, make x and it*s
functions to vanilb, and we (hall have — = ^ . . In this equation, if
the value of p given by z (ball be fuch, that the quantity ~ may be reduced
to the diiferential of a circular arch by due fubftitutions ; and that, making the
neceffary integrations, the two circular arches (hall be to each other as number
to number ; then the curve Qiall be algebraical, and we (hall find it's equation
to the axis by a formula, after the manner of Cartefms, In every other cafe the
curve will be tranfcendental.
EXAMPLE.
Let the equation of a curve referred ta a focus be = It* We
'vtc — zbz ■— Kz
fhall have, in this cafe, p = —r====r=z ; and in the equation ~ zr -rJr.
cc — zbz—zz » yhh — qq
fubftituting the value of ^, it will be ^ - == = ■ . ^ . Make
Vfc T- zbz — ZZ \ hh — qq.
^ -}- z zz /, then hb + ^'bz + zz = //, and bb — // r= — ^bz — zz ;
wherefore, makiniz; the fubftitution,, it will be • . t= = . ^
• *^ V a- -1- ^^ — ^ // s/hh —
For
$ECT. nr.
ANALYTICAL INSTITUTIONS,
15^1
For a particular cafe, let it be cc -{■ l^i^ ~ hh, on which fuppofition it will
be / = y, that is, ^ + z =: ^ = ~ . Therefore b% •\' zz — bx, and, in-
flead of 2, fubftituting it's value, the equation of the curve will be b\^xx -h yy
+ XX -\- yy zz hx.
88. The affigned canon alfo teaches us the manner of pafling from the
differential equation of a curve to the axis to that of the focus, in the way
following.
it will be ~ =
EXAMPLE I.
Let it be propofed to find the equation to
a focus in a circle, taking the focus in a
point of the circumference A.
Make AH -=. h, AG z= a?, AC = ;s
iz s/hx, Refume the formula — n
hx
s/ hh
n
, where is taken ^ z: — . Be-
Vhh
caufe, by the local equation of the circle, it
H O IS hx -=z zzy it will be ^ == z. Then mak-
ing q to vanilh, by fubftituting it's value Zj
= . Therefore, in the formula
or p —
'/hh
pz =: Uj if, inftead of p, we fl^ould fubftitute it's value i^ow found, it will be
^y,'^^ = j{, an equation of the circle to the focus, which is taken in A,
a point of the circumference.
EXAMPLE II.
89. Let it be propofed to find the equation of a conic fcdion, referred to
It's umbilicus M, that is, to it's focus (Fig. 102.)
Z 2 " Make
1 78 ANALYTICAL INSTITUTIONS-, BOOK 1 IF.
Make MB — .v, BD izjy ; the general equation, which comprehends all the
Ccdions of a cone, will be this; a ± -r- = ^/ xx + yy ', to the parabola with
the parameter 2/?, when c ":=:, h \ to the ellipfis with tranfverfe axis ~
rahh
bb - (C '
if h be greater than c ; to the hyperbola with tranfverfe axis rz
lahh
conjugate axis
, diftance of the vertex from the focus zr
c - bb"*
ah
^ + f '
with
if
'vcc — bb
h be Icfs than c. If ^ rz o, it will be to the circle with diameter := za. Put
z. And befides, hx = 2;^ ; then
ahchz
\/ XX + yy zz z 'r therefore <2 dt -r-
czq
IT
and gq
aahhhh
= 2;, or
bbhh
c
lablhh
a%
zz q. And taking the fluxions,
+ aa^>bhh ^ i i 11 bbhh
^ , and hh — qq zz hh f-
ccz,-z
lahhhh
= f«
cczsi
Hence
1 —
Jh ahch'%
^hh -
n
therefore p z:
cz^ hhcc%% — bbhhzz -j- zabbhhz
± abb
= — , and
*/ hhcc%7i — bbhhzz. + zabbhh^ — aabbhh
aabbhh
And as it is pi, zz 11, we
fliall have the equaiion required, u
-\r abhz
S^ bhiczz — bbhhTt.* -f- labbhhz — aabbbh
The negative fign ferves when the abfcifles are taken from the focus towards*
the vertex, and the pofitive are the contrary way.
90. I faid we ought to reduce the equation of the curve to the focus, to-
another referred to the axis; not becaufe this is abfolutely neceflary for the
complanation of fuperficies, or for the cubature of folids ; for the whole may
be obtained by means of this known theorem : The periphery of the curve^
drawn into the line defcribed by the centre ot gravity of that periphery, is equal
to the fuperficies of the folid which is generated by it's rotation. And the area
qf the curve, drawn into the line defcribed by the centre of gravity of the faid
area, is equal to the faid folid. But here wc muft not fuppofe our readers fa
flcillful as to be acq^uainted with the theory of Centres of Gravity.
Now, to have a competent notion of curves referred
to a focus, I fhall make an attempt at finding out
their conftrudion. Let BCD be one of thefe j the
co-ordinates infinitely near are AC, AE» which pro-
ceed from the point A, and may be called ;2, their
difference FE = i;, and the little arch CF, defcribed
with centre A, may be =: //. The nature of the
curve is commonly expreffed by the differential equa-
tion.
Bg. 105..
SECT. III. ANALYTICAL INSTITUTIONS. 1 73
tion pz zz ?V, In which p is any how given by z. Wherefore it mufl: be
obfervcd, that the firft member pz, having the variable Zy all which take their
origin from the pole A, is integrable either algebtaically or tranfcentlencaily.
But the other member li cannot be integrated without falling into a parall^gifiTij
as not being yet the complete fluxion of the arch «. For that element it in-
creafes or decreafes in a double refped, that is, in itfejf, and alfo by th&
increafing or diminifl-iing of the ordinates AC, AE. To proceed, therefore,
with accuracy, with any radius at pleafure, AI zr r, let a circle IGH be
defcribed, and in the periphery let any determinate pomt I be tiken, from
which, as from a fixed point, the increafing arches IG, Ifl, have their origin.
And producing, if neceflary, the variables AC, AE, to G and H, the fedors
ACF, AGH, will be fimilar, and therefore it is z . ii :: r . GH, which rnay
be called ^. Then -^ — u. But, by the general equation of the curve, it
1% pz '=■ iii then — =: pz, and therefore ^ zz ^. Now, by finding the
fluent, it will be /— = ^ r=z IG. The adding or taking away of the con-
ftants in the integration, will have no other effed, but to diverfify the fituatioa
of the point I.
EXAMPLE I.
Let the logarithmic fpfral be to be conftrudted, the equation of which is
-^ zz «. But « = -^, therefore -^ = -^ . Or, becaufe the radius AI
b r or
is aflumed at pleafure, making b zz r, and taking a as unity, it will be
— z=. q. And by integration, Iz zz q, the geometrical conflrudlion of which
is tranfcendcntal, but yet is very fimple*
EXAMPLE IL
Let it be the hyperboHcal fpiral, with the conflant fubtangent rz a, and
therefore the equation is — = iu But u zz ~ , therefore — ~ ^; and by
integrating, it will be ^ — — =: 2. ,
la
174 Analytical institutions. book iit.
In fi-ch conflrudlons we have always the circular arch IG, which forjns the
homogeneum ^omparalionis ', the other member/-^ may be analytically inte-
grable, as in the fecond example, or tranfcendentally, by means of the quadra-
ture of the hyperbola, as in the firfl, or by any other method more compounded.
Whence, in one cafe only, our curves may be algebraical, and that is, when the
quantity f— may be reduced to the redification of an arch of a circle, which
to it's correfpondent IG is as number to number. If the proportion happen
to be furd, then the curve will indeed be mechanic, as BCED, but not de-
pen lent on the quadrature of the circle, being reduced to a different problem,
confilf ing in the dividing circular arches in any given ratio ; which may be
obtained by means of the helix or fpiral of Archimedes, or of the quadratrix of
Vimjiratus,
The things afore-mentioned furnidi us with another manner of pafTing from
expreffions of curves to a focus, to thole which are referred to an axis, or on
the contrary. For, becaufe — =: q — — — , making the tangent IK ~ /,
(§ 26.) this tangent / will be given analytically or tranfcendentally by z»
But AI zi r, AK zz v/rr ■+- //, AM = ^, MC = y. Therefore -^ =
i/rr + //, and, after due redudions, ^—^ — ^ = / = -^. But / is given
X
by z, and z — \/xx + yy ; fo that we are arrived at the curve in refped to
the axis, which may foonbe reduced to the ufual co-ordinates x and y. By
oing the fame fteps backwards, we may pafs from the equation to the axis, to
at in refped of the focus.
th
=. zz u re-
I refume the example of 6 87; that is, the curve .
vcc — 2hz — zz
ferred to a focus, to reduce it to the axis. Now, if fz zz u be taken for a
general equation of curves referred to a focus, it will be, in this particular cafe,
p = ^ . So that, fubflituting this value, inftead of p, in the
V ctf — zbz — zz
equation ^ = ^ = ^ . it will be ;^-;^ = ^r Make
h -\- z zz Sy z — s, I'lf -{- 2l^z -\- zz zz ss ; whence — liz — zz = i^i^ — ss.
And fubftitutin^ thefe values, it will be ; _. ^' -^ r= = - . .. . Making
° Vcc + bi> - 'ss ^^ + "
cc -^bb :;z hh, it will be /' = - f'' = ~^. But the integral of
the
SECT. Ill, ANALYTICAL INSTITUTIONS. I75
the firft member will be the arch of a circle, the radius of which is ^, and s is
the fine of the complement (§ 37.) multiplied by the conftant fraflion —;
and the integral of the fecond is an arch of a ci'-cle vvirh radius = r, and
tangent. equal to t. Wherefore the firft arch will be to the fecond as b to r,
or they will be to each other as their radii refpeilivelv ; then they will be
limilar, and therefore their tangents alfo will be in the fame ratio as their radii.
Therefore the tangent of the firft arch is — v/M — ss ; and it will be
— ^/bb — ss . t 11 b , r, or / zz — \/bb — ss. So that, reftoring the value
of X, and puttine; — inftead of /, we fliall have — — ^-^— — r —■ — ^»
which is an equadon reduced to the axis, and which may be exprcHed by the
co-ordjnates x and y only, by putting, inftead of 22, it's value xx + yy^
Then the equation will be hy -\- y\/x:< \-yy — x ^Z bh —bb ^^zh s/ xx-Yyy ^xx—yy^,
which is the fame as chat found at § 87, as before cited.
To pafs from equations to an axis, to thofe belonging to a focus, I take
Example I. at ^ 88, the equation of which to the circle is 2 1= Vhx (Fig. 104 )
The tangent given by z of the arch OQ* defcribed with centre A, and radius r,
was found to be — ^ — "^ — zz /. Then, in the canonical equation q zz — ^- — ,
'inftead of / and f, fubftituting their refpedive values, we fliall have — 9 ^^
- . I put it — ^, becaufe, as AC n: z increafes> the arch OQjz: q
will diminidi. But g 21: — : wherefore — = — ; that Is, — ■■
zz ii, which is the fame equation as that found at § 88.
91. The particular formula?, which are found in the cafe of curves having
their co ordinaies at oblique angles, are not lefs ufeful, becaufe fuch equations
may always be changed into others, which have their co-ordinates at right
angles ; and after that we may make ufe of the ordinary formulje.
To fliow this, make HG = />, (Fig. 103.) AG zi q , then it is /> =: — ,
J = * + — , naming, as before, AB = Xy BD = ^, and the ratio of the
whole fine to the right fine that of m to <?. Therefore it will be jy =:
~, and A^ = j— ~ z=z q ^, Wherefore, inftead of x and y» fubfti-
8 tutiag^
ANALYTICAL INSTITUTIONS,
BOOK 111.
176
tilting, in the propofed equation, tbefe values given by p and q^ we (hall have
the equation of the curve with the ordinates at right angles to each other. But
it will often happen that the primitive equation will be fimple ; and yet, by
transforming it, it may become fufficiently compound. AHb, though the
variables are feparate in the propofed equation, they may not be io in the
transformed equation; and what may increafe the difficulty, they cannot be
feparated by the ordinary rules of Divifion, Extradion of Roots, &c. How-
ever, in many particular cafes, perhaps it may not be amifs to try each method,
that we may make choice of that which, in the given cafe, fliall be moft
convenient.
But now it will be time to proceed to Examples, in which it is always
underftood, except when warning is given to the contrary, that the co-ordinates
are at right angles to each other.
EXAMPLE I.
Theqnadra- fig. lo5.
ture of curvi-
liaear fpaces.
92. Let ABC be an Apollonian parabola, with
the equation ax "=. yy^ any abfcifs AD zi x, it's
ordinate DB = y, and the fpace ADB is to be
fquarcd. Therefore it will be jy zz \/ax ; and
this value, being fubftituted, inftead of ^, in the
general formula for fpaces ^x, it will bcx^/ ax \ and
by integration, it will be ^xV ax + b. The quan-
tity b is the ufual conftant, which, in the integration,
ought to be added, and which now ought to be
determined. In the point A, that is, when .v = o, the fpace is nothing, and
therefore the integral ^xVax -f- b, which exprefles this fpace, ought alfo to
be nothing. Therefore, making ^v z= o, it will be ^o x \/ a X o + b = o,
that is, ^ = o • which is as much as to fay that, in this cafe, no conftant
quantity is to be joined to the integral. Therefore the fpace ABD = ^xV ax.
But i\/ ax — y. Whence ABD = \xy, that is, is equal to two third parts of
the reftangle of the abfcifs into the ordinate.
Now, if we ihould require the fpace comprehended by an affigned and
determinate abfcifs and ordinate, for example, when it is «• ~ 2^ ; as, by the
•equation of the curve, it is in this cafe y zz \/iaa^ this fpace will be =: ^aa^2.
If the abfcifles of the parabola fliould not begin at the vertex A, but at fome
given point D \ making, for example, AD -iz a, any line DE =: Xy the para-
meter = /, the equation will be a/ -\-fx = yy, and y zz \/a/ -h/x, Subfti-
luting
SECT. lit. ANALYTICAL INSTITUTIONS. X?7
4 -i
tuting this value in tbc formula yx^ it will be x\/ af -j- jx^ and by integrating,
-t X ^ + A' X y/af +fx -J- b will be equal to the fpace DECB. But, to
determine the conftant quantity b, it muft be coniidered, that at the poinc D,
where at zz o, the fpace will alfo be zz o , fo that, in the integral, making
X -^ o, it will be -1-^ v' ^/ -f- b z= o, and therefore the conftant b ir — \a*/ af.
So that, to have the integral complete, inftead of adding by we muft fubtratfl
^a*/ of\ and therefore the fpace required will be DECB = 4 x <2 + *? x
Let AE = a, and let x begin at E towards A, and take any line ED zz xi
the equation will be af '-' fx rz yy, and y =: \/ af — fx. Whence yx zi
x\/ of — fx^ and by integration, it will be —■ I X a — .v X s/af ^ fx ^ b.
But when x ~ o, the fpace alfo r: o. Therefore, in the integral, making
X zz Q>, it will become — \ay/af + ^ = o, or /» = \as/af. Therefore the
fpace EDBC = \a^/ af — ^ a — x\/af — fx.
It may be obferved, that, in general, the parabolical fpace AEC = ^AE x
EC ; wherefore the fpace ADB = ^AD X DB ; fo that the fpace DECB will
be = 4AE X EC — 4AD x DB ; which agrees with the calculus in both
cafes, when the origin of x is in the point D towards E, and in the point E
towards D.
I -take the general equation. to all parabolas, of what degree foever,
m H
it it = y i whence It will htyzza^x^^ and therefore the formula yx =
m n
a X X ; and, by Integration, the fpace will be 3= — — ■-- — — + ^. But,
taking a; z: o, it is found that b zi o ; fo that there is no conftant quantity to
be annexed to it, but the Integral before found is complete. Now, putting /
m n
inftead of ^ '^ ;v *^ , it will be -^^^ = to the fpace required.
Yoi.II. A a EX-
178
ANALYTICAL INSTITUTIONS,
BOOK III,
EXAMPLE II.
93, Let the curve he y zz ^/x -f a ; therefore it will be yx x: x^x + ^j
I
and, by integration, the fpace will be jj-^ X ^ + ^ X x + a\ ^ + ^. But,
making x =: o, it will be ^ =: ^ X ^\/a. Therefore the complete
integral or fpace required 1= — ^ x x -i- a x Z/x -\- a — ——-• X as/^*
EXAMPLE IIL
94. Let FED be the hyperbola between the
afymptotes, and make AB zz x, BE iz;y, and
the equation is xy z= aa. Then y zz — , and
therefore yx = — j and, by integration, the
fpace will be zz alx + b, taking the logarithm
from the logarithmic curve with fubtangene
zr a. B^jt, putting .v, zi o, the logarithm of
o is an infinite negative quantity, and therefore
the fpace is infinite which is contained by the-
curve EF continued in injinitum^ by the afym-
ptote, and by the co-ordinates AB, BE.
Let there be a liyperboloid of this equation a^iz xyj; then y =1/— , and there*
hre yx zr x\i^ — ; and, by integration, the fpace wi^U be n 2s/a^x + l>.
Now, putting X zz o. It is B'ZZ o ; therefore no conftant quantity need be
added to complete the integral. So that the fpace ABEF, infinitely produced
upwards, will be the finite quantity z*/ a?x, or from the equation of the curve
Let there be a byperboloid of this equation, a^ z: xxy ; then y zi —^ , and
j^v = ~ ; and, by^ integration,, the fpace will be = — -^ + ^. But, putting
^ ~ o.
SECT. Til. ANALYTICAL INSTITUTIONS. 17^
X zz o, it will be — , an infinite quantity, and therefore ^ is infinite. Where-
fore, to have the integral complete, an infinite quantity ought to be added to
ir, and therefore the fpace itfelf is infinite.
Let the equation be tf'"^ := x^y", which is to all hyperboloids in general ;
then y :=. a x "* , and therefore fyx ~ — \-b. Now, ifn^rr i,
I? =: I, that is, xy z= aa, we (hould have fyx = — +^, an infinite quan-
tity ; vvhence the fpace will be infinite, as was feen before.
If » n I, »j — 2, that is, a^ =z xyy, then fyx z= 2\/a^x -f- b. But,
putting a; zr o, it will be alfo b zz o ; therefore the complete integral, or the
fpace required, will be zz 2\^a^x ~ xxy^ by the equation of the curve; which
is therefore finite, though infinitely produced upwards towards F.
If « zz 2, m zz I, that is, a} z=z xxy, it will be/j;c = — ~ -f- b. Bur,
making x =: o, b will be infinite; fo that an infinite quantity is to be added ta
the integral, and the fpace itfelf will be infinite.
If n =: I, m zz 2-, that is, a'* = xy' ; it will be fyx = In'Tx^ -{, b, Buf,
making .v = o, it will be <^ =: o, and therefore the integral is complete.
That is, the fpace will be =: '^a*xx = ^xy, a finite quantity, however
infinitely produced upwards.
If « = 3, m=z ti that is, a* = x^y ; it will he fyx zz — +3. But,
making x "z: o, b will be infinite, and therefore the fpace is infinite.
If « = I, m — 4, that is, a^ = xy* ; it will be fyx zz -*- */a^x^ -f b. But,
making x := o, it will be ^ — o ; fo that the integral is complete, and the
whole ipace — *■ ^aKx^ z=z ^xy, a finite quantity-.
If « = 4, m z= I, that is, a^ =z x*y ; it will be fyx zz — -{ b. Now
making .y = o, b will be infinite, and therefore the fpace is infinite. In the
fame manner we might proceed to other cafes, as far as we pleafe.
Now jet us take the abfciffcs from the point B, to find the fpace BCDE.
Make AB = ^, BC = ,v, CD zz _y, and let it be the fame Jpllonian hyper-
bola, whofe equation is ly -^ xy ^=- aa. Then it will be y =r 7-—- » and
b -^ X
A a 2 therefore
iSo ANALYTICAL INSTITUTIONS. BOOKIIf.
therefore yx — - — '— . Then, by Integration, fyx zz al b + x 4* /, taking
t'he logarithm from the logarithmic with fubtangent =r a. But, re determine
the conflant quantity /, making a; = o, it ought to be / zz — alb-^i lo that
the complete integral or fpace BCDE will ht al b -^^ x — alb.
If we take x negative zi; BA = — i-, then al b ■\- x is equal to a multi-
plied into the logarithln of o. But the logarithm of o is. an infinite negative
<]nantity ; fo that, in this cafe, the fpace is negative, that is, towards M, and
a!fo infinite, as has been feen above; and therefore the fpace between the
Apollonian hj>perbola and. it's^ afymptotes is mfinite, being infinitely produced
both ways. , .
Let it be the cubical hyperboloid whofe equation is hyy + xyy = c^. It will
be J' = v/ , , whence jj'a: = '^^r-T~> and by integration, Jyx zz
Q.s/ a^t? ■\- a^x +/. Bat, making x zz o, it will be /= — 7.^a^b\ fo that
the complete integral or fpace EBCD will be = 2^a^b + a^x — 2*/a^b', and
taking X infinite, the fpace EBCD, infinitely produced towards C, will be
infinite alfa.
Taking .v negative = BA =z — h, the integral will be — 2*/ a^b, fo that
the fpace will be negative ; that is, it will be FEBAM, and will be finite,
however infinitely produced towards M ; as is alfo feen before.
Let it be the hyperboloid of this equation b + x\^ x y — a^' It will be
y. = ==7I J whence yx = T==r, . And, by integrating, /yx = — -^ +/.
Now, putting a; zz o, it will be /== -^, and therefore the complete integral,
or the fpace EBCD, will be -7— — t-^- — . Taking x infinite, the term
— ~ — will be = o ; fo that the fpace will be finite, though infinitely pro-
duced towards C. Let x be negative zz BA = — ^; the integral will be
-T -~ , But — • ~ is infinite and negative, and therefore the fpace
towards M will be infinite. By proceeding in this manner, we may find that
the fpace between the Apollonian hyperbola and it's afymptotes, produced both
ways infinitely, will be mfinite ; between the firft cubical hyperboloid and it*s
afymptotes, it will be finite towards M, and infinite towards C; between the
fecond cubical hyperboloid and it's afymptotes, it will be infinite towards M,
a>^d finite towards C •, between the firft hyperboloid of the fourth kind and it's
afymptotes;^ .
52CT. nr*
ANALYTICAL INSTITUTION ♦•
iSl
afymptotes, it will be finite- towards M, and infinite towards C; between the
fecond liyperboloid and it's afymptotes, it will be finite towards C, and infinite
towards M. And fo on.
Now, to have recourfe to infinite feries : I take the expreffion of the fpace..
BCDE, of the aforefaid 4P^//c;?w« hyperbola, that is, j^^ . Th'is, reduced
into a feries, will be z:
KiS y
hb
b^
7^ > &c. And, .by integra--
tion,
+
,, . -^ 77 , &c. ; which feries, infinitely continued, wilt
be accurately equal to the fpace BCDE. And if it were fummable, if would
give us the fpace required in finite terms, that is, algebraically: and this would
be the true quadrature of the hyperbola. But as this is- not fummable, the
more terms we take of it, beginning with the firll, the nearer. approach we
ihall make to the juft value of this fpace,.
Now I take the abfcifs BT on the negative fide, and the equation: of the
curve will be By — xy =: aa^ and. therefore yx =: y— — ; and, reducifig to- a
j^, &c. And by in-
feries, it will be J'a; = -7 — h -r: ^
aPx
a^x3
bi
H n r
b*
tegration./yx = T" + IF + "P^ + ?i^ "^ 5F ' ^^' "^^''^^ '^ ^^''^^ ^° ^^^•
fpace BTPK. Taking BT = BA, the fpace FKBAM, infinitely produced
towards M, will be =z aa -\- \aa -^ ^aa + ^aa + ^aa^ &c. ; the- value of:
which feries.bejng infinite,, the fpace it denotes will be infinite alfo.
EXAMPLE IV..
H^..ioa*
95. Let OC be an equilateral hyperbola,
between the afymptotes AS, AB, and make*
AB = BC = ^, BI = — A'. Let the me-
chanical curve BEF be conceived to be de-
fcribed, fuch, that the redangle of AB into
any ordinate IE may be equal to the corre-
fponding hyperbolical fpace BCOI. . The
indeterminate fpace SABEF is required. .
Make the ordinate IE ■=. z. It has beeiv
found already, that the fpace BCOI ..is equal
t(x>
1^82
ANALYTICAL INSTITUTIONS.
BOOK III.
#J
X*
to the ferles ^;tf + t^* + -^ + -^ + —-, &c. making a and h equal. Then,
by the property of the curve, it will be 2; =: .v +
therefore zx
+ ITT + -;r > ^c. and
XX +
3 a* 4fl
4. fll -j- fZl , &c. And finally, by integration, the
3^'
4^3
fpace BIE will be = ^ + ^+-f^ + -^4-:;^,&c. Now, taking
a: := ^ — BA, as to the whole fpace SABEF infinitely produced, it will be =:
^aa ■\- ^(,aa 4- -r\aa -f ^V^ + tV^^> &c. which feries is fummable, and is
z=: aa-y fo that it is algebiaically quadrable, and the fpace SABEF, infinitely
produced, is equal to the fquare of BA.
EXAMPLE V.
Fig, 109;
96. Let ATC be a hyperbola, it's
tranfverfe axis AD = 2a , the parameter
= p, EB = X, BC = y, and therefore
the equation xx -— aa zz: -^, and let
the fpace ABC be required. It will be
^ '\'l\
tr» —
pa^
and the
therefore 5^ = \/
formula will be yx = x\/^-^ ~,
Now, if we proceed to integration, we
fhould find, after the ufual manner, that
the integral is partly algebraical, and
paj-tly logarithmical ; fo that the fpace ABC of the hyperbola depends on the
defcfiption of the logarithmic curve.
If we would have the fpace ACHE ; making MT infinitely near to BC, it's
element will be the infinittfimal fpace ITCH; and therefore the formula will be
xjy in which, inftcad of .v, fubftituting it's value given by y from the equation.
It will be xy z=:yV^^ ^ , the integral of which, in the fame manner, de-
pends upon the logarithmic curve.
And, as well in the formula yx of the firft fpnce, as in xy of the fecond, if,
inftead of x in that, or of j^' in this, we fhould fubflitute their refpedtive values
given from the equation ,; we fliould likewife find integrals of the fame nature.
. I^ow,
SECT. HI. ANALYTICAL INSTITWTIONS. iSj
Now, to return to infinite feries. I take the formula of the fpace AG'IEA,
that is, xy. Then xy ~ j -/ ?^^^2L_^ i and, for greater facility, making laizip^
(for the conftants make no alteration in the me: hod,) that is^ fuppofing the
hyperbola to be equilateral, it will be xy r=:y\^yy -t aa; and, reducing the
radical to an infinite feries, it will be xy = ay -{- — ^rr + '^ ^^-^ »
&c. And by integration, fxy, or the fpace AC HE A, zn ay + -f- — -^
+ — ~-7 n J , &c. a feries, the fummation of which is unknown^.
7 X ita^ 9 X i^Sa? ' »
And fubtrading this feries from the redlangle xy, we flioiild have the fpace
ABC.
From the centre E let the lines ET, EC, be drawn infinitely near, and let AKP
be a tangent at the vertex. With centre E let the little circular arches KQ>_TR,
be drawn. ItwiU be AK = -^ , KP = fiZ-Zj^ ET := ^/xx + yy, EK =
. And, becaufe of fimilar triangles PKQ^ KEA, or TEiVf, it will
be KQ^= ^^-^ "L21 . And^ becaufe of fimilar fedors EKQ», ETR, It will be
TR = -^=^y4= ; and therefore it will be t^T x TR = 'Z_Z2± t^g
element of the fedor ETA. And, inftead of ^ and j), fubflituting their values
given from the equation of the curve y zz Vxx — aa„ (fuppofing the hyperbola
to be equilateral,) it will be — -^-^ ; and by integration, / fi*" i\^^^
2yf^xx — aa lV XX — aa
is, the fedor ETA, will he equal to — - \alx — \/ xx - aa in the logarithmic:
with fubtangent n a ; which Ipace is therefore expreffed by a negative quan-
tity, becaufe it is affumed on the negative fide.
By reducing the formula into^a feries, we (hall find ^— - ^ + -i
%.Vxx-aa. 2x 4^i
aVxx + v^'
Now, to integrate the firft term of the feries,. there wouW be occaf^on, firf^,
to reduce it to an infinite feries. Therefore it would be better to do it more
expeditioully after the following manner. Make EM =: .v, MT = y, AK — z^
then KP =: z. Make KE zz ^, AE = a, the iranlverfe fcmiaxis* and the
8i ftinuf
,j§4 ANALYTICAL l-K S T I T U T I O N'^i BOOK 111.
femi-€onjugate •= h. Therefore it will be KQ^= —^ , ET = ~ , TR =:
'—, and therefore fET x TR = — . But, by the equation of the curve,
^ ' 'aa; and by fi mi lar -triangles EAK, EMT, it will be jy i=
~ . Therefore zx = l^^^x — -aa, and xx ~ jfzT^ » ^^^ confeqtiendy the
.formula will be rr^^ , which, reduced to a feries, will be -^ +' ^ +
f^ + ff!j + ^, .&c.; and by integration, f-^^, that is, the fpace
iz:5 az} az^ a'xP a%^ q
'ETA, will be =-^ + ^+7^ + 7:^^ + 78^. &c.
It IS y =— v/aw
EXAMPLE VI.
F/V. lio« ► 97» Let ABD be a circle defcribed with
diameter AD zz a, and let the area of any half
fegment AHE be required. Make AE = x,
EH zz yi the equation will be ^ = \^ax — xx,
and therefore yx = xy^ax — xx. Here it would
be to no purpofe to free the formula from it's
radical, or to try any other methods, in order
to change it into fome other formula, which
may admit of an algebraical integration, or by
1> means of the logarithms. For this would be
an ufelefs trouble, becaufe we fhould flill be
brought to a formula of quadrature or redilicatiop of the circle ; as has been
obferved at § 27* And therefore we (hall thus proceed by way of infinite
feries.
Refolving the formula into a feries, it will be x\/ax — xx = a^x*x — •
I-
za"
x^x
i:.
x'x
8a* i6tf^
S 7
, &c. And by integration, f^x^ or the fpace AEH =:
, J- i *- X- ^ o ^
5^*
z%a^
,72a'
J^aw
$ECT. III. ANALYTICAL INSTITUTIONS. > iSj
Now make the radius CA n <2, and let CE = x^ EH zi _y, and the equa-
tion will be ^ =: \/aa — xx. Therefore yx =: :x\/aa — xx ; and reducing
this to a feries, ;,i = ^i — ^ - gl ^ ^ _ ^^ , &c. And by inte-
gration, fyx, that is, the fpace CEHB z=. <7a; — -^ — -^ — .
1152^7
6a 40a' 1 1 2«^
, &C.
And making x -iz a, m refped of the whole quadrant, it will be aa — -|^«
— -^^aa — T-TTT*^^ — -t-ctt:^^* &c. the quadruple of which feries will be the
area of the whole circle.
Now, by means of a fedor. Make CA = «, AQ^n at, and drawing CK
infinitely near to CQ»^ it will be QK zz Xy QQ^zz \/ aa + xx \ and with
centre C defcribing the infinitefimal arch QS j becaufe of finiilar triangles KSQ»_
QAC, it will be QS =r "'" , and therefore MN = -ffl- . Whence
the little fedor CMN, the element of the fc6lor CAM, will be =: —"^ ,
2 X aa-i- XX
which, reduced into a feries, will be = — — ^-^ + f--^ — aixx ^
And by integration, it will be/ 'V^^ " > or tbe fedor CMA = ~ ^
+ 7^ — -^TT + TTT ' ^c* ; ^"^ fuppofing the arch AM to be half the
quadrant, that is, taking; x := a. the feries is -^ + — — — , &c. :
* 'o » 26' 10 14'*
and the double of this, or aa — ^aa + ^aa — -f «^, &c. will be the
quadrant ABC.
Inftead of taking the radius CA =z a, if I had taken it iz V^aa, the
quadrant would have been ABC := ~ + ^ ~ , &c. ; and
^ 8 3x85x87x8'
actually fubtrafting every negative term from the pofitive term before it, [and
multiplying the refult by 4,1 it would be -^ \. fl. ^ ^ ^ ffL &c, r=the
area of the whole circle j] which is the fame feries as is inferted by Mr. Leibnitz
in the Leipfic Ads, for the yelr 1682,
Vol. II. Bb EX-
i86
ANALYTICAL INSTITUTIONS.
BOOK III.
EXAMPLE VII.
Fig. III.
98. Let BCD be an ellipfis, the trand
vertc femiaxis AB = <?, the femi-con jugate
AC =: b, AE = A*, EH — y ; whence the
bh
equation will be — y^ aa — xx :=:yy, and
aa
bk
therefore y^c = — »/ aa^xx^ the element
'' a
hx
of the area AEHC. But x\/ (.a — xx is
the formula for fquaring the circle BOD,
the diameter of which is the tranfverfe axis
of the ellipfis ; fo that the quadrature of
the ellipfis will depend on that of the circle.
And becaufe f-^\/aa — xx = EHCA, Sindi fx^aa — xx — EMOA, any
fpace of the ellipfis to the correfpondent fpace of the circle on the diameter
DB, will be as b to ay that is, as the conjugate femiaxis to the tranfverfe femi-
axis ; and confequendy the whole ellipfis to the whole circle will be in the fame
ratio. But, as circles are to each other as the fquares of their diameters or
radii, if we take a circle the radius of which is iz \/ab, that is, a mean prc^-
portional between the two femiaxes of the ellipfis BCD, this circle will be to
the circle BOD as ab , aa \\ h , a. But the area of the ellipfis BCD is to the
fame circle BOD, in this very ratio. Therefore the area of the ellipfis will be
equal to the area of the circle, the radius of which is a mean proportional be-
tween the two femiaxes of the ellipfis.
Now, by the help of feries. The formula — \/ aa — xx^ being reduced
to a feries, will be =z — into a r-r- ?-r ^J-r , &c. And by
ix
bx^
hxi
bx^
integration,/ — \/aa — xx, or area ACHE, •=. bx ^^ -r-^ — ^
^-^ , &c. And making x :^ a, the area ACB, or a fourth part of the
ellipfis, will be z; ab into i — 4- — ^V — txt — tttt» &c.
In the fame ellipfis, taking any arch DS, let DP be a tangent in D, AI -=r *»,
IS =1 J' ; and through the point S drawing AP infinitely near AK, which cuts
the ellipfis in T. With centre A let the little arches of a circle KQ»_ TR, be
defcribed.
SECT. III. ANALYTICAL INSTITUTIONS. 1 87
_ °y
defcribed. Then it will be AS ^ V xx ■\- yy == AT, DP =: -^, AK=AP
jV^l+_tv , KP iz IL£fZ_+-^2i, PK being a negative difference. And by
— axy At ay'x
X '^ XX + yy
— x} Jf y'x
the fimilitude of the triangles PQK, PAD, it will be KQ^.zz
And by the fimilitude of the fedors ATR, AKQ^ it will be TR _ ^ ,
'vxx + yy
and therefore ^TR X AT, that is, ^^-^ — — , will be the formula for the
fpace ACT. This will be finally -— -^ '' , by fubflituting, inftead of y and
2'vaa — XX
J/, their values given from the equation of the curve.
Rnf f— --^ is the redificacion of the circle, as was feen at § 37, and
w aa — XX
as will be here feen alfo. Therefore the quadrature of elliptical fedors will
depend on the rectification or quadrature of the circle. It would fignify nothing
to take pains to free the formula from it*s radical, becaufe, notwithilanding this,
we fliould flill fall upon a formula, which would depend on the fame circle.
Now, by the means of feries, we fhould find it to be — r=~=T =: ~ -f
3£^ 3^ 53^ 35^ ^^^ ^^^ integration, the fpace ATC =
il + ±L + 1^ + i^ + lifil , &c. And making * = «. in refneft
to the whole fpace ADC, a fourth part of the entire elliptical fpace, it will be
2 12 80 ' 224 2304'
If we would free the formula from the radical vinculum, making the fubfti-
tution of \/aa — xx = ^ — — , it would be changed into this other, -^^ ^ • ,
which, being reduced into a feries, would be found to be bz ^ + -fi
bz^z bz^z „ * J I • • 7 bz* , b%^ bz) , bz*
- -i^ + — ' ^"- ^""^ ''y >ntegrat.on, hz -—+ — - — + --
&c. ; and making x = a, in which cafe it is alfo z zn a, it will be ab — ^a^
4. *.al^ — ^ab + -^al^, &c. in refpeft of a quadrant of the ellipfis.
B b 2 And
i88
ANALYTICAL INSTITUTIONS.
BOOK III,
And if we fuppofe a ■=■ h, the ellipfis becomes a circle with radius n a,
and the feries will be as at § 97, which will exprefs the quadrant. And there-
fore, from hence it may alfo be feen, that the area of the ellipfis is to the area,
of the circle, the diameter of which is equal to the tranfverfe axis of the
ellipfis, as the conjugate axis is to the tranfverfe axis of the fame ellipfis.
EXAMPLE VIII.
Fig, 112.
^^, Let NAM be a cycloid, it's ge-»
nerating circle ARH, and make AH nr^jr,
AB zz ;f, BC = i, BE zz y, DF - y.
The equation will be j/ = —~==^^ =:
V ax — XX
But the little fpace QRFP is
x'va —
the element of the fpace AEQ>, and therefore FP X PQl that is,
fcs/ax — XX will be it's formula. But fx\/ax — xx is the circular fegment
ASB; therefore the cycloidal fpace AEQ^will be equal to the correfpondent
circular fpace ASB, an5 the whole fpace AMK will be equal to the femicircle.
But the redangle AHMK is quadruple of the femicircle, becaufe it is the
produ<5l of the femiperiphery into the diameter. Therefore the fpace AMH
will be triple of the femicircle, and therefore the whole cycloidal fpace will be
triple of the generating circle.
If we would have the fpace AFC immediately; as the little fpace FCBE,
that is, yx, is it's element, and from the equation of the curve we have y zz
Xn/a — X
I let the homogeneum comparationis be reduced into a feries, firR: multi-
plying the numerator and denominator by Vx ; whence it would be
x\/av — XX
x'^x
1,
x^x
I.
x^x
— — J- — — ^ , &c. ; and therefore, by integration^
2a*
/
xtJax —XX i X
— —-J-, &c. Whence yx zz
3^
20a'
56a'
aa
{SECT. rtr.
AITALTTICAL INSTITUTIONS.
109^
3<»'
2oa'
56a^
, &c* And laflly, by integration, Jyk s:
ABE = i££ - ^ - ^ _ -^, &c.
3 15a*
100}
2^za^
EXAMPLE IX.
F*S' "3
CGI, CDF, it will be DF =
GI =
___ aax + aaz
100. Let ADR be the conchoid, CB :z:
BA = a, CM = X, MD rz j, and let the
fpace ADGB be required. Make CO = 2,
which will always be given by x and y of
the propofed curve, as is plain enoughs
Let CE be infinitely near to CD, and with
centre C, intervals CO, CD, let the two
little arches GI, DF, be defcribed. It will
be HI zz z, and the trapezium tDGI will
be the elenDent of the fpace required. By
the fimilar triangles HIG, BGC, it will be
md by the limilar fedors
Vzz — aa
DF T GI X tGD =
2^*22; + a^z "
But the trapezium FDGI =;:
ia^zz + a^z
Therefore /
that is,.
2zVzz — aa 2.z^zz — aa
ala -{■ \a x arch of a circle of which the radius =
[alz -t \/zz — aa
and the tangent — \/zz — aa^ (taking the logarithm in the logarithmic with
fubtangent ~ a,) will be equal to the fpace required,
Alfo, the whole fpace may be had of the fame conchoid, and likewife the
parts, by confidering the curve in relation to it's axis. In the fame Figure
make AB z= DG =: BC = a, BM zz x, MD = y ; and from the point G
let there be drawn GO perpendicular to the ordinate MD ; it will be DO =
^ ad ^ XX, becaufe of the right angle GOD; and by the fimilitude of the
triangles CBG, GOD, it will be BG = djl-rjll ^ MO. Therefore MI>
^90
z= \/ aa — XX 4-
ANALYTICAL 1 NS TIT U T IONS.
BOOK IIT.
aa — XX
= y. Whence ;'i,. that is, the element of the
fpace, will ht ks/ aa — xx +
ax
'J aa —
The fluent of the firft term de-
pends on the quadrature of the circle, and of the fecond on that of the
hyperbola.
Fig, 114.
EXAMPLE X.
loi. Let AMI be the ciflbid of Diodes^ the equation
of which is yy = ^ . Therefore, fubftituting the
value of y given by the equation, the formula will be
~-^=^= , the integral of which depends on the quadra-
Va — X
ture of the circle. To have the relation of the whole
fpace of the ciflbid to that of the generating circle, it
mufl: be confidered, that, the equation being yy =
, it will be alfo yy x ax '^ xx =z x*, and there-
fore y\^ax — XX — XX, This fuppofed, by differencing the propofed equation
ayy — xyy = a:', there arifes 2ayj — ixyy — yyx = ^xxx, that is, ay x
a '— X '— yic zz. ^f^ , And, becaufe xx zz y\/ax — xx, therefore 2y X
a — X — yx ::: ^x\/ax — xx. But y x a — ^ is the element of the fpace
AMQB, and yx is the element of the fpace AMP ; then, by integrating, as
to the whole fpace, \t\% Jy Y^ a — x — /yx. Then, in this circumftance, it
will be 2/y x a — x — fyx — /y x a — x, and therefore fy x a — x
zr. ^fxs/ax •— XX J and becaufe, in the cafe of the total fpace of the ciflbid,
fks/ax — XX is the area of the femicircle ABN ; thence the fpace of the ciflbid,
infinitely produced, will be triple of the generating circle.
EX.
SECT. ni.
ANALYTICAL INSTITUTIONS.
191
EXAMPLE XI.
Fig, 115.
102. Let HBD be the logarithmic to this
afympiote 1VIQ> and let AB zz a — fub-
tangent, KH = ^, AK zz Xy and the
equation ~ = x. Then the formula will
be yx z= ay^ and by integration, Jyx = ay
+ bb. But, fuppofing y zz a, it will be
bb zz -^ aa\ fo that the integral complete,
or fpace AKHB zz ay — aa. Taking any
other ordinate MN = z, it will be alfo
AMNB = ^2 — aa, fo that MKHN zi
ay — az. Let there be an ordinate EF lefs than AB, and equal to y^ AE =
— ;tf ; in the fame manner, the equation will be — =: x, becaufe, it being
negative, it's difference will be negative alfo. But the abfcifs x increafing, the
ordinate y decreafes, and therefore y mud be negative. Now, becaufe the
element of the fpace will alfo be negative, this element will be — yic, that is,
— ay\ and by integrating, — ay -\- bb. But when y = a^ it will be hb = aa\
therefore the complete integral, that is, the fpace AEFB, will ht "=. aa — ay.
And making y zz o^ that is, when it is infinitely produced towards Q»^ it will
be n aa. And confequeiitly the fame fpace, infinitely produced towards Q,
but which begins from any ordinate EF zi y, will be = ay.
EXAMPLE Xir.
Fig, 116.
Q/
A
*'k. •■
C ^^}
^
~-^
^l"'
•••'1
•
...11
K N
D
103. Let the curve ABF be the
tra^rix, the primary property of which
is this, that the tangent BP, at any
point B, is always equal to a conftant
right line given. Make any abfcifs
ED ~ X, the ordinate DB iz y, the
arch of the curve AB =: «, and the
given right line =1 a, Becaufe, as the
abfcifs ED increafes, the ordinate DB
diminifhes, it*s element will be nega-
tive.
•gl
ANALYTICAL INSTITUTIONS.
BOOK IIJ.
tive, that is, — j/. Whence, from the property of the curve, we fliall have
3" _
the equation — -^ r: a; and, inllead o{ u, putting it's value \/xx + yj.
It IS .V ss
— yy/aa — yy
This being done, in the formula for areas yx, inftead
of Xy putting it's value given by the equation of the curve, we fliall have
— yy/aa — yy for the element of any fpace ABDE. But, fuppofing the firfl
of the ordinates AE = a, and with radius EA defcribing the quadrant AQM,
and drawing BQ^parallel to MH ; becaufe Y)^ = EC = jy, and, by the pro-
perty of the circle, CQ^= s/ aa — yy, the element of the circular fpace CQA
will alfo be — yt/aa — yy» Whence the fpace CQA will be equal to the
fpace ABDE i and fo of others. And confequently the fpace infinitely pro-
duced, comprehended by the tra^rix ABF, b^ the afymptote EH, and by the
right line AE, will be equal to the quadrant AME.
EXAMPLE XIII.
Fig, 117. 104. Let ACB be a fpiral, and AB = a the
radius of the circle BMD, the periphery of which
= ^, any arch BD = x, AC = y \ the equation
will be by — ax. Drawing AE infinitely near to
AD, it will be ED = xi and with centre A
defcribe the infinitefimal arch CH. Becaufe of
fimilar fedors ACH, ADE, it will be CH =
^ , and therefore the fedor ACH, the element
a
of the fpace ANCA, will be = ^ . But, by the
equation of the curve, it is jy = -^ ; therefore
ihatelement will be"=: ~^, and by integration, and omitting the conftant
qiiantity as fuperfluous, the fpace ACN will be ^ ; and making a; = /», in
refped of the whole fpace ANB, which will be = {ab,^
Let
•SECT, III. ANALYTICAL INSTITUTIONS. 1^3
Let the equation be general to infinite fpirals d^x" z: fy"i then It will be
yy
2«
aax ^^
b «
2n-f ff»
2»
, and the formula of the fpace will be — — ^, and by integration,
2h »*
max f"
7.71
; and making x = ^, the whole fpace will be z:
mah
40 + 2m
It is eafy to perceive, that the fpace ABMDCNA, ,terminated by the radius
AB, the circular arch HMD, and the portion of the fpiral ANC, will be
— — gjT- ; becaufe it is equal to the fedor ABMDA, diminiflied by the fpace
ACN. But if we would have it by means of the differential formula, it is
enough to obferve, that it's element will be the infinitefimal trapezium ECHD,
which is known to be zz D£ + CH x 4^CD, that is, ;c- + — X
a — y
aax — yyx
2a
. And, inftead oi yy, putting it*s value — ^ given by. the equation,
it Will be — — -^^-i and by mtegration, — ^, omittmg the fuperfluous
conftant quantity.
EXAMPLE XIV.
Fig, iiS.
105. Let ABM be the parabola, whofe equa*
tlon is ax HZ yy, and make AC = x, CB = y-,
and let the ratio of the whole fine to the right
line of the angle BCD be that of ^ to ^; to the
fine of the complement be that of a to /; then
it will be BD = ^, and CD = A. Let
a a
CH = X, then CH x DB = CHMB, the
element of the fpace ACB, and therefore the
— . And, inftead of y, put-
formula will be
Vol. n.
ting it's value given from the equation, that is,
\/ax, it will be -^^^ ; and by integration,
2ii^, or ^> = lAC X BD.
Cc EX-
I^:
AKAL y TIC AL I N STIT U TTO N 8.
BOOK 111.
EXAMPLE XV.
