SIR ISAAC NEWTON'S
ENUMERATION OF LINES OF THE THIRD ORDER.




LONDON:
Printed by G. BARCLAY, Castle St. Leicester Sq.




SIR ISAAC NEWTON'S
ENUMERATION OF LINES 0F THE
THIRD ORDER,
GENERATION OF CURVES BY SHADOWS,
ORGANIC DESCRIPTION OF CURVES,
AND
CONSTRUCTION OF EQUATIONS BY CURVES.
TRANSLATED FROM THE LATIN.
WITH NOTES AND EXAMPLES.
BY
C. R. M. TALBOT, M.P. F.R.S.
LONDON:
H. G. BOHN, YORK STREET, COVENT GARDEN.
MDCCCLX.








PREFACE.
IT is the object of the present volume to place within
reach of the student a work to which he is frequently
referred in the course of his reading, but which he
will seldom have the opportunity of consulting, unless
he happens to be possessed of the quarto edition of
Sir I. Newton's work; for the " Enumeratio linearum
tertii ordinis," originally printed as an appendix to the
treatise on Optics, and subsequently in the " Opuscula,"
has not hitherto been published in a separate form.
In 17I7, Mr. Stirling, of Balliol College, Oxford,
published his commentary on this work, entitled II1 -lustratio tractatus D. Newtoni de enumeratione linearum tertii ordinis;" but this book, though reprinted
at Paris in I797, is now scarce, and beyond the reach
of the general reader.
To have published a translation of the mere text
of Newton's treatise, without the addition of any explanation or illustration, would have been a barren and
useless undertaking, for its style is concise and abrupt
almost to obscurity, and its enunciations are unaccoma2




Vi                   PREFACE.
panied by any proofs.   The Abbé de Gua, in his
" Usage de l'Analyse de Descartes," in remarking on
the difficulty of following the thread of the argument
made use of by Newton, writes thus:-" Ce géomètre
dont tous les ouvrages portent un caractère singulier de
sublimité, paroit en particulier dans celui-ci s'être élevé
à une hauteur immense, à laquelle toute autre génie
moins pénétrant et moins fort que le sien, auroit tenté
vainement d'atteindre: mas la route qu'il a tenue dans
une entreprise si difficile, se dérobe aux yeux de ceux
qui apperçoivent avec étonnement le degré d'élévation
auquel il est parvenu. On doit en excepter quelques
legères traces qu'il a eu soin de laisser sur son
passage, aux endroits qui avoient mérité qu'il s'y arrétât plus long tems.  Ces endroits, au reste, sont
presque toujours assez distants les uns des autres. Si
l'on se propose donc de suivre la même carrière, on
est obligé se guider soi-même dans de longs intervalles."  And Cramer has gone so far as to accuse
Newton of purposely withholding the demonstrations
of the method employed by him in this work; for, in
speaking of it, he says, " Il est facheux que M.
Newton se soit contenté d'étaler ses découvertes sans y
joindre les démonstrations, et qu'il ait préféré le plaisir
de se faire admirer à celui d'instruire."
There is, however, little doubt that Newton's object
in hastily publishing this treatise, was to vindicate the




PREFACE.                 Vii
priority of lis own discoveries from the piracy of
writers who had appropriated sonie of his theorems;
and, in the advertisement prefixed to the "l Optics," he
complains that "having lent out his manuscript containing such theorems, he had since met with some
things copied out of it, and therefore took occasion
to publish the tract." 1Nor can it be doubted that
the work of Stirling, which supplies the omitted demonstrations, was sanctioned and assisted by Newton
himself, with whom  the author was on terms of
intimate friendship.
But although the criticism of the French mathematicians would appear to be ill founded, it cannot be
denied that to those who have no previous familiarity
with the subjects treated in the "' Enumeratio," some
explanation and illustration is wanted in order to
enable them to arrive at the author's meaning; and it
is for this reason, and with a view of supplying the
desideratun, that notes to elucidate the text have been
added in the present volume.
Occasion has also been taken in the following
pages to give a short account of the application of
Newton's analytical parallelogram to the investigation
of curve-lines,-a method which seems to have undeservedly fallen into disuse. The writer fully concurs
in the truth of the remark of Stirling, I' Hâc methodo
universali procedendo, scilicet argumentando à naturis




Viii                 PREFACE.
equationum, patescunt non solùm sectionum coni proprietates, quas tanto labore adinvenerunt Veteres, et
tot ambagibus demonstratas dederunt, idque methodo
quæ ad alias curvas extendi nequit; sed et proprietates
curvarum omnium ordinum superiorum."
A few examples and problems relating to lines of
the third order have been added at the end of the
volume, in order to render the work more useful to
the student.




CONTENTS.
Page
SECTION I. THE ORDERS OF LINES...  7
SECTION II. THE PROPERTIES OF CONIC SECTIONS ARE ANALOGOUS TO THOSE OF CURVES OF HIGHER ORDERS..   8
1. Of Curves of the Second Genus, their Ordinates,
Diameters, Vertices, Centres, and Axes..  8
2. Of Asymptotes and their Properties...   9
3. Of Latera Recta and Transversa....   9
4. Of the Ratio of the Products of the Segments of
Parallel Lines..                         10
5. Of Hyperbolic and Parabolic Branches, and their
Directions......10
SECTION III. THE REDUCTION O ALL CURVES OF THE SECOND
GENUS TO FOUR CASES OF EQUTIONS....11
SECTION IV. THE ENUMERATION OF CURVES..       14
1. Of the nine Redundant Hyperbolas, having no
Diameter, and three Asymptotes...14
2. Of the twelve Redundant Hyperbolas, having but
one Diameter....15




X                        CONTENTS.
Page
3. Of the two Redundant Hyperbolas, with three
Diameters.......17
4. Of the nine Redundant Hyperbolas, with three
Asymptotes converging to a common point. 17
5. Of the six Defective Hyperbolas, without a Diameter 17
6. Of the seven Hyperbolas, Defective, but having
Diameters.......18
7. Of the seven Parabolic Hyperbolas, having no
Diameter....... 19
8. Of the four Parabolic Hyperbolas, having a Diameter 20
9. Of the four Hyperbolisms of the Hyperbola..21
10. Of the three Hyperbolisms of the Ellipse.. 22
11. Of the two Hyperbolisms of the Parabola.. 23
12. Of the Trident....... 23
13. Of the five Diverging Parabolas.... 24
14. Of the Cubic Parabola.... 24
SECTION V. THE GENERATION OF CURVES BY SHADOWS. 25
SECTION VI. OF THE ORGANIC DESCRIPTION OF CURVES. 26
SECTION VII. THE CONSTRUCTION O0' EQUATIONS BY THE
DESCRIPTION OF CURVES....          29
NOTES TO SECTION I. Orders of Lines.           33




CONTENTS.              xi
Page
NOTES TO SECTION II.  Diameters......  35
Ratio of the Products of Ordinates and Abscissas..38
Hyperbolic and Parabolic Branches...  41
Asymptotes....42
NOTES TO SECTION III. Infinite Branches... 45
Hyperbola with six Branches...  48
Diameters of Redundant Hyperbola... 51
Absolute Diameters......53
NOTES TO SECTION IV. Redundant Hyperbola without a
Diameter.........  54
Hyperbolas with a Diameter....56
Hyperbolas with three Diameters... 57
Hyperbolas with Converging Asymptotes..58
Defective Hyperbolas.......58
Cissoid.....         60
Parabolic Hyperbolas....61
Hyperbolisms of the Conic Sections...   63
Witch.......            66
Trident.....                67
Semicubic Parabolas....          68
Cubic Parabola......       70
NOTES TO SECTION V. Generation of Curves by Shadows. 72
NOTES TO SECTION VI. Organic Description of Curves.. 84
NOTES TO SECTION VII. Construction of Equations by Curves 85




xii                  CONTENTS.
Page
NOTES ON THE ANALYTICAL PARALLELOGRAM...88
EXAMPLES..........104
LoCI OF THE THIRD ORDER..112
PROBLEMS. 125




AN ENUMERATION         OF LINES 01F' T1 llI 1IRD
ORDER.
SECTION I.
The Orders of Lines.
GEOMETRICAL lines are best divided into orders, according to
the dimensions of the equation expressing the relation between
absciss and ordinate, or, which is the same thing, according to
the number of points in which they can be eut by a straight
line. So that a line of the first order will be a straight line;
those of the second or quadratic order will be conie sections
and the circle; and those of the third or cubic order will be
the cubic Parabola, the Neilian Parabola, the Cissoïd of the
ancients, and others we are about to describe. A curve of the
first genus (since straight lines are not to be reckoned among
curves) is the same as a line of the second order, and a curve
of the second genus is the same as a line of the third order.
And a line of the infinitesiiial order is one which a straight
line may cut in an infinite number of points, such as the spiral,
cycloïdl, quadratrix, and every line generated by tle infinitely
continued rotations of a radius.




8          NEWTON S LINES OF THE THIRD ORDER.
SECTION II.
The Properties of Conic Sections are analogous to those of Curves
of higher orders.
The chief properties of conic sections have been much
treated of by geometers, and the properties of curves of the
second and higher genera are very similar to them, as will be
shown in the following enumeration of their principal properties:1. Of Curves of the Second Genus, their Ordinates, Diameters,
Vertices, Centres, and Axes.
If parallel straight lnes terminated by the curve be drawn
in a conic section, the straight line bisecting two of them will
bisect all the others, and is called the diaineter, and the
bisected lines are called ordinates to the diameter, and the
intersection of all the diameters is the centre; the intersection
of the diameter with the curve is called the vertex; that diameter, whose ordinates are rectangular to it, being called the
axis. In like manner, in curves of the second genus, if any
two parallel straight lines are drawn, meeting the curve in
three points; the straight line which cuts these parallel lines,
so that the sum of the two segments meeting the curve on one
side of the secant, equals the third segment meeting the curve
on the other side of the secant, will cut in the same ratio all
lines parallel to these, provided they also meet the curve in
three points; that is, so that the sum of the two parts on one
side of the secant, shall equal the third part on the other side.
These three parts, thus equal, may be called ordinates, and the
secant or cutting line to which the ordinates are applied, the
diameter; the intersection of diameter and curve, the vertex;
ard the intersection of two diameters, the centre. The diameter having rectangular ordinates, if any exist, may also be
called an axis; and where all the diameters meet in a point,
that point will be the general centre.




ASYMPTOTES AND LATERA RECTA,              9
2. Of Asymptotes and their Properties.
A hyperbola of the first genus will have two asymptotes;
that of the second genus will have three; of the third, four,
and no more; and so on for the rest. And as the segments of
any straight line intercepted between the conic hyperbola and
its asymptotes on each side are equal, so in hyperbolas of the
second genus, if any straight line be drawn cutting both the
curve and its three asymptotes in three points, the sum of those
two segments of the secant, which are drawn from any two
asymptotes on the same side to two points of the curve, will be
equal to the third part, which is drawn from the third asymptote
on the contrary side, to the third point in the curve,
3. Of Latera Recta and Transversa.
And in like manner, as in the non-parabolic conic sections,
the square of the ordinate-i. e. the rectangle contained by the
ordinates on each side of the diameter-is to the rectangle of
the segments of the diameter of the ellipse and hyperbola, terminated at the vertices; as a certain given line called the latus
rectum, to the part of the diameter which lies between the vertices, and is called the latus transversum; so in non-parabolie
curves of the second genus, the product of three ordinates, is to
the product ofthe abscisses ofthe diameter between the ordinates
and the three vertices of the curve, in a given ratio; in which
ratio, if three lines are taken to three segments of the diameter
between the vertices of the curve, each to each, then these three
lines may be considered as the latera recta of the curve, and the
three segments of the diameter between the vertices as its latera
transversa. And as in the conic parabola, which has but one
vertex to a diameter, the rectangle under the ordinates, is equal
to the rectangle under the absciss between the vertex and ordinates, and a given straight line called the latus rectum: so in
curves of the second genus, which have only two vertices to the
same diameter, the product of the three ordinates is equal to
the product of the two parts of the diameter eut off between
the ordinates and the two vertices, and a certain given straight
line, which may be called the latus rectum.




10         NEWTON'S LINES OF THE THIRD ORDER.
4. Of the Ratio of the Products of the Segments of Parallel Lines.
Finally, as in the conic sections, where two parallel chords
are eut by two parallel chords, the first by the third, and the
second by the fourth, the rectangle of the segments of the first
is to the rectangle of the segments of the third, as the rectangle
of the segments of the second, to the rectangle of the segments
of the fourth: so also where four such straight lines occur in a
curve of the second genus, each being cut in three points, the
product of the segments of the first line will be to the product
of the segments of the third as tlie product of the segments of
the second to the product of the segments of the fourth.
5. Of Hyperbolic and Parrbolic Branches, and their Directions.
All infinite branches of curves of the second and higher
genus, like those of the first, are either of the hyperbolic or
parabolic sort. I define a hyperbolic branch as one which constantly approaches some asymptote, a parabolic branch to
be that which, although infinite, has no asymptote. These
branches are easily distinguished by their tangents; for supposing the point of contact to be infinitely distant, the tangent
of the hyperbolic branch will coincide with the asymptote, but
the tangent ofthe parabolic branch being at an infinite distance,
vanishes, and is not to be found. The asymptote to any branch
is, therefo fore, found by seeking for the tangent to a point in
that branch at an infinite distance. The direction of the
branch may be found, by determining the position of a straight
line parallel to the tangent referred to a point in the curve
infinitely distant; for such straight line will have the same
direction as the infinite branch itself.




THE FOUR CASES OF EQUATIONS.            11
SECTION III.
The Reduction of all Curves of the Second Genus to four Cases
of Equations.
All lines of the first, third, fifth, seventh, or odd orders, have
at least two infinite branches extending in opposite directions;
and all lines of the third order have two branches of the same
kind, proceeding in opposite directions, towards which no other
of their infinite branches proceed, (except only the Cartesian
parabola).
Case I. (fig. 1.)
If the branches be hyperbolic, let G AS be their asymptote,
and let CBc be any line drawn parallel to it, meeting the
curve on each side (if possible); let this line be bisected in X,
which will be the locus of a hyperbola, say X(c, one of whose
asymptotes is AG. Let its other asymptote be AB, and the
equation defining the relation between absciss AB and ordinate
BC, if A B = x, BC = y, will be of the formi always
xy2+ey = ax3- +bx 2+ x+ d
where the terms b, c, d, a, e, designate given quantities, affected
by their proper signs + or -, of which any may be deficient,
so that the figure, by reason of their absence, be not changed
to a conic section. It may be, however, that this conic hyperbola coincides with its asymptotes; that is, the point X may
fall in the straight line AB; and in that case the terni + ey is
deficient.
Case II. ( fi. 2.)
But if the straight line CBc is not bounded by the curve at
each end, but only meets the curve in one point, draw A B
any straight line given in position, meeting the asymptote AS
in A, and draw another line BC parallel to that asymptote,
and meeting the curve in C; then the equation expressing the
relation of the ordinate BC and absciss AB always assumes the
forn 
xy = ax~ + bx2 + ex + d




12         NEWTON'S LINES OF THE THIRD ORDER,
Case III. (fig. 3.)
But if the opposite branches are of the parabolic sort, let
the straight line CBc be drawn, if possible, meeting each
branch of the curve, and being bisected in B, the locus of B
will be a straight line. Let this straight line be AB, terminating at any point A, then the equation expressing the relation
of ordinate BC and absciss AB always assumes the form
y2 = ax3 + bx2 + Ccx  d
Case IV. (fig. 4.)
But when CBc only meets the curve in one point, and,
therefore, cannot be bounded at both ends, let that one point
be C; and at the point B let CBc meet another straight line
given in position, AB, and terminating at any point A, then
the equation expressing the relation between ordinate BC and
absciss AB always assumes the form
y = ax3 + bx2 + cx + d
The Names ofthe Curves.
In the enumeration of these cases of curves, we shall call
that which is included within the angle of the asymptotes in
like manner as the hyperbola of the cone, the inscribed hyperbola; that which cuts the asymptotes, and includes within its
branches the parts of the asymptotes so cut off, the circunmscribed hyperbola; that which, as to one branch, is inscribed,
and, as to the other, circumscribed, we shall call the ambigenous
hyperbola; that which has branches concave to each other,
and proceeding towards the same direction, the converging
hyperbola; that which has branches convex to each other, and
proceeding towards contrary directions, the diverging hyperbola; that which has branches convex to contrary parts and
infinite towards contrary sides, the contrary branched hyperbola;
that which, with reference to its asymptote, is concave at the
vertex, and has diverging branches, the conchoïdal hyperbola;
that which cuts the asymptote in contrary flexures, having on
both sides contrary branches, the serpentine hyperbola; that




REDUNDANT HYPERBOLAS.               13
whicl intersects its conjugate, the crufciform hyperbola; that
which intersects and returns in a loop upon itself, the nodate
hyperbola; that which has two branches meeting at an angle
of contact, and there stopping, the cusped hyperbola; that
which has an infinitely small conjugate oval, i. e. a conjugate
point, the punctate hyperbola; that which, from the impossibility of two roots, has neither oval, node, cusp, or conjugate
point, thepure hyperbola.
In the same way we shall speak of parabolas, as converging,
diverging, contrary branched, cruciform, nodate, cusped, punctate, and pure.
In the case of the first-mentioned equation, if ax3 is positive,
(fig. 5), the figure will be a triple hyperbola with six hyperbolic branches progressing to infinity alongside of three
asymptotes, no one of which is parallel to another, two alongside ofeach hyperbola, on contrary sides. And these asymptotes,
if the term bx2 is not deficient, will cut one another at three
points, making a triangle (Dda); but if the term, bx2 is deficient, all the points will converge to one point. In the former
-b                 b
case, take AD = 2 aandd A  =A     2 = -   oin Dd, Dô; and
za              2Va
Ad, Dd, DD, will be the three asymptotes. lIn the latter case
(fig. 6) draw any ordinate BC parallel to the principal ordinate
AG, and in it produced in each direction take BF, Bf, equal
to each other, and in the ratio to AB of /a: 1, join AF, Af;
and AG, AF, Af, will be the three asymptotes. This hyperbola we call redundant, because it exceeds the conic hyperbola
in the number of its hyperbolic branches.
In every redundant hyperbola, if neither the term ey be
deficient nor b2 — 4ac be equal to + 4ae V/a, the curve will have
no diameter. If either of these cases occur, the curve will
have a single diameter; if both, it will have three diameters.
Now, the diameter always passes through the intersection of
two asymptotes, and bisects all straight lies which are terminated by these asymptotes on both sides and are parallel to the
third asymptote.  And the absciss AB is a diameter of the
curve wherever the terin ey is deficient. I use the word dianieter called absolute, here and hereafter, in its common acceptation; namely, as the absciss which has everywhere two equal
ordinates to the same point, one on each side of it.




14         NEWTON'S LINES OF THE THIRD ORDER.
SECTION IV.
THE ENUMERATION OF CURVES.
Of the?ine redundant Hyperbolas, having no Diamneter, and
three Asymptotes, making a Triangle.
VWiEN a redundant hyperbola has no diameter (fig. 5), let
the four roots be found of the equation ax4 + bx3 + ex2 + dx
e2
+   = 0. Let these be AP, Aw, Aer, Ap; draw the ordinates PT, r, e-, pt; then these will touch the curve in so
many points T, r,, t, and, by touching, give the limits of the
curve, by which its species is determined.
For, if all the roots A P, Aw, Ac, A p (figs. 5, 7), are real
and unequal, and of the same sign, the curve consists of three
hyperbolas (inscribed, circumscribed, and ambigenous), and of
an oval. One hyperbola lies towards D, another towards d,
and the third towards ô; and the oval always lies within the
triangle Ddô within the limits r7, in which it is touched by the
ordinates ls and rr.
This is the Ist species.
If the two greatest roots A., Ap (fig. 8), or the two least
(AP, A, fig. 9), are equal to each other, and all of the same
sign, the oval and circumscribed hyperbola will coalesce, their
points of contact 1 and t, or T and r, coming together, and the
branches of the hyperbola, intersecting one another, run on into
the oval, making the figure nodate.
This is the 2d species.
If the three greatest roots Ap, ATr, Am (fig. 10), or the
three least roots A c, A, AP (fig. 11), are equal to each
other, the node becomes a sharp cusp; because the two
branches of the circumscribed hyperbola meet at an angle of
contact, and extend no farther.
This is the 3d species.
If the two middle roots Aw, Ar (fig. 12), are equal, the
points of contact r and 1 coincide, and, therefore, the oval
vanishes to a point; and the figure consists of three hyperbolas,




ENUMERATION OF CURVES.                15
inscribed, circumscribed, and ambigenous, with a conjugate
point.
This is the 4th species.
If two roots are impossible and the other two unequal, and
of the same sign, (they cannot be affected by contrary signs),
three pure hyperbolas result, without oval, node, cusp, or conjugate point; and these hyperbolas will either lie at the sides
of the triangle made by the asymptotes, or at its angles;
thus forming either the 5th (figs. 12, 13) or the 6th species
(figs. 14, 15).
If two roots are equal, and the other two, either impossible,
(figs. 16, 18) or real, and with different signs from the equal
roots, (figs. 17, 19), the figure will be cruciform, two of the
hyperbolas intersecting one another, either at the vertex of the
asymptotic triangle, or at its base, (figs. 16, 17).
These are the 7th and 8th species.
Lastly, if all the roots are impossible (fig. 20), or all real
and unequal, (fig. 21), two being positive and two negative;
then there will result two hyperbolas at the opposite angles of
the two asymptotes, with a serpentine hyperbola round the
third asymptote.
This is the 9th species.
The above are all the possible cases of roots. For if two
roots are equal to each other, and the other two also equal, the
curve will be a conic section with a straight line.
2. Of the twelve Redwndant Hyperbolas, haviiny but one
Diameter.
If a redundant hyperbola has but one diameter, let its
absciss be A B, and obtain the three roots or values of x in the
equation ax3+ bx2 + cx +d = 0.
If all these roots are possible and of the same sign, the
figure will consist of an oval lying within the asymptotic triangle, (fig. 22) and of' three hyperbolas at its angles, viz. circumscribed at the angle D, and inscribed at the angles d and 3.
This is the 10ith species.




16          NEWTON'S LINES 0F THE THIRD ORDER.
If the two greater roots are equal, and the third of the saiie
sign, the branches of the hyperbola lying towards D (fig. 23),
will intersect one another in a node, on account of the contact
of the oval.
This is the 1 lth species.
If the three roots are equal, the hyperbola will be cusped,
without an oval (fig. 24).
This is the 12th species.
If the two lesser roots are equal, the third being of the same
sign, the oval vanishes to a point (fig. 25).
This is the 13th species.
In these last four cases, the hyperbola which lies towards
D  includes its asymptotes; the other two lie within their
asymptotes.
If two roots are impossible, three pure hyperbolas will
result, without oval, intersection, or cusp. Of this case there
are four examples; viz. if the circumscribed hyperbola lies
towards D, (fig. 25); if the inscribed hyperbola lies towards D,
(fig. 26); if the circumscribed hyperbola lies beneath the base
d  of the triangle D d ô, (fig. 27); if the inscribed hyperbola
lies under the same base, (fig. 28).
These are the 14th, 15th, 16th, and 17th species.
If two roots are equal, and the third differing in sign, the
hyperbolas will be cruciform; viz. two of the three intersecting
each other either at the vertex of the asymptotic triangle or at
its base, (fig. 29, 30).
These are the 18th and 19th species.
It two roots are unequal, and of the same sign, the third
being of a different sign, two hyperbolas will result in the
opposite angles of the two asymptotes, with an intermediate
conchoïd, which will either lie at the same side of its asymptote
as the asymptotic triangle, (fig. 31), or on the contrary side,
(fig. 32).
These are the 20th and 21st species.




