CONI, a strong town of Italy, in Piedmont, and capital of a territory of that name, with a good citadel. This town being divided into two factions, it surrendered to the French in 1641; but was restored to the duke of Savoy soon after. It is seated at the confluence of the rivers Gresse and Sture, E. Long. 7. 29. N. Lat. 44. 23.
A R E 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.
Plate LXXXIV. 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.
“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 stopt 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 A, 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.
“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 latus rectum; and any other line passing through D, a diameter.
“If to the point A, fig. 6. in any plane, one end VOL. III.
“of the rule AB be placed, in such a manner, that Plate LXXXIV. 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 stopt 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
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, and , in a hyperbola, there be drawn lines parallel to the asymptotes SQ and dc, the rectangle under and will be equal to the rectangle under QS and ; also the parallelograms , , and SQGb, 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 GSV cdW, are equiangular, Wd is to SV as cd to Sb, and RS is to DY as SQ to da; wherefore, dc is to Sb as SQ to da: and the rectangle dc, 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 dc, 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 , in the hyperbola, and draw , , 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.
THE area of a parabola is equal to 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. Plate LXXXIV.
If two lines, and QS, be drawn parallel to one of the asymptotes of an hyperbola, the space , bounded by these parallel lines, the asymptotes and the hyperbola will be equal to the logarithm of , whole module is , supposing equal to unity.
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.
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 so return after a great number of years; or whether they describe parabolas and hyperbolas, in which case they will never return.
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.