I^ig. lipa
rc6. Let ACM be a parabola referred to^
the focus B, the equation of which will be
zi.u, making BC = z, CD'n «,
2 vzaz — aa
z + a
'^2a% — aa
an infinitely little arch of a circle, and the
parameter ~ ia. Then the infinitefimal
fedor BMG, or BDC, will be the element
of the fpace ABC, and therefore ^zu\ or
, will be the formula ; the integral of which will be found to be
^/2az-~aa + nvn. Now, taking 2 =: BA = f^, in which cafe th'e
fpace ought to be nothing, it will be mm z: o, and therefore tlie complete
integral, that is, the fpace ABC, is ^—r^\/2az — aa*
And in fa(5l:, from the point C letting fall CQ^ perpendicular to AQ^ the
fpace BCA is equal to the fpace QCA lefTened by the triangle BQC. But,
making BQ^= x, Q£^ zn y, it will be QCA — QCB = 4- x t« + ^ X y-
^i-xy =
3 a •{■ x
X y^
Therefore BCA = ^^4^ x y. But, by the pro-
6 '^-^ ' 6
perty of the parabola, BC == AQ_+ AB = a: + /?, that is, z =: ^ + ^, and
y zz ^aa'\'.2ax^iz \/2aZ'^aa, Therefore, thefe values being fubftituted;
inftead of ;!? and y, we fliall find- BCA w '^ Xj = '^-~^'s/2az^aa] as
above.
EXAMPLE XVL
10.7. If the fourth part AC of the periphery^,
of a circle be conceived to be flretched out
into a right line (ac), and taking. any portion
{ae) equal to the arch AE, let there be raifed
the perpendicular (ed) equal to the right line
DE;.
SECT. lU, ANALYTICAL INSTITUTIONS. ig^
DE ; the curve (af) which pafles through all the points (d) Co determined, k
called the line of right fines. Producing {ac) till it be equal to the Temicircum-
ference of the circle, the curve will have another branch beyond {ci), fimilar
and equal to the firft.
Let the radius be zr r, any arch AE =s x :=z {ae), the correfponding fine
DE — jy r: [ed) ; becaufe the fluxion or differential of the arch, exprefled by
means of the fine, is found to be -r==-, vve (ball have x =:= —-^— ^ which
Vrr — yy '^' rr —yy
is the equation of our curve. Therefore the formula yXy by flibftituting the
value of X., will be "^^ ; and by integration, — r\/rr — j/ + ??. But,
■»</rr — yy
putting jy rr o, it is n zz. rr. Therefore the complete integral is rr — r»/rr —yy
zz fpace (ade) ; and making y zz r, k will be rr = to the whole fpace (ale).
Whence, making TH the fquare of the radius, and producing the fine DE
to M, the fpace (ade) will be equal to the redlangle DH, and the whole fpace
{ate) equal to the fquare TH.
ic8. The Examples now produced may fuffice to Ihow the ufe of the method.
It only remains toobferve, that dften the equations of the curves, the areas df
which are to be fquared, (and this is alfo to be underllood in refpeft to redtifi-
cations, quadratures of -fuperficies, and cubatures,) may be fuch, that they
have not the variable quantities feparate, nor can they be feparated by divifion
only, and confequently are not reducible to the formulas required. Such would
be the curve, whofe equation is x^ -^ y^ zz: axy, for example.
In thefe cafes there is occafion to take the advantage of fome proper fubfti-
tution, by means of which the equation may be transformed into another, in
which the variable quantities arc feparate, or at lead are feparable. But it
cannot be determined, in general, what thofe fubflitutions ought to be. Therp
is need of practice, and perhaps many trials, to know when this may be fuc-
cefsfully performed.
As to the propofed equation x^ + y = axy, make y zz — ; and making
the iubltitution, the equation will be x^ H 5- = , that is, x' zz. 5 — .
Then, by difFerencins;, x'x zz. 111^^—. Wherefore, taking the formula
for fpaces, which is yx^ becaufe, by fubftitution, it is j^ zi ^ , this formula
axx'x
zz
4..z3i - 6zsi^^ it will be yx = Ifffirl^. and by integration, /yx = ^zx
will be — - ; and fubflituting, inftead of xxx, it*s value now found,
— r , It Will be yx =:
CC2
]j^6 ANALYTICAI* INSTITUTIONS. BOOK 11 J*
— ^j and, inftead of zz, reftoring it's value — , it will be finally /jy^c* =:
Zjaxx a*
3sy ^yy '
EXAMPLE XVII.
109. Let the curve be aVj* — x^ zn a^y, whofe area is required. Put
y zz ^ y and the equation will be transformed into this other, a^z — x^z^
— c^^ from whence we have x = — — — ~— ^; and therefore x ~
3»X a<2a— «']?
-, and y = r . Hence we Inall have the ele-
Or'k a}% , a^k 2a^z
ment of the area yx "= —r X aaz — «' :^ — r- — — r- ; and therefore,.
by integration, /^;c' := —^ + ^. And, inftead of 2;, reftoring it's value
— , the area will be 4- + -r-^" •
To this purpofe may be feen the Method of Mr. Craig, in his Book De
Calculo Fluentium,
EXAMPLE XVIII.
The reftifi- ^'°' ^^^ ^^^ Apollonian parabola be given to be rectified ; that is, to find a-
cation of right line equal to any arch of the fame parabola, the equation of which is-
ax '=^ yy» It's fluxion will be ax = lyjy and xx = ffi^. Now the formula
for re6tification is >/xx + yy ; (o that, fubftituting here, inftead of xx,
it's value given from the fluxional equation, it will be x/xx + j/j/ z:
= ~v/4j'jy + aa, the element of the Jpllonian parabola
ax =::
9ECT, m. A^NALYTICAL INSTITUTIONS. tqy
ax — yy. Proceeding to the integration; by making the fubnitiiiion of
s/ ^yy + aa zz iy -f z, in order to take away the radical, we Qiall find it to be
~v 4y^ -\- aa •=. — 8^ "" "7^ ^ gj' ^"^ integral of which we may fee is
partly algebraical, and partly logarithmical \. and therefore the redification of
tlie parabola depends on the quadrature of the hyperbola ; which truth may be
difcovered after this other manner, Lee
ADE be an equilateral hyperbola, with
femiaxis ~ ^, BC zz x from the centre,
CD = 2y, the equation of which will be
XX -— aa zz j\yy. Drawing GE infinitely
near to HD, then^HGED will be the -
element of the fpace ADHB. But we ■
know HGED to be ly^/^yy -{- aa^ which
is the fame formula as that for the redifi-
cation of the parabola, excepting the conftant denominator ia. Therefore, &c.
By the help of infinite feries. I take the above-written formula for the
redification of -the parabola, that is, -^s/^yy + aa, which, being reduced to
a feries, will be y + -^^
2vV 4V y lory o a i 1 •
~- + ^ - -^» &c. And, by integration,
2y3
zyi
4^7
whatever.
9a®
, &c. will be any arch
In the general formula s/ xx + yy, inftead of fubftituting, in the place of x,
it's value given by y from the equation of the curves if we fhould fubftitute,
in the place of y, it's value given by x, it would be b^+fl or fv£i±il"
which is not indeed more manageable than the other.
If the parabola was not that of Apllonius, but the fecond cubic, the equation
of which is axx z^y ; by taking the difference, it would be xx -zz 2E^ ^ ^^d
therefore the formula ^xx+yy = y\/^ — —, the integral of which is
^3LjLJff.^gay + ^aa + m. But, putting _y ~ o, it will he m =. — ^V^>
therefore the complete integral, or the length of the arch, will be
I»j
198
A-NALYTICAL I N S T I T U T I OIT'S.
300K III,
In the Apollonian parabola ADM, if it (hall
be AC z= ^ay and taking any line CK = y,
the parameter 1= |<j ; it will be AK ■=. ^a ■\- y^
KM - V'lffJl^'. Whence the element of
4
the area MKCD will be i^i^^-±-^, which \s
-^ 4
the fame with the element of the length of the
fecond cubical parabola, except the conftant
quantity a. And therefore the redification of this, and the quadrature of that,
is the fame thing. Whence, becaufe the quadrature of that may be found
algebraically, this is alfo algebraically reftifiable. And hence, in general, if
the expreffion of the element of any given curve, divided by the diiference of
the unknown quantity, be put for the ordinate, and the unknown quantity be
put for the abfcifs ; a new curve will thence arife, the quadrature of which will
give the rectification of the ^iven curve.
EXAMPLE XIX.
■Fig, 1^3.
X .— i
^a'^x *Ar-f
x**r
x +
III. Let AEM be a circle, it's diameter
AM =: a, AB = ;c ; it will be BF := j
z= \/^ax — XX. Then y zr , yy zz
"y ax — XX
\aaxx ^ a.kx ^ xxkx ^ And therefore the ele-
ax — XX
\ ment of the curve FH r= V xx + yy =
2V ax —
, and reducing it to a feries, it will be
+
J
+
105* -.V
2 X2a^ 2x2x4a* 2X2X4x6^'* 2X2X4x6x8^^
~, &c. And
X i
T +
3x'
+
i^x^
by integration, it will be a'^x^ 4- — ^
a X 3fl* 2 X 4 X ^a^- 2 X 4 X 6 X 7a*
'^ — T* &c. Or, becaufe it is xx = /-^ x ^^ -^ ^^^^ . ^ ^^^^
X4X6x^XQa* *
aX4
4lituting yy inftead of ax — xx, xx = , ^^-^^ , ■-, then putting this valu^ in-
ilead
siecT. II r.
ANALYTICAL INSTITUTIONS'.
J 99
flead' of XX in the general formula, it will be \/xx -^ yy zz
2 V iaa — VI
which, being reduced to a feries, will be found to be — y + ^^ + ^^ -f
—^ — h J 9 &c. And by integration, it will be finally the arch FA =: y
3^2* 5fl* ;a^ ' 9a' '
But if the radius were made = a, the feries would be y + — - — +
2 X 4 X 5<2+ ^ 2 X 4 X 6 X 7a^ ^ 2 X 4 X 6 X 8 X 9a8 '
Laftly, if it were DB = x, the radius DA =: a, it would be ^ = */aa xx,
and J/ :=: "" '^'^ - ; therefore -/^a: +_yj/ =: ^ and, reducing to a
yC v-
feries, it will be ^ = x -{ f- -^ r -\ — -r-z + — --^-- — .
Vaa — xx 2fl^. 2 X 4a+ 2 x4x6a'' 2x4x6x8a"
&c. Whence the arch EF = ,r +
3*5
i^xf
loc^x^
2 X 3«» ~ a x 4 X 5«* 2 x 4 X 6 X 7«*
-h
2 X 4 X 6 X 8 X 9a'
, &c.
EXAMPLE XX.
Fig» 124,
hhxxxx
1X2. Let ADC be an ellipfis with
tranfverfe femiaxis BA — «, and conju-
gate femiaxrs ^D := ^, BE = x, EO zzy ;
the equation will be '^'^
^^
= ^«
xx.
^ and therefore jyy = ""^^'^'^ , *nd yj/^ =
aa
aa y. aa — xx
and the general formula Vxx ■\' yy =: v
• • , hbxxx'x
XX + -T
a* — aaxft.
X^a* — a*«* + ****
l\/<?fl —
A:.;f
Inftead
aoo ANALYTICAL I N S T I T U Tt OIT S. BOOK ITI.
Inftead of fubllituting the value of y given by h from the equation, if wc
-, - . , r y\/aayy — bbyy + ^^ ,
mould fubflitute the value of x, it would be s/xx + yy "^ ■ -■ /,, = *
-^^ hvbb—yy
But both of the expreffions fo found would want one of the conditions of § 38,
without which it may be ieen, that thefe formulas cannot be freed from radical
figns, and fo prepared for integration. Then to proceed to feries, 1 take one
of the two formulas, for inftance tJ l J "" ^ , which alfo may be thus
ay aa — xx
exprefled, Xs/ 1 + -r^ — ; and this being reduced to a feries, will be =:
^ + \>hxxx_ __ ^M^x^', ^ _J££^ _ ..jM£L=-^ , &c. And
aay.au'-xx u^Xaa—xx^ a^ y. aa — xx\ ' a^ X. aa — aaV
again, reducing every term of this into a feries, beginning at the fecond, it witi
into -- + ^ 4. — + -^, &c.
I , 4*» , 10** , 20^« 5
into _ + Ij. + -^ + .^, &c.
i^*;clv
+
/ hhx'x
And by integration, the arch DO will be fx^ i + ^^ _ ^^^^ =
i* . ^ dpS , 2*' , 3*9 , 4.1?" 5
■ + -7-£ into — r + ^ H r? > ^^»
i--5 mto — ^ + -^r, &C.
128a' 9a* ' ila*°'
And laftiy, reducing the homogeneous terms into the fame denomination, wc
{hall find DO -
. 3*«' . 4a«3*^V f-- 8a*^» - 4^^^^ -hi* „7
"** 9 X 128«*^ ' . .
6 Now,
SECr. lif*
■A N A L V 1" i fc A V h N'fe 4? "I t- u 1 1 6 » d.
201
Now, if we fliould fuppofe a :=. h, in which cafe the ellipfis would become a
circler we (liall have the arch DO = x -\- -^ 4- ~ {- -^^ + — ^^-o~8) ^^»
juft as was found before, at § iii.
Aftef another manner, thus. In the formula — - — -=~= — ^, if we makft
avaa — xx
, the two radicals being rc-
AVa4 _
hb -^ cia tz. ^ ic, (o that it may be
a'^' aa — xx
folved into feries, it will be
X . - c*.r» c^j;* c*^^
— mto a^ — __. — -
aw aa — xx
%a
$*'
I28«'
, &C.
; and ai.*l:aally making the
divifion of the numerator by the denominator, after a very long calculation we
fliall find another feries, which, being integrated, and the value of cc rcftored
in it's place, will give us the fame feries as above, which exprefles the value of
the arch DO.
EXAMPLE XXI.
Fig, 125.
A
113. Let BD be an hyperbola with tranf-
verfc femiaxis AB n a, conjugate femiaxis
AE ZL b, CD =^, AC z= x \ the equation
will be XX "^ aa "=. ^~ . Then, by taking
the fluxions, it will be :v = '^-^^
by bb + yy
whence vAT+i; = yj . + ^-^ = i- X "^^^^g^- Therefore.
this being reduced into a feries, after either of the ways before made ufe of for
the ellipfis, we fliall find it's integral, or the arch BD z: ^ + ^^ — "^^-^^y^
ZaH' + ,a^' + a' _ 6,.^^ + ,MM + M^'^' + 5^' . g,,. „,,;,,, i3 j^e fame
feries as that for the ellipfis, excepting the figns, and the change of the con-
flants aj b.
Vol. II.
Dd
EX-
aoz .
ANALVTICAL INSTITUTIONS.
BOOK III.
EXAMPLE XXII.
Ftg. 112.
114. Let it be the cycloid of Ex*
ample VIII. of Quadratures, the equation
of which we know to be j/ zz xV "^ ;
therefore the formula will be \/xx -\- yy
= x\/ — , and therefore, by integra-
tion, it will be the arch EA = 2\/ax, or the double of the chord AS of ibe
correfponding circular arch AS. And putting x :=. a, AM will be the double
of the diameter of the generating circle, and therefore the whole cycloid will
be quadruple.
EXAMPLE XXIII.
Fig* 116. / 115. Let ABFbethe traMx^ whofe
equation is (§ 103.) — ^ = a. There-
fore « zr — — , and, by integration,
any arch AB =z a = — /y ± », in the
logarithmic curve with fubtangent i:: a*
But, making « zr o, \x.\'=,yzz.a, and
/y zz o ; therefore « = o, and the
complete integral will be « rz — ly. Therefore, if the logarithmic AKS be
defcribed through the point A, with the fubtangent AE, to the afymptote MH;
taking any point B in the traSirix, and drawing to the logarithmic BK parallel
to the afymptote, and letting fall the perpendicular KN, the intercepted line NE
will be equal to the arch AB.
EX-
SECT. III.
ANALYTICAL INSTITUTIONS.
203
EXAMPLE XXIV.
Ftg. 117.
1x6. Let ACB be the fpiral of Archimedes of
§ 104. the radius of the circle =: a, the circum-
ference = bi the arch BMP = Xy and AC z= y.
Let AE be infinitely near to AD, and therefore
DE = X, With centre A let the arch CH be
defcribed; then it will be CH = — , and OH=:»y.
Therefore CO, the element of the curve, is equal
to — »-y^ oayy ^ g^^ ^^ equation of the curve
is ^tx = by, and therefore xk =: -^j whence,
making the fubftitution, it will be CO ==
■^\/a* + bbyy. The integral of this, after a long calculation, which, to
aa or
avoid being tedious, I Ihall omit, will be found to depend on the logarithms,
or, which is the fame, on the quadrature of the hyperbola.
Now, by infinite feries. Firft, I make a* zz bbmm ; whence the formula will
by
be this, —y/mm + jyjy, which, being reduced to a feries, will be ~ into
m
5/
+ i^ — ^ + 16^ -iS^' ^^^ ^"^ therefore, by integration, the
arch AC = ^ 4- i^ -
'''' + nSb- - rrnb^ > «'<=• ^nd making
y =a, the whole curve ACB =— + #-- ^ + -^ Si'l— ,
&c. Then, inftead of w, refloring it*s value ~ , it will be ACB = « +
6a
h*
b^
5^'
i Aoa^ U2«5 Q X 122a'' ^
Dd2
If
ANALYTICAL INSTITUTION S.
BOOK III,
C_ If the curve ABC were the logarithmic
^ fpiral, whofe equation \s ay zz Ifx ; making
RB 13: yy and the infinitely little arch
BD — X ; putting, in the general formula
Vxx + yj/y the value of .v given from the
equation, it will be -, , and by in-
tegration, the curve AB = -y-v/<2<2 + ^-^.
Let the curve ABC be the hyperbolical fpiral, in which the fubtangent is
always conflantj and therefore, retaining the fame names as above, the equation
will be yx zz ay. Therefore it will be \/xx -^ yy zz ^^s/aa + yy \ the
integral of which formula, freed from the radical fign, will be found to depend
on the logarithmic.
By means of feries we (hall find —s/aa + yy' = jl into —
<*6^* g7, &c. But jf \}re would proceed to the integration, the fiirft term-
cannot be integrated, but by the help of another infinite feries. Wherefore,
*the fum of the faid feries being integrated, all but the firft term, together
with the integral of the feries expreffing that firft term, will form a feries which.
will be the value of the curve propofed.
II: V f. ^ ^
EXAMPLE XXV.
Tig, 115,
111 — r
117. Let HBD be the logarithmic, AB
the fubtangent — a^ AK =: .r, KH zz j^,
and the equation ~ = x. The value of
X being fubftituted in the general formula^
yyy of which the in-
I
it will be -^s/aa
y
tegral depends on the fame logarithmic.
fliall forbear to apply infinite feries, becaufe
their ufe may be fufficiently feen in the
former Examples.
EX-
SECT. m. ANALYTlCAt IN S T ! l" U T I N S. 20^
EXAMPLE XXVI.
ii8. ,Let the curve be the ApUonian parabola, with it*s co-ordinates at any
oblique angle, and whofe equation is ax-^iyy. This being differenced, and
fubftituted in the general formula for rectifications, when the ordinates are at
oblique angles; that is, in the formula sj ^x -^ yy ■\- -^^^, indead of x^ it's
value given by y being fubftituted, we fliall have —\Jyy + — ^ + -^aa-, the in-
tegral of which will be partly algebraical, and will depend partly on the qua-
drature of the hyperbola.
EXAMPLE XXVII.
119. Let the equation be x — jy, which is to infinite parabolas, and ta
infinite hyperbolas between the afymptotes. By differencing, it will h^ x x
"= y, and X ~^xx -n yy \ whence \/ xx -\- yy% or the element of the curve,
will be x>J x^^^^ 4- I. Proceeding to the integration, I (hall have recourfe
to the method of § 61, and fhall exhibit the formula in the following manner.
==— — ; . The canonical formula of the faid article, or . » is
X +1' X + a
algebraically integrable when ^ "" "^ — ^ is an integer affirmative number; and if
it be an integer negative number, it will be reduced' to known >fimple quadra-
lures. Now, bv comparing this formula - — 7 with the canonical, we
X +1'
have « = o, 2/ — 2 = m, and a = i» By which it will be neceffary that
^-^^ (hall be an inteeer, which I call b. Then ^-— ^ , that is,
LZ-E — by and confequently ^-M^ = /, the determining exponent of the
infinite curves.
Let
^C% ANALYTICAL INSTITUTIONS. BOOK III.
Let h be a pofitive integer, beginning from o. Now, if ^ i= o, it will be
t z= ±; ']{ h z=: i, it will be / =: 4. j ii b zz 2y it will be / = ^, &c. Let b
be any one of the feries of natural numbers, o, i, 2, 3, 4, 5, &c. the innu-
merable values of the exponent / will be exprefled by the following progreffion,
/ — !•> -4, ^i I, x-o-» -!-T» ^c. the law of which feries is manifeft ; and in all thefe
cafes rhe parabolical curves will be algebraically redifiable j the firfl of which is
the fecond cubical parabola.
Let h be equal to an integer negative number; and, firft, make b := — o,
in which cafe the fame cubical parabola arifes, becaufe — o and + o are the
fame thing. Make ^ n: — i, and the exponent / becoming rz 4-> it is con-
fequently infinite. Make b zz — 2, then / z= i. Make b :=z — 3, then
i zz *.. And Co on. Therefore the infinite values of the exponent / will be
exprefled by this progreflion, t := t> t> h tj tVj &c. and the parabolical
curves thence arifing will be redifiable by means of known quadratures.
The firft curve which prefents itfelf is the conic parabola, the redtification of
which requires the quadrature of the hyperbola, § no.
The other cafe, in which the general formula of § 61 is either redifiable
algebraically, or by means of known quadratures, is when a — i
is an integer. That is, by fubftituting the particular fpecies of this example,
^^-~- — ^ = b, and therefore '^ = /, the determining exponent of the
infinite curves.
Let b be a pofitive integer number, beginning at o ; we fliall have the fol-
lowing progreffion, / = 4, ♦, % 4, 44, &c.
Let ^ be a negative integer, and, firft, ht b = — o. Then the fame
exponent / = •§• returns upon us, becaufe — o is equivalent to 4- o. Let
^ =— I, the exponent / becomes equal to the fradion 4j ^ind confequently is
nothing. Let b zz -- 2, b zz — 3, &c. and we fliall have this following
progreffion, / = *, 4, 4-, |, %^, &c.
The fradion which gives the value of the determining exponent /, is the
fame in both cafes, only that in the fecond it is the reciprocal of the firft; fo
that the progreffions ought to come out reciprocal, as in effect they do.
Therefore the curves determined by means of each formula are the fame, but
with reciprocal exponents, that is^ they are referred to two different axes. As
for example, the two exponents t and 4- belong to the Apollonian parabola,
which offers itfelf in two manners^ x^ —y, that is, x '^yy^ and likewife x'^ zzy^ or
XX zz y ; both local equations to the parabolical trilmum.
Wherefore
SECT. in.
ANALYTICAL INSTITUTIONS.
207
Wherefore thefe curves, which are included in the foregoing progreffion'',
are either algebraically integrable, or do not require quadratures beyond the
circle or hyperbola. But the other curves, infinite in number, require quadra-
tures of a higher order.
It appears from our progreffions, that the value of the exponent / is never
negative. Hence no hyperbola admits of a rectification, either algebraical, or
depending on the forementioned firapie quadratures.
EXAMPLE XXVlir.
Fig* 127.
DE — y — -7- 9 and therefore, fubftituting this value inflead
chhxxx
-, It Will DC
ar 2aar
120. Let ACGKA be an eredb cone, AB zr a, BC = ^ ;0f cubature«.
let AD zi ^ be any portion of the axis AB ; it will be
bx
a
of y in the general formula, ^^^, it will be i_f^ ^^d by
integration, ^-^ , in refped to any portion taken from the
vertex ; omitting the conflant quantity, which here is need-
lefs. And making x '=. a, the whole cone ACGKA will be
=: ^ zi ^^ — X ~ , that is, equal to the produd of the
bafe into a third part of the altitude.
And, becaufe the folid content of a cylinder is the produd of the bafe into
it*s height, the cylinder will be to the infcribed cone as 3 to i.
The cone ACGKA is therefore ^^ and the cone AIEMP = ~ ; there-
or taar '
fore the fruftum of the cone IMCK will be ^ x « — -^, and therefore will
Or aa '
be to the whole cone in the ratio of a^ — x^ to a}. Whence, for example, if
we fliould make AD = t^B =: t^, the fruftum will be to the whole cone as
d^ — 4«% or -^«', to a^, or as 7 to 8 ; and to the cone AEMPI, as 7 to i.
- Therefore, as often as we are to meafure any folid, it is neceflary to confider,
of what elements we defign to have it compofed, according to the different
fedions that may be adapted lo it; varying it fometimes one way, fometimes
another, as circumftances and conveniency may require. Tht^n, among the
aforefaid elements, to choofe thofe which may be managed with the greateft
facility.
208
•ANALYTlCAt INSTtTUTIONS.
BOOK iir.
facility, and to which the calculation may be mod naturally adapted. In the
ere(5t cone for example, of which we are treating, we have as many circles as
we pleafe parallel to the bafe.; and alfo as many triangles, which have their
vertex the fame as the cone, and for a bafe the parallel ordinates of the circle
CGK. We may alfo cut the cone according to fo many parabolas, which are
equidiftant from each other, and with axes parallel to the fide AK j and many
other fedions may be made. . .
Neverthelefs it is true, that, to find the folidity of the cone, fuch means as
thefe are to be confidered as not to the purpofe, as being too compounded for
the cafe propofed. But it may be propofed to cut the cone, or other folid,
according to any plane whatever, and then to meafure the two fcgments into
which it is divided ; and, in this cafe, it is convenient to make ufe of fuch
elements as (hall correfpond to that fedtion; as may be feen in Examples XXXVII.
and XXXVIII. following.
EXAMPLE XXIX.
Fig, 128.
, 121. Let CDK be a femicircle, which is converted
about a fixed radius DB, by which a hemifphere
will be produced; and make DB = a, DA = at,
and it will be AE zz y zz \/iax ~xx. Then,
fubfl.it uting this value in the general formula, it will
be — X '2.ax — xx ; and, by integration, the foli-
dity of the indefinite fegment AEM will be =c
And making x = a, the folidity of the hemifphere will be =
— , and it's double, — — , will be the whole fphere.
And becaufe the cylinder, the height of which is equal to the diameter of the
bafe, or la^ is -^ ; the cylinder circumfcribed will be to the fphere infcribed,
And confequently the half cylinder will be
to the hemifphere in the fame ratio. But the cone alfo, whofe height is equal
to the radius of the bafe, (or equal to a, the radius of the fphere,) is = ~;
therefore the hemifphere will be to the cone infcribed as 2 to i.
8 Furthermore,
as
ca
IS to , or as 3 to 2.
SECT. III.
ANALYTICAL INSTITUTIONS.
209
Furthermore, as it is known that
^/laa
is the radius of the bafe of an equi-
lateral cone infcribed in a fphere, the radius of which is = <7 ; and the height of
_ 3«
^cai
ica^
the fame being = -^ , the cone will be — —3—, and the fphere will be
o 2 ' 48r ' . t 3r '
and therefore the fphere to the cone as ^ to /-j., or as 32 to 9. In like manner
may be demonftraied as many Theorems oi Archimedes as we pleafe, which are
of a like nature.
Hence the manner is plain, of obtaining any fedlor of the fphere, which is
generated (for example) by the feftor of the circle BEDM. For to the
fegment of the fphere generated by the figure AED, which we know to be =
icaxx -_cx^ ^ ^^^ y^^ added the cone generated by the triangle EBA, and which
6r
caax
is found to be = -r— X zax — xx X « — a\ and the fum, -^i^^^^, will be the
.6/ 3r '
feftor required.
EXAMPLE XXX.
Fig» 129.
ca
2m— z 2
X "" X
m ^ m
122, Let there be a parabola of any order,
whofe equation \% y — a x ; which, being
converted about the axis AM, generates a para-
W— I I
bolical conoid. Then it will be y = a '^ x '" ,
2OT— 2 2
and yy ~ a "* x "* ; and therefore, fubftitut-
ing this value, the general formula will be
im — a i» + 2
2r
j and, by integrating,
mca '" X
m AM fit
%r X CT+2
will be the folid content
2-\-m
of the indefinite conoid. Or elfe, becaufe x"* :=. • ^^ 7 , and therefore .v m
zm~-2
a ^
xyy
2m — z
3-, by fubflituting this value in the integral now found, it will be
a ^
mcxyy
ir X m + Z
Vol. II.
Eg
Make
»IO ANAl-TTICAl. INSTlTUTlOilS, BOOK III,
Make « = 2, that is, let it be the Apollonian parabola ; the conoid will be
=: -— , that IS, the produft of the bafe into half the height ; and, by confe-
quence, the faid conoid will be half a cylinder of the fame height, and of the
fame bafe.
If we would have the folid content of the di(h, or of the folid generated by
the figure ACD, converted about the axis AB j from the cylinder defcribed by
the redangle ABCD, which we know to be == ^^ , we muft fubtrad the pa-
rabolical conoid ""^^ ^^ , the remainder, — '^ "^■^ • , wilF be the content of the
zr X m + z r X « + 2
di(h. And making w zz 2, in refped of the Apollonian parabola, the dilh will
be ^^ , which is half the cylinder, juft as it ought to be, the conoid being alfo
half of the fame cylinder.
Let the figure move about the ordinate MO,, and make AM = ^, MO =/,
AB zi *, BC = ;^, CiC = ^ — X, KO z= / — y. The circle, with radius CK,
will be = — X ^ — j^") % and therefore the produd of this circle into y will be
the differential of KM ; that is, — Y^ bby — zbxy + xxy will be the element
of the folid generated by the figure MACK. Therefore, by integrating,
and, inftead of x, putting it*s value given by yx it will be ~ x
ih — ^ ^ + -^ . , equal to the indefinite folid. Or,
putting X in the place of -^> it will be — x % — ~^ + ^^ *
a
Now, putting X =. hf y = fy'm refped to the whole folid generated by the figure.
ACOM. ic will be -£-XW/-^ + -^, that is, .^^BL^ x -^.
And if we would have the parabola to be that of Apolloniui^ that is,, if m =: 2,
then the folid will be = ^-^ •
It is eafy to perceive, that, in the Apollonian parabola, a cylinder on the fame
bafe, and of the height of the faid folid, Ihall be to the folid as 15 to 8 ; and
that the folid generated by the figure OAP Ihall be = ^.
^ Ljft
»5CT. Ill, AKALYTICAi INSTITUTIONS. 2U
Let the figure move about the right line AP, and let it be, as before,
AB = X, BC = y ; then ~ will be a circle with radius DC, and ^^ will be
the element of the folid generated by the figure ACD. And, inllead of x,
putting it's value given by jy, and then integrating, it will be — X
■ ■ ^ , lh:it is, — X— — ^» equal to the indefinite folid. And
making x — h^ y ^ f^ it will be ^ — , in refped to the whole folid,
2r X 2;« + 1
generated by the figure AOP.
But the cylinder on the fame bafe and altitude is = — - ; therefore the folid
zr
generated by the figure AMO is = -^ X ^^ .
But fl:ill, in another manner, we may obtain the folid generated by the figure
AOM, revolving about the axis AP. Make AM = b^ MO =/. A circle
with radius DC will be = — , and the circle with radius DK will be equal
to ^- — . Therefore — Y. bb -^ xx will be the annulus defcribed by the line.
ar ar '
CK, and — X bb — xx will be the element of the fohd generated by the
figure CKMA ; and, inftead of x, putting it's value given by j, it will be
aw. aw + i
— X bby ^ ^ , and by integration, ~ X bby — ^'
— : — w 2«— 2
zm+i X a
Lafl:ly, making 7 = /, in refpefl of the whole folid, generated by the figure
AMO A, it will be — X bbf — -^ . But, when 7 = /, becaufc
of the parabola, it will be x — h ~ — — , and hh = -^ . Therefore, in
a
the integral, fubflituting the value given by ^, the folid will be — x ^^
W c imhhf ,
rr-T", — T~ X — 7- » ^s above.
E c a EX-
212
ANALYTICAL INSTITUTIONS,
BOOK III.
EXAMPLE XXXI.
Fig, 124.
123. Let ADC be an elllpfis, ABrz^,
BDrzr^jAEzr^CjEOzi)'; and therefore the
hh
equation is — X 2ax—xx=yy. There-
fore, in the general formula, fubftituting
the value of J' given from the equation,
it will be - — X 2axx —xxx ; and by in-
cbh
2aar
tegratlon, it will be X axx — -x^, equal to the indefinite lolid generated
by the figure AEO, turning about the axis AC. Making x :=: a, it will be
^ — , half of the fpheroid ; and putting x = 2a, it will be , the whole
fpheroid.
And, becaufe the cone of the fame altitude AC, and of a bafe the radius of
which is the conjugate fcmiaxis BD, is = , and the cylinder is = ,
the fpheroid will be two third parts of the cylinder, and double to the cone.
EXAMPLE XXXII.
cb
124. Let AD be an hyperbola, which is
converted about BC, and let it's tranfverfe
femiaxis be BA =: t«, the centre B, and
it's parameter = If, AC = x, CD = jy.
and the equation is ax -i- xx "X — = yy,
Subftituting the value of y in the general
cbx
formula, it will be — X «^ + xx ; and
2ar
by integration, it will be ~ X faxx + ^x\ equal to the indefinite hyperbolical
conoid, generated by the figure ADC.
Make
SECT. III.
ANALYTICAL INSTITUTIONS.
215
Make BC = x, and the reft as above. The equation will be — x
ckx
XX — iaa — y)\ and therefore the formula will be — X xx — ±aa, and by in-
tegration, — X t^^ — t^^^ + /• ^ ^^^ ^^^^ conftant quantity/, which, in
this cafe, will be fomething. To determine what, it muft be obferved that in
the point A, when x = '\a, the folid ought to be nothing. Wherefore, inftead
of X, putting ^<3 in the integral, it ought to be / +
Q,ar
and therefore / n
l^r
ch
Therefore the complete nitegral will be x
^4^^ — i«^ ~ o,
i
■w
^a'x -t- VT^^
Let the hyperbola be converted about the conjugate femiaxis HB, and make
the tranfverfe femiaxis AB rz a, the conjugate femiaxis — h^ BC r= x,
CD = y. The circle with radius HD will be = -^ , and therefore ^^^ will
2r
be the element of the folid generated by the plane or figure BHDA. And,
inftead of xx, fubftituting it's value given from the equation of the curve, we
Ihall have ~ x °^^^ 7, '^^ > and by inteerration, — X ^^ + ^^y '> and
Zr hb "' ° 2r xbb
making y zi h, the folid will be =
icaah
EXAMPLE XXXIir.
Fig. 130.
125. Let KHF be an hyperbola between
the afymptotes; AD zz a, DE =: b,
AP =: Xy PH = y, and the equation
xy = ab. Let the curve revolve about
the afymptote AB. Then the circle with
radius QH will be = — , and therefore
^^^^ will be the element of the folid gene-
rated by the figure AQHFMA, infinitely
produced towards M. And, inftead of x^
putting it's value given from the equation^
it
£14 ANALYTICAt INSTITUTIONS, BOOK III.
k will be f . "*^ ^ , and by integration, /— ^^^. Now, to determine/, it may
be obferved, that, when it is y =■ o, the folid ought to be nothing, and there-
fore / = -^ — , an infinite quantity, and therefore, the complete integral will
be — ^-^ j- 00 : fo that the folid is of an infinite value.
zry
Inftead of fubftituting in the formula the value given by y from the equa-
tion, in the place of xx, if we fliould fubilitute the value of ^ ; it would be
•— ^-^ , and by integration, — ^-^ + /. But the folid cannot be nothing
except when x is infinite, and then the conftant quantity /to be added ought to
be infinite, and therefore the folid will be infinite.
To have the folid generated by the plane or figure BAPHK infinitely pro-
duced towards B, it will be enough to confider, that as ~ is the periphery of
the circle whofe radius is QH z: ;v, then — will be the fuperficies of the cy-
linder, generated by the plane AQHP, and confequently -^^ will be the folid
content of the hollow cylinder, generated by the infinitely little redlangle
IPHO. Therefore the fum of all thefe, or /-^^ , will be the folid required.
Therefore, inftead of ^, putting it*s value — • , the integral will be ^^, a finite
quantity, although the folid be of an infinite altitude,
Jn the expreffion — — of the folid, infiead of ab putting it*s value xy, given
from the equation ; it will be -^^ . But -^-^ is the cylinder generated by the
redangle APHQ^ Therefore the hyperbolical folid will be double to this
cylinder. And therefore the folid generated by the figure BQHK, infinitely
produced, will be equal to the cylinder which ferves it for a bafe. Therefore,
taking x — a, and confequently y - h, this cylinder will be =: -^^ , which is
equal to the folid ereded upon it.
£X^
8£CT. Ill*
ANALYTICAL INSTITUTIONS,
a«5
EXAMPLE XXXIV,
Fts* t^i*
126. Let HCD be the logarithmic curve*
it's fubtangent CA = «, AB = x, BD = y^
and it*s equation ;c* = — . Let it be con-
verted about the afymptote EB, In the ge-
neral formula, inflead of Xf putting it's value
given from the equation, it will be ^^; and
by integration, it will be ~ +/. But when
it is _y = AC = a, the folid will be = o. Therefore it muft be / = ;
and the complete integral, that is, the folid generated by- the indefinite plane
ABDC, will be zz
cayy -^ ca^
4^
Let the abfcifs AE be negative, and therefore rr — x ; and it's fluxion alfo
will be negative, or — x. And becaufe, as the abfcifs increafes, the ordinate
will diminilb, therefore the fluxion of EH will alfo be negative, or — j/^ fo<
that the equation of the curve will be ftill the fame, x zz -^. But,, becaufe xr-
is negative, the general formula will be negative alfo, or — i^,, Subllituting.
cayy
therefore, the value of x, it will be -^ -^ , and by integration, — -^^^ + /.
But when the folid is nothing, it will be y = a ; therefore /= ~, and the-
4»*
complete integral will be — ^^^yy ^ equal to the folid generated by the plane-
ACHE. Putting ji' = o, that is, fuppofing the folid to be infinitely produced «
towards M, the integral will be = ~ , and then the folid itfelf, infinitely pro-
duced, will be =: — . But the folid generated by the plane ACHE we have
feen to be — f^v-y
LEMH, is/-^,.
4r
5 then the folid infinitely produced, generated by the plan*:
Now
>»
2l6
ANALYTICAL INSTITUTIONS.
BOOK III,
Now, becanfe the cylinder, the radius of whofe bafe is AC = «, and it's
height alfo = a, is — ; .the foUd of the logarithmic curve, infinitely produced
towards M, on the bafe with radius AC zz a, will be to .the faid cyHnder, in
the ratio of -^ to tj oi" ^s i to 2.
EXAMPLE XXXV.
Fig. 114.
1/ !H 127. Let the curve AMI be the cilToid of Diodes,
which, by revolving about the right line AB, defcribes
a folid. Make AP = x, PM ~ jy, AB zz a, and the
equation will be yy = -^T"* Therefore, the value of
Q yy being fubftituted, the general formula of folids will
be-
_-JB
— . , and by integration, — —
2r X a—x ^'* 4^
2r
caaa
2r
X / a — X + f. But, making x = o, the
caaa
folid ought to be nothing, and therefore / = — la.
cx^
And the complete integral — la I a — x — -y- is equal
*■ ° 2r 2r ar ^r or ^
to the folid generated by the figure APM. And making x — a, the whole
folid will be =: '^— la ^/o ^. But the logarithm of o is an infinite
ar zr izr o
quantity and negative, which, multiplied into — — ", makes an affirmative
quantity ; fo that the intire folid will be infinite. It is to be obferved, that the
aforefaid logarithms are to be taken from the logarithmic curve, the fubtangent
of which = a.
By the help of infinite feries, it will be
i"3»"
eS V
2r X a—x
2ar
iraa
+
^^- , &c. ; and by integration, the folid generated by the plane APM will
zra
be = ^
■^ oar lora^
cx^
CSC) ex
+ 7— -r + —-7 , &c. And making x = a, in refpe<5l of
if
the intire folid, it will be — into ^ + f -f- -^ + f , &c. the total value of
which feries is infinite.
4 EX-
SECT. III.
ANALYTICAL INSTITUTIONS,
aiy
EXAMPLE XXXVI.
Fig, 1x6.
A
n..
'■ K,--"
C
^S
'
&:■■•
^
M
14
1
I,
1
)
P
128. Let the tra5irix ABF be con-
verted about the afymptote EH. In
the general formula 2^^ fnbflituting
the value of x given from the equation
§ 103, we fhall
^ _. yy^a^ - yy
have — ^^-^ — ^ . And by Integra-
tion/it will be ~ X aa^yy\'^ , equal to the folid generated by the figure
AEDB, omitting the addition of a conftanr, which is here unneceflary.
Wherefore,- making ^ = o, the folid infinitely produced will be zz ~, But
the folid content of the fphere whofe radius is AE •=. a, (§ 121.) will be zi:
; and therefore the folid infinitely produced will be a fourth part of that
fphere.
EXAMPLE XXXVn.
129. Let QBMCPT be a cylinder,
from which, by a plane through the
diameter BC, and in the diredion AP,
a portion or ungula, BMCPB, is cut off;
the folid content of this is required.
Make BC z: QM = 2j, MP = QT
= ^, AD = x, and DH being drawn,
(hall be an ordinate in the circle zz
x/aa — XX. From the point H let the right line HO be drawn parallel to MP
or QT, which (hall be in the fiiperficies of the cylinder. Then from D to the
point O let the right line DO be drawn, which (hall be in the plane BOPC.
Then we (hall have formed in the folidity of the mgula the triangle DHO,
Vol. II. F f which
2l8
ANALYTIC Air INSTITUTIONS.
BOOK I!I«
which Is fimilar to the triangle AMP, and therefore it will be HO z: — i^^ff—ff .
But the aggregate of all thefe triangles, DHO, is juft the folidity required of
half the ttngula^ and therefore it will be =: J-^ x aa ^ xx ; and by integra-
zz ^aab^ and the whole z: \aah^
Fig* 133-
tion, ^ — ~- , And making x zz a, the aforefaid half tmgula will be finally
In another manner, and more ge-
nerally, thus. Let DiVCHEG be half
of a cylinder, which, through the dia-
meter CD, is cut by a plane in the
diredtion DE, whence arifes the ungula
DBCEAD, the folidity of which is
required. Make BA = a, AE n b,
BQ^= Xy QM =7; it will be QK z=
, and therefore the redangle PONM
X, or
2iyxx
a
a
And this being drawn inta
, will be the element of the folidity of the ungula^
Let the curve D AC be a femicircle j then y = ^aa — xx^ and the formula
7,hxx
will be ^-^s/aa — xx\ and by integration, — — X aa^ — xx]'^ + m. Now,
3a
by putting x = o, the conftant, w, will be found to be = ^haa, and therefore
the integral of the folid complete will be \haa X aa—xxS^ ; and male-
ing X :=. a, iti refped of the whole ungula, it will be ^baa, as before^
Let the curve DAC be one of the parabolas ad infinitum^ and it's equation
y* -zi a — X, Subftituting the vahe of yy the formula will be ?-^ X a^^x^ ^
which being integrated according to § 29, and a conftant being joined, and
making xzi a\ it will give
3»» + I X OT + 1
for the iblidrty of the whole
Hftgula, And taking m zz 2, or the Apllonian. parabola, it will be — ^,
Kow, fuppofing that, when a? = o, it is BC =:^ = f j it will be a^ zzc, and
therefore
SECT. Ill,
ANAIfYTlCAL INSTITUTIONS;
21^
therefore the ungula n -^^abc» After the fame manner we may find the ungula
of the elliptical cylinder to be ^abc, fiippofing the tranfverfe femiaxis zz a,
afld the conjugate femiaxis zz c.
EXAMPLE XXXVIII.
Fig. 134.
130. Let the parabolical conoid BAG be cut by any
plane lEH, perpendicular to the circular bafe BICH ;
it is required to find the meafure of the fegment,
comprehended by the fedion lEH, and by the plane
parallel to it, through the axis AD.
Make the parameter zz ^ of the generating parabola
BAG, the given abfcifs AD zr h, then the ordinate
DB zz \/ab. Let the co-ordinates be DF ii= x^
FE = jy, and therefore the equation of the aforefaid
curve BAG will be ab — xx iz: ay. By the nature of
the circle BICH, the redangle CFB zz ab -' xxj equal to the fquare FH = 2Z.
But ab — XX zz ay ; therefore ay zz 22:, and confequently the fedion lEH
will be a parabola, with the fame parameter as the principal. Wherefore the
redangle EFH remains fixed, zz yz z= yy/ay ; and becaufe this is to the area
lEH, as 3 to 4, this area will be zz ^y\/ ay^ and the produd of this area lEH
into the infinitely little height x, the fluxion of DF iz: a*, will be the element
of the folid in queftion, that is, ^yx\/ay. But y zz — — — j therefore the
dement will be \x X — '^^ ^ ab — xx^ or \bxV ab — xx —x^x^/ab-^xx.
3*
The fluent of the firft term depends on the quadrature of the circle BHC j
the fecond is reduced to known quadratures, by means of the firft formula
of § 61.
131. I forbear from giving examples of folids generated by curves with the
co-ordinates at oblique angles to each other j becaufe, the formula for thefe cafes
being different from the ufual and ordinary ones, only by conftant quantities,
no difficulties can be met with of a different nature from thefe already
produced.
Thus, alfo, I omit examples of folids generated by curves which are referred
to 2i,focuSy becaufe I am not willing to introduce the Theory of the Centers of
Gravity, as i have faid before. The given curves may be reduced to others
referred to an axis, about which I have already treated.
N. B. The letter D is omitted In the center of the bafe of Fig. 134,
Ff* EX-
!32ia
ANALYTICAL INSTITUTIONS,
BQOK III,
The com- Fig, 12^
planation of /.
curved fur~
faces.
EXAMPLE XXXIX.
132. Let ACGK be an ereft cone, AB =: ^, BC rz h,
any portion of the axis, as AD — x ; it will be DE z: y
= —^ and y=z—,yyzz
aa
Therefore this value
of j/j/, being fubftituted in the general formula -^:-^xx-{-yy^
.,, , cv , aa'..x + hhxx cyx'^aa + bb
It Will be -^v/ = J
X^ y being fubftituted, that is, — , it will be
chxxs/ aa + bh
; and the value of
I'M •. '11 i_ ^ chxxvan + bb
■j and
by integration,
2aar
, in refpeft of the fuperficies
chv^aa + bb-
ar
of the cone AEMPI. And making x = a^ it will be
, in refpedl of the fuperficies of the whole cone, and therefore it
is equal to the redangle of half the circumference of the bafe into the fide AC^
The fame conclufion would have been had, if, inftead of fubftituting in the
general formula the value of ^, we had fubllltuted the value of xx.