ENUMERATION OF CURVES.               17
3. Of the two Redundant Hyperbolas, with three Diameters.
The redundant hyperbola, which has three diameters, consists of three hyperbolas included within their asymptotes,
either at the angles of the asymptotic triangle (fig. 33), or at
its sides (fig. 34).
These are the 22d and 23d species.
4. Of the nine Redundant Hyperbolas, with three Asymptotes
converging to a common point.
If three asymptotes intersect one another in a common
point, the 5th and 6th species are changed to the 24th, (fig. 6);
the 7th and 8th to the 25th, (fig. 35); the 9th to the 26t-,
(fig. 36), where the serpentine curve does not pass through the
intersection of the asymptotes; and to the 27th, where it passes
through that point, (fig. 37), in which case the terms b and d
are deficient; and the intersection of the asymptotes is the
centre of the figure equally distant from all its opposite parts.
These four species have no diameter.
The 14th and 16th species are also changed to the 28th,
(fig. 38); the 15th and 17th to the 29th, (fig. 39); the 18th
and the 19th to the 30th, (fig. 40); the 20th and 21st to the
31st, (fig. 41). These species have one diaineter.
Lastly, the 22d and 23d species are changed to the 32d
species, which has three diameters, all passing through the
intersection of the asymptotes, (fig. 42).
All these changes are readily understood by conceiving the
asymptotic triangle diminished till it vanishes to a point.
5. Of the six Defective Hyperbolas, without a Diameter.
If, in the case 1 of the equations, the term ax3 is negative,
the curve will be a defective hyperbola with one asymptote,
and only two infinite branches, extending by the side of that
asymptote. This asymptote is the first and principal ordinate
AG. If the term ey is not deficient, the figure will have no
diameter; if deficient, it will have one diameter only. In the
former case the species will be as follows:e




18         NEWTON'S LINES OF THE THIRD ORDER.
When the roots of the equation
ax4 = bx3 + cx2 + dx + -
namely A r, A P, Ap, A w, are all possible and unequal, the
curve will be a serpentine hyperbola, surrounding its asymptote
in contrary flexure, together with a conjugate oval, (fig. 43).
This is the 33d species.
If the two middle roots AP, Ap (fig. 44), are equal, the
serpentine and oval become joined, intersecting each other so
as to make a node.
This is the 34th species.
If the three roots are equal, the node becomes a sharp cusp
at the vertex of the serpentine hyperbola, (fig. 45).
This is the 35th species.
If the two greater of three roots with the same sign, (Ap,
At, fig. 47,) are equal to each other, the oval vanishes into a
point.
This is the 36th species.
If any two roots are imaginary, there will only remain a
pure serpentine hyperbola, without oval, intersection, cusp, or
conjugate point.  If this serpentine hyperbola does not pass
through A, (fig. 46), the species is the 37th; but if it passes
through A, (which happens when the terms b and d are deficient), then that point A will be the centre of the curve, and
will bisect all straight lines terminated at each end by the
curve, (fig. 47).
This is the 38th species.
6. Of the seven Hyperbolas, Defective, but having Diameters.
In the case where the term ey is deficient, and when on
that account the curve has a diameter, if all the roots of the
equation ax3 = bx2 + cx + d, viz., AT, At, Ar, (fig. 48), are
real, unequal, and of the same sign, the curve will be a
conchoïdal hyperbola, with an oval at its convexity.
This is the 39th species.




ENUMERATION OF CURVES.               19
If two roots are unequal, and of the same sign, the third
being of a contrary sign, the oval will lie on the concave side
of the conchoïd, (fig. 49).
This is the 40th species.
If the two lesser roots AT, At (fig. 50), are equal, the third
Ar being of the same sign, the oval and conchoïd coalesce,
intersecting each other in a node.
This is the 41st species.
If three roots are equal, the node becomes a cusp, and the
curve is the cissoïd of the ancients, (fig. 51).
This is the 42d species.
If the two greater roots are equal, the third being of the
same sign, the conchoïd will have a conjugate point on its
convex side, (fig. 52).
This is the 43d species.
If two roots being equal, the third have a contrary sign,
the conchoïd will have a conjugate point on its concave side,
(fig. 53).
This is the 44th species.
If two roots are impossible, a pure conchoïd results, without
oval, node, cusp, or conjugate point, (figs. 52, 53).
This is the 45th species.
7. Of the seven Parabolic Hyperbolas, having no Diameter.
Whenever, in the first cited case of equations, the term
ax3 is deficient, the term bx2 being present, the carve will be
a parabolic hyperbola with two hyperbolic branches to one
asymptote S A G, and two parabolic branches converging
towards one and the same side. If the term ey be present, the
curve will have no diameter; if it be deficient, then it will
have one diameter.
In the former case the following species will result:When three roots A P, A, Ar (fig. 54), of the equation
bx3 -cx +dx + - e  02
bx + cx2 + dx+    = O




20         NEWTON'S LINES OF THE THIRD ORDERo
are unequal, and of the same sign, the curve will consist of an
oval and two other curves, part hyperbolic and part parabolic:
that is, the parabolic branches will run into the hyperbolic
branches next to them.
This is the 46th species.
If the two lesser roots aie equal, the third being of the
sane sign, the oval and one of the two hyperbolo-parabolic
branches beconie joined, and intersect each other in a node
(ig. 55).
This is the 47th species.
If three roots are equal, this node becomes a cusp, (fig. 56).
This is the 48th species.
If the two greater roots are equal, and the third of the same
sign, the oval vanishes into a conjugate point, (fig. 57).
This is the 49th species.
If two roots are impossible, the two hyperbolo-parabolic
curves will be pure, without oval, intersection, cusp, or conjugate point, (fig. 57, 58).
This is the 50th species.
If two roots are equal, the third of a contrary sign, the
hyperbolo-parabolic curves will intersect one another crosswise, (fig. 59).
This is the 51st species.
If two roots being unequal, and of the same sign, the third
is of a contrary sign, the curve becomes a serpentine hyperbola
about its asymptote, (fig. 60), with a conjugate parabola.
This is the 52d species.
8. Of the four Parabolic Hyperbolas, having a Diameter.
In the other case, (that is, where the term ey is deficient,
and the figure has a diameter,) if two roots of the equation
bx2 - ex - d = 0




ENUMERATION OF CURVES.               21
are impossible, two hyperbolo-parabolie curves will result
equally distant from each side of the diameter AB, (fig. 61).
This is the 53d species.
If in this equation two roots are real and equal, the hyperbolo-parabolic curves unite, intersecting one another crosswise,
(fig. 62).
This is the 54th species.
If the roots are unequal and of the same sign, a conchoïdal
hyperbola will result, with a parabola on the same side of the
asymptote, (fig. 63).
This is the 55th species.
If the roots are of different signs, the conchoïdal hyperbola
and the parabola will be on different sides of the asymptote,
(fig. 64).
This is the 56th species.
9. Of the four Hyperbolisms of the Hyperbola.
Whenever, in the first cited case of the equations, botll the
terns ax3 and bx2 are deficient, the curve will be a hyperbolism
of some conic section. I call it the hyperbolisin of a curve,
when the ordinate is found by applying the product of the
ordinate of that curve, and a given straight line, to a common
absciss. In this manner a straight line is turned into a conic
hyperbola, and every conic section is turned into some curve,
which I here call the hyperbolism of a conic section.
For the equation to the curves of which we are speaking,
(that is to say, xy2 + ey = cx + d,) gives
e ~ V/e2-+4dx +4cxc2
Y            2x
which is formed by applying the product of the ordinate of the
conic section
e - /e2 + 4 dx - 4 c x
2m
and a given straight line m, to the common absciss of the
curves x.




22          NEWTON'S LINES OF THE THIRD ORDER.
From whence it appears that the curve will be a hyperbolism of the hyperbola, ellipse, or parabola, according as the
term cx is positive, negative, or zero.
The hyperbolism of the hyperbola has three asymptotes, of
which one is the prime and principal ordinate Ad, (fig. 65); the
two others are parallel to the absciss AB, equally distant from
it, and on each side of it. In the principal ordinate Ad, take
Ad, Ab, on each side equal to the quantity V/c; and through
the points d and ô draw dg, ay, asymptotes parallel to the
absciss AB.
Where the term ey is not deficient, the curve has no diameter: in this case, if the two roots (AP, Ap, fig. 65) of the
equation                       e
cx2 + dx+ t    0
are possible and unequal (they cannot be equal unless the curve
is a conic section), the curve will consist of three hyperbolas,
opposite each other, of which one lies within the parallel
asymptotes, and the other two outside of it.
This is the 57th species.
When these two roots are impossible, two opposite hyperbolas result outside of the parallel asymptotes, and a hyperbolic
serpentine curve inside of them. This curve is of a double
species. For when the term d is not deficient, it has no centre,
(fig. 66); but when the term d is deficient, the point A is the
centre, (fig. 67).
These are respectively the 58th and 59th species.
But if the term ey is deficient, the curve will consist of
three opposite hyperbolas, one of which lies within tie parallel
asymptotes, and the other two outside then, like tle 57th
species; and, besides, it has a diameter, which is the absciss
AB, (fg. 68).
This is the 60th species.
10. Of the three Hlyperbolisms of the Ellipse.
The hyperbolism of the ellipse is expressed by this equation
xy2 + ey = cx + d




ENUMERATION OF CURVES.               23
and has but one asymptote, namely, the principal ordinate Ad,
(fig. 69). When the term ey is not deficient, the curve is a
serpentine hyperbola without a diameter, and even without a
centre, if the term d is not deficient.
This is the 61st species.
But if the terni d is deficient, the figure has a centre, but
no diameter, its centre being the point A, (fig. 70).
This is the 62d species.
If the term ey is deficient, and the term d not deficient, the
curve is a conchoïd to the asymptote AG, (fig. 71), and has a
diameter without a centre, its diameter being the absciss AB.
This is the 63d species.
11. Of the two Hyperbolisms of the Parabola.
A hyperbolism of the parabola is expressed by the equation
xy2 + ey = d
and has two asymptotes, namely, the absciss AB and the
prime or principal ordinate A G.  The hyperbolas in this
curve are two, not lying within the opposite angles of the
asymptotes, but in the adjacent angles, on each side of the
absciss AB, and either without a diameter, where the term ey
is present, (fig. 72), or with a diameter where ey is deficient,
(fig. 73).
These two are the 64th and 65th species.
12. Of the Trident.
In the second cited case ofthe equations,we had the equation
xy = ax3 + bx2 + cx + d
In this case the curve will have four infinite branches, of
which two are hyperbolic about the asymptote AG, (fig. 74),
extending on contrary sides, and two are parabolic, converging
and making with the other two a sort of trildent-shaped figure.
And this is the very parabolic curve by means of which Des
Cartes constructed an equation of six dimensions.
This is the 66th species.




24         NEWTON'S LINES OF THE THIRD ORDER.
13. Of thefive Diverging Parabolas.
In the third cited case of the equations we had the equation
y2 = ax3 + bx + cx + d
which indicates a parabola, whose branches are divergent with
respect to each other, and extend to infinity towards contrary
parts. The absciss AB is its diameter, and the species are tle
five following.
If the roots of the equation, a X3+ bx2 + cx + d = 0, are
ail possible, and unequal, A-, AT, At, the curve will be a
divergent bell-shaped parabola, with an oval at the vertex,
(figs. 74, 75).
This is the 67th species.
If two of the roots are equal, the parabola becomes either
nodate, by running into the oval, (fig. 77), or punctate, on
account of the oval becoming infinitely small, (fig. 78).
These two are the 68th and 69th species.
If the three roots are equal, the parabola will be cusped at
the vertex, (fig. 80), and this is the Neilian parabola, commonly
called the semicubic.
This is the 70th species.
If two roots are impossible, a pure bell-shaped parabola will
result, (figs. 78, 79).
This is the 71st species.
14. Of the Cubic Parabola.
In the fourth case cited of the equations, if the equation be
y = ax3 +bx + cx+ d
a parabola is indicated, having branches in contrary directions,
and conimnonly called the cubic parabola, (fig. 81).
Thus, altogether, the species are 72.




GENERATION OF CURVES.               25
SECTION V.
The Generation of Curves by Shadows.
If the shadows of curves caused by a luminous point, be
projected on an infinite plane, the shadows of conic sections
will always be conic sections; those of curves of the second
genus will always be curves of the second genus; those of the
third genus will always be curves of the third genus; and so
on ad infinitum.
And in the same manner as the circle, projecting its shadow, generates all the conic sections, so the five divergent
parabolas, by their shadows, generate all other curves of the
second genus. And thus some of the more simple curves of
other genera might be found, which would form all curves ofthe
same genus by the projection of their shadows on a plane.
Of Double Points in Curves.
We have remarked that curves of the second genus can be
cut by a straight line in three points. Sometimes two of these
points coincide; as in the case when the straight line passes
through an infinitely small oval, or through the intersection of
two parts of the curve cutting each other, or meeting in a
cusp. And whenever all the straight lines, extending in the
direction of any infinite branch, cut the curve in only one
point (as occurs in the ordinates of the Cartesian parabola, and
of the cubic parabola, as well as in the straight lines of the
absciss of the hyperbolisms of the hyperbola, and in the parallel lines of the parabola), we must conceive that those
straight lines pass through two other points in the curve at an
infinite distance. Two intersections of this sort when they
coincide, whether at a finite or at an infinite distance, we shall
call a double point.  Now the curves possessing a double
point may be described by the help of the following theorems.




26         NEWTON'S LINES OF THE THIRD ORDER.
SECTION VI.
Of the Organic Description of Curves.
Theorem I.
If (fig. 82) two angles of given magnitude PAD, PBD,
are made to rotate about the poles A, B, given in position, and
their legs A P, B P, at their concourse P, move along a straight
line, the concourse of the other two legs AD, BD, will describe a conic section passing through the poles A, B; except
when the straight line in question passes through either pole A
or B, and except when the angles BAD, ABD, vanish together; in which cases the point D will describe a straight line.
Theor. II.
If the legs AP, BP, at their point of concourse P, pass
along a conic section running through either pole, A, the two
other legs AD, BD, at their point of concourse D, (fig. 83),
will describe a curve of the second genus, passing through the
other pole B, and having a double point in the first pole A,
through which the conic section passes; except when the
angles B AD, A B D, vanish together; in which case the point
D will describe another conic section passing through the pole A.
Theor. III.
But if the conic section, which the point P passes along,
runs through neither of the poles, A, B, (fig. 84), the point P
will describe a curve of the second genus, or one of the third
genus, having a double point. This double point will be found
at the point of concourse of the legs AB, BD, which describe
the curve, when the angles BAP, ABP, vanish together. But
the curve described will be of the second genus, if the angles
BAD, ABD, vanish together, otherwise it will be of the third
genus, and will have two other double points at the poles A, B.




ORGANIC DESCRIPTION OF CURVES.          27
To describe Conic Sections by five given Points.
A conic section is determined when five points in it are
given, and by them may thus be described. Let the five points
be A, B, C, D, E (fig. 85). Join any three of them, A, B, C,
and of the triangle ABC let any two of the angles CAB,
CBA, rotate about their vertices A, B, and when C, the intersection of the two legs AC, BC, is successively applied to the
two other points D, E, let the intersection of the remaining legs
AB, B  A, take place at the points P, Q. Let the straight line
P Q be drawn and infinitely extended, and let the moveable
angles be so rotated as that the intersection of the legs AB,
B A, may run along the line P Q; then C, the intersection of the
remaining legs, will describe the conic section proposed by the
first theorem.
The Description of Curves of the Second Genus having a Double
Point, by means of Seven given Points.
All curves of the second genus, having a double point, are
determined where seven points in them are given, and may be
described through those points as follows, one of these points
being the double point:-Let any seven points ABCDEFG
(fig. 86) in the curve to be described, be given, A being a double
point. Join A, and any two other of the points, say B and C,
and of the triangle ABC, let the angle CAB revolve about
its vertex A, as well as either of the remaining angles ABC
about its vertex B. And when the point C, being the concourse of the legs AC, BC, is successively applied to the four
remaining points D, E, F, G; let' the concourse of the other
legs AB, BA, fall on the four points P, Q, R, S. Let a conic
section now be described passing through the four points, and
the fifth A; and let the aforesaid angles CAB, CBA, so
rotate, that the concourse of their legs AB, BA, shall run
along the said conic section, then the concourse of the remaining legs AC, BC, will describe the curve proposed in the
second theorem.
If, instead of the point C, the straight line BC is given in
position, which touches the curve to be described in B, the




28         NEWTON'S LINES OF THE THIRD ORDER.
lines AD, AP, will coincide; and, instead of the angle D AP,
there will be a right line to be rotated about the pole A.
If the double point A be infinitely distant, the straight line
must be drawn continually extending in the direction of that
point, and be moved in parallel motion while the angle ABC
revolves about the pole B.
These curves may also be described somewhat differently
by means of Theorem III.; but it is enough to have explained
the more simple method here given.
In like manner, curves of the third, fourth, and higher
genera may be described; not, indeed, all curves, but so many
of them as by reason of some convenient relation, may be drawn
by the motion of their loci. But to describe any curve whatever
of the second or superior genus conveniently, when it has no
double point, must be considered a very difficult problem.




CONSTRUCTION OF EQUATIONS.             29
SECTION VII.
The Construction of Equations by the Description of Curves,
One of the uses of geometrical curves is, that by means of
their intersections, the solution of problems may be effected.
Let an equation of nine dimensions be proposed for construction,
x9 * + bx7 + C6 + dx5 + ex4 +fx3 + gx  + hx + k = 0
+ m
Where b, c, d, &c. signify any given quantities whatever,
with their signs + and -. Let the equation to the cubic
parabola x3 = y, be assumed, and the first equation becomes,
by the substitution of y for x3,
y3 + bxy+ cy2J dx2y + exy +   y + x + f gx2 +   hx + h = 0
which is an equation to a curve of the second genus. Where
m, orf, may be deficient, or may be assumed at pleasure. By
the description of these curves, and their intersections, the roots
of the equation to be constructed will be given.  It will be
enough to describe the cubic parabola once only.
If the equation to be constructed be reduced to one ofseven
dimensions by the deficiency of the two last terms hx and k,
the other curve, by destroying m, will have a double point at
the beginning of the absciss, and hence may easily be described
by the above rules.
If the equation to be constructed, by the deficiency of the
three last terms gx2 + hx + k, be reduced to one of six dimensions, the other curve, by destroyingf, becomes a conic section.
And if, by the deficiency of the six last terms, the equation
be reduced to one of three dimensions, we come to Wallis's
construction by means of the cubic parabola and straight line.
Equations may also be constructed by means of the hyperbolism of the parabola with a diameter. Thus, if an equation
of nine dimensions wanting the last term, be to be constructed,
a + cx2 +dx3 + ex4 +fx5 +gx6 + hx7 + kx8 + lx9 = O
+t- 




30         NEWTON'S LINES OF THE THIRD ORDER.
Let the equation to the hyberbolism in question be assumed
X2y = 1
Then substituting y for x, the equation becomes
ay3 + cy2 + dxy2 + ey +fxy + m+g + hx + kx2 + lx3 = 0
which indicates a curve of the second genus, by whose description the problem will be solved. Here either the quantity m
or g may be deficient or assumed at pleasure.
By means of the cubic parabola and curves of the third
genus, equations may also be constructed of all dimensions up
to twelve, and by the same parabola and curves of the fourth
genus, equations may be constructed up to fifteen dimensions,
and so on ad infinitum. These curves of the third, fourth, and
higher genera, may always be found by means of description
through points ascertained by geometrical means. Thus, if the
equation to be constructed be
x12, * +axlo+bx9+cx+8 dx7 +ex6+fx5+gx +hx3+ix+kx+l = O
and a cubic parabola be described, let the equation to that
parabola be x3 = y, then substituting y for x3, the equation to
be constructed becomes
y4 + axy3 + cx2y2 +fx2y + ix2 = 0
+b     + dx   +gx   +  x
+e     +h    +1
which is an equation to a curve of the third genus, by the
description of which the problem may be solved. This curve
may be described by finding its points by means of plane
geometry, because the indeterminate quantity x does not rise
to higher dimensions than two.
Lon-don:Printed by G. BA.RCLAY, Castle St,. Leicester Sq8




NOTES TO THE FOREGOING TEXT.
c




In the following notes the undermentioned propositions are
taken as proved:I. In any algebraic curve, if one of the co-ordinates becomes
equal to zero, so that one variable in the equation which defines
the curve vanishes, the roots of the equation so reduced are the
values of the corresponding co-ordinate.
2. Infinite branches can only occur in pairs, the number of
such branches is therefore always even.
3. If an algebraic curve has no infinite branch, it is necessarily
an oval, that is, it returns upon itself.
4. The general indeterminate equation of the nth degree
includes all the lines of the nth order.
5. If n denote the dimensions of a curve, "  3n will be the
2
number of constants in the general equation, defining all the
curves of the order n.
Thus a straight line is determined by two points, its order being
I; a conic section by five points, its order being 2: a cubic line
by nine points, its order being 3, &c.
6. Every algebraic curve may have as many asymptotes as
there are dimensions in the equation expressing it.




33
Notes to Section I.
On the Ordesrs of Lines.
Newton here makes use of the word ' geometrical' in a limited Note to
sense in accordance with the meaning assigned to it by the inos.
mathematicians following Descartes, viz. as distinguishing lines
which can be always indicated by an algebraical equation from
those of which the equations are transcendental. There are many
curves which can be eut by a straight line in only two points,
but only one class of geometrical curves which can be so cut,
namely, the conic sections. Curves, such as the catenary, &c.,
which admit of a straight line cutting them in only two points,
but which cannot be expressed by an algebraical equation in
finite terms, were formerly called mechanical curves; but this
distinction of geometrical and mechanical, originally proposed
by Descartes, has fallen into disuse, the division of curves into
algebraical and transcendental being considered more appropriate.
Neither has the distinction adopted by Newton between lines
and curves been generally followed by other writers, it being
found more convenient to treat the straight line as a curve whose
radius of curvature is infinitely great, so that the classification
of curve lines may proceed according to the dimensions of the
equation expressing the relation of their co-ordinates. A curve
of the first order will thus be indicated by a simple equation
between two variables, a curve of the second order by a quadratic,
a curve of the third order by a cubic equation, and so on for the
higher orders.
".. according to the number of points in which it can be cut by a No;o te
straight line."                                           lin'7.
A line of the nth order is expressed by an equation of n dimensions which can have no more than n roots. A straight line
therefore cannot eut a line of the nth order in more than n points




34                   NOTES TO SECTION I.
for let the general equation representing a line of the nth order
be assumed, and let the equation to a straight line referred to the
same axes be
y= ax + b
then at the points where the straight line intersects the curve, the
co-ordinates of the line and curve will be the same. Substituting
the value of (y), viz. a x + b, in the general equation we have a
resulting equation of the same dimensions, and therefore having
the same number of roots as before, and these roots when real
represent the abscissas of the points of intersection. If some of
the roots are imaginary, there will be fewer points of intersection.
Note to     "A  line of the infinitesimal order is one which a straight line
Ule 17.  may cut in an infinite number of points, such as the spiral, cycloïd,
quadratrix."
It is obvious that the spiral and the cycloïd (if we imagine its
generating circle to be perpetually rolling on) are curves which
may be cut by a straight line in an infinite number of points. It
is not so immediately obvious that the quadratrix of the circle,
which it is to be presumed is the curve alluded to by Newton,
can be so cut. Indeed this curve, as it was known to the ancients,
cannot be eut by a straight line in more than two points, because
its inventor and those who followed him did not contemplate the
continuance of the curve beyond the limits of its generating semicircle. It was first shown by Vincent Leotaud, a Jesuit professor
of mathematics in the college of Dôle, about the year I65o, that
by a more general consideration of the genesis of the curve it may
become susceptible of two infinite branches, extending below the
axis of x, and limited by asymptotes parallel to the axis of y, at
distances - a, and 3 a from the origin, (a) being the radius of the
circle used. (See his " Cyclomathia sen de Multiplici Circuli Contemplatione," lib. iii.) If we imagine the rotation of the radius
beyond 360~, the branches of the quadratrix become infinite in
number as well as in extent: the curve is therefore one of infinitesimal order.
There is another quadratrix of the circle, invented by Tschirnhauser in the seventeenth century, which has no infinite branches,




NOTES TO SECTION II.                 35
but still is a curve of infinitesimal order. Its equation is, (a) being
the radius of the circle,
y=a sin 
2a
From which we infer that the curve is composed of an infinite
number of sinuosities, meeting the axis of x at intervals equal to
the diameter of the circle; for when x = 2 n a, y = o, whatever
value may be assigned to n; and thus a straight line may eut the
curve in an infinite number of points.
Notes to Section II.
Of Curves of the Second Genus, their Ordinates, Diameters, 'c."
The proposition regarding the rectilinear diameters of lines of Note to
the third order is, that if two parallel chords be drawn and be lie I8
eut by a straight line, so that their sum on one side of it be equal
to their sum on the other, then all lines drawn to meet the curve
parallel to these will be eut by the straight line in the same
manner.   The proposition may be shown to be true for all
algebraic curves thus, let
y - (a + b) y"- + (c   d + e)    — &c. = o
Then, since in every equation the coefficient of the second term
with its sign changed is equal to the excess by which the sum of
the positive exceeds the sum of the negative roots, where this term
is deficient, it is an indication that the sum of the positive is equal
to the sum of the negative roots, that is, in the curve represented
by the equation, that the sum of the positive equals the sur of the
negative ordinates.
Let x and x' represent any two values of the absciss corresponding to intersections with two parallel ordinates, then the
equation in the two cases becomes
yn-(ax J b) y -1 +... &c.=o, and y -(a' + b) y"' +... &c. o




36                   NOTES TO SECTION II.
And since by hypothesis, in each case the suing of the positive
equals the sum of the negative ordinates,
a, + b = o, ancd ax' + b = o.. a (x-') = o.
Hence a and b also = o, showing that whatever value be assigned to the absciss, the coefficient of the second term in the
equation vanishes; in other words, the sum of the ordinates on
each side of the rectilinear diameter is equal for all values of z.
" The Intersection of two Diameters is the Centre."
Noteto      This definition is open to objection, for there may be a centre
ine 30.  to a curve which admits of no rectilinear diameter. For instance,
the curve No. 38 of the enumeration, fig. 47, and some of the
curves whose asymptotes intersect in a point, as fig. 37. A curve
may be said to have a centre when every value of a radius vector
has its equal and opposite value. In the case of curves of uneven degree, this can only take place when the equation referred
to the centre contains no term of even dimensions in x and y, the
variables; in the case of curves of even degree the central equation
can contain no term of uneven dimensions in x and y.    To
ascertain, therefore, whether a curve admits of a centre, it is
sufficient to see whether the terms of odd (or even) dimensions of
the variables can be exterminated by transforming the equation.
The conditions of a centre existing, and its position, may be
readily obtained by differentiation; for instance, in the circle
whose equation is
y= 2 a x -_ v2
If this equation be differentiated and the terms made separately = o, the result gives x = a, y - o, the co-ordinates of the
centre. In the case of a line of the third order, if the equation be
twice differentiated, and the resulting terms made separately = o,
the values obtained for x and y will indicate the conditions which
determine the existence of a general centre. Tlus, in the first
case of the Newtonian equations,
xy2 + ey = ax' + bx2 + cx + d




NOTES TO SECTION II.                37
From the second differentiation we obtain
(I) 2=o, (2) 4 =o, (3) (6ax +2b)=o.. b =o.
And since d = o, also, the existence of a general centre depends
on the conditions b = o, d = o, and the centre must be the origin
itself.
c The sum of those two segments of the secant which are drawn Note to
page 9,
from any two asymptotes on the same side to two points of the ine8.
curve, will be equal to the third part which is drawn from the third
asymptote on the contrary side to tate third point of the curve."
In the figure, let AP be the diameter to the ordinates PM,
PM', PM"; therefore, P M + PM' = P M".
Now at an infinite distance the curve coincides with its
asymptote; therefore the points M  and H, M' and i', M" and
H" coincide, when the secant is supposed at an infinite distance;
so that PH' + PH = PH", which equation will be true for any
position of the secant parallel to itself;
M:8.'..PM- PH + PM     - PH    =
PM/ — PH/", and
a> ---JL.
HM - -I'M' = H"M", or
\ i '     HM = H'M' + H"M".
Where there are three rectilinear asymptotes, the three
branches of a curve cannot be all situated on the same side
relatively to such asymptotes; but
3:I      if two branches lie beneath two
asymptotes, the third lies above it.
For instance, such a case as is
represented in the figure is impossible; for the segments of the
secant being all affected by the
same sign, their sum will not be
equal to o, as it should.




