Are curve lines formed by the intersections of a cone and plane.

If a cone be cut by a plane through the vertex, the section will be a triangle ABC, Plate LXXIV. fig. 1. If a cone be cut by a plane parallel to its base, the section will be a circle. If it be cut by a plane DEF, fig. 1, in such a direction, that the side AC of a triangle passing through the vertex, and having its base BC perpendicular to EF, may be parallel to DP, the section is a parabola; if it be cut by a plane DR, fig. 2, meeting AC, the section is an ellipse; and if it be cut by a plane DMO, fig. 3, which would meet AC extended beyond A, it is an hyperbola.

If any line HG, fig. 1, be drawn in a parabola perpendicular to DP, the square of HG will be to the square of EP, as DG to DP; for let LHK be a section parallel to the base, and therefore a circle, the rectangle LGK, will be equal to the square of HG, and the rectangle BPC equal to the square of EP; therefore these squares will be to each other as their rectangles; that is, as BP to LG, that is DP to DG.

Sect. I. Description of Conic Sections on a Plane.

1. PARABOLA.

Let AB, fig. 4, be any right line, and C any point without it, and DKF a ruler, which let be placed in same plane in which the right line and point are, in such a manner that one side of it, as DK, be applied to the right line AB, and the other side KF coincide with the point C; and at F, the extremity of the side KF, let be fixed one end of the thread FNC, whose length is equal to KF, and the other extremity of it at the point C, and let part of the thread, as FG, be brought close to the side KF by a small pin G; then let the square DKF be moved from B towards A, so that all the while its side DK be applied close to the line BA, and in the mean time the thread being extended will always be applied to the side KF, being kept from going from it by means of the small pin; and by the motion of the small pin N there will be described a certain curve, which is called a semi-parabola.

And if the square be brought to its first given position, and in the same manner be moved along the line AB, from B towards H, the other semi-parabola will be described.

The line AB is called the directrix; C, the focus; any line perpendicular to AB, a diameter; the point where it meets the curve, its vertex; and four times the distance of the vertex from the directrix, its latus rectum or parameter.

2. ELLIPSE.

If any two points, as A and B, fig. 5, be taken in any plane, and in them are fixed the extremities of a thread, whose length is greater than the distance between the points, and the thread extended by means of a small pin C, and if the pin be moved round from any point until it return to the place from whence it began to move, the thread being extended during the whole time of the revolution, the figure which the small pin by this revolution describes is called an ellipse.

The points AB are called the foci; D, the centre; EF, the transverse axis; GH, the lesser axis; and any other line passing through D, a diameter.

3. HYPERBOLA.

If to the point A, fig. 6, in any plane, one end of the rule AB be placed, in such a manner, that Plate about that point, as a centre, it may freely move; and if to the other end B, of the rule AB, be fixed the extremity of the thread BDC, whose length is smaller than the rule AB, and the other end of the thread being fixed in the point C, coinciding with the side of the rule AB, which is in the same plane with the given point A; and let part of the thread, as BD, be brought close to the side of the rule AB, by means of a small pin D; then let the rule be moved about the point A, from C towards T, the thread all the while being extended, and the remaining part coinciding with the side of the rule being kept from going from it by means of the small pin, and by the motion of the small pin D, a certain figure is described which is called the semi-hyperbola.

The other semi-hyperbola is described in the same way, and the opposite HKF, by fixing the ruler to C, and the thread to A, and describing it in the same manner, A and C are called foci; the point G, which bisects AC, the centre; KE, the transverse axis; a line drawn through the centre meeting the hyperbolas, a transverse diameter; a line drawn through the centre, perpendicular to the transverse axis, and cut off by the circle MN, whose centre is E, and radius equal to CG, is called the second axis.

If a line be drawn through the vertex E, equal and parallel to the second axis GP and GO be joined, they are called asymptotes. Any line drawn through the centre, not meeting the hyperbolas, and equal in length to the part of a tangent parallel to it, and intercepted betwixt the asymptotes, is called a second diameter.

An ordinate to any section is a line bisected by a diameter and the abscissa, the part of the diameter cut off by the ordinate.

Conjugate diameters in the ellipse and hyperbola are such as mutually bisect lines parallel to the other; and a third proportional to two conjugate diameters is called the latus rectum of that diameter, which is the first in the proportion.

In the parabola, the lines drawn from any point to the focus are equal to perpendiculars to the directrix; being both equal to the part of the thread separated from the ruler.

In the ellipse, the two lines drawn from any point in the curve to the foci are equal to each other, being equal to the length of the thread; they are also equal to the transverse axis. In the hyperbola the difference of the lines drawn from any point to the foci is equal, being equal to the difference of the lengths of the ruler and thread, and is equal to the transverse axis.

From these fundamental properties all the others are derived.

The ellipse returns into itself. The parabola and hyperbola may be extended without limit.

Every line perpendicular to the directrix of a parabola meets it in one point, and falls afterwards within it; and every line drawn from the focus meets it in one point, and falls afterwards without it. And every line that passes through a parabola, not perpendicular to the directrix, will meet it again, but only once.