Wherefore the furface of the fruftum of the cone IlVrfvCG 'will be rr
— r-\/ QA ■\' bb k/ aa H- bb, that is, --^ ; and thereiore
%r ^ zaar traa
it will be to the furface of the whole cone, as aa — xx to aa^
Fig, 135
133,. But if the eone be fcalene, it i&
neceflary to proceed after another man-
ner. Let PAFBM be a fcalene cone,,
the bafe of which is the circle AFBM ;
and on the diameter produced (if need
be), let fall PD perpendicular to the plane
of the circle, or the bafe. Let two
points F, /, be taken in the periphery of
the circle,, infinitely near to each other,
and let the two fides of the cone FP,/P,,
be drawn. It is plain that the infini-
tefimal triangle PF/will be the difference
0£
SECT, III, A N»A LYTICAL INSTITUTIONS, 2*1C
or element of the fiiperficies of the cone. Then to the point F let the tangent
TFQ^be drawn, to which let DQ^be perpendicular, and let the points Pj C^
be joined by the right line PQ^
Now, becaufe the plane of the triangle PDQ^paffes through the right line
PD, which is perpendicular to the plane of the bafe of the cone, the plane
PQP will alfo be perpendicular to the fame plane of the bale. But the
tangent TQi which is alfo in the plane of the bale, makes a right angle with
op, the common leftion of the two planes, and therefore will be perpenr-
dicular to the plane PQD, and confequently to the right hne Ql* ; and there-
fore the triangle PF/ zz £S=^^.
Make the radius CA z: r, CD =: ^, CE =: ^ ; it will be FE =: ^rr^xx;
and becaufe the angle CFT is a right one, TF being a tangent to the circ!?,
the triangles CF£, TCF, will be fimilar. Whence it will be CT =
X
But CT . CF :: CF . CE :: TD . Dq^ Therefore DQ^z: ^^-±if. Make
the given line PD =z p. Therefore it will be PQ^= ^ pp + ^"^ "^ ^ '
But
the element of the circle F/ we know to be ^^ ; therefore -^F/" X PQ»_
rx I
rr + ^^*
the element of the foperficies, will be s/ pp + '■ — -^ V^^^ — xx;
a formula which hitherto has not been reduced to the known quadratures of
the circle or hyperbola, becaufe it cannot be freed from radical figns, as has>
been feen at § 38, and as we have alfo feen, in our attempt to rcclify the
ellipfis.
Tf we have recourfe to infinite feries, the numerator muft be reduced, to^
a feries, and alfo the denominator ; then we muft proceed in the fame manner
as was done in the fecond method concerning the elhpfis, in Example XX>
§ 112.
EX-
ZiZ
A-NALYTICAL INSTITUTION! S.
IJOOK III,
EXAMPLE XL.
Fig. 128.
134. Let there be a Iiemifphere, the generating
femicircle of which is CDK, which is converted
about the radius DB = a^ and make any line
DA zz A? ; it will be AE zz ^ zz \/ zax — xx^ and
therefore yy =
zax — XX
And making the fub-
ftitutions in the general formula, it will be zz. — %
and by integration, ~ z: to the fupcrficies of the fegment of the fphere,
generated by the arch EDM. And making x zi a, the fuperficies of the
hemifphere will be =z ~ , and therefore -^ will be the fuperficies of the
whole fphere. Therefore the fuperficies of any fegment will be equal to the
produft of the periphery of the generating circle of the fphere, into the alti-
tude of that fegment ; of the hemifphere, equal to the retflangle of the fame
periphery into the radius ; and of the fphere, equal to the redtangle of the
periphery into the diameter ; and therefore thefe fuperficies will be to each
other in the ratio of their refpeftive altitudes, the radius, and the diameter.
And becaufe the area of the generating circle of the fphere is = — , the
fuperficies of the fphere will be to the fame area as 4 to i, that is, quadruple
of the greateft circle.
And becaufe, alfo, the fuperficies of the cylinder, (excluding it's bafes,)
which is circumfcribed to the hemifphere, is equal to the produd: of the
periphery of the bafe into the height ; it will therefore be z= — , and confe-
t^uently the fuperficies will be equal to that of the hemifphere. Now the cone
infcribed in the hemifphere has alfo it's fuperficies =: ^^^i— f j therefore the fu-
perficies of the cylinder, or of the hemifphere, to the fuperficies of the in-
scribed cone, will be as la to V 2.aa^ that is, as the diameter to the fide of
the cone.
EX.
i£CT. Ill,
ANALYTICAL INSTITUTIONS,
Hi
EXAMPLE XLI.
Frg, 129* 135. If the parabola ACO of the equation
ax :^yy, turns about the axis AM; it will be
ax = 2jy, and xx zz "^-^-^-^ , and therefore, making
the fubftitution, the formula will be —\/4yy'\-aa,
and by integration, — ^ X 4yy 4- aa)^, equal to
the [fuperficies of the] indefinite parabolical conoid,
equal to the fourth propo'tional of 6^, \/'4yy -{- aa^, and the area of the circle
whofe radius is n s/ ^yy 4- aa^.
r
136. More generally, I«t ~ =j> be the equation of the parabola AGO,,
{Fig. 129.) with it*s abfcifs AB = Xy and with it's ordinate BC =jv; whicb
r
equation for the/r;7«/tf«;?« ACD will belf/l ' =: _y, if we take AD zz a as ab-
fcifs, and DC = ;; as ordinate. At § 119, Example XXVII, it has been
feen, thai the element of the curve, which I call //, was = ■ — - ; and'
the differential formula for the fuperficies is ~ . Then it will be — =
But, by the local equation, it is -y- = jr. Then it willbc^
cyx
V 2/ — 2 . ,>■
/.
cyu ___ ex X
rt XX 4- 1/
To proceed to the integrations or quadratures, I Oiall make ufe of the
method explained at § 61, and applied to the aforefaid Example XXVII;.
Bur, fird, it is to be obfervedj that c, being the periphery of the circle whofe
radius is r, the integral/— will give us the furface of the conoid. But if c
reprefcnts any right line whatever, we (hall have the meafure of the furface of
the mgula, wlien a cylindroid is erected upon the baie GAB^ which i^ cut by a
plane
tH ^ ANALYTICAL I N S T I T U T I O JJ S. BOOK:lIJ.
plane paffing through the axis AB, and with the fubjea: bafe CAB forms an
angle, of which the right fine is to that' of the complement, as c is to r.
Then tlie lingular fuperficies is to that of the round folid, as a given right line
ts to the circumference <;.
Operating, therefore, as explained above, at § 6i, that our formula may
be algebraically integrable, or reducible to known quadratures, we (hall find
that it mud be / iz j-^-^ » or clfe / = ^^~~ , where h denotes any integer
number, pofitive or negative.
The firfl: condition, or / zi • . ' / > making h any integer number, -fit ft
pofitive and then negative, will give us thefe two progreflions :
if 3 5 7 S> iX ^p TT/— li «^» S 7 * firp
J* J — Xj T» T> T3 9 > ^^^» ^i» * — i.> T> T' T> TJ T i * CV,^«
The fecond condition, or / = ^ , making h any integer number, firft
pofitive and then negative, will give us thefe other two progreflions :
m* X 2 3 4 5 5lrf. TV / — » 3 ♦ 5 6 «^f
' To this I (hall fubjoin a few fhort obfervations.
I. As the two progreffions, the firft and the third, contain the exponents of
all thofe parabolas, which, by circulating about the axis, generate conoids, the
fuperficies of which are analytically quadrable, fuppofing only the redification
of the circular periphery ; and confequently all the unguis above defcribed, of
a given altitude, admit an algebraical quadrature: So, in the cafes of the
fecond and fourth progreflions, fomething more is intended, as they require the
quadrature of the hyperbola.
II. It is obfervable that, the firft feries being com|)ared with the fecond,
and the third with the fourth, the exponents are reciprocal, and belong to the
fame curve. This fliows that, as the parabolical area may be converted, cither
about the axis AB, or about the axis AD, and in each cafe may produce very
different fuperficies ; if, in the firft cafe, it generates a fuperficies that is abfo-
lutely quadrable, at ieaft confidered in the ungula ^ in the fecond cafe, on the
cojitrary, the values being reciprocal, the above-faid fuperficies will arife, which
are only hypoihetically quadrable. For example, the conoid formed from the
firft cubical parabola being turned about AD, furniflies us with the furfaceof an
««^a//j which is algebraically quadrable, and alio that of the- round folid, pro-
vided wehave a rightr line equal to the circumference. But if it be converted
about the axis AB, then quadratures are required. The fame thing obtains in
the fecond cubical parabola, and quite the contrary in that oi Apllonius^
9 III. Com-
fiCT. m.
AKAtYTXCAL IMSTITVTiONi.
«*S
III. Comparing thefe feries with thofe of § 1 19, we may difcover, that amono-
thefe there is no parabola of the firft or fecond feries, that is redifiable either
analytically, or by the means of known quadratures; on the contrary, thofe of
the third and fourth are ail re(5tifiable, and comprehend all that are contained
m the progreiTions of § 119.
IV. Among the hyperbolas, the common one only between the afymptotes
admits a iuperficies reducible to the quadrature of the faid hyperbola ; becaufc
no other negative exponent appears in the progreflTions, except — i.
V. The exponents which are not found in the faid feries are thefe, / iz 4,
5, 6, &c. I-, -|-> &c. for which higher quadratures are required, to meafurc
the conoidal furfaces thence arifing.
EXAMPLE XLII.
Fig, 124.
137. Let ADC be an ellipfis, which
is converted about the axis AC, and make
AB = ^, BD t= l>, AE zr x, EO = j;
and the equation is —^ = lax — xx.
Therefore, by differencing, it will be
, and therefore xx zz
X zz
— __^yy
lb X a — x
'^^^^^ ; and, inftead of — ^ax + xx, putting it's value — ^ given by
i* X a—x\
the equation, it will be xx =:
aayyyy
bl X bb —yy
Then fubftituting this value in
the general formula, wc (hall have 2L_i_Z_£f5-r_^ j and, for brevity-fake,
rbybb — yy
making aa — bh zz Jf, fuppofmg a to be greater than b, or that the axis about
which the ellipfis circulates to be the greater axis (for, if a were lefs than ^, we
cyy\^b'* + ffyy
ought to make aa — hh = — ff)^ the formula will be ^ — -=~, which, for
rbv bb -~yy
reafons already mentioned in their place, may be freed from radicals; and the
integral of which, by means of the canon of § 56, we (hall find to depend on
the quadrature of the circle. But if a fhall be lefs than b, or the axis about
which the ellipfis turns be the lefler axis, the fuperficias of the fpheroid will
Vol. II. G g . depend
226
ANALYTICAL INSTITUTIONS,
BOOK III,
Fig. 136.
depend on both the quadratures, that of the circle and that of the hyperbola.
Wherefore the fuperficies of the ungula, in the firft cafe, is equal to a portion
of the elliptic fpace, which is eafiiy determined by means of the perpendicular
to the curve. Bur, in the fecond cafe, thefe perpendiculars will give us an
hyperbolical fpace equal to the lame fuperficies of the imgula. That this may
be plainly feen, let ACF be the curve on which a
cylindroid is fuppofed to be eredted, which is to
be cut by a plane which pailes through the axis
AB, and forms with the fubjacent plane CAB
half a right angle. It is evident that, making u
the element of the curve, fy'u will be the fuper-
ficies of the lower ungula, and J~ will be the
fuperficies of the conoid, generated by the con-
verfion of the figure CAB about the axis AB ;
and therefore the fuperficies of the iingula will be
to that of the conoid, as radius to the circum-
ference of the circle.
Now let the two ordinates BC, DF, be infinitely near, and drawing the
perpendicular FG at the point F, let it be put in DH, and reprefent the ordi-
nate of a new curve MIH drawn by the method prefcribed. I fay that the
area MABI is equal to the fuperficies of the ungula, which has for it's bafe the
arch AC.
The two triangles FCE, GFD, are fimilar ; then it will be FC . CE ::
GF . FD. Therefore FD X FC z= GF X CE = DH x DB. But FD x
FC (y'u) is the element of the fuperficies of the ungula^ and HD X DB is the
clement of the area IMAB. Then, thefe elements being equal, their integrals
will be equal alfo ; that is, the aforefaid areas. This being. premifed, let
the figure ACB be a fourth part of the ellipfis, the equation of which is
^^ = zax — XX. Then the perpendicular will be FG =
• — s/^ia^x — aaxx + bbxx — ^abbx + aabb. Then, making the ordinate
BI = 2, it will be z zz — J xx — lax x ^* — ^* + ^*^% an equation to the
curve MIH, which will be another ellipfis when a is greater than h, or if AB
be the greater axis of the ellipfis ACB ,• and on the contrary,- an hyperbola,
when a is lefs than ^, that is, when AB is the lelTer axis.
Laftly, in the middle cafe, or when the ellipfis degenerates into a circle, we
know already, that the faid furface of the ungula is quadrable, as being equal
to a rectangle.
EX.
SECT, III.
A K A L Y T I C A L INSTITUTIONS,
2.27
EXAxMPLK XLIir.
pig. 125.
138. Let BD be an hyperbola, which cir-
culates about the tranfverfe axis BA. Let A
be it's" centre, BA zz «, the conjugate femi-
axis AE = i?, AC — x, CD = y. The
equation will be xx — aa zz ^^, and there-
bxx
fore y ^ez — \/xx — aa^ and y-iz . ,
Therefore the general forrmik, when the fubftitutions are made, will be
ch
ar
s/xx -^ aa X V ■- , that is, ^— \/aaxx + bbxx — a* ;
or, making t^a + hb - jf, it will be "^J^xx — ^f^ the integral of which,
aar ff ' o »
when it is freed from it's radical fign, we (hall find, in like manner, to depend
on the quadrature of the hyperbola.
EXAMPLE XLIV.
Fig, 137.
139. Let MD be an equilateral hyper-
bola, between it's afymptotes, and let it
turn about the afymptote AC, of which the
equation is ay -\' xy "=• aai making AB ~ a,
BC r: iv, and CD =: jy. Then it will be
— — <7, and X zn •— -^ , xx =:
Thei"efore, making the fubftitution,
X zz
*^yy
y
zz — a\
the general formula will be — x/y"^ + a*,
° ry -^
Put vO'* 4- «* = z, and therefore y* =z
y :=: —r. Make thefc fubftitutions, and we fliall have the formula
G g 2 transformed
228
ANALYTICAL INSTITITTIONS*
BOOK III.
transformed into this other,
ezzz
, which is free from radical figns ; the
2r X »a — fl*
integral of which depends partly on the logarithms, as is eafy to perceive*
Therefore the fuperficies required, defcribed by our hyperbola, will alfo depend
on the quadrature of the hyperbola.
EXAMPLE XLV.
Fig. 1 1 6.
140. Let ABF be the folid generated
by the tra^rix^ as in Example XXXVI,
§ 128, of which the fuperficies is re-
quired. In the general formula -^^
(where u reprefents the element of the
curve,) inftead of ii, -fubftituting it's
value — obtained from the equation
of the curve, we fhall have — — , and by integration, — ~ + ». BuC
when the fuperficies is nothing, we have y zz a; therefore the conflant
ji — ^ , and therefore the complete integral is — — — , equal to the fur-
face of the folid generated by the figure AEDB. And making ^ = o, then
— will be equal to the furface of the folid infinitely produced. But the area
of the circle, whofe radius is Viaay was found to be zz ~; then the furface
of the folid, infinitely produced, is equal to the area of the circlCj whofe radiu*
is equal to the diagonal of the fquare of AE«
EX-
lECT. IXX«
AKALYTICAL 1N8TITVTIOK9.
2251
EXAMPLE XLVI.
Fig- 133-
141. Let CNEODAC be the ungula
whole fuperficies is required. Impof-
rng the fame names as at § 129, it will
be QK - -^ =: MN. But M/, the
element of the curve, is \/xx + yj/,
and therefore it will be — ^xx + _yy»
equal to the infinitefimal quadrilineum
M.im'N, the element of the fuperficies
of half the ungula.
Let the curve DAC be a femiclrcle ; in this cafe it will be \/xx + yy
And by integration
ax
V aa—i
and therefore the formula is
V aa — XX
(according to § 31), it will be — h\/aa — xx + /. Bur, making x zz. o,
it wiU be / rr ab ; therefore the complete integral will be found to be
ah — h\^aa — AW. And making x :=: a, in refpedt of the whole fuperficies
of the half ungula, that fuperficies will be = ab.
Let the curve DAC be the parabola of the equation j'j^ iz <? — ^ j it will be
\/xx 4- yy =: f^y^if-Ulf L and therefore the formula is — /llHl^ — i,
the integral of which depends on the quadrature of the hyperbola ; fo that the
fuperficies of the ungula will depend on the fame quadrature.
EX.
»3®
ANALYTICAL INSTITUTIONS.
BOOK III.
EXAMPLE XLVII.
142. Let there be a parabolical conoid,
generated by the rotation of the parabola
ADH with the co-ordinates at an oblique
angle, about the axis AC, whofe eqiution is
ax zzyy. In the formula for the fuperficies
belonging to this cafe, that is, the formula
cny
J
yy -\- XX +
lexy
let there be fubfti-
tuted the value of x given by j/, from the
differential equation of the curve, and it will
be transformed into this other, ^f!l^^ yy + --^ + ^aa j the integral of which
arm ^ ■'■' m 4 » o
Tvill be found to be partly algebraical, and partly logarithmicaL
143. In purfuance of the method already explained, for quadratures, reAi-
fications, &c. this would be the proper place to give alfo formulas for centres of
gravity, of ofcillation and percnffion ; but 1 rather choofe to omit them, as they
neceffarily require fome knowledge of the prmciples of Staticks and Mccha-
nicks, which I fhall not fuppofe my young readers to underftand at prefent.
SECT;
SECT. III. ANALYTICAL INSTITITTIOKS* 23»
SECT. IV.
The Calculus of Logarithmic and Exponential ^antjlies.
144, Exponential Qiiantities, (of which, as alfo of logarithmic quan-
tities, we have treated elfewhere,) are thofe which are raifed to any indeterminate
power. Such would he a ^ y , &c. the exponents of which, at, z, are inde-
terminate or variable quantities. And therefore the method of computation,
which is converfant about fuch quantities, is called The Exponential Calculus,
145. But exponential quantities are of feveral degrees, Thofe are faid to
be of the firft degree, the exponents of which are the common indeterminates,
as are the quantities a", y . Thofe are of the fecond degree, the exponents of
t p
which are the faid exponential quantities ; fuch would be « , j , where .v is
raifed to the power /, and 2 to the power p, Thofe are of the third degree,
which have an exponential of the fecond degree for their exponent, An4
fo on.
146. Now here we fhould call to mind what is already faid at § 11, that
/— = /y, in the logarithmic curve, the fubtangent of which iz a. There-
fore the differential of ly will be — multiplied into the fubtangent of the
logarithmic, from which the logarithm is taken. Thus, the differential of
l\/ aa — XX will be — — in the logarithmic. In which the fubtangent
aa — XX °
1= I. And, in general, the differential of any logarithmic quantity whatever
will be a formula, compounded of the differential of the quantity itfelf, multi-
plied into the fubtangent, and divided by the fame quantity,
147, This
£^2 AHAtYTICAt INSTITUTIONS. JBOOK III.
147. This fuppofed, let it be required to difference the logarithmic quantity
/'"x, where m is the exponent of the power of the logarithm. Make l'" x =
y'**, then it will be Ix zr y, and ~ = j/. But the differential oi I x will be
my^^^^yi and it is^'""'' iz l"*~ x. So that, inftead of j and j/, fubftiiuting
their values given by x, the differential of J x = ml " x X -^ y fuppofing
the fubtangent of the logarithmic z: a. Or othcrwife, = ml'"'' x X ~, fup-
pofing that fubtangent =1.
148. If we were to difference / x", making x" ~ 2, it will be l"^z ; and
the differential of this will be ml"'" z X ~. But z =. nx^^^Xj and by
fubftitution, tlie differential of the propofed formula / x^ will be nml^"^ x^
AT
149. Let it be propofed to difference the formula llx. Make Ix z:^, and
tl^reforc llx =: /y. But it will be -^ n y, in the logarithmic whofe fubtan-
gent zz I (which is always to be underftood, whenever thefe fubtangents are
not particularly exprelTed). But, becaufc llx zz ly, the differential of llx will
be—. Therefore, inftead of jf and y, putting their values given by x, it
will be ~ for the differential of the formula propofed.
But, more generally^ let it be required to difference l"^ Ix. Put Ix ziy, and
therefoi'e / Ix •=. I y, and — — JK» ^^t the differential oi I y is mf*" y
X — ; therefore, fubftituting the values of y and y given by x, the differ-
ential required will be mt " Ix x -y»
Still more generally. Let it be required to difference l^l^x. Make /**^
:== y'", and therefore Ixzz. y^ and — = j. Then it will be l^ f'x zi /"^^.
6 But
SECT. IV. AN A L Y TICAL rNSTITlTT IONS. 23^
But the differential of l" y"* is mnl^^^ y^ X. — . So that, making the fub*
ftitutions, mnl " / x X -r- will be the differential required.
150. Now for the method of differencing exponential quantities. Let the
quantity to be differenced be z"*^. Make z" z: /, and confequently it will be
/z* r: //. But, by § 14, it is /z* — xlz^ and therefore it will be xlz z: //,
and therefore, by differencing, xlz -f — — -—- . But / iz: z^ y whence xlz
+ — = — , and finally, / =. z'xlz + .V2*~ Zy which is the differential
required*
151. Let It be required to difference the exponential quantity of the fecond
P P
degree, z* . Make 2'' = /, and therefore it will be x^lz = //, And,J)y
differencing, the differential of x^ X Iz + x^ x -^ will be = ~ . But,
by the foregoing article, we know the differential of x^ to be x^plx -^ px^^^x;
and therefore it will be x^ plx + px^^ x X Iz -\ = — - . But tzr: z'' .
p p p
Therefore it will be f = z* x^ plxlz + z* px^^^xlz ■\- z 2""^v^s for the
differential required.
In the fame manner, we may proceed to exponential quantities of any other
degrees,
152. Likewife, in the fame manner, we may have the differentials of quan-
tities, which are the produdls of exponential quantities 5 as, for example, of
X y'** For the differential of this will be the produfl of a?"^ into the differential
y", together with the prod ud of jy" into the differential of ;; . But it has been
(hown how to find the differentials of x^ and y^.' Therefore, &c.
153. From the order in which logarithmic differentials proceed, we may
derive rules for the integration of logarithmic differential formulas. And, firft,
thofe canons which ferve for the integration of common differential quantities,
will alfo ferve for logarithmical differentials which are like to them ; becaufe
Vol. II. H h thefe.
Ji34 ANALYTICALINSTXTUTIONS. BOOK III.
thefe are divided alfo by the variable, and the integrals of thele will be the
fame as the integrals of thofe, putting only in thefe, inftead of the variable or
it*s power, the logarithm or power of the logarithm of the fame variable ;
dividing the whole by the fubtangent of the logarithmic.
Thus, becaufe the integral oi mx " x is x , alfo the integral of w/**" x
X — will be — ^.
X a
In the fame manner, becaufe /a?"" x z;! lx\ fo likewife fl^^x X -^ , or
/^ will be llx-y fuppofing the fubtangent = i.
And, becaufe /jyyv^^^ -f- j^ z= | X aa + j'jl » ; it will be alfo fly^aa-^l*y
y
,m—'i, X
Let ml ~ /a? X -7- t>e given to be integrated. Make Ixmy ; then — zzjf.
And making the fubftitution, it will be ml^" y x —* But we know the
integral of my^'^^y to be y*, and therefore the integral of ml^^^y x -^ will
be l^ y. But y = Ix, and therefore ly = llx, and l^y zz l"*Ix, Therefore
Let it be nml"~ x"* x -^ • Make x = y, and therefore x = —
mx
And making the fubftitutions, it will be nml " y X , that is, «/^ y
ffix Xx
X — , the integral of which is /*y. Then reftoring the value of^, it will be
jnmi XX — :=: I X ,
Let it be nml^" l^x X ~ . Make Ix = y ; then ~ =z jf, and i"'x :s
y'". Making the fubftitution, it will be nml^^^y*" X — . But the integral of
this is l^y'". Therefore, reftoring the value, it will be /»/»/""* l"^x X — =
I X.
154. To
SECT. IV. ANALYTICAL INSTITUTIONS. 2^^
154. To this I (hall add. a general rule for the integration of the formula
y i y^y* and fay, in general, it will hcfy I y X y zz - — - — ■ . — ~
H- =4:^::^ ^ ■/ =^ , &c. And thus
the feries may be continued in infinitum, by obferving the law of it's progreffion,.
which is manifeft of itfelf.
Hence, if the exponent n fhall be a pofitive integer number, it is eafy to
obferve, that the feries will break off of itftlf, and confequently the integral of
the propofed formula will be given in a finite number of terms.
For example, make w = 2 ; then it will be » — 2 zz o, and therefore the
co-efficient of the fourth term will be nothing, and of all that follow, becaufe
every one is multiplied by n — 2. So, if « zr 3, the feries will break oiF at
the fifth term ; and fo of others.
Make » = 2, ;w == i ; then the formula to be integrated will be yl*y Xy,
Therefore the fourth term, and all the fubfequent terms, will be nothing.
Therefore the integral will be ^^ - ^^ + ^ . *
Now, if it were w = — i, the feries would be of no ufe, becaufe it would
be m + I •= o, which makes every term infinite. But, in this cafe, there
would be no need of a feries, becaufe we know already how to integrate fuch
formulas, by what has been faid before.
It remains to give the demonftration of this rule. To do which, make
/y = 2, and therefore ^ zz z. Then making the fubftitution, it will be
y.lyy=^yzy. ^^t y z y ^ y z y -{-^^^j z z ^ -^j z ay
— ■ - ■ y z , az -{- ■■■ ■ y z ay, &c. And fo on m infinitum ;
m + iJ tn-\-\]
becaufe, in this manner, every term, except the firft, will be deftroyed by that
immediately following, becaufe it is i; zr ~. Now, becaufe fuch an infinite
feries is integrable, by taking the terms two by two j for the integral of the firft
and fecond term is - — •— , of the third and fourth is — f^L__-f Qf
the fifth and fixth is «--ix^ g_^ . ^^^^ ^^ ^^ ^^^ ^^^ ^
•H h 2 integral,.
7-^^ ANALYTICAL HJSTXTUTIONS. BOOK Xlt*
integral, inftead of 2, reftoring it's value fy, we fliall find it to be at laft^
fi" i^'yy zzi ^ -^ — J!2-__._£, &c. as before.
1 55» %The artifice of finding the aforefaid feries is this. We may conceive
m + 1 ,n
the integral of jy /"^j/ to be '^- i as it really would be, if /*jy were not
a variable quantity j but, fuppofing the fubtangent zz a^ the differential of
m ,n— I
m ,n—l
at
,« — I .
ought to be, by the integral of — ^ , and therefore the integral of this
ought to be fubtradcd from the fuppofed integral.
m ,« — I . m + I ,« — I
And here again I conceive that the integral of "-^ ^ . — ^ is ^ ^ — ^;
whence the integral of the propofed formula will be -^ ^ _.
«> ^ _y,. But, by differencing -^ — "1, -^, we Ihall have ^^-^ '^ ..JL
77? ,n— I •
this integral \s y" I yy + - "-^ ^ — ^. This is found greater than the pro-
pofed formula by -^2LiL. — :H ^ fo that the integral aflumed is greater than it
m , jW — 3 . m ,«•- I
+ — ~ ^ • Therefore the mtegral of -^ — ■ — -^^ is not
m jti — i
-^^=^^-~- , but is greater than it ought to be by the integral of "~^ x
y a'i " yy. Therefore too much is fubtraded, and this integral is to be
added, which again I imagine to be - — -^^^^--^ — -L -3 -* So that the
integral of the propofed formula will be —^y"* l^y — ~-sz y^ ^ ^^^"^ y
+ ■ — L3 y ^V"~ J, See. And thus proceeding in the fame manner, the
feries may be continued in infinitum,
'' 156. We
SECT. IV^""'
ANAtYTlCAl- IKSTITUX IONS.
^37
156, We may alfo have the integrals of logarithmic differential formulse by
the help of feries, which fliall not contain logarithmic quantities, but only
common quantities j which feries, therefore, will never jjreak ofl', but are
always infinite.
Let xlx X X be propofed to be integrated. Make x zn z + a ; then, by
fubllitution, it will be z -i- a x Iz +a X z. But, by § 70, k'ls Iz-i- a ~
— — — r H ^r — r » &c. Suppolins: the fubtaneent zz i. Then, by
adually multiplying, we fliall have z -i- a X Iz + a x z zi zz + - —
+
rAi-.
la^
a 2a*
z*z
— - , &c. r Y + — r > Sec. ; that is, zz -\
z^z
-\ r r- , &c. ; and, by integration, it will be h -^ ;
12a' 20U* » » > 7 b ' 2 6a 2aa*
+ -rrr — TZr^^ ^^' = /z + ^ X Iz-t a x z.
6oa^
120a*
So, if the formula were x Ix X x, that is, z + a] x iz + a Xz, wemuft
multiply the feries expreffing the logarithm into the power z + a\ . And
moreover, if the logarithtti alfo were raifed to a power, as x'*'/"x x x, that is,
z + a) X l" z -{- a X z, there would be occafion, befides, to raife the
infinite feries, expreffing the logarithm, to the power », and to do the reft,
as above.
157. Differential formulas, or equations afFeclcd by logarithmic quantities,
very often admit of integrations which are geometrical, and which depend on
quadratures of curvilinear fpaces, which may eafily be defcribed, fuppofing the
logarithmic curve to be given.' ' Here are fome examples feledted out of the
more fmiple Ones. . o.... ..;, ^.iiz....^
Fig* 138. ■ 'tertKfc t(5uaTi8n"'be yfy zz x, and in the
logarithmic defcribed let CD zz y ; and taking
JV the fubtangent for unity, we fliall have AC iz
E HD zi ly. Whence the infinitefimal reiftangle
DG, of which the bafe is GH iz FE zi j, will
he zz yfy. But this retlangle is the element of
the increafing area BDH, and therefore the fum
or integral fyly is equal to the faid area. In
- fad, the area itfelf is equal to the re<5langle AD,
fubtrading the logarithmic fpace ABDC. But
this
.v_._
a
1
i\\ - 1 -
^
.^
ij
K. ^
D
■^
L ii:ij:?o
■
A
•
J
G
2$t ANALYTICAL INSTITUTIONS.
BOOK III.
this fpace, as is known, is meaftired by the re61angle AB X CD i= y, There-
'fore I he area BDH =■ fyljf := yly — y, as may be found by the way of
analyfis.
I fliall confider another formula, yPy ~ x. The firft member is no other
than the folid generated by the fluxion HG, muhiplied into the fquare of the
ordinate GF ; which folid is analogous to the element of the conoid, generated
by the area BDH, revolving about the axis BG. Therefore the integral /yly
•^ y^y "~ V'fy + ^y ^s to the faid conoid in a given ratio.
More generally, let us have jl y. Raifing the ordinate HD to the power w,
(the index vi being either an affirmative or negative number, either whole or
broken, it will fuffice that the ordinate HM may be made equal to the dignity
HD , and that through the point M, and infinite others to be determined in
the fame manner, the curve BMN may pafs ; in order that the area BMH =
/MH X y niay be equal to, or analogous to, the integral Jyl y.
The difficulty will not be greater, even though the logarithms of logarithms
fliould alfo be found in our expreffions. Let there be propofed ylly — x.
Whereas AC is the logarithm of CD ; if, in the logiftic, the new ordinate IL,
equal to the abfcifs AC, fhould be adapted ; Al will be the logarithm of IL,
and confequendy the logarithm of the logarithm of CD. Let the right line IL
be prolonged, fo as to cut HD, parallel and equal to AC, in the point K ;
through which and infinite others, determined in the fame manner, let a new
curve pafs, drawn relatively to the logiftic. I fay, that the quadrature of the
fpace belonging to this curve will give us the integral of the formula ylly = x.
After another manner. I takd the fluxion of the quantity y/y, that is,
.ylly + y- , and adding the term ~ to both fides of our expreflion, we fliall
have jl'/Zy + ^ =: i + •/"■> ^^^ by integration, ylly = a? + /4— • There-
fore, to the abfcifs AH annexing the correfponding ordinate in the reciprocal
ratio of HD m /y, a curve will be produced, the quadrature of which will
exprefs the integral /y-^ And this will be enough to fliow how the method
proceeds.
158. I (hall now go on to the integration of differential formula, which
contain exponential quantities ; and let us integrate x'x. Put x zz. i + jr,
(taking unity for any conftant quantity,) then it will be .v :v == 1+7' y*
4 This
SECT. IV, .ANALYTICAL INSTITUTIONS. 2^
This fuppofed, make alfo i 4->] ^ -^ :=z i + «, and then it will be i -^ y x
1 1 +JK :=: 1 1 +u. Now let the two logarithms be converted into feries, by
^ 70 ; and making an adlual multiplication of the firft feries by t .+ y» we
Ihall have ;^ + f/ — f_)>' + ^'^y^ — -%y\ &c. = a — fw* + ^«» — ^u*
+ ^«*, &c. Then make a fiditious equation, fiippofing it to be a = ^ + Aj*
+ B>' + Cy* + Dy*, &c. (where A, B, C, D, 8cc. are quantities to be deLer*-
mined by the procefs.)
Therefore a« = / + iKy^ + A^* + 2ABj*, &c.
4- 2B>* + 2C/
«'!=/ + sAy + 3 Ay, &c.
+ 3%* a* z:y + 4AjyS &c. n^ zz y\ &c.
Whence u — fa* :t- ^«» -^ t«* ,+- 1«% &c. =:
jy + A/ + By + Cf + Dy, &c.
— tr' — A>' — t Ay - AB>*
— By — Cy*
J- {;'' + Ay* + Ay . > = J' + iy - ty' + tV^* — Tvy, &c.
— tr —Ay
Now, by comparing homologous terms, we (ball find the values of the
aflumed quantities to be A z: i, B = f , C = -^, D = t^, &c. ; fo that,
putting thefe values in the places of the capitals, we (hall have i 4- a , ss
r+3^'"^^ = I + ^^ + y + t)'' + i:>* + tt/» &c. Whence 7+3^ '■'"•^i =
y + yy + yy + iy^ + t^^j &c. ; and laftly, by integration, /i -^y) ^ ^ Xy
= y -^ iy" + "^y' + ry* + tV>' + tV/, &c.
159. We may find the integral of the formula x'^x thus, in another manner.
Make x* =z 1 + y, then xlx zz. ^ i -{■ y. Reduce / 1 +jy to a feries, and it
will be / I +y •=: y — f_y* + ^y'^ — ^y -f ^y^, &c. This fuppofed, make
V =: l\ +_y + AivT^T^ + B/' 7T} + 0*7^7 + D/*'iT3', &c. (where
A, B, C, D, &c. are quantities to be determined,) and it will be
y =
S4®
J
kN AL YT
ICi
IL IKS
r 1 T I
J T I N S.
BOOK IIT<
y
= /•
1 -i-y
+ 2A/3 7
+y
+ AV*7
+ 2BIU
■^y
+ 2AB/'
I -\-y>
I -i-y
&C.
y
/'i
+y
J.- 3 A/* I
■ty
+ 4A/^ "
1 +y
/
+j
f
efore
f + t/'
y' + fj'S
&c. :
P
D/57
her
= /i +jy
/i
+ jy
I ^-^^ 4-
I +^ +
I +y +
+yy
&c.
-t/'
1 +;' -
+
A/'
1 -ty — ,
iAV^
B/*
A/^
1 +;' —
AB/*T
A*/*l
+ y
+y
I +^ —
1 +y +
I +y +
+y
+
B/»^
-^y
—
-/^
1 H-j —
A/5 I
+ y
+ T^' ^ +y
Now, by the comparifon of homologous terms, we fhall find A = f, B =
C = ^, D == -rici &c. ; whence 1 +^= i +/i4-y + t^*i+j'
+ l/^rrj + -^T^*^ +y + T^/* I + y, &c. But /T^Ky = a:/:v, and
I +j = flf*3 therefore, making the fubftitutions, and multiplying by i, it
will be x'x z: ^ + xxlx + ^x^xl^x + -^x^xPx + -^^x^xl^x + -r^^x^xl^x, &c. j
and integrating, by the known rules above delivered, it will be fx^x =: x +
j/i-. 139.
160. Now, to add fomething concerning the con-
ftrudion of curves expreffed by logarithmic and ex-
ponential equations. Firft, let it be required to
3
defcribe the curve of the equation x = — =^ . Let
BD (Fig. 139.) be the logarithmic, in which we are
to take the logarithms of the propofed equation,
whofe fubtangcnt (for example) is = « = AB. This
9 fuppofed.
(
M
SKCT. IV, ANALYTICAL INSTITUTIONS. 24I
Fig, 140. fuppofed, taking y ~ a :^ AB, the logarithm of y will
/ be rz o, and therefore x =r o. Making, then, M^J rr^
/ zz ^ (F'g- '4°)' N will be a point in the curve. Taking
/ ' y Itfs than AB, ly will be a negative quantity, and there-
X fore /"^y will be an imaginary quantity, becaufe the even
^^^-^"^ number 2 is the index of the root of a negative quantity;
X whence x will be imaginary whenever y is lets, than a.
Taking y greater than AB, fuppofe zz: CD, it will be
AC rr/y. But, by the given equation, it is a^- . I'-y :;
ly , X, ox a , s/ aly :: ly . x \ and therefore, making
K^P zr CD, we muft take PH equal to the fourth pro-
portional of AB, a mean proportional between AB and
AC, and the faid AC ; which fourth proporiional will be zz .r, and H will be
a point in the curve. After this manner we may find as many points as we
plealc, and fo defcribc the curve, which will go on ad infinitum^ as is eafy to
perceive.
To have the fubtangent of the given curve, I take the differential formula
-rr- of the fubtangent, find the difference ofthe equation of the curve, which is Af zz
\l^y X — . Making the fubftitution in the place of ^c", we (hall have the
fubtangent zz i./^_y x <?* = ^ = \a^Xi,
Alfo, our curve will have a contrary flexure; to find which I take the
fecond fluxion of the given equation, fuppofmg x conftant, and I find
|.5^/^x>.H.^/-^, -1.^/^ zz o J and therefore y zz
— . . I. i . . ^i
\(^ yy y -^ i^ yy 1 ^ g^^j.^ I^y ^^ fnethod of contrary flexures, it ought to
la-yl-y
be y zz o. Therefore it will be ia*yyhy .-^ la^yyl'~*y zz o ; that is, hy —
^al^'^y = 0, or /y = ^a. Therefore the point of contrary flexure will be
there, where ii \s ly zz ^a.
If the curve propofed to be defcribed were xlx = y, refolving the equation
into an analogy, it will be i , Ix :: x ,y, which may be conftrucled in a like
manner.
Vol. II. I i If
S42
AlIALYTICAL INSTITUTIONS,
BOOK III.
If the curve were *V.v = y, or xHa — y, or jc'^lx — y, or, more generally,
x^lx ~ y, fuppofing n to denote any power of a", whether integer or fraction ;
this equation being likewife lefolved into an analogy, i , h :: ^v" .jy, and
taking in the logarithmic any line CD zz x, whence AC z: Ix ; the multiple
of AC, according to the number », if it be an integer, the fubmukiple, if a
fradion, will give the correlponding ordinate in the logarithmic itfelf, which
fliall be A", by the property of the logarithmic.
Fig. 139.
d/
* •
I
/
A
a
C
If the curve fliould contain quantities that are
logarithms of logarithms, fuch as x'^llx zijy, we fliouId
eafily have in the logarithmic the line expreffed by llx^
by taking any line CD zz x (Fig. 139.). whence it
is AC == Ix', and then putting AC for an ordinate
in {ae). For Aa would be the logarithm of (^<?),
that is, llx-j as has before been taken notice of at
§ ^S7^
Fig, 141. 161. Let It be required to conftruft
the exponential curve of the equation
X zz y. Now, taking the logarithms,
it will be xlx = /y > and defcnbing the
logarithmic curve PAB, (Fig. 141.) with
the fubtangent AD zz i, and taking any
line CB zi DE = x, it will be DC zzlx.
Then, becaufe the equation may be re-
folved into the analogy, i . x :: Ix , ly ;
the fourth proportional to AD, BC, and
DC, which fuppofe is DM, will be ^ ; fo that MN zz y. Therefore, if it be
made EF = MN, it will be DE zz x, EF z: y, and F will be a point in the
curve to be defcribed.
The curve will cut the afymptotc HM in H, making DH = DA. For,
putting X zz o^ it will be ly zz o, that is, y zz DA, Making, therefore,
AG zz DH, G will be a point in the curve.
From the point H drawing HP, an ordinate to the logarithmic, and drawing
POR parallel to HD, then OR will be the leaft ordinate, j, to the c\irve. For,
taking the difference of the equation, it will be ;£■ + xix zz — , that is,
yx •+• yxlx zz J/. But, by the method de max'tmis et minimii^ it muft be j/ zz oj
therefore yx + yxlx zz o, and therefore — Jx zz in HD iz DA.
8 Becaufe
SECT. IV. ANALYTICAL INSTITUTIONS. 243
Becaufe -^ is the general formula for the fubtangent, and. having ^ =
from the given equation of the curve, by fubftituting this value in
jy
the formula, the fubtangent belonging to any point of the curve will be =r
; and for the point G, in refped of which it is a: =: AD, and confe-
quently Ix = o, the fubtangent will be = i = AD, which is the fubtangent
of the logarithmic.
As to the area, take the general formula yx ; but y zz x", in the
equation of the curve. Therefore, fubftituting the value of y in the
formula. It will become x^x^ and therefore /x'^'x is the indefinite area
HOFEADH; which, being integrated according to § 159, will be n ;tf 4- -1-1
— i* H p- — + — -i h -T- 9 &c. And taking x zz
♦ 6 9 27 24 32 04 .
AD = I, it will be Ix zz.o, and therefore the area HOGAD zz 1 ^ j — L
•} ::!l ,/■; 4 27
•l. jij, &c. J that h,;zS:^^-L. + J^ _ Jj. + ^ , &c.
162. Let x^ ::! a be the equation of the curve. Then ylx = la, ajpd
therefore it may be conftruded by means of the logarithmic. By taking the
fluxion of the equation, we (hall have — + jflx zz o, making the fubtangent
of the logarithmic = i. And therefore it will be a" = — ^-^; and therefore
the fubtangent = — xlx,
163. Let it be x^ zz a"^ ; therefore xlx zz yla, which may be conflruded as
ufual. Taking the fluxion, it will be ;tf + xlx zz yla ; and the fubtangent
xlx
I + Ix'
Here, becaufe y — -7I, it will be yx, or the element of the area, zz -4—;
"^ la "^ la
J . ^ . 1 ,- . • IXxlx — XX
and mtegrating, by § 154, it is 7 = area.
164. Other queftions may be ftill propofed, relating to exponential equations;
as, for example, in exponential equations compofed of only known quantities,
but with variable exponents, to find ihofe exponents. So, let it be <r = ^^*~'j
the value of the unknown exponent, x, is required, ^, ^, f, being given,
I i 2 Becaufe
^44 ANALYTICAL 1 N S TI T U T 1 O I* S. BOOK Ml.
X
c ,x—\
Becaufe — r=. h , it will be at/^ — • /^ zr ^ — i Ih, and therefore xlc — xlb
la --Ih
•zz. la — lb. Whence x zz
Ic - lb'
165. Another queftion (liall be this. To find fuch "a number x, as that it
may be ;?"* zz^, and aifo x" '' — b. Now, by the firfl Condition, we Qiall
have xlx — la^ and therefore x = y— , or Ix ~ ~. By the fecond condi-
• ■ 7a ^ i>lx
tion, we (hall have x -\- p Ix :zz lb. Therefore it will be at = — j—^, or /a; =:
Ix
, Then it will be — :r , that is, xla + p/^ = xlb, ov x zi
A- + / AT X -\- p
,, \ ' ^'" ^^^"^ T~ "~ — /~> ^hat is, Ix =z — ^^^. This fuppofed, I (hall
propofe to myfelf to refolve the following Problem.
166. A veflel being given of a known capacity, full of any liquor, fuppofe
wine, out of which is drawn a draught of a given quantity, and then the veCCei
is filled np with water. Of this mixture of wine and water another draught is
drawn equal to the former, .and the veffel is again filled up with water. Again,
of this mixed liquor another fuch draught is drawn out; and the fame operation
is conrinually repeated in the fame manner. It is demanded how many fuch
di^aughts maybe drawn .our, or how many times the operation mud be repeated,
that a given quantity of wine may be left in the veflel.
Let the capacity of the veflel be = a, and the quantity of each draught
= b. Therefore, at the firft draught, will be drawn fuch a quantity of wine as
will be exprefled by b ; and as much water will be poured in again ; whence,
after the firft draught, will be left in the veflel the quantity of wine = ^ — b.
At the fecond draught will be drawn out the quantity b of the mixture ; fo
that, to have the quantity of pure wine contained in it, we muft make this
analogy ; as the capacity of the veflel (a) is to the quantity of the draught (b),
fo is the wine which is in the veflel {a —• b) to a. fourth proportional — ~ — ,
which will be the quantity of pure wine which is drawn out at the fecond
draught. Then there remains in the vefTel the quantity of pure wine,
aa — 2ab + bb ^l ^ • a—S)*'
, that is.
Therefore, for the third draught, making alfo this analogy ; as the capacity
of the veflel {a) is to the quantity of a draught [b), fo is the wine in the
vefle],
4
SECT. IV. ANALYTICAL INSTITUTIONS. I45
veffel, ^-^ — , to a fourth, ^-^ — X — • This will be the quantity of pure
wine, which was drawn out at the third draught ; fo that there will remain in
the veilcl the quantity or pure wine, X > or . And
thus, after the fourth draught, there will be left in the veffel the quantity of
pure wine, ^-^^^ — ; and, in general, after a number of draughts denoted by n,
^ __ A»
there will be left in the veflel the quantity of pure wine iz -. Therefore,
a ""
if we would know how many draughts muft be taken, fo that there (hould
remain in the vefTel a given quantity of pure wine, fuppofe, for example,
— part of the whole ; we muft make the equation ^ "" ' =: — } which, be-
caufe « is an unknown number, will be an exponential quantity. Wherefore,
the equation being reduced to the logarithms, it will be / n / — , that
a
is, nl a — b •=. la — Im -^ n — i la, or nl a — ^ = — Im + rtla, and there-
Itn
fore n = ; fo that it will be eafy from hence to find the number n,
la — I a — b
by the help of a Table of Logarithms.
END OF THE THIRD BOOK.
ANALYTICAL INSTITUTIONS.
BOOK x^^
THE INFERSE METHOD OF TANGENTS.
I. AS, when any cufve is given, the manner of finding it's tangent,
JTx fubtangenr, perpendicular, or any hne of that kind, is called the
Diredt Methbd of Tangents ; To, when the targenc, fubtangenr, perpendicular,
or any fuch line is giveft, — or when the redification or area is given, to find
the curve to which fuch properties belong, is called the Inverfe Mechod of
Tangents.
In the fecond and third Books are found the general differential expreffions of
the tangent, or other lines auulogous to it j as alfo, of rectifications and areas.
Therefore, by comparing the given property ot the tangent, rectification, &c.
with the refpedtive exprelhon or general differential fornjuia, there will arife a
differential equation of the firft degree, or of a fuperior degree, which, being
integrated, either algebraically, or reduced to known quadratures, will give the
curve required, to which belongs the given property. For example, let the
curve be required of which the lubtangent is double to the abfcils. Calling the
abfcifs A", and the ordinate y, the formula of the fubtangent is ~- , and ttiere-
y
fore the equation will be 4- zz 2^. Again, let iis feek the curve, the area of
which
248 ANALYTICAL INSTITUTIONS.
BOOK IV.
which miifi: be equal to two third parrs of the redangle of the co-ordinates ;
the element of the area is ^x, and" therefore it ought to bt fyx =4*7, or
2xy, -i- 2yx
~ 3
arch taken from the vertex (hall be equal to the refpedive fubnormal ; the
^x zz -^ — — ^ . If we would find the curve whofe property it is, that any
expreffion of the. arch \s J\/xx + yy^ and that of the fubnonnal is ~; fo that
we (hall have fs/xx -\- yy =. -^ , and therefore \/xx -i- yjf =. ^^-^ . . ^-^'-^
(taking x for conftant,) which is a difTerential equation of the fecond degree.