38                   NOTES TO SECTION II.
Note to     "In non-parabolic curves of the second genus, the product of three
lin-oz9.  ordinates is to the product of the abscissas of the diameter between
the ordinates and the three vertices of the curve, in a given ratio."
Let the absciss AM and ordinate MP each eut the curve in
three points; the product of the ordinates, viz., Mr. M Ia.MP
will be to the product of the abscissas M K. MI M H in an invariable ratio; the inclination of the axes being given.
In the general equation representing all lines of the third
order, the last term, or that in which no power of y is involved,
viz.,
fx3 -g x + h x - k,
is, by the theory of equations, equal to the product of all the
roots. Therefore,
MlI.M-z. MP =fx3 - gX + hx -k.
W/                  `1R
/r'
Let the three points in which the absciss cuts the curve be
H, I, K: then A being the origin, AH, AI, AK, will be the
values of x when y = o, and will represent the three roots of the
equation,
_g2    hx   k
3-y + -          -0.
This equation is formed by the three factors, x - AH, x - AI,
x- AK, multiplied together; that is,
MH.MI.MK =- x         g- 9 + -hx  k     (M.M.MP);
f     --  if    f
or, the product of the ordinates between M and the curve is to
the product of the abscisses between M  and the curve in the
invariable ratio f: i.




NOTES TO SECTION II.                  39
A consideration of the general equation, without referring to
geometrical ideas, leads to the same result, for in the equation
ay'   bxya +ccx  + dx3+ ey    fy +~. q + 1 + iy + k c= o
if x aiid y are successively assumed = o
ay3+ ey2 + iy + k = o      (I)
dx3 + gx + 7ix + k = o     (2)
The product of the roots of (I) is k
a
and the product of the roots of (2) is k
They are therefore to each other in the invariable ratio d: a,
which ratio will not be affected by any change of origin, but
will therefore remain the same for all axes of co-ordinates
parallel to the axes assumed.
We have only had occasion to notice the case in which both
absciss and ordinate meet the curve in three points: but suppose
the absciss cuts the curve in only one point, and let that point be
the origin, we have
M.M.MP=                   ~h f: + -  +{(-$) + f  24f2)}
Take AC towards M     =g — and draw CB perpendicular to
AM    and =   V4fh -        then since (x-     )2=CMM      and
AM and =/     2f g
h    g" =CB2
f   4/2
r




40                  NOTES TO SECTION II.
/. AM (CM" + CB2) orf. AM. BM2 = MP. M. MX.
Therefore when the absciss meets a line of the third order in
only one point, the product of the ordinates is to the product of
the absciss multiplied by the square of a given line BM, in an
invariable ratio. That the perpendicular BC will always have a
real value, appears from the consideration that whenever the
absciss AM cuts the curve only once, the roots of the quadratic
fx —gx +h = o
are necessarily imaginary, and therefore 4fh > g2, so that
V/4f h - g2 is real.
Even when the absciss does not meet the curve anywhere,
in which case the equation for determining the abscisses is
gx - -hx + k = o, h2 being < 4gk,
the theorem  still holds good, for a line may always be found
such that its square shall be to the product of the ordinates in an
invariable ratio. Thus let the absciss AM to the ordinate MP
be assumed so as not to meet the curve.
Mt. My. MP =         gx- -x + k =      g  (  -  +  -- -lTake AB = - -, and draw the perpendicular B C H.  With
2g
centre A, and radius ~k, describe a circle determining the point
\-. — -  g   C, then CM    will be the required
I\ """        line. It is evident that the point C
'`..   1is fixed, whatever inay be the posi" -----  tion of the point M  in the absciss
CB ___ ----        >M A M.
/^. ---^':         <See Cramer, Analyse des Lignes
Courbes, ch. v., Maclaurin, Lin. Geom.
and Stirling's Commentary.
Note to   c Of the Ratio of the Products of the Segments of Parallel Lines."
page Io,
lino     We have seen that where the absciss and ordinate each cut
the curve in three points, the product of the three ordinates is to
the product of the three abscisses in a given ratio. In the curve
ABCD let the absciss and ordinate cut the curve in the points




NOTES TO SECTION II.                41
IKLBHD, and draw the parallel ordinate ANC. The product
/          GBIGB. GH. GD: GI. GK. GL::
/MA. MN. MC: MI. MK. ML
-GU/  I    and drawing R O parallel to IL, we have
for the same reason
i,/  G ^    _MA. MN. MC: MI. MK. ML::
/Q r  O0         PA. PN. PC: PR. PQ. PO
therefore
GB. G  H. GH GD: GI: GK. GL::
PA. PN. PC:PR. PQ.PO
which is the proposition in the text.
"I define a hyperbolic branch as one which constantly approaches Note to
some asymptote, a parabolic branch to be that which, although ininite, iiUe 15.
has no asymptote."
Here it is to be understood that rectilinear asymptotes are
spoken of, and the definition suggests a simple method of distinguishing hyperbolic from parabolic branches, for if we express
the ordinate to an infinite branch in the form of a series of
descending powers of the absciss, and take all the terms of this
series in which the exponent of x is positive or zero, then if the
sum of these ternis indicates a straight line, this straight line is an
asymptote to the curve, which is therefore hyperbolic. If the sum
of the above-mentioned terms does not indicate a straight line the
curve is parabolic.
Thus in the equation
y3 +- axy - x3 = o
the series for y will be
a    a3
y=x-       +    3 —  &c.
t/y  {t3     34X2
where the sum of the ternis of the series in which the exponent
of x is positive or zero, viz. x - -, indicates the ordinate of a
straight line, and the curve is therefore hyperbolic as to its branch.
So in the equation
yf2 - x3 + axz  = o




42                   NOTES TO SECTION II.
the series for y will be
y -= x + &c.
which is not indicative of a straight line, and therefore the branch
will be parabolic.
If the series for y commence with a negative power of x, as
a2
y =    - &c.
the branch of the curve will be hyperbolic, for when x is supa2
posed infinitely great 2 becomes equivalent to o, and the branch
coincides with the straight line y = o, that is, the absciss is itself
an asymptote.
Since several different series may be derived from one equation,
and these series may commence with different terms, the same
curve may have both hyperbolic and parabolic branches.
Noto to     " The asymptote to any branch is therefore to be found by seeking
page xo,
line 23-  for the tangent to a point in that branch at an infinite distance"
This is the principle on which is founded the method
usually given in modern treatises, for drawing the asymptotes to
a curve whose equation is given; for the limiting position of the
tangent, when the point of contact is infinitely remote, becomes
asymptotic: and if its position is unlimited there will be no rectilinear asymptote.
There are, however, sometimes more convenient methods for
determining the existence of the asymptotes of curve lines than
this. The following is the method proposed by Stirling in his
"Lineoe tertii ordinis Newtonianoe."
Let the ordinate be reduced to the form
y    = Ax' + Bx-' + Cx-2' + &c.,
converging for large values of x.
Assume a new ordinate, y', equal to all the initial terms of
this series which are not diminished by the increase of x. Then
the difference of these two ordinates as x increases must continually diminish, and ultimately vanish; therefore the branches




NOTES TO SECTION II.                43
having the ordinates y and y' to the same absciss x, will continually
tend to coincidence, and y' is the ordinate to the asyrptote, whose
equation is therefore given.
This method enables us to perceive not only the nature of the
asymptote, whether rectilinear or curvilinear, but also on which
side of the curve it lies; for if the first term in the series which is
diminished by the increase of x, that is the first term which
contains the inverse power of x, is positive, the asymptote lies
between the curve and absciss, if not, then the asymptote lies on
the other side of the curve from the absciss.
And it will not be necessary in every case investigated, to
have recurrence to this particular series. For we may assume
one general series, applicable to all the curves included in the
general equation belonging to any order, and then we may
construct a general canon which will be sufficient for the investigation of the asymptotes of all such curves. Thus let the
general equation of the second order be
Ay2 + Bxy + Cx2 + Dy +- Ex + F = o.
Let y = ax + b + cx-l + &c.
Determining coefficients, we shall have a =  a root of the
equation
Aa2 + Ba + C = o, whence a is given
Da +E          Ab + Db+ F
a A     -t B'       2aA+B
Again, if we assume the general equation of the 3d order,
Ay3 + Bxy2 + Cxy + D  3+Ey +F     y +      Gx2+Hy — Kx+L == o
which includes all lines of the 3d order whatever, then a will be
a root of the equation
Aa3 + Bar + Ca +D = o
Aa2 + Ba + C
3Ea2 + 2Fa + C
3Aab2 + Bb2 + Eab + Fb + Ha + K
3Aa2 + 2Ba + C




44                  NOTES TO SECTION II.
From which expressions may be found the asymptotes of all
these curves without further recourse to series. For a always
gives the inclination of the asymptote to the absciss; b gives the
distance between the origin and intersection of the asymptote
with the absciss; and c shows on which side of the asymptote
lie the branches of the curve.
The asymptotes of a curve may also be found by means of
Newton's parallelogram, of which examples will be given in a
subsequent part of this volume.
Asymptotes of lines of an order superior to the conic sections
may be cut by the curve, and the number of points in which
they may so intersect the curve will be less by two than the
dimensions of the equation of the curve. This may be shown as
follows: -
Let the equation to the curve be
myn + (ax + b) y1-l +   2      +  + +  e)  -2 + &c. = o
If the curve has an asymptote parallel to the axis of y, the
term myn will vanish, since m = o in this case, and the x to
which this asymptotic ordinate corresponds, will be determined
b
by the equation ax + b = o, or x = —. Substituting which
value in the equation to the curve, the term (ai + b) y~-1 will
also vanish, and then the term
cb — dba +ea2
yn-2
a2      Y
becomes the first term in the equation. Therefore for the absciss
b
-, the equation whose roots determine the points where the
asymptote and curve intersect, will be of n - 2 dimensions, and
will not have more than n - 2 roots. The asymptote and curve
can therefore intersect in n - 2 points only.
Cor. Thus a line of the third order can intersect its asymptote
only once, i.e. each of the asymptotes may be once intersected by
a branch, but only once.
This theorem admits of a simple geometrical illustration, for
let AB C represent a line of the third order, which the straight
line G B D cuts in three points. If the straight line be made to
revolve on the pole B until it becomes parallel to the asymptote




NOTES TO SECTION III.               45
H L, it evidently will intersect the curve in only
two points, the point G going off to infinity. And
if the straight line continuing parallel to the asymp/  tote be now conceived to move towards the asymptote H L, until coincidence takes place, the point of
intersection D will also go off to infinity, and
there will remain only B, in which the curve cuts
the asymptote.
From the property of the asymptote here adg Ey   verted to, it follows that if we draw the ordinates
to the curve parallel to it, they will cut the curve
in one point less than the dimensions of the equation defining it.
Thus an ordinate parallel to an asymptote of a line of the
third order will only rise to quadratic dimensions, so that by a
simple transformation we are enabled to describe any species
whose branches are hyperbolic with facility, since to find the
values of the ordinate to any point in the absciss, only the
solution of a quadratic is needed.
Notes to Section III.
" Al lines of the Ist, 3rd, 5th, 7th, or odd orders, have at least Note to
two infinite branches extending in opposite directions."  line4
Since impossible roots can only enter equations by pairs, it
follows that every equation of an odd order, and therefore having
an odd number of roots, must have one of those roots real.
Therefore, however great the absciss belonging to such a curve
be assumed, it must always have a real ordinate indicating a point
in the curve, whose branch is consequently infinite. Also since
the absciss may be assumed negative as well as positive, there
will be a corresponding infinite branch on the negative side, and
the curve has therefore at least two infinite branches.
One pair of infinite branches, then, is the least number which
a line of uneven order must necessarily have, but it may have
more than one pair, in fact as many pairs of branches as there are
unities in the dimensions of the equation.
Lines of the second, fourth, sixth, and even orders, on the




46                  NOTES TO SECTION III.
other hand, will frequently have no infinite branches, the ordinates
becoming impossible for any value of
the absciss, positive or negative. Take,
for example, the equation
y4 + 2 y2X + X4 - 6 axy- 2 ax3
+ a2C2 = O
whence
y = +- ax + 2ax- x2/
the four values of y, corresponding to
any given absciss of a line of the fourth
order, are here shown, and are evidently made up of the sums and differences of the ordinates of the parabola y -=  V/ax, and of the circle
y  -= -    v2 ax - x2, referred to the
same axis and origin.
Now since the circle cannot extend beyond its diameter, it is
evident that the curve cannot extend beyond the extremity of
the circle's diameter, and must therefore return upon itself; consequently it has no infinite branches: the annexed figure represents the course of the curve.
Again the equation
2y =- V /6x - x2 ~ V/6x + x      / 36 - xZ
represents a line of the eighth order, having
four positive and four negative ordinates to each
absciss. And it is evident if x is assumed either / 
to be negative, or greater than 6, that then the
ordinates will be imaginary, and that there are  ( 
no infinite branches. The curve may be easily 
described by points, for we have the values of y
When    =0o   =I  =z 2  =3 c=4     =5  2=6  / 
+ + + y =  3'00  5 40 6 25 6-70 68I  6'48  4'24 
- + +y=   3-00  3'I6 3'41 3'70 3'98  4'25  4'24
+- + y=   3T00  2'75 2'25 I'5o 050 -0'93 -4'24
+ + - y = -3'00 -o-51 '59 1'50 2 35 | 3'16  424




NOTES TO SECTION III.               47
The figure shows the course of the curve; the example is
taken from Euler's Analysis Infinitorum.
Case I. of the Equations.
"If the branches be hyperbolic, let GA S be their asymptote," 8'c. Note to
line Io.
The reduction of all lines of the third order to four cases of
equations, is the basis upon which Newton has founded the whole
system of the subsequent enumeration; he has, however, given no
demonstration of this fundamental principle of his classification of
these curves.
To demonstrate conclusively that it is always possible to
reduce the general equation of the tlird order to one of the four
cases proposed, would be to exceed the limits of a note. The
French mathematician Nicole, has taken the trouble to discuss
all the equations, thirty in number, which result from assuming
each of the terms and combinations of terms in the general
equation to vanish; see "Mémoires de l'Academie," for the year
1729, p. 194. It will be enough here to state that he arrives at
the same conclusions as those of Newton.
Assuming then that the reduction in question can always be
effected, we may proceed to show that the asymptote G A S and
any chord ordinate C N parallel to it being given, the point which
bisects this chord ordinate is the locus of a conic section, or else a
straight line.
When the general equation is referred to axes one of which is
the given asymptote, any point in it, S being taken as the origin,
it appears in the former
xy2 + (b ^ + cx + d) y =fx3 + gx2 + hX + k,
whence
bx2 + cx+ d ~If x3+ + g    1X2 + hxb        c + k  bx +cx + 
=       2X          _        X                 +      IX
Y                       xc
or for brevity write y = P + Q.
E




48                    NOTES TO SECTION III.
Let S K be the absciss to the ordinate KNC.
ti\    ~    ThenKC= P + Q,KN =P-Q,CN = 2Q.
\ \^,       Bisect C N in D, and D will be the locus of a
\     yperbola, for K D = P, (D C & D N = + Q)
bxa + cx + d,  i
_  -—: _ that is, y =   +, +  a hyperbola.
When d = o, y = b, an equation to
a straight line, which explains the meaning of
the author when he says, "It may be that this conic hyperbola
coincides with its asymptotes, that is, the point X may fall in the
straight line AB."
When the term     x is not found in the denominator, and
bx + cx + d
y       = +  +   +- &c., the bisecting line is a parabola, and the
curve is a parabolic hyperbola.
If the hyperbola Xe> be referred to its asymptotes as axes, its
a         e
equation will appear in the form y' (= B D) =      where - is
2 X         2
some constant quantity: and if the proposed curve be referred to
the same axes, the sum of the positive and negative ordinates
BC- -BD+DC
(BCandBN)=2BD. For B                 BD + D
FBN=BD —D0
Therefore in the equation to the curve, -, will be the coefficient of y' in the transformed quadratic, and the form of it
will be
y,'2 -   = A e2 + BD' + C +
or,
x'y'-   ey' = Ax3 + Bx'2 + Cx' + D
which agrees with the first case of the equations given by Sir
Isaac Newton.
Note to      "In the case of the first-mentioned equation if axn is positive
e                    curve  ill be a trle     bola  wit six herbal
fie 12-  (fig. 5) the curve wvill be a triple /iyperbola witl. six hyperbolic




NOTES TO SECTION III.               49
branches progressing to infinity, alongside of 3 asymptotes no one of
which is parallel to another."
The proposed equation is
xy2 - ey = ax3 + bx      x + cx + d
whence is obtained
e                      d
d     e2
y  - -   /a2 + bx + c +- +       -2
2x                     x     4 
Expanding the irrational portion of the equation in a series,
we have
4- e -4-     &c.,
2X    e
the two values of the ordinate will therefore be,
e   d
Y=       + AAx-+Bx3 + &c.     (I)
y =-d - Ax —Bx       - &c.      (2)
e
Examining the curve in the neighbourhood of the origin,
from (I) we infer that when w = o, y is infinite and therefore
the axis of y is an asymptote.  From (2) we infer that the
curve cuts this asymptote at a distance - - from the axis of x.
This asymptote has two infinite branches of the curve alongside it, one on each side, in contrary directions.
If x is continued infinitely in a positive direction, there will
always be a positive and a negative value for y, whence we infer
the existence of two more infinite branches.
If x is negative the equation becomes
- -xy - ey = - ax3 + bx2 - cx + d,
or,
xy2 + ey = ax'3 - bX' + cx, - c.




50                   NOTES TO SECTION III.
Here also, when x is continued infinitely, there will be two
values for y, one positive and the other negative. The number
of infinite branches will therefore be, in all, six.
To show that no two of the asymptotes are parallel to one
another, let y be reduced to a series converging for large values
of x, there will result
b        4ac - b~ + 4ae Ja +
y-=x,,/a 4 27"a '           --          4+ &c.
2 yVa         8 ax,/a
-  b     4ac - b - 4ae /a
y      -=-  x,/a-   -          — &c.
2,/a         8ax/ a
the first two terms of these series indicating the equations to
two straight lines, viz.,
y     (          2 + a)
show the existence of two rectilinear asymptotes, cutting the axis
of x at a distance -   from the origin, and making angles with
2a
this axis whose trigonometrical tangents are / a and - V/a.
b                   b
Thus (in fig. 5) if A D =- and A d, A ô, = —, joining
2 a                2 Va'
D d, D, and producing Dd, D 3, d, indefinitely, the three
asymptotes of the six hyperbolic branches are apparent, and form
a triangle.  When the term  b 2x is deficient in the equation,
b = o, whence the term         vanishes, that is, the distance
2 Va
between the intersection of the asymptotes and the origin disappears, and the triangle becomes evanescent, the three asymptotes
intersecting at a point.  When this case occurs, the equation to
the asymptotes will be represented by
y-= x   a, y =- x ^/a
or, as in the text, B F or Bf: AB:: /a: I.




NOTES TO SECTION III.               51
The geometrical signification of the series for y will be evident
by an inspection of fig. 5.
+ x /a is that part of B C or B c which = d.
~- b is the part of B C or B c extending from the last-named
point to the asymptote.
The remainder of the series consists of terms containing descending powers of x, and is represented by the intercept between
asymptote and curve, so that when x is made infinite the third
term of the series vanishes, the curve and its asymptote coinciding.
"In every redundant hyperbola, if neither the term ey be deficient, Note to
page 13,
nor b2 - 4ac be equal to ~- 4a e,/a, the curve will have no line 8.
diameter."
The proposition put affirmatively is that all redundant hyperbolas
have a diameter (I) when e =, (2) when b2 - 4 ac -    4 aeV/a
Case I. when e = o.
In this case
y' =~4-  ax2 + bx + c +x
The absciss here evidently bisects parallel chords, since the corresponding positive and negative ordinates are equal. There is
therefore a diameter. When x is assumed infinitely small, y = - d
and the exponent of x in the denominator being even, the infinite
branches of the curve lie on the same side of the asymptote to
whicli the ordinates are taken parallel, and therefore cannot
intersect it.
It is a test of the existence of a diameter, that the curve does
not intersect the asymptote which is assumed to be parallel to the
bisected ordinates.




52                  NOTES TO SECTION III.
Case II. when b2 - 4 ac = --- 4 ae Va.
b + 2ax
Here the ordinate to the asymptote is y - 
z /a
and the ordinate to the curve is
e +    l ax4 + b 3 + cx2 + dx + -
4
yl =
And these ordinates are equal at the point where the curve and
asymptote intersect, therefore when y = y'
62           \         be
-e V/a-c x=d+        be
4a                     2 Va
or,
4ad + 2,/abe
b2- 4ac —4aeVa
is the value of the corresponding absciss. Here the condition
b2 - 4ac = 4ae   /a renders the denominator = o, and x is
therefore infinite, that is, the curve and asymptote have no point
of intersection. There is, consequently, a diameter in this case
bisecting ordinates parallel to the asymptote assumed.
It appears, then, that the redundant hyperbola admits of a
diameter in the following three cases,
when e = o
4ac - b2
when e -   - 
4 ab2 - 4ac
when e
4aJ
and it is evident that if the ordinates parallel to two of the
asymptotes have a diameter, the ordinates parallel to the third
asymptote will necessarily have one also, since of these three
equations of condition, two cannot be true unless the third is also.
There will therefore belong to the redundant hyperbolas, either
one diameter, three diameters, or none.