Every line passing through the centre of an ellipse is is bisected by it; the transverse axis is the greatest of all these lines; the lesser axis the least; and these nearer the transverse axis greater than those more remote.

In the hyperbola, every line passing through the centre, is bisected by the opposite hyperbola, and the transverse axis is the least of all these lines; also the second axis is the least of all the second diameters. Every line drawn from the centre within the angle contained by the asymptotes, meets at once, and falls afterwards within it; and every line drawn through the centre without that angle never meets it; and a line which cuts one of the asymptotes, and cuts the other extended beyond the centre, will meet both the opposite hyperbolas in one point.

If a line GM, fig. 4. be drawn from a point in a parabola perpendicular to the axis, it will be an ordinate to the axis, and its square will be equal to the rectangle under the abscissa MI and latus rectum; for, because GMC is a right angle, GM² is equal to the difference of GC² and CM²; but GC is equal to GE, which is equal to MB; therefore GM² is equal to BM² - CM²; which, because CI and IB are equal, is (8 Euc. 2.) equal to four times the rectangle under MI and IB, or equal to the rectangle under MI and the latus rectum.

Hence it follows, that if different ordinates be drawn to the axis, their squares being each equal to the rectangle under the abscissa and latus rectum, will be to each other in the proportion of the abscissas, which is the same property as was shewn before to take place in the parabola cut from the cone, and proves those curves to be the same.

This property is extended also to the ordinates of other diameters, whose squares are equal to the rectangle under the abscissas and parameters of their respective diameters.

In the ellipse, the square of the ordinate is to the rectangle under the segments of the diameter, as the square of the diameter parallel to the ordinate to the square of the diameter to which it is drawn, or as the first diameter to its latus rectum; that is, LK² fig. 5. is to EKF as EF² to GH².

In the hyperbola, the square of the ordinate is to the rectangle contained under the segments of the diameters betwixt its vertices, as the square of the diameter parallel to the ordinate to the square of the diameter to which it is drawn, or as the first diameter to its latus rectum; that is, SX² is to EXK as MN² to XE².

Or if an ordinate be drawn to a second diameter, its square will be to the sum of the squares of the second diameter, and of the line intercepted betwixt the ordinate and centre, in the same proportion; that is, RZ² fig. 6. is to ZG² added to GM², as KE² to MN². These are the most important properties of the conic sections: and, by means of these, it is demonstrated, that the figures are the same described on a plane as cut from the cone; which we have demonstrated in the case of the parabola.

**Sect. II. Equations of the Conic Sections**

Are derived from the above properties. The equation of any curve, is an algebraic expression, which denotes the relation betwixt the ordinate and abscissa; Plate the abscissa being equal to x, and the ordinate equal to y.

If p be the parameter of a parabola, then \( y^2 = px \); which is an equation for all parabolas.

If a be the diameter of an ellipse, p its parameter; then \( y^2 : ax - xx : : p : a \); and \( y^2 = \frac{p}{a} \times ax - xx \); an equation for all ellipses.

If a be a transverse diameter of a hyperbola, p its parameter; then \( y^2 : ax + xx : : p : a \), and \( y^2 = \frac{p}{a} \times ax + xx \).

If a be a second diameter of an hyperbola, then \( y^2 = ax + xx : : p : a \); and \( y^2 = \frac{p}{a} \times ax + xx \); which are equations for all hyperbolas.

As all these equations are expressed by the second powers of x and y, all conic sections are curves of the second order; and conversely, the locus of every quadratic equation is a conic section, and is a parabola, ellipse, or hyperbola, according as the form of the equation corresponds with the above ones, or with some other deduced from lines drawn in a different manner with respect to the section.

**Sect. III. General Properties of Conic Sections.**

A tangent to a parabola bisects the angle contained by the lines drawn to the focus and directrix; in an ellipse and hyperbola, it bisects the angle contained by the lines drawn to the foci.

In all the sections, lines parallel to the tangent are ordinates to the diameter passing through the point of contact; and in the ellipse and hyperbola, the diameters parallel to the tangent, and those passing through the points of contact, are mutually conjugate to each other. If an ordinate be drawn from a point to a diameter, and a tangent from the same point which meets the diameter produced; in the parabola, the part of the diameter betwixt the ordinate and tangent will be bisected in the vertex; and in the ellipse and hyperbola, the semi-diameter will be a mean proportion betwixt the segments of the diameter betwixt the centre and ordinate, and betwixt the centre and tangent.

The parallelogram formed by tangents drawn through the vertices of any conjugate diameters, in the same ellipse or hyperbola, will be equal to each other.

**Sect. IV. Properties peculiar to the Hyperbola.**

As the hyperbola has some curious properties arising from its asymptotes, which appear at first view almost incredible, we shall briefly demonstrate them.

1. The hyperbola and its asymptotes never meet: if not, let them meet in S, fig. 6.; then by the property of the curve the rectangle KXE is to SX² as GE² to GM² or EP²; that is, as GX² to SX²; therefore, KXE will be equal to the square of GX; but the rectangle KXE, together with the square of GE, is also equal to the square of GX; which is absurd.