2. The equations which arife by proceeding after this manner, will always
have (as is eafy to perceive,) the indeterminares and differentials intermixed and
blended with each other, fo that at prefcnt they cannot be managed, in order 10
proceed to til eir integration, fo a-s to-obtain, the curves required; and much
more if they -contain differentials of the fecond, tiiird, and higher degrees.
For, in the third Sedion aforegoing, the differential formulae have always been
fuppofed to be compounded of one indeterminate only, with it's difference or
fluxion. Therefore other expedients are neceffary, to try to reduce fuch
equations to integration, or quadratures, which is called the Conftrud:ion of
Differential Equations, of the firfl, fecond, &c. Degrees, And, as to the
Con(lru(5lion of thofe of the firft degree, we ma-y- proceed, two ways ; one is, to
pafs immediately to integrations or quadratures, without any previous feparaiion
of the indeterminates and their differentials ; the other is, firft to feparate
the indeterminates, and fo to make the equations fit for integration or qua-
drature.
I (hall proceed to (how feveral particular methods for both the ways, by
which we may attain our purpofe in moft equations. But \ery often we (hall
meet with others,' which will be found fo ftubborn, as not to fubmit to any
methods hitherto difcovered, or which have not the univerfality that is
neceflary.
/p> srff .Slot
SECX.
SECT. I. ANALYTICAL INSTITUTIONS. 249
SECT. r.
Of the ConfiruEHon of 'Differential Equations of the Firft 'Degree , without
any previous Separation of the Lideterminates,
3. The moft fimple formulae which have the two variables mixed together, are
thefe two, xy + yx, and — "" ^ . The integral of the firft is xy, and of th&
fecond — , as is manifeft. To thefe, therefore, we (hould endeavour to reduce
y
the more compounded, and that by the ufual helps of the common Analyticks,
by adding, fubtrading, multiplying, dividing, &c. by any quantities that will
make for the purpofe, which will be different according to different cafes. We
fhall here fee fomething of the pratlice.
Let it be j^c" =: xx — xy. By tranfpofing the laft term, it will he yx -{- xy
zz XX, and therefore, by integration, xy zz -^xx ± bb. Let the equation be
x*yjf -|- 2x^yxy zz a*xx — xxyyxx ; then tranfpofing the laft term, and dividing
by .v^, it is x^y^ + 'ixyxy + y'^x^ zz — ^ ^ and extrading the fquare-root,
xy + yx zz — ; and by integration, xy zz alx ± b, in the logarithmic with
fubtangcnt z: a. Let the equation be yx zz y^y + y'^jf -f- xy» that is, yx •— xy
zz yy + /j/. The firft member would be integrable if it were divided by yy;
therefore 1 divide the equation, and it will be ^-^LlUL :z: yy -\- y, and, by in-^
tegration, it is — zz ^yy + y ■±. b, .
4. Let the equation be jy'j/ zz myx + xy. If there was not here the co-
efficient w, the matter would be eafy, becaufe the integral of the fecond member
Vol. II. Kk would
250 ANALYTICAL INSTITUTIONS. BOOK IV,
would be xy. The operation would not fucceed any better, by tranfpofing the
member xy to the other fide, or by writing y y -- xy :zl myk ; yet I obferve,
III
that the differential of w^ w is mxym -{■ xym j), different from that pro-
I
pofed, myx + ^j^ only in this, that it is multiplied by ^ »» . Therefore, to
make the quantity iJtyx + xy become integrable, it will be fufficient to multiply
it by J w , and, to preferve the equality, to multiply alfo the correfpondino"
member of the equation y^y ; therefore it will be ^ *» y zz my m x +
I .11
I . , , . . . r -^i 1 . ,
icym y^ and, by mtegration, fy m y zz mxy m a- ip^
Let the equation be the fame, but with a different co- efficient in each of
the two laft terms } that is, let it- be jj/ — myx + nxy. The fecond member
« «
is not integrable; yet I obferve, that the differential of mxy m is my ^ x +
n ,
nxy^ y. Therdove the bomogejieum comparatiofiis v/ou\d be integrable, if it
11
were multiplied by ^ »» . Therefore I multiply the whole equation, and it
n n n
will become^ « y zz my m x •\- nxy f» y, and the integral will be
n n
r + I . ,
jy f» JK — f^^y "* dt^'
5. The differential of x'y is x"y + fyx*^~ x. This fuppofed, let the equa-
tion be y^y =z x^y + yx^^^x. If the laft term had n for it's co-efficient, the
integral of the fecond member of the equation would be x''y. I obferve,
therefore, that the differential of x^y" \s nx^y "j^ + ny^x " x -, therefore, mul-
tiplying the equation by ny^"^ , there will arife ny^ ""^y rz nx'^y^^^y +
ny^if^x, which is found to be integrable, it*s integral being /»/"'""'" j/ iz
But if the lafl term, inftead of the co-cfRcient », had any other, or, in
general, if both the laft terms were affeded by different co-efficients 3 or if the
equation
SECT. I. ANALYTICAL INSTITUTIONS. 2^1
« — I
equation were JVJ^ ~ ^^y -\- eyx x; 1 obferve, that the differential of
en en en
— x»y ^ is cx^'y ' y ■\- ey ^ x x. Therefore multiply the equation by
CM . ca en en
I ,r+ I, '-I. „_,
y " , that It may hQ y ^ y =. cxy ' ' y ■{• ey ^ x x, which is
en en
»■ + ^ . e
mtegrable, and it's mtegral is /y . ^ y = — xy ^ ± b*
Here make r •=. %, c nz ^, »=z i, ^zri, that is, the equation yjf zz 3.vy
4- yx I the integral will be ^y* zi xy^. Make c = 2, <? zr 3, « — i, r z: i,
that is, the equation will be jyy = 2xy -{- ^yx, and the integral will be
— :r = Z^y^i or ^y-r zz '^xy-T, Make <r = 2, tf = 2, « = 3, r 1= 3, or
the equation y^y zz. zx^ + lyx^x j and the integral will be ^y := ^x^y^,
en
If the equation were exprefled thus, jy ^ x x zz cxy + eyx " x, it is
en
eafy to fee, that it would be integrable. For, multiplying by j « , it would .
en en
be .« . .x" ZI f Jtf"j ^ y + ey " X y. But the -integral of the fecond member -
en
is known to be — x''y ^ ; &c.
6. Now let the equation he y^y ^ ^^^-—^ . If it were not for the co-
efficient 2, the integral of the fecond member would be — . But it will be
to no purpofe to tranfpofe to the other fide the term jfx, and to write it y^ —
^ _. ^LUii. But I obferve that the differential of ^ is !^LZliL^ fo that
XX XX X XX
if the propofed equation be multiplied by y, that it may be ^'^ y zz — ,
it will be integrable, and it's integral will he Jy y •=. — ± ^. But, more
generally, let there be any co-efEcient », and therefore the equation \% y y zz
«iLzJf. I obferve that the differential of -^ is • "''•^""'^"^'"^ ; therefore, if it
XX X XX
Kk,2 be
25* ANALYTICAt INSTITUTIONS, BOOK IV,
be multiplied by jy*"*', fo that the equation may be ^ j/ = — — ^2JLj
It will be integrable, and it*s mtegral will be /y y zz -^— + ^.
Thus, let both the laft terms have different co-efficients, and let the
n
equation be y^y .= ^^-^ " ^^"^ . I obferve, that the differential of .^LfL is
^•^ ~ -^ "" ^.y ^ -^ . therefore, if the equation be multiplied by ym "" , fo that
n n
It may be;' »» y = -^^^ , it will be mtegrable, and it's
n
n
V integral will be /)» w» y = ^^ ^ + h.
, : If the equation were y m a^x = "^-^ "" '^^'^ ^ Jt would alfo be inte^rable.
^ _» »
For, multiplying it by y~^ "" ', it will be xx = ^^^^^-^ j_j2_:2JUL^ But
the integral of the fecond member is known to be ^^ ^ ^ therefore, &c.
Let the denominator j^;^ be wanting in the aforefaid equations, and let the
equation be y^y = nxy — yx» To integrate the fecond part of the equation,
there would be occafion to multiply it by y"\ and to divide it by xx. But
r+n—i.
as this mufl be done alfo in refped: to the firft part, it would be ^ Z
which cannot by any means be integrated. Therefore let the figns of the
equation be changed, and it will be —-y^y zz yx — nxy. I obferve that the
differential of ~ is ^-^ — —■ — ^ . Therefore, if the equation be multiplied
«— I J .1 -„ J- -j-^j 1 -.««
by y"" 9 and then divided by ;' , fo that it may be — ^
__r+n^i.
y _-
/"
SECT. I. ANALYTICAL institutions; 255
n— I • __ r + « — I •
j^^x - nxy y_ j^. ^jjj ^^ integrable, and the integral is / — ~ =^ =
Let the equation have both the laft terms with a co-efficient, and let it be
yy z=. nxy '— tnyx. Let the figns be changed, and it will hQ -^ y y •=z myx
— nxy, I obferve that the differential of — ^ — i
n n
my m X -^ nxy m j
in
my m mmy m
n 2«
Therefore, if the equation be multiplied by y f» , and divided by mmy m ,
« n ' n
. I — V f" y my m X — nxy m y . -n i_ • ^
fo that It may be. — ^ — = -^ ^ , it will be inte-
^ 2» 2«
mmy tn mmy m
grable, and the integral is / — ^ ^ = — - — ± K
mmy m my m
. 7. Let the equation be y^y zz x^y — nyx^^^x. Change the figns, and it
will be — y'y ^ ^yx "" ^ — ^ y* I obferve that the differential of — is
!^^ — ^JLJL, Therefore, dividing the equation by yy, it will become
yy
J-y y = T-
But if the co-efficient n had been wanting, and the equation were y*^y zz xy
I— yx"~^x ; change the figns, and it will be — y'^y iz yx^^^x — xy. It may
y^^^y = — ^> which will be integrable, and it's integral is
« « «— I • K ?7 — I
be obferved, that the differential of ~ is -^^ ^^^^ ^, Therefore,
n 2a '
y y
multiplying the equation by wj""', and dividing it by /^, it will become
2,54 ANALYTICAL INSTITUTIONS. BOOK IV.
— == — =^, which wiU be integrable, and us
r-\-n— I . «
integral is / '— z: V b.
^ y
But if, inftead of the co-efficient », there (hould be another of a different
nature i or if both the lafl terms were affefted by a different co-efFicient, as if
the equation were y^ zz ex y — eyx " x -, change the figns, and it will be
— y*^y zz eyx " X — ex y, I obferve that the differential of — - — is
nc »c ey €
_f2: — f " "^"^ ■^ ^ — =^ . Therefore, multiplying the equation by
eey e *
•^ . _ nc
ne 2nc f j^ — _ j
ny e , and dividing it by eey e ^ it will be — ny e y^ ___
nc nc eey e
.f2 — f X ^ ncx y^ y^ which Will bc integrable, and it's integral will be
eey e
nc
r+ I .
/•— ^>^ i ^ =: —^1- -4- ^.
eey e ey e
ne
1 —
But if the equation were thus expreffed, y e x^x r: cxy — eyx^^^x ;
nc
without changing the figns, I obferve, that the differential of ^ ^ is
X
nc ne
.— •. T — — , nc
Kg ' e ''■"*A —.—I
jafLZ L-ZJL2 — J therefore, multiplying the equation by ^y ' ,
X
fic ne
T' n '~p '• , n — I .
,,..,. . , 2/1 ii 11 L nx X ncx y ^ y — ney ' k x ...
and dividing it by a; , we mail have — — z: ^ ^ i. , which
X X
nc ,
r . """^ ■
nx X ev ^
will bc integrable ; for it's integral is / —^ = — ^ ± h,
X ^ X
8. I have
SECT. I. ANALYTICAL INSTITUTIONS. 255
8. I have already faid, in the foregoing Book, § 17, that as often as the numerator
of a fracflion, compofed of only one variable and conftants, is the exadl differential
of the denominator, or proportional to that differential; the integral of fuch.a
formula is the logarithm of the denominator, or in a given proportion to that loga-
rithm. This alfo obtains when the formula contains two variables, intermixed with
each other and with their differentials. Therefore the integral of - — ^ = z, (z,
* -i- y ' \ '
after any manner, being given by x or by y,) will be I x + y iz 2 -h ^. The
X +y
integral of — -—— n z will be l\/x -f- jy = 2: + h. The integral of
^xx - 4)7 __ ^ ^.jj ^^ ^^^^ —-yy ^z X -\r h. The integral of ^- «7 — ajyv _
XX — yy "^ — ° ixy — zyy
zz z will be l\/xy — yy zz z ± if. And, in general, the integral of
nm— 1. «z«— I- - , — ni-^-n — l
my X ^ -f- ;i;f y y _ m + « v ' y -. mi u l I ^ ^ m-\-n , r
-^ -^ — — z: z will be />/ x y -^ y z:z±:fj.
m n m-\-n
r X X y —y
And fo of any other equation whatever, which (hall have the condition afTigned.
9. Wherefore many equations, though they have not the neceflary condition,
yet may eafily be made to acquire it, with the affiftance of Ibme calculation.
Thus, the equation — — — = — y, has not the required condition in the firft
member ; but it will have it if it be divided by y. Then it will be '^ — ^ =
^ •" xy
— — ; and therefore, by integration, I xy — ly" ± lb.
Let the equation be axy + 2ayx zz xyy, I divide it by axyy and it will be
'^ — ^ = -=^. This would be integrable if it were not for the co-efiicient 2
xy a °
in the fecond term of the firft member ; therefore I fubtradl the quantity -1-
xy
fr6m each member, and it will be — — =^ =: — ~, that is, — — —
xy a xy xy
— •— — ; and therefore, by integration, Ixy zz — Ix ±:, lb.
Let the equation be yxx zz x^yy 4- y^y X vO* "" y^j' ^ divide it by j, and
it will be XX = x'y + y'-y X s/y — yy, that is, xx -\- yy zz x^y ^ y'^y y, ^y,
XX + yy
XX -^ yy
And dividing again by xx -\- yyj it will be ^^ ^-^-^ = ys^y. And therefore.
by integration, l\/xx -j- yy =. I^"^ ± b,
5 " 10, From
256 ANALYTICAL INSTITUTIONS. BOOK IV,
10, From § 31, 32, of the faid Book III, we may gather, that any formula
compofed of one variable only, if it be the produfb of any complicate quantities
raifed to a pofitive or negative power, integer or fradion, into the exa(5t differ-
ential, or into a proportional of the differential of the terms of the quantity ;
it will always be integrable. And the integral will be the fame quantity, the
exponent of which will be that as at firfi:, but incrcafed by unity, and multiplied
into the fame exponent fo increafed, but taken inverfely : Or, which is the fame
thing, divided, by it j or elfe this integral (liall be proportional to it. Never-
thelefs the rule obtains when the differential formula are hkewife compofed of
two variables and their differentials promifcuoully, provided they have the
condition required.
Thus, the integral of x + y x \/x -^ y =: z, (where z is any how given-
by X or by y,) will be 4 x x+y)'^ = 2; ± ^. The integral of t^* -h ty X
\/x + y =■ z will be t X I- X x -\- y^'^ ^ z ± by that is, f X x +/)* = z
-f- k The integral of ^iJ:^4±:J^±l!^ - z> will be s/WTYp =
z ± b.
The integral of xy + yx -\- zyy x b x ^y + yy^ '« =: z will be — ^ — X
m + n
^^i+]yUr - .z ± b. The integral of ''^ +-^^ + ^1^ = z, will be
h X -YA- -{-yy] m
m—n
m
^ ""-^ ^^ zz z ±: bi And fo of inSnite others of the like kind.
m — n X b
But fome equations of this kind will firft have need of fome preparation*
Let the equation be xxx + xyy + yyx ~ z, (where z is any how given by Xj)
I multiply it by x, and it will be x^x + x^yy + xy'^x = xz, or xx X xx -^ yy
•{■ XX X yy -zz xz, which has not yet the neceffary condition. But it would
have it if yy were alfo multiplied into yy -, therefore I add to each member the
term y^, and it will be xx x xx -\- yy -}• yy X xx -t- y^ =z xz + y^y, that
is, XX ^ yy x xx -Y yy "=■ xz •{• y^y, which is capable of integration, and it's
gral is -^ X »?* + yV z: ^^y* ± A + f^z.
mte
But it is not; always eafy to perceive, what quantities are to be added or
fubtraftcd, or what other alterations muft be made in the equations, that they
may be brought under the foregoing method ; efpecially when the equations are
fomethi ng compounded. In this way, to arrive at a folution is rather the work.
of
SECT. II. ANALYTICAL INSTITtTTlONS. 2^7
of chance than of art. In fuch cafes, therefore, we miiH: have recourfe to the
Methods of Separation of the Indeterminaies, which (hall now follow.
SECT. II.
X)J the Confiru^ion of DiffererJial Equations, hy a Separation of the Indeterminates,
11. The Separation of the Indeterminates in fome equations, although but
few, may be performed by the firft operations only of the common Algebra.
Such would be the equation x'^x^ -f xyxy — a^y"", in which I obferve, that the
firfl: member is a formula of an affedted quadratick, which would be made a
complete fquare if the term ^^^ were added to it. Therefore I add this quan-
tity on each fide, and the equation will be xxxx + xyxy + ^yyyy n aayy
+ i^yyyy* And extrading the root, it will be xx + ^yy zr y\^^yy 4- aa, in
which the variables are feparated, and therefore, by integration, ^xx -f- ^yy z:
fy\/aa + ^^yy ± b. The integral of the fecond member depends on the
quadrature of the hyperbola.
12. But moft frequently it will be convenient to make ufe of fubrtitutions.
Let the equation be aax zr xxy + 2xyy + yyy. Make x + y -zz Zy aflliming
z as a new indeterminate ; and therefore x -\- y zz z, and xx -f- 2xy + yy
zz zz. Then making the fubftitutions, it will be aaz — aay zz zzy, that is,
— — — =_y> an equation in which the variables are feparate. The integration
of the firfl member depends on the rectification of the circle.
Let the equation be xy -^ yx x \/a* — xxyy z=z xx + yy — ^
's/ XX ■\'yy X "y XX -\-jy
Here I obferve in the firfl member, that th^ integral of ^ + yx is xy^ and that
the fquare of this integral is found exadly in the quantity \/a* — xxyy ; there-
fore, if I put xy zz z, in the firft member the variables will be feparated, and
it will be z\/a'^ — zz. I obferve further, that, in the fecond member, the
^ • ■ sex II I ■ -uv •
integral o\ xx •\' yy is ^ , and that the quantities in the denominator are
2
like to this integral. Therefore, by the fubftitution xx ■\' yy zz 2p, the inde-
terminates of the fecond member will alfo be feparated^ and the equation will
be z\/a^ — zz zz f — ,
Vol. II. L 1 Let
2^B ANALYTICAL INSTITUTION S4 BOOK IV,
Let the equation be ^^^^ ~./"^ = z, (where z is any how given by .v or y ;)
the integral of xy — yx will be had, if we divide by xx, and it will be — .
Let us fuppofe, then, ~ z= — , and therefore "^^ ""^'^ = -^, and ^"^^ ~ y^
r= — , and 2xy — lyx = £f^^ Making, therefore, the fubftitutions, it
will be ^'^^ - zz 2;, and dividing the numerator and denominator of
a Y. XX — ixy -^ yy
the firft member by xXy it will be — = ' z. But it was put
« X I — V —
.*■ XX
— = -^- , and ^ r: — ; therefore it will be -■ ''•^ , ^^ =: z. And, be*
X a XX aa aa — lap + pp
eaufe the integral of this equation is algebraical, I will go on to the integration*
Make, therefore, a -^p zz q, and it will be — — = ir, and by integration^^
— ±h z=: z. But q/zi a — p, and p := -^ ; therefore it is ^ t= a^ - ay ^
Now, reftoring this value, it will be — h ^ = z, which is the curve be-
o ' X -~ y "^
longing to the differential equation propofed. If, inftead of making a — p
zn q, I had made p — a zz q, another integral would have been found, but
differing from this only in the figns.
13. The above equation gives me an occafion of making an ufeful obferva-
tlon J which is, that fometimes curves do not only change their nature by taking
their integrals, either fimply or with the addition of conftants, which has beea
already obferved from the firft original of infinitefimal quantities j but fome-
times alfo prefent us with fuch formula, as admit of integrations which are
really different, and fupply us with curves of various kinds, even without ther
addition of any conftant quantity ; which is a matter deferving confideration.
By means of the fuppofition — zz -^ , the equation - ^^ "^f^ - = i^ is pre*,
X a X — y]
fently integrated, and the integration is found to be _^f_ zz 2, omitting the
X — y
conftant. Now I make the fuppofition of — zz — , and attempt the inte-
gration. It will be, therefore, •^'^ ~ ^ zz -^ , and thence 2xy — 2yx zz
•" ^ • And, by fubftitution, the equation will be — ■■■ " — = Zm,
XX 2X
a X + 1
yy y
But -^ = -|- ; therefore -^--~ifl — = z. And making p -^ a zz q, it
J. fi pp — zap + aa ° * ■*
will
SECT. II. ANALYTICAl* INSTITUTIONS'.
259
will be ^ zz X ; and, by integration, if. == 2. Now, reftoring the va-
lues, it is -^ = z, the integral of the propofed differential equation, which
is different from the firft.
Another integral of the propofed formula, different from the two firfl:, is
1±^ = z. For, by differencing, it is ^^-.y^ + ^v -ly - -- -j>'^+^ + ^
=z %^ and ftriking out the terms that deftroy one another, it is ''^' "^^ '1 — z
X —y\'' *
which is the equation at firfl propofed.
Make z =: y, and the propofed equation is ^^-^ "" ^{ - =z y. If I make ufc
of the fecond integral found above, there arifes the equation — ^ =: y^ and
K '- y
therefore 1 + y iz x, which is a locus to a triangle. Then, if I make ufe of
the firfl, and of the third integral, by putting — ^ z= _y, or "—^ = _y, the
curve will be of the fecond degree.
In general, let it be ^'Z-Z^J =: y^y. The firft and the third integration
being performed, the curve thence arifing will afcend to a degree denoted by
m + 2, li m be 3. pofitive number. But, making ufe of the fecond, the curve
will flop one degree fliort.
14. But, however, the'method of fubftitutions is neverthelefs univerfal the
greateft difficulty of which is, that it is often very hard to know what fubfti-
tutions ought to be made, that we may not work by chance, and beftow much
labour unfuccefsfully. However, we fhall proceed with the greateft fecurity in
all fuch equations, in which the fum of the exponents of the variable quantities
is the fame in every term, and the feparation of the indeterminates will alwavs
fucceed. It matters not that thefe equations are affedied by radicals, or byfradions
orby feries, and that the co-efficients and fignsare of any kind. The fubflitution
to be made in all thefe equations will be, by putting one of the variables equal
to the produd of the other into a new variable, fo that, if the equation be given
by X and y, we muft make x zz -^^ or elfe ^ = ~ , (where by the denomi-
nator a is underftood any conftant quantity at pleafure,) and therefore y as
iii-^; and, making the fubftitutions, we fliall arrive at another equation
which will always be divifible by as high a power of the indeterminate x, as
was the fum of the exponents of x and y in every term of the propofed equa-
L 1 2 tion.
26o ANALYTICAL INSTITUTIONS. BOOK IV,
tion. Wherefore, making the divifion, the letter x will not exceed the firft
power, and will always be multiplied by z ; whence the equation will be Co
reduced, that on one fide there will be -^, and on the other fide z, with only
X
the fun(5lions of z ; and thus the variables will be feparated. For, calling A
all thofe terms which are multiplied into y, and B thofe which are multiplied
into Xy the equation will be Ay ~ Bx, and A and B will be given promif-
cuoufly by x and y. Now, becaufe the dimenfions of the letter jy, together
with the dimenfions of the letter x, in every term make the fame number ; if.
inftead of j, we put — , ifwill follow from thence, that in every term of the
quantities A, B, the letter x will have the fame dimenfion which, at firfl:,
X and y had together. Whence, if this dimenfion be called », the equation
will be divifible by x", there only remaining z, a, j), x» Let it be fuppofed,
that after the fubftitution of -^ , and after the divifion by ^", that which
remains in the quantity A may be called C, and that which remains in the
quantity B may be called D ; the equation will be Cy 2= Dx, and C and D
will be given by z and by conftants. But y ss ; ther&fore the equa-
tion will be ^^ — — = Dx, that is, Dax — Czx = Cxz, and therefore
a
~ =. 7- — %r-. And thus the indeterminates, with their- differentials, will be
fe Da — Ca
feparated, and the equation will be conflrudlible, at leaft by quadratures.
It is indifferent whether you put y — ~, or .v rz — ; for, in either of the
two ways, the Indeterminates will always be feparated. But fometimes one
fubftitution will give a more fimple equation, and of fewer terms, than the
other, and the conftrudion will be more eafy and elegant. Wherefore it will
not be ainifs to try them both, and, a; lail, to make choice of that which
fucceeds beft.
EXAMPLE!.
Let the equation be xxy = yyx + xyx. Make ^ = — ■ , and therefore y =3
^--11 — . Makme the fubltitutions, it will be z: .
a ° <* aa a
And reducing to a common denominator, and dividing by xXy it will be
axz + azx zz zzx H- azx ; that is, axz = zzx, or — s: — - .
^iV 25J5
EX.
SfiCT. II»
ANALYTICAL INSTITUTIO N^S. 26l
EXAMPLE II.
Let the equation be xxy = yyx + xx>:. Putting y z:l ^ , it will be
y — — - — . And, making the lubliitutions, it will be — ^^
4- x^'x. And, reducing to a common denominator, and dividing by xxj it will
be axz + azx = zzx + ^^;r, that is, zzx — «z;c' + aax r= ^.y2;, and there-
fore — = — . Now, making another fubaitution, x rr ^'^-, it will
X zz — az -i- aa " "
be x = ^t±JL, and therefore ^^ = ^-1^^ + -^i^^-^'^^ ; and, di-
viding by yy, it is- ^/)/j)> = ^^j/*' + aapy + jy/>p/ + ^^_>"', that is, appy — aapy
^py = aayp + yppp ; and therefore -^ = ^^^.^ tf- p- '
EXAMPLE IIL
Let the equation be ys/xx -\- yy =: jyx. Make y = ~, and y iz "^^^ — ^3
and, making the fubftitutions, it will be fi±fi x ^^"^^ "^ ^^-^^ =: ^ , that
a a
is, xxz + ZAJ';c' X \/ aa 4- zz = ^zxx, and, dividing by x^ it will be
xz"^ aa + Z2; 4- zxy/aa + »2 = <?zx, or xz^/aa -\- zz zz azx — zx^/ aa 4- z^g
Therefore aa _j± — zz — . If I had made x zz ^, I fhould have had
az -- z '^ aa 4- as *" ^
this equation, — =
Vfla -v pp — p
15. But fometimes the differentials themfelves, x and j, afcend to higher
dimenfions, the condition mentioned before being, however, in the equations.
In which cafes, the fubftitution of — , inftead of j, being made as before, not
meddling v.'ith y at prefent, will make every term of the equation divifible by
the
263 ANALYTICAL INSTITUTIONS. BOOK IV.
the fame power of x, apd there will remain In the equation only 2, x, and y,
with the conftants given or afllimed, but not x. Now, becaufe, inflead of y,
we mufl put ^ — ^ , by which the letter x will again be introduced ; make
—^ = /, and, inflead of j), write ^ — —, and the equation will have only
Sj /, X, with conftant quantities given or aflumed, but na longer x. Now, if
we make a . u '.'. x , t , and if, inftead of /, we put every where — , we (hall
have an equation free from differential quantities, in which will be only u, z,
and conflants, for an algebraical curve. By means of this curve, we may find
the real values of u. Let there be, therefore. A, B, C, &:c. fo that it may be
u zz A, « zz B, a = C, &c. and A, B, &c. will be given only by z, and by
conftants, and it will he x — -^ , x zz ~, &c. ; and therefore / = — will
be ^ zz -J-, AT zz ^ , &c. ; whence, laflly, ~ = -^ » ~ ^ ~^» ^^*> ^"d
the logarithms of ;t' will be diredly proportional to the fpaces comprehended by
the curves, of which, the abfciffes being z, the ordinates will be reciprocally
proportional to the values of the quantity ti before found. And the curves
fatisfying the purpofe will be fo many, as are the real values (different from each
other) of the letter u j ftill obferving, that the adding of a conftant quantity in
the integration of the equations ~ zz ~, ~ n -^j &c, may again diver-
iify the curves that fatisfy the demand, and will often double their number.
Then Ix will be equal to the area of that curve, which has z for it's abfcifs, and
-J- , -Tjr- , &c. for it's ordinate 5 that is, it will be equal to the integral of
-J-, -^ , &c. Wherefore, taking z at pleafure, the logarithm of ;ii: will be
given, and confequently the correfponding ordinate x in the logarithmic will be
given alfo. Then, v being given, by means of the equation y zz ■ — will y be
given alfo, that is, both the co-ordinates of the differential equation propofed,
or of the curve required. Then, in reference to the different values which will
be given to z, fo will be the different points alfo, which will be found in the
fame curve required.
I {hall apply the rule to an example. Let the equation be xxyy + xyxy =z
xxxx. Make, therefore,^ zz — , and, putting this value in the equation,
inftead of y, we fimll have axy* + x^zxy zz ax^^x*, and dividing by xx, it will
be ^y^ + 2^7 =: ax""* Here we fee, that x and it's fundions entirely difappear,
there
SECT. II» ANALYTICAL IKSTITUTIONS, 26^
there remaining only z, x, y, with their funclions. But, becaufe, by fubfti-
tuting, inftead of j/,, it's value ^^^-^-— -» we fliall again introduce x into the
equation 3 make — = /, and therefore j> zz —^ — , and the equation will
be "'^^^ + ^^^"^ +^^''' -f ''^^ "^ ^''"' = ^;^i', that is, CLZZxx + 3^2;^/ + aalt
a «
rz aaxx\ in which only enter 2, ^, i, with their funftions. Again, fuppofing
/ — — , and making the fubftitution, we fhall arrive at an expreffion which
a
is purely algebraical, 222 + yz.u + uu zz aa, fo that we fliall have the value
of u given algebraically by z and conftant quantities. But / ~ j^ __ *x_ ^
whence — =1 ~ , in which equation, « being given by z, the variables will
be feparated. Therefore the curve being defcribed, of which the abfcifles
are 2,. and the ordinates reciprocally proportional to the values of « ^ we ftiall
have X, and thence y^ by making the fubftitution of
xz
a
16. Now, from this and other examples, it will fucceed alfo, without making
life of this method, that they may eafily be reduced by the method of § 14.
And, indeed, if to each of the members of the aforefaid equation, xxjy + xyxy
zi xxxx, there be added the fquare ^yyxx, it will be xxjj + xyxy + ^yyxx
= xxxx + "^yyxx, and extrading the root, xy 4- iyx zi x\/xx + -^yy} where
now it is reduced to the aforefaid general method of § 14. Or elfe, tranfpofing
the term xyxjf, and adding the fquare "^yyyyy it will be xxjjr + -^yyyy zz xxxx
— xyxy + -iyyyy ; and, extrafling the root, it is ys/xx -i- -^yy = xx — j;yj^
now reduced to the fame method,
1 7. Equations which contain differentials mixed together, and raifed to any
power, may not only be conftrudled in the cafe confidered at § 15, which
fuppofes the fum of the exponents of the variables to be equal in every term ;
but, in general, in what manner foever thofe equations are, provided one of the
two indeterminates, x or y, be abfent. This is done by making x zz — , if
x be wanting, or > = ~, if j^ be wanting j 2 being a new indeterminate, and
a any conftant quantity. For, by fuch a fubftitution in the propofed equation,
of — inftead of x, it is plain that another will arife, which will be divifiblc
by the power of i ; fo that it will be compofed of finite quantities only, and
3 therefore
a64
ANALYTICAL INSTITUTIONS.
BOOK IV.
therefore will have 2, given by y and conftants only, and the relation ot y to z
will be exprefled by an equation, or an algebraical curve. Therefore, in the
equation x = -^ , inftead of ^, putting the value that will be derived from
fuch algebraical equation, we fliall have the variables feparated.
EXAMPLE I.
Let the equation be yy^x — ax^ + lax'^y'- + ay*. Make i n ^ • and,
making the fubftitutions, inftead oi x and it's powers, we fliall have the equation
^ _ ^ _^ 2^ _^ .,. ^^^^ dividing by y\ it will be -5L = 4. + !£.
^ z' aa \ ' 3ZZZ , • aa'z. _,, _
+ ^. Or y rr ~ + 2z + — > and jK = ^ -f 22; — — - . Therefore
2L — X - ^ + i^ ^ . If we go on to the integration, it will be
a a^ a a
X z=. ^ -{- ^ Iz, taking the logarithm from the logarithmic with the
fubtangent — a. Whence we have the values of the two co ordinates x and y
of the propofed differential equation, by means of two curves, which have z
for a common indeterminate. Now, &s to the conftrudUon, we may proceed
thus.
Fig, 142.
Taking the abfcifles in the axis <^,
defcribe the curve DAH of the equation
y zz ~ — H 22 4 , and the curve RIK
•' aa 2;
of the equation a? = ~ 4. ~ — /z,
^ 4«3 a
Then EH =: ^, and EK = x, will be the
cO'Ordinates of the propofed differential
curve ; by the conftru<5lion of which, mak-
ing CM parallel to EK, then KM is pro-
duced to N, whence it will always be
MN = EH ; and the curve NBN will be
that required.
EX.
SECT. Ilk
ANALYTICAL I NSTI T U T I O N S.
Si6.S
EXAMPLE" IL
. Let the equation be /i* + aayjx* =: a^yK Make ^i- = -^ ; and, making
the fubftitutions, we Ihall have —f^ + fff^ ~ a'j/'. And, dividing by;/',
it will be «'/ + a^z^y = <2'. Therefore z will be given only by y and con-
ftants, and therefore, in the equation x zz ^ , the variables arc feparated.
^'ir*^43' Now, to have the curve of the propofed
differential equation j to the axis CE let there
be defcribed the curve IK of the equation*
zy + tfVjT r= «% it being CM = y, and
MK z= z. In KM, produced, take MN"
equal to the area CMKI, divided by a. Then;
will it be MN = /— = *, and. the point
N will be in the curve. .
1 8. The method of ^ 14. may be rendered flill more general, by tran^
forming the equations which have not the condition required, of the fum of
the exponents being equal, into others which ih all have thofe fums equal, and
coniequently ihall come under the rule of that article. This may be done two
ways. One will be, to make ufe of convenient fubftitutions, for which there
can be no rule, and it muft be by examples alone that this artifice can be
acquired. The other is, by changing the exponents of the propofed formula .
or equation, that it may be determined, at kaft, in what cafes, and with what
fubftitutions it may fucceed, to transform the equation into one equivalent to -
it,. in which the condition required may be found. Thus, though the (epa-
ration of the variables cannot be univerfally performed, yet infinite cafes may,
be aifigned, in which that feparation will be effected.
Vol, II,
Mm
EX.
-9106 ANALYTICAL INSTITUTIONS, BOOK IV%
EXAMPLE I.
Kow, as to the firft manner. Let the equation be x\/aaxx -f- az^ — zzz,
which has not the neceffary condition. Make z^ = ajy, and, taking the
fluxions, zzz ±z r^yj- Therefore, making the fubftitutions, x\^aaxx + aayy
rn ^ayy ; an expieflion that may be managed by the method of § 14. Wc
may alfo have our defire, by putting \/aaxx + az^ n <?«, and therefore
aaxx + az' zz aauu^ and, by difFerencing, Q.aax>c + ^azzz zz zaauliy that is,
zzz =: ^auh — *axx j and, making the fubftitutions, it is ux zz """ i ^ *
-«!
EXAMPLE II.
Let the equation be x^x + f.^ =; y. Make y^a +y = «, and therc-
fore a -^ y zz zz, and y zz 2zz* And, by fubftitution, x^x + 2xxz z= zzz.
But this ftill requires a little further reduction. Therefore make xx zz u, or
U^ zz uu, and 4;^'^ = 2uu - whence, thefe values being fubftituted, it will be
finally ^m + 2uz = izz, &c.
19. I (hall go on to the fecond manner of altering the exponents, and
therefore I (hall take a general equation of three terms, ay^x^x + by^ifx -f-
cxyy = o J in which the figns may be as we pleafe, cither pofitive or negative.
If it were n-\-m'ziq-\-'pzzr\-5, it would be the cafe of § 14. But,
fuppofing fuch an equality (hould not be found between the fums of the
exponents ; make ^ =: 2 , whence y zz tz^^z, y zz z ^ ^ zz. z^y y" z:z z ,
and making the neceffary fubftitutions in the propofed equation, it will be
az^sTx + hT^ifx -f" tcxz sj = o. But, by the condition of the afore-
faid § 14, it is neceffary that it fliould be «/ + ;» = j/ + ^ :s r + j/
-f- / — I. From the firft equation, therefore, nt + m ^ qt + p, we muft
derive the value of the affumcd exponent / = ^-3^ 9 which, being fubftituted
in
SECT. II, ANALYTICAL INSTITUTIONS. 267
in the fecond, ^/4-p = r-{-j/ + / — i, orj--^4-i X/=:p — r + i,
will give J — q + I X/» — »»=^ — ^+1 X» — ^; which is the con-
dition that the exponents of the propofed equation ought to have. To verify
which, it will always be reducible by the rule of § 14; and the fubftitution
J> — m
to be made will be y zz z^-g.
Inflead of making y = z, if I had made a: = z'', I (hould have found the
fame condition to be verified in the exponents, but it would have been / =
«-y
** *" ^ , and therefore the fubftitution to be made is x =: z^ '*.
f — m .
p—m
It may happen, that the fubftitution of y ■= 2»-? may become impoffible,
that is, when p zz tn, or n s=z q. But it may be obferved, that, in thefe cafes,,
the indeterminates are feparable without need of redu(5tion.
In the canonical equation ay\^x + hy^scx + cxyy =r o, if, befides the
fuppofition of ^ — 2', we fhall alfo make x ■=. u^ ; making all the fubftitu.
tions, we (hall find awz u w + bwz^ u ^ u + ctu z z=zo.
By the comparifon of the exponents of the firft and fecond terms, we (hould
have nt 4- wm 4- «; — i z= qt -^ wp -^ w — i, that is, / = w x ~ *" .
From the comparifon of thofe of the fecond and third, we (liall have wr + st-
+ i — I =: qt + wp -j- w ^ J, or t X s — q + i :=. w x p — r+i.
And, inftead of /, putting it's value, w x p — »» X^— ^ + 1 =wx
n '- q X p —- r + i, which is the condition the exponents of the propofed
equation ought to have. But the letter iv vanilhes out of the condition;.
therefore the fecond fubftitution of x zz u is altogether fuperfluous j whence-
it may be inferred, that all the formulas, in general, cannot be reduced to the
rule of § 14, but only fuch, in which the condition /> — w x J — ^ -h i
zz n — q X p—" r + I may be verified. The fame thing is to be concluded
of others, when compounded of a greater number of terms, which I thail now
proceed to treat of.
20. As the number of terms increafes beyond three, (o, in like manner, the.
number of conditions increafes, which the exponents of the eq^uation muft have,
M m a ia
268 ANALYTICAL INSTI TV T I N S, BOOK. IV.
ia^oi'der to be reducible by the method of § 14 . I will take this canonical
equation of four terms, aiTfx + hx^ y^x + cxy'y + dxy'y = o. Putting
jr — 2;,j) = /2 ""'i;, and making the fubflitutions, it is az^x^y + bz^*xtx
+ tcx z z -\- aix z z "=2 o. Therefore it ought to be «/ + w =r
qi -h p. Whence we mfy derive the value of the aflumed exponent / = ^^.
Alfoj it ought to be r + J/ 4- / — 1 r: ^/ + p, or J/ — ^/ -f- ^ rr jp — r
H- I ; and, fubftituting the value of /, it will be j — ^ -f i x p — m •=.
/» — r + I X n — qt the firfl: condition. And, befides, it ought to be
e -\- iu -{- t — I zz qt + p, or iu — qt -{- t = p — t -h i, and, fubftituting
the value of /, a — ^ -1- i x /> — »» = ^ — ^ + i x » — ?» the fecond
condition. If, therefore, the exponents of a propofed equation fliall be fuch,
as that both thefe conditions fliall be found therein, it will be reducible to the
cafe of § 14, and the fubftitution to be made will be y ^ 2;«— f .
If the equations fliall have five terms, the conditions to be verified witi be
three ; and fo on to more terms.
EXAMPLE.
Let the equation be ay'^xx + hyyx^x =z cxy. This, being compared with
the canonical equation, will give ^ = 3, w=i, ^ = 2, /)=:^, >'=i,
J — o. And, becauie, in the prefent cafe, the condition is verified of
J — J + I X p ^m zzp — r+i x« — ^ giving — iX— t = i-Xi*
which is true; the equation will be reducible to the method of § 14, and the
fubflitution to be made will ht y zz zn~q r= z . Therefore I make y r=:
z *> J/ = — iz ^2;, >' = z *, y = 2 ; and, making the fubftitutions,
I find az *i* -\' bz **i = — -^cxz ^z; which is now reduced to the cafe
of the faid article.
21. But^
8
rBCT. lit AKALYTICAL INSTITUTIONS. 269
21. But, without applying particular equations to canonical ones, perhaps it
may be more commodious to manage them by this method only.
EXAMPLE I.
Let tlie equation be ay'^a ^ x — bx^y y =: cx^yy* Make jc z= zr, x rr
iz^^Zi making the fubftitutions, it will be iay'^z' z — bz^^y^^y zz
cz^yy. But it ought to be ^ + Y' ^- ^ — i = 3' — i> whence I obtain
/ = 2, which, being put inftead of /, gives me this equation laf^z 3 z —
bz^y^^y =: cz^yy, which is juft the cafe of § 14. Therefore the fubflitution
10 be made, x = 2*.
EXAMPLE II,
Let the equation be x^x -^ yx + x^yy = y^y. Put y zz z', y :=: tz^^z,
and, making the fubftitutions, it will be x*x -^ z^ x + tx'^z ^^z = tz^^^z.
But it ought to be t = |/, whence I have / = 4 > which value, being put
inftead of /, gives me the equation x^x -{• z^x •\- ^x^z 'z = ^z^z, which
is juft the cafe of § 14. Therefore the fubftitution to be made is j^ = zK
EXAMPLE III.
Let the equation be <i)>V.v + ^jf + cyxx + dx^y'^jf == o. Put^ =: 2,
y •=! tz^^Z'y making the fubftitutions, it will be az*x*x + ^^ + czxx +
tdic*^*''^z = o. Now it o*ight to be 2f + 2 =: / -f- I, whence ^ = — i ;
and.
270 ANALYTICAL INSTITUTIONS. BOOK IV.
and, putting this inftead of /, gives me the equation ^^ ^ l,x + — ^ -^
=: o, which is the cafe of § 14. The fubftitution to be made is ^ s — ^ .
22. The method of § 14 being thus made more general, I (hall proceed to
another, which is alfo general in it's kind. This comprehends all thofe equa-
tions, in which neither the indeterminates, nor their differentials, exceed the
firft dimenfion*
Wherefore let the general differential equation, which includes all pofhble
cafes wherein the variables and their fluxions do not afcend beyond one dimen-
fion, be axx + ^y + cyx + gxy +/v + i>y = o. The co-efficients a^ b,
c, &c. may be pofitive, or negative, or nothing, as the circumflances of the
particular equation may require, which is propofed to be conftruded. As to
this equation, I obferve, in the firft place, that, if it fhall bt e — g^ both of
them being pofitive, or both negative, the equation may be integrated. For
then it will ht ^^ c x yx -{■ xy zr — axx — hyy ^ fx — hy, and, by inte-
gration, ^i cxy -:=. — \axx — '\hyy — fx — by. But, it not being ^ zz ^, I
make ;; = p -f- A, j n ^ -J- B, where ^ and q are two new indeterminates,
and A and B are arbitrary conftants, to be determined as the fequel may
require. It will be then ;c == /, j/ n ^, xic r: />/j + A/>, yy •=z qq -{■ B^.
Thefe values being fubftituted in the principal equation propofed, there will,
arife this following.
ap^ + (i^P + bqq + h^q + cq^ + gpi = O.
+ fP\ + i^
.bii!v/ • A zjli y,
In this equation, if the fecond and fourth terms be made to vanilh, this
will be the cafe of § 14 ; and we fhall know how to feparate the indeterminates.
But the fecond term will vanifh, if it be made « A 4- r B -f / z: o, and the
fourth, if it be /'B + ^A + Z* =r o. Whence, from thefe two equations, the
values of the aflumed quantities A and B wjll, be determined, fo as that the
new equation will be a cafe of the aforefaid § 14. Then it will be A =
JlilrZ , B = ^^4-^'. that i,, A = '1^ . B = '±^. If, there-
tf ' ' ' eg — ab eg — ab ^,
fore, we make the fubflitutions oi x •=. p + ■' " '^\ ^ and of v :s ^ + "^ , ,
' '^ eg ^ ab ^ i * cg — aV
an equation will arife, whrch may be managed by the method of § 14.
If it fhould happen, in a particular equation, that it fhould be ^ = ch, or
th ^ fgi fo that either of the aflumed conftants fhould be nothing; it would
be
SECT, II. a^NALYTICAL INSTITUTIONS. Zfi
be a fure token, that we might obtain our defire by one fubftitution only.
For example-fake, let ^ __ ^^ = A = o. In this cafe, omitting the quan-
tity X with it's fluxion, it will be enough to fubllitute ^ + B inftead of ^, and
to proceed in the manner above explained.
Now, if both the quantities A and B fliould be nothing, in this hypothefis
we fliouId have l^f = cB, and ah = fg\ and confequently -^ = — =: /.
Then eg =z ah^ by which we fliould no longer have any need of thefe fubfti-
tutions. Therefore, as often as it is eg r: ah^ make the fubftitution ax + cy
— z, and take y Jfnd y out of the equation. It will be then y = LZ^, y =
y "" ^ . Make thefe fubfl;itutions in the principal equation, and we fliall have
axx + .- \- zx — axx + V fx ^
CC C '' c
-zz o. That is, fl:riking out the firft and feventh terms, and, reducing all to a
common denominator, hzx — abxz — ahzx + aabxx + cczx + cgxz — acgxx
4- ccfx + ehz — achx = o. But, becaufe gc zz ah, the fecond term will
defliroy the lixtb, and the fourth the feventh, fd that there will remain only
bzz — ahzx + £czx -f ufx + chz = achx^ or x = ^z + cjz
aliz — cez — ce/' + ach
EXAMPLE I.
Let the equation be axx + layx + hxy — ahy = o. Make a? = ^ + A,
y '=' q -^ B, X ssp, y =: ^; and, making the fubfl:itutions, the equation
will be
app + aAp + la^ + bp^ + bA<i = o.