NOTES TO SECTION III.               53
In the diagram the lines A B, C D, E F, represent the three
diameters of a redundant hyperbola bisecting
ordinates parallel to the three asymptotes.
The curve which has but one diameter           / 
cuts the two asymptotes through the intersection of which the diameter passes, but not
the third asymptote. The curve that has three
diameters will not cut any of the asymptotes.     v
Diameters called ' absolute.'
It is evident from the definition here given of an absolute Noteto
page I3,
diameter, that the conic sections are the only curves whose lne36.
diameters must always be straight lines. It is essential to an
absolute diameter that the algebraic sum of its ordinates should
vanish for any given absciss, the positive and negative values
being equal: and the equation to any conic section may always
be so transformed as that this may be effected, by making the
uneven powers of y to disappear.
But it is not so with lines of the 3d order; for although we
may always make one of the uneven powers of y to disappear, by
changing the direction of the ordinates, we cannot always make
the other terms containing uneven powers of y to vanish contemporaneously. But a curve line may be always drawn which shall
so divide the ordinates between it and the curve as that their
algebraic sum may equal o, or that the sum   of the positive
products may equal the sum of the negative products of the roots.
In the case of a line of the 3d order this curvilinear diameter
will be a conic section. In that of a line of the 4th order it will
be a line of the 2d or 3d order, &c., as may be inferred also from
the consideration that when the equation to any curve is arranged
according to the powers of one of its variables commencing with
the highest, the coefficient of the 2d tern equated to o represents
a straight line, that of the 3d term a conic section, that of the 4th
term a line of the 3d order, &c.
A curve may therefore have a diameter of any order inferior
to its own, according as one or more of the coefficients in its
equation duly arranged is zero.




54                  NOTES TO SECTION IV.
Notes to Section IV.
Note to     " Wlen a redundant hyperbola has no diameter, let the four
page 14,
lile 5.  roots be found of the equation
e2
ax4 + bx3 + cx2 + dx +  -o o
4
It is necessary to find these four roots in order to obtain the
limits within which the curve exists.  Now the solution of
Newton's first case of equations where all the terms are present
gives
ax4 + bx'3 + cx + dx + -
e                             4
y  -- -
= 2<x xc
which furnishes the two values of the ordinate yo
The first term of the right-hand side of this equation indicates
a hyperbolic diameter, while the second term  represents the
ordinates bisected by this hyperbolic diameter.  So that the
existence and position of the curve depend upon the nature of
the irrational numerator; hence the necessity of finding the four
roots of the biquadratic.
When the quantity under the quadratic vinculum = o,
nothing but the intersection of the curve with the hyperbolic
diameter is indicated in the equation which becomes
e
y = -     o,
but if the irrational quantity is greater than o, the ordinates have
a real value, and corresponding points in the curve will exist;
and when it is less than o, since the irrational quantity becomes
imaginary, there will be no value of y corresponding to the value
of x, and consequently no curve, for, as De Lagny remarks, the
imaginary term "rend toute la formule imaginaire par une espèce
de contagion qu'elle communique à la partie réelle."
When we have ascertained these four roots we may proceed
to determine the limits within which the curve will be found, and
thence to assign the species to which any individual equation
refers; for proceeds the text,




NOTES TO SECTION IV.                 55
"Let the roots be AP, Am, Am-, Ap, draw the ordinates PT, Note to
page 14,,T, 1, pt, then these will touch the curve, and, by touching, give lile 7.
the limits of the curve."
That is, since the irrational quantity already referred to, may
be positive, zero, or negative, and cannot pass from the positive
to the negative category, or vice versa, without intermediately
becoming = o, we see that the point in the curvilinear diameter
where the ordinate passes from real to imaginary is the limit
between the existence and non-existence of the curve; so that the
curve will be found on one side of this point, but not on the
other. The ordinate through this point, therefore, touches the
curve.
"If all the roots are real, unequal, and of the sarne sign, the pote to
page 14,
curve consists of three hyperbolas, inscribed, circumscribed, and am- lile I.
bigenous, and of an oval."
Such are the conditions necessary to constitute the first species.  Let AP, A-r, Am, Ap, represent the four real unequal
roots which, measured in tle same direction from the origin A,
will be of the same sign.  Through P, r,,, p, draw lines parallel
to the axis of y. These lines will divide space into regions, and
also will touch the curve.  If the curve exists in the region
APT, it will not exist in the region P z1, it will be founc in the
region ~qr r, but will become imaginary in the region zp t, beyond
which it will accompany the asymptotes to infinity.
Hence the portion of the curve included in the space ermr is necessarily an
oval, for if it had infinite branches, it would
be possible to draw a straight line to cut
the curve in four points, in which case it 
would not belong to the system of lines of 
the third order.  Since each of the asymptotes intersects the curve once, and only
once, and since there is no rectilinear diameter, it necessarily follows that one hyperbola is circumscribed and one ambigenous, the third being
consequently inscribed.
F




56                   NOTES TO SECTION 1V.
When the two greatest roots, or the two least, are equal, the
oval becomes joined to the circumscribed hyperbola, and forms
a node or folium, as it has been sometimes called.  Nicole,
who commenced writing an elaborate commentary on Newton's
treatise on the enumeration of lines of the third order, which
is printed in the "Mémoires de l'Académie," has fallen into
the mistake of supposing that the oval is capable of uniting
with the ambigenous hyperbola. It is evident from the diagram
that this cannot be the case, for if it could occur, it would follow
that a part of the oval might exist outside of the asymptotic
triangle, which is impossible; and it is equally impossible that the
ambigenous hyperbola should enter that triangle, its branch being
below the asymptote, to which it is inscribed.
The circumscribed hyperbola, when it joins the oval, will
always have its branches convex towards each other, otherwise a
straight line might cut the curve in more than three points.
There is nothing in the remaining seven species of redundant
hyperbolas sufficiently remarkable to call for particular notice.
When two roots are equal, and the other two are also equal,
the curve is no longer one of the third order, but is a combination
of a straight line with a conic section, for the quantity
ax4 + bx3 + cx2 + dx + -
is in this case no longer irrational, and therefore the equation,y2 + ey = ax3 + bx'2 + cx + d
is divided into two equations, one representing a straight line, and
the other a conic.
Note to     "( Of the twelve redundant hyperbolas with one diameter."
page x5,
line 6.
The term ey being deficient in the equation to these curves,
the ordinate will be expressed by
=  ax3 + bx + cx + d
Here the absciss itself is a diameter, as appears from the
double sign.




ENUMERATION OF CURVES.               57
Newton has enumerated twelve of these hyperbolas, and two
more have been added to the number by Stirling.  If in the
figure an oval or conjugate point be supplied, these two additional
curves will be described. The case is that in which the two
lesser roots are real, while the hyperbolas consist of one inscribed
and two ambigenous curves.
There are, therefore, according to the principle laid down by
Sir Isaac Newton himself, fourteen redundant hyperbolas having
one diameter.
In the " Ladies' Diary" for I788 a prize question is proposed
in the following terms: -" Fig. 20 in Newton's Catalogue consists of two ambigenous hyperbolas at d and a, and one inscribed
at D, without oval or conjugate point; but between these two
there are five more curves, essentially different from either; two
of which have been described by Mr. Stirling. It is, therefore,
required to determine the other three, with an example of a
numerical equation for each." In the answers which were given
to this challenge, the writers have been misled into the assumption that a difference in figure constitutes a difference in species.
By assuming the roots so that the vertex of the hyperbola (fig.
20) falls to the left of D, they argue that three additional curves
may be added to the catalogue. From the numerical examples
given, viz.,xy5 = _: -3 + 4x4 + I74 + 20 (roots - 2, - 4, - Io)
xCy2 = - x3 + 4x2 + I74 x + 22- (roots -3, -3, -  o)
XY2 2=- x    3   + 4 i2   + 8  30 (roots -  3, -  /- 3, - o)
4
all of which belong legitimately to Newton's species already enumerated, it is evident that these three proposed curves are not
"essentially different" from the enumerated species.
Of the two hyperbolas having three diameters."  Note to
page I7,
line I.
According to what has already been shown, the hyperbolas
admitting of three diameters are represented by the first case of
the equations, where e = o, and b2 = 4ac.




5 8                        NOTES.
The equation in this case, by the substitution of -  for its
4a
equal (c), becomes
xy2 =- a  +   bx2    x + -x d
4a
Sir I. Newton appears to have omitted to consider all the
cases of the roots of this equation, and has only taken notice of
the curves which result from the case in which two of the roots
are imaginary. Stirling, however, has pointed out that all three
roots may be real, and that then the curve admits of an oval, or
of a conjugate point, according as the two lesser roots are unequal
or equal. No other case can arise with reference to the roots of
the equation, for it is not possible that its two greater roots should
be equal, neither is it possible they should be of different signs.
Ngoe t      " Of the nine redundant hyperbolas having three asymptotes
liUe 7.  converging to a point."
When the three asymptotes converge to a point, the triangle
formed by their intersection with one another is evanescent. The
coefficient, b, in the first case of the equations then becomes = o,
and the equation resulting is
xy2 - ey = ax3 + cx + d
The principal difference between these redundant hyperbolas
and those which have been hitherto discussed is, that they, under
no circumstances, admit of an oval, conjugate point, node, or
cusp. For these can only occur within an asymptotic triangle,
in the case of redundant hyperbolas. The nine species of this
curve enumerated by Newton present no subject for remark.
Note to        " Of the six defective hyperbolas without a diameter."
page 17,
line 27.
In the first case of the equations, when the term ax3 is supposed to be negative, the equation solved for y, gives
V    - a4 + bx    x +  + dx +e
e                               4 -y   2   -     - __
î) ~  x  ~ ~




ENUJMERATION OF CURVES.              59
From which it appears that the curve admits of only two
infinite branches, and it is, therefore, called defective by Newton;
for the quantity under the quadratic vinculum becomes impossible
as soon as the first term exceeds the remainder of the expression
in the numerator, and cannot again become possible for any value
of x.
Since the curve which has no rectilinear diameter cuts its
asymptote in one point, the infinite branches of these six defective
hyperbolas lie on different sides of their asymptote. It is evident,
also, that they admit of a hyperbolic diameter bisecting the ordinates terminated by the curve, on which there may occur an
oval, node, &c., according as the roots of the irrational portion of
the equation are equal or unequal to one another.
The case where two of the roots are impossible, and at the
same time b and d = o, is remarkable, for then the curve has a
centre, though it has no diameter. The same curve when its
equation by a transformation of axis assumes the form
y3 = ax2 -_x3
has been called by some writers a circle of the second order,
because its equation may be conceived to be derived from  a
property analogous to that of the circle. Thus, as the equation
to the circle whose diameter is (a), is derived from the proportion,
x' y     y  a-x,
so in circles of the higher orders the equations are derived from
the proportion,:    ' y:: y: a-x.
If n = 2, we have,
2: y: y:: a-x, or y3 a x-x3,
an equation to the defective hyperbola under discussion. If
n= 3, y4=a 3 —4, an equation to the lemniscate, and generally
if (n) is an even number the curve will be infinite, if n be odd, the
curve will be closed.




6o                         NOTES.
Note to         " Of the seven defective hyperbolas with a diameter."
page I8,
line 27.
These curves differ from the last six species only by the
deficiency in the equation defining them of the term e y. Since
they possess an absolute diameter, their infinite branches will not
intersect the asymptote, and will therefore lie on the same side of
it. The cissoid of the ancients, which is one of these defective
hyperbolas, is a curve of historical interest, having been invented
either by Diocles or Hippias, for the purpose of finding two
continued mean proportionals between two given lines.
In the Arithmetica Universalis, Newton has given a method of
describing this curve by the continued motion of a sliding ruler.
The curve may also be described by the vertex of a conic
parabola made to roll upon an equal fixed parabola, their vertices
being supposed coincident at the commencement of the motion.
It is worthy of remark that any other point than the vertex in the
rolling parabola, will trace out one of Newton's defective hyperbolas. (See fig. 45), having a double point in the like point of the
fixed parabola.
The cissoid may also be described in the following manner by
points.
D,           The infinite line AB,/^^[  ~   being drawn perpendicular
/ ~/'/1  \    to the diameter C D, and
-/ ^  /   ) ~ touching the circle in C,
let any number of lines be
drawn from D to cut A B.
Then measuring from the
points of intersection with A B, lines equal to the corresponding
chords, cut off the circle, as A F, G N, &c., the extremities of
these lines will lie in the curve of a cissoid.
It is evident that all cissoids referred to the same axis are
similar figures.
Note to       C Of the seven parabolic hyperbolas having no diameter."
page 19,
line 23.
The équation to these curves differs only from that already
discussed in the case of the redundant hyperbolas, inasmuch as the
term a x3 is deficient. It then becomes,




ENUMEIRATION OF CURVES.              6I
xy2-ey=bx +- cx + d... (i)
whence,
y      ~-V- +/bx3+c  +d +-... (2)
2x                     4 -We draw the same inference from the expression for y when x
is very small, as in the curves referred to, viz., that the axis of y
is an asymptote having two hyperbolic branches, one on each side
of it, in contrary directions, and that this asymptote is intersected
by one of the branches at a point above or below the origin,
according as (e) is positive or negative, at a distance =e
But if we expand y in a series for large values of x, we shall
no longer find rectilinear asymptotes, although the possibility of
values for y continues to infinity. For it is evident, since the
irrational quantity in the second member of the equation, when x
approaches infinity becomes +~ /bx, that the branches of the
curve admit only of a curvilinear asymptote, and are consequently
parabolic.
If in equation (2), x is supposed to be negative, the ordinate
becomes imaginary as soon as the irrational portion of the equation becomes negative, and hence there are no more infinite
branches than the four already spoken of, namely, two hyperbolic
having a rectilinear asymptote, and two parabolic, having a conic
parabola for their asymptote whose parameter is b, and whose
diameters are parallel to the axis of x.
Expanding y in a series we have,
c
y=+ /bx-+ 2         &c.,
which shows that the asymptote parabola will be inside or outside
of the parabolic branch, according as c is positive or negative.
That this asymptote can only cut the curve twice, may be thus
shown. At the point where the curve and asymptote intersect,
y= + /b x, whence -~ =-, is the corresponding value of the abscissa, substituting which value in the equation (I) we obtain
another equation,
-be ~ 2/ b2e2 - 4becd
2c




62                         NOTES.
which does not give more than two values for y, and therefore
there will be no more than two points of intersection. When
4 c d >b e2, the equation determining the points of intersection
becomes impossible, that is, the parabola will not intersect the
curve, but will lie entirely within its branches. When the parabola intersects the curve twice, the points of intersection will lie
on the same or on opposite sides of the absciss, according as c is
positive or negative in sign.
If the equation (J) be solved for x, it shows a parabolic
diameter, bisecting chords parallel to the axis of x.
Note to Co Of the four parabolic hyperbolas 7iaving a diameter."
line z8.
The hyperbolo-parabolic curves which have a rectilinear
diameter, as enumerated by Newton, are four in number. Their
species being furnished by the cases in which the roots of the
equation,
bx2 + cx + d= o,
are (i) impossible, (2) equal and of the same sign, (3) unequal and
of the same sign, (4) of different signs.
It is remarkable that neither Newton nor his commentator
Stirling should have noticed that when the roots of this equation
are possible, and (whether equal or unequal) of the same sign,
the curve will differ according as that sign is positive or negative.
So that there are in reality six hyperbolo-parabolic curves with a
diameter.
Thus when the roots are positive the equation will be of the
form,
y=+      bxb -c+-,
an equation giving possible values for y, only on the assumption
that x is measured on the positive side of the origin, for when x is
taken negatively, the values of + y are all imaginary.
But if the roots are negative,
y   -==~  bx+c+,




ENUMERATION 0F CURVES.                63
and when x is negative this becomes,
y =~V — bx+    c —,
indicating certain limits between which a closed curve will exist
on the negative side of the origin. The figure will be as shown.
When the negative roots are equal, the
oval becomes a conjugate point.                      -
This curve is the subject of a paper in 
the Transactions of the Royal Society in the
year I736, by Mr. Edmund Stone, a learned
mathematician of the time, who seems not to
have been aware of the previous notice of the same curve by
others. It appears to have been first observed upon and figured
by Nicole in an article on Newton's Theory of Shadows, published in the Mémoires de l'Académie for the year I73I. The
saine curve was also known to N. Bernouilli, and in his " Usage de
l'Analyse de Descartes," the Abbé de Gua remarks the omission
of it by Newton in his enumeration.
Of the hyperbolisms of the conic sections.     P.o to
line I6.
When the first case of the equations becomes still further
simplified by the deficiency of the terms a x3 and b x, and is of
the fornm,
xyV 2- ey =c+d...   (I)
Newton calls the curves which are defined by it, hyperbolisrs of
the conic sections, because they may be generated from the ordinate of the hyperbola ellipse or parabola; thus if A B represent a
given line, the locus of the extremity of the ordinate tracing out
the hyperbolism is found by multiplying the ordinate M P by the
given line A B, and dividing the product by the common absciss
A M.
"In this manner a straight line is turned into a conic lhyperbola," Not to
line zz.
thus, if the straight line be represented by y = x + c, and this
G




64                        NOTES.
ordinate be multiplied by the given line a, and divided by the
a (x +- c)
common absciss x, the equation to the new curve is y -   + 
which represents a hyperbola.
In like manner where the ordinate of a conic section is,
-e — V/e2 +4dx + 4cx
2 M
this being multiplied by a given straight line m, and divided by
the common absciss x, becomes
-e ~ v/e2 + 4rdx + 4cx2
2X
for the ordinate to the hyperbolism.
Case I. Where e > o, and c is positive.
The axis of y is an asymptote having an infinite branch lying
on each side of it, extending in contrary directions, as appears
from expressing y in a series for small values of x. If y be expanded in a series converging for large values of x, we have,
y  +      de/ c +  2 / + &c..   (2)
which indicates two parallel asymptotes above and below the axis
of x, at distances + V/c, and - /c.  Whence it is evident that
when c is negative these parallel asymptotes become imaginary,
and when c= o, that they coalesce and form one line.
The equation (1) solved for y giving,
e, /     d     2 
4=_   e + c+  - +  * * * (3)
2x         x   4 X2
shows a hyperbolic diameter having infinite branches of the curve
on each side of it, whatever be the sign of x, also when x = o, the
ordinate is an asymptote having a branch on each side of it.
There are therefore six branches, and the curve is consequently a
hyperbolism of the hyperbola. Since the parallel asymptotes are
also parallel to the axis of x, they can never be intersected by the
curve.




ENUMERATION OF CURVES.                65
Case 2. Where e = o, and c is positive.
Here the ordinate = ~   c +.
The curve, since it has an absolute diameter, will not intersect the asymptote  d (fig. 68); and the infinite branches will
always lie on the outside of the parallel asymptotes when x is
positive, but on the inside of then when x is negative. This
is evident, because the equation to the asymptotes is y= V/c, and
/c is always less than  c +  greater than Vc —.
Case 3. Where e > o, and c is negative.
The ordinate becomes  --     ~     -c +    -+
2x             x   4x9'
There is here only one pair of infinite branches cutting their
asymptote; the parallel asymptotes become imaginary; the curve
is a hyperbolism of the ellipse, having hyperbolic diameters.
Case 4. Where e = o, and c is negative.
The infinite branches lie on the same side of the asymptotes,
for there is now an absolute diameter. The form of the curve
is capable of an infinite variety of figure, from a straight line
coinciding in the asymptote to an elongated cusp-like curve,
which occurs when d greatly exceeds c.
If the foci of the ellipse from which this curve is conceived
to be generated coincide, so that the ellipse becomes a circle, the
resulting hyperbolism is the same as the curve called the Witch,
the invention of which has been generally attributed by modern
writers to the celebrated female professor of mathematics in
Bologna, Donna Maria Agnesi, A.D. I740. It, however, does
not appear, in the Istituzioni Analytiche of the authoress, that
she advances any claim to the invention; nor does she appear to
have been responsible for the unmeaning name attached to the
curve, for she speaks of it as the curve " volgarmente detto la
Versiera." Her work, which abounds with examples of interesting
and curious problems, was translated into English, and published,




66                        NOTES.
after the death of the translator, Mr. Colson, by Baron Maseres,
in I80I.
According to Agnesi (Inst. Anal. Book I. Sect. 4), the Witch
is thus generated:-" The semicircle A DC being given, required
a point outside it, such that, drawing a perpendicular MB to the
diameter, cutting the circle in D, we may have A B  B D 
AC: B M. And because there will be an infinite number of
points that will satisfy the problem, the locus of these points is
required."
Thus, if AC = a, AB = x, BM = y, BD/ax-x,
^" — ^x: C        ax-x:: a: y, therefore
WR  D....... _''-       -          y  = a ax;
which equation is the same as Newton's, where d = a3 and
c =(t a.
Case 5. When e > o, and c = o.
The equation is now
~y +} ey = d; whence y   = e           - - ~  - 
2x      4L2   ~
When x is positive, it is evident there are two infinite branches
on each side of the axis of x, extending in the same direction,
and at an infinite distance coinciding with it.  This apparent
anomaly is explained by the consideration that since c = o, the
two parallel asymptotes defined by y = ~ -/ c, have come together, and the axis of x is consequently a double asymptote.
The axis of y is also an asymptote, having infinite branches
on different sides of it; the curve, which is a hyperbolism of the
parabola, is the only one of the third genus which has four
branches only, all of the hyperbolic sort.
Case 6. When e = o, and c = o.
The equation is xy' = d.
Unlike the hyperbolism of the ellipse, this curve is susceptible
of but one figure when referred to axes of given inclination. The




ENUMERATION OF CURVES.               67
simplicity of its equation renders it useful in the construction
of equations Ly means of the intersection of curve-lines. The
hyperbolism of a parabola with a diameter is called by some
writers the cubic hyperbola.
The Second Case of the Equations.           Note to
P. 23,
li:le 24.
The curve indicated by the equation
xy,= ax3 + bx2 + cx + d
furnishes no subdivisions into species, and is easily discussed;
for, since it gives
d
y = ax2 + bx + c + -,
it is plain that for any value whatever, positive or negative, of
the absciss, only one value of the ordinate can exist, on the
supposition which is assumed, viz., that the ordinate is parallel
to a rectilinear asymptote.
When x = o, y is infinite, and therefore the axis of y is an
asymptote, having a branch extending in one direction; and
when - x = o, there will be another branch in a contrary direction, but these branches will not unite.
When x =     oo+, y = a2, or the two remaining branches of
the curve approach to a conic parabola, and as the values of x
are continuous from ~ o to + infinity, the parabolic branches
unite with the hyperbolic.
The ordinate to the parabolic asymptote is
ax2   bx + G,
that to the curve is
ax~ ~ bx + c ~ -,
x
the upper sign being taken when the roots are negative; whence
it appears that the asymptote on the positive side lies above the
curve, and on the negative side below the curve, its position




68                          NOTES.
being governed by the sign of -, the first term  containing x in
the denominator. It can therefore never cut the curve.
Newton calls this single instance of a curve, corresponding
to his second case of equations, the" Cartesian parabola," because,
as he says, Des Cartes by its assistance constructed an equation
of six dimensions.
The trident, as it is also called, may be readily described by
observing that the ordinate consists of the sum or difference of
the ordinates of the parabola
y = ax2 - bx + c, and the hyperbola y = -- d
Des Cartes gives a method of describing the trident by continuous motion.  (See also Maclaurin's Geometria Organica.)
Note io                    Thitrd Case of the Equations.
p. 24,
line i.
The five species enumerated as referable to the equation
y2 -= ax3 + bx + cx + d
differ only in the finite part of the curve; for, since
y = ~    ax3 + bx2 + cx + d
when x is very great, the terms following ax3 may be neglected;
and it follows that the   semicubic parabola represented   by
y = ~ V/ax3 is the asymptote to all the divergent parabolas.
Newton calls this curve the "Neilian parabola," in compliment to Mr. W. Neil,* who first showed how to rectify the
curve by means of the quadrature of a parabolic space.
- Neil was born in 1637, was educated at Wadham College, Oxford, and
died in 1670. His discovery of the method for rectifying the semicubic
parabola was made when he was only nineteen. (See Phil. Trans. for 1673.)
Neil's priority in this discovery was disputed, but in Newton's opinion
without reason.




