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(5.) The section is an hyperbola, when the cutting plane
makes a greater angle with the base than the side of the cone
makes, and if the plane be continued to cut the opposite
cone, this latter section is called the opposite hyperbola to
the former.

(6.) The vertices of any section are the points where the
cutting plane meets the opposite sides of the cone. Hence,
the ellipse and the opposite hyperbolas have each two vertices,
but the parabola only one.

There is no work ofantiquity which professedly treats of the
history of conic sections, that has reached our time, and
there is little to satisfy curiosity in this inquiry, excepting
some incidental notices collected from different authors. The
discovery of the curves, denominated the conic sections, is
attributed to the philosophers of the school of Plato, or even
to Plato himself. The theory of these curves probably grew
up gradually from small beginnings, increasing in magnitude
and importance, by the successive improvements of many
geoinetricians. The histor}' of the malliematics mentions two
problems, famous in ancient times, and both of them so



CONIC SECTIONS. 495

difficult as to surpass the limits of plane geometry. These
problems were the duplication of the cube, and llie trisection
of an angle ; and there is no doubt, but that the theory of the
conic sections received great additions, and was enriched with
many new properties, by the researches that were undertaken
for resolving these problems. Two solutions of the former
problem, derived from the conic sections, are preserved by
utochius, in his commentary on the works of Archimedes,
which are attributed to Menechmus. Some solutions of the
latter problem, by means of the conic sections, are likewise
extant in ancient authors, for which, science is thought to be
indebted to the ingenuity of die followers of Plato. Hence
it has been inferred, that great progress must have been made
in investigating the properties of the conic sections before the
time of Archimedes. This conclusion is confirmed by the
writings of that celebrated mathematician, the best and most
splendid edition of whose works was printed at the Oxford
press, in 1792. In these works many principal propositions
are there expressly said to have been demonstrated by preced-
ing writers, and are spoken of as truths commonly known to
mathematicians. Archimedes himself, - perhaps the greatest
genius of antiquity, and deserving to be ranked with Galileo
and Newton, enriched the theory of the conic sections, with
many noble discoveries. After a lapse ofiwo thousand years, the
quadrature of the parabola is even yet the most remarkable
instance in the science of geometry, of the exact equality of
a curvilinear to a rectilineal space. To this discovery must
be added, the determining of the proportions of the elliptic
spaces to one another, and to the circle ; and likewise the
mensuration of the solids generated by the revolution of the
conic sections about their axes.

We are principally indebted to the preservation of the writ-
ings of Apollonius, for a more perfect knowledge of the
theories of the ancient geometricians, on conic sections. He
was instructed in geometry in the school of Alexandria ; and
under the successors of Euclid, he there acquired that superior



496 MATHEMATICS.

skill ill the science which distinguishes his writings. Besidt#
his great work on conic sections, he published many smaller
treatises, relating chiefly to geometrical analysis, which have
all perished, and are known to us only by the account given of
them in the seventh book of the collections of Pappus.

The treatise of ApoUonius on the conic sections, is writ-
ten in eight books, and it was a work in such high estimation
among his contemporaries, that it obtained for him the title
of " The great matliematician."

The four first books of the conies of ApoUonius, is the
only part of that work that has come down to us in the origi-
nal Greek. But in the year 1658, Borelli, passing through
Florence, found an Arabic manuscript in the library of the
Medici family, which he judged to be a translation of all the
eight books of the Conies of ApoUonius. Transported with
joy, he had interest enough to prevail on the Duke of Tus-
cany to entrust him with the manuscript, Nvhich he carried to
Rome, where he published a translation of it in 160 1. The
manuscript found by Borelli, was entitled " Apollouii Pergaei
Libri Octo," and was at first supposed to be a complete
translation of tlie work of the ancient geometrician ; but on
examination, it was found to contain the first seven books
only. Two other Arabic translations of the conies of Apol-
lonius, have since that period been discovere<i, but both these
have the same defect as that found at Florence. Hence it is
imagined, that since all the three manuscripts agree in wanting
the eighth book, that it was not in existence when the Arabic
translations were made. It cannot be ascertained when the
original of ApoUonius 's work disappeared, but it is certain
that it was extant in the time of Pappus of Alexandria, as in
his " CoUecliones Mathematicze," is given an account of the
contents of the eight books, and he has even added the lem-
mas required for the demonstrations of the propositions which
they contain: and this circumstance enabled Dr. Halley to
annex to his edition of the conies of ApoUonius, published in
1710, a restoration of the eighth book, executed with so



CONIC SECTIONS. 497

much talent as to leave little room to regret the want of the
original.