+ 2«B/> — «^^
The laft term will vanifli if it be ^A -?- tfB = o, or A = a. The fecond
will vanifli if it be zaB + aA. = o, or B = — fa. Therefore the
fubftitutions are * = /> + a, and y =^ q — ia ; and the equation will be
ccduced to the cafe of § 14,
The aforefaid terms vanifliing out of the equation, it may be integrated by
iiieans of § 4, without having recourfe to § 24,
EX-
aya analytical iNSTixuxxaNs, book iv»
EXAMPLE II,
Let the equation be 2axx — 2lyy — j^ayx ^- i^xy — aax -= o. In this the
co-efficient 2a correfponds with a in the canonical equicion, — 2/^ with ^,
— 4.a with t, h with j? ; and gives us the cafe, that it is eg = ai^, in refpe(S
to the eonflants of the canonical equation. Therefore I make the fubftitution
2ax — » • zax — »
2ax — 40^ = z, and therefore y = , y = ; wherefore, eliminate
ing y and y^ we Ihall have laxx — ^ — r^-i laxx +
ayf 4. ^^ -y'^ - -*g — ^^^ _, Q 'j'jj^j 1 5^ Aahzx — 2^zz + iSaazx — i6tf*,v
. __ 2^22;
aj. Equations of this kind, as alfo thoTe of a higher degree, may be thus
managed by the help of one, but a more compounded fubftitution. I refumc
the canonical equation above, axx + lyy + cyX -^ gxy + fx + by z=z o,
becanfe thofe of higher degrees would involve us in too long calculations-; and
what I fhall fay concerning this, will be fufficient to (how us how thofe others
are- to be treated. Therefore 1 make a; = Ay + /> + B, in which fubfidiary>
equation p is a new indeterminate, which has no conftant prefixed to it, becaufc
that would be unnecefTary, as the operation will fliow. A and B are two-
conftants, to be determined as occafiOn may require. Making, then, x = Ajr
+ /> + B, it will be :v = Aj) + />, xx ■=■ AAyy -i- Afy + ABy + Ayp
-H pp + Bp i fo that, thefe values being fubftituted in the canonical equation,,
it will be transformed into this following.
a AAyy + aAfy + a Ayp + app + aAEy
+ hy + gpy + (yp + g^y
+ cAyy + JAy
+ gAyy + hy
+ fp K
r
=: O..
Now we muft contrive to make fome of the terms of this equation to vanifh,
by conveniently determining the afTumed arbitrary qtiantities A and B, and to
make it capable of the end propofed j when fome of the conditions are to be
verified, which arife from the values of A and B. If, therefore, the lecond and
third terms could.be deltroyed, the variables would be feparated, and the
equation would become integrable. But, that thefe two terms may becon^e
nothing, it is neceifary that it be <iA + ^ =: o in rcfped of ^he fecond,
and
SECT. It, ANALYTICAL INSTITUTIONS. 1^^
and aA •{' c — o, in refped of the third ; and confeqiiently g ^ c. Bur,
fuppofing this, the principal equation will be already integrable, without the
help of any operation.
If the two lafl: terms were nothing, the equation would be reduced to the
canon of § 14. But, that they may vanifli, it will be neceflary that a^ A- f
=. Of or B zi — -^, in refped of the laft, and aAB + ^B + /A -f ^ rz o,
in refped of the fifth. Bur, fubftltuting the value of B, it will be — A/
— — + A/ + /^ =z o", that is, a/j =. gf. Therefore the laft two terms
cannot be made to vanifli, fo that by them the equation may be reduced,
except in the particular cafe, in which is verified the condition of ah zr gf.
If we endeavour, then, to take away the firft and fifth terms, by which the
equation will be reduced to the cafe of § 4 and § 6 \ then, in refped of the
firft term, it will be MA + ^ + <:A + ^A = o, or A* + ^-i-^A z=. ~,
from whence we may deduce the value of A. This being found, the value of
B will be difcovered from the fifth term, and will be B zz — 7 . And
the new equation will become aA -h g x py + ^A -f- ^ x yp =: — app
— aBp — fp, which may be conftruded by means of § 4, if the co-efficients
of the two firft terms are both pofitive or negative ; but, by means of § 6,
if one be pofitive, and the other negative.
But, to obtain the fcparation required, it will be fufficient to make the firft
term of the fubfidiary equation to vanilh, by making it ^AA + cA -\- gA
+ Ip zz o. Now, putting the afl^umed conftant B — o, which, in this cafe,
will be unnecefiary, there will remain the equation — a^p — fp = aA -f- g,
X py + /A -\- b X y + aA -\- c X ypi in which the variables may be fepa-
rated by the method, which (hall be explained in the following article. Or
elfe, by the foregoing, with the help of an eafy preparation, that is, making
Aa + g X p +/A + b z=: q^ and taking the fluxion Aa + g x p :=^ ^'
Then, by fubftitution, — app — fp zz qy •{ V^ "^ ^ ^ * But we ought to
confider, that, in making ufe of fuch formula, very often imaginary quantities
will infinuate theiiifelves, arifing from the extradion of the root A out of the
affeded quadratick equation ^AA •{■ c -\- g X A -{- b zz o. And thefe will
not only obtrude themfelves into the co- efficients, but will often pafs from
thence into the exponents. And, becaufe as yet we have not found out the
. Vol. II. N n ways
274 ANALYTICAL INSTITUTIONS, JIOOK If,
ways of managing them, k is ncceffary to avoid them as much as poffible j
and, among various methods, to adhere to that which (hall be found mo(i
convenient.
•For an example ; let the equation be abxxx -f ihyxx + ahx + aahyj 4-
^a^xy zz o. Make y =1 Ax -\- p -{- B, whence y zz Ax + p. Here I choole
to fuhft'tute inQead of ^y rather than .v, becaufe I forefee the calculation will be
fliorter. Subftituting, therefore, we fiiall have this equation following.
ah^x + bifxx + l>bBxx + a^px -\- a^Bx + a^bAxp
bbAx^x + 2n^Axx + a'bA^X + a^bAEx + ah-p
4- a^bA^xx
+ a'bpp + a'bBp no.
Here I obferve, that, in this equation, if I make the firft, third, fifth, and
lixth terms to vani(h, we Qiould have the in determinates feparable; for it would
be bhpxX + a^px + a^bApx + a^bpp + a^hBp ^ o. And, dividing by ^,
IhxX + a^X + a^bAx = — d'bp — - — —^ Now, that the firfl may vanifli, it
p
is neceflary that .3 + ^A = o, or A = 7- . And, together with this will
alfo vanifh the fifth and fixth, without any condition arifing from thence.
That the third fhould vanilh, ' it is necelfary that bbB + 2^'A -{- aahAK — o.
And fubftituting the value of A, it is hbB ^ — f- ^ = o, that is, B i=4r.
b to ^ o*
Therefore the fubftitution will be ^ iz r — h /> + -77* » ^"^ ^^ equation
thence arifing will be hbxx zz. — aabp — ~- ,
24. The method of this article confifls, firfl:, in difpofing the propofed
equation in fuch manner, as that the fluxions may continue accompanied with
their indeterminates refpedively, and that a half-feparation (as I may fo fay)
may be made, by throwing into the common multipliers, or divifors, fuch
quantities as hinder the operation. Then taking the integrals of the differential
thus prepared, compounded of two variables, it muft be made equal to one
afibmed variable, and, by means of an auxiliary equation, it muft give a new
form to the principal equation. Laftly, taking obfervation by that which
fucceeds, the operation muft be repeated till the defired feparation is com«>
pleted, or till we fee the formula eludes all our endeavours.
This method has this advantage above the others, that in trying thefe fiibftitu-
tions, at the fame time it informs us, which will be fuccefsful and which ulelefs.
But it muft be obferved, that there are fome equations which will not admit of
8 the
SECT. II» ANALYTICAL INSTITUTIONS. 275
the artifice of the prefent method, unlefs they are firft prepared according to
art. The whole will be better underftood by the following Examples.
EXAMPLE L
Let this equation be propofed, " ^ ^ '^ =: x, in which ii.
XX + yy X V XX ■{■ yy — xxyy
ftands for any fundion of * or ^ whatever. 1 fet afide the denominator, which
is an affedion common to the two terms which compofe the firft part of the
equation, and the bare differential x^y + y^x will remain. I divide x by a;',
and y by y, and then it will be x^y •{■ yKx n x^ x ™ + -7- . Hence the
propofed equation will take this new form, " ^ JL jl. J-^
XX ■{- yy X ^xx-i-yy—xxjy x^ y*
= z. Having obtained this half.fcparation, in which the fluxions X, y, appear
combined fimply with the fundions of their variables x^, y^, and the other
terms conftitute, as it were, a foreign quantity, which has the appearance of a
multiplier $ I make -^ 4- -^ = — ' —j- , and then, by integration, ~ +
— =: p. Now, finding the value, fuppofe of <tf, which will be ^t = ^ ,
^yy >/zyyp-^a^
and fubftitutinsr this inftead of at, and ^ inftead of -4- + -4- in the
, <^' x^ y*
equation, it will be ^ , ^— ^ = z. Wherefore, &c.
It may be recolleded, that, taking a quantity at plcafure any how given by p,
<z' . •!! I tf' a^ tf' , . XV
as p = — -, It will be — 1= — ~ + — , that is, q zz - _ ■ - - ■ ;' bv which,
in an inftant, we may perceive infinite fubfiitutions, which will promote the
defired feparation of the variables. All the other paffible ones will be u(elefs,
and will leave the variables as much blended and intermixed as before.
Moreover, let it be obferved, that it often happens with the fubflitutions
here explained, that in one member of the equation there may remain fome
fondion of one of the variables .v or jy ; in which cafe, if z were given by the
variable whofe fundion remains, one fimple divifion would anfwer the purpofe.
N n a EX-
276 ANALYTXCALINBTXTVTIONI. BOOK IV.
EXAMPLE II.
Let the equation "be — '^ -^'^ =r 2^, in which z is any how given by v.
To reduce this equation to the method, I take the integral of the numerator of
the fraftion, that is, yy -f xy, and make it equal to p. Now, making x and x
to vanifh out of the equation, by fubftituting their values^ I fliall have a new
equation — = z, which is reduced to the following, yp — pz zz ayz.
y
And this, being prepared according to the method, will be found to be
f) X — =^ ^^' I make — — -^ ::r -~ » and therefore Ip — /—
^ p y ^ P y 1 r J y
zz Iq. I make alfoy — zi ulm, where Im is fome conftant logarithm. Then
it will be ip -^ Iq ^=. ulm. And going on from logarithmic quantities to expo-
nentials, it will be -^ = m , Therefore, in the reduced equation, making
the fubftitutions of — inftead of -^ ^ , and of tn*q inflead of p, it will
3 P y ^ ^
be w ^ = az, that is, ^ = — ; in which the variables are feparated, becaufe
m
both z and m^ are given by >
EXAMPLE IIF.
_ , . , 2XXX + xyy + yyx xx •\- yy t> /• ^ , ,
Let the equation be — ^ ^-^ ^ jj — /•:■. .. Before we attempt this
T ;»;♦ + xxyy + a'> ^ ^x + jy
formula, it will be beft to reduce it. I obferve that the fecond member is
integrable, and it^^s integral is ^^xx -t yy (§ 10). Wherefore I make ^xx -\-yy
z=. Xy and making y to vanilh, finding that it's powers afcend to the fquare,
and putting zz — xx inftead oi yy, and zz — xx inflead of jj/, we fliall have
the equation ^ r— 1 = ^> ^"^^ is, -7— r = z; which.
being
SECT, II, ANALYTICAL INSTITUTIONS, 277
being prepared as ufual, will be -r:rXT4 X xz ^-zx zz z, I make xz -k- xx
-zz p, and, by integration, xzzzp-, and, making x to vanirti, we (hall have
— -^^— - zz z^ and, finally, — - — - zz ~ .
EXAMPLE IV.
Let it be the laft equation of the foregoing article, — app — fp = aA -}- g
X Py + f^ -\- h X y + «A + f X yp, which I undertook to condrud.
This equation being prepared according to the method, and, for brevity,
making aA -^ g zz e, /A -^ h = m^ aA + £•=:», it will be reduced to this,
_ cjt±ft ^ ^ 1.j^JlL, Therefore I put i- + -^ = -^ ;
ep -^ m -^ y ep + m ^ y ep ■\- m q '
and, by integration, ly -\ 1 p •\- — z:z Iq. And therefore y z= •
T
m\ e
And eliminating ;', we (hall have — ^-itlJl = £ that is, — ^il±Jl
P+ —
ep-^m.
X p ■¥ — — f .
EXAMPLE V.
Let the equation be this already prepared, y Y. xx -Y yy "iz x x yx — xyy.
m — z ________
which I write in this manner, -^ „ x xx + yy = ^ — ^ , in order to make
V
the fecond member integrable. In this I make ufe of a double fubftitution,
and therefore 1 put xx -i- yy =z pp, and, by mtegration, xx + yy zz pp. I put
alfo — — — = q, and by integration, — = q. Making the fubftitutions, we
m — 2
fliall have —-r" X pp =: q. But yy = pp — xx, and xx =: qqyy^, fo that it
will
BOOK IV»
Wl — 2
^^8 ANALYTICAL INSTITUTIONS,
mW hcyy-pp - qi^yy, that Is, yy = 7^^ . and f*"^ =z ■— ^;n» ^"^
X* = — 2-^ • Wherefore, fubflituting thefe values of y^" and ;v", we
Ihall have p"''^'''^p = /^ X 7"+^ a
EXAMPLE VI.
Let the equation be ^ ^-^^ ::= 2; j in which i; is any how given by x or y.
I obferve that the numerator of the firft member is Integrable, if it were divided
by ATA", and that it's integral would be ~ , and therefore I thus difpofe the
equation, c=Lrr X ^^ ~ ^^^ = ~, Put -^ = />, whence it will be
2xy — iyx _. • ^^^j ^.^^ equation will be changed into this following, ~=rz zz
~ . But 2v := />j?, and yy = :J:/'/»A;;f 5 fo that, making the fubftiiutions, it
will be — ; '=■ — ; and, multiplying by xx, it is ^- = z,
in which the variables are feparated. I go on to the integration •, and therefore
it will be — ~ — V c zz fz •, and, reftoring the value of p, it is — - — ^- c
X
zz fZi and reducing to a common denominator, ic is _ ■ "" '^ ~ Jz* If
we make the conftant ^ = o, we fhall have __ =* fz ; and, making c =
— 2, it will be — ^ = fz, which is another integral of the propofed formula
different from the firft. Laftly, putting f = — ^> a third integral will arife,
"Lll^fz. .
X " y "* •
*5- The
SECT. Ilk ANALYTICAL INSTITUTION Si 279
25. The method I now undertake to explain, although much limited and
confined, is yet of great ufe in fome particular cafes. By this the variables
may be feparated in the canonical equation ay rr ypx + by'^qXy in which the
quantities /», ^, are to be underftood as any how given by x, Tlje quaatities
d, b, are conltant ; the figns may be pofitive or negative at pleafure, and the
exponent n may be integer, fradlion, pofitive, negative, or even nothing. In
this equation, then, make y zz z^, where x and u are two new variables j
and, by taking the fluxions, it will be j/ = zli + uz ; and, by fubitituting,
inflead of j/, y, and j«, their values z'u -f uz^ zw, -and u^z", we Qiall have the
equation a^li -}- auz = uzpx + bz^'u^qx, in which, if two terms Ihall vanifh,
the indeterminates will be feparated. To do which, let us feign an equation
between the two terms auz = uzpXj then ~ = fXy and, by integration,
ah zz.fpxi and, proceeding from logarithms to exponential quantities, it is<
•fJi
z = nr^^^ or 2; = w ** , fuppofing Im = i. This laft equation fhows us the
value of z, and informs us, that, to reduce the equation propofed to twc
terms only, and to caufe the other two to deflroy each other, inflead of y iz zu,,
we ought to put y — urn ^ ^ that is, — = w <* , or /y — lu =/— . And,.
by differencing, ^ _?!L = px, and therefore ay == ypx + — . Therefore, .
in the canonical equation ay — ypx + byqx, inftead of/, 1 fubftitui^^e it's
value now found, and it will be ypx 4- ^ = ypx + by^qx^ that is, — =:
hy^qXf and therefore — =: by"~ qx. But y = zu, und y"~ = z"~^u"~'')
whence, finally, it will be -^ z=z bz " qx; in which equation the variables will
be feparated, becaufe z is fuppofed given by x. When we came to the equation
alz = fpx, it is plain, that if p given by x is fuch, that the integral fpx
depends on the quadrature of the hyperbola, that is, on the logarithms, and the
quantity a is any number whatfoever, the relation of 2 to a; will be algebraical,,
and in all other cafes tranfcendental.
And here it may be obferved, that, in order to have a given equation come
under the cafe of the canonical formula, it is ncceffary that the following con-
ditions fhould take place. Firft, that the fluxion y may be alone, or, at Icaft,
multiplied by a conflanr, on one fide of the equation. Then, that, on the other
fide, the firft term may contain the fluxion x, multiplied by any function of x.
cxprefifed by p, and by the indeterminate y. Then, that, in the other term,
the quantity qx given by x may be multiplied by a power of j. In a word,
makings
28o ANALYTICAL INSTITUTIONS. BOOK IV,
making the dlvlfion by y, it is required, that, on one fide of the equation,
there may remain the logarithmical fluxion -^ , and, on the other fide, the
firft term may be free from the indeterminate^, and the fecond multiplied by
the dignity jy "^ . If any one of thefe requifites be wanting, this method cannot
take place ; as we fliould not have them in the following equations, ay z=: yypx
+ hy^qx, Sind ay = ypx + ayy -{■ y^ X, qx»
But fome formulas are very eafily reduced to the canon, by a little preparation
only. For example, take this equation ay z=. ypx + l^yjx + yyqx. Confider
that thcTjuantity px + l^qx, multiplied by jv, and that the binomial p -j- l^q is
given by x, fo that in it's place may be fubftituted the quantity r, alike given
by;f; the expreffion then will be changed into the following, ay =r yrx 4-
yyqx, in which the method here explained will take place. And this will be
lufficient to fliow the way of operation in all like cafes.
EXAMPLE I.
Let the equation be ay zz — + yyx. Make y — zu, and therefore ay zz:
azu 4- auz. And, making the neceflary fubftitutions, we (hall have azii + auz
fuzx . . T • fuzx , ^ . az fx J • .
= ^ h zzuux. Let auz — "^ — , that is, — ~ ^^— > ana integratmg, it
will be alz = fix, and therefore z^ :=■ x^.
If the conftants a, f, (hall be rational numbers, whole or fraded, afHrmative
or negative, 2 will be given algebraically by x. For example, make « = 1,
f zz z, [q that it may be 2; zi xx. Then eliminating the terms auz, ■^—^ ,
there will remain the two, azli =z zzuux. Bwt z n: xx^ therefore it will be
— = xxXf an equation in which the variables are feparated.
In proceeding to the integration, it will be -j- ^ = ja,-'. But u =
— = -7-, and therefore — — + c =z ^-x^ j that is, ^cy — o^axx = x^y ;
M/hich is the algebraical equation concealed under the propofed differential.
EX-
SECT. II, ANALYTICAL INSTITUTIONS. zBl
EXAMPLE II.
Let the equation be y = -^ 1- ^ . Make, as above, y zz zu, and
y zz zu + uz; then, making the fubftitutions, we (hall have zu -f uz zz'
azux z>u^x A J r r ^J^z"*" \ • ^ «*
r-* And, luppoiins: uz zz , that is, — — , or
XX — aa A-3 ' i r & xx — aa^ ' % xx — aa
XX — aa
. ax
J vv — n/i Out I . . ^U^X W %%X •
2 = «; -^^ aa^ ^,£ Q^^H j^^^g ^j^g equation zu zz — j— , or — j- r: — p , m
which the variables are feparated, z being given by x. Bat it may be ob-
ferved, that the quantity — — — may be reduced to a logarithmic fluxion, by
making x =: l±-lif_f ; wherefore, makins; the due fubftitutions, it will be
^ a — n ' °
r:-r~~Z. — "f" • Whence ~ = -^ and therefore zz = « = " — t-T"- •
XX — aa 2n z aa x + a
And, putting this value, inftead of zz, in the final equation, we (hall have
K axx — aa'x
»' X* ■\- ax^ * •
Without making the fubftitution oi x zz t.^Ll2Ll the quantity — ^f —
=» a — n *■ ^ XX ■— aa
may be reduced to a log^ithmical fluxion, by means of § 21, Book HI; and
we (hould have =: H = — , and confequently
XX ^ aa 2 X AT+tf z X x-^a «
X — a
ZZ ZZ
* + tf
EXAMPLE IIL
Let the equation he y zz — ^ + y x. Make y zz zu, y = zu + uz :
therefore, fubftituting, it will be zu ■{- uz z=: — ^ + tT z" x. Suppofing
uz ZZ —y or — = _, and, by integration, z =z — ; we (hall have
X
_«— I.
. • • m m . . . u m-~i . « ax
the equation zu zz z u x, that is, — = z x. ox — ~ .
Vol. H. O o EX-
282 ANALYTICAL INSTITUTIONS. BOOK IV,
EXAMPLE IV.
Sometimes a two-fold operation is neceflary ; as in certain equations which
have more than three terms. Wherefore, let the equation be xy + yx := au
4- xuy and let u be any how given in the terms of y, I difpofe the equation
,/.,,. ... . . ail ^ xu xy
in the rollowmg manner, au ■\- xu — xy zz yx^ or 1 =^ •= x*
Make x z: />^, and x zz pq + qp ; then, making the fubftitutions, it will be
f!L + fJ!L _ &L — pq ^ qp. If any one would reduce the formula by one
operation only, he muft put ^ — — = pq, that is, -^^ — = — ; by
which we find q given by y. But the operation will be performed more neatly
in the following manner. Make — — rz pq^ then — -^ =: -^ , and, by
integration, — = ^- Taking, therefore, the other terms of the equation
-JL -|- <31. ~ qp, and, inftead of q, fubflituting it's value ~ , it will be
— +^ zz — , that is, w + -^ = />. Make p zz mn, then p iz mn
+ »«, and making the fubftitution, it will he u -\ — ^— zz. m'n •\- rim. Sup-
pofe — ■ = w'n, that is, — zz -2-, Therefore n will be given by y, and in
the remaining equation, after the terms — , mu) have been eliminated, that is,
in the equation k z: n'm^ the variables will be feparated, and it will be" — = m,
Q.6» Still, after another manner, the variables may be feparated in the cano-
nical equation y z= pyx + qy'x. Make px zz , x =
Making the fubftitutions, it will be y = — — + — — ; that is, y =
•^ I— «Xa 1— aX/«
ttvz •4" ov^-^ ' " ' " • . .
1 — , or I — n X pzy = pyz + qyz ; and therefore, dividing, by /)>*„
it is - — " — ^^- — — = ~ . Laftly, dividing by zz, it will be
SECT. II. ANALYTICAL INSTITUTIONS. 283
= -2— , and, by integration, -^ = f-^— , that is,
7*""* rr 2/—. And, becaufe ^ and ^ are fuppofed to be given by .v j and
2 alfo, by the fubftitution o( px ■=: , is given by x -, the variables will
1 —a X a
bc/eparated, at leaft tranfcendentally.
Refuming, therefore, the equation of the firft example, ^ rz — + /i,
that is, jK = :^ + :^, it will be /) = -^ , ^ = -1. , « = 2. So that, fub-
(lituting thefe values in the final equation y'"" — zf-^, it will be ~ —
z/^ , and the fubftitution px =: ^ will be ^ z= ^ . And
making /=z 2, a = i, we (hall have — z= ^, that is, z = ~. And
therefore — = — / — xxx. And, by integration, ~ = ^ X ^x^ + c
y •*■'*' y XX ^ *
that is, 3fy — ^t^x = ^^^y, as before. And fo we may proceed with the other
Examples.
EXAMPLE V.
Let the equation be ax^yy — hx^yy =z ayyv?x — hyyx^x + a^x — x^Xy which
<3ivided by ax^y — hx'^y^ will be found to be j/ = — + ^ '^ "" * "^ , which is a
X *ty
cafe of the canonical equation. Therefore it will be /> ~ — , q zz. -^ '^ ,
* a~3Xx*
» z: — I. And, by fubftitutlon, px = ^ will be — = ~^ whence
z =: XA^. Then, putting thefe values in the final canonical equation, y^"^ zr
^/£" » ^^ ^^^^ ^^^^ ^y = ^^/=^^-^. in which the variables are fc-
parated.
O o 2
27. If
*8-4 . ANALYTICAL INSTITUTIONS. BOOK IV.
«--I
27. If the canonical equation were y " y zz px ■{■ qy^x, where p and q, m
a like manner, are any how given by x -,, the indeterminares may be fepaiated
a , • a
by making gx in — , and x == — . For, making the fubftitutions, it will
i>n^
« — I . «.
^^y y = ^-^1^^ ^^'^^ ''* r--^ = ^' ^"^' dividing by 2,
n — I . _ . ^ . „
,1 — — = £- ; and, by integration, -^ = /£- , that is, ;"' i= z/^~- ,
an equation in which the variables are feparated.
For an example, let the equation be la'xyy — aayyx -f- ihx^x^ that is, yy zz
,£ff J. ?11 , It will be « = 2, i> = — , ^ = , and therefore we fliall
have -^^ zi / -. But qx = — zz — , and x = z. Therefore it will be
JL. — yi-^, and, by integration, — = — ± ^ ; an algebraical curve.
Alfo, the general formula y y zz px + qy^x might be conftruded, ai\d
confequently the particular example, by means of the method at § 24.
a8. Before I finifli this Sedlion, I fhall add one obfervation, that fometimes
the indeterminares are involved and mingled with differential quantities, when
it may be allowed to modify the co- efficients ; and this fiiccecds efpecially when
the exponents are formed of the co-efficients ; and thus rriaking a kind of
circuit in the redudion. This artifice chiefly takes place in Phyfico-mathe-
maiical Problems, in which magnitudes of very different kinds mingling
together, we are more at liberty to make ufe of fuch conftant quantities, as
bcft ferve the prefent purpofe.
For an example, I fhall propofe to myfelf this equation, x x + by -k- yy X
L = yy^ which, being prepared according to the method of § 24, will be
m^ bcyx _ y_^s!L, Make, then, J-. - fl. zz ^ , and we fhall
'^ * X "^ y X y " p
have the value oi y — px ^ and yy = ppx . Thefe values, conveniently fub-
ftituted, will give the equation x^x + hcpx'^ x = x '^pp; and, dividing by
ie^\ it will be x^^'^'x + bcpx^'^^^x — pp. Here it is plain, that, an equality
being given b'etween the exponents of the indeterminate x, that is, between
ffi 2.C and — r — i, the variables will be feparate, the homogeneum compa-
ratmis ph being only to be divided by the binomial i + kp. Now, putting
m — 2^
6
SECT, Illt ANALYTICAL INSTITUTIONS. 285
ffi zc — — ^ — I, it follows m -\- I zz c ; (o that, expounding the con-
ftant c by m + i, we (hall have our defire. If c reprefents unity, which we
are at hberty to fuppofe, it will be m zz o ', and if ^ — 2, it will be m = j.
And fo we may go on. . - ^
The artifice here explained may be applied to all other equations of a like
, «• r .
kind ; for example, to this following, x x -{■ —^ V -^ — =1 y y. i*or,
putting / =: r — i, or — n — i, the formula will be thence abbreviated by
making ufe of the logarithms.
SECT. III.
Of the CotiJiruBion of more Limited Equations j by the Help of various Subfiitutions,
29. In the equation x^x ± ay^y x p ^=^ xy — yx x g, the indeterminates
are always feparable j where p and q are promifcuoufly given by y and x after
any manner j algebraically, when, in every term of the quantity/), the Sum of
the exponents of x and y is the fame, and thus likewife in every term of the
quantity q j but it is not required that the fum fhould be the fame in p and q.
a 1
The fubftitutions to be made are y z: tz»-\-i, and x ^ t x a^ :p ^zzV + i.
Thefe being fubftituted, refpecSlively, inftead of x, x, y, y, and making the
neceflary operations, after a very long calculation we (hall come ta this equation,.
I— «
3 «+I . a
— -—Z Z X -^
^« — 2; _ " + 1 p
Now, becaufe it is known, that, in every term of />, the fum of the expo-
nents of X and y is equal, as alfo in every term of q ; making in them the
fubftitutions of the values given by / and z ; in every term of />, t will have
the fame power, as alfo in every term of ^f a fame power j that is to fay,, that
the homogeneum comparationis will be multiplied by a pofitive or negative power
of /, or the firft memher will be multiplied or divided by that power, and
therefore the variables will be feparated.
a86 ANALYTICAL INSTITUTIONS, BOOK IV.
As, for example, let the equation be xx -|- ayy x -/j = xy — yx x V^ »
it will be « = I, /> = v/y> q - s/^i and therefore --- = - , ■ ^ A .
But y = iz; therefore it will be -— ■ = z^a __ _^
In the fame equation the indeterminates may be feparated, when alfo the
exponent « is negative ; that is, when the equation is this, x''"x ± ^j" j) X p
= xj •- yx X q j and the fubftitutions are y z=: fzi -«, and x =. t x
a I —n • a
z z X
'^n — z. I— « p
V =P tf2;2V-«. Thefe will give the equation /
the fame as that above, only with the figns of n changed. And though the
equation were alfo thus expreffed, y'x ± axy x —^ — xy —yx X q\ it
follows that this alfo is cpnftrufiible by the fame fubftitutions.
—n— I —c
30, Let the equation be more general, x^x ±^;' t y x p = xy + cyx
X q* The variables will always be feparated by making the fubftitutions of
y = /2;"+^, and x = t '^ x a ± acz *" " , where s and r are numbers
affumed at pleafurc j fuppofing, however, this condition, that the quantities
pt q^ are given algebraically, and in fuch a manner, that, in every term of the
quantity /», the exponent of y, taken as often as the number c denotes, may
exceed, or be exceeded by, the exponent of x in the fame excefs ; and fo in
eVery term of the quantity q ; but it is no matter that the excefs in p fhall be
the fame as in q. Thus, for example, if t = 3, it may be /> ~ hy'^x'' -+-^V%
i £3 .
&c. ; and it may be q zz gyx^ — by^°x » , &c. It is eafy to perceive, that it
cannot be ^ = o.
Making the due fubftitutions, inftead of x and y, in the propofed equation.
— sn—c~cs xz «+i z X -^
«+i p
we fhall have this following, — ~ / c / = " . — — ^ - — t^ ,
a ±: acz c l«-i-i
For
SECT, III, ANAL-YTICAL INSTITUTIONS. 28/
For example, let it be xx -{- ay''^y x — = xy + yx X x. Make j = i,
r = 2 ; it will be«=i, c = i, p = ~, q = x ; and, making the fubfti-
tutions In the laft equation found above, we (hall have — /'~^/ = ===—=3-—,
a + az^^^'
But, by the fubftitutions made, .v = / ^ x a -^ az ^/*, and j' = /z. There-
fore xy = z X a -^ <22~^'^. Whence we (hall have -j- = zz^
31. But let the equation be fl:ill more general, x'^x ±, ay c y x p =■
fxy + cyx X qy which comprehends, as particular cafes, the two canonical
equations of the foregoing articles; that is, that of § 30, wJien it is/— i ;
and that of § 29, when it is / — i, and r z: — i.
The indeterminates are feparated bj' means of the" fubftitutions jy-zs/Zz/^w+t^
and X = t X a ■±, — - — ; the condition concerning the quantities
p and q being fiich, that, in- thefe, the exponent of y being multiplied by r,
may exceed, or be exceeded by, the exponent of x multiplied by /, by the
fame cxcefs in each term. The fame quantities />, q, may alfo be fradions, or
mixed with fradions, and rational or irrational integers, whatever they may be.
And the indeterminates will always be feparable in the equations, provided that
p and q are given by x and y in fuch a manner, that, the affigned fubftitutions
being made, fuch quantities may arife in their place, that they may be the
produd of two, one of which Ihail contain 2, and not /, the other / ani
not z.
The faid fubflitutions being made, we (hall have this formula,
r-/«-/
EX.
a88 ANALYTICAL INSTITUTIONS, BOOK IV.
EXAMPLE I.
Let the equation be xxx + ay^ X J>' ~ — 3*^ + yx x ax. Let it b6, as
before, j ~ i, r ~ 2, it will be / z= — 3, c zz i, » zz 2, ^ n: ^.a?, /) — jy ;
and, making the fubftitutions in the laft formula found above, we (hall have
/"
— -- ax
8 *2 ^!i X
. 3 y
1 / — —
5 / «_ ^.1
efc
irc it will be = -
, But J = / ^z ', A? = / X a -^^az^^^^i
Q,az
3«
X a — iaz *^
- ; as was to be found.
EXAMPLE II.
Let the equation be x'^x + ay jf x ay^x -\- yyx* = 2xj + ^yx Xy^x-^yxx,
Let J = I, r = f i it will be <r = 3, / = 2, « = t> P = ^y*^ + yyx'*', q zz
yix — yxx. And, making the fubftitutions, it will be
- \t __ |z~^g X a + laz^^ ^ -^g^^g X tf + |az~3 ^
. '** az'^ X a + ^az 3 i^ _|. 2;T X a+laz~ i^ <*
in which the variables are feparated, as was required.
32. In the equations (i) pxy ~ j)r = /j«i -f- 5-;^,
(2) />Ary "" J/ = —-pyx + ^^c",
(3) ^/'^J' ~J>^ — ^/>.v";c' + y^,
• (4) apxy " y zz ~~bpy^x + qx,
where f and q are any how given by :cj the indeterminates may be feparated,
by putting, as to the firft, y = xzi as to the fecond, y = — ; as to the third,
-A. --L
^ = a; <2t z ; as to the fourth, j = a^ « z.
* This equation evidently admits of a Ampler form. Editor.
As,
^ECT, 111. ANALYTICAL I N S T I T iTT I O N S, 289
As, for example, let the equation be ihhicyyy — ix^yyy n hx^x — ^hhy^x
+ ^xxy^x, which I write thus, hh — xx X zxyyy rz hx^x -{•hh'-xxy. — 3/x,
This being referred to the laft of the four canonical equations, it will be
p ■=. hh '^ XX, a — 2, n zz 2i ^ = 3, ^ = kx\ Therefore we muft put
y ziz -j-y y :=: -^ , yy ^ -^, y^ zn -^ , And, making the fubRitu-
i . -
tions, we fliall have ihhx — 2.«' X -^ = i&^*^ + ^hh — ^xx x
— ^ i that is, zhh — 2xx x ^^22:2; — ^z^x zz hx ^ x -+• 3M — ^xx x
— z^x ; and, making the ufual multiplications, it will be ihbxzzz — ix^zzz^
A* 4 a:
1 7 ,
r= ^2 » ;c, that is, zzz = ,, _^ *
33. Let the equation be axy + l?yx 4- O*"^*""" x ■\- fx y '^ y rz o. In this
the indeterminates may be feparated, in general, by putting x zz u " z ^ ^
and y •=. z^ \ for, making the neceflary operations, we (hall come to the
.^_^^___ — ' » -^
equation i — w x az + /« 2; + » — i x ^2; + i:« ^
;?;
= » — I X — ^2« u — czu u, that IS, — =
^ , w 1 "~ ' • 'WW — m — it'
„ , J- mn — m-'n + l . 7 , m« — m — «+l
z — OT X a +fK T^*+« — ix3 + c» ^
As, for example, let the equation be a^xy — PyX = cyyxx -^fxxyy. Then
it willbe « = 2, w = 2. Therefore I put x zz — , and y zz — , that is,
X ■=. — ^ and therefore x = 2!LlLfI^, Whence, making the due fubftitu-*
y yy
tions, we fhalll^ave ^ - 3' x "^^-^^^ = f^2!lillfff-2 _-^ff^ ^hatis,
y y y y .
^*uy + aPuy + aacuuy •\- faamy zz aPyh + aacyuu, and therefore — =
ab'^u + aacuii ^
a*u ■+■ ab'^u + aacuu + aafuu
34. Let the equation be -^"^ — j l o^> "^o^^ generally, r^
w/— I .
ox + dy X ' pr •\-ay'^x '
=: j). The indeterminates will be feparated by putting hx + ay^a zz zx^^^
Vol. IL P p Whence
* Sec the Note at the bottom of the preceding page,
290 ANALYTICAL INSTITUTIONS. BOOK IV.
J— a
Whence y = , and therefore y = ;j; x
a" a'
m
f — f 1jL_ — — f ^ putting tliefe values of y and x in the propofed
general equation j and dividing by , it vviU be
^^
I 1 t—r ^—i. . I t—r — l. - t — r—l.
X X z ^ + t — r X z X x + t — rX-^ox x _i.
' = , that IS.
A t—r 7 t—r z ' *•
z' X — ex
-^z"z + / — r X z'"'^^x''^x + / — r X — bza^^x = nz"'x~^x — nhx^^xi
m
X,
and therefore — 7 — ■ ' — ^ — = — •
If you fhould have terms with negative figns, you muft proceed after the
fame manner, and in the final equation there would be no other difference, but
that of the figns themfelves.
35. Alfo, taking a more univerfal equation,, as - ^ ^ —
v.t—mnt'~t-\-r-\-n—ur
77T «7)»*
ax + ay X '
cx « y ', the indeterminates would be feparated by the fanic
fubflitution*
EXAMPLE I.
Let the equation be . ^ ■ — s= ly. Make s/ bbxx — a^y = xz^ and
\r bhxx — fl'j; ,
therefore y = ^ , and y zz — . And, making
the
SECT. III. ANALYTICAL INSTITUTIONS. 29!
the fubltitutions, — X r = \ > that is, aahhxx
xz a^ a^
«— aazzxx,'=z iPzxx — ibz^tcx — ihxzzz, or 2^xxzzz = 2Pzxx — ilfz^xx
ibzzx X
+ aazzxx — aabhxx -, and therefore
2^'» — ibz^ 4" <**2;» — aabb ^
EXAMPLE ir.
Let the equation be -^ z= -4-« Make v^— ^^v* 4. a^xyv z:
V - ^^** + a^xyy ^ •'•^
2;;if;i^, and therefore y = v 7 , and y =: -i-^ — -i-7^ . Where-
flV r-
a*
fore, making the fubftitutions, we (hall have i^v'^^^^— =
k'«z + Izzjfjri + ^bbxxx i«f •.»••• ,• ,. ,.
TzTm^T^ » '"^' '^' ^22A?AfAf + ^'^ATAT = X^ZZZ + ^z'aTAT* + |.teyA?Ar,
a^h^
or hzzxxx + ^^■«'^;c: — \z^kxx — ^bbzxxx zz. x^zzz ; and therefore — =:
b»z - |a3 _ 3^^a + 3J *
36. By the fame fubftitution as above, the indeterminates in this equation
alfo may be feparated.
«. tu—n — tmn — r* + '— »*
bx + ay «^l
^ =Z CX
/ft
X. Make bx+^^x^"^ - x^'^z, it will
i—m
be;'= ; then^y = + —--z'"x x +
mn
I— a
t—r ^ _^i,J''*\~ir
LlL^bx^^^^X into ~ — t ^ ; and, making the fubftitutions, we
1—m
n 11 1 X* *" z "' z , t — r ^ /— r—i . . r — ?,f— r — I. .
(hall have •— z x x + bx ^ into
ma » n
Pp2
492 ANALYTICAL INSTITUTIONS. BOOK IV»
«— «+r tu—n-^tmft"~ru-\-t—r
J "''J'' - bJ~') " rz ^a: " X. Wherefore, dividing the-
a n X z
numerator and denominator of the firft member of the equation by x%
, a+t— «
»+i ir^-^ —
and muhiplying the whole by ^ « z ; and, inftead of a; ^^z'^-'l>x '^ ' " ,.
tu'-tn + t — ru + nr — r ^ u + i—n
writing x « X 2^' — ^ » , which is the fame; and, uniting
the dimenfions of the letter x, we (hall find the equation to be divifible by
lit — n—tmn — ru-\-t — r .
X n ^ and that being divided accordingly, it will be — -^ — ■
a+ I— «
« + l
+ -z'"x + '^■—'hx into z" — ^ " z=^ ca n zx, Andlaftly, dividing
n n
«+T —n 1— iw
again by ?"^ " , it will be ''±Jt±. = Lzi x ^-^J- + ^-=-^ X ^^ +
«+l -T X
-"T"-. • v> -.^ /.I »
mn. n
n — «— X
^tf « z;v X 2" — ^ " , that is, — =
a+ I
I_L_: "i ) — ~ i ________
mnca n z X zT -^ b^ + mr — mt X «^ + mt — mr xh
EXAMPLE.
Let the equation be ^ ^ ■ = ~ . Put >/bbxx — aaxy — abxy
ybbxx — aaxy — abxy '^
= ^2;, and therefore y = -^r^rj- > and j)/ = -^^^ ; Makmg,
therefore, the fubftitutions, it will be r-nr X ■ — — .
aa-): bb ^ aa + bb X xz '
And, inftead of bbx — zzx^^', writing x^ x bb — zz\\ and multiplying the
whole equation by aa + ab)* X %9c, we (hall have x^ x bb — zz] \ x
SECT. Ill* ANALYTICAL INSTITUTIONS,
^9S
zx^x
bbx — ^^^ — 2Af2^ = aa + abY x -y- . And, dividing by x^ X bb^zz]*.
it will be l^i^x — zzx — 2x22; =: aa + abY x bb ^ zz")""^ X — i that is
^^;c' — ^zx + ^«+a^)* X ^^— 22:1"^ x — — = zxzz. And therefore
37. The fame fubftitution will ferve, in like manner, for a more general
equation, -"-^ .J^ = f^ ^ i. Alfo, it will ferve
Ox •\- ay X ^
a— I
for the equation ^ ^ — =/v' ^"^ ^* x,. by making.
bx + fA''^ + ay^x^^*" = x* Z', which, if w z: i, will be a particular cafe of
S 27 ; and if it be r = o, will be a particular cafe of § 36. Moreover, we
may alio conftruct the equation - — ==r = y*^ ^,
when it is cb :z ^/&, making ufe of the fame fubftitution, ax + bx + cy*/^^
mt
Now, if it (hould be alfo i?> = o, ^ z= o, the equation will be a particular
cafe of the firft equation of this article.
38. Thefe equations may be conftrudled, --^ =. gy " x, and
b + cy+fx]"
«— I.
ay y «» — i-
= gx^^^x^ by putting, for the firft, 0'" +/^1* = z, and for
the fecond, 9* + ^*"r zs z. And, as for the firft, it will be then y zz
""" 'I* ' , and J/ z= — X ^^^^=^4 X "T^ * z -^ fx ; and therefore.
I— »
making the fubftitutions, we ftiall have <?» » ij = ^a^f^;^ + nucgzx + ^»/2v,
that
294 ANALYTICAL INSTITUTIONS. BOOK IV.
I— «
that is, — ; — °^ " - ^ ^ = X, As to the fecond, we (hall have y =
' nubcg + nucgz + ^HJ
i—n
I — a
llr> , and therefore y zz — X ^-Ili^ X -^z « z^m/x"''x;
I— «
•« — I . fl« a «
. and, making the fubllitutions, x x = •— - — -
Likewife, if w£ take a more general equation, — i5L_-rI— _ — ^^^^ where p
and 5^ are any how given by x and conftants ; if it be ^ z= -4- , the indetermi-
nates may be feparated, by putting, in like manner, cy" + /> = z. For it
.1" X K — a'
I—n
will be ^^ = -j-i^ , and j = Jt x — ^ u z -^ p ; and,
making the fubftitutions, the equation will be nhcguq^x + ncguzqx + aup =
i?2 « i;. But if we fuppofe p =: gx, then it will be -r — ^^— ^- — = qx
EXAMPLE I.
a^J
Let the equation be ^j^" = 6^'^ — '^hhxs/ cy + ^^^ or "^ ===•—: 3^^^",
%% — bx . 222! — ^;t
Make \/ cy + Av ir z, it will be ^ = — ^ — , j/ z= -^ ; and, making
the fubftitutions, ^^ TiT^g ~ %hhx^ or 2^'z;j = (^h^cx — ibhczx + i?'^;^,
and therefore ^,. . ^^ . .. = ^.
EX-
SECT. Ill, ANALYTICAL INSTITUTIONS. 295
EXAMPLE II.
Let the equation be __ _±yv --. ^^^ ^ %hxx. Make
b + \y^ + aax •— bxx
{/y^ -t a^x — ifx'' r= 2 ; it will be y zz z' — aax + bxx^^y and y =
zzz — I f ^"^ ' ^ ^ . whence, making the fubftitutions, the equation will be
»3 — d'^x + bxx T
azzz — ya';f j^jw __ ^^^ __ 2^;v:v ; that is, 7azzz zz ^'^c* — 25*^;^;^ + 7aal?x
— S^^A-^c- + laazx — dbzxx 5 and, dividing by a ■{• ^b -{■ 32, it will be
— ^—r-; — = aax — zbxx,
39. The equation, or canonical formula, ax x + cyyx*x zz j/, has not it's
indeterminates feparable in general, whatever the exponent m may be ; yet
they are feparable in an infinire number of cales ; that is, the exponent m may
receive infinite values, in which the defired reparation will fucceed.
To determine which I make ufe of a method not unlike to that of § 23.
Make y zz Ax^ + x^i i where the quantity A, and the exponents p^ r, are
arbitrary conftants, to be determined as exigence may require, and / in a new
indeterminate quantity. Therefore it will be y zz pAx^^^x + rtx^^x +
//, and ^jy = AAaj^^ -^ ^Ax^'^^ t -\- itx^^ , Wherefore, fubftituting thefe
values in the propofed formula, they will give this following, ax"^x +
fAA/^+"^ + 2cAtx^''"''''k + r///"+";^ =z pAx^'^x + rtx'^'x + //.
Let us fuppofe ^AA zz /)A, 2p -{• n zz p — i,rzz 2^ A j that is, p z=z
— n — I, A = ~ " ~ ) r zz — 2« — 2. By thefe, in the laft formula,,
will vanifh the fecond, third, fifth, and fixth terms, and it will be reduced to
ax X + citx X z=z X t. That is, (dividing by ;tf ,) ax ^ ^ x
+ cttx" " X z=L t i or (D) ax X -\- cttx x zz ty making »i + 2n + 2 =K,.
and — « — 2 zz X.
I refume
4
%^6 ANALYTICAL INSTITUTIONS, BOOK IV,
I refume the propofed equation ax x + cyyx''x = j/, which, putting y is
, is transformed into this other, azzx x + cx^x =. — z -j in which is put,
as above, z rz Bx^ + x^u, where B, q, v^ are conftants, to be determined as
before, and « is a new indeterminate quantity. Therefore it will be ;s =
qB>^''^x + vux^^^x 4- x^'ut zz n: BBx*^ + iBx^ ^u + uux'^'". And thefe
values being fubftitutcd, we (hall have ^BBa; ^ "*x + laBux^ x +
auux X -{- ex X "=. -— qox^ x — vux x — x u. Now, if we fuppofe
aBB = — Bq^ 2q A- tn =. q — i, — v z=z loB ; that is, ^ + /« = — i,
B = ^ , V zz — 2» — 2 ; with thefe in this lad formula will vaniQi the
firft, fecond, fifth, and (ixth terms, and it will be reduced to auux~'^^'''^x +
XX X z=. ^ X u ; that is, (dividmg by a: j) cx x •{■
•auux'~""'^x = — «, or (G) ex x -i- auux x zz '^ it ; making 2«i + « + 2
=: J, and — • zw — 2 zz w.