ENUMERATION 0F CUWRES.               69
"If two roots are impossible, a pure bell-shaped parabola will Note to
result (figs. 78, 79)."                                   lne zz.
It may, however, happen that when two roots are impossible, the figure of the curve in question is not bell-shaped
(campaniforinis).
Thus, suppose the equation made up of the factors
X2 + h2 = o, x + k = o,
so that it is
y2= x3 + k 2 + h2X + kh2,
differentiating and assuming dy = o, we have
2        7 2
3,2 + 2kx + h2 = o, or x2  - k,
3        3
solving the quadratic,
x= —k k +      _ k    h2, -
3        9     3
which, when real, shows that there is a pair of maximum and a
pair of minimum values for y, and therefore the
curve cannot be bell-shaped. The values obtained
for x are those of the abscisses belonging to the
points at which the tangent to the curve is parallel to the axis. Whenever, therefore, the irrational
quantity is real, the parabola will not be bellshaped. If - k   _ h2, there will be a point of
9     3
inflexion when x = - 1k, and the tangent at
this point will be parallel to the axis.
If a solid generated by the revolution of a semicubic parabola
about its axis be cut by a plane, each of these five parabolas will
result from the sections; for when the cutting plane is oblique to
the axis, but falls below the vertex, the section is a diverging parabola with an oval. When the plane passes through the vertex,
this oval becomes a point; when the plane both cuts and touches
the surface of the solid, the section is a nodate parabola; when
the plane falls above the vertex, either parallel or oblique to the
axis, the section is a pure parabola; and when the plane passes
through the axis, the section is the semicubice




70                         NOTES.
Note to                  Fourth Case of the Equations.
p. 24,
line.5.
y =ax3 + bx.2 + cx + d.
Here, again, as in the case of the trident, we can have no
more than one value for y, whatever may be the value of x,
positive or negative. This case evidently admits of no subdivision
into species, for when thle equation
y = ax3 ~+ bv2 + ex +- d = o
has real and unequal, or real and equal roots, &c., nothing is
indicated except that the curve cuts or touches the axis of x in
certain points. The branches are evidently parabolic, and extend
on each side of the axes; the curve is called the cubic parabola.
As the semicubic was the first curve which was rectified, so the
cubic parabola will probably be the last, a solution of the problem
of its rectification being still a desideratum.
Note to                 Thus altogether the species are 72."
p. 24, last
line.
De Gua has remarked, in his review of Newton's enumeration,
that before commencing it some definition should have been given
to fix the meaning intended to be attached to the word " species;"
and it would seem that the want of some such definition has been
the cause of much criticism on the part of subsequent writers
which might have been avoided. It would, perhaps, have been
better to have classified the curves into species and subdivisions
of species, in which case it might be easily shown that the species
are at most not more than six or seven; provided that Newton's
own principle of classification be adopted, namely, the relation to
one another of the roots of the equations considered, as referred to
axes one of which is parallel to an asymptote. De Gua further
observes that Newton has proceeded solely on the principle of considering the number, position, and equations respectively of the
maxima and minima of x without adopting the same course
relatively to the maxima and minima of y, although the axis of x
and that of y are equally determinate straight lines, and in a sin



ENUMERATION OF CURVES.                71
gular position. And that if he had made the same observations
on the maximum and minimum values of y as he has done upon
those of x, the species 1, 2, 3, 6, 10, 14, I5, I6, 22, and 28,might
have been further subdivided, and consequently a much larger
number of species enumerated.  It is also a common observation
of modern writers on algebraic geomnetry and the differential calculus, that Newton's enumeration is far from complete, and might
be extended so as to include many more species. To which it
may be replied, that his principle of classification is entirely arbitrary, and that if we adopt that principle, it is in the highest
degree improbable that the number of species could be increased,
seeing that the ingenuity and research of mathematical writers for
more than a century have failed to add more than six additional
curves to the total number. It is worthy too of remark, that
these six curves are different from already enumerated species,
only by the addition of an oval or conjugate point.
Both Cramer, and Euler in his "Analysis Infinitorum," have
given a classification of lines of the third order, depending upon a
principle entirely different from Newton's, namely, upon a consideration of the nature and direction of the infinite parts of the
curves. According to Cramer, there are fourteen different species,
and, according to Euler, sixteen different species. Neither of them
considers the existence of a conjugate oval or point as sufficient to
constitute a separate species. This view of the subject may be
less liable to logical objection than that which Newton has adopted,
but it is not so well calculated to give a general idea of the varieties
of which the different curves are susceptible.
In more modern times the same subject has been taken up with
much elaboration of research by mathematicians in Germany,
especially Plucker, Mobius, Peters, Krause, Beer, &c., but it is
impossible to institute a comparison between the classification of
curves adopted by these writers and the enumeration of Newton,
because the former proceeds upon the assumption of a different
system of co-ordinates, and a different notion of the genesis of
curve lines. Plucker gives a catalogue of 219 curves of the third
order. According to the German writers, a curve is to be considered as the envelope of a moveable right line, and is classed
according to the number of tangents to it which can be drawn
through any given point. A line of the third order is thus a curve
H




72                         NOTES.
of the sixth class, because six tangents may be drawn to it from
one point.
Dr. Waring undertook the herculean task of the enumeration
of lines of the fourth order upon the Newtonian system, and the
result of his labours is published in his " Miscellanea Analytica,"
printed at Cambridge, 1762. He reduces all lines of the fourth
order to twelve cases of equations, beginning with the biquadratic
parabola, of which there is only one species, and ending with the
curve possessing four asymptotes, of which no less than 72,480
varieties are specified. The total number of species he calculates at
84,55I. An interesting work on curves of the fourth order, containing many plates, entitled, "Tabule curvarum quarti ordinis
symmetricarum   asymptotis rectis et line fundamental recta
preditarum," was published in I852 by Professor Augustus Beer,
at Bonn, which is more likely to be useful to the student than
Dr. Waring's catalogue.
Notes to Section V.
Note to                TThe generation of curves by shadows.
P. 25,
line z.
The remarkable theory so briefly enunciated in this fifth
section appears to be substantially the same as that which is
discussed at greater length in the 22d lemma of the Principia,
in which it is proposed to ' transmute' any rectilinear or curvilinear figure into another of the same analytical order by means
of the method of projections. This theory, as Newton has remarked, is of considerable value in the solution of problems~
which scarcely admit of being otherwise solved, anc he has himself made use of it in Prop. 25, 26, Book I. of the Principia.
We premise the discussion of the principles on which the
genesis of curves by shadows is founded, by the following
definitions.
Inservit autem hoc Lemma solutioni difficiliorum Problematun transmutando figuras propositas in simpliciores. PRTN.




GENERATION OF CURVES BY SHADOWS.           73
I. A plane of infinite extent being supposed, on which is
delineated the curve which is proposed to be projected, this plane
is called the " plane of the base."
2. A second plane parallel to the plane of the base, is called
lhe " horizontal plane."
3. A third plane cutting both these planes at any given angle,
is called the " plane of projection," from its receiving the projection
of the curve delineated on the first-named plane.
4. A fourth plane is to be supposed, parallel to the third plane,
at a finite distance frorn it.
5. The line of intersection between the horizontal plane and
the plane of projection is called the "horizontal line."
6. The line of intersection between the plane of the base and the
fourth plane is called the " directrix,"  Its position with regard to
the curve to be projected determines the species of the projection.
7. The line of intersection between the plane of projection and
the plane of the base, is called the " line of the base."
Cor. The horizontal line is parallel to the line of the base.
8. If an infinite straight line is supposed to revolve about a
fixed point P, situated in the line of intersection between the
horizontal plane and the fourth' plane, and is guided along the
course of the curve delineated on the plane of the base, it will, by
its intersection with the plane of projection, describe the projection
of the given curve.
9. A plane supposed to be drawn through this pole P, and
perpendicular to the before-mentioned planes is called the "vertical
plane," and its intersection with the horizontal plane is the " axis
of projection;" the point in which the axis of projection meets the
horizontal line is called the "centre of projection."
Thus in the annexed
G-   __________       figure A B C D repre/... —     -........ --- —. ---- ------  /  sents  the  plane  of  the
/  base, GIHEF the horizontal plane, D C E F the...... /.. —. —. —            /     plane of projection, AB
/-\-/ C  /   \/         H G the fourth plane,
'B..   t-      - -- / ^ aA    B   t he  directrix,  C D
~/ (/ /      t/the line of the base, E F
l/  ~ /     the horizontal line, or, as
=..        it is sometimes called, the




74                         NOTES.
anti-directrix, P the pole, PN the axis of projection, and N the
centre of projection.
It is evident that several points in a straight line can only be
projected on a plane by as many points also in a straight line,
and thence it follows that the outline of the shadow of any curve
may be cut by a straight line precisely in as many points as the
curve, of which it is the shadow, may be cut by a straight line;
and therefore the proposed curve and the projected curve are of
the same degree. Moreover, the projection of the tangent or the
asymptote, to any point whatever, is always either a tangent or
an asymptote to the projection of this point; and the projection
of the axis of any curve will be the axis of the new curve.
The curve may lie on either side of the directrix, or partly
on one side, partly on the other. When this latter case arises,
or when it touches the directrix, the point or points projected
which cut or touch the directrix, go off to infinity. Thus, for
example, suppose a circle delineated on the plane of the base, it is
obvious that while it is placed to the right of tle directrix, its
projection will be thrown on the plane of projection below the
line of the base DC. When placed to the left of AB, its projection will be thrown on the plane above FE, in either case
being an ellipse. If, however, it touches the directrix, the projected curve will have infinite branches ultimately becoming
parallel, and the curve will be a parabola; and if it is eut by the
directrix, there will result four infinite hyperbolic branches, of
which two will be above and two below the line of the base.
Suppose now the curve delineated on the plane of the base is
a semi-cubic parabola, lying wholly between the directrix and
the line of the base, but without touching either, and having its
axis at right angles to these lines, its ordinates parallel to them.
The projection will take the form of a cissoid. If the cusp of
the semi-cubic touch the directrix, the projected curve will
consist of four hyperbolic branches, converging to one double
and one single asymptote, viz., the cubic hyperbola; and if the
curve cuts the directrix in two points, each point of intersection
will be projected as an asymptote, while the two branches of the
semi-cubic itself will be projected towards a third asymptote, and
the projection will constitute a redundant hyperbola, two of the
hyperbolic curves being inscribed, and the third circumscribed
and cusped (fig. 24).




GENERATION OF CURVES BY SHADOWS.           75
We have here supposed the simplest case which can occur,
namely, that in which the axis is normal to the ordinates and to
the directrix, the curve itself having no points of inflexion or
conjugate oval. The case in which the curve is more complicated and the directrix oblique to the axis, will be hereafter
adverted to.
We now proceed to show how the equation to the projection
may be obtained from that of the generating curve, or vice versc.
Assurning rectangular co-ordinates, let it be required to compare
the curves (either of which may be considered as the projection of
the other,) ATB, AEB, where S represents the pole, LSCI,
____A /
I~~D-     ~
^..^ ----------------------— ^
the vertical plane, Q, Q', the points in the two curves to which
the ordinates PQ, P'Q', belong. Let I be taken as the origin in
one curve, and L in the other. Let the axis C E of one curve be
the projection of the axis CT of the other. Then by sim. tri.
SPI, CPP', and the equality SI = LC, CI = SL,
(LC+CP')LP': SI:: SL          IP,
and calling the known lines S I and S L, mn and n,
x: +m:: -n: x'.
Again, by sim. tri. SPQ, SP'Q', and the equality SI = LC,
SP': SP: (LP': SI):: P'Q': PQ,
or                  x: + m: y    y',
whence we have the two values x' = -  y' = —
OS        Q




76                         NOTES.
which, substituted in the equation of the given curve, will produce that of its projection. For example, let the equation given
be
y2 = ax'3 + bx'2 + cx' + d,
the equation to the projection will be
d      cn
yx = - x+ +-    xa + bn2x + an3 m.
This form of the equation shows that the projection will
be some curve of the third degree, not one of the divergent
parabolas.
The proposition we are now to discuss is, that as the projection of the circle generates all conic sections, so that of the
five divergent parabolas will generate every one of the lines of
the third order.
Assuming that the curve whose projection is sought is delineated upon the plane of the base, the inclination of which, with
the plane of projection, is given, and that its position, with reference to the directrix, is fixed in each case; then it will be seen
that it is sufficient to know this position, in order to determine
the species of the projected curve. It is evident that all the
points of the curve will be projected on the plane of projection,
save and except those points which are in the directrix itself,
which latter will not be projected at all. The points infinitely
near this line will be projected to an infinite distance, that on one
side, in the direction of the positive x's, that on the other, in the
direction of the negative 's. Therefore, when the directrix cuts
the curve, the projections of the points immediately adjoining this
line take the form of infinite hyperbolic branches to the right
and left of a common asymptote, which is the projection of the
tangent at tle point intersected.  But if the directrix only
touches the curve, the infinite branches will be both positive or
both negative in direction, and (except in the case where the
point touched is a cusp) parabolic.
In the application of the foregoing remarks to the case of the
projections of the five divergent parabolas, we shall first consider
all the projections which can be furnished by the divergent parabola with an oval. (Fig. 75.)




GENERATION OF CURVES BY SIADOWS.          77
Case I. Where DX, the directrix, is parallel to the ordinates
of the curve.
I. Let DX eut the axis outside the curve; the projection
will take the form of hyperbolic branches on the same side of
their asymptote, and with an oval at their convexity. (Species 39,
fig. 48.)
2. Let DX touch the oval; the last projection will remain,
with the exception that the oval will be converted into a parabola.
(Species 55, fig. 63.)
z  a   4       6   7   8    9     10   r1
1   e  3   4   5   6   7   8    9     o
3. Let DX cut the axis in unequal parts, as at line 3. The
conchoidal hyperbola remaining as before, the oval will be
projected as two hyperbolas in the opposite angles of the two
asymptotes. (Species 21, fig. 32.)
4. Let D X eut the oval in half; if tangents be drawn to
the points where DX  intersects, they will be parallel to each
other; therefore their projection will be that of two asymptotes
intersecting each other in the third asymptote; three hyperbolas.
(Species 3I, fig. 41.)
5. Let DX cut the axis at 5; the asymptotes will formn a
triangle. (Species 20, fig. 3I.)
6. Let DX touch the oval. (Species 56, fig. 65.)
7. Let DX cut the axis between the oval and vertex of the
curve. (Species 40, fig. 49.)
8. Let DX touch the parabola; the projection takes the form
of a curve not enumerated by Newton or by Stirling, having an
oval on one side of the asymptote and a hyperbolo-parabolic
curve on the other.




78                        NOTES.
9. Let DX eut the parabola, but in the part where the curve
is concave to the axis; the projection gives a species which has
been also overlooked by Newton, though supplied by Stirlingnamely, a curve comprising three hyperbolas, with an oval in
the asymptotic triangle, two of the hyperbolas ambigenous and
one inscribed. To obtain the projections of the asymptotes, draw
a pair of tangents from the points where D X intersects the
curve; these will intersect in the axis, and also each cut the
curve once.
Io. Let DX cut the parabola through the points of inflexion;
the projection will also be a curve omitted by Newton, viz., three
inscribed hyperbolas with an oval.
1. Let DX cut the parabola where the curve is convex to
the axis; the projected curve will consist of one circumscribed
and two inscribed hyperbolas. (Species IO, fig. 22.)
Case 2. When the directrix is oblique to the ordinates of
the delineated curve.
i. Let DX cut the parabola (without cutting or touching
Q~6,~~~ ~the oval) in any point
except one of inflexion;
a tangent to that point
will be projected as an
asymptote to the pro-:3,4^2a ^  V^~.~       ~jected curve, which will
be a defective hyperbola
without a diameter, hav' ing hyperbolic branches
on  each  side  of this
\6 5  asymptote,  the   point
where the branches intersect their asymptote being one of inflexion.  (Species 33,
fig. 43.)
2. Let DX touch the oval, the other conditions remaining
as in the last case; the serpentine hyperbola remains, the oval
becomes a parabola. (Species 52, fig. 60.)
3. Let DX cut the oval and the parabola also, and conceive
tangents drawn from the points of intersection with the oval, so
as to meet the tangent to the point in the parabola; the tangents




GENERATION OF CURVES BY SHADOWS.          79
will be projected as asymptotes, and the projection will consist
of two inscribed hyperbolas and one serpentine hyperbola.
(Species 9, fig. 20.)
4. If the two tangents to the oval meet each other in the
saine point as that in which each intersects the third tangent, this
point of intersection will be projected as a point of intersection of
the three asymptotes; and the species in this case is the 26th,
fig. 36.
5. Let DX, still cutting the parabola, also touch it; the
projection will consist of both hyperbolic and parabolic branches
and an oval, and the curve will have no diameter; the species
will be the 46th, fig. 54.
6. Let D X cut the parabola in three points; tangents drawn
through these points will be projected as three asymptotes, making
a triangle within which will be the projected oval; the resulting
species will be the first (fig. 7), because it will consist of one
circumscribed, one inscribed, and one ambigenous hyperbola; the
infinite branches of the parabola being projected in the form of
a circumscribed hyperbola, the part of the parabola which cuts
neither of the tangents being projected as an inscribed hyperbola,
and the remaining part being projected as an ambigenous
hyperbola.
7. If DX coincides with the axis of the curve, the asymptotes of the projected curve intersect in a point, and the species
is the 27th, fig. 37.
We have next to consider the projections of the punctate
parabola.
Case i. Where the ordinates of the delineated curve are
parallel to the directrix.
i. Let DX cut the axis outside P, the conjugate point; the
species will be the 43d, fig. 52.
2. Let D X eut the axis through the conjugate point; the
species will be the 63d, fig. 7I.
3. Let DX cut the axis in a point between P and the vertex;
the species will be the 44th, fig. 53.
4. Let it touch the parabola; species not enumerated by
I




80                         NOTES.
Newton or Stirling: a hyperbolo-parabolic curve with conjugate
point.
5. Let it cut the parabola; species 13, fig. 25, and 25, fig. 35.
Case 2. Where DX is not parallel to the ordinates of the
given curve.
Let P be the conjugate point, T a point in the parabola on
which DX is conceived to revolve as on a pole.
i. If DX passes through P, the conjugate point, P is not
projected at all; the equation consequently loses two dimensions,
and the species is a hyperbolism  of the ellipse. (Species 6I,
fig. 69.)
2. If DX passes on either side of P without cutting the
curve, the projection will be a serpentine hyperbola. (Species 36,
fig. 47.)
3. If DX touch the parabola, species 49, fig. 57.
4. If DX cuts the parabola in three points, species 4, fig. I2.
5. If the axis coincides with DX, the species is 62, fig. 70.
Projections generated by the pure Divergent Parabola.
Case I. Where the ordinates of the given curve are parallel
to DX.
i. Let DX cut the axis outside the curve; the projection
will be species 45, figs. 52, 53.
2. Let DX touch the parabola in the vertex; the projection
will be species 53, fig. 6I.
3. Let D X cut the parabola in two points where the curve
is concave to the axis, but so that tangents to the points of intersection meet in the axis, and also eut the infinite branches; the
projection will 'consist of two ambigenous and one inscribed
hyperbola. (Species I5, fig. 26.)
4. Let DX eut the curve through the points of inflexion;
the projection will consist of three inscribed hyperbolas, corresponding to species 22, 23, or 32, figs. 33, 34, 42.
5. Let DX cut the curve beyond the points of inflexion;
thé projection will be species I4, fig. 25.




GENERATION OF CURVES BY SHADOWS.          81
Case 2. When the directrix is not parallel to the ordinates.
I. Let DX cut the curve only once; the projection will be
species 37, fig. 46.
2. Let DX touch and cut the curve; the projection will be
species 50, figs. 57, 58.
3. Let DX   eut the curve in three points.   (Species 5,
figs. I2, I3.)
4. Let DX coincide with the axis. (Species 38, fig. 47.)
Projections generated by the Nodate Parabola.
Case I. When the directrix is parallel to the ordinates of the
given curve.
I. Let DX cut the axis outside the curve. Projection, Species 41, fig. 50.
2. Let it touch the folium. Projection, Species 54, fig. 62.
3, 4, 5. Let it cut the folium, the projection will be three
inscribed hyperbolas, two of which intersect one another. Species
18, I9, 30, figs. 29, 30, 40.
6. Let DX eut the point of intersection of the given curve.
The projection will be a hyperbolism of the hyperbola. Species
60, fig. 68; for the tangents to the double point will be projected
as two parallel asymptotes.
7. Let DX cut the curve inside the node. Species II, fig. 23.
Case 2. Where the directrix is oblique to the ordinates.
Suppose DX to revolve about T, a point in the curve, as a
pole.
i. Let it cut the curve once. Species 34, fig. 45.
2. Let it touch the folium. Species 5I, fig. 59.
3. Let it cut the folium.  Species 8, fig. I7, or Species 7,
fig. I6.
4. Let it cut the double point. The tangents at the double
point will be projected as parallel asymptotes on the plane of
projection. The figure will be a hyperbolism of the hyperbola
without a diameter. Species 57, fig. 65.




82                         NOTES.
5. Let DX be a tangent passing through the double point
of the node, the projection will consist of two parabolic and a
hyperbolic branch extending in one direction, and of a hyperbolic branch in the opposite direction. Species 66, or the trident,
fig. 74.
6. Let DX   cut the branches in three points.  Species 2,
fig. 8.
7. Let DX touch one and cut the other branch. The projection will consist of two hyperbolo-parabolic curves, one of
which has a node. Species 47, fig. 55.
8. Let DX pass through the double point, and cut the folium,
tlhe species will be 58 or 59, according as the directrix bisects the
folium or not. Figs. 66, 67.
Projections generated by the semi-cubic parabola.
Case I. The three projections which are the result of the case
in which the ordinates are parallel to the directrix, have already
been noticed.
Case 2. When the ordinates of the given curve are oblique to
the directrix.
1. Let DX cut the curve in one point, the projection will be
a serpentine cusped hyperbola.  Species 35, fig. 45.
2. Let it also pass through the cusp. The projection will be
a hyperbolism of the parabola, without a diameter.  Species 64,
fig. 72.
3. Let it cut the curve in three points. Species 3, fig. Io.
4. Let it eut and touch the curve. Species 48, fig. 56.
5. Let the directrix coincide with the axis. The projection
will be a cubic parabola. Species 72, fig. 8I.
In the foregoing catalogue nearly all the Newtonian curves
are accounted for, the exceptions being the i6th, I7th, 28th, and
29th species, all of which may be generated from the divergent
parabola whose equation has two impossible roots, the curve which
generates these projections is, however, not bell-shaped, but rather
of the form by heralds called nebuly, the case in which this
variety of parabola is produced las already been noticed.




GENERATION OF CURVES BY SIIADOWS.         83
Whenever it may be required from the given equation of
any line of the third order, to find the equation of its projection,
the inquiry will be assisted by recollecting that when the ordinates of the proposed curve are not parallel to the directrix, the
projection will have no diameter, that whenever the directrix
cuts the curve in three points, or in two points of which neither
is a double point, the projection will be a redundant hyperbola,
that whenever the directrix both cuts and touches the curve, the
projection will be a hyperbolo-parabolic curve, that whenever
the directrix cuts the curve in but one point, the generated
projection will be a defective hyperbola intersecting its asymptote;
finally, that whenever the directrix does not cut the curve at all,
the projection will be a defective hyperbola having branches on
the same side of their asymptote.
Clairaut in his treatise, "Sur les courbes que l'on forme en
coupant une surface courbe quelconque par un plan donné de
position," has treated this subject in a somewhat different manner.
He supposes an immovable pole or luminous point, whence rays
of light proceed, forming a cone of which the given curve is the
base. And if one of the five divergent parabolas be assumed as a
base, he shows that all the sections which a plane can make with
the cone thus formed, will be curves of the same order.
To find all the different species which may be thus generated
from the five cones of the third order, we must form the equation
of the conical surface, and substituting in it the general equation
to a plane,
x + my+nz=q,
thence deduce all the different curves which may be expressed by
the transformed equations.
The subject has also been discussed in an elaborate paper by
Nicole, in the Mémoires de l'Académie des Sciences, An I73I,
under the title " Manière d'engendrer dans un corps solide toutes
les lignes du troisième ordre." The conclusions at which he
arrives are similar to those of Clairaut, although dependent on a
different mode of reasoning.




