# HYPERBOLA

, one of the conic sections, being that which is made by a plane cutting a cone so, that, entering one side of the cone, and not being parallel to the opposite side, it may cut the circular base when the opposite side is ever so far produced below the vertex, or shall cut the opposite side of the cone produced above the vertex, or shall make a greater angle with the base than the opposite side of the cone makes; all these three circumstances amounting to the same thing, but in other words.

1. Thus, the figure DAE is an Hyperbola, made by a plane entering the side VQ of a cone PVQ at A, and either cutting the bafe PEQ when the plane is not parallel to VP, and this is ever so far produced; or when the angle ARQ is greater than the angle VPQ; or when the plane cuts the opposite side in B above the vertex.

2. By the Hyperbola is sometimes meant the whole plane of the section, and sometimes only the curve line of the section.

3. Hence, the cutting plane
meets the opposite cone in B,
and there forms another Hyperbola
*d* B *e,* equal to the former
one, and having the same
transverse axis AB; and the same vertices A and B.
Also the two are called Opposite Hyperbolas.

4. The centre C is the middle point of the tranverse axis.

5. The semi-conjugate axis is CL, a mean proportional between CI and CK, the distances to the sides of the opposite cone, when CI is drawn parallel to the diameter PQ of the base of the cone. Or the whole conjugate axis is a mean proportional between AF and BH, which are drawn parallel to the base of the cone.

6. If DAE and FBG be two opposite Hyperbolas,
having the same transverse and conjugate axes AB and
*a b,* perpendicularly bisecting each other; and if *d a e*
and *f b g* be two other opposite Hyperbolas, having the
same axes with the two former, but in the contrary
order, viz, having *a b* for their first or transverse axis,
and AB for their second or conjugate axis: then any
two adjacent curves are called Conjugate Hyperbolas,
and the whole figure formed by all the four curves, the
Figure of the Conjugate Hyperbolas. And if the rectangle
HIKL be inscribed within the four conjugate
Hyperbolas, touching the vertices A, B, *a, b,* and
having their sides parallel and equal to the two
axes; and if then the two diagonals HCK, ICL, of
the parallelogram be drawn, these diagonals are the
asymptotes of the curves, being lines that continually
approach nearer and nearer to the curves, without
meeting them, except at an infinite distance, where
each asymptote and the two adjacent sides of the two
conjugate Hyperbolas may be supposed all to meet; the
asymptote being there a common tangent to them both,
viz, at that infinite distance.

7. Hence the four Hyperbolas, meeting and running into each other at the infinite distance, may be cons<*>dered as the four parts of one entire curve, having the same axes, tangents, and other properties.

8. A Diameter in general, is any line, as MN,
drawn through the centre C, and meeting, or termi-|
nated by the opposite legs of the opposite Hyperbolas.
And if parallel to this diameter there be drawn
two tangents, at *m* and *n,* to the opposite legs of the
other two opposite Hyperbolas, the line *m*C*n* joining
the points of contact, is the conjugate diameter to MN,
and the two mutually conjugates to each other. Or, if
to the points M or N there be drawn a tangent, and
through the centre C the line *mn* parallel to it, that
line will be the conjugate to MN. The points where
each of these meet the curves, as M, N, *m, n,* are the
vertices of the diameters; and the tangents to the
curves at the two vertices of any diameter, are parallel
to each other, and also to the other or conjugate
diameter.

9. Moreover, if those tangents to the four Hyperbolas, at the vertices of two conjugate diameters, be produced till they meet, they will form a parallelogram OPQR; and the diagonals OQ and PR of the parallelogram will be the asymptotes of the curves; which therefore pass through the opposite angles of all the parallelograms so inscribed between the curves. Also it is a property of these parallelograms, that they are all equal to each other, and therefore equal to the rectangle of the two axes; as will be farther noticed below. Farther, if these diagonals or asymptotes make a right angle between them, or if the inscribed parallelogram be a square, or if the two axes be equal to each other, then the Hyperbola is called a right-angled or an equilateral one.

10. An Ordinate to any diameter, is a line drawn parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter produced and the curve. So MS and TN are ordinates to the axis AB; also AD and BG are ordinates to the diameter MN (last fig. but one). Hence the ordinates to the axis are perpendicular to it; but ordinates to the other diameters are oblique to them.

11. An absciss is a part of any diameter, contained between its vertex and an ordinate to it; and every ordinate has two abscisses: as AT and BT, or MV and NV.