Now, in the propofed equation, the indeterminates are feparable when
m =: «. Wherefore, alfo, in the formulas marked D, G, the indeterminates
will be feparable, when itis«?4-2« + 2zz — n — 2j2iw + «4-2i=:
— m — 2j becaufe m obtains two values, that is, w = — 3« — 4, m zz
— -^-— ^i which being fubftituted, the feparation of the indeterminates will
iucceed. For then, in the propofed equation, the indeterminates will be
feparated when it is w iz j alfo, they will be feparated in the formulas
D, G, when it is K = ~ "" , ^ =: ^^J^LZ-i becaufe tliere are other two
' 3 3
values of w, tliat is, m zz — — , m = "" ^"~ . . -
' . ' 3 5
By the fame way of argumentation, we may have infinite other values of m ;
— 7« — 12 — C« — 1.2 — g« _ 16 — •^w — 16 -
as m = — '— , m = — 2— , m = — ^ , m zz -^ , &c. ;
.and, in general, m zz — ~_Z-Llli_^ taking /& any integer pofitive number,
beginning from unity. Putting any of thefe values in the propofed equation,
we ftiall have the indeterminates feparable.
It
SECT. Iir. ANALYTICAL INSTITUTIONS. 29*
It may be added, that the indetermlnates will alfo be feparable in the pro-
pofed equation, when the exponent m is fuch, that, by the method of § ly,
it may be reduced to a cafe of § 14.
This would be the place to make ufe of two Dlflertations of the very learned
Mr. Euler, infcrted in the Memoirs of the Academy of Peterjhurg, Tom. VI.
Bur, becaufe of the fubtile manner in which that author proceeds, I (hould be
objiged to exceed thofe limits which I had fixed to myfelf, intending only a
plain and fimple Inftitution. I Ihall therefore leave the curious reader 10 feek.
them in the book, itfelf.
PROBLEM I.
40. To find the curve, the fubtangent of which is equal to the fquare of the
ordinate, divided by a conftant quantity.
Making the abfcifs equal to .v, the ordinate equal to y, the fubtangent Is
mlways — , which therefore ought to be equal to ~ . Therefore we fliali
have the equation ^ =: — , or <7;^ z: yy, and, by integration, ax rz fyy^ or
2/7^ z: yy, which is the Jpollonian parabola.
If the fubtangent ought to be equal to twice the abfcifs, we (hould have the
equation -^ = 2x, and therefore — = — , and, by integration, {h -f- \la
zz ly, (where the conftant \Ia is added, to fulfil the law of homogeneity,) that
is, l\/ax iz ly j and, returning from the logarithms, y/ax = y, or ax ^z: yy^
which is alfo the fame parabola.
If the fubnormal is to be conftant, it will be -^ zz ^, that is, yy zz axy
and, by integration, fjjy = ax, or yy zz lax, which is again the fame pa-
rabola.
Let the fubtangent be triple of the abfcifs; it will be -^ = 'ix. ox — zz,
or y ^ ' sx
J^ , and, by integration, l^aax =: ly^ or aax = y^, which is the firft cubical
parabola.
Vol. II. Q^q Let
298 ANALVTI-CAL INSTITUTIONS. BOOK IV*
Let the fubtangent be a multiple of the abfcifs, according to any number m ;
it will be ^ =z mx, that is, — =: — , and, by integration, Hya"" x "^tly^
or a*''^x zz /", a curve of the parabolic kind.
Let the fubtangent be ^"^"^ ^ '"' i then the equation is -^ =: ^-^^^ — ^ , that
IS, ayx 4- yxx = o^axy 4- ^xy^ or ■ ^ , ^^ = -r- • And, by integration, it
will be /y =: -l^ax -f- ;c.v, and therefore xx + 25.v = jj'j', an equation to the
hyperbola.
. Let the fubtangent be if^LH^- then the equation will be ■—. = — ^.^ * ,
° fly + 3x;s' ^ i' ^y + i^x
that is, ^jjy^t" 4- 3^^:^:;^ zi 2^rvy — 3a;'j/, According to what has been already
delivered at § 18, I endeavour to reduce this equation to a cafe of § 14,
Therefore I make y — —, y z= — ; and, making the fubftitutions, it will
be z^x + ^zzxxx = 4^:2;'^; — 6x^zZy where now it is reduced to the faid
cafe. Wherefore the indeterminates will be feparated, if we put 2 z: -^ ,
^ __ xp + px ^^ ^^^ making the fubftitutions, it will be ^—^ 4- — "^ = ^—^ X
a " Or aa a*
fft+iZ tA. X ^I±£f , that is, gaapx — ^P^x =: ^xppp — 6aaxp, and therefore —
/^ppp - aap ^^j , integration, Ix = ., "* — • And, reftoring the value
of />, that is, tf>/-^ , it will be «• =: '—^s^^J;^„^ that is, finally, ^y —
3 (3*^ A?* = mx, ^^
The two fubftitutions made of ^ iz ~, and 2 zz -^j in order to feparate
the indeterminates, plainly fhow us that it would have been fufiicient if, at
firft, we had madp but one of them, or y =: ^^ .
But w€ miglit have obtained our defire fomething more expeditioufly, by
writing the equation thus : $yxxx + 3flf'y zz laxyy — ayyx; which, divided by
XX, will be ^yx + 3^ = ?^*x>' ^ /'y ^ . ^^^^ ^^y integration, 3^ =: ~ , that
is, ^ay zz xx^ the 4polioman parabola, when we omit the conftant «;.
» Let
SECT. Ill, ANALYTICAL INSTITUTIONS. 299
Let the fubtangent be ^- "" ^''' - ; the equation will be ^' " ^"'^ •=. ^ ,.
'^ 3xx — ay ^ ixx —ay y
that is, 4flf'j — Gxyy = 'i^xxyx — ayyx^ which I write in another manner, thus :
^^'j) — i.yxxx r= axyy — ayyic. I obferve that ihe fecond member would be
integrable, if it were divided by x'a^/; I divide, therefore, the whole equation,
whence it is "^ "" =: '"^ " "^-^^ . I fuppofe the integral of this fecond
member — n: z-, and, making jy to vahilh out of the equation, it will be
^ y. xz + g^ — 3K^ __ - ^|.^^ J l^^LZ_fi — 2; which may be conftrudled by the
ZX ' 5S ■' •'
method of § 14, or elfe prepared according to the method of § 24, it will be x'A
— + — zz z. Therefore 1 make — + — =:—, and,, by integration,
lz*x zz la*py or z*x zz a*p ; and therefore, making at 10 vanifh out of the final-
equation, we fh all have, laflly, iA X ~ = i;, that is, a*p z= z^z, and, by,
integration, a*p ir -j-z* ; in which, refloring the value of />, then that of z, it-
will be XX — ^y, which is the ApQllonian parabola,.
T 1 r \ ^ *i a •\- X y. la -^-x , . .,,, a-\-x X la + x
Let the fubtangent be = — ;. the equation, will: be —^ — ■
a + la-^x. a + /tf+i*
•^, that IS, -^^ r: = . Ih order to proceed to the integration, I
J y n + xXia+x ^ ^
make a -^ x x la -[- x — z, and therefore z -:= x x ia -\- x + ax; (fup-
pofing the logarithmic with the fubtangent = a.) Thefe values being fubfli-^
Uited in the equation, it will be -=^ = -^^ and integrating, it is j^^ = z, that
is, y — a -^ X X ia -\-x, a tranfcendent curve, but which is eafily defcribed,
ibppofing the logarithmic.
PROBLEM IL
41. To find the curve, the area of which is equal to two third parts of the
re6Vangle of the co-ordinates.
The formula for the area is yxy and therefore we fhall have fyx •= l^ic j
whence yx = ^xy + ^yx, that is, yx = ixy, or ^ = -^j and, by integra-
Q;^q 2 tiop>
300
ANALYTICAL INSTITUTIONS.
BOOK ly.
tion, as before, it is W avi m /y, ax =^> The curve is the fame Apollonian
parabola.
Let the area be equal to the fourth power of the ordinate, divided by a
conftant fquare ; then it will be Jyx = — , that is, yx =. > , or aax zz
4i^yy ; and, by integration, \aax — j% the firfl: cubic parabola.
Let the area be equal to the power denoted by m of the ordinate, divided
tn m—i.
by a conftant ; it will be /yx = f - , that is, yx zz ""-^ _ / , a curve of
m — z
a
m — z
the parabolic or hyperbolic kind, according as »> — i (hall be pofitive or
negative.
PROBLEM III.
42. In infinite number of parabolas being given, of any the fame kind ; to
find what that curve is, which cuts them all at right angles.
. Let the equation of the curve required be
fig, 144. ^
.m~-n n
P' X =y , which, (p being confidered
as arbitrary, and fufceptible of infinite values,)
exprefles infinite parabolas ; and (confidering
m and n in the fame rnanner,) exprefles any
kind of parabolas. And, firft, let them all
belong to the fame axis AB, (Fig. 144.) with
vertex A, and different only in their para-
meters. Let AC be one of thefe infinite
parabolas, in which AB zz at, BC =: y.
From any point C let the tangent CT be drawn, and the normal CP. It is'
known already, that it will be BT = — . Let DC be the curve required 5
and, becaufe this ought to cut the parabola perpendicularly in the point C, in
an infinitefimal portion it muft coincide with the normal CP in the point C.
Therefore CT, the tangent of the parabola AC, will be likewife perpendicular
to the curve DC in the point C, and confequently, at the fame rime, BT will
be both a fubtangent to the parabola, and a fubnormal of the curve required,
DC. What is laid of the parabola AC agrees with any other of the fame
kind. Therefore the problem conlifl:s in findmg, of what kind is the curve
DC,
SECT. III.
ANALYTICAL INSTITUTIONS.
301
DC, whofe fubnormal is =
ntje
Now the general expreflion of the fubnormal
is =^, which, in this cafe, ought to be taken negative, becaufe, in the curve
DC, as AB, or x, increafes, at the fame time BC, or y, decreafes ; and there-
fore the differential equation will be — = — —-; and, feparating the vari-
ables, — zz ^ yy } and, by integration, — = — -lyy + aa, or ^^ =
-^ — - xx, which is an equation to the ellipfis. And, becaufe the parameter
p does not at all enter here, the folution will be general for the infinite parabolas
that may be thus defcribed.
If the exponent n of the equation p"''"x^ z: y^ is fuppofed to be negative,
fo that the equation may be x^y"* rr p" y in which now it is pofitive; it will
belong to infinite hyperbolas of the fame kind between the afymptotes, the
fubtangents of which are — — , and the fubnormal of the curve DC ought
alfo to be equal to thefe. Then it will be — ^ zz — ~ , or ^^ =: j;j.
And, by integration, ^^ r: ^yy + aa, or xx
the hyperbola.
Fis. 145
inaa nyy
=■ -— , an equation to
If the infinite parabolas AC, QC, &c.
of the equation p " z =jy , Hiall have
all the fame parameter, but each a different
vertex in the fame axis ; that is to fay, if
one of them be conceived to move always
upon the axis parallel to itfelf; from a
fixed point A (Fig. 145.) making any
abfcifs AB = x, and taking any curve
QC, whofe abfcifs is QB = z, and ordi-
nate BC "=. y ; then will alfo — ~- be
the fubnormal of the curve DC required, and therefore equal to the fub-
tangent BT of the parabola QC, Whence the equation — ^ — ~ ; but.
D P
by the equation of the parabola we have z = ■ _
p n
, and therefore — ^ sa
302 ANALYTICAL INSTITC7TI0NS. BOOlt IV,
m
«/— H m
, that IS, A^ = p '^ y « JK; and, by integration, x zz
m—n m
np n
, the equation of the curve required, DC.
m X zn '— m
If the parabolas arc the Apollonian, that is, w = 2, « = i, the integrated
equation would not be of ufe in this cafe ; for, making the fubllitutions of the
values of m and x, we fiiould have ;r zz ~ -^ , But, taking the differential
equation, it would be :v =: — fp x —> an equation to the logarithmic.
Therefore the curve which cuts the infinite Apollonian parabolas at right angles
will be the logarithmic MCN, the fubtangent of which is equal to half the
parameter of the parabola.
Let the parabolas be the firft cubics, that is, ;« = 3, » =: i j it will- be
^ — — ^ — , or AJy zi ~pp, and the curve DC will be the hyperbola between
it's afymptotes.
Let the parabolas be the fecond cubics, that is, w = 3, » = 2 ; it will be
* — — TX^pyi 01^ *^ = ^^Py* ^"d the curve DC will be the common-,
parabola. Taking other values for m and », we (hall have other curves.
If the parabolas AC, QC, tec. befides having a different vertex on the fame
axis, fliould have their parameter variable, that is, equal in each to the
refpedive diftances of the vertex from the fixed point E; taking any one of
them, QC, make EB z: x the abfcifs of the curve required DC, the ordinate
BC zij, EQ^zi p z: parameter; it will be QB z=i x — ^, and the equation
of the infinite parabolas i& "" X x — p^" :=: y , and the fubtangent BT a=s — i
^ X -~-p, and therefore the equation — ■— = — X* — p.
If the parabolas be Apollonian, that is, ;w = 2,. » =r i, it will be ^ :z f;r
+ y/^xx-^yyi whence, making the fubflitutions in the equation — ~- =r:
— y, X — p, it will be — ^ zz x ^ x\/\xx -^yy, which may be reduced'
to a feparation of the indeterminates by the method of ^ 14 i and then we may
go on 10 the integral, which will be algebraical*
l£
ANALYTICAL INSTITITTIONS.
303
If the infinite parabolas AC, QC, &c.
of the equation p "' z" :=: y fhall have
the fame conftant parameter, the axes
parallel, and the vertices variable in the
perpendicular to the axes ; that is to fay,
if one of them be moved in fuch a
manner, as that every one of it*s points
may defcribe perpendiculars to the axes :
M D P Taking any one of them, EC, (Fig. 146.)
and calling AM - EB = 2, BC zz y,
MC -= x; and, drawing to the parabola EC the tangent CT, produced to V,
then MV will be the fubnormal of the curve DC required. Now, becaufe it
is BT = — , it will be MV = — j whence we (hould have the equation
m—n n
ny
= — 4^ ; and, inflead of y, fubflituting it's value p "* z"* , given by
mzx
the equation p "" z" =: ^ , it will be, finally, ^__^ — — = ^ , that is,
tip m Z m
= ^^c", and, by integration, a? = —
mmz m
m — n n
ftp m zln"
equation of the curve required, DC.
m—tt
, the
n X 2m— a X p m
Let the parabolas be the Apollonian, that is, m zn 2, n zz i ; it will be
^ =r — ^ , or -^^pxx = z' ; and therefore the curve DC will be the fecond
cubic parabola, of which the laius return will be to that of the parabola AC as
9 to 16.
Fig, 146. It is to be obferved, that, in this cafe, the
pofition of the curve DC will not be that
marked in Fig. 146, but will have it*s vertex
in A, cutting the inferior part of the Apollonian
parabola at right angles ; that is, meeting the
convexity, as in Fig. 147.
Another kind being pitched upon for the
parabolas AC, alfo the curve DC will be a
parabola of another kind.
PRO.
304
ANALYTICAL IjrSTItUTIONS.
BOOK ir.
Ff^. 148.
PROBLEM IV.
43. Upon the right line AD let the right
line AC infift at half a right angle ; the equa-
tion of the curve AB is required, the property
of which is, that the ordinate BD may have to
the fubtangent DF, the ratio of a conftant line
a, to BC.
Make AD = Af, DB = ;^ j it will be CB =:
y -^ X, Whence, by the condition of the pro-
blem, we (hall have y . -^ :: a , y -^ x ; and
therefore the equation ax zz yy ^ xy. Now, to feparate the indetermi nates,
I make ufe of the method of § 23. Wherefore, putting ;tf zi Ay + /> + B,
and X zi Ay + p ; and, making the fubftitutions, it will be aAy + ap •=. yy
— hyy — py '-' By* Now, in this equation, the indeterminates will be
feparated, if the firft and fecond terms of the homogeneum comparationis be
made to vaniQi ; that is, if A r: i, and B remains arbitrary, which, for
brevity-fake, 1 will make B =: o. Therefore the fubftitutions to be made will
be X •=. y -\- p^ X z=z y + p, and the equation will be ^/ = — ^ — py,
that is, ^ - = — J/, a tranfcendent curve, and which depends on th^
logarithmic.
PROBLEM V.
44. To find the curve, the area of which is axy + l>xY i where the abfcifs
is X, and the ordinate y, as ufual.
c c
Therefore it ought to htfyx z=. axy '\' Ix y ; and therefore ^ji- ziaxy-\-ayx-\^
cby^x^^x 4- ehxy ^ y '» or, making <? — i = w, it is myx + axy + chy x^" x +»
eh^y^^y = o. To feparate the indeterminates in this equation, we may make
tife of the method of § 33, putting a:- = a "" 2 "" , and y -zz z ""^j whence
making
SECT. III. ANALYTICAL INSTITUTIONS. 305
making the fubftitutions, we (hould obtain an equation much compounded,
and which would require a very long calculation.
To come, then, to the point with brevity ; refuming the equation fyx =
Ix'y + axy, put x^y n y, whence the equation will be fyx "zz bq '\- axy, and
therefore yx zz. bq '\- axy + ayx. This fuppofed, I make ufe of the method
of § 24, in the form of which I write the equation thus, axy X -^ x — — -^
=2 bq ; then I put X -"^ = -^ ; and then integrating, it will be
} " ^ iJL z: Ipj or "—^ = p. Wherefore, making the neceflary fubftitu-
X
tions, we (hall have the equation — - zz bj. Now, to exprefs the quantity x"
by the affumed quantities /, ^, we muft conlider, that x^y' = ^, that is, y^ —
X I— a I —a ^
-^, or y = -^. But we have alfo — zz y; therefore — z: -^,
e — ae-\-ac 1
or x"^' = q^pi and, laftly, / = j'-^'+«^ x p'-'"+^'. Then, makr
e
— 2
ing this fubftitution in (lead of x", we (hall have the equation ap''~'^^'^^^ p
lae-—2ac — e — i
— b. ; that is, ap"-^-^'"' p z=. bq'"'^^-^^' q\ and, by integration.
ae—ac ^ e—af+ac—i
— X p'-^^-r^'^ zz. X g f-^erac • g. which is the
ae — ac ■* e — ae -^ ac — I ■* • o '
equation of the curve required.
It is plain that this curve will be algebraical, at leaft when the quantities
a^ f, e^ (hall be rational j and, on the contrary, it will be tranfcendental when
one of thefe fball be irrational. 1 fay at leaft, becaufe, making a, c, e, ra-
tional, the curve, however, will be tranfcendental if <? = ^^ or if ^ = ~ ;
or if f rz I, and at the fame time ^ =: i ; or ^ = o, and alfo e ■= i. And
in feveral other cafes, which it is not neceflary to enumerate.
Vol. II. Rr SECT.
306 ANALYTICAL INSTITUTIONS. BOOK 1 V
■g "i:j.j.^-xn- i'
S E C T. IV.
Of the ReduSlion of Fluxional Equations, of the Seccnd Degree, i£c.
45. When the differential equations of the fecond degree are fuch, that the
rules here explained for integrations may be adapted to ihem, as well in cafes
of feparate variables, as in thofe that are mixed ; nothing elfe remains to be
done, but to apply the faid rules, and thus, by means of mtegration, to reduce
them to firft differentials ; therefore there is no need to add any thing further
about this matter. If, after the formulcc thus reduced to the firft degree, the
indeterminares will not then be feparable, as is often the cafe, nor fhall be in
any wife conftriw51:ibIe; it is not the method that is in fault, by which the
fecond differences ar£ refolved, but rather that by which the firft differences
.ar<e managed.
Therefore we ought to employ our induflry about the reduflion of the
diiferentio -differential equations, that, by the rules already taught, they may
be made fit for integration, which may be attempted feveral ways.
46. One way will be, to make ufe of the common expedients of vulgar
Algebra, by tranfpofing the terms, by multiplying or dividing them by fome
quantity, and fuch like. But, firft, before any other thing, it is neceliary to
recolleft, or to know, if, from pafiing from firft to fecond fluxio/is, there be
^ny fluxion that was taken for conftant, and what it was. And befides, that
.as, in the integration of firft differences to finite quantities, there is always
added fome conftant quantity ; fo, likewife, in the integrations of fecond to
firft differences, fome conftant quantities (hould be added. This fuppofed,
let us proceed to fome Examples.
EX-
SECT, IV, ANALYTICAL INSTITUTIONS, 307
EXAMPLE I,
Let this equation be propofed, ■^— zz. — ^ . . "^ , in which u = ^/ xk + yy
c -^
is the element of a curve, and is fuppofed conftant. I write it thus,
by yu •■ , • •
■ - -^ ■ zz 2ayx + axy.
c
As j^ is conftant, the firft member will be integrable, even though it (liould
be multiplied or divided by any fundlion of y ; and I obferve, that the fecond
would be (o alfo, if it were divided by 2 '/y. Therefore I divide the whole
equation by 2V'y, and it will be -^ -^" z=. i^:f_^_i2Lj and, by integration, it
will be —^ = ax\^y + aiWa, which equation is now reduced to firft
fluxions.
In the integration I have added « for this reafon, becaufe it is conftant ; and
I have multipUed it by av^a, to preferve the law of homogeneity.
EXAMPLE IL
Let the equation be / = " ^^.^/-^ , in which yx. is taken for a conftant. I
muhiply it by 21/, and it will be 2fy = ^ffZJI-5^ that is, 2/y = -^ ^•
and, by integration, becaufe of yx being conftant, it will be /2/y zr
I yy
-^^^ + nyyxx,
yy yyxx
R r 2 E X-
3o8 ANALYTICAL INSTITUTIONS, BOOK IV,
EXAMPLE III.
Let the equation be/zz ^^^^tt— , in which let x be conftant, and it the
element of a curve, that is, \^ xx +yy :=z iu Therefore, becaufe x is conftant,
it will be yy — m ; and therefore, fubftituting the value of j' in the equation,
it will be / z= i£Lz^ ,. and, multiplying by zy, it is ify = ^^^y^jtfj^y^
that is, 2fy zz ^^^^~^^^^^; and, by integration, zffy z= — ^ + nxx.
Again, after another manner. Tnftead of u, putting it's value in the equa-
tion, it will be / =: -^-^ +->;^ ""-^ j and, multiplying by 2)|y, it is 2fyjf zr
m^l±ML^lm., that is, ^fy = ^^^ ^^ - ^;j>nd. by integration.
/-. XX •{■ yy , . .
&jfy = — rr^ + flXX,
J •/■/ yyxx
EXAMPLE IV.
Let the equation be ^x r= ^ '^•^ , in which let x be conftant. Multi-
plying by X, and dividing by x, it will be ^^ = j^' + j)yj and, by integration,
becaufe :v is conftant, it is axlx + Ax z= ^j/. Now, if we fhould make the
affumed conftant A = ^z, we (hould have axlx + ax zzyy ; and, proceeding
£0 integration, axlx z: ijj^.
EX.
SECT. IV. ANALYTICAL INSTITUTIONS, 30JJ
EXAMPLE V.
Let the equation be /n: ^^ — ~^^ — yxuu^ j^ which u is the little arch
^ -^ . yxytt
or element of a curve, / is given by x and ^, and no firft fluxion is yet taken
for conftant. .1 divide it by jy^;c% and multiply it by 2, and it will be
if ___ i'xyu'u + lyuu'x ^ zylcuu ' ify'tt ij/.vaJk'J + ayy^iiMi'jf — zyy'xx'u'u , ,
jf^AT* ^'^AT^/i ' yy'xx y'^'x*' ' ' ^
rfy'ti , iiu , •
inteeration, 2/ -^-^ ., ■ = — -^^ 4- »,
But it may truly be faid to be a thing iinpofiible, to make ufe of this method
in fuch equations, in which the quantities are intricate and compounded, when
we do not know the integrations pretty nearly before-hand, which we are to
make. Wherefore I fhall go on to other methods.
47. In the folution of problems, when we are to proceed from firfl: to fecond
fluxions, it may be much more convenient not to aflume any fluxion for
conftant, though we are at liberty to do it : that vve may be able the better,
when the formula is under our infped:ion, to determine that to be fuch confl:ant,
by which the expreflion may be much abbreviated, and mod readily integrable.
The Examples will beil make this method to be underftood.
EXAMPLE I,
Let the equation be / n ^- — — ~^ — ^^ , which may arife without
liaving taken any fluxion for conftant. To (horten this formula, I confider,
what may be that fluxion which, taken for confl:ant, will deftroy two terms of
the bomogeneum comparattGnis , and leave only two in the equation ; and I find
there may be two, that is, xy and — . Therefore make xy — c, and taking
the difference, it is *y + ;t^ ~ o. Then multiplying by x» it is xxy 4- xxy
= o, by which means the fecond and fourth terms of the bomogeneum difappear
out of the principal equation, fo that we (hall have / = ^-^^^ ' But, as it is
4 xy +
3IO ANALYTICAL I N S-T I T U T I O N S, BOOK IV.
xj + xjf :=z o, it will be y = -r- i whence, by fubftitution, f = ^^
4^ , that IS, / = — -'-^^ , ■ • , 9 or / = — - ^ ;.. . But xy ~ Cy and
therefore / =2 -«^L__flj and, laftly^ /v z: — -^^^ ^; and, by integration,
//;^ = _iL±if ^_ ^ or ffx = - i^-±ii ± «. When I came to the
equation / zz ^ " /^^ , we might more briefly have gone on to the integration,
by multiplying by x, and difpofing it thus, /x = -^ ^— r, where, becaufe
xy is conftant, it will be /fx zr — — — --^-t-:- + », as before.
Now let us make conftant the quantity — . Such a fuppofition giving
= o, and alfo — i^yx + •'^•^ =: o, takes away the fecond and third
XX — XX
XX
terms from the principal equation, and changes it into this, /=:-Ljtf^. and,
multiplying by .V, it is fx — ^^."^^ , the integral of which, (becaufe of
• • • ' _ • •
— , or ~ conftant,) will be found to be ffx zz ^^ 4_ ;^ as
above.
48. But, to know nearly what fluxion may be taken for conftant, it may be
obferved, if, in the propofed equation, there be two, three, or more terms,
which, being multiplied or divided by a quantity which is common to them,
they may be reduced to be integrable ; then making the integration, their
integral may be taken as conftant, and fo proceed in the manner fpecified. If
not always, yet fometimes, at leaft, we Qiall fucceed in our attempt.
I refume the equation / z: ^' """^ 7^^^ — "^ » ^"^ obferve, that the two
terms x^y + xxy^ being divided by jr, will become icy + xy^ which is an
integrable quantity, and that it's integral is sy. See, then, upon what account
we may take this quantity for conftant. In like manner, I obferve, that the
two terms x^y — xyxy if they be divided by — xxy^ will give us "" -^-^ + -^-^ ^
an integrable quantity, the integral of which is —\ therefore the fluxion —
might alfo be taken as conftant.
For
SECT. IV, ANALYTICAL INSTITUTIONS, 3II
For example, let the formula xy x ^y — yx "=■ yyx'^ — ^"^* -*- xxy"^ be
propofed, in which the variable z is any how given by y, I difpofe it thus,
xyxy + yyzy"^ zi yxyx + yyx^ — xx^, and obferve, that, if the homogeyieum
eomparationis be divided by yyy, it \yill be -^ — 2!1-Z_f^^ tjie integral of
which is — . Therefore I take ~ for conftant, and make ~ =: f, and
y y y
thence ^ — >i — xxy __ ^^ Whence the propofed equation will become
xyxy + )7%K* =: o, that is, s n — ^ ; and, by integration, becaufe of —
conHant, it will be 2. = — r- ± «.
yy ^
49. In an equation of the fecond degree, when either of the two indetermi-
nates are wanting with all it's fundlions, and only it's firft or fecond differences
enter in the formula, any how compounded and raifed to any dignity ; the
integration, or reduction to firft fluxions, will always be in our power, by help
of a fubftitution. This will be, to make the firft fluxion, which is flowing or
indeterminate, equal to a new variable multiplied into a conftant aflTumed
fluxion, or which may be affumed at pleafure, in cafe that no other be ap-
pointed conftant. For example, in a given equation, let ^, at firft, be fup-
pofed variable, and y conftant ; make x — pj), and taking the fluxions, on
the fuppofition of j/ being conftant, it will be x "^-py. Making this fubfti-
tution inflead of ic, and the equation being managed by fubftituiing the values
taken from the equation x =1 £)/, it will always be reduced to firft fluxions.
Or, perhaps, it may be more convenient to make the firft fluxion of the
variable, which is wanting in the equation, equal to a new indeterminate,
multiplied into the firft fluxion of the other. Making the neceffary fubfti-
tutions, and having a due regard to the fluxion which, at firft, was taken for
conftant, we Qiall have the propofed equation reduced to firft fluxions.
EXAMPLE I.
Let us take again the equation of the firft example of § 46, -—- = !21±f2^
in which u is fuppofed conftant. Make, therefore, x = />», and by difFer-
encing, k •=. pk Then, fubftituting this value, we fliall have -^ =:
312 ANALTTICAL INSTITUTIONS, BOOK ly.
'Jii+Jt!t, that is, ^ = f^4^, and therefore ^ = 2a?p + oty,
which equation, divided by 2/^? is integrable, and the integral is — --
m+
= ap\/y ±: g. But ^ z= -4-, therefore — ^ ^ =: ax^y + g^*
m + iX2c"
EXAMPLE IL
Let the equation be fyyyxx •=■ — - uuy where / is given by y\ u is the element
of a curve, and yx is the fluxion taken for conftant. Therefore I make u —
pyx, and, by differencing, it is 'u = ypx ; and therefore, making the fubfti-
tutions> it is ^^'^^ = — y^pp^^i that is,/)/ zz — pf. Whence, by integra-
tion, iffyzz — pP + '^m. But pp :=. -^ zz "-^L—ZL^ Wherefore, makinst
the fubftitutions and the redudion, we fhall have x :r — ^ -rr. .
Now I reduce the fame equation by means of the other fubftitution men-
tioned before. Make, therefore, x = pu, and x = pu + pu^ whence u =z
t "" ^". . Making the fubftitutions, the equation will be fyyppym = ^^^^ — ^.
But the fluxion yx is afl'umed as conftant, whence we fliall have yx + yx = o>
that is, if =: — — , or ic" = — -^ . And, fubftituting this value again in
in the equation, it will be fppyyy — ~ + — . This fuppofed, we may go
on, and make — — \- -^ zr. -2— , whence py — q, and therefore fqqjf = -^,
or /j/ = -^ . And, by integration, ffy = ^-^ + w. But qq = /);>)^ =
^ =: ..-y-y^^... Therefore it will be '2//)/ = — iL+ii + 2m -, from
whence we may derive, as above, x = ^ zmyy -i - 2yy//y '
EX-
IICT. IV, ANAJ[,YTICAl. INSTITUTIONS. 3,13
EXAMPLE lir.
I refume the equation of Example III, § /^6y fy^xx zz xx -{• jfy ^-^ yy, in
which X is conftant ; and make y = px, and therefore y -rr px. Making the
fubftitutions, it will be fy^xx zz xx •{• yy -- ypx j and, making x to vaniOi^
by it's value -=^, we (hall have *^^ ~ "5" "^-^-^ — ^' ^^^^ ^^» -^'-^-^ ~-^-^
+ />^j)/j/ — yfyp. And, dividing by /i, it will be /y = -=^ -f
And, by integrration, ffy zz — -^ -^ + m. And, inftead of ^, fubfti*
tuting; it*s value -—^ it is /"/y zz — — — •^•^. . -' + m, that is, zffy zz — -
j- 2m I and therefore x = "^
»»
50. If, in the propofed equation, no fluxion has been taken for conftapt
one may be taken at pleafure, and the operation may be performed, as is done
at § 48.
As, for example, the equation of Example V, § 46, being given, in which
no* fluxion is afTumed as conftant, that is, fy^yx^ — xyu'u -j- yuux -^yxim,
(putting yx inftead of /,) if x be made conftant, it will expunge the term
yuUx, and the equation will become j3'5''^* ~J^^* — y^^^' N<5W, to reduce it,.
we muft put u zz px^ whence u z= px, Thefe values being fubftituted, we
Ihall have Jy^yicx = ppyxx —-yppxXy that is, fy^y z= ppy — ypp j which equa-
tion, in order to proceed to integration, I write thus, fy^y =r fpy X ~ — — •
Therefore, integrating by the method of § 24 aforegoing, ffy =z — — + m\,
and, refloring the value of p, ffy z: rr- + m.
If u be taken as conftant, the term yxm will be expunged, and the equation
will be fy'^yx} = xy'm + ymx^ and therefore we muft put x zz pu, x zz pu,.
Thefe values being fubftituted, we (hall have fy^y X />'^' = py'u^ -^ ypii^, that-
is, j&ly zz- ^-^ +.y/* . jj^gn, by integration, it will be ffy zz — -^^ + m;^
and reftoring the value of p, it will be //> = — j^ + /»•
Vol. II, Sr 5U To
314 AKALYTICAL INSTITUTIONS. BOOK IV,
51. To affbme at pleafure any fluxion as conftant, in equations wherein
there is none already fo taken, may make fome equations fubjed to the
method of § 49, which are not fo already, becaufe of having both the indeter-
minates finite quantities. And this by affuming fuch a fluxion for conftant, as
may make all the terms to vanifli, in which is found one of the finite indeter-
minates, thofe only remaining which include the other.
■ 6^
For example, let the equation be x^ — xyy zz yxx + 2jryj/, in which no
fluxion is taken as conftant. If we make x conftant, the firft term of the
homogeneum comparationis will vanifli ; and if we make y conftant, the laft term
will vanifli ; and, in either cafe, there remains only one of the indeterminates.n
Therefore, appointing x to be conftant, the equation will be x^ — xyy — 2xyy,'^
Put y n ~, y = — , and tpakine; the fubftitutions, it will be x^ — ^^ z=
— ^— , that is, aax — ppx = 2xpp, or — = ^^ _ ; then, by mtegration,
it will be /y = — laa ^ pp + ^«^> and therefore x zz , And, inftead
* •* aa — pp
c£ p, refl:oring it*s value -^ , it will be x zz .. , that is, x = — J^ — ,
•^ ° X ' aayy ' aax*—aaf'*
js'x
or mOQ^ zz aaxx* — a*xj/*.
52. But when the taking at pleafure a fluxion for conftant, does not fucceed
in eliminating one of the two finite indeterminates, or if the conftant fluxion
be already fixed, fo that both the indeterminates remain in the equation ; there
is no general method as yet difcovered, how to proceed further.
The methods here explained may fometimes have their ufe, as alfo the ufual
expedients of common Algebra, fuch as multiplication, divifion, &c. As, for
example, in the equation xxyyy = jc if — xx, which, being divided by xxy will
be yyy zz ^l-H^, and therefore is integrable, (fuppofing y to be conftant,)
and the integral is ^yyy = ~ + ^.
Sometimes a fubftitution may make the propofed equation within the reach
of the method of § 49, And, indeed, the equation x'"x zz yy -^ yy +
j^yyy, which is not fubjed to the canon of the aforefaid article, will however be
fo, if we make yy zz Z', whence it will be x*^x zz z + zz,
53. Wherefore, in cafe that in the equation there ftiould be already a conftant
^uxion, it may be of good ufe to change the propofed equation into another
equivalent
SECT. IV. AKALYTICAL INSTITUTIONS.. 3X5
equivalent to it, in which no fluxion is conftant. To do which, let there be a
general equation y zz pxy where p is a. quantity any how given by x and jy,
and let x be conft'ant. By taking the difference, it will be y — px. But it is
^ = -4- ; then, by differencing, without making any conftant fluxion, it will
be p =: '^ T".-^^ . Wherefore, the value of/ being fubftituted in the equation
y zi px, we fliall have y rz "^ ~jy^ ^ ^ 5q ^j^^^^ Jj^ ^j^y propofed equation In
which X is conftant, inftead of y, if we put it's value, "^LJZZf ^ it will be
changed into another that is equivalent to it, in which there is no conftant
fluxion.
But, becaufe often other more compound fluxions may be alfumed as
conftant, or have been at firft affumed, it may be of ufe to render this method
more univerfah
•
Let us take this general equation j/ zz mpx, where p is likewife given, m.
any manner, by x and y, and m is any fundlion whatever of x or of y, or of
both together. Let mx be conftant; then, by differencing, it will be y zz:
mxp. But p =: ~; and bydifferencing, without affuming any conftant, it is pzz
mx
tnxy — mxy — myx
__ mxy — mxy —myx
Wherefore, fubftituting this value in the equation y zz mxp^
inftead of />, we fliall have y = —^ — -^. • — . Wherefore in any propofed
equation, in which mx is conftant, if, inftead of y, we pul it's value noW'
found, it will be changed into another which is equivalent^ in which no fluxion
is conftant.
After this manner equations being made complete, that is, fuch as may have-
no conftant fluxion, in proceeding to the redudion, we fhall be at liberty to take
that for conftant, by the afliftance of which we may beft attain our purpofe.
EXAMPLE L
Let it be propofed to reduce this equation, xxy — j/' =z axy + xxy, in-
which X is conftant. Therefore, inftead of j', putting it's value "^ T-^^ , (foe
S f 2 ia^
3l6 ANALYTICAL INSTITUTIONS. BOOK IV.
in this cafe w = i, and m = o,) it will be xxy — >^ = ^xy — ayx 4 xxy
— xyXy in which no fluxion is conftant. Whence, making y condant, it will
be found to he- xx -\- xx + ax :=: J>J> ; and, by integration, xx -^ ax ^ yy,
which is an equation to the hyperbola.
EXAMPLE ir.
L.et the equation be — ^ — ^ — ^ = fffJUf^ jn which the fluxioa
'■ yy aa + xx
yX is aflbmed as conftant. To transform it into another, in which there is no
confl:ant fluxion, becaufe in this cafe it is m zzy, the value of j' to be fubfti-
tuted will be -^^' "" "^Z " ^^ ^ ^nd therefore the equation is — — — ^ —
J- '
t^xy — xxyy — xyy x ___ aax — xxx ^ y^ rcduce this, making xy a conft:ant
yxy aa + AX c» •
fluxion, in confequence of which it will he xy + xy =i o, that is, — j' z: — ;
, , . I i- t rt* • • • ^y ' , • , ^ , *^ aax— XXX
then makms: the fubftitution, it is — -^ X + x -\ — r- — ; ,
o y y ^ aa + xx '
, ' . X XXX — aax i t • ^ • ;«^ , « a + arv , .
that IS, r- = ; — ~i and, by integration, — /a; :=: / — Ixy,
' X aax +x^ •' ° X •'
Here I fubtrad Ixy^ becaufe it is a conftant quantity. And, taking away the
logarithms, -4- = ^^ — ~ ^ that is, x^y z: a'x + x^k.
EXAMPLE IIL
Let the equation be — -^ — =^ = ^ — =^ , and yx a conftant fluxion.
Therefore, inftead of j', I put it's correfponding value, - ^^ " ''y^ ~-^'* ^ and it
will be — -^ + ~ =: — ^ — — . in which there is no conftant fluxion.
Wherefore, taking^ conftant, it will be xx = xx + yy. Which equation is
the cafe of § 49, and therefore ii*s reduction is known.
54. The
SECT. IV. ANALYTICAL INSTITtTTIOlTS. 317
54. The method explained in the foregoing Sedion, at ^ 24, maybe alfo
of ufe in difFerentio-differential equations, by proceeding nearly in the manner
there purfued. Here is the practice in fome Examples.
EXAMPLE I.
I refume the formula of the firft Example of this Sedion, — — n ^^'^^. . ^^ ,
in which u z^ ^/ xx -\- yy is aflumed'conftant. It will be ^-^^ zz lyx + xy*
m.
t "*• •
I prepare it after the following manner, -1-4- JL x x :=: — ^ " , where I ob-
•^ ac X 2y
ferve, that the two quantities under the vinculum are integrable, by means of
the logarithms. Therefore 1 make -4- + — rr ~, and therefore Ix -f- l^y
"zi Ip -\- I'w, (I add /«, becaufe of u conftant,) that is, x\/y z: fu. Where-
fore, in the propofed equation, inftead of -: j- — , fubftituting it's value
— , and, inftead of x, it*s value -^, it will be ~- zz -^ --^3 or p ±:
-^ ^; and, by integration, b + p zz . But p zz ^~, and
2ac m + i X. 2ac
therefore, laftly, hit + x\/y zz — -. — , as in the Example quoted above.
m + i X 2ac"
EXAMPLE IL
Let. the equation be "" "* "^^^ _, J^ ./^ ■> in which j'^c — xy is conftant.
The fecond fluxion x, divided by the conftant xy -- xy^ will give us an
integrable quantity, and therefore I write the equation thus, ■ ^ _ . ==
9
3I0 ANALYTICALINSTITUTIONS.
BOOK IV.
X y yx — xy
== .y ^^^ ^ obferve, that, in the fecond member, the quaniity
XX -j- yy X ^ XX -j- yy ■ ••^ '
yx — xj is fnmtnable when it is divided by yy ; therefore I prepare the
equation according to this method, and it will be ."" "^ . . rr ^1
y ^y XX +yy X S/ XX + yy
X -^'^ ~ "^^ . Make -^^ ~ ^-^ ~ p^ and, by integration, — z= ^. Whence,
making the fubftitution, we fhall have .~ "^ . z: . "^^^ ^ from whence
yx -^ xy xx+yyxVxx+yy
we can expunge x or _y, by means of the equation — — p. Expunge x from
the fecond member, by putting it's value py in it*s place, and we fhall have
^x__pp
. , ; and, proceeding to the integration, it will be
yx - xy \ -If pp X ^ I ■\- pp ^ ° '
."" *^ . =: , , that is, -: — '-^ :z -t==z= , inftead of />, by re-
yx ^ xy \/ I +PP y^-'^y \^iyy + xx ^' ^
ftoring it's value — . .
y ■
In this integration the conftant yx — xy might have been added ; but
whether it be added or omitted, the reduction of firft differences to finite
quantities, in each cafe, will always give the conic fedions.
g^, I faid before, at § 52, that when the differentio-differential equations
contain both the variables, there is no general method to reduce them. One^
however, may be affigned, which, though it does not ferve in all cafes, yet is
very general in it's kind, and comprehends all the infinite number of equations,
which may be referred to thefe three following canons. By the help of this
method, the given equations are transformed into others, in which one of the
two variables is wanting, and confequently they may be managed by the method
of § 49.
The firft canon comprehends thofe which are of two terms only, and are
expreffed by the general formula ax"'^^ =z yy^^^y, in which let x be taken
as conftant. To reduce this equation, make x zz c , and y = c"f, where ch
a number, the logarithm of which is unity, and b is an arbitrary quantity to be
determined afterwards, and », /, are two new variables. Now, fince x =. c ^,
and y = f*/, by the rules of the exponential calculus it will he x zz be ^u.
lu
if = ^f X « + hiiii, y zz ( i + c*tu, y s: c* y, i -\- liU 4- titu + /«.
But^
SILCT. IV, ANALYTICAL INSTITUTIONS.
3»9
Bur, making A' conftant, h is x = o, and therefore hc^" X u + h'uu zz o, or
u zz — bi(u. This, being fubftituted, inftead of u, in the value of y, will be
y zz c" X / + 2}u 4- r^TTj X i^^'^' In the propofed equation, fubftituting
the refpedive values inftead of x, y^ and their differentials, it will be changed
into this other, ac x » X ^ '^ = ^ ^^ X f / 4- f /« ■ x c x
t -\- 2tu + J .-b X tiiu, that IS, ac "h'^u'^ zz c ^^ t^ X t -\- tuf
•-X / + 2/« + I -h X tuu.
Now, to free this equation from exponential quantities, that is, to take c out
of it, it will be neceffary that n + p — i = bm + hp^ by which the value of
the affumed quantity h will be determined, that is, h zz " ^^ " ' . Whence
the equation will be -^J^JHE^LUL - ^ x IIH^^-" x
m 4- /.K
'i + 2/« + ^ — " ^ X /^V/V, which, becaufe it contains only one of the finite
m — « + I
« +/
variables, that is, /, will now be fubjed to the above-cited rule.
Now, lince we have found the value of Z> z= "^ » ^^ eafily appears
what fubftitutions might have been made at the beginning, that is, .v =
c m+p , and y = tf% in order to obtam our intention.
To go on with the operation according to the method of ^ 49, make
« = 2/, and therefore u z: z'i + zL But the fuppofition of x con-
ftant has eiven us « — — km, that is, u = — ^ X zztt» Therefore
" m + p
we {hall have ' ~ ? 7 ^ X 22// =2/ + /i;, whence / = ■ '""'" -^
X 2// — ~ . Wherefore, fubftituting in the equation their refpedive values,
inftead of « and /*, it will be « x ^-= ^-Z' X z^i^ = /* X / + zt'i^^^ X
ir^ X zU - -^ + 22// + ^^ X 22///i or, dividing by i^'\,
and multiplying by 2, it will be i? X ^-44=-^^ x z^"^^} zz /« Xsi +/«/"*
X
^20 ANAXYTICAL INSTITUTIONS. BOOK IV»
V I + ^^ •- « + / ^ 22/ + r.""" "^ ' X /2'/ — z ', which equation is now
reduced to firft fluxions. It is eafy to perceive, that, to reduce the equation,,
it would be fufficient to ipake x rz c m+P , and y -^ c x t.
In this general equation, which I have now reduced, t fuppofed the fluxion
jc to be conftanti yet it would make no difficulty in the method, that, in any"
propofed equation, fome other fluxion different from x (hould be made
confl:ant. For, by § ^^^ the propofed equation may be changed into another
equivalent to it, in which no fluxion is conftant, and then the faid x may be
made confl:ant.
EXAMPLE I.
Let the equation be xxj — jyy, in which x is conftant. I write it thus,
XX = yy" y. This being compared with the canonical equation, it will be
a ■=. I, m :=. i, />— i, n zz i; whence, thefe values being fubftituted in the
general differential equation of the firft degree found above, we fliall have ^zzt zz
— - — X izzi + iiz^i — z.
EXAMPLE II,
—I- ..—I-— »•
Let /> =: I, « zi — I, 7« n — I, or the equation ax" x = y'' y"^ j/r
or — = -=^, in which ,y is a conftant fluxion. In refpcdi: of this, the method
will be of no ufe, for we ftiall have p + m = c, and confequently every one
of the terms, of the general differential equation of tiie fiul degree, except the
laft, will be infinite.
But, in this cafe, the reduction is eafy, without any further artifice. I write
ike equation thus, xy n ayyx. Now the integral of the firlt member is
Ky — yx, that of the fecond is jrayyx. Therefore the equation is xy — yx =z
iayyx ± <^.v. .
56. The
SECT. IV. ANALYTICAL INSTITUTIONS, 3^'
56. The fecond canon comprehends all thofe equations, in which the fiim of
the exponents of the indeterminates, and of their differentials, is the fame in
every term. Suppofing x and y the two indeterminates, aiid x to be conltant,
thefe are reduced to the cafe of § 49, by putting x ~ c , and y — c"i ; c being
flill the number, the logarithm of which is uniiy, and «, /, are new indeter-
minates. To (how the method, I (hall take the equation ax*y~'^^^x''y^~^ -f-
hx'y" "" icy "^ = j* which, though it be but of one dimenfion only, and of
three terms only, yet the method is general notwithftanding, and will ferve for
any number of terms and dimenfions, if the conditions be oblerved.