84                         NOTES.
Notes to Section VI.
Note to
p. 6,                " On th/e organic description of curves.
line i.
The demonstration of the first theorem in this section is given
by Newton in the Principia, sect. vo, lemma 2I, and the subject
is discussed further in the same section, prop. 22, &c. The reader
may also with advantage consult Leslie's I" Geometrical Analysis,"
Book II., Props. 5 to I2, where the description of conic sections by
means of Newton's rotating angles is investigated at length. Or if
he desires to know how the problem may be solved by the help
of the anharmonic properties of conics, he will find tho subject
discussed in Mr. Salmon's " Treatise on Conic Sections," p. 281,
3d edition.
We may remark that the species of the conic section which is
traced by the intersection of the legs AD, BD (fig. 82), will
depend on the position of the given guiding line. If the angular
points are joined by a straight line AB, and on it the segment of
a circle is described such as may contain the supplement to 4 right
angles of the given angles PAD, P BD.
Then if the guiding line cuts the circle twice, the conic section
traced will be a hyperbola; and this hyperbola will be equilateral
in the case of the given straight line passing through the centre of
the circle. If the guiding line touches the circle, the result will
be a parabola, if it falls altogether outside of the circle, an ellipse,
which becomes a circle when the guiding line is infinitely distant.
Maclaurin has given a method by which curves may be traced,
dependent on the same principle as that of the rotating angles, but
which has the advantage of greater simplicity of construction.
Instead of angles he makes use only of three straight lines made
to revolve about fixed points as poles.
By this method a conic section may be described which shall
pass through any five given points (no three of which lie in the
same straight line).  See Maclaurin's Algebra, and his papers
in the Phil. Transactions.  See also Leslie's " Geometry of
Curves," Prop. I I.
We may observe that the number of the poles assumed is not




CONSTRUCTION OF EQUATIONS.             85
necessarily restricted to three, for if any. number of poles be
assumed having straight lines revolving on them, provided al the
intersections but one are carried along given straight lines, that
one will describe a conic section. The same observation holds
good in the case of the rotation of angles about fixed poles, that is
the curve described by the rotation of several angles, will not exceed
a conic section, provided all the intersections of the legs except
one, are carried along straight lines.
The Theorems II. and III. of this section are not of considerable importance in the investigation of the properties of curves of
the higher orders, and the reasoning on which the demonstration
of their truth depends, does not admit of condensation within the
limits of a note. Newton has given no demonstration ofthem, but
an elaborate exposition and illustration of the subject is to be
found in the Geometria Organica of Maclaurin, who has also
given two other methods for the description of these lines in the
same work.
As regards lines of the third order, his methods may be generalised by the enunciation of the following theorem.
Two points in a plane C, and S, being given; if a straight
line SN, making with the straight line NL a         T
given angle SNL, revolve round S as a pole,        /
and if the given angle F CO revolve about the
pole C, then, if any two of the three points   d   _q
Q, P, N, in the quadrilateral CQPN are guided       /    0
along straight lines given in position, the third
will describe a line of the third order, having a double point
in C or S.
Note to Section VII
On the construction of equations by thte description of curves."
" By the description of these curves, and their intersections, tte Note to
p. z9,
roots of the equation to be constructed will be given."    line 1.
Suppose, for instance, it were required to find the roots of the
equation of the fifth degree,
V + a32 -5 = o        (I)




86                        NOTES.
Assuming the parabola x2= ay, the new equation is,
xy2+axy-a3=o          (2)
Let the curve indicated by this equation be described as shown in
the figure, it will be a hyperbolism of the hyperbola referred to
OX, OY as axes. Let the parabola 2x = ay be constructed to
~Y  ~     the same axes and origin, it will cut
the branch of the hyperbolism in B.
Then drawing the perpendicular A B,
-.  ^  _ /._. OA is the only real root belonging
to the proposed equation ( ).
+~< — o          A   x   In the same manner might the root.........   ---- - be found by the assumption of the
equation to the hyperbola xy=a2,
and the substitution of the value of x thus obtained in equation (I).
In this case however the second equation becomes
x3 + a2x-ay2= o,
which indicates a semicubic parabola, and if this curve and also
the hyperbola be described, as in the
figure, the root is the same as before,
viz., OA=x.
Our author remarks that there are                   X
many probleins whose solution may
be effected by the intersection of curve 
lines. Among these, is the celebrated
Delian problem of the duplication of the cube, which depends on
the finding of two mean proportionals between two given quantities,
viz., the side of the given cube, and the side of its double; the
first of the mean proportionals being the side of the double cube,
as was observed by Hippocrates of Chios 2ooo years ago.
The intersection of curve lines enables us to find any number
of mean proportionals between two given quantities a, and b: for
if x be the first of them they will form the progression,
2e  33  34  "5
a, x, -, -     -&c.,
a     a- a — a4 sc.
If now two mean proportionals are wanted, the fourth term of
the progression must be = b, and therefore we have,




CONSTRUCTION OF EQUATIONS.            87
æ3
— =, or x3 = a2 b, or x4 = a2 bx,
a
by which the problem may be solved either by the intersection of
two parabolas, by that of a parabola and hyperbola, or by that
of a parabola and circle.
Let it be required to find four mean proportionals, in which
case b should be the sixth term ofthe progression, and the equation
to be solved is
X- a4b = o,
assuming the parabola x2. ay, the substitution gives for the locus,
y2 —a2 b  o,
which is that of the cubic hyperbola. This curve being described,
will necessarily cut the parabola in one point P, and the absciss
which corresponds to the ordinate of the point of intersection
P, will be the root required, that is the first of the four mean proportionals, whence the others may be found.
The root might also have been obtained through the means of
the hyperbola xy = a", by substituting in the original equation
e2y2 for a4, giving for the locus
3 = ay2,
a semicubic necessarily intersecting the hyperbola in one point.
If it be required to insert six mean proportionals between
a and b, the intersection of two lines of the third order will solve
the problem. Thus let x be the first of them,..   7 = a6 b,
On the axes AC, AD with vertex A, describe the cubic
parabola A GL, whose parameter AB    a, and the semicubic
It       7
K




88                        NOTESo
parabola A G M, whose parameter AF = b. Let G be the point
where the curves intersect, and draw the ordinates E G,  G.
In the cubic parabola, if AH = EG = x, AE =   G = y
x3         z9
AB. AE=EG3.. y.   /.  = a
In the semicubic, if AE = HG, AH = EG = x
AF.AH2=HG3.-. y3=bx9
o, x9= a6  2, or x7-a6b
whence we have E G = x = the first of the required means, and
A E will represent the third, whence the rest may be found.
By the intersection of curve lines many other problems may be
solved, but as it is evident that the correctness of the result must
depend on the graphical accuracy with which the curve lines can
be described, the method must be classed in the very numerous
category of subjects which are more curious than useful.
On the Analytical Parallelogram.
Having brought to a conclusion the notes to the text of
Sir 1. Newton's enumeration of lines of the third order, we now
propose to add some observations on the best method ofdistinguishing the species of any curve of which the equation is given.
In ordinary practice it is usual to have recourse to the assistance
of the differential calculus for this purpose, but another method
founded upon the properties of the "analytical parallelogram,"
invented by Newton, may frequently be used with advantage,
especially if the nature and direction of the asymptotes and infinite
branches are the object of inquiry. It is indeed on this method
that Cramer has founded most of the reasoning of his great work
on curve lines, modestly entitled "Introduction à l'Analyse des
Lignes Courbes," in which curves of great intricacy up to the
sixth order are discussed, and in which not only the infinite parts
of the curves, but their singular points and maximum and minimum co-ordinates are found, without any reference to principles
beyond those of ordinary algebra. The first notice of this invention appears in Newton's letter to Oldenburg, "de Seriebus




ANALYTICAL PARALLELOGRAM.              89
condendis et invertendis," and was afterwards more fully explained
iii the Geometria Analytica. We proceed to consider this invention
as applicable to the investigation of curve lines, althougli that was
not the original object its inventor was aiming at when he first described it.
It frequently happens that the equation of a curve is presented
in a form requiring tedious transformations in order to enable us
to ascertain the species to which it belongs. Whenever such a
difficulty occurs, the analytical parallelogram will be found a
useful coadjutor.  The peculiar value indeed of this ingenious
invention appears to have escaped the notice of modern elementary
writers, who when they mention it, usually content themselves
with describing its construction without adverting to the assistance
it is calculated to afford the student in the discussion of the
properties of algebraic curves, particularly of those whose equations
are not easy solvable. What the differential calculus is to the
finite parts of the curve, the analytical parallelogram may be s-aid
to be with respect to the infinite portions of it, namely an important
aid in the abridgment of laborious inquiry; not that it supplies
any information absolutely unattainable by other means, but that
it enables a much more speedy conclusion to be attained than
could by other methods be made available.
The diagram annexed, which is taken from Sir I. Newton's
Geometria Analytica," is sufficiently explanatory of the conc.           _-..__
x6   x6y  x6 y2  Sy3  64 I  y5 xy6   6y7
XI5  5y   5y'   5y3 Y  5y4  y5 s5y6  5 y7
X4   r4y 4 4y2  4y3 a ^4y4  X4y5       7,vy6  X4y'
Xa3  x3y   3 y2  3y3  x3y4  3 y5  x3 y6 x3 y7
~2   o2y   2yl2 x32y3 |S  2y4  2y5  ^2y6,Zy7
xe  y    y2   y3    y y 4  x y5; y6, y7
i    y    y2   y3    y 4  y5    y    y7
Al. _                         _______ B




9o                         NOTES.
struction of the parallelogram. It will be seen that the terms are
placed in the centre of each square and arranged in geometrical
progression, not only in the vertical columns, as well as the
horizontal, but also when taken in any straight line which passes
through the centres of the cells or squares, for in any such cases
the indices of x and y will be in arithmetical progression. E. g.
In the terms y7, y5x, y.X2, yx3, the indices of y decreasing by the
common difference 2, while the indices of x increase in the progression of the natural numbers, the common ratio of the terms is
2-  It follows, that if any two terms be supposed equal, then
all the terms in the same straight line with these terms will be
equal, because. by supposing these two terms equal, the common
ratio is supposed to be a ratio of equality, and, therefore, if we
substitute for x, its value expressed in terms of y, the dimensions
of y in all the terms in the straight line will be equal, while the
dimensions of y in the terms above the straight line will be greater,
the dimensions of y in the terms below the straight line will be
less than its dimensions in that line. E. g. by supposing y7 =y x3,
we find y6 = x3, whence x = y2. Substituting therefore y2 for x,
the dimensions of y in the terms y7, y5x, y3'x, yx3 will be 7, but
the dimensions in all the terms above that line will be greater
than, in all the terms below that line less than, 7.
Hence it follows that if we write all the terms of an equation
f (x y) = o in their proper squares, and form a convex polygon
by lines joining these terms, those sides of the polygon which are
uppermost when the parallelogram is placed with A B horizontal
will represent the equation when x is assumed infinitely great,
those sides which are undermost representing the equation when
v is assumed infinitely small.  And if we desire to know the
consequences of assuming y infinitely great or small, we have
only to place the parallelogram with the side AC horizontal, and
then to draw similar inferences. For if in an indeterminate
equation, x, or y, is supposed infinite, the supposition renders
certain terms of the equation infinitely greater than the remaining terms, which latter may, therefore, be neglected, and
thus the greater terms will constitute the whole equation.  The
polygon enables us to ascertain which are the terms to be
selected as being effective for the required purpose, and which of
them may be neglected as irrelevant to the object in view.




ANALYTICAL PARALLELOGRAM.              9 
We may further observe, that when there are several upper
or several lower sides to the polygon, each of such sides will
have its peculiar significance, and by forming separate equations
from each of them several results will be obtained for the same
curve.
But as the use of the parallelogram is better shown by examples than by description, we shall proceed to give some
illustrative cases.
Suppose the equation is xey + ay2 - a2x = o,
and it is required to ascertain the species to which
this equation belongs. Let the terms be marked on
their proper squares on the parallelogram, as shown
in the figure: if we desire to know the result of the
hypothesis that x = oo, let the diagram be placed
with the column containing only y's horizontal. The polygon will
then show two upper sides, froin which we form the equations
xa2y- aQx = o, or xy -a2    o      (I)
x2y + ay2 = o, or x2 + ay = o      (2)
From (I) we obtain the hyperbolic branches which form the
curve at infinity, positive and negative.
(2.) No branch in the direction of the axis of x.
Let the parallelogram be now placed with the row of x's horizontal, and the polygon then shows only one upper line, showing
the consequence of y infinite, viz.,
x2y + ay2   o, or x + ay = o,
which, when y is negative, indicates a
parabola, the curvilinear asymptote
to which the branch    approaches.
— )-' \"       We therefore infer the curve to belong to one of the seven parabolic
hyperbolas.  Its figure will be as
shown.
Example 2. Let
x0 - a2 - b-3 = ao. 
1,oeZy




92                        NOTES.
Placing these terms in their squares as before, there are two equations obtainable, one for x infinite and one for y infinite, viz.,
xy2 - b3 = o, representing the branches of the cubic hyperbola when x is infinite,
x y2 - a2 y = o, or x y - a2 =  o, representing hyperbolic
branches when y is infinite.
The curve is, therefore, a hyperbolism of the parabola without
a diameter, Species 64.
Example 3. Let
y2 - I2y + a3- IOX2 + 255X = 0.
When x is infinite there is only one upper side to the  1
polygon, viz.,
xy2 +  3 = o      (I)
shewing only imaginary branches.
When y is infinite, in addition to (I),
Vy2 — Izy = o, or xy- 12 = o         (2)
The curve is evidently a defective hyperbola, species 33 of the
enumeration.
Example 4. Let
xy2 - ay2 - 3axy + 2a2e = O,
x being supposed infinite we have
xy2-3axy + 2a2x = o, or y2-3 ay + 2a2= o,
whence y = a, and 2 a. (I)
For y infinite the equation is xy2 - ay2 = o, whence x = a. (2)
Equation (I) marks the existence of the two parallel rectilinear asymptotes corresponding to the ordinates a and 2 a.
Equation (2), x = a, likewise indicates an asymptote parallel to
the ordinates at a distance = a from the origin. The curve is
a hyperbolism of the hyperbola.




ANALYTICAL PARALLELOGtAM.              93
In general, whenever the curve admits._  _ __  oof infinite branches and the side of the
i.....   polygon is parallel to one of the sides of
-_:  o Zx    the parallelogram, we may infer the existence of rectilinear asymptotes parallel to
-o^ \       one or other of the axes.
Example 5. Let
y2 - a3y - b3x = o.
The upper side when x is made infinite gives
2 2 -_ b3x = o, or My2 = b3,
and when y is made infinite
2y2 - a3y -    o, or x2y = a3,
_there are therefore four infinite branches,
one on each side of the positive axis of
abscisses, and one on each side of the
positive axis of ordinates.
Example 6. Let the equation be
y4 + 2 2y2 + x' - 6 axy- - 2   a  4+ a2 2 = o,
x being infinite, the equation y4 + 2e2y2 + xæ = o gives only
imaginary results; and the same remark being true for y infinite,
it results that the curve has no infinite branch. See figure at
page 46.
In order more fully to exemplify the power of the analytical
parallelogram, a few examples are here added, involving equations
of higher dimensions.
Example 7. In the equation
y4X - y'3x  + 3yx3 - y2x + 4y - 2z = o,
it is required to find the number and nature of the infinite
branches. First, assuming x infinite,
(i) y4x - y3X2 = o, or y = x




94                          NOTES.
shewing two infinite branches accompanying an asymptote cutting
the axes at an angle of 45~.
(2)  3x3 - y3x2 =, O or 3x =y.. y =_    /3x
two parabolic branches, above and below the axis of x, on the
positive side.
(3) 3y3- 2x = 0, or 3y2 - 2 = o,. y = 3         X
two hyperbolic branches of the third order.
Next assuming y to be infinite, in addition to the equations
(i) and (2), we have
y x +4y =, ory3 x+4=o.-. x — -
whence the axis of y is itself an asymptote, having branches of
the curve on each side, for the term -  is negative for y positive,
and positive for y negative. There are therefore eight infinite
branches.
Example 8. Let
y5 3        -- y46 + y25 - 4y33 + 2y4X + X4y — 5 2 +- axy-3x = o.
L _ \             7     _I
-a J           ir-z" ---




ANALYTICAL PARALLELOGRAM.             95
The equations obtained when x is supposed infinite here, are
(I) y5x3 - y46 = o, or y= x3
(2) y x5 -yx26=o,   oy=, o.'. y=.    + 
3
(3) yX5 +$4y   =o, or y 
(4)   4y -3x — =O, ory=     -
When y is assumed infinite, besides (I),
(5) y5X3+2y4x=o, orx=+- 4 -(6) 2y4x-5y2=o, orx=       5
(i) Indicates a cubic parabola, the
# ]7   asymptote of two branches of the curve.
'/ I(2) A hyperbolic asymptote furnish_____              ing two branches on the positive side.
(3) Positive and negative branches.
(4) Two hyperbolic branches coni/ \1,-verging more closely to the axis than
ilf          those of (2) or (3).
(5) Two hyperbolic branches in the negative region of y.
(6) Two hyperbolic branches.
The curve has therefore apparently twelve infinite branches.
Thus far we have not remarked upon the significance of the
lower sides of the polygon, which give the consequences of x or y
infinitely small, on the same principle that the upper sides shew
the result of x or y infinitely great. However ingenious is the
theory by which the singular points, maxima and minima ordinates, &c. of curves, are inferred from the equations obtainable
from the lower sides of the polygon, yet, since the differential
calculus is in general better adapted for the discussion of infinitesimal quantities, we shall not notice this use of the parallelogram,
but proceed to advert to the case in which neither x nor y is
supposed to be infinite.
The parallelogram is now supposed to be placed with its
L




96                        NOTES.
angle downwards, and in that position it is evident that each
row of terms read horizontally is composed of terms of the same
dimensions. The lowest cell is occupied by a constant, viz., the
absolute term; the first row consists of the terms of a simple
equation, the second row consists of the terms of a quadratic,
the third row of the terms of a cubic, &c. This disposition of
the terms furnishes several useful hints for the formation of rules
for ascertaining the nature of the curve near the origin. Thus,
when the lowest cell is vacant, we see that the curve passes
through the origin; when the first row also is vacant, there is
a double point at the origin; when the second or quadratic row
is also vacant, there is a triple point at the origin; when the third
row also, a quadruple point; and so on.
Again, to ascertain the direction of the curve at the origin,
that is, to draw a tangent there, we have only to equate to zero
the lowest row, and construct the straight line or lines which
this equation indicates.  Such straight line or lines will be
tangent to the curve at the origin.  When the equation so
determining the tangent shows that y=o, the axis of x touches
the curve; if the roots of the tangential equation are imaginary,
the series derivable therefrom are also imaginary, and in this
case the origin is a conjugate point; and if the roots of the
equation are equal, there is a double point at the origin, formed
by the contact of two branches having a common tangent, or
rather the coincidence of two tangents, indicating (where the line
is of the third order) a cusp.
For example, take the equation
y4 -2y2"ix + Xz + 2ay2x-5 ax3=o
The first and second rows are vacant when this equation is
referred to the analytical parallelogram; there is, therefore, a triple
point on the origin. The lowest row is
zayx - 5 a x3,
which equated to o gives
X=o, y /2-x /5=0, yV/2+- V/5=0




ANALYTICAL PARALLELOGRAM.              97
There are, therefore, three tangents, one of which is the axis of
y, and the other two are determined by taking on the absciss </2,
the ordinates + 2/5, and drawing straight lines froni the origin to
the extremities of the ordinates.
If the tangential equation is such as that it will divide one of
the rows of terms superior to it, another inference may be drawn,
viz., that the origin is a point of inflexion, and of a degree
dependent upon the number of superior rows which each root of
the tangential equation will divide,
Example. For the curve denoted by x3-axy —by=o, the
tangential equation is -b2y-=o, or y=o
This root, substituted in the second row, makes it also =o,
whence the origin is a point of inflexion; but the third row is not
divisible by y=o, therefore the point of inflexion is simple.
Example 2. In the equation
y4 + 2 y2 x2-4ay3-4a x2y + 8 a8y2-8 a3y-o
(Cramer, Méthode des Tangentes, ~ 185.)
The first row -8 a3y, gives y= o, which value, substituted in
the superior rows, makes the second and third disappear, but not
the fourth. The origin is, therefore, a point of double inflexion,
or serpentement.
If the point in question be not the origin, it is always possible
to transport it thither by a transformation; and then, by a mere
inspection of the analytic parallelogram, we can ascertain the
simplicity or multiplicity of any assigned point, the direction of
its tangent and the degree of its inflexion. This use of the
Newtonian invention was first pointed out by the Abbé de Gua,
in his Usage d'Analyse de Descartes.
Another use which may be made of the Newtonian parallelogram consists in the development of equations by infinite series
to be effected by its means. This, indeed, was the object which
the illustrious inventor had in view when he originally designed
it, as may be seen in the explanation of its use to be found in his
letter to Oldenburgh, and in the Geometria Analytica.  This




98                         NOTES.
use may be made available, as being supplemental to that which
we have been discussing, for the investigation of curves, we shall
therefore here shortly notice it.
An algebraic equation in x and y being given, to find the
value of y in terms of x, expressed in a descending series, the
parallelogram is to be placed with the row without x horizontal, and the first term of the series is to be obtained as has
been already shown, the second and subsequent terms being
obtained as is shown in the following example.  And to find
the first term of an ascending series, the equations must be formed
from the lower sides of the polygon.  Similarly, to obtain a series
giving the value of x in terns of y, it is only necessary to place
the row without y horizontal, and then proceed as before shown.
For example, in the equation
y2-2_xy+x-2 2ay+ax+a-=o          (I) 
to find the value of y in terms of x in an ascending series, the
lower side of the triangle gives but one equation, y2-2 a ay+= o,
whence y=a, the first term of the series.
To find the second terin, substitute this value + u, and apply
the transformed equation to the analytical triangle arranged in
powers of x and u                                    -.'. u'-ax-22tX+X =o              (2)   ~~_
and this, applied to the triangle, gives one lower side, viz. ax=u2,
whence u= -- ax, which is the second term of the series.
To find the third term of the series, substitute - a2l4  + t for u
in (2), and there results
- 2 al t x + t2 - 2 a2X -- 2 tx +  = o  (3)
Before applying this equation to the triangle it is 
necessary to insert the terms in which the exponenrt of x  LII1}
is 2, midway between the squares x ancd tx, and that in which the
exponent of x is 3 midway between the squares containing x2 and x.
The lower side will give - 2zat4~- 2aXI-'= o, or t =X, which
is the third term of the series.




ANALYTICAL PARALLELOGIAM.              99
If to find the fourth term of the series we substitute x + s for
t in the equation (3) we only obtain an exponent for s whose value
is less than that of the preceding one. There is therefore no
fourth termn, and the series is terminated, being
y =a ~    ax + x
Again the equation
x2y + ay2 - 2axy + ax2 = o,
following the same process, furnishes the series,
X3 3~~
a   2a
possible only when x is negative.
And the equation
x3 + x2y + ay2 - 2a y + a2 =,
furnishes a series whose second terni is imaginary, showing that
it is not possible to express y in terms of x by an ascending series.
If the equation be
y3 + a2y - 2a3 + axy - x3= o,
the series will be
X    X+ I3I   3
y = a- -  + 4   -       4 &c.
4-   64a + 5 I 2a2 
which is the example given by Newton.
When it is desired to express x in terms of y by a descending
series, the process for obtaining the first and subsequent terms of
the series required, is precisely the same as that for obtaining the
ascending series, with the exception that the upper sides of the
analytical polygon are to be chosen, wherefrom to form the
separate equations.
Thus in example 7, from equation (i), the first term of a
series descending by powers of x, and therefore converging the
faster in proportion as x is assumed greater, was found to be y = x.




I00                       NOTES.
To find the second term of the series let x + u be substituted for y,
where u is supposed to represent all the remaining ternis of the
series, and the transformed equation being placed on the parallelogram, gives one effective upper side, viz.
UX4 + 3 4 = o, whence u = -3
which is the second term of the series.
Then assuming u = - 3 + t and again applying the transformed equation as before, and assuming x infinite, we obtain the
third term of the series, which proceeds
27
y =x- 3 - -L7.... &.
whence the equation to the rectilinear asymptote is y = x - 3,
the negative sign of the third term shows that the branch accompanying this asymptote is to be found on the lower side of this
line while x is positive, and the upper side when x is negative.
In the same example from equation (3) the first term of the
descending series is y —~ 3x,  and proceeding in the same
manner as in the last case, the second term is found to be + 3.
4
If another term of the series be calculated, the term x~ appears in
the denominator. The series therefore shows that the branches
approach a parabolic asymptote whose axis is the line y = -.
4
So again in example 8, the first term of a descending series
being found by equation (I), viz. y = x3, substituting in the
2
equation x3 + u for y, the second term is found to be- -2,
whence it appears that the two branches of the curve converge to
the branches of a cubic parabola, that in the positive region below
the asymptote, that in the negative region, above it.
The following rule will be found to abridge the labour
necessary to adapt the transformed equations to the analytical
parallelogram.*
Multiply each term of the equation which contains y by the
exponent of y in such term, and also by A x the ascertained value
of y when x is supposed infinite, and divide such term by y. A
+ Cramer, Analyse des Lignes Courbes.