The mathematicians who followed Apollonius, seem to
have been content with the humble task of illustrating his
treatise. We shall not attempt to enumerate all the commen-
tators who, at different times, have written on his work, and
have endeavoured to render the important truths contained in it, '.
of more easy access to the general class of mathematicians.
Among the number, however, was the learned and accom-
plished Hypatia, whose praises we have already recorded ; and
we are still in possession of the lemmas of Pappus, and of
the pomnilntary of Eutochins on the four first books. Since
tlie revival of learning, the theory of Conic Sections has been
much cultivated, and is the subject of a great variety of
works.

Th^re is, says a writer on this subject, a relation subsisting
between all parts of human knowledge, which frequently con-
nects speculations the most abstracted, and seemingly the
most barren, with inquiries that are highly interesting to us,
and most fertile in its consequences. In studying the proper-
ties of the Conic Sections, the followers of Plato sought
merely to gratify a contemplative mind, without the most dis-
tant regard to practical utility ; but, among the^ moderns,
they have been employed to explain many of the most re-
markable phenomena of the material world. The doctrine
of the Conic Sections has been found of utility in the science
of optics, as well as in determining the path of a projectile
body. This branch of mathematics, however, derives its chief
importance from the application that is made of it in modern
astronomy.

There are different modes of treating this branch of sci-
ence ; before Dr. Wallis, all writers on Conic Sections fol-
lowed the ancient geometricians in making the solid cone the
common origin and foundation of their theories. But that
great mathematician, in a work entitled " De Sectionibus
Conjcis," published at Oxford in 1655, deduced all the pro?
vo):.. 1. 2 jf



498 MATHEMATICS.

perties of the curves from a description of them on a plane ;
since which, authors have been divided as to the best way of
defining the curves, and demonstrating their elementary pro-
perties. Many, in imitation of the ancient geometricians,
make the cone the foundation of their theories, while others,
equally respectable for talents and learning, have followed
;the example of Dr. Wallis.

The latest and most approved writers, who have deduced
tlte properties of the Conic Sections fix>ni the cone itself, are
Dr. Hugh Hamilton, whose work, originally published in the
Latin language in 1?^, was translated in 1773, under the
title of " A Geometrical Treatise of the Qonic Sections, io
which the properties of the sections are derived from the na-
ture of the cone, in an easy manner, and by a new method ;"
and Dr. Abraham Robertson, professor of Oxford, whose
work was published in the Latin language in 1792. Dr.
Hamilton, in treating on this subject, explains the nature of
the conic surface in such a manner, that, as soon as the seve-
ral curves are defined by the intersection of a plane with this
surface, it appears what their principal properties are. " Ac-
cordingly," he says, " before the sections are defined, certain
properties of the Conic Sections are demonstrated, which
contain in them most of the fundamental properties of the
Sections. Then, since it appears from their definitions, that
these sections are curves, all the points of which are placed
on a conic surface, it is manifest that every right line which
any way meets these sections, must, in the same manner, and
in the same points, meet the conic surface ; and, therefore, all
the properties which are proved to agree with right lines
meeting the conic surface, are immediately transferred to
those which meet the Conic Sections. Thus the principal
and most general properties of the sections being laid down
in the beginning, he expects that their particular properties
will be more easily deduced.

Besides these, we have, as has been already observed, many
treatises on the Conic Sections, in which the cone is entirely



CONIC SECTIONS. 499

laid aside, and the curves are given and defined from descrip-
tions in piano. M. De la Hire, in his treatise entitled
*' Nouveaux Elemens des Sections Coniques/*" published in
Paris in 1679, is the first author who successfully treated on
Conic Sections in this new view of the subject. " He derives
his description of the parabola from the equality that subsists
between two lines meeting in any point of the curve, one of
which is drawn to the focus, and the other is perpendicular to
the directrix ; and he describes the ellipse and hyperbola from
the analogous properties, that in the fornjer, the sum of two
lines drawn from any point in the curve to the two foci, and,
in the latter, the difference of two such lines, are equal to
their transverse axis." In Uiese fundamental properties, De
la Hire is followed by most of the later writers, who have
treated of the Conic Sections independently of the cone ;
and hi particular by Dr. Robert Simson, of Glasgow, who
published in 1735, an extensive treatise on this subject, en-
titled " Sectionum Conicarum," lib. v. a work which is en-
titled to high commendation, on account of its elegance and
geometrical accuracy. An English translation of the first
three book* of this work, in 8vo. was published in London in
1775, and again in Glasgow in 1804, entitled " Elements of
the Conic Sections."