Therefore I make x — c^^ y — "i; it will be ^i* =: A*'; and, becauTe x is
conftant, we fliall have c% + c li'u = o, that is, « z= -— u'u. It will be alfo
y zz "i + c"lu, and y zz <" X ^ 4- 2.ut + t'l'u -j- /'/. But 'it zz ^ im-y there-
fore y ^:z c* X / -f- I'u'i. Wherefore, thefe values being fubftituted in the
propofed equation, it will b^ at" ~ ii^ X i-^i-tlh ^ -\r bt it X
i 4- tu^~^ — 'i-{- I'ut. And, becaufe in this the indeterminate u is wanting,
we may proceed by the method of § 49.
Make u zz zi ; it will be u — zi + zL But u zz — m zz — . zzli ;
therefore '/ zz ^ ~ zti . Wherefore, fubflituting thefe values, we thall
have at-'"-\fif X /Ti^l^-'' + br'-'z^l'' X FTTdf-'' =-±
Z
+ ztt^ or ct z^t X i-hzO ^ + ht z^ t x ^-{-z^ ^ =
+ 2;/, a differential equation of the firft degree. From hence it may be feen,
that the propoled equation might have been reduced at the beginning, by
fzi y fid
puttmg ;? — r , and y "=. c t.
For example, let the equation be xxy — yxx zz yyy. To bring this to the
canonical equation, I write it thus, xy xy — y xx zz y. Then it will be
a zz I, m — i,pzzi,nzzoyb = — i, q zz 2. Wherefore, thefe values
being fubltituted in the diflfereniial canonical equation, here before found, we
(hall have the equation reduced, t~^zi x 1 -t- ^^ — f" zzt zz — -^-j-z/j
or — — = -^ — . that IS, zzi — zziU zz — tf%,
Vol.11. Tt if
321 AHALYTICAL INSTITUTIONS, BOOK IV,
If we proceed on to the integration, it will be — ~— zz — - , and therefore,
by integrating, / H z: + /, (where/ is a conftant to complete
the integral,) that is, //z + 2 = — / + ftz. But, by the fubftitutions,
2Z=— , X :=. c , y zz c i, \t will be?< = ~,/ = -^,/= - ^ / , and
therefore z = ."^^^ . : wherefore, fubflituting the values of / and z, we ihall
yx •'
57. The third canon comprehends all thofe equations, in which one of the
two variables, whatever it may be, together with it's differentials, always makes
in every term the fame number of dimenfions. But we mud here diftinguifti
two cafes. One is, when the differential of that variable is conftant, which
forms the fame number of dimenfions. The other cafe is, when the differential
of the other is conftant.
As to the firft cafe, let the canonical equation be Vx^^y" +
Qx"''"x'y'" =z x"y, in which the fum of the exponents of x and x
is the fame in every term. P and Q^are any functions of y, and x is conftant.
To reduce this equation, make x zz c , where alfo ^ is a number, the logarithm
of which is unity, and « is a new variable. Therefore it will be iv =r ru; and
differencing again, making x conftant, it will be f ^ + c uu zz Oy that is,
» =1 -^ itu, Thefe values being fubftituted in the equation, we ftiall have
"Py" ^ ^ Qhy'" ^~* = ^'"ji which, becaufe it does not contain a, will be
iinder the canon of § 49.
Therefore I put u =z zy, and it will he u = zy -\' zy ; but u zz '•^ uu zz
»— z*y ; therefore we (hall have zy •\- zy zz -— zzyy ; and thence y =3
. T ^^^^ " '^ . Wherefore, thefe values of u and y being fubftituted in the
equation before found, it will be Vy^ + Qz^"*^* n: — z" y*" -^
z^^^y" z ; and, dividing by ^'" ', it will be Py + Qzy = — ^*^V
— % " ;z, an equation of the firft degree. Therefore we might at firft have
made x zz f , and thus have reduced the equation at one ftrokc.
3 For
SECT, IV» ANALYTICAL INSTITUTIONS, 323
For example, let the equation be zaxxy + axxy "=. ixxyy + 2xxyy, in
which let x be conftant. Put x n r-^, and therefore x r: zyr^, and x zz
c '^^•^ X 2^* + zy + j/z. But jv is conftant, and therefore zzyy + ^ + >'^
= o, whence j/ zi Z-^^^I^. Now, the values of x and x being fubfti-
tuted in the equation, we (hall have lazy^ + azyy r= 22)}' -f- lyy ; and,
fubftituting the value of j", it is 2^2^ + azy x ~ '^•^ ~" ^"^ = 22y^ + 2j/ X
azz — 2z
, that is, dividing by yy, az^y — azz = — 22?, or ay =
. And, by integration, ay •=. ■\- — . Laftly, reftoring the
value of z, which is given from the fuppofition made of ;v — c^"\ that isj
% = — T- , we (hall have the equation reduced, ayicx =: xxyy — tf;ejt;y.
58. As to the fecond cafe, let the canonical equation be YiTy +
Q*'""""^^^**'"" = ^'"""'if, in which let y be conftant, and P, (^ any
fund ions of j'.
Put, as above, x = f*, and therefore x =: c\, x -=. c^'u -\- c^'uu. Make the
fubftitutions in the canonical equation, and we (hall have Vy" -{-Quy'"^"
= u"* + ii"^^u, which, becaufe it does not involve k, is ful:)je(fl to the
canon of § 49. Therefore I put u in zy ; and, as j/ is conftant, it will be
u -^ zy ; and then making the fubftitutions, we fliall have Vj^^^ + Qz^y"^^
— - J'+z ^2;; and, dividing by 7 , it will be Vy + Qz"^
= 2
m+l . , m— I
ZZ 2 " • -
y + z 2;, an equation of the firft degree; which might have been
reduced at once, by putting, as above, x n r -''.
For an example, let the equation be 2xy zi ax — yx, in which let y be
conftant. Therefore, putting x zz r^^, thence x zz zy X r-^, and x zz c-^^-^
X zy + zy -{- yz. But j) is fuppofed conftant, and therefore j^ =: o, and
thence^ ii* = r -^ x zzyy + zy. Wherefore, making the fubftitutions in the
propofed equation, we (hall have 2zyjf zz azzyy + azy — zzyyy — yyz ; and,
dividing by y, it will be 2zy zz azzy -\- az — zzyy ^-yz, which is a diifer-
ential equation of the firft degree. '
T t 2 To
324 ANALYTICAL INSTITUTIONS, BOOK IV.
To go on to the integration, 1 divide the equation by az •— jyz, whence it is
-^ z: zy 4- — , or — — — — := zy. And now, if you pleafe, making
ufe of the method in § 24, by integrating, we (hall have
a-yV X a a —j^
+ m ; and, laftly, by reftoring the value of 2 zi -^, we fliall have the equa-
tion reduced, yx -{■ xy ■=: ax, where the confhant m is negleded, which was
introduced in the integration.
This example has ferved to (how the application of the method ; for other-
wife fo many operations would have been unnecefTary. Indeed, the equation
itfelf, ^xx — ax — y'x^ might have been reduced in an inftant, by only tranf-
pofing the term jvic*, and writing it thus: ixy -{- jyi^ = ax \ for, asj/ is conftant,
the integral of the firft member is yx + xy^ as plainly appears.
59. To what has been already faid, concerning difFerentiodifFerential equa-
tions, in which no firft fluxion was taken for conftant j another method may be
added which is more univerfal, and which will ferve for all fuch as are compre-
hended under this canonical formula, z^ x x •{ y — y"^y ; in which
z is any how given by the functions of x and y.
To reduce this, appoint the fluxion — for conftant, where q is any how
given by the functions of ss and y. Then make — =: />. Now, becaufe ~
• . .. . . .. xq
IS conftant, it will be, by differencmg, qx — xq z:. o, that is, x zz -^ ; or,
inftead of ~ , writing it's value p, it will he x z= qp. Belides, make y — «/>,
and taking the fecond fluxions, fuppofing / conftant, as being equal to — ,
which is conftant, it will be y = up* Therefore, in the canonical equation,
fubftituting the values thus determined, inftead of x, x, y, and y, we fliall
have the equation z q qp -| 1=: u up ; and, dividing by
jf , It Will be 2 ^ ^ + =: u u, or q q ^ — ^ . And,
«+ 1 «/+ 1
by integration, -^ + f =1 — , and therefore a = 2 x
lECT. IV. ANALYTICAL INSTITUTIONS* 325
a +/7>-I-iXp- . But « r: 4- =: -^^ . Then -^ zz 2 x
Iw + l
applying this to the canonical equation, it will be ;» — i, 2; rr — j therefore
the reduced equation is ~- = "— X ??~~Hh~2^ *• I take q rz x \ it will be
^' +»«-t-i^>an equation reduced to firft fluxions,
^o. Concerning this laft equation we are to obferve, that, if the quantity z
be given by x and y in fuch manner, that to the qiiai)iity q fuch a value may
be afllgned, alfo given by x and _y, that the indeterminates may be feparable in
the equation, and therefore that it may be conftrudlible, either algebraically,
or, at leafl, by quadratures, we may have the curve, on which the differentio-
differential equation depends. And, becaufe the values are many which may
be a/Tigned to q^ the curves may be many alfo, and every value of q will fupply
us with a different curve, either tranfcendent or algebiaical, which will fatisfy
the queftion. Let the equation be • •" ^ + — -^-^ iz aayy. Now,
xxy
aa
qy __ xxy
aa
-^ r: — \/;va: + 2^^,' that is, — =: xx*/xx + %g\ the integral of which
plainly depends on the- quadrature of the hyperbola, and the curve will be
tranfcendent,
61. In paffing from firfl: to fecond fluxions, either we alTume no fluxion for
conftant, or we afTume fuch an one as is moft eligible, as faid before. Where*
fore, in finding the integrals of formulae of the fecond degree, becaufe we
know what fluxion had been fo taken, we know alfo how to proceed, and the
rules for it have been explained.
But there are an infinite number of problems, which require fecond fluxions,
without our knowing what conftants are involved in the formulae thence arifing.
It often happens, that we cannot arrive at the analytical expreffion without the
afTiftance of the conftants; and likewife, it fucceeds fometimes, that the equation
may be refolved without recurring to the conftants. Thefe two cafes, therefore,
ought to be exan\ined, and we (hould leek for fome criterion, to diftinguifh one
from the other. And, becaufe examples will perform this better than any
thing elle, I (hall take this following.
It is required to find fuch a curve, that it's abfcifs, raifed to any dignity, may
be direclly as the fecond difference of the ordinate, and reciprocally as the
fecond difference of the fame abfcifs. Therefore we Ihall have this analogy,
m
X
326 ANALYTICAL INSTITUTIONS, BOOK IV.
X • Ar X*, a * h. And confequently hx x ~ ay. In this equation I find the
fecqnd differences both of the abfcifs and of the ordinate ; but I cannot know
what conftanc was affumed, or whether any conflant was affumed or no i lo that
I cannot know what courfe I am to purfue.
I fay, in the cafe of this equation, that no poffible curve will fatisfy the
Problem, fince we pafs from firft to fecond fluxions, without the afTiftance of
conllants. On the contrary, the conftants being determined, we may find
curves that will fulfil the conditions of the Problem, but they are infinite in
number, and different in their nature, as varying by ihe change of the arbitrary
conflant which is affumed.
To diftinguiQi one fpecies from another of thefe equations, we may make nfc
of the method, or canon, which will arife from the following Examples, and
which will ferve in all Inch cafes, wherein the Integral Calculus does not tor-
fake us.
EXAMPLE I.
Let this equation, 2 x x + -^ X y z:zy y,ht propofed. I fay, this
is one of thofe formulae to which we may attain, without taking any quantity
by way of a conflant. Let the variable z be any how given by x and y.
The demonflration will be made general, as far as that can be done, by
taking the fluxion ~ as conflant, in which y is a fundion of x and j, any
how combined. Wherefore I put JL — ^'j and, becaufe the firfl member of
this equation is conflant, the fecond p will be fo too. And, as h is x zz. qp,
if we pafs to fecond fluxions, it will be x zz. qp,
Now make J) n up; and, taking the fecond fluxions, on the fuppofition of
p being conflant, we fliall have y z: up. Wherefore, fubftituting, in the
principal equation, the values thus determined, there will arile the equation
z q qp •\ zz u Up J and, dividing by p , an equa-
tion will arife which is free from the unknown quantity p, and from it's
funftions, that is, z* q" q + ^ = ti^u. Taking the fluent, therefore,
by
SECT. IV. ANALYTICAL INSTITUTIONS. 947
by the rules before explained, not omitting to add .the conftant g, it will
+ ^ = ^T7» which equation gives us « z: 2 x
be ?■
n + I X «
9"*^^ + ^^ + ^ "'"*" '• And, becaufe y — up z=. — , making the neceflary
fubilitutions, we (lia'l have the equation reduced to it's limpleft ftate, that is.
• ^x .. ' m-^l \ '. D'''+'
From the foregoing manner of operation, we may deduce the following
Corollaries.
T. The quantity z being determined, if the laft equation can be con{lru6!:ed,
even by qiiad(atures, fo that it may bur be executed, it is plain that infinite
curves will agree to our formula,- which will change their nature by changing
the affumed conftant fluxion — . And every value of the quantity q will
fupply us with a new local equation, either algebraical or tranfcendental.
II. Although, if the value of the fymbol q be altered, different curves will
arife ; yet it is certain, that, if we make the additional conftant g zz. o, we
(hall always have the equation y := zx. In which cafe, it matters not what
fluxion ~ is taken for conftant ; becaufe, the given quantity g vaniQiing, the
variable q alfo vanilhes.
III. Here, then, is a token by which it may be known, that wc (Iiall arrive
at our primary equation, without aliiiming any fluxion as conftant, and that, in
fuch a fuppofition, it's integral is zx zz y. tor, recalling to our view the
cxpreflion z x x + — x y — j'j'^o, and agam dinerencing the
integral zx — j, without affiiming any conflant ; thence we (hall have zx 4-
ZX — j' i il> by meins of thcfe two laft equations, we Qiould mike to vanifh
out of he piincipal ti>rmula, firft y^ then x, with their tunc-tions, we (hall (ind
z .^ X -\- z z,. -- z X X — z zx =1 Of and^ y ^y ^f
IV. The
328 ANALYTICAL INSTITUTIONS. BOOK IV.
IV, The primary formula being managed as above, and the equation being
found reduced to the firft degree, that is, y zz — X ^'"^^ + gm -i- gi ^ ,
we fliould pafs on to the integrations, which (bmetimes will be out of our
power, according to the various values oi the exponent m of the fradion z
given by x and by y, and of the quantity — , which is taken for conftant.
However the reft may proceed, the aforefaid values being determined in infinite
particular cafes, the local equation of the curve is alfo difcovered in finite
terms ; when we proceed to the firit, and thence to fecond differences, keeping
flill the conftant — , which our principal formula will prefent us with. Bur,
changing the conftant, different f)rmulse will be found. I can affare nothing
further, but this is very maniteft, by turning back again the fteps of the Analyfis.
V. The fame thing happens by taking the firft fluxion — for conftant. For,
making the operation according to the method, (which 1 fliall omit for the fake of
brevity,) we (liould arrive at the reduced equation xz:z -^ ^ Xmg + g^" ;
in which it may be obferved, in like manner, thar, making ^ — o, it concludes
by reftoring the equation x =z —~ , exprefted by firft differences.
VI. Afl!\iming fome limitations that are more fimplc, that is, m "zz j,
z =: XX, and q zi x ; if we make ufe cf the conftant — , as in Cor. IV, the
formula j) ir — - X q"^^ -h ^'« + g'"^^ will be changed into this following,
y zz xxy^xx + 2g, which admits of analytical integration. Now, making ufe
I
of the expreflion contained in Corol. V, that is, x rz — — — - X t/^g -r ^T ',
arifing from the affumed conftant — , and keeping ftill the limirations of
m "= If z = xXi and q "=■ Xy there refulis the expreflion '— — rri/, which
is not inregrable without the help of the logarithms, and confcqueutly gives us
none but tranlcendent curves.
Therefore
SECT, IV. ANALYTICAL IKSTITUTIONS, ^2^
Therefore it is plain that we may arrive at the differential formula of the
fecond order, 2; x x -^ XJV — JKJ* without takmg any conltant ;
in which cafe the integral zx zz y will take place j or, fixing for conflant the
made as before, that were found in thefe fuppoficions.
fluxions — , — , for example-fake, and then the fame integrations will be
9 9 ^
EXAMPLE II.
Let us take the equation x^x zz j^ + yy, I fay, we cannot arrive at it,
without taking fome conftant, except in one cafe, in which it is ;;z — — i<
To (how this plainly, I (hall manage the formula in the manner following.
Firft, I take x for conftant, and thence x zz o. Then •— -4- zz y, and by
ilitegrating, /~r- z=^*, or -7- = r. Make c =z z, it will be yk =■ Izy and
therefore J/ zz — . and, inftead of y, fubftituting this value, we (hall have
~ zz c^. But €^ = z, therefore x = z, and x zz z zz c^ ; and therefore
— zz J/, an equation to the logarithmic.
Secondly, I propofe to inveftigate how it may fucceed on the fuppofition of
another conftant, jf for example, whence y zz o, I make x := sy -\- cy,
where 5 is a new variable, and c a given quantity. I go on to fecond differ-
ences, and it will he x zzsy ; and, making the fubftitution, it is ^ sy := yy, or
X*" s zrj^. But J? = -~- 5 then ss + cs zz ji?""'"jf; and integrating (omitting
— m+i / —m+i
to add a conftant), j-ss + cs iz ~ — ; — , or x + f zz v — \- cc. But
" * — Z«+I ' — OT+l
y-W + I
y z:j/V V cc\ therefore ' ■ =^.
* Sec $ 46. Editob.
Vol. II. U u I proceed
33^ ANALYTICAL INSTITUTIONS. »00K IV.
I proceed to inquire if poffibly the logarithmic curve maybe concealed under
the lad formula, which being found above, in the hypothefis of x being
conftalnr, it may likewife have place in the other fuppofition of y being
— m+l
conflant. Making f z: o, it is neceflary that the equation s/— zz x
fhould be verified, or elfe ix"" zz — m -]r i x ^x. And, that the equa-
tion may be found, the fame quantity — m -|- i, both in the co-efficient and
the exponent, ought to be = 2 j for this to obtain, it follows, that it muft be
w = — I.
Therefore, in the formula x x zz: y -^ yy, by limiting the value of the
exponent to ;7i zr — i, we come to a differential equation of the fecond degree,
without affuming a conftant, the integral of which is the logarithmic expreffion
. — z= y. In any other cafe we could not obtain the forefaid expreffion,
without fixing upon fome infinitefimal quantity of the firft order as a conftanr.
EXAMPLE III.
It remains that we (hould propofe a differential equation of the other clafs,
at which we cannot arrive without affuming a conftant.
I refume the problem : To conftrud a curve, in which any dignity whatever
of the abfcifs may be in a dired ratio of the fecond fluxion of the ordinate,
compounded with the inverfe ratio of the fecond fluxion of the abfcifs.
The equation h i?x x z=. ay. Make x zr c/p, y -zz up; and perform the
operations, as in the iirft Example. Taking the fecond fluxions, we fliall have
X =: p^, y = «/; ^^^f fubftituting thefe values, it will be l>x"^^ = <?»; and
by integration, Z^;";"*^ = «^ ± j:- But y = up — — ,• then ay = JLjtx^q
q: i!fL. Making ^ = o, in this cafe, whatever be the value of the fymbol q,
it gives us a different curve, if alfo we do not put the exponent »; z: o, by
which the hypothefis will be deftroyed, and the problem changed. The
fame thing may be faid if we make conftant the fradlion ~; and from
8 hence
SECT. IV. ANALYTICAL INSTITUTIONS. ^JI
hence we may conclude, that it is not poffible a differential equation of the firft
degree, without the benefit of a conftant, (hall reftore our formuize, when it is
differenced again ; for, if it were fo, it would be manifefted in any affumption
of a conflant j and alfo, the analyfis evidences the contrary.
PROBLEM I.
62. The radius of curvature being given, any how expreffed by the ordinate
of a curve, to find the curve itfelf.
As, when the curve is given, to find it*s radius of curvature, it is called the
Direct Method, or Problem of the Radii of Curvature, of which we have
treated already ; fo, when the radius of curvature is given, to find what curve
it is to which it belongs, is called the Inverfe Problem of the Radii of Curva-
ture. Wherefore, let the radius of curvature be zz r, and be any how given
by _y, the ordinate of the curve ; and we may take any one of the formulae for
the radii of curvature, which we pleafe ; but, firft, for the curves referred to a
focus y as, for example, -rr^ — — , in which x is conftant, and i is the element
• ' ' ^ ' xss —yxy
of the curve. Then we fhall have the equation r n: -rr^ — rr.; or clfe, it being;
■» xss ^ yxy ' ' o
ss zz XX ■¥ yjy it is s's "=. yy, becaufe of X conftant, and r n:
xys-» yxs
To reduce this equation, I make ufe of the method of § 49 j and therefore
I make s =: px, whence }' zi px» Then, making the fubftitutions in the
equation, it will be r = .''^^ . , or elfe ^-^ ~-^^ ' =: —,* and then, by ince-
gration, becaufe r is given by y^ it will be — =: /~ ± h. But p z=i —r-
r: \^^ ^'^ ^^ . therefore the curve will be -^^ ir /^ ± ^, an equation
Vk^+Jy "^ »"
reduced to firft fluxions, becaufe, r being given by y, the integral /— may
always be had, at leaft tranfcendcntally.
U u 2 Another
ANALYTICAL iNSTITtJTIONfl.
BOOK IV.
Another way. I write the equation, r ==
ys^
-rr, in this manner,
Then, from the point B, (Fig. 149.) from whence
proceed the cdinates BE of the curve required
AEC, I draw BF perpendicular to EB, terminated
at the radius of curvature E Q^J and, making
BF = />, EF z= J, by the known formulae of the
oormal and fubnormal, it will be ^ n: ■—, p zz
^r- , QX y zz. — . And, by taking the fluxions,
X y
on the fuppofition of x being conftant, it will hQ y zz - ^-^^ "" P^> ^ And, mak-
ing the fubftitutions in the principal equation, it will be — r: xl* — px^ +
i.»«
■ yyp + fyy. And, becaufe it is
^^ . But i zr — ; therefore — n aax
y y y iJ
X zz — f it will be ^ z= qqy + ppy — ypp. But, becaufe of the right angle
EBF, it is pp zr gq, — yy, and pp =■ qq — yy. Wherefore, making the
fubftitution, we (hall have ~ iz iqy — yq ; and, multiplying by y^ and
dividing by qq^ it will be ~ zz ■■ ""-^ y . and, by integration, it is /-^
± h ^ ^. But ^ = 4^; therefore f^ ±b - ^^ .
It may be done thus more limply, by avoiding fecond fluxions.
Taking the infinitely little arch EC, let the chord CED be produced, to which
let BD be perpendicular. Now, if we make BD z: />, by what has been faid at
§ 115, Sed.V, B. II, QE — rzzM- , and therefore ^ zz p; and by integra,-
tion, becaufe r is given by ^, it is /^ ■±, b zz. ■ "^^ ; for p zz -——===.,
by the place now quoted.
hy
Let it be r = -^s/ aa -h bb -, then it will be /
w aa + hb '^ XX -^-yy
and by aftual mtegration, (omitting the confl:ant h for greater fimplicity,),
h X
' = ■ - , and therefore b^X*" + bY zz «V + b*x\ that is,
'^ aa -^ oP ^xx +yy
by ZZ ax, which is the logarithmic fpir^l of Example V, § 128, Book II.
Inllead
9
SECT, IV, ANALYTICAL INSTITUTIONS, 33 J
Inflead of the radius QE, let the co-radius HE zr z be any how given by
the ordinate > Becaufe of fimilar triangles, EBD, QEH, it will be EB . BD-
:: QE . EH : that is, y . /> :: ^ . z, and therefore z =. ^, or — = -^;
and by integration, /— ± ^ — ip» Make z — y, then /-^ ± h =/-^ i
» y p
and by integrating, /y ::: /p + /-^ *, that h, y z=z ^ , But p = j!f . ,
then hs/xx •\- yy -zz mx, and therefore Ay = X\/fnm — /y^, whicli is th©
logarithmic fpiral ; and, when h — h, m :=. \/ aa + bb is the fame as the
above-cited.
^'},» For curves referred to an axis, the formula of the radius of curvature ij
rr-, putting X conftant j and therefore the equation will be r =:: _ ■■ •• .
I put y =r qx^ whence y n qx ; and, making the fubftitutions, it is r ir
XX + '-y * . and, inftcad of i^, putting it's value -^ , it will be r =: — :7"^^>
that is, -^ = — ^!^, . And> by integration, /— ± ^ = -; = . But
J = 4- J therefore /— ± -& z: -~
Let r =: 4VL±!f^i then it will be /. ^''"\, ± b :=: — -l==r • And,
by adual integration, omitting the eonftant i>, it is — — "^ ,
that is, 2yy z= ^^c ; and by integration, yy zz ax^ which is the parabola of tlie
firft Example, § 122, Sed. V, Book II.
Inftead of the radius, let the co-radius be given, which make zi 2, the
fornmia of which (fuppofino X to be coi)lUnt,) is "^^ ~-^ Then 11-22! — z\.
and making y — qxy y zz qx, and making the fubftitutions of thefe values of
y and x, it will be -LiLLii? = z, that is, -^ = r^ ' •^"^' ^^ Integra-
tion,/— + /6 — — /y'l -r ^^. Whence, if 2, or the co-radius, be in fucb
manner given, by y, as that /— be a logarithmic expreflion, we (hall have a
* This equation, as well as the fubfequent work, would have been clearcc and topler, if m had
been put for the conliant number oi which the logarithm is b, ii.mroiu
dii&reniial
33+ AKALYTICAL INSTITITTIONS. BOOK IV.
(litferential ecjiiation of the fiifl; degree expreffed after the ufual mannerj in any
othei cafe, it will be expreffed by logarithmic quantities.
Let it be % = ^ —^i we fliall have the equation / — t^ ± ^ zz
— 1\/ 1 i (^q. And, by adual integration, (omitting the conftant ^,) it is
/ ^ zz I ■ ^ r and therefore — ^, — = — ; . And, fubftituting
the value of q, it is 2yy ~ ax, and, by integration, it is yy = ax, the fame
parabola as before.
64. In the fecond place, let the radius, or co-radius, of curvature be any
how given by the abfcifs x ; it is plain that, in this cafe, we cannot make ufe of
the fame redudions we did in the firft, becaufe we cannot have the fluents
/Z-, or/~, if r and z are given by x»
Taking, therefore, the formula of the radius of curvature, in which x is
• • , • •■\ 3.
conftant, that is, '^'^ '^^' "' for curves referred to an axis, (for, in thofe referred
- xy
to a focus y the radius, or co- radius, cannot be given by the abfcifs,) it will be
r — ^^ ''"•f j'^^ , and therefore, in the fame manner as before, I put J) s= qx^
^ xy
whence y — qx, yy = qq^x » and, making the fubftitutions, r = -^-y + y ^ t^
that is, — zz , JlL.j , and, by integration,/-^ ± ^ = — li. , which is
an equation reduced to firft fluxions ; becaufe r, being given by Xy the fluent
jJL. niay always be had, at leaft tranfcendentally. And, fubftituting the value
of ^, it is /— ± ^ = ■■ /.- . •• •
Let it be r = ^\/ \aa — zax -^ then it will be / -= ± >& =
2y j^aa — tax
"•^ =::., And, by adual integration, omitting the conftant h, it will be
. And, by fquaring, and reducing to a common
Va-^ ■\- yy
^ \^^aa — 2ax "-y
2a t^'x^ +J'*
denommator, it is ^aaxx — laxxx — 2axyy zz o, that is, y =z xv ,
equation to the cycloid of § 131, Se(5t, V, B. IL
2a — X
an
Inftead
SECT. IV. ANALYTICAL INSTITUTIONS. 335
Inftead of the radius, let the co-radius be given ; then z =r — — ^ ,
And putting, in like manner, y = qX, it is y — ^x, yy ~ q^XX -, and making
the fubftitutions, inftead of j/' and yy, it will be 2; iz; ^.^-—4—^, that is, -^
zz — "" ^ : and, by iniesratlon, /— ■±,h zi f — — ^. But the integral of
the homogeneum comparalionis is the arch of a circle ; therefore, if the co-radius
fhall be given in fuch manner, as that f — is alfo the arch of a circle, and
thefe arches fliall fo correfpond, as to be to each other as number to number,
we fliall have the equation reduced to firft fluxions, and expreffed in common
quantities.
Let z =: i\/2ax — xx; then it will be / — ; — n: / "" ^ ■ . But the
integral of the firft member is the arch of a circle, the tangent of which is
< ' : and of the fecond, is the ^ch of a circle, the tangent of which
X ' o
is a. Then it will be ~ = ^ = -4-; therefore y r: xk/ ^'* " "^ *
•* X * X "^ X
an equation to the fame cycloid.
PROBLEM II.
65. The radius of curvature being given in any manner, in a curve referred
to ^n axis, to find the faid curve.
«, , " • .
'The formula for the radius of curvature is ~, making 's the element of the
curve conftant; whence the equation will be r =: 4^. Call the tangent of tKe
curve /, and the fubtangent p. It will be ^ =: /, and, differencing in the
hypothefis of i conftant, it will be / = ^^^ 7". ^'-^ , that is, y = -^-^"^ "7-^-^ .
Wherefore, making the fubftitutions, it will be r = rr^^^^^^-rrr. Bur, becaufe
we
336 ANALYTICAL INSTITUTIONS, BOOK IV.
we have p -=1 K- ^ and / zi ^ , it will ht x zz. -^, i 1= --- . Then, fub-
^ y y y y
ftitutlng thefe values in the equation above, we fhall have r zz - . ^ . . But
p = \/// — JJi therefore r = — ^ A. or — =: ^ -
^ -^^ r f^tt -
yy
The iirfl: member of this laft equation is in our power, at lead tranfcen.
dentally, becaufe r is a fundiion ol s. Then, in the fecond, the indetermi-
nates will be eafily feparated, if we make ^ = — , by which we fliall have a
very fimple equation, ~ z: '^
t V I -• qq
In the formula r zz -J^ — , if, inftead of/, we had taken it's value VPp-\-yy^
ty -yi
we fliould have found r zz -^iJliLiU-.; and, making — =: z, we Qiould alfo
py -yp f
have had a very fimple equation, — = — -
zz
The two differential quantities — ■ ^ anc^ — -- — are the exprefllons of
the element of the arch of a circle. Whence, if the integral /— fhall b-e
algebraical, or fhall depend on the logarithq:is, or on higher quadratures, the
redification of the curves required, and the value of the radius of curvature,
will fuppofe the quadrature of the circle. But, on the contrary, each of them
may be algebraical, if the integral/— agrees with a formula of the circular
arch.
* •
Retaining one of the two equations, for example the fecond, -^ == — ^ — ;
becaufe i = -|- = JL^^p ^ yy^ and ;> =: -^j it will be i = ■i-i/i+zz.
Then, fubflituting this value into the equation, we fhall have y zz
===== y . Now, it being s = -^\/i + zz, we (hall have alfo ss zz
XX -^ yy ZZ P "^ '^-^-^ , and therefore X zz —:
Make
$ECT. IV. ANALYTICAL INSTITITTIONS. 337
Make the given radius of curvature r zz i + jtj. Then the equation
• — - — n -^ will be changed into this, — ; — ~ : from whence we ob-
i + zx r ° ' 1, + zz I + «
tain z = J, and therefore r — i + zz. Subftitute this value in the equation
J' =
, and it will be j/ n And, by integration,
i+zz X s/i+zz V'l +
omitting the conftant, it is j^ = v^i + zz, whence z =. \/yy — i. Then,
becaufe I retained x ^=. — ^ it will be finally x — -^ , an equation of
the curve required, on the aflumed fuppofition of the radius of curvature. It*s
conftru(ftion depends on the quadrature of the hyperbola.
. I take the formula of the radius of curvature, ~ r= ^^ "/"^ , in which no
firft fluxion is conftant. I difpofe the equation thus, -^ Y^ —. 4- = -^ .
The integral of ~ — A- is iy — Is, which I make equal to Ip, Then it
will be '^. — zz — -, and 4- zz />, and then the equation will be ~ zz
•4- X 4- • But p zz —-, and ^ zz ss := :)^x + yy ; therefore .v z:
^ — LH^, And, fubftituting this value, it will be — r: ^ -, an equa-
/ ° ' r */i^pp^ ^
tion in which the variables are feparated, and confequently may be treated in
the manner made ufe of before.
Let the formula of the radius of curvature be -^ =: — 4^, in which v is
r sx ' y
conftant. Make s = qy, and therefore 's zz qy. Then — zi — ^4- -, but
'si zz XX -\- yy "=. qqyy. Whence we have x iz ys/qq.^ i, and xs "zi
^i^v/^* — !• Wherefore, making this fubftitution, it will be ~ zz
_f
Vol. II. Xr Laftly,
$^t ANALYTICAL INSTITUTIONS. BOOK IV.
Laftly, let the formula of the radius of curvature be -^ =: — S-, in which
X is conftant. Make z ~ -^ , and therefore z =z — -?-. Then -i- z= -^ .
But X n 2y, and ii zz xx + yy zz zzyy -\-jiy. Whence — =: — ^ — ,
Therefore, after whatever manner we opera'ie, the integral/ — will always
be brought, either to the reftification or quadrature of the circle.
Let the co-radius « be any how given, to find the curve. Take one of the
three fojrmuia^ before, that, for example, ip which y is taken for conftant \ that
s
IS, — iz — — , in which it is put i =i qy* The radius will be r =:
-^ ; and, putting this value in the formula, we (hall have — = — — 7===.
•* * qxw <iq — I
But i = qy, and x =. y\/qq — 1. Whence, making the fubftitutions, it
will be — =: ^ — , But u is given by s 5 therefore, &c. *
Here it may be obferyed, that, as the integral /— is equal to an expreffion
of a circular arch ; fo the other integral /— will be referred to the quadrature
of the hyperbola, or to the logarithms.
66. By like artifices and expedients, or but little different from thefe, many
equations, or formulas, may be reduced to fecond differentials, which are
exprefled by third, fourth, or higher degrees of fluxions. And, firft, the
method of ^ 49 may be extended, (yet within certam limitations,) to differ-
ential equations of the third, fourth, &c. order. That is to lay, equations of
the third order may always be reduced to the firft order, provided that either
one or the other of the finite variables, x or y, is wantmg in them. 1 hofe of
the fourth order may be reduced, if, befides one or other of the two finite
variables, x or y, one or other of the firft fluxions, x or y, be wanting, together
with their refpeftive fun(5tions. Thofe of the fifth may be reduced, if .both the
finite variables, and both their firft fluxions, be wanting in them. Thole of
the fixth, if, befides all this, one or other of their fecond fluxions be wanting.
And fo on.
Let the equation be xy + xxy = ;t* -I- >*, in which x is taken for conftant,
I make, as ufual, ^x = y, and therefore px = y, and px = j/. Wherefore,
making
«ECT, IV, ANALYTICAL INSTITOTIOMS. ^j'^
making the fiibftitutions, we (hall have xxp + i(}§ — x*' ■{■ Jr*. Bbt >* =
f'x*'^ therefore ir will ht p + xp — xx -{■ p*xx, an equation reduced to the
fecond order. Make funher qX = />, retaining t)C as conftanr, and therefore
^X = p. Then, by fubftitution, ic will be qx + pX := xx -^ p^xx, that is,
q ^ p zz ijc -f p*x, felit ;f = -^ ; therefore 5? + »' = JL -L J^ . which
q ■ i
equation is now reduced to firft fluxions.
Let there be a fluxional equation of the fourth order, y + xy — xxy = o;
in which let x be conftant. Therefore I make pX =: j?, and thence pX ~ j'i
and px zr j/, and px n y. Therefore, making the fubftitutions, we ftiall have
p •\- xp — xxp 1= o ; an equation which is a cafe of the foregoing Example^
and which therefore we know how to manage ; and which will eafily be reduced
to firft fluxions.
The method of § 49, found fomc tiine ago by S. Count Jatnes Riccati, was
row firft known to me ; but the foregoing application, as alfo the fecond inverfc
Problem concerning Radii of Curvature, I have learned of him only fince the
fecond Tome of the Commentaries of the Jnftitute of Bolonia is fallen into my
hands. And, indeed, fomething too late for me, becaufe I was now at the
clofe of the impreflion of this my Work ; nor could I take the advantage of the
other learned Differtations, neither of P. Vincent Riccati^ fon of the aforefaid
gentleman, nor of S. Gabriel Manfredi^ therein inferted. Therefore it mult
fuflice that I have juft named them to the readers^ that they may there find
them, and be improved and inftruded by them.
(>"], Having (hown the aforefaid application, or improvemertt of the method
of § 49, I fhall go on to other equations, and to other expedients. Therefore
let the equation be -pyyy z: pxxy — ipxxy — fxxjy in which p is any how given'
by X and ^, and now the element of the curve, i, is taken for conftant. Becauf(^|
} is conftanr, it will ht xx ■=! '^ yy \ then, fubftituting this value inftead of
icx^ it will be pyyy = pxxy + zpyyy — fxxy^ that is, ftriking out the
fuperfluous terms, pxxj = py'j'j + pkxy^ or — =: 4~ + -4- . And, inftead
of yyt putting it's value — xx^ it will be — = ^ + 4- . And laftly,
integrating by the logarithms, Ip =r /y — Ix — /i, i being conftant i and
therefore p zz 4^: which equation is reduced to fecond fluxions.
' Xxz Let
340 ANALYTICAL INSTITUTIONS. BOOK TV.
Let the equation be hzx — '2>^zx — hzx = o, in which b is any how
given by x and z. Let us affume the following fiduious equation, b^z^x =i
conftant ; where m, n, r, are unknown exponents of powers, lo be deter.nined
by the procefs. Then, by taking the fluxions, we fhall have rh^^k^'x"^ x -f-
/>»..r-« — I.. ,m — J. . ..r i • i i • i- • i i i ,« — i .« — j..r — i
nb X z z •\' mb bz^x = o, which, being divided by b z x ,
will be reduced to rhzx + nhxz + f^/^i^ii* iz o. This equation being com-
pared, term by term, with the principal equation propofed, we fliali have
r = I, » ::z — 3, «; = ~ i ; wherefore, inftead of the fiditious equation
b z'^x zz conftant, we (hall have the true one, 7.— zz conftant, which is the
integral of the propofed equation.
Alfo, by the way of the logarithms, we may obtain the fame integration.
I refume the equation bzx — ^bzx — hzx zz o, I divide it by bzx -, it will
be -~ — — ^ J- zz. o, and by integration, Ix — Iz^ — lb zz to x
conftant logarithm. Therefore r^ is equal to a conftant quantity.
ADVER TISEMENT.
, . 68. .1 SHALL finifti thefe Inftitutions with an Advertifement, which is
this; that the ingenious Analyft muft endeavour, with all his ikill, in the
folution pf Problems, to avoid fecpnd fluxions, and much more thofe of a
higher order j and this by means of various expedients, which will offer
themfelves commodioufly on the fpot. Such artifices may be (een, as they
are made ufe of by famous Mathematicians, in the Problems of the Elaftic
Curves, the Catenaria, the Velaria, in that of Ifoperimetral Curves, and in
others of this kind ; the folutions of which may be feen in the Leipftc Ads
and other works of this nature : by which a learner may acquire fuch fkill and
dexterity, as will be very beneficial to him.
END OF THE FOURTH BOOK.
AN
AN ADDITION
TO THE FOREGOING
ANALYTICAL INSTITUTIONS;
Being a Paper of Mr. Col/on' s, containing a; Specimen of the Manner i»
which Two or more Perfons may entertain themfelves, by
propofing and anfwering curious Queftions
in the Mathematicks.
'T'HE Manufcript of this little piece appears to be a firft draught, and only
a part, of what Mr. Col/on intended to draw up : yer, I perfiiade myfelf, it
is fufficient to point out to the readers of it the way in which' fevera) perfons
may amufe themfelves with propofing and anfwering Queftions of rhis kind.
Tho'e readers, who wilh to fee more of this, may find it in the Vlth Sedion
of Mr. Coijon's Comment on Sir Isaac Newton's Fluxions, They may alfo,
with a little attention, propofe and folve, in the fame manner, any of the
C^ieltions in thefe Volumes.
*' A Problem is fuppofed to be managed between two perfons, the ^eriji and
the Rejpondent : the Data are fuch numbers or quantities as are given or
fuDplitd by the Qiierift; the Affumpta or ^ajita are fuch as are affumed or
found by the Refpondent."
PROBLEM I.
" Querist. / givt you three numbers^ 4, 5, and 10 ; I require a fourth.
Respondent. I afTume x to denote that fourth,
(^ 5o ihaty if from the froda^ of this into the third, the firfi he fuhtra5ied,
R. Then
542^ AN ADDITION TO THE FOREGOING ANALYTICAL INSTITUTIONS,
R. Then the remainder will be denoted by loy — 4,
Q^ Jnd if the remainder be divided by the firji^
R. The quotient will be denoted by '°^ " ^ j
Q. I'he ^otient will be equal to the fecond number.
lOA?
R. Then the equation is
= 24, and ;? = 4rt" == ^"4-
10* — 4
= 5i whence 10 a? — 4 =: 20, and
PROBLEM II.
^* Q^ A certain number of JhillingSf
R, That number fhall be denoted by a* ;
Q^ Was to be dijlributed among a certain number of poor people ;
R. The number of poor (hall be y,
Q^ Now if three fhillings were given to each, there would be 8 wanting 3
R. Then jf = 3;f — 8.
Q. But if two were given to each, there would be 3 to fpare.
R. Then ;? = ay + 3 == 3^ -^ 8, ox y z^ it, the number of poor; and
thence ;tf = 2jr + 3 == 2a + 3 = 25, the number of (hillings.**
PROBLEM HI.
Fig. 4;
*'' Q, In the triangle ABC, 1 give ydu the fides
AC rrr at BC = b, and the bafe AB = c ; you are to
find in this fuch a point D, R. I will afiume
AD = X ; then DB = « — j? ; (5[. ^hat drawing
DH parallel to BC, R. Then it will be AB (<r) .
BC {}?) :: AD [x) . DH = -^ -, Q^The fquare of
B DH may be equal to the reSlangle of AD and DB.
hhxx
bbx
R. Then = ^ X a — x, and — = «
ec
ate — ccx^ and bbx + ccx = jff , and jp =
6
flIfC
*^ +«*
PRO-
AH ADDITION TO THE rORE«OrN6 ANALYTICAt f SSTlTITf lOifS* J^J
P R O B L E M IV.
Fig^ 5-
** (^ / give you in pojit'nn the tw ri^bt
lines AF, At, and a point C m njtiher ofthofe
lines ; R. Then I can coniinue hb to Dy
and draw CD parallel to AE; and as AD
will be given, I (hall make AD — a. And
I can let fall the perpendicular CB, which
will be given alfo ; and therefore 1 will make
CB =: b. Q^ Tcu are to draw the line CEF
in fuch a manner, as that it Jhall cut off the triangle AEF equal in area to the
given plm^ cc. I will let fall the perpendicular EG, and make the bafe K¥ ~ x.
And then, by fimilar triangles, it wiUt be DF {a. + x) . AF (^) :: DC . AE ::
CB {b) . EG = -^ . Bur the area of the triangle AEF is t AF X EG =
t* X ^ ^ ^ ' Therefore -^— — = cc.'* [From which quadratick equatioa
the value of x is eafily obtained by ^ 74, Sed:. 11, Book L]
Fig. 6, D
PROBLEM V.
" Q^ / give you the ifofceles triangle CDB ;
R. Then I will make CD — a, BC =z b; I will
bifedl CD in E, and draw the indehnite line BEA,
Q^ T^he d ameter of the circle is required in which it
may be injcibed. R. Let AB = a; be the dia-
meter, and the circle ACBD. Now, becaufe of
fimilar triangles, itis AB {x) . BC {b) \\ BC {b) .
BE=— . But BEzii/BC^-^GE^ = y/bb^^aa.
Ih
Therefore — = y/bb—\aa, and x> zz
hh
Vbb - Xm
PROBLEM VI.
%• 7«
*' Q^ In the triangle ABC, I give you the three
JideSy AB = a^ AC = b, and BC =: ^ j and letting
fall the 'perpendicular AK, I require the fe^^n.ents of
the bafe y BE <j«^ EC. R. 1 make BE zi x , then
is EC ^c — x. But ABj — ^Eq — At^ =
ACj — ECj' i ihat IS, aa — xx = (AEj; = bb
— tff -4- 2CX — XX ; from which x zz ^^ .**
PRO.
344 ^^ ADDITION TO THE FOREGOING ANALYTICAL INSTITUTIONS,
PROBLEM VII.
" Q^ /« the quadrant al arch AM, defer i bed with
center C, and radius AC, the tangent A I of the
arch AH, and alfo the tangent HK of the arch HD,
are given ; R« I will make AC = ^?, AI = h,
and HK = c Q^Tou are to find AB, the tangent
of AD, which is the fum of thoje two arches, R. I
will make AB i= Xy and let fall the perpendi-
culars DF and DEj and then, from limilar tri-
angles, I Ihall have
CB {^s/aa + XX) . AB {x) :: CD (a) . DE =
and CB {^s/ aa + xx) . CA {a) :: CD (a) . CE =
w aa + XX
aa
'^ aa + XX
ab
and BC (s/aa + xx) . DC (a) :: AC . EC :: AI {b) , EO = , ' ."
^ Vaa +XX
[and BC (x/aa + ;v^) . DC (tf) :: IC (v/^^ + bb) . OC = ~^^:p=^.
and KC (x/tjtf + cc) . DC («) :: KH (c) . DF = -
fltf H- ce
.a^aa + bb
ew aa +bb
and CE ( — °- ).CQ f ;i1:L^ ) :: DF ( "^ ) . DO = ^-7^=;=.
V^^ + -^^^ 'vaa +XX Vaa + fc v aa + cc
But DO = DE — OE; therefore I have f^fl+i^ = JfLZ^, an equation
V tffl + cc vaa + a?;tf
which differs from that in § 108, Se6l. II, Book I, only in notation, and
which therefore may be folved in the fame manner.]
AN
AN INDEX,
POINTING OUT
THE MATTER CONTAINED IN EACH ARTICLE, OR MINOR SECTION;
OF THESE TWO VOLUMES. '
VOLUME I, BOOK I.
THE ANALYSIS OF FINITE QUANTITIES.
Sect. I. Of the firfl: Notions and Operations of the Analyfis of Finite Quantities.
1. THE operations of j4lgehrai what • ■ i
2. Pofttive and negative quantities dijilnguijhed — 2
3. The figns of pofitive and negative quantities 3 and other marks explained ibid.
4. ^antities divided into fimple and compound • ■ ibid.
£. Addition of fimple quantities • 5
6. Subtra^ion of fimple quantities • — — — ibid.
7. Multiplication of fimple quantities < ■ 4
8. Notation of fmple powers • — — — ^
9. Names of the powers, and their difiin5iion into pofittze and negative ibid.
10. Divif on of fimple quantities < - ' S
1 1 . When quotients are to he reprefented as fr anions ' • ibid.
12. The Jign of the quotient J what ■ • ^
13. Signs reciprocal in fimple fr actions — — • ibid.
14. Roots of ftmple fraSlions extracted — — — — — -— ibid.
15. Signs of roots y and impofftble roots • • • ' 8
16. Roots extracted of imperfe£f powers —— ■ " ibid,
1 7. Addition of compound quantities «,__ . ibid.