12. The Parameter of any diameter, is a third proportional
to the diameter and its conjugate.—The Parameter
of the axis is also equal to the line AG or B*g*
(fig. 1), if FG be drawn to make the angle AFG =
the angle BAV, or the line H*g* to make the angle
BH*g* = the angle ABV.

13. The Focus is the point in the axis where the ordinate
is equal to half the parameter of the axis; as
S and T (fig. 2) if MS and TN be half the parámeter,
or the 3d proportional to CA and C*a.* Hence there
are two Foci, one on each side the vertex, or one for
each of the opposite Hyperbolas. These two points in
the axis are called Foci, or burning points, because it is
found by opticians that rays of light issuing from one of
them, and falling upon the curve of the Hyperbola,
are reflected into lines that verge towards the other
point or Focus.

*To describé an Hyperbola, in various ways.*

14. (1st *Way by points.)*—In the transverse axis AF
produced, take the foci F and *f,* by making CF and
C*f* = A*a* or B*a,* assume any point I: Then with the
radii AI, BI, and centres F, *s,* describe arcs intersecting
in E, which will give four points in the curves.
In like manner, assuming other points I, as many other
points will be found in the curve. Then, with a steady
hand, draw the curve line through all the points of
intersection E.—In the same manner are to be constructed
the other pair of opposite Hyperbolas, using
the axis *ab* instead of AB.

15. (2d *Way by points, for a Right-angled Hyperbola
only.)*—On, the axis produced if necessary, take any
point I, through which draw a perpendicular line,
upon which set off IM and IN equal to the distance I*a*
or I*b* from I to the extremities of the other axis; and
M and N will be points in the curve.

16. (3d *Way by points, to describe the curve through
a given point.)*—CG and CH being the asymptotes,
and P the given point of the curve; through the
point P draw any line GPH between the asymptotes,
upon which take GI = PH, so shall I be another
point of the curve. And in this manner may any
number of points be found, drawing as many lines
through the given point P.|

17. (4th *Way by a continued Motion.)*—If one end
of a long ruler *f*MO be fastened at the point *f,* by a
pin on a plane, so as to turn freely about that point as
a centre. Then take a thread FMO, shorter than the
ruler, and fix one end of it in F, and the other to the
end O of the ruler. Then if the ruler *f*MO be turned
about the fixed point *f,* at the same time keeping the
thread OMF always tight, and its part MO close to
the side of the ruler, by means of the pin M; the curve
line AX described by the motion of the pin M is one
part of an Hyperbola. And if the ruler be turned, and
move on the other side of the fixed point F, the other
part AZ of the same Hyperbola may be described after
the same manner.—But if the end of the ruler be sixed
in F, and that of the thread in *f,* the opposite Hyperbola
*xaz* may be described.

18. (5th *Way, by a continued Motion.)*—Let C and
F be the two foci, and E and K the two vertices of
the Hyperbola. (See the last fig. above.) Take three
rulers CD, DG, GF, so that CD = GF = EK, and
DG = CF; the rulers CD and GF being of an indefinite
length beyond C and G, and having slits in them
for a pin to move in; and the rulers having holes in
them at C and F, to fasten them to the foci C and F
by means of pins, and at the points D and G they are to
be joined by the ruler DG. Then, if a pin be put in
the slits, viz, the common intersection of the rulers CD
and GF, and moved along, causing the two rulers GF,
CD, to turn about the foci C and F, that pin will describe
the portion E*e* of an Hyperbola.—The foregoing
are a few among various ways given by several authors.

*Some of the chief Properties os the Hyperbola.*

19. (1st) The squares of the ordinates, of any diameter, are to each other, as the rectangles of their abscisses; i. e. .

20. As the square of any diameter, is to the square of its conjugate; so is the rectangle of two abscisses, to the square of their ordinate. That is, .

Or, because the rectangle AD . BD is = the difference
of the squares CD^{2} - CB^{2}, the same property
is,
,
Or ,

That is ,
where *p* is the parameter of the diameter AB, or the
3d proportional *ab*^{2}/(AB).

And hence is deduced the common equation of the
Hyperbola, by which its general nature is expressed.
Thus, putting *d* = the semidiameter CA or CB,
*c* = its semiconjugate C*a* or C*b,*
*p* = its parameter or 2*d*^{2}/*c,*
*x* = the absciss BD from the vertex,
*y* = the ordinate DE, and
*v* = the absciss CD from the centre:

Then is | , |

or | , |

or | , |

or | ; so that |

*d*

^{2}to

*c*

^{2}, or of

*d*to

*p,*exceeds that of

*2dx*to

*y*

^{2}; that ratio being equal in the parabola, and defective in the ellipse, from which circumstances also these take their names.