Besides the method of De la Hire, another way of defining
curves in piano has been proposed, founded on a general pro-
perty of the directrix, first discovered, or at least given, by
Pappus, viz. the ratio that subsists between the distances of
every point in the curve from the focus and the directrix.
The Abbe Boscovich has drawn his definitions of the curves
from this fundamental property, in his treatise entitled " Ele-
menta Matheseos Universae ;" and we have a work in our own
language, which is founded on the same primary definitions,
viz. " A short treatise on the Conic Sections, in which the
three curves are derived from a general description on a
plane, and the most useful properties of each deduced from
a common principle/' by the Rev. T. Newtou, This is an

2 K 2



500 MATHEMATICS.

admirable little performance, and may be recommended to
those students who wish to get a sufficient insight into the
subject without much expense of money or time. The de-
monstrations are neat, and frequently elegant. The author
has compressed into about a hundred pages all those proposi-
tions with which every one ought to be acquainted, previously
to his entering on the Principia of Newton, and the other
branches of natural philosophy; and he has taken care that
the demonstrations should be strictly geometrical, and siich as
the young student will find no difficulty in understanding, pro-
vided he be well acquainted with the Elements of Euclid,
and plane trigonometry. It should be observed also, tliat
!Pr. Hamilton intended the work, described above, as an In>
troduction to the Newtonian philosophy ; and he gives fre-
quent references to certain propositions in the Principia.
With a similar view, the following work, still more concise
than that of Mr. Newton, was published, of which the first
edition was entitled " The Elements of the Conic Sections,
as preparatory to the reading of Sir Isaac Newton's Principia,"
by the Rev. S. Vince. In the subsequent editions, Dr. Vince
has dropped the latter part of the title, and, in its stead, has
inserted " Adapted to the Students in Philosophy." It will
be found an useful treatise, as introductory to larger works.

" Elements of Conic Sections, in three books," &c. by
Richard Jack, we remember to have read many years ago
with satisfaction.

The propositions of Trevigar's work on this subject in
L^tin, and those in Steel's, in English, are demonstrated al-
gebraically. Besides the above works, there are many others,
viz. one by Milne, in Latin, others by Muller and Emerson,
and one of the latest by Dr. Hutton, adapted chiefly to the
students at Woolwich. By some modems likewise, the sub-
ject has been treated of in a very elaborate manner, as by
Euler, Prony, and Lacroix, who have inferred the chief pro-
perties of the Conic Sections from the different modifications
pf the general algebraic equations of lines of the second order.



FLUXIONS. 501

and the establislied analogies between the properties of equa-
tions, and those of curves.

FLUXIONS.

To forlh a proper idea of the nature of fluxions, all kinds
of magnitudes are to be considered as generated by the conti-
nual motion of some of their bounds or extremes, as a line
by the motion of a point, a surface by the motion of a line,
and a solid by the motion of a surface. Every quantity so
generated is called a variable, or flowing quantity ; and a
fluxion may be defined as the magnitude by which any flow-
ing quantity would be uniformly increased in a given portion
of time, with the generating celerity at any particular instant,
supposing it from thence to continue invariable, which is thus
explained by Mr. Thbnias Simpson, in his work entitled
" The Doctrine and Application of Fluxions :" Thus, let
the point m, plate 1, fig. 1, be conceived to move from A,
and generate the variable right line Aw, by a motion any how
regulated ; and let the celerity thereof, when it arrives at any
proposed position R, be such as would, was it to continue
uniform from that point, be sufficient to describe the distance,
or line Rr, in the given time allotted for the fluxion ; then
will Rr be the fluxion of the variable hne Am, in that po-
sition.