2. If a line be drawn through a hyperbola parallel to its second axis, the rectangle, by the segments of that line, betwixt the point in the hyperbola and the asymptotes, will be equal to the square of the second axis.

For if $SZ$, fig. 6, be drawn perpendicular to the second axis, by the property of the curve, the square of $MG$, that is, the square of $PE$, is to the square of $GE$, as the squares of $ZG$ and the square of $MG$ together, to the square of $SZ$ or $GX$: and the squares of $RX$ and $GX$ are in the same proportion, because the triangles $RXG$, $PEG$ are equiangular; therefore the squares $ZG$ and $MG$ are equal to the square of $RX$; from which, taking the equal squares of $SX$ and $ZG$, there remains the rectangle $RSV$, equal to the square of $MG$.

3. Hence, if right lines be drawn parallel to the second axis, cutting an hyperbola and its asymptotes, the rectangles contained betwixt the hyperbola and points where the lines cut the asymptotes will be equal to each other; for they are severally equal to the square of the second axis.

4. If from any points $a$ and $S$, in a hyperbola, there be drawn lines parallel to the asymptotes $aSQ$ and $Sbde$, the rectangle under $da$ and $de$ will be equal to the rectangle under $QS$ and $Sb$; also the parallelograms $da$, $Ge$, and $SQG$, which are equiangular, and consequently proportional to the rectangles, are equal.

For draw $YW$ RV parallel to the second axis, the rectangle $YdW$ is equal to the rectangle $RSV$; wherefore, $WD$ is to $SV$ as $RS$ is to $dY$. But because the triangles $RQS$, $AYD$, and $GSVcdW$, are equiangular, $Wd$ is to $SV$ as $cd$ to $Sb$, and $RS$ is to $DY$ as $SQ$ to $da$; therefore, $de$ is to $Sb$ as $SQ$ to $da$; and the rectangle $de$, $da$, is equal to the rectangle $QS$, $Sb$.

5. The asymptotes always approach nearer the hyperbola.

For, because the rectangle under $SQ$ and $Sb$ or $QG$, is equal to the rectangle under $da$ and $de$, or $AG$, and $QG$ is greater than $aG$; therefore $ad$ is greater than $QS$.

6. The asymptotes come nearer the hyperbola than any assignable distance.

Let $X$ be any small line. Take any point, as $d$, in the hyperbola, and draw $da$, $de$, parallel to the asymptotes; and as $X$ is to $da$, so let $aG$ be to $GQ$. Draw $QS$ parallel to $ad$, meeting the hyperbola in $S$, then $QS$ will be equal to $X$. For the rectangle $SQG$ will be equal to the rectangle $daG$; and consequently $SQ$ is to $da$ as $AG$ to $GQ$.

If any point be taken in the asymptote below $Q$, it can easily be shown that its distance is less than the line $X$.

**Sect. V. Areas contained by Conic Sections.**

The area of a parabola is equal to $\frac{1}{2}$ the area of a circumscribed parallelogram.

The area of an ellipse is equal to the area of a circle whose diameter is a mean proportional betwixt its greater and lesser axes.

If two lines, $ad$ and $QS$, be drawn parallel to one of the asymptotes of an hyperbola, the space $aQSd$, bounded by these parallel lines, the asymptotes and the hyperbola will be equal to the logarithm of $aQ$, whose module is $ad$, supposing $aG$ equal to unity.

**Sect. VI. Curvature of Conic Sections.**

The curvature of any conic section, at the vertices of its axis, is equal to the curvature of a circle whose diameter is equal to the parameter of its axis.

If a tangent be drawn from any other point of a conic section, the curvature of the section in that point will be equal to the curvature of a circle to which the same line is a tangent, and which cuts off from the diameter of the section, drawn through the point, a part equal to its parameter.

**Sect. VII. Uses of Conic Sections.**

Any body, projected from the surface of the earth, describes a parabola, to which the direction wherein it is projected is a tangent; and the distance of the directrix is equal to the height from which a body must fall to acquire the velocity wherewith it is projected: hence the properties of the parabola are the foundation of gunnery.

All bodies acted on by a central force, which decreases as the square of the distances increases, and impressed with any projectile motion, making any angle with the direction of the central force, must describe conic sections, having the central force in one of the foci, and will describe parabolas, ellipses, and hyperbolas, according to the proportion betwixt the central and projectile force. This is proved by direct demonstration.

The great principle of gravitation acts in this manner; and all the heavenly bodies describe conic sections having the sun in one of the foci; the orbits of the planets are ellipses, whose transverse and lesser diameters are nearly equal; it is uncertain whether the comets describe ellipses with very unequal axes, and to return after a great number of years; or whether they describe parabolas and hyperbolas, in which case they will never return.

**Sect. VIII. Uses of Conic Sections in the Solution of Geometrical Problems.**

Many problems can be solved by conic sections that cannot be solved by right lines and circles. The following theorems, which follow from the simpler properties of the sections, will give a specimen of this.

A point equally distant from a given point and a given line, is situated in a given parabola.

A point, the sum of whose distances from two given points is given, is situated in a given ellipse.

A point, the difference of whose distances from two given points is given, is situated in a given hyperbola.

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