21. The distance between the centre and the focus,
is equal to the distance between the extremities of the
transverse and conjugate axes. That is, CF = A*a* or
A*b,* where F is the focus.

22. The conjugate semi-axis is a mean proportional
between the distances of the focus from both vertices of
the transverse. That is, C*a* is a mean between AF
and BF, or , or .

23. The difference of two lines drawn from the
foci, to meet in any point of the curve, is equal to the
transverse axis. That is, *f*E - FE = AB, where
F and *f* are the two foci.

24. All the parallelograms inscribed between the
four conjugate Hyperbolas are equal to one another,
and each equal to the rectangle of the two axes. That
is, the parallelogram OPQR = AB . *ab* (fig. to
art. 9).

25. The difference of the squares of every pair of
conjugate diameters, is equal to the same constant
quantity, viz, the difference of the squares of the two
axes. That is, , (fig. to
art. 6)<*> where MN and *mn* are any two conjugate
diameters.|

26. The rectangles of the parts of two parallel lines, terminated by the curve, are to one another, as the rectangles of the parts of any other two parallel lines, any where cutting the former. Or the rectangles of the parts of two intersecting lines, are as the squares of their parallel diameters, or squares of their parallel tangents.

27. All the rectangles are equal which are made of
the segments of any parallel lines, cut by the curve, and
limited by the asymptotes, and each equal to the square
of their parallel diameter. That is, HE . EK or
or CP^{2}.

28. All the parallelograms are equal, which are formed
between the asymptotes and curve, by lines parallel
to the asymptotes. That is, the paral. CGEK =
CPBQ.—Hence is obtained another method of expressing
the nature of the curve by an equation, involving
the absciss taken on one asymptote, and ordinate
parallel to the other asymptote. Thus, if *x* =
CK, *y* = KE, *a* = CQ, and *b* = BQ the ordinate
at the vertex B of the curve; then, by the property in
this article, *ab* = *xy,* or ; that is, the
rectangle of the absciss and ordinate is every where of
the same magnitude, or any ordinate is reciprocally as
its absciss.

29. If the abscisses CQ, CK, CL, &c, taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates QB, KE, LM, &c, parallel to the other asymptote, be a like geometrical progression in the same ratio, but decreasing; and all the rectangles are equal, under every absciss and its ordinate, viz, , &c.

30. The abscisses CQ, CK, CL, &c, being taken in geometrical progression; the spaces or asymptotie areas BQKE, EKLM, &c, will be all equal; or, the spaces BQKE, BQLM, &c, will be in arithmetical progression; and therefore these spaces are the hyperbolic logarithms of those abscisses.

These, and many other curious properties of the Hyperbola, may be seen demonstrated in my Treatise on Conic Sections, and several others. See also Conic Sections.

*Acute* Hyperbola, one whose asymptotes make an
acute angle.

*Ambigenal* Hyperbola, is that which has one of
its infinite legs falling within an angle formed by the
asymptotes, and the other falling without that angle.
This is one of Newton's triple Hyperbolas of the 2d
order. See his Enumeratio Lin. tert. Ord. See also
Ambigenal.

*Common,* or *Conic* Hyperbola, is that which arises
from the section of a cone by a plane; called also the
Apollonian Hyperbola, being that kind treated on by
the first and chief author Apollonius.

*Conjugate* Hyperbolas, are those formed or lying
together, and having the same axes, but in a contrary
order, viz, the transverse of each equal the conjugate
of the other; as the two Conjugate Hyperbolas *Pee*
and EEE in the last figure but one.

*Equilateral,* or *Rectanglar* Hyperbola, is that
whose two axes are equal to each other, or whose
asymptotes make a right angle.—Hence, the property
or equation of the equilateral Hyperbola, is , where *a* is the axis, *x* the absciss, and *y* its
ordinate; which is similar to the equation of the circle,
viz, , differing only in the sign of the
second term, and where *a* is the diameter of the
circle.

*Infinite* Hyperbolas, or Hyperbolas *of the higher
kinds,* are expressed or defined by general equations
similar to that of the conic or common Hyperbola,
but having general exponents, instead of the particular
numeral ones, but so as that the sum of those on one
side of the equation, is equal to the sum of those on
the other side. Such as, ,
where *x* and *y* are the absciss and ordinate to the axis
or diameter of the curve; or , where the
absciss *x* is taken on one asymptote, and the ordinate
*y* parallel to the other.

As the Hyperbola of the first kind, or order, viz the conic Hyperbola, has two asymptotes; that of the 2d kind or order has three; that of the 3d kind, four; and so on.