The fluxion of a plane surface is conceived in like manner,
by supposing a given right line mti, fig. 2, to move parallel to
itself, in the plane of the parallel^ and immovable lines AF
and BG ; for if Rr be taken to express the fluxion of the
line Am, and the rectangle RrsS be completed, then that
rectangle, being the space which would be uniformly described
by the generating line mn, in the time that Am would be uni-
formly increased by mr, is therefore the fluxion of the gene-
rated rectangle Bw, in that position, according to the true
meaning of the definition.

If the length of the generating line mn, fig. 3, continually
varies, the fluxion of the area will still be expounded by a



502 MATHEMATICS.

rectangle under that line, and the fluxion of the abscissa, or
base : for let the curvilineal space Amn be generated by the
continual and parallel motion of the (now) variable line mn,
and let Rr be the fluxion of the base, or abscissa, Am (as be-
fore) ; then the rectangle RrsS will, here also, be the fluxion
of the generated space Amn; because, if the length and velo-
city of the generating line mn were to continue invariable
from the position RS, the rectangle RrsS would then be uni-
formly generated, with the very celerity wherewith it begins
to be generated, or with which the space Amn is increased in
that position.

From what has been hitherto said, it will appear, that the
fluxions of quantities are always in proportion to the celeri-
ties hy which the quantities themselves increase in magnitude;
M'hence it will not be difhcult to form 9 notion of the fluxions
of quantities otherwise generated, as well such as arise from
the revolution of right lines and planes, as those by parallel
motions.

In the application of algebra to the theory of curve lines,
we find that some of the quantities which are the subject of
consideration, may be conceived as having always the same
magnitude as the diameter of a circle ; the parameter of a
parabola ; and the axis of an ellipse, or hyperbola ; while
others are indefinite in respect of magnitude, and may have
any number of particular values ; such are the ordinates, &c.
of a curve line. This difference in the nature of the quanti-
ties which are compared together, takes place in other theo-
ries, and on other subjects, in pure and mixed mathematics,
which suggests the division of all quantities whatever, into
those that are constant, and those that are variable.

A constant quantity is that which retains always the same
magnitude, although other quantities with which it is con-
nected, may be supposed to change ; and a variable quantity
is. that which is indefinite in respect to magnitude, according
to its position. Thus the radius of a circle is a constant
quantity, while the sine, co-sine, tangent, secant, &c. of an



FLUXIONS. 503

Urc, are variable quantities, depending upon the magnitude of
the arc for their relative value. In Conic Sections likewise,
the axes, and the parameters of the axes, are constant quanti-
ties, and the abscissas and ordiuates are variable quantities.
To determine the values of these variable quantities, the doc-
trine of fluxions, invented by the illustrious Newton, when he
was only twenty-three years of age, is perfectly adapted ; and,
by this doctrine, as it has been explained and illustrated by
various authors, m^ny difficulties, insurmountable by any other
known method, are solved with expedition, ease, and elegance.
We shall just give our reader an insight into the science, by
shewing, in a few of the simplest cases, which the young al-
gebraist will readily understand, how fluxions are adapted for
determining the maxima and minima of bodies for drawing
tangents to curves, &c. by which he will readily understand
in what way the science may be made to extend to the inves-
tigation of the most abstruse and difficult problems in the
various branches of mathematics and natural philosophy.

Constant quantities are usually denoted by the early letters
of the alphabet, as a, b, c, &c. and those that are variable,
by the latter letters, as u, w, x, y, and z. The diameter of
a given circle may be represented by a, and the sine, tangent,
&c. of an arc or angle of it, by ar.

The fluxion of a quantity represented by a single letter, is
usually expressed by the same letter with a dot over it; thus
the fluxion of x is represented by x, of y by y, of z by z,
and so on.

The rules for finding the fluxions of any flowing quantities
are as follow, x being the fluxion of x.