1 8 . Subtraction of compound quantities ■ ■ ^ ^ 9
19. Multiplication of compound quantities — . ibid.
ao. Multiplication how infmuated • —— ■ ^ _IQ
21. Powers of compound quantities J how injinuated) how usually performed ibid.
Vol, II. y y 22* Poivers
34^ INDEX. — VOL. I. BOOK I, S E C T. ir.
§ Fage
22. Towers raifed hy Sir Ifaac Newton's Binomial Theorem ■ ii
23. Divifi07i of compound qiiantities ■■ ■ 12
24. Procejs of divlfion ■ — — — — * i^
25. The Jquare-root extraofed ^ < ■ " 15
26. T^tf cube-root extra^ed • < > — — • 16
27. The biquadratick root extraBed 17
28. The fifth and higher roots extracted — i8
29. Notation of algebrdick fractions — — jbid.
30. Fr anions, how reduced — — . . ibid.
3 1 . Fradiojjs reduced to a common denominator • ■ — — i g
32. Fractions, how added and fubtra^ied «— - 21
33. — -— , /'ow multiplied - . — ibid.
34. ■ , how divided • 22
2S- Roots of fractions J how extruded , 23
36. Greateji common divifor^ hozv found < 24
37. Surds J how reduced • ■ 27
38. Radicals, how reduced to the fame denomination ibid.
39. Surds, how added and fubtra^led ■ 28
40. , how multiplied — — 29
41. Surds multiplied by furds — — — — . ibid.
42. having rational co efficients ibid.
43. --—— fomet.mes mny become rational • jbid.
44. n:ay have their rational co-efficients Irought under the vinculum 30
45. Different fu. ds, how multiplied ibid.
46. , how divided •— — 3 [
47. , wl. en tht index is the fame » ibid.
48. ■ , w! en (he index alfo is different ^ ■ ibid.
49. Ihe fquare root of furds extratled 32
50. Powers, how calculated, when the exponents are integers < 3 ^5
^i, — — > wlen they are fradions > — — ibid.
52. , how multipLed or divided — — 34
53. Pozvers rafcd or roots extracted, by their exponents ■ 35
54. This extended to compound quantities ■ ■ ^6
55. Si?nple divifcrSy hozv found; as alfo compound divifors ibid.
5^. Compound (ormulx, how to rejche • 38
57. How to remove the co- efficient of the firfi term — 40
Sect. II. Of Equations, and of Plane Determinate Problems.
§ _ ?age
58. Equations and tleir affe^ionSi what — — — 40
59. «r^ problem, what »■ ■' " ■ ' . ■ ■ 41
60. JVhen
INDEX.— VOL. I, BOOK I. f E C T. II. 347
60. iVhen problems are determinate y when indeterminate — 41
6 1 . Known and unknown quantities y how dijiinguijijed ■ — — . ibid.
62. Equations, how derived — — 42
63. Some lines in a figure to he denominated by inftrence — — 4j
64. Sometiines neiv lines are to he drawn ' ibid.
65. Equations, hozv formed from different values of the fame quantity 44
fid. When angles are concerned, bow to proceed ■■ 46
67. Equations y how reduced — — 4-7
68. — by multiplication •■ ibid.
69. hy divifton " ■■ 48
70. by raifmg powers ibid.
71. Equations, hozv refolved — — — — - 49
72. further refolved — — <jO
73' having ftmple powers — — - — .— ibid.
74. Affe^ed quadraticks, how refolved 5 1
75. Ambiguous figny its ufe ibid.
76. Imaginary quantities, their ufe — — 51
77. Identical Equations y what to be learned from them ■ ■ ■ 53
78. Equations and problem Sy how divided — — .— — ^4
79. Equations may fometimes he depreffed to a lower degree — — ibid.
3o. Several unknown quantities often reqiured in a problem 55
81. How thefe are to be eliminated — — 56
82. ' fometimes hy comparifon — •— — — ibid,
83. how when there are feveral equations — — 57
84. Sometimes the number of equations is infufficient — — 58
85. Sometimes more than fufficient, and thence the problem may become impoffible 59
86. How to conftru^ ftmple equations geometrically — — ibid.
87. if they corififi of feveral terms — — 60
88. The terms of an equation may be transformed, and fo fitted for conjiru^ion 61
89. How complicate terms may be transformed — — — — ibid,
90. Other fractions conftruEled — — • — — 62
91. Radicals, how conJlru5fed — — ■ — — 63
92. How radicals may be transformed in order to 'confiru&ion 64
93. ^adraticks conflru6led without transformation ibid.
94. Affected quadraticks, how confiru6led independently of their folution 66
95. — — • otherwife confiruSled — — — — 6S
96. An arithmetical problem — — . — — 70
97. A problem of equable motion — ibid.
^"i, A famous problem of hx^xmoAfi 7^
i^(). An arithmetical problem — ^-— ■■ ■ 73
100. A geometrical problem — — — .. . .■ ibid.
10 !. Another — — — 74
102. Another . ■ . 75
10^, Another, producing an identical equation — — ^ •— — 76
Y y 2 104. yi
348
I J? D B X. — V O L. I. B O O K I. SECT, lit.
104. u4 geometrical J or rather arithmetical problem
105. A geometrical prollem •
106. Another
107. Another ■■'
108. A trigonometrical problem, with a general foluiion
109 A geometrical problem • —
1. 10. The trije^ion of an angle •
77
79
80
82
83
87
8^
Sect. III. The Conllru6^ion of Loci, or Geometrical Places, not exceeding.
the Second Degree.
^
II.
12
14.
15-
16.
^7-
18.
19.
ao.
21.
22.
23-
24.
25-
26.
27.
2S.
29.
32.
33-
34.
Variable quantities ivhat^ ayidwhat the law by zvhich they vary
General precepts for the conflrziSiion of loci, with examples
Different equations require different loci, and vice versa
When the locus will be a right line —
When a conic fediion " . ■ — —
Loci, or curves ^ difiinguifhed into orders „ . , ,
The loci to a right line conftru6ted — — ^— -
The locus 'vjhen one variable vanifhes
The loci to a circle confiruded
The fimplefl loci to the parabola confiru5led "
to the hyperbola
'- to the hyperbvla between its afymptotes — —
to the ellipfis — — — —
How the diumeters may be founds if not given — —
To find the loci when referred to a parameter — —
The loci to the conic fe6lions diflributed into three fpecies — .— .
Loci of the firfl fpecies confiru6fedy with examples
of the fecond fpecies confiruded — — — —
of the third ipecies conjiru^ed — —
Complicate loci of any fpecies reduced to fmple by fubflitntion^ with
examples
General confiruSlion of the loci to the hyperbola between the afymptotes,
with examples — —
A geometrical problem^ confrucfed by the parabola •— —
Another, conflruded by the hyperbola between the afymptotes
A problem with three^ cafes, confiruded by the parabola, elUpJis, and
hyperbola
A locus to the conic fedions confiruded — ^ —
Another —- ^' . ■ 1
i37<
Page
90
91
92
ibid.
ibid.
ibid.
ibid.
94
95
97
98
99
loa
101
102»
103
ibid.
ic5
108
1 09
119
123
ibid.
124
127
128
An-
I N D E X.— V r-. I. B O K I. S E C T. I\r. 349
I'^j. Mother — ^— . — — . .. — — 130
138. Am I her — — — — ■ i-^i
139. Another — — — — — 133
140. Another — ■ — 134
141. A method to determine majority and minority^ in complicate quantities ^37
142. A geometrical -problem ibid.
i^^^ A fpecimen of the demonjiratiofi of thefe examples ■ 13S
Sect. IV. Of Solid Problems and their Equations.
§ ^ ' T'age
J 44. The roots of equations, what — — — — — 140
145. Or other-wife J the fever al divif or s of an equation — — 141
146. Equations may be refclved by divifion, when their roots are known ibid.
147. Jriow to know the conjiitution of the fever al co- efficients — — 142
148. When the fecond term will be wanting ibid.
149. How t?e absence of a term is denoted — — ibid.
150. Surd roots, and imaginary roots , ahvays proceed in pairs • ibid.
151. AffcPAons of the roots, how d/jiinguijhed ■ 1 43
152. Affedlions of the roots of equations of the third or fourth degree ' 1 44
153. The pcfitive roots may be made negative, and vice versa — — it)id.
154. The roots of an equation may be increafed or diminifjed At pleafure r45"
155. Or they may be multiplied or divided at pleafure 147
156. The reafon of thefe operations ' — — - ibid.
157- ' and their life — — — — ibid*
1^8. How equations are freed from f rations or furds — — 148
I <;q. Conditions for expwtging radicals — • — — — 149
160. The fecond term of an equation may be taken away ibid.-
161. — — or the third term — — - — — 15O'
162. or tl'e laji but or.e, if the fecond is wanting — ■ ~ ibidi-
163. or any other, on a certain condition — — - — — 151
164. Or an equation may be completed, or raifed highr ibid,
■165. PrcbLms may often le reduced to a I wer degree by fimple divif or s ibid^
1 6 6. and fomc times by compound divifoi s 152
167. Equations of the fourth degree may often be reduced by two quadratlck-
divfors — — ibid.-
168. Th'.s redudion performed by a general canon — — — 154
169. Somet.rnes a biquadratick may be reduced to a quadratick — — 156'
170. Sometimes h.gher equations may be rejclved by the fame method ibid.^
171. Ihi.! exemplified in equations of the fifth degrte — — 158-
172. Equations of the Jixth degree refohed —— — .— 160
6 ^ ITS. Ihe
;50 I K D E X."— V t. I. B K I. SECT. IV.^
§ ... ^^«?^
73- The fame method may he applied to higher equations — — < 163-
74. applied to the folution of an arithmetical prohlem 164
75. to a geometrical problem — — 1 65
76. to another — — 166
77. How higher equations may Jometimes he avoided — — 167
78. How otherivifey by finding two values of the fume quantity 168
y9. " exemplified in a geometrical problem ibid.
80. Solid problems may be folved by Cardan^ s rule, or by confiru^ion 171
81. How by the four cafes (f Cardan's rule — — ibid.
82. by the fecond cafe of the fame rule — — ibid.
83. by the third — — • 172
84. by the fourth — .— ibid.
85. Other expreffions of the fame roots — — — — - ibid,
86. To diftinguijh when thefe roots are real, and when imaginary 173
87. A compendium by the three cubic roots of unity — — 174
88. Example Df this reduction — — ibid.
89. Examples without the formula — — • I75
90. Equations of the fourth degree refolved - 176
9 1 . Equations refAved geometrically, by a combination of loci — — 177
92. When ti^o of the roots will be equal, when nothing, when imaginary 178
93. The loci fuoidd be fuch, as afford the fimplefi confirudion ibid.
94. To diftinguifh thefe loci i 79
^i). Cautions lo be obferved ibid.
96. Example of the confiru5lion of a cubic equation — — ibid.
97. by two parabolas — — 181
98. by a parabola and an equilateral hyperbola ibid.
^(). by a circle and hyperbola — — ' ibid.
2,00. Thefe equations confiruded by various loci, with examples 182
201 . by given loci, or fuch as are fimilar to given — — 185
202. Confirudion of equations of the fifth and fixth degree — — 191
203. — — ^— . of the feventh or eighth degree — — . ■ ■ 193
204. ■ of higher degrees — — • 1 94
205. j4ll equations may be confiruded by a locus of the fame degree ibid.
206. Ufe of this method — — — — 195
207. exemplified by a problem — _ ibid.
208. The fame otherwife confiruded — — . — — 196
209. A fimpler cafe of the fame problem — - — — — ibid.
210. The fame fiill purfued — — _-— ibid.
211. confiruded otherwife — — '■ 197
2 12. ' extended to higher cafes — — _— ibid.
213. Thg loci exemplified by other problems — — ibid.
214. — — — by another problem ■ 199
21^. — — by another problem, for angular Je£}ions — 200
216. Other cafes of this problem confiruded — ~ — — 203
ai7. The
INDEX,— VOL. I. BOOK I. S E C T. V. 551
§ ?age
a 1 7. The fame conftruBed another way — — — — 205
218. ■ raifed higher — — — — 206
>2i^. higher Jill — — ibid.
Sect. V. Of the Conftrudlion of Loci which exceed the Second Degree.
§ . ^ Page
2^o. Two ways to conflruB the higher loci — — 207
221. The firjl manner is by finding an indefinite number of points ibid.
222. The ordinates at right angles to the abfcifs — — 208
223. An example of deCcribing t^e curve by points — — ibid.
224. The Jign of the axis is ambiguous in even powers — — 209
225. To find where the curve cuts the axis — — ibid,
226. The more points are found the better — ■— ' ibid.
227. To find when the curve can have an afymptote ibid.
228 Afymptote found by changim^ the equation — — 2 to
229. Cautions to be obferved in finding afymptotes ibid.
230. To find whether the curve be convex or concave towards its axis 'ibid.
23 1 . Further to determine the forms of the curves^ with examples 211
232. Examples to determine when the ordinates are real — — 214
233. To de: ermine I '^e fame when the equ.Jicns are irreducible 218
234. which may be done by finding points , 219
235. An obje^iion obviated — — — -— ibid.
236. Example fr determining the forms of the loci from the equation 220
237. Another example for the fame purpofe 221
238. Example of the curve called the Witch 222
239. Another example, being the Conchoid of Nichomedes — — 223
240. Another cafe of the fame curve 225
241. A third cafe of the fame curve ^ — — • — — 227
242. The method imp/ovcd <f dejcribing curves by points 228
243. improved by the conic feciions — — 231
244. by parabolas of higher degrees — — ibid.
245. The fi:fi cubical parabola conJlfvBed • 23a
246. Tie firjl parabola of the fourth degree conftru^ed — — ibid.
247. • • of the fifth degree confirucied — — 233
248. of any degree — .— 234
24.9. ConjhuSlion of other fuiceeding parabolas ibid.
250. Reduplication of the curve produced by fquaring the equation ibid.
251. Confirudfion of hyp:rbolc.ds 235
252. — of higher hyperboloids — — — — -30
253. Other hyperboloids confirucied — — — — ibid.
254. Ob"
352 I N D E X.— V O t. I. B K I. SECT. VI.
254. Ohfervattott on the forms of the firjl paraboloids — — ^237
255. of higher paraboloids and hyperbokids — — ibid.
256. Curves of feveral terms confiru^edy in three cafes ^2,38
257. Example of the firjt cafe — '— — — ' ibid.
258. Jnother example — — "— — ^^39
259. Another " ^ ' *bid.
260. The CO ordinates may make any angle — — 240
261. The fecond cafe of curves conjlrudled — — 241
262. The third cafe confiru5fed, vAth a general example ibid.
263. To fparate the indeterminates when involved ibid.
264. Example of the confiruSion of thefe loci — -— — — ^2,42
265. Another locus conflru6led — — ibid.
266. An obfervation — -" — — . — ;— 243
267. Conclufton of the examples — — - — - i^id.
Sect. VI. Of the Method De Maximis ct Minimis^ of the Tangents of Curves,
of Contrary Flexure and RegreiTion ; making ufe only of Common Algebra.
§ Tags
268. To find the maxima and minima of quantities by comparing the
equation with another which hath equal roots 244
269- To find the fame by multiplying the equation by an arithmetical pro-
grefjicn 245
270. Tangents and perpendiculars to curves ^ how found 247
271. Example of this .-ik— ■ 248
272. How to choofe a convenient progreffion — — 249
273. The problem folved another way — — . ibid.
274. Points of contrary flexure and regreffiony what, and hoiv found 249
275. To diflinguijh contrary flexures from regreffions, and maxima from
minima — 251
VOLUME
INDEX.^VOL. II, BOOK II. SECT, I.
VOLUME 11. BOOK IL
THE ANALYSIS OF QUANTITIES INFINITELY SMALL.
ZSi
Sect. L Of the Notion or Notation of Differentials^ of feveral Orders, and the
Method of calculating with the fame.
§ . . . ^^S^
1. Variable quantities, what
2. Conjlant quantities, what. Notation of each 2
3 . A fluxion, or differential, what ibid.
4. Fluxions, hoijo exprejfed. A proof, from the incommenfur ability of fome
quantities, that there are infinitefimals of feveral orders — — • 4
5. Fluxions of the higher orders, how reprefented c
6. Theorem I, with its corollaries -, fijowing the exiftence and fome pro-
per ties of infinitefimals of feveral orders ibid.
7. Theorem II. Other properties of infinitefimals 7
8. Theorem III. The verfed fine of an infinitefimal arch is an infinitefimal
of the fecond degree ; and the difference between the right fine and the
tangent of that' arch is an infinitefimal of the third degree 8
9. Coroll. I. In an infinitefimal arch, the tangent, arch, chord, and right
fine, may be affumed as equal ibid.
to. Coroll. 2. If the radius of an infinitefimal angle be alfo an infinitefimal,
the arch and its right fine will be infinitefimals of the fecond order, and
the verfed fine zvill be an infinitefimal of the third order ibid.
1 1 . Coroll. 3. Infinitefimals of the firfi and fecond order in curve lines g
12. ScHOL. If the firfi fluxion of either the abfcifs, ordinate, or curve, be
taken con ft ant, the fluxions of the other two will be variable. The
fuppofition of a conftant firft fluxion ftoortens and facilitates calcU'
lations — . — ibid,
13. The foregoing conclufions are not affe5led by the angle of the co-ordinates 1 1
14. A lemma. What is the ratio of angles to each other ibid.
15. Theorem IV. 16. Coroll. Some properties of the involute, evolute,
and radius of curvature 1 2
17. Theorem V. 18. Coroll. 19. Coroll. 2. Properties of three per-
pendiculars to as many points in a curve, infinitely near to each other,
and of the angle at the middle point — — — _- . 13
Vol. II. Z z ao. The-
354 INDEX. VOL. II. B O K II. SECT. II.
20. Theorem YI. Tbs difference of the verfed fines cf two equal inf nit efimal
arches of circles^ the diameters of which differ only by an infinitejimali
is an infinit efimal of the third 07'der 13
21. TheiRc.m Vil. 2 2. Coroll. Properties of infinit efimals when the
curve is referred to a focus 14, 15
23. Scholium I. ji difficulty obviated — r— 16
24. Scholium II. Some further obfervations on infinit efimals. Two ifn-
fortant circumftances to be conf.dered. Caution to be obferved in the
ufe of them ibid.
25. Rule to find the fluxions of fimple quantities — — 17
26. when the quantities are multiplied together 18
27. for finding the fluxions of fractions i with examples ibid.
28. for finding the fluxions of powers 19
29. Finding of fecond fluxions i or higher orders — — 21
Sect. II. The Method of Tangents.
^ Page
30. Finding tangents to curves by a general formula for the fubtangent 24
31. Second fluxions have no place in finding tangents 25
32. Several formulae for the tangefjtj fubtangenty normal, tffr. ibid.
33. The angle made by the tangent and fubtangent may be found ibid.
34. The fame things may be done when the curve is referred to a focus 26
35. Example 1. To find all thefe lines in the parabola ibid,
36. Example 2. To find the fame in parabolas of all orders 28
37. Example 3. To find the fubtangent for the Apollonian hyperbola, and
all others J between the afymptotes 29
38. Example 4. To find the fame for the c\rc\t ^o
39. Example 5. To find the fubtangents in ellipfes, and hyperbolas of all
orders — — ibid.
40. To find the afymptotes — — 32
41. Example in a general equation — •— — ibid.
42. j^nother example to find afymptotes — — . 33
43. When the angle of the co-ordinates is not a right angle — — - 34
44. When the curves are not algebraical but mechanical J5
45. Example in the cycloid 36
46. Another way more general 37
47. Example, when the given curve is a circle — — 38
48. l^xample 2. When the given curve is a parabola 29
49. The fubtangent found from the generation of the curve ibid.
5o> S^*
INDEX. VOL. II. BOOK II. S E C T. III. 35^
§ , Page
50, 51. j^noiher example of this — — 40,41
52. Another way — — 41
53. More generally ibid.
54. Objervation when the curnjes become right lines — — — ^— ibid.
c,c^y 56. Tangents drawn to fprals \ with examples 42> 43
57, 58. The (ormv\2i of the fubtangent more Jimple y with an example 44.
59. Tangents drawn from the generation of the curve 45
60. Particular ca'es of this 46
6 1 . Drawing tangents to a curve by means of another curve — — • ibid.
62. An example of this in the ciflbid 47
65. The fame thing done more expeditioufly ibid.
64. Drawing tangents to a curve from its relation to another curve — — . 48
65. Example in the quadratrix 49
66. Drawing tangents to a curve from its relation to two other curves ibid.
67. Example in the logarithmic fpiral 50
68. yf difficulty fiarted in the bufinefs of drawing tangent Sy when the ex-
preffion cf the fubtangent becomes ■ — • . 5 1
69. This may be removed by multiplying by arithmetical progreffions 52
70. This method confirmed by recurring to the fir ft principles of fluxions ^^
71. The fame difficulty removed in the conftruBion of curves 56
Sect. III. Of the Maxima and Minima of Quantities.
^ . .... ^"^^
72. The foundation of the maxima and minima, and their formulae for
ordinates — — 58
73. Applied to curve lines — — — — — _— ^^
74. The ufe of this method — ■ — 60
75. Exemplified in the circle — — — — ibid.
76. 77. More examples — — 61
78. To diftinguijh a maximum from a minimum 62
79. Another example — — — — ibid.
80. A difficulty removed — — — — ibid.
81. An example , — — 6'^
^1. A difficulty folved — — 64
83, 84, 85, 86. Other examples 64 — 68
87, 08, 89. Problems to find mz-)L\m2. or VMrnrnz. 68 — 70
90. To diftinguiJJj between a maximum and minimum — — 70
91, 92. Other problems — — — — 71
93. A problem with a conftruEiion — — — 72
Z z 2 Sect.
^^6 1 N D E X.— V O L, II. BOOK II. S E C T. V,
Sect. IV. Of Points of Contrary Flexure, and of Regreflion.
94. FormuljE for points of contrary Jlexurey or fegrejjionj when the curve
is referred to an axis , 74,
95' when the curve is referred to a focus 76
gS.'How, l>y thefe fovmulx, to find the points required 77
97. To diftinguijh contrary flexure from regrejjion — — 78
98. Of another kind of regrejfton — — ibid.
99. 100, loi, 102. Various examples — — — — 78 — 81
103, 104. Examples with conJiru5fions — — — — Z'^ — 86
Sect. V. Of Evolutcs, and of the Rays of Curvature.
§ Page
05. Of involutes and evolutes . 87
06. Fundamental properties of thefe curves — — ibid.
07. Another property 88
08. 109. To determine the center of curvature of the involute ibid,
10. The co-ordinates may make an oblique angle 89
I \ . The co-radius, whatj and how to find it 90
1 2. When the co-ordinates are at oblique angles — — ibid.
13, 114. Other ways of finding the formula of the radius of curvature 91
1 5. Formula for curves referred to a focus — — ibid.
1 6. Thefe may become curves referred to an axis — — 93.
17. The fame otherwife — — — — ibid.
i8. Otherwife for the co-radius — ibid.
1 9. Thefe curves can have but one evolute — — — — 94
20. A corollary — — = ibid.
2 1 . When the radius of curvature may change from pofitive to negative ibid.
22. Example in the common parabola ■ 95
23. To find the equation of its evolute ^ — — ^y
44'. The evolutes of algebraical curves will be algebraical and reSlifiable ibid.
25! Example in the common hyperbola^ and to all parabolas and hyperbolas 98
26. . in the ellipfisj or hyperbola 102
27. ■• — in the logarithmic curve — — ■ 103
28. in the logarithmic Jpiral ■ 104
29. in the hyperbolic Jpiral ■ ibid.
30: in all fpirals in general 105
31. — in the cycloid — — — . 106
21* Points of regrefiion of the Jecond fpecies ■ 107
BOOK
INDEX.— VOL. II. BOOK III. S E C T. I. 357
BOOK III.
OF THE INTEGRAL CALCULUS.
Sect. I. The Rules of Integrations exprefled by Finite Algebraical FormuU, or
which are reduced to fuppofed Quadratures.
§ - Pagf
I. To find the fluents offimpk fluxions, when multiplied hy any power of the
variable quantity no
■2. when multiplied alfo by any confiant quantity ibid.
3 . — — — when both multiplied and divided by any powers of the unknown
quantity ^ ibid.
4. J conftant quantity fhould be added to the integral 1 1 1
5. To find the fluents of complicate fluxions when they can he rejolved into
fimple ones ibid.
6. ■ if raijed to any power 112
7. except when the index of the variable ^ quantity is a negative unit ibid.
8. In this caje^ we have recourje to logarithms ibid.
9. Conftru5iion of the logarithmic curve — — ibid.
10. Another defcription of the logarithmic, with confe^aries — — 113
11. Fluents reduced to the logarithms, or logarithmic curve • 114
12.. The Notation of logarithmic quantities • 1 1 5
1 3 . The logarithm of a negative quantity • 1 1 6
14. The logarithm of powers or roots — — .. ibid.
1.5. -. of produ5ls or quotients ■ 117
16. Thefe fluents require alfo a conftant quantity to be added 118
17. Some cafes in which the fluents of fraSlions may be found — — ibid.
18. 19. When the fluents of other fratlions may be reduced to logarithms ibid.
20. Fluxionary exprefjions prepared by reduction 119
2 1 . Complex fratlions prepared by fplitiing them into fimple ones 1 20
22. when the denominator of the formula is the produ£f of equal and
unequal roots — — 121
23. Redu^ion by a partial divijion • ibid.
24. If the roots of the denominators cannot be found algebraically, jet they
may be found geometrically ■ — — - 123
25. Some of thefe roots may be imaginary ' ■ 1 24
26. Fluents
35o INDEX. VOL. II, BOOK III, S E C T. I.
§ Page
26. Fluents reduced to the arch of a circle 124
27. Formula; reduced partly to a circular arch^ and partly to the loga-
rithmic curve -— — — — 12^
28. Radical formulie irhich admit of algehraic fiuents 126
29. Redu^ion of a formula with a general exponent — — 1 -27
30. • if that exponent were ?tegative 128
51. Other algebraic integrals found ibid.
3 2 . ■ jnore generally y 'with fever al examples — — i -^ o
33. Other formu]^^ algehdicidly integrate 131
34. ¥orm\}lie Jbrnetimes algebraical, fometimes logarithmical 132
315- Certain formute freed from the radical quantity by fubjlitution ibid.
36. Other examples — — •■ 133
37. Formulas requiring the re5lificntion of the circle 135
38. FormuIiE containing two radical quantities freed from them by fub'
Jlitutim — 136
39. Conditions requifite in formulse which may be freed from radicals ibid.
40. Rational fraSlionSy having complex denominators j refohed into others 138
41. when the numerator is multiplied by any pofitive power of the
variable quantity 139
42 . when the denominator is n>ultiplied by any power of the unknown
quantity 140
43. A convertible (ormiA^. ibid.
44. Certain binomials refohed into their real component parts ■ 141
45. Other binomials rejolved 144
46. Binomials refohed into trinomials 14^
47. The integrals cf thefe formulae may be had by the quadrature of the circle
and hyperbola 146
48. If not other wife, by geometrical conjlruolions — — ibid.
49. Trinomials refohed — — — ibid.
5.0. Trinomial integrals of other formulas obtained by logarithms and circular
arches — — — — — 147
51. — When the index is negative, reduced to the former cafe — — ibid.
52. When the numerator is multip.ied by any power of the variable
quantity 148
^^, When the denominator is multiplied by any power of the variable
quantity ibid.
54. Fr anions in the exponents may be removed — — 149
55. Other fraBional exponents chan'^cd into integers ibid.
56. Another formula integrated by the circle and hyperbola < 150
57. Obfervations on this ibid.
58. When the exponent of the multiplier is negative 151
59. 60. I'Vhen the other exponent is negative, or both are fractions 1 52
6i. Other cafes confidered — — — — ibid.
^2. Integration of a formula in which the denominator is a multinomial 153
63. — — When
INDEX. — VOL, II. BOOK III. SECT. III. 359
§ Page
6 ■5. TFhen the numerator is multiplied hy any poiver of the variable
quantity jca
64. Count James Riccati's method of integrating fra^ional formulse, of
which the denominators are multinomials ' ibid.
Sect. II. Gf the Rules of Integration, having Recourfe to Infinite Series.
§ _ ... ^^'Z^
65. ^antities reduced to infinite feries by divifwn leo
66. by the extraction of the fquare-root 160
67. Infijiite feries found by a cznon ibid'.
68. Jn infinite feries raifed to any power by the fame canon — — 161
69. The logarithmic formula integrated by a Jeries — — 16a
70. The fame more explicitly \h\d.
71.^ radical formula integrated by a Jeries - i (^ a
72. j4pproximations by thefe feries ibid.
73. Reference to James Bernoulli /or certain properties of feries ibid.
74. A general canon for the fluents of binomial formulje ibid.
Sect. III. The Rules of the foregoing Se6iions applied to the Rediification of
Curve-Lines, the Quadrature of Curvilinear Spaces, the Complanation of
Curve Superficies, and the Cubature of their Solids.
§ Fage
75. ^ formula for finding the areas of curves referred to an axis 1 66
76. for curves referred to a {qzw^ 167
^y. _— for curves referred to a diameter when the angle of the co-
ordinates is oblique ibid.
78. y^ formula for the verification of curves, the co-ordinates being at right
angles . ibid.
79. — — - when the curves are referred to a focus 1 68
80. when the co-ordinates are at oblique angles — — ibid.
Si. In each of thefe cafes to r edify the curve • — ibid.
82. A formula for the fluxions of the fuperficies of a round /olid ibid,
ii^. . of the round folid itfelf . ibid.
84. ■ of the fuperficies when the co-ordinates make a given oblique angle 169
85. for the folid in the fame cafe ibid..
86. How to proceed when the curve is referred to a focus ibid.
2)^. Reduction
360 INDEX. VOL, II, BOOK III. SECT. III.
o
§ Page
87. ReduB'ion of a curve from a focus to an axis > ibid.
88. Reduuiion from an axis to a focus ■ 171
89. Example in a conic fedion in general ibid.
9D. A general method of this re dud ion, with examples ■ • 172
^i. y^ fubjiitution ivhen the co-ordinates make an oblique an^le — — — i '■j^
92. The quadrature of the Apollonian parabola^ and of all parabolas 176
93. Jinother general example 178
94. Several other examples, fome by logarithms, fome by infinite feries ibid.
95. The quadrature of a mechanical curve — — . 181
96. of the hyperbola ■ 182
97. , of the circle, by feveral feries 184
98. of the ellipfis, by feries 186
99, of the cycloid, by feries 188
00. of the conchoid, reduced to the circle and hyperbola 189
01. of the ciffoid, reduced to the circle < 190
02. of the logarithmic curve — — — 191
03. of the tratlrix, reduced to the circle ibid.
04. of fpirals 192
05. of the parabola, when the co-ordinates form an oblique
angle 193
06. of the parabola referred to a focus > 194
07. • of the figure of right fines ibid.
08. ^adrature of curves by means of new fubjlitutions • 195
oc
Another example of this 196
10. The r edification of the Apollonian parabola, and of the fecond cubical
parabola .— — . — : — ^ . . ibid.
1 1. of the arch of a- circle 198
12. of the arch of an ellipfis 199
1 3 . of the hyperbola 20 1
1 4. of the cycloid 202
15. of the tradrix ibid.
16. of the fpiral of Archimedes, and of the logarithmic
Jpiral 203
17. of the logarithmic curve . 204
18. ' of the Apollonian parabola, when the co-ordinates make
an obi que angle ■ ■ 205
19. ■ of infinite parabolas and hyperbolas — — ibid.
20. The cubature of the cone 207
21. of the fphere — — 208
22. of parabolic conoids of any order 209
2'3. of the Jpheroid 212
24. of the hyperbolic conoid • ibid.
25. • ■ between the afymptotes ■ 21 3
26. ■' of the conoid generated by the logarithmic curve 215
127.
INDEX.—VOL. II. BOOK IXT. SECT. IV, j6i
§ Page
127. The ciilature of the folid generated hy cifToid — 2 1 6
128. • by the tradlrix — — — . 21 y
129. • of Jeveral forts of ungulas __. jbid.
130. -— — of a fegment of the parabolic conoid ■ ^19
13 r. Ohfervation — ibid.
132. Complanation of curve furfaces I and Jirji of the cone • 220
133. when the cone is fcalene ■ — — . jbid.
134. of the fphere — — — — — — . 222
^2i5* ' ^f f^^ parabolic conoid — •— i223
I J 6, of various parabolical conoids, which are quadrable^ and which
are not — — ibid.
137. of the fpheroid • 225
I J 8. of the hyperboloid — — — — 227
^Z9' ^—— of the equilateral hyperboloid — — •^'- — ibid.
140. The fuperficies of the folid generated by the revolution of the tradtrix 228
141. The fuperficies of an ungula of a paraboloid 229
142. • — of the parabolic conoid, when the co-ordinates form an
oblique angle — — 230
143. Obfervation ibid.
Sect. IV. The Calculus of Logarithmic and Exponential Quantities.
§ Page
144. Exponential quantities, what — — ■ 231
145. of feveral degrees '- ibid.
146. To find the fluxion of a logarithmic quantity < ibid.
147. - • of any power of a logarithm 232
148. ■ of any power of the logarithm of any power ibid.
149. ' of any power of the logarithm of a logarithm ibid.
1^0. —————— of an exponential quantity — — 233
i^i. ___-_—_— , of exponentials of the fecond degree ibid.
1^2. ■ ' of produ5ls of exponentials ibid.
153. To find the fluents of logarithmic difl'erential formulEe ibid.
154. The ifitegration of a general logarithmic (ormuh. 235
155. The artifice of finding the preceding feries i'3,S
156. Integrals of logarithmic ^ovrnvXis. found by different feries — — 2,37
157. found by quadratures ibid.
138. Exponential {ormu\2£^ integrated by feries • — — 238
J 59. The fame thing done in a different manner 239
160. Logarithmic and exponential curves conftruSfedy their fub tangents
foundy &c» —.— — . — — . — 240
Vol. II. :; A 161. Con-
3^2 INDEX.— VOL, II. BOOK JVt SECT. If,
§ Page
j6i. Ccnftru^ion and quadrature of an exponential curve — — • 242
162. The fubtangent found of another — 24 j
163. Another exponential curve conJiru5iedy and its area found — — ibid,
1 64. Variable exponents found, the reft of the quantities in the equation being
conftunt ' ibid.
165. 166, Two exponential "problems — — 244.
BOOK IF.
THE INVERSE METHOD OF TANGENTS.
§ ■ _ Pagf
lo Definition and illuftration 247
2.. Further explanation of this matter. Two ways ef proceeding in it 24S
Sect. I. Of the Gonflru6lion of Differential Equations of the Firft Degree,
without any previous Separation of the Indeterminates.
§ _ Page
3. Redu5fion and integration of differential equations 249
4. Other examples more compounded ibid.
5. Other examples of reduction to integr ability 25a
6. More examples of this reduction ' 251
7. Other examples -~ — — — 252
8. Redu5lion to logarithmical forms — « — — . ^ 255>
9. Other expreffions reduced to forms of that kind ibid*
10. Other more complicate examples of integration — — 256-
Sect. II. Of the Conftrudion of Differential Equations, by a Separation of the
Indeterminates.
§ Page
II. Example of the feparation of the variables — ■ — . 257
11. The redu^ion of differentials by fubftitution — ^ ibid.
13. Some
INDEX.— "VOL, ir. BOOK IV. SECT. IV. '^6;^
S Page
13. Some amhigmties in integrntions — — . 258
14. Some difficulties in the choice of Juhjiitutions — 259
\ ^, Differentials eliminated by fubjiitutions — — 261
16. The fame example otherwife reduced — — 263
iy. The Jeparation of the variables ; and defcription of the curves ibid.
1 8. More examples of the feparation of the variables 265
\g. The variables feparated by altering the exponents ■ 266
20. Separation of the variables by a canonical equation — — . 267
21. without the canonical equation — 269
IT. A canonical equation^ or method, for fome fimple cafes 270
23. y/ general method of feparating the variables — ^— 272
24. A tentative method of doing the fame, with examples 274
25. Another method of feparating the variables, of ufe in particular cafes,
with examples ■ - 279
26. Another method of feparating them in a canonical equation 282
27. Another canonical equation 284
28. ^ redu6lion by the exponents — • ibid.
Sect. III. Of the Conftruftion of more limited Equations, by the Help of
various Subflitutions.
§ Page
29. The feparation of the variables in a general formula by fubflitutions 285
30. in a more general equation — 286
31. — in an equation flill more general ■ 287
32. — — in four other equations ■ ■ 288.
33j 34> 7) Si 3^j 37> 3^. Examples of feparation in more complex equations 289 — 293
39. Other fubjiitutions for feparating the variables in a canonical equation 295
40. From the property of the fubtangent, to find the curve 297
41. From the area given to find the curve — — , 299
42. A problem concerning parabolas cut at right angles by a curve 300
43. 44. Two other problems — — 304
Sect, IV. Of the Redudlion of Fluxional Equations of the Second
Degree, &c.
45, Rules for the redu8ion of equations containing fecond fluxions — 306
46. Examples of paffmg from fecond to firft fluxions • ibid.
3 A 2 47. Integration
364 I N D E X. — y O U II. B O O K IV. S E C T. IV.
47. Integration of fecond fluxions, without ajfuming a conjiant at fir ft 309
48. To know what fluxion may be taken for conft ant ■ 310
49. Redu^ion to firfl fluxions by fubftitutions • — . 311
50. JVhen no fluxion has been taken for conftant, one may be fo taken at
pleafure — ■ — 313
51. By this ajfumption Jome equations are brought under the method of § 49. 314
52. Other inethods fuggefled for this — - — — — ibid,
53. ReduBion by changing the conftant fluxion — ibid.
54. Exam-pie by a method before explained • ■ 317
5^. Reduhion of fecond fluxional equations by a canon — 31 g
56. Integrations by another canon ' 321
jfy, ■ hy a third canon > ■ - — _ 322
58. Second cafe of the canonical equation ■ — — . 323
59. jlnother method, more general — • ■ 324
60. An cbfervation — • 325
61. Difficulties in thefe reductions, arifing from conflants — — ibid.
62* A problem in the Inverfe method of the radius of curvature y when the
curve is referred to a focus •— — 331
63, when the curve is referred to an axis 333
64. — . when the radius, or co-radlus, is given by the abfc'ifs 334
6^. -— the radius being referred to the axis, to flnd the curve 335
66. The foregoing methods extended to equations in which there are higher
orders of fluxions ■ —— 338
67. The fame fubjed continued ■ 339
68. Conclufion . 340
An Addition — . _-^ >■ 341
FINIS.
ERRATA.
ERRATA.
Note. When the letter b is joined to the number of any line, it is counted
from the bottom of the page.
VOLUME I.
In the Plan of the Lady*s Syftem of Analy ticks.
Page, Line.
xU II. After the word branch, irifert a comma*
In the Body of the Work.
41, 3.^, Dele as.
12^. 7. Injiead of 2aaccxy read laacx.
And in the head-lines ^ on the right-hand pages^ from p. 209 to p. 223,
infiead of Sect. IV., read Sect. V.
VOLUME II,
Page. Line.
9. In fig, II, the ferpndicular to AC u drawn from the pint G,
injiead of E.
II. The fmall letter i is wanting in fig, 15.
15. 4.^. Injiead of Gs.ch, read onQ ihc,
16. 9. Injiead of EG, read EF.
24, In the head- line, infiead of Book I,, read Book JI.
64. 7.^. After the letter a, injiead of — , read zr.
113. Injiead of art, 9, read 10. N. B. All the articles from 9 to 22 are
numbered too httle by i.
125.
366 ERRATA.
125. 20, towards the end of the line, after the word radius, dele the comma i
and injlead of adding, read added to,
189. 9.^. After -zZi infert the letter a.
t . J
205. 8. Infiead of x , read ^ — . ,
216. 6.h. After zz, infiead of a, read i.
295. 13. Jnflead of in, read is.
2 1 7. 7,b. Injlead of •^— — ^ , read ^^ — ~^ '
339. 3. Infiead of qx, read qx.
N; B. The name of the city Bologna is in a few places printed Bolonia, as it was
found in the Tranllator's Manufcript, but I take it to be erroneous.
Editor.
A LETTER
A LETTER
FROM
PHILALEl'HES CANrABRlGIENSIS,
Reprinted from the Gentleman's Magazine for November i8ou
TN the Gentleman's Magazine for November lafl, pages 997 and 998,
is a Letter ligned Philakthes Cantahrigienjls^ the dellgn of which is
fo laudable, that I gladly embrace this opportunity of contributing my ,
mite to it by reprinting the Letter ; conceiving that it cannot fail of
the approbation of all the fober and difcerning part of mankind, and
that, if the fuggeftions of it be duly attended to, it vvill prove very
beneficial to thofe who are of a different charadter, as well as to the
public in general. Editor.
Dec. 10, 1801.
* Mr. Urban, Oa,l,
' The following paflage, taken from the preface to the fourth volume
of the *' Scriptores Logarithmici," lately publilhed by Mr. Baron
Maferes, appears to be written with fo benevolent a defign, and points>
out
368 A LETTER FROM PHILALETHES CANTABRIGIENSIS,
out to the Great obje61s fo worthy of their attention, that I wifh it were
more generally known ; and therefore fhall be glad to fee it in the
Gentleman's Magazine.
* The paiTage begins in the ixth page of the preface, where, fpeaking of
Dr. James Wilfon's " Hiftorical Diflertation of the Rife and Progrefs of
the Modern Art of Navigation," the Baron fays,'
*' It is full of curious hiftorical matter, and has fuggefted to my mind a wi(h
that fome perfon of affluence, fond of the fubje6t of navigation, and who (hould
have been indebted to it, perhaps, for his rank or fortune, would caufe a
colledtion of all the authors on that fubjed, whofe works are mentioned in this
Diflertation, to be made, and reprinted in a handfome manner in a fet of
quarto volumes, pf the fize of thefe volumes of the Scriptores Logaritbmici^
5nder the title of Scriptores NauticL Such colledions of learned tracts on
particular fubjeds, under various titles fuited to the feveral fubjeds of which
they treated, would be very convenient in the prefent ftate of fcience ; which is
extended to fuch a variety of fubjefls, and difperfed in fuch a number of
different books, that it is very difficult and very expenfive for a perfon, fond of
apy particular branch of fcience, to procure himfelf all the books that relate
to it. Befides the colle6tion called Scriptores Nauticiy relating to navigation,
there might be a colledion called Scriptores Statici, relating to the dodlrine of
JlatickSj or bodies at reft that form an equilibrium, or counterpoife to each
other ; under which head all the books of merit that treat of the leverf the
inclined plane, and the other mechanical powers, would be comprized, and thofe
that treat of the catenary curve, and of the partial immerfion and the pofitions
of bodies floating in liquids of greater fpecifick gravity than themfelves, and of
many other curious fubjeds of the like nature. And there might be another
colledion called Scriptores Pboronomici, relating to the dodrine of bodies in
motion ; under which head would be comprized Galileo's Mechanical Dialogues,
of which the 3d and 4th contain the dodrine of the fall of heavy bodies to the
earth with the law of their acceleration, and of their motion on inclined planes,
6 and
A LETTER FROM PHILALETHES C ANTABRICIENSIS. 369
and of the motion of pendulums in circular arches, and of the motion of pro-
jedliles, which (abflrading from the refiftance of the air,) would defcribe
parabolas; and under the fame head would be comprized Mr. Huygens's traft
on the motions of perfedly elaftic bodies flriking againft each other, and his
admirable treatife De Horologio OJcillatorto, or on the motion of a pendulum-
clock, and his traft on central forces ; and all Sir Ifaac Newton's moft pro-
found, but very difficult work, called the Principia^ or Mathematical Principles
of Natural Philofophy, with the feveral commentators on it, and Herman's
Phoronomia, and Euler's work De Motu. Another colledlion might relate to the
finding the centres of gravity of different bodies ; which is, I believe, a more
fubtle and difficult fubjed than is generally fuppofed. This coUedlion might
be called Scriptores Centrobarici, And another colleflion might confift of all the
writers on opticks, under the title of Scriptores Optici, This coUedion fhould
comprize the work of Euclid, or that which has been afcribed to him, on this
fubjedl, and thofe of Alhazen, and Vitellio, and Roger Bacon (the learned
Englilh monkj, and Antonio De Dominis, and Willebrord Snell, and Des Cartes,
and Huygens's Dioptricks, and his treatife De Luminey and other works of his
on the fubjeft of opticks, and James Gregory's Optica Promota, and Dr.
Barrow's LeStiones Optica, and Sir Ifaac Newton's Le^liones Optica, and his
Treatife of Opticks, or Experiments on Light and Colours, and Molineux^s
Dioptricks, and Dr. Smith's Compleat Syftem of Opticks, and Harris's Opticks,
and many papers in the Philofophical Tranfadions relating to the fame fubjed.
If fuch feparate colledions of authors were publifhed, every perfon who was
devoted to any particular branch of thefe fciences, (and no man can attend to
all of them, or even to many of them, with any great profpe(51: of becoming
mafter of them,) might buy the coUedion which related to his particular
branch at a moderate expence."
* On this occafion I beg leave to make another remark or two,
* The importance of the art of navigation to this ifland, in times of
peace as v^rell as of war, is generally acknowledged ; yet it may be juftly
doubted whether it has been encouraged here in a degree fuitable to its
Vol. II. 3 B importance,
370 A LETTER FROM PHILALETHES CANTABRIGIENSI8.
importance, or equal to what it has received, in the laft fifty years, from
other nations.; certainly not fo as to excite equal emulation amongft men
of fcience *. In fupport of this aflertion, I might enumerate the prizes
which, from time to time, have been given by foreign academies for
improvements in navigation and aftronomy, and recount the learned
rra<fts which have been produced in confequence of that encouragement ;
but I fhall at prefent wave this fubjed.
* In all civilized nations, arts and fcienccs have been confidered as
making a part of the education of the Great, and as being under their
patronage. Amongft the men of rank in this country, in former ages,
are to be found the names of Napier^ Bacon ^ Boyle ^ Newton, Macclesfield,
and Stanhope ; men who excelled in fcience, and patronized it in others.
May I then be allowed to fuggeft to the nobility and gentry who, of
late, have made a confpicuous figure in Wejlminjler-Hall, and to all
others of rank and fortune, who, although their names have not yet
graced the columns of the London news-papers, are wafting their time
and money in the fedudtion of the wives and daughters of their friends,
or in other idle and vicious amufements, that, if they would exchange
thofe vicious amufements for the innocent and rational ones purfued by
the men whofe names I have mentioned, and, inftead of fquandering
away thoufands on courtefans, lay out a few hundreds in printing fuch
fcienfific trails as the worthy baron has mentioned, and in the fupport
of Geniui firuggling with poverty y it would undoubtedly be much more
* I am aware of the rewards which have been offered by ads of parliament for the
difcovery of the longitude at fea, and not unacquainted with the manner in which 20,oool.
has been beftowed. * ^ ' ' ' ' '
for
A LETTER FROM PHILALfiTHES CANTABRIGIEKSIS.
371
for their prefent honour and future fatisfaftion, as well as for the good
of mankind.*
* Philalethes Cantabrigiensis.'
"^ Omne anim't vitium tanto confpe6l'ius in fe
Crimen habet, quanta major, qui fee cat, habeiur^
Tota licet veteres exornent undique cera
Atria, KGB 1 LIT AS foJa eft at que unica virtus.
Juv,
Printed by Taylor and Wilks, Chancery-lane.
A
L
It"'