*Obtuse* Hyperbola, is that whose asymptotes form
an obtuse angle.

*Rectangular* Hyperbola, the same as Equilateral
Hyperbola.

Hyperbolic *Arc,* is the arc of an Hyperbola.

Put *a* = CA the semitransverse
axe, *c* = C*a* the semiconjugate,
*y* = an ordinate PQ to the axe
drawn from the end Q of the
arc AQ, beginning at the vertex
A: then putting , &c;
then is the length of the arc AQ expressed by
&c;|
or by , nearly; where
*t*
is the whole transverse axe 2CA, *c* = 2C*a* the conjugate,
*x* = AP the absciss, and *y* = PQ the ordinate.

These and other rules may be seen demonstrated in my Mensuration, p. 408, &c, 2d edit.

Hyperbolic *Area,* or *Space,* the area or space included
by the Hyperbolic curve and other lines.

Putting *a* = CA the semitransverse, *c* = C*a* the semiconjugate,
*y* = PQ the ordinate, and *v* = CP its
distance from the centre; then is the
area ;
sector ;
area ; or
nearly.

Let CT and CE be the two asymptotes, and the ordinates DA, EF parallel to the other asymptote CT; then the asymptotic space ADEF or sector CAF is or or &c; and this last series was first given by Mercator in his Logarithmotechnia.

See my Mensuration, p. 413, &c, 2d edit.

Generally, if be an equation expressing
an Hyperbola of any order; then its asymptotic
area will be ; which space therefore is always
quadrable, in all the orders of Hyperbolas, except the
first or common Hyperbola only, in which *m* and *n*
being each 1, the denominator *n* - *m* becomes 0 or
nothing.

Hyperbolic *Conoid,* a solid formed by the revolution
of an Hyperbola about its axis, otherwise called an
Hyperboloid.

*To find the Solid Content of an Hyperboloid.*

Let AC be the semitransverse of the generating Hyperbola, and AH the height of the solid; then as 2AC + AH is to 3AC + AH, so is the cone of the same base and altitude, to the content of the Conoid.

*To find the Curve Surface of an Hyperboloid.*

Let AC be the semitransverse, and AB perpendicular to it, and equal to the semiconjugate of ADE the generating Hyperbola, or section through the axis of the solid. Join CB; make CF = CA, and on CA let fall the perpendicular FG; then with the semitransverse CG, and semiconjugate GH = AB, describe the Hyperbola GIK; then as the diameter of a circle is to its circumference, so is the Hyperbolic frustum ILAMK to the curve surface of the Conoid generated by DAE. See my Mensur. p. 429, &c, 2d edit.

Hyperbolic *Cylindroid,* a solid formed by the revolution
of an Hyperbola about its conjugate axis, or
line through the centre perpendicular to the transverse
axis. This solid is treated of in the Philos. Trans.
by Sir Christopher Wren, where he shews some of
its properties, and applies it to the grinding of Hyperbolical
Glasses; affirming that they must be formed
this way, or not at all. See Philos. Trans. vol. 4,
pa. 961.

Hyperbolic *Leg,* of a curve, is that having an
asymptote, or tangent at an infinite distance.—Newton
reduces all curves, both of the first and higher kinds,
into Hyperbolic and parabolic legs, i. e. such as have
asymptotes, and such as have not, or such as have tangents
at an infinite distance, and such as have not.

Hyperbolic *Line,* is used by some authors for what
is more commonly called the Hyperbola itself, being the
curve line of that figure; in which sense the surface
terminated by it is called the Hyperbola.

Hyperbolic *Logarithm,* a logarithm so called as
being similar to the asymptotic spaces of the Hyperbola.
The Hyperbolic logarithm of a number, is to
the common logarithm, as 2.3025850929940457 to 1,
or as 1 to .4342944819032518. The first invented
logarithms, by Napier, are of the Hyperbolic kind;
and so are Kepler's. See Logarithm.

Hyperbolic *Mirror,* is one ground into that
shape.

Hyperbolic *Space,* that contained by the curve
of the Hyperbola, and certain other lines. See HYPERBOLIC
Area.

HYPERBOLICUM *Acutum,* a solid made by the
revolution of the infinite area or space contained between
the curve of the Hyperbola, and its asymptote.
This produces a solid, which though infinitely long and
generated by an infinite area, is nevertheless equal to
a finite solid body; as is demonstrated by Torricelli,
who gave it this name.

HYPERBOLIFORM *Figures,* are such curves as
approach, in their properties, to the nature of the Hyperbola;
called also Hyperboloides.