ANALYTICAL PARALLELOGRAM.               0 I
new row of terms being thus obtained, multiply each of them by
half the exponent of y, by AXh, and divide by y. A third row is
to be subjected to the same process, only that I the exponent of y
3
is to be used, and so on until y no longer appears in the remaining
terms. Collecting these terms and referring them to the analytical
parallelogram, one of the upper sides will indicate the equation
which gives the second term of the series, and this new value of y
is to be used for finding the third term, the process being repeated
according to the accuracy required. It is necessary to remember
that here by y is intended " the remainder of the series."
For example, in the first case of the Newtonian equations,
xy2 + ey-ax3-bx2-c x —d =         o 
The first term of the series being ~_ /ax, A =  _~ /a, and
h = i, then the process is as follows:xy2 + ey - ax3 - bx -cx - d = o
2     I    O    O     O    O
42 /axZy ~ e V/ax
2         o
ax3
Collecting the terms, the equation is
y2 + ey - bx- (c I e Va) x -2 2z   ax2y    o
whence y =    2 ~  --- tlhe second term of the series.  I
2 V/a'
xy2 + ey- bx2 - (c ~_ e / a) x + 2 / ax2y
2     I     O           O       I
bxy       be _
Va     2Va~ --- bx2
2       o                       o
b2x
4a




102                   NOTES.
Collecting the terms, the equation is
xy + 2 / axy +  - - c +- e /a  x = o
4a ai
= b2 -4.ac ~ 4àae V/ a
whence   - 4 a-  4 ae /a, the tliird term of
8 ax ~/ a
the series.
Example 2. Let x3-axy +y3=-o 
Here the first term is -x,.. A  - I, and h = i,
x3- axy + y3
O    I   3
ax2 - 3 Y2x
3 yxe
O I
3
- X3
Collecting terms, the equation becomes
y3 - axy + ax2 - 3 y2x + 3 y+ Z2 = o
whence y = -, the second term of the series.
3
y3 - axy + ax2 - 3y"x + 3yâ2
3    I   O     2      I
a2
ay2 -   ax  + 2 axy - ax2
3
I    O 
a2y          a2x
3            3
3     '      0
a3
27




ANALYTICAL PARALLELOGRAM.              I03
Collecting terms, the equation gives for the highest terms,
3yx2 
3
2 a~
whence y = -, the third term of the series.
9 x
Therefore,   y = -      a -  + 2a..   &c.
3    9 x
The reader who may desire further information relative to
the important theorems deducible from the Newtonian parallelogram, will find the subject discussed at great length in
Stewart's "Analysis by Equations Explained," published in I745;
and he is further referred to a paper on " Newton's Method of
Co-ordinated Exponents,"  by Professor De Morgan, in the
Transactions of the Cambridge Philosophical Society, vol. ix.
part 4, as well as to the work of Cramer already cited.
The method given above of finding the value of one co-ordinate
in terms of the other by a descending series, is useful for determining the asymptotes of curves, since the first two terms are
generally all that need be computed. Previously to making use
of this or any of the rules derived from the analytical parallelogram,
it is desirable to notice whether the given equation is resolvable
into factors, for in such a case it will represent only a combination
of lines of an inferior order.
It is also necessary to observe whether the sum of the indices
in the equation be the same in every terni, for then lines joining
the centres of the squares containing these terms on the analytical parallelogram, will not form a polygon or triangle, but
will lie in one straight line. In this case the locus consists of
only straight lines. Thus if
y3 - 2 Xy2 + aC3 =-,dividing by 3,
(xJ )    ( )
2+  J5 - - 1/5
The roots of which are I,  ---   - -   5
'  2  n  2
M




I04                      EXAMPLES.
_  +_/  5       I —/5
therefore y  x,-                    2,   yx 
2             2
showing three straight lines passing through the origin, making
the respective angles indicated with the axis of x.
Examples.
The examples which follow are intended to illustrate the application of the foregoing rules, to the discussion of lines of the
third order, whose equations are given.
Ex. i. The equation to the curve is
y3 -3 -  a3 = 
Referring these terms to the analytical parallelogrami  we
obtain for x infinite,
a3                         _'  __
y =         + * *....   &c
3 t'
The asymptote therefore passes through the origin and cuts the
axis at an angle of 45~. The branches always lie above the
asymptote, since the second term of the series is always positive.
The curve has therefore a diameter. There being no multiple
point indicated by the equation, and two of the roots being
imaginary, the curve can be no other than the 45th of the
enumeration.
Ex. 2. The equation is x y2 + y - s3 =O o
The parallelogram here gives one equation for x infinite anc
one for y infinite, viz.
x y2 '-  = o, ory y =        (I):c      - y  + / =o, or x ' /= - i  (2)




ANALYTICAL PARALLELOGRAM.            105
The series obtainable from (i) are
I                  I
y =  -   - &c.   y   -----    &c.
2x                 2x
showing two rectilinear asymptotes intersecting each other at right
angles at the origin. From (2) the axis of y is itself an asymptote.
Solved for y, the equation gives
y       -  4-   X2 +         n- and forx negative, =   - +  Y+ -
2x -        4d2Y                    2 z        4 -the curve is therefore similar on each side of the origin.
When x = o, y = o and -- oo. There are two real ordinates
for every value of the absciss positive or negative. The curve
will lie in the first and fourth quadrant below the asymptote, and
in the second and third above it. There is a point of inflexion on
the origin, and the four branches which do not pass through the
origin are inscribed.
The species is No. 27 of the enumeration.
Ex. 3. The equation is x y2 + 2y - a3 = o
The axis of x is an asymptote, as also is the axis of y, the infinite branches being in each case on the positive side of the axis.
There is a third asymptote passing through the origin, having
branches below it. The equation shows there is no multiple
point or point of inflexion. There are three diameters.
The species is No. 32 of the enumeration.
Ex. 4. The equation is a y2 - b x2 -- 3 = o
There is one pair of parabolic branches. The origin is a
double point, which as the curve is possible on each side of it is a
point of intersection. When x = - b, y = o, therefore there is a
node on the negative side of the origin. There are maximum
2b
ordinates when  = - -.  The equation giving the tangents at
3
the point of intersection is
y/b-            __  +&c.
-a                    2b       2
y = -_ -- -          &-c..ba   2V ab




o6                       EXAMPLES.
The first term gives the direction of the branches, and the second
shows they are convex to the axis.
The curve is a nodate parabola. Species 68.
Ex. 5. Trace the curve whose equation is
4y3 — 6xy+ + 2x3 + 2ay2 + 4a,2 - b3 = o
Applied to the analytical parallelogram, whether for x or y
infinite, the resulting equation is
4.   -6xy2+- 23 = o
which is resolvable into the factors
4y+2 x= o, and y2-2y+- x= o
whence            y =-     y =     / =y,
are the first terms of the series, showing the infinite branches.
Calculating the second and third terns of these series according
to the rule given, p. Ioo.
x   a   2 a2
_,___,&c.  ( )
y=^~^/-ax~              -a-,&C.(2
From (I) we see that an asymptote cuts the axis of x at an
angle tan1' (-  ), and at a distance — a from the origin. The
uneven power of x in the denominator of the third term shews
that on the positive side the curve will be above, on the negative
side, below the asymptote,
From   (2) we infer that the parabolic branches indicated
exist only in the negative region, for the series is imaginary
when x is positive.  The branches are outside the parabolic
asymptote.  The equation y = x -~ -/- ax, shews that the
diameters of this parabola are inclined to the axis of x at an
angle of 45~.




LNES OF THE THIRD ORDER.             I07
The curve is a parabolic hyperbola without a diameter.
Species 50.
Ex. 6. The equation is
y3 -   x   + xy - + y- +ay- 2axy + 3ax - 2a2_ x = o
The first term of the series obtained cn the hypothesis that x is
iininite, is y = x, and proceeding according to the rule given at
page  oo, the second term is found to be - a, but the third
term is either o or imaginary, so that when x is - co the locus
is the straight line
y = x - aC
There is no branch of a curve either for x or y infinite, and
therefore the equation refers to no curve belonging to the enuneration. Between o and 2 a there are three values for the
ordinate to any assumed absciss, by calculating one or two of
which it is easy to see that the remainder of the locus consists
of the circle touching the axis of y at the origin, and having
the radius a.
Ex. 7. To compare the curves expressed by the equations
y ( - a) =       (x + a)... ()
y~ (a - x) x= (x + a).   (2)
Expanding for the asymptotes in (I) the result is
a2
y _=  (x +a + -+ &c.)
also y is infinite when x = a. There are therefore three asymptotes, of which two intersect each other, and the thirc is
parallel to the axis of y.  If the equation is referred to the
analytical parallelogram with the angle downwards, the first two
rows of terms are vacant, therefore the origin is a double
point, which in this case is obviously a conjugate point, for y is
imaginary on each side of it. The species is the  3th of the
enumeration.




x08                     EXAMPLES.
For the equation to determine the asymptotes in (2) we
obtain
xtyif x3 = o, or y2 +  = o
showing that the two asymptotes which were real in (I) are
imaginary in (2), but the third asymptote indicated by x = a
still remains.
The origin is in this curve, as in the former one, a double
point, but it is now a point of intersection, since y is possible on
each side of it. The species is No. 41 of the enumeration.
Ex. 8.
x3 + IOv + x-y2 + Io = o
The asymptote is here a semicubic parabola. Since this equation has but one real root, there is no oval node or conjugate
point.  When x is positive the values of ~ y increase to
infinity; when x is negative and greater than o, y is imaginary.
Between the origin and the vertex there are both maxima and
minima values for y; the figure of the branches is represented
at page 69.
Ex. 9.
Xy2 -  3 - 1  - +8 -  2 = 0
There is a double point when x = 2, the curve has six infinite
branches, and is a cruciform hyperbola. Species I8, fig. 30.
Ex. Io.
xy2 + x3 + 92 - 12  -20 = 0
For x infinite, the branches are imaginary.  For y infinite
there are two hyperbolic branches to the axis of y as asymptote.
The absciss is a diameter; when x is negative there is a conjugate oval.
Species 40, fig. 49.
Ex. iI.
y3 _   4c + 9 x - 18 =x  o
The asymptote cuts the axis at the point x = 3, making an angle




LINES OF TIE THIRD ORDER.            IO9
of 450 with it. There are three points of inflexion, all in the
axis of x, viz., x = o, x = 3, x = 6. There is a general centre
at the point x = 3, for the even power of x disappears when the
origin is transported thither. The axis of y is a tangent at the
origin. The curve is a defective hyperbola with a centre, but
no diameter. Species 38.
Ex. 12. The curve is
y3- 2xy2 + a2y - a3 = O
When x is infinite there are two equations derivable,
éey-a-=o     (I) and, y3-2Xy2 + x2y=         (2)
From (I) we infer that the absciss is an asymptote, for the series
for y is
a3
a3.
y= _  -+ &c.,
the curve being always above the asymptoteo  From (2) we
infer that y2 — 2Xy + x-= o, the factors of which are x - y, x - y.
There is therefore a double asymptote cutting the axis at an angle
of 45~.  The series obtainable from (2) are y =  x  -&c.,
and therefore the curve exists above and below the double
asymptote, but only in the positive region of ordinates.
Since the curve does not intersect either asymptote, it has a
diameter, and therefore the species is the 65th.  The double
asymptote is, in this case, the diameter belonging to the ordinates
drawn parallel to the axis of x.
Ex. 13. The curve is
xy - x2 + x -4 - + 4 =0
The series for y commence with
y=, and y= +     x
There are therefore hyperbolic as well as parabolic branches.
Solved for y, the equation gives y = - for the curvilinear diamete



II0                     EXAMPLES.
bisecting chords parallel to the axis of y, and solved for x, the
parabola 2 = y2 + 1, is the curvilinear diameter to chords parallel
to the axis of x.
The curve is species 52 of the enumeration.
Ex. 14.
xy -- -2 - 3x - 2y- 3 = 
The curvilinear diameters are the hyperbole xy = I, and the
parabola y2 = 2x + 3.  There is a cusp at the point x= —,
y =-. The species is numbered 48, fig. 56.
Ex. I5.
ey~' -- x2 - 6x - 4y - 9 = 
The turve is a parabolic hyperbola, species 47. There is a
point of intersection when  =- I, y = - 2, the axis of x is a
tangent at the point x= -3. There are curvilinear diameters
corresponding to 2X = y2-6, and y=- 2. (Fig. 55.)
Ex. I6.   To investigate all the curves expressed by the
equation
y - x3 + ax2 - b2x = o
The equation to the asymptote here, is y =     a —, when
a = o the asymptote passes through the origin.
There are only two infinite branches, which when b2 is greater
or less than - a, will be found cutting the asymptote, and there3
fore extending on each side of it. When b2 -a2 the ordinates
3
parallel to the asymptote will have a diameter, and in this case
the branches lie wholly on one side of it, for when 3 b2 = a2, the
distance determining the intersection of the curve with its asymptote is infinite.
There will be a cusp in two cases, when b = o, anc when
r   - b,- in the latter case the cusp will be on the axis of x, at a
2




LINES OF THE THIRD ORDER.            I I I
distance =  a from the origin. In the former case the cusp is
2
at the origin.
There will exist a general centre in two cases, ist, when
a=o, when the origin itself is a general centre, 2dly, when
9 b2= 2a2, in which case the general centre is on the axis at a
distance = a from the origin.
3
When b > o, there is a point of inflexion at the origin.
Since the proposed equation has two imaginary roots, except
when the three roots are equal, the curve does not admit of oval
or conjugate point. There are therefore only species 35, 37, 38,
42, or 45 of the enumeration to which it can be referred.
Ex. 17. Trace the curve
y2 - xy - x2y + x3 + a2 = o
The equation referred to the analytical parallelogram gives
two effective equations for x infinite, viz.,
x3 - x-2y= o, and y2 - x2y = o
whence x =y, and y = x, indicating hyperbolic branches approaching an asymptote which cuts the axis at an angle of 45~,
and also parabolic branches approaching a parabolic asymptote,
of which the axis of y is a diameter.
Expressing y in a series, the terns are
a2
y= x + 7i &c.,
therefore the curve lies above the asymptote;
and since when x = o, y = v- a2, showing that the positive and
negative hyperbolic branches are not joined together, it follows
that the parabolic branches join the hyperbolic branches.
In this example if a2 = o, the equation becomes
y2 - Xy -_ 2y + X3 = 0 = (y - _) (y/ -  2)
and the locus is a combination of the straight line and parabola.
The curve is the 53d of the enumeration.
N




112                       EXAMPLES.
If the term - -2y is also deficient in the equation, the curve
is a divergent parabola whose axis cuts the axis of x at an
angle of 222~, the branches lying on the negative side of the
origin.
If the terms y2 and - xy are deficient, the curve is a trident.
Examples of Propositions producing Loci of the
Third Order.
Ex. i. At equal distances from the centre of a circle, ordinates to AB, the diameter, as NR, MP, are drawn.     Then Q,
the intersection of AP with NR, is the locus of a line of the
third order.
R   ---,        By similar triangles,
AN: NQ:: AM (=-NB): MP
A;.. MP2= NQA2 NB2
A-AN- Mp2===NB.AN
ANA N
multiplying by ANB           N Q2. NB = AN3
or (if AN = x, NQ = y, AB = a),
(a - x) y2_= x
the locus of a cissoid, whose asymptote passes through B, and
whose vertex is at A.
Ex. 2. From one end of the diameter of a circle, straight
lines as BA are drawn to meet the tangent at the other end. In




LOCI 0F THE THIRD ORDER.                I 13
BA take AH always equal to the corresponding chord BI.     To
find the locus of H,
Let B G= a 
GD==IGD =x
DH=y                         /          \a
Since B I = AH, drawing IC, H D parallel to the tangent,
we have BC = DG,
and                 BC: CI:: BD: DH
or                x:V/ax -:: a-  x: y
whence                  Xy2= (a -   )3
which represents a defective hyperbola having a diameter, and
the three equal roots show that the locus is a cissoid proper.
Ex. 3. Q, A, N, are three flagstaffs standing on a plane.
A person at a distance views them apparently at equal distances
apart, although they are in reality at unequal distances.  What
must be his path on the supposition that he moves towards
the middle flagstaff, always preserving equal apparent angular
distances between it and the other two?
If Q, A, N, the three flagstaffs be in a straight line, draw P M
perpendicular from P a point in the
path.  Join P Q, PA, PN, the angle
// \      QPN is bisected by PA.    Let
çQ _  -' -i         QA = a, AN =b, AM = x,MP=y
(byEuclid,VI.B.) QP. PN = QA.AN +AP2 =          Zv + ab + y/ (I)
also,     QP2=y2+(a +x)2, and PN2=y          + (b -x).f. from (I)




I14                      EXAMPLES.
QP2.PN2= (Zx+ ab + y2)2=   y + (a + )2} {ty2 + (b -x)2} (2)
which equation reduced is
a-b              a2 —6ab +- b(
XY2 +    Z ~~  y  --  x    (a   b b   X + abx      (3)
In the particular case supposed, viz., that the flagstaffs stand in a
line with one another, the equation (3) represents a combination
of a circle and straight line, but in any other case the locus will
be a defective hyperbola passing through the origin, and having a
loop passing through Q or N.
Ex. 4.   Two straight lines, A B, H D, intersect at right
angles in C. The radius BI revolving on B cuts CD in H.
C Q perpendicular to B I being drawn, to find the locus of the
point which is determined by taking on B I, HP always equal
to QB.
1t  -    Let BC=CA=a.
f t\      ~\       Then B being the pole, the polar
c____,__      I _   equation is evidently
n:<D~ r = asec ô + a cos 0
To find the equation referred to rectangular axes, let AM =x,
MP=-y, then by similar triangles,
BI =BC. B P a      /(2a-)2 + y
BM            2a-x
cHCM. BP _(a-x) w/(z2a.-x:)' - 
also,       H   = CP          (a-X)/(2a-x)2+y2
- BM             2a —x
but                  BH.HP = BC2. (a- ax) {(2a-x)2 + Y2t = a'(2a-x)2




LOCI OF THE THIRD OIDER.              I 15
whence            (a-x)y2= x3-   4ax 2+ 4a2x
The equation is that of a conchoidal hyperbola, whose asymp2
tote passes through C. There is a point of inflexion when x -=a,
5
and a conjugate point at B, therefore the species is 44, fig. 53.
Ex. 5. A point D is taken on the diameter AB of a given
circle, and any perpendicular NN' to AB being drawn, AN. AN'
are joined. Through D, parallel to AN, AN, are drawn DP, DP'.
Required the locus of P.
/  / P\  Let AG=ox        AB=a
GP=y        AD=b
Since AG2: GN2:: DG2: GP2,
\.. X2: ax-x.: (x-b)2: y2
or        Xy2 + x3-(2 b-a) x2+ (b2+ 2ab)x-ab2=o
The curve passes through B and twice through D, and its asymptote is the axis of y. Species 4I.
Cramer Analyse, p. 44I.
Ex. 6. The side BC of a right-angled triangle being produced, a radius revolves on A as a pole. Required the locus of
a point in the revolving line whose distance from the point of
intersection, D, is equal to the distance of D from C the vertex of
the triangle.
AB =a
_-_                BC =b
AF = 
FP =y
`__________       Draw FP perpendicular to AB
PE                to BD




I 6                       EXAMPLES.
By sim. tri.
BD==a ~.CD=ay
BD           -— C,.  D = 'a_- b
x            X
But                PD2    =   CD2
/    \2   /a      \2 a  ay-bx 2
which equation reduced gives as that of the locus,
y2- 2ay2+ 2aby =- -3+ 2ax + (b2-a2) x
from which it appears that the curve is a serpentine hyperbola
without a diameter.   When x = a, y = b, therefore the curve
passes through C, which is a double point, and there is a loop
between A and C. There is an asymptote when x = 2a.
The polar equation to this curve is obviously
r = a sec ô j a tan 0 ~ b
Ex. 7. The vertex of a right angle slides on a straight line,
AB, while one of its legs passes through a fixed point, H; to
show that the intersection P of a perpendicular from a second
fixed point, K, with the other leg, will be the locus of a line of
the third order.
Let HNQ be a position of the right angle; draw HA perpendicular on AB, KP
x/      ~~\    perpendicular  on  N Q,
IKM   parallel to AB,
/    MPD    perpendicular to
IKM.
I   /         _......./ Let KM=x, MP=y,
_/____           S__- M  Q   KI=a, AI=b, HA=c
ur/.'. D-DP=b-y.
Bysim.tri.KMP, NDP, HAN, NA=             D, ND=b       y
y            X..NA+ND=(a+)=~L+bYY^
Iy     x




LOCI OF THE THIRD ORDER.             I 17
Hence         by2-y3-xy + axy-cx~
This equation, applied to the analytical triangle, gives
(I) cx2-x~2y=o, whence y=c, the equation to the asymptote.
(2) -x2y-y3=o, two impossible asymptotes.
Hence the curve is a defective hyperbola; its species will
depend on the relative position of AB, H, and K. If the line H K
is drawn, and a semicircle described to the diameter HK, then,
if AB intersects the semicircle twice, the curve is No. 34, a
nodate hyperbola. If AB touches the semicircle, the locus is a
cusped hyperbola (No. 35); if AB is outside the semicircle, the
curve is a punctate hyperbola. Species 36.
Maclaurin, Geom. Org. p. 36, cor. 7.
Ex. 8. The corner of a page being doubled down, so that the
sumn of the edges is constant, to show that the locus of the angular
point is a line of the third order.
Take the edges for axes, and their intersection as the origin.
Let KPD be any position of the doubled part. Draw OCP,
KCD, and let fall CB perpendicular to OM, CI to OK.
P
OM=x, OB==       x 
MP=y, BC= y 
K /
Since O C D, O C K, are right-  1 
angled triangles,
B C= OB. B D,
CIP =.OI.IK.             0  B                  D
X2+y 2                         + y.. OD=(OB+BD)=         2Y, and OK=(OI+IK)=         2+
2x                          2y
but O D +  K = constant = a,.X.3 +X2y+xy2 +y3=2 axy




I 18                     EXAMPLES.
The locus represents the loop of a defective hyperbola, having
a double point at 0, and the axes touch the curve at O. Referred
to polar co-ordinates, if OP=r, the equation is
2a
sec 0 + cosec à
If O D. O K is constant, the locus represents a loop of a line
of the fourth order, viz., a lemniscate.
If, instead of the sides, it is the crease which is constant, the
locus of the point P will be the loop of a line of the sixth order;
for if r=OC, CO D=
Since KC + CD=constant=a,
r cot 0 +r tan 0=a
a
cot 0 + tan 0
or in rectangular co-ordinates (x2 + y2)3 -a2 y2
This is one of the curves called " the roses" by the Abbate
Guido Grandi, in his book called Flores Geometrici, where its
equation is given in the form
r=sin 20
Ex. 9. Tangents are drawn to a circle, cutting the diameter
produced (as IB). From a point H, which divides the tangent
in a given ratio m: n, a perpendicular to the tangent is drawn,
which cuts the ordinate or ordinate produced, in P. To find the
locus of the intersection P.
I      ~\E                          Let A be the centre
\ `s      ~, of the circle.
I//      \  \     \^               Let AM=x, MP=y,:VA   M            `/ AI =a.
pY-Â-M       — ^       -----     g




LOCI OF THE THIRD ORDER.           I 1
Then, since A M: M I:: AI: IB, x:/a2-x2: a \a-/a2 —
also BH: HI:: m: n,.. IB: HI:: m+n: n
n a V/a2 - x2..   _  _.  _, -.. -
(m + n) 
and by sim. tri. AM: AI:: HI: IP-=na /a
(m + n) x2
MPy="TaD /~-       na2V/ac-x2
MP —y-/= V/a1-x a       -- - X 
(m+n) x
If x = a, or a /,5 in either case y=o; and if,x=o,
y  + mom+n 
y= ~oo.
If x > a, y is imaginary. The curve is therefore a defective
hyperbola whose diameter coincides with AB, and whose asymptote passes through A.
Ex. Io. The base and altitude of a triangle being given, to
find the locus of the intersection of the bisectors of the base
angles.
Let ABC represent a position of the triangle; let AP, BP,
bisecting the angles A, B, meet in P.             c
Take the middle point of the base, O,
as origin. 
Let OQ=x                       A                 B 
OR=x'
PQ=y=(a-x) tan PBA=(a+x) tan PAB
CR=y'=(a-x') tan CBA=(a+x') tan CAB
AO=OB=a.
Since by hypothesis C A B=2 P A B, and C B A=2 P B A,
2 tan PAB                   2 tan PBA
Itan CAB=,PA             and tan CBA=    ta
-ta A  oi - taP
o




120                      EXAMPLES.
from whence, by substituting these values,
_y _    2y (a+42 )  a  _ y      2y (a-_)
a +c      (a4 +,)2 —Y'  an -,'- (a -,)2 -y2
eliminating x' and making y', which is the constant altitude of the
triangle = I, we have
2 X2y-y2-X2-2 a2y +a2= o
Here, when x = +~ a, y = o, whence the curve passes through
A and B; and if x = o,
y = - a2 - %/a4 + a2
showing that the curve exists below as well as above the axis
of x.
Solving the equation for y,
y = Xe - a2 +4 v 4 - (2a2 + I) 22 + a4 + a2
which shows a parabolic diameter bisecting chords parallel to the
axis of y, and since the radical is always possible, if x is less than
a, it follows that the curve is an oval, belonging to a parabolic
hyperbola.
The portions of the curve outside the triangle are the loci of
tlhe intersection of the bisectors of the external angles of the
triangle made by producing its sides, since the expressions
already used for the co-ordinates of intersection still hold good.
To find the infinite branches, let the equation be referred to
the analytical parallelogram; on the hypothesis of x infinite, we
have
2Xwy = x2, whence y =             (I)
2w2y = y2, whence 2x2 = y         (2)
From (i) it appears there is a rectilinear asymptote parallel to
the axis of x, at a distance =  y'.
From (2) there is an asymptotic parabola, the axis of which is
the axis of y.