1 . To find the fluxion of a given fluent, in which there is
but one variable quantity. Rule. Mark the letter that re-
presents the variable quantity with a dot over it, and you
have the fluxion required. Thus the fluxion of ax is a x.
For, if we suppose the variable rectangle AS, fig. 2, to be
generated by the given line RS, setting out from the situation



504 MATHEMATfCS.

AB, and moving along \vith a parallel motion between tfi
parallel and indefinite lines BG and AF, it is evident that the
velocity with which the rectangle flows, is equal to the gene-
rating line RS multiplied into velocity with which the point
R moves along the line AF j that is, tlie fluxion of the rect-
angle AS is equal to the invariable line RS, multiplied into
the fluxion of the variable quantity AR. Therefore, if AB,
or its equal RS, be denoted by a, and AR, or BS x, then
thef fluxion of the rectangle a t, will be equal a x x-=.a x.
2. To find the fluxion of the product of two or more flow-
ing quantities multiplied into each other. Rule. Multiply
the fluxion of each quantity separately, by the other, or the
product of the rest of the quantities ; and the sum of these
products will be the fluxion required. Thus the fluxion of
X yrrx y + x y; of x y z=:x yz + xyz + xyz, &c. The
reason of this and the following rules, is given in all the intro-
ductory books to the science, to several of which we shall
presently direct the attention of the reader.

3. To find the fluxion of a fraction. Rule. Multiply the
denominator into the fluxion of the numerator; from the pro-
duct of which, subtract the numerator multiplied into the
fluxion of the denominator ; then divide the remainder by the
square of the denominator, and the result is the fluxion of the

X XV XV

fraction required. Thus the fluxion of - =. 1 ; and

X- G X xy' 3 x'^ y^ y_2 X X y 3 x' y
the fluxion of -r= 1 ~i .

yS y6 y*

The following examples will shew the application of flux-
icms to the solution of problems de maximis et minimis, and
to Ae drawing tangents to curves.

Ex. 1. To divide a given right line A B in two such parts,
A C and B C, that the rectangle of their parts may be the
greatest possible.

Suppose the whole line A B
- the variable part equal x
then the other part will be a x



J'LUXIONS. 505

of course the rectangle required will be a x x xr:a tx',
of which the fluxion is ox 2 x x which being put equal to

o ; we have a\ zz 2 x x or a =: 2 x and x ^ . Therefore

2"

the rectangle will be the greatest possible, when the line is
equally divided.

2. To determine the greatest rectangle that can be inscrib-
ed in a given triangle. See fig. 4.

Put the Base AC of thfe given Triangle b, and its Alti-
tude BD =. a ; and let the Altitude (BS) of the inscribed
Rectangle mc (considered as variable) be denoted by x: Then,
because of the parallel Lines AC, and ac, it will be as BD

(fl) : AC (6) : : DS (a j) : ^^~^^ zzac : Whence the Area

of the Rectangle, or ac x BS will be= ^^^~^^^ : Whose

a

Fluxion '^ being put = o, we shall get x^z^a.

Whence the greatest inscribed Rectangle is that whose Alti-
tude is just half the Altitude of the Triangle.

A tangent is a right line which coincides with a curve in a
point, and there shews its direction, that is, the inclination
which it bears to the axis, or the angle it makes with the
ordinate. Now, in general, what is requisite in order to
draw a tangent to any point, is to find the right line, called
the sub-tangent, or the distance of the point from the ordinate
through which the tangent must pass.

Ex. I. To draw a Tangent to a Circle. See fig. 5.

Put the radius EA or ED a, absciss AC zz x, and
ordinate C B r= 3/ ; then, CD = '2 a x. Now, by 35
E. 3. A C X C D z: C B^, that is, Q a xx"^ =f ; and
this equation put into Fluxions ie 2 ax 2xa: 2^y;

which, divided by 2 a 2 x, makes x =. V V ; which

a X

substituted for x in the general expression found by other means,



606 MATHEMATICS.

for die Subtaiigent, viz. \^' makes the Subtangent C T =

n

-^ (which, by writing lax x" for its above value, viz.
a X

V', is) = ~^ = \-7^' Wherefore, if the distance

a X LC

signified by this expression be set off from tlie poirvt C, in
the diameter D A produced, we shall have the point T
through which the Tangent to the point B must pass.

Construction. Through the point B describe the semi-
circle E B T : tlien will T be the point from which the Tan-
gent to tlie point B is to be drawn. For, by 3 1 E. 3. the ai^le
E B T w ill be right ; and therefore, by 8 E. 6. the triangles
EC B and B CT will be similar, and by 4E. 6. EC : C B

: :BC:CT; -.CT^^l

EC

Ex. II. To draw a Tangent to a Parabola. See Fig. 6.

Suppose F to be the focus ; and P R the parameter, which
put = a ; also, put the absciss A C z= x, and ordinate
C B = ^., Now, by a well known property of the curve,



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 41 of 44)