LOCI OF THE THIRD ORDER.            121
Solving the equation for x, we have
_y2 + 2a2y-_a
4-a2y —I
which being always possible for large positive values of y, shows
the existence of branches of the curve accompanying the asymptote to infinity.
The curve here considered is not contained in Newton's enumeration nor in Stirling's commentary, but is noticed in Nicole's
paper on the projections of the divergent parabola with an oval,
in the Mém. de 'Acadéemie for the year 1731. See fig. at p. 63.
Ex. i. The ordinate to the diameter of a circle is produced to a point P, so that the rectangle -under the absciss
and such produced ordinate, shall equal the rectangle under the
diameter of the circle and its ordinate. To show that the locus
of P is a line of the third order,
\P
Let    AM = x
MQ=y' /.,
AB =a 
MP =y                            _ 
By hypothesis,          y = ay'
and               y' =_    /ax - x2
hence            y2x + a2x - a3 = o
The curve is Species 62, fig. 7I.
Ex. I2. Required the locus of intersections of the tangent
to a parabola with the perpendicular on it from the origin at
the vertex.
The equation to the tangent of the parabola being
/=,.- ( +   )       (I)




122                      EXAMPLES.
that of a perpendicular on it is
y =     -a.x        (2)
where x, and y, are the co-ordinates of the perpendicular on the
tangent, and at their common intersection we have x   x,, and
y' = y,; therefore
aJ
*y,=  r (v, + x)
And since frorn (2),
- = —      u we find x= -'
whence equation (I) becomes
x,~(     ay)~   ory
y/, =  _ _ z.  + -   ) or y/e  3 —    -
Y/        (/                + t/
representing a cissoid whose cusp is at the origin, and whose
asymptote is the line x =- a.
If the origin be taken on some other point in the parabola
than the vertex, the curve will be the 35th species, fig. 45.
Ex. 13. From any point K of a right line AK given in
position, straight lines KD, KC, are drawn to two fixed points
C,D. In K1D take KP, KP', each equal to KC. The points
P, P', will trace out a line of the third order.
Draw   DB (=a),                        A 
CA   (= b)
perpendicular to AK. 
Let AB = c,                                    \
DM=, MP=y.                                          \
From the similar triangles D M P, DB K,
BK    - ayAK=c+ aY         and CK    - (c        + b
x              x                \- x +x




LOCI OF THE THIRD ORDER.            123
Also   PK2 =(     - y y  (a - x)2, and CK2 == P K'.  3- 2ax2 — (b2 +- c -a2) x = 2ay2 -xy2 + 2acy
which equation belongs to a defective hyperbola without a diameter, having a conjugate oval passing through C and D.
Equating the cubic terms to o, we have
2ay2 — xy = o, or 2a = x
which gives the rectilinear asymptote, passing through F, if
BF = a.
It is a property of this curve that the straight line A K bisects
all lines terminated by the curve, provided they all meet in the
point D. If the fixed points do not lie on the same side of the
given straight line, the infinite branch will pass through one of
the fixed points, and the oval through the other.
The curve is species 33 of the enumeration.
Ex. 14. If from any three points, lines be drawn to intersect in a fourth point, so that one of them shall be always an
arithmetic mean to the other two, to find the locus of the point of
intersection.
The locus required is the 29th, 33d, 34th, 4oth, 4Ist, or 44th
species of the enumeration.
On the same hypothesis to find the locus, when the lines are
drawn so that the rectangle under two of then shall equal the
square of the third.
The locus required is the 33d, 34th, 37th, 40th, or 45th
species, according as the relative position of the three points
varies.
Leybourn's Mathematical Repository, vol. iii. p. 227.
Ex. I5. A    straight line parallel to the diagonal of a
rhombus is drawn through one of its angular points. Required
the locus of centres of a system of conic sections, each of which
touches the line parallel to the diagonal, and passes through the
remaining angular points of the rhombus.




124                       EXAMPLES.
The required locus is a redundant hyperbola whose asymptotes form a triangle, with a cusp at the centre of the rhombus.
Species 12, fig. 24.
Hearn's Researches on Curves of the Second Order, p. 45.
Ex. I6.  From a given point S in the diameter AB produced
of a given circle, let any line be drawn cutting the circle in N
and D, draw AD, BN intersecting in H. If in AD produced
there is always taken DP = DH, to determine the curve traced
by P.
From  S draw the tangent S T, then the perpendicular T G is
given in position, and is the locus of H.
S    ~-          A     /   {.. 3/
LetAB = a, AG= b, PAM            0.
Since AP - AD + DP = AD + DH             2 AD - AH
we have if AP =, AD = r
g       = 2- b sec...   ()
but by the equation to the circle, r = a cos 6
Hence (I) becomes
= 2a cos   - b sec 0
in rectangular co-ordinates, if AM = x, MP =
(b + x) y2 = (2 a-  ) xz - 
(Species 41.)
Ex. 17. To determine the species of the curve traced by the
extremity of a portion of the linear sine of an angle 0, which is
taken always equal to the linear tangent of half that angle.




LINES OF TIIE THIRD ORDER.            125
By the hypothesis, y = tan -
2
-     / I - cos 6
I + cos 0
which equation represents species 63 of the enumeration.
Miscellaneous Problems having reference to Lines of
the Third Order.
Prob. i. To describe lines of the third order by continuous
motion.
Take a grooved ruler of sufficient length and let it revolve
on the fixed pin A, while one end of it slides along a straight
line B C. Let one end of a string of equal length with the ruler
be fastened at the point R, pass over the sliding pin P, and being
stretched along the ruler's edge, have its other end attached to
the ruler's farther end. Then the point P will describe a line of
the third order, whose species will depend on the position of the
point R.
Draw   the perpendiculars AB,
P PG, PM, RH,
R I..^^^ "''^  \Let AM = x,
',, i-          MP = y,
AH  = a,
AB      c




126               MISCELLANEOUS PROBLEMS.
By sim. tri. APM, PCG,
AM: AP:: PG: PC = PR =          /(y-b) + (a -)
/ ---   C_    (c- x) v2 +y2
x: V/'x 4-y:: c-x:. (c - X) (X2 + y) =,2 {(y_ -b) + (a- x),c y~_b       a -c      c -  ~2-  - b2
and     xy2 -- 2            a-c X3 +- 
and J XY  2 y  c x~y  c         2C
is the general equation to all lines capable of description by this
method.
First let us suppose that R is a point in AB, in this case
b = o; if R coincides with B, a = c, and the curve degenerates
into straight lnes.  But when R   is between A  and B, the
curve is a conchoidal hyperbola with a conjugate point at A,
Species 44. When R falls on the left of A, so long as its distance from A is less than AB, the curve is still a conchoidal
hyperbola, but with its convex side towards the conjugate point,
Species 43. When this distance is exactly AB, the curve is a
cissoid; if the distance exceed AB, the curve is Species 41.
Next, supposing R to fall on the right of B, at a less distance
than AB; the curve is now a cruciform hyperbola, Species I8.
When R is exactly at the distance AB, the curve is the 3oth
species; when at a greater distance, the Igth species.  The
asymptote in all these cases is the same, viz. the line bisecting
A B and parallel to BC.
But if R lies in the line BC, the locus degenerates into a
conic hyperbola with a straight line.
Suppose a conic parabola described with AB as axis, B as
the vertex, and A as the focus; and suppose also a circle on AB
as diameter, and that R falls between the parabola and B C.
The curve will be a redundant hyperbola; when R is on the
parabola the curve is a parabolic hyperbola; when within it, a
defective hyperbola, which becomes the 34th, 35th, or 36th
species, according as R is without, upon, or within the circumference of the circle: in all these cases the curve has no
diameter.
But R may fall on the riglit of BC, in which case the curve




LINES OF THE THIRD ORDER.             127
will be of the 7th, 8th, or 25th species, according as its distance
from BC varies. In fact, by this method all the non-parabolic
curves which have a double point may be drawn.
Prob. 2. BE represents the hyperbolic branch of a conchoidal curve, species 44, of the enumeration, A its conjugate
point, C the intersection of the asymptote with the axis. If now
BF the vertical tangent, be drawn, and on AB, as diameter, a
circle be described, and a line A F be supposed to revolve on A,
cutting the circle in D, the branch in E, and the tangent in F, to
show that the ratio DE  E F is constant.
Join D B
Draw DI, E M, perpendicular to AB.
By the property of the curve
MC. ME2 = MB. MA2. M.:MB: (MA2: ME2:: AD2: DB):: AI: IB
comp.       AB: IB:: CB:MB
alt.        AB: CB::IB: MB:: DF: EF
div.        AC    CB::::DE: EF
And the ratio AC: CB being constant, so also is the ratio
D E: E F, whatever the position of the line A F.
Prob. 3. If from the cusp of a cissoid, radii be drawn to
intersect the branches, perpendiculars through the points of intersection will envelope a parabola.
P




128               MISCELLANEOUS PROBLEMS.
The equation to the cissoid being (a + x) y2 =- - x3
x2   ay
y ==-      -            (I)
y    x
Let the equation to any assumed radius be
my =    x              (2)
Then that of a perpendicular on it will be
y = mx + b            (3)
Since at the intersection these three equations are simultaneously true for that point,
- x2   ay                         x         a
--  --       mx +b, and m= —.        b=
y    x                         y          m
Hence (3) becomes y = mx + -
m
which is an equation to a line touching the parabola
yf = 4ax
the asymptote and directrix coincide, the cissoid lying in the
negative and the parabola in the positive region.
Prob. 4. In the semicircle AGB, of which M G is an ordinate, the area AMP is always taken equal to the corresponding
segment ANG.     Required the curve generated.
<<7                    -e ALet AM  - x
MP =y
P/  /                               MG=y
rad = I
A-1




LINES OF THE THIRD ORDER.              129
Thenjydx = ANG = ANGM-AGM
fi' d  - ___
=-f/2x-x2     dx - x    /2x —'
/     2        2 (IX-X).. ydx = _ V2~_   -           2-~       2  2x-   Y2
Hence y =2 2 —         is the equation to the curve, which
represents a hyperbolism of the ellipse, whose asymptote passes
through B, and whose vertex is A. It is evident that the area
of this curve between the vertex and asymptote is equal to the
circle described on AB as diameter.
Prob. 5. Required the curve in which the area is always
equal to the fourth power of the ordinate.
We have       y4    ydx
4y3dy = ydx
4y2dy = dx, and integrating
y = 3_   + C
whicli is a case of the cubic parabola.
Prob. 6. Required the curve in which the area is always
equal to twice the rectangle under the co-ordinates?
Since          s ydx = 2xy
ydx + 2xdy = o
2xydy + y2ldx - o, and integrating
2xyn = C




130               MISCELLANEOUS PROBLEMS.
the curve is a hyperbolism  of the parabola, the origin of coordinates being taken at the intersection of the asymptotes.
Prob. 7. Required the curve in which the area is always
equal to three-fifths the rectangle under the co-ordinates.
Since      jydx = 3 xy
5
3        3
ydx   ==   ydx + -xdy
5        5
2 dx      3 dy
5x        5 y
2         3
- log x=    log y
5   ~     5.   a c  y3
the curve is a semicubic parabola.
Prob. 8. Required the curve in which the area is always
equal to three-fourths the rectangle under the co-ordinates.
I          3l
We have here - log x=3 logy
4r         4
x   c   y3
the curve is a cubic parabola.
Prob. 9. To find the area of the hyperbolism of the hyperbola
included between the curve and parallel asymptotes.
The equation being
(a2- x2) y - a3
where 2a is the distance between the parallel asymptotes,
=r ra3dx         a2     a+x
Jd      J / ---   = -a log - x = the area required.
e te wo   ara w    n 2   a, i x
Hence the whole area, when x = a, is infinite.




LINES OF THE THIRD ORDER.            13I
Prob. o. To find the area of the hyperbolism of the ellipse
taken between the curve and its asymptote.
The axis of the ellipse being taken as the parameter of the
curve, the required area is equal to four times the generating
ellipse.
Prob. I I. To find the area of the cissoid between the curve
and its asymptote.
The required area is equal to three times the generating
circle.
Prob. I2. To find the area of the loop of the folium of Des
Cartes.
The equation being
X3 - 3axy + y' = o
the required area will be =  a2.
2
Prob. 13. To find the point of greatest curvature in the
semicubic parabola,
ay =x3
the radius of curvature being =  4a  (  +
& 3       4a/
which is o when x = o; the required point is the vertex, where
the curvature is infinite.
Prob. I4. To find the point of greatest curvature in the
cubic parabola
X3     a 2y
the required point occurs when: y:: 3 V/5: I
In the cubic hyperbola
y2 = Sa3




32               MISCELLANEOUS PROBLEMS.
the point of greatest curvature occurs when:y:: +   /5: I
Newton, Anal. Geom. p. 457.
Prob. I5. To find the points of inflexion in the curve
xy2 + ax x+ b3 = o
When a and b are of the same sign, the curve is a parabolic
hyperbola, species 53 (fig. 6 ).
There are points of inflexion when
x=     bt(3 + v/2)}-, y=~{4+              )a 3}
When a and b are of contrary signs, the curve is species 56
(fig. 64); and there will be points of inflexion when
I= Zb{(3 -v/12)j y, ==~{(4-,           3) ab3} 
Prob. I6. To find the point of inflexion in the curve
xI y  bxy - ax2 + aby -   ax = o
the tangential equation is here aby- a2x = o; and since this
will divide the row of terms superior to it (p. 97), which is,
bxy - axI = o, there is a point of inflexion at the origin.
Prob. 17. A perpendicular to the axis of a curve is drawn
through a fixed point in the axis, and intersects the tangent to
any point P, in the point Q. The length of this perpendicular
and the intercept PQ together is double the length of the curve
between the vertex and P. Required the equation and species of
the curve.




LINES OF TIIE THIIRD ORDER.          133
If the distance from the fixed point to the vertex is called a,
the equation is found to be
y = + --    (4 a-x) /x - a
which equation represents a nodate parabola, species 68 of the
enumeration.
Leybourn's Mathematical Repository, New  Series, vol. iv.,
p. 100.
Prob. I8. A straight line being supposed to revolve about a
fixed point P in the absciss of a line of the tliird order, so as to
cut the curve in three points, and tangents to these points being
drawn to meet the absciss, to show that the sum of the reciprocals
of the subtangents is constant.
Let the equation be arranged in the form
3- (ax + b) y + (cx2 + dx + e) y &c. = o
Let y, y', y", be the three values of ordinates corresponding to
a given absciss, then (p. 38)
yy'y" =. f3 - gx + h  - k......   (I)
and differentiating
y'y"dy + yy"dy' + yy'dy" = 3fxfzdx - 2gxdx + hdx... (2)
dividing (2) by (I) and also by dx,
dy    dy'    dy" _     3/x2 - 2gx + h
ydx   y'dx   y'idx - fx3 - gx2 + hz - k
which proves the proposition, because the fraction is constant,
x being invariable while P is fixed, whatever the angle of ordination, and therefore the sum of the reciprocals of the subtangents
is also constant.




I34                MISCELLANEOUS PROBLEMS.
Thus if AP = x,
/      PM, PN, P O, - the
/^ 1    three values of y, y', y,
N^^^        MR,     NS, OT, the
I        I       I
>M/-   _.~fit)\    /       three tangents, the sum
P_        li \__ c_   \S     T/ \A     p     +  Ps + PT
constant.
Prob. 19.  The same things being supposed as in the last
proposition, if the fixed line AP also cuts the curve in three
points, D, C, B, the sum
I I          I I     I             I
--.                -,-   + -    +
PR     PS       T +PDT PD    P + PB
that is, the sum of the reciprocals of the subtangents, will be
equal to the sum of the reciprocals of the segments of the absciss
between the fixed point P and the curve.
This is evident from the consideration that when P O coincides
with PB, PD = PR, PC = PS, and PB = PT; therefore
I      I      I       I      I      I
PD + PC + P-B =PR + PS + PT
which sum is constant at whatever angle the ordinates are inclined
to the fixed axis. When the segments are on contrary sides of P,
the terms will be affected by contrary signs.
Prob. 20. A straight line PA revolving about P cuts a line
of the third order in three points, B, C, D; and in PA, Q is
taken so that PQ is a harmonie mean between PB, PC, PD.
To show that the locus of Q will be a straight line.
In the case of a right line PA, drawn from  a fixed point P,
which cuts any number of right lines given in position in the
points E, F, G, H, &c., if in P A there is taken a point K such
P   J1a        E     p I  I I  I    I
that -p 'I a  always = pE + PI + PG -     PH    &c., then, as is
well known, the locus of K   will be a straight line.




LINES OF THE THIRD ORDER.              135
Assuming PMNO (see the last diagram) to be fixed, and
PA   to revolve, then the tangents MR, NS, OT, are given in
position; and if a point K is taken so that
i      I     I        i      i      i       i
PD + PC+ PB (P          R    P S    P T )
the locus of K will be a straight line; and if we take another
point Q so that PQ = 3PK, we have
I      I      I     3
1 1    1     3
PD     PC P+B       PQ.. whên PQ is a harmonie mean between the three lines PD,
PC, PB, the locus of Q is a straight line.
Cotes on the Nature of Curves; Maclaurin, Lineae Geometrice,
p. 460; Salmon's Higher Plane Curves, Art. 56.
Prob. 2I. Through three points in a line of the third order
in the same straight line tangents are drawn which also eut the
curve. -These points which they cut will also be in a straight
line.
This and the six subsequent propositions are deductions from
the three preceding propositions.
Prob. 22. If from a point in the curve a pair of tangents be
drawn, and from either point of contact another pair of tangents
be drawn, the line joining the last-named points of contact will
pass through the remaining point of contact.
Maclaurin, Lin. Geom., Prop. 7, Cor. i.
Prob. 23. If from a point of inflexion a pair of tangents be
~,                      drawn, as A F, AG, and the
(yyG ^-" —^   ^^    line joining the points of conX 7/ _....ll ^       tact be produced to meet the
curve in 1H, AH  touches the
Pr   ^                         curve in H.
Not more than three tangents can be drawn from a point
Q




136              MISCELLANEOUS PROBLEMS.
of inflexion (besides that which touches and cuts it at the same
point), and the three points of contact are in the same straight
lne.
Prob. 24. If from a point of inflexion A (see the last figure),
three tangents are drawn, the straight line joining the points of
contact will eut h-armonically any straight line drawn through
the point of inflexion and terminated by the curve.
For let F GH, the line joining points of contact, be produced
to meet the line CAB drawn through the point of inflexion in
P, then, since the three tangents all meet in one point,
I     I      I     3       I      I     2
PA+ PB       PC    PA' or    B + PC = PA
ttlat is, PA is a harmonie mean between PB and PC.
Prob. 25. A straight line joining two points of inflexion
either passes through a third point of inflexion'or is parallel to
the ultimate direction of the infinite branches. For draw PQ
joining P and Q, two points of inflexion, and produce it to meet
the curve in R, then, to show that R must be also a point of
inflexion, let a tangent be drawn through R, cutting the curve in
another point S, if possible; and by Prop. 21 P, Q, and S are in
the same straight line with R, whicl is impossible in a line of the
third order. Therefore R is a point of inflexion.
Hence we infer that the infinite branches of the divergent
parabolas are ultimately parallel to the line joining their points
of inflexion.
Prob. 26. A line drawn through a point of inflexion A, and
parallel to the asymptote, meets the curve again in B.  If a
tangent at B is drawn cutting the curve in C, the line AC
passes through the point where the curve and asymptote intersect,
Maclaurin, Lin. Geom., p. Io.
Prob. 27. If four tangents are drawn from P, a point in the
curve, lines joining the points of contact will intersect in some




LINES OF THE THIRD ORDER.            137
point of the curve, and any straight line drawn from P will be
eut harmonically by the curve and the lines joining two points of
contact.
Maclaurin's Fluxions, art. 402.
Prob. 28. The equation to a line of the third order being
given, to draw a chord parallel to one of the axes of co-ordinates,
so as to be bisected by the curve.
The equation being arranged in the form
x3+ (a + by) x + &c. = o
or              y3 +(a+ bx) y2 + &c, = o
according as the chord is required to be parallel to one or other
of the axes, let the locus be constructed of the equation
3  = — a - by
or                    3Y =-a-bx
If the straight line thus drawn cuts the curve, it intersects it at
a point which bisects the required chord. If it does not cut the
curve, no such chord can be drawn. When the coefficient of
x2 or y2 = o, the required chord coincides with the axis.
Probo 29. Given the three asymptotes of a redundant hyperbola intersecting so as to form a triangle; required the locus
of the points of osculation within the triangle.
The condition for a point of osculation is (Gregory's Exvamples,
ch. x. p. I66),
/d2u d2'    (d2 U\  d2 u
dx dy     \dJ x    dy
Referring the curve to two of its asymptotes -ls axes, their
intersection being the origin, its equation takes the form
u = ax2y + 2 bxy - cxy2 = hI




138               MISCELLANEOUS PROBLEMS.
hence            (d ) = 2 axy+2by+cyc
(d    = 2     + 2 b x+ ax2
dy
(d2u             d2u\             du \ u                \
\dx2)     ay,   )dy2J =2c       \d ddy)      /\)
Therefore for a point of osculation,
(ax + b +  cy)2 - acxy = o
or    a2x2+ acxy +   y2 + 2 abx + 2 bcy+ b =o... (I)
This equation represents an ellipse, the position of which is
determined by its touching the middle points of the three sides of
the triangle. For it appears from (i) that it touches the axes at
distances -, and - b from the origin, and the equations to the
a        C
three asymptotes are
x = o, y =o, ax+2 b+ cy =          o...  (2)
2b        2b
giving -    and - -   as the intercepts cut off by the third
a         G
asymptote. If the values - - and -    are substituted for x
a        c
and y in (I), the value of d/ derived from    the resulting
dx
equation coincides with the value derived from the equation to
the third asymptote, viz. - -, and as these values of x and y
satisfy both the equation to the ellipse and that to the asymptote,
the ellipse touches the third side of the triangle in its middle
point.
Prob. 30. To investigate the properties of the logocyclic
curve.
The name logocyclic las been given by Dr. Booth to a pecu



LINES OF THE THIRD ORDER.             139
liar combination of Des Cartes' folium with a common parabola.
The equation to the former being for rectangular co-ordinates,
y2 (2 a - x) = x (a - x)
F being the origin, and F  being taken = a, the course of the
curve will be as shown in the diagram, where QO represents the
branch of the parabola whose focus is at F, and whose vertex is
at O.
If F be made the pole and FO   the prime radius, r and 0
being the polar co-ordinates, since   =  tan 0, -    sec 
r =- /x2 + y2, the polar equation is
r = a (sec O ~ tan O)
Thus, if the radius vector cuts the curve in the points R, R',
FR = a (sec O + tan 6), FR' = a (sec  - tan O),
and the product FR. FR' = a2 (sec2   - tan2 0), is constant,
and = a2.
If through R and R' nor- 
mals RQ, R'Q be drawn to
the curve intersecting one another, they are equal.
If tangents be drawn at                 "^
R, R' meeting in V, they also
are equal, and equally inclined       ^
to the chord RR'. These three      /____         _
properties of this curve are also  k       ~\
properties of the circle. 
The locus of the points Q 
is a parabola whose focus is F,
whose vertical focal distance is 
a, and whose directrix is the             / 
asymptote of the curve. By 
means of this parabola and this
its allied curve, Dr. Booth has,
in a recent communication to the proceedings of the Royal




140               MISCELLANEOUS PROBLEMS.
Society, exhibited the whole theory of logarithms in a geometrical form singularly comprehensive and beautiful.  From the
analogy of its properties with those of the circle, and from its use
in illustrating the logarithmic theory, he has called it the logocyclic curve.
To represent numbers and their logarithns by the logocyclic curve and its conjugate parabola, Dr. Booth gives the
following rule - A parabola whose focal vertical distance is
a = i being drawn, and also its logocyclic curve, let a radius
vector be drawn to the latter equal to the number n. Then
n = sec ô + tan ô.
Let this line meet the vertical tangent in T, the parabolic
arc OQ-QT is the logarithm of n.
It is clear that the infinite branch of the curve from + oo to o
will give radii vectores of every magnitude from oo to I, and
parabolic arcs from oo to o.  Hence, while the numbers range
from oo to I, the parabolic arcs range from oo to o. When the
number lies between I and o, the radius vector representing it
is drawn below the axis; its extremity will be found on the loop,
and the corresponding arc of the parabola will be negative.
Hence the logarithm of a positive number is equal to the logarithm
of its reciprocal, with the sign changed; for the magnitude of the
parabolic arc depends upon 0, and 0 is the same in sec à + tan ô
as in its reciprocal sec ô - tan 0.
Hence, while the infinite branch of the logocyclic curve from
+4o through R, O, S, to F, may by its radii vectores represent all
positive numbers from +oo to +o, the two infinite branches of
the parabola will be used in representing the logarithms of positive
numbers from + oo to +o; that is, the upper or positive branch
of tle parabola will be " used up" in representing the logarithms
of positive numbers from +oo to + I, and the lower or negative
branch of the parabola in representing the logarithms of positive
fractional numbers from + I to +o.  Hence there is no construction by which we can represent negative numbers or their
logarithms; therefore such numbers can have no real logarithms.




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