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1. By Conic Sections are understood the sections produced from a cone by cutting it with a plane.
For example, the circle is a conic section, because if we cut a right cone by a plane, parallel to its base, the section will be circle.
The triangle is also a conic section, because if we cut a right cone by a plane perpendicular to its base, and passing through its vertex, the section will be a triangle.
But the name of conic sections is more particularly applied to three other sections of a cone, of which we shall explain the origin and properties, after having made known the manner of treating them analytically.
Descartes first conceived the idea of applying Algebra to Geometry. The utility of this application soon became apparent, and the geometers who succeeded him, availed themselves of this discovery to such an extent, that it will for ever be celebrated for its great fecundity. The principal uses of this doctrine are perceived in its application to the theory of curves, the study of which is indispensable to those who desire to obtain a thorough knowledge of the Physico Mathematical sciences.
2. The object of this theory is to express, by equations, the laws, according to which we suppose any given curves have been described; and reciprocally, to direct the Analyst, either in the description of the curves of which he knows the equations, or in the investigation of the properties of those curves.
For this purpose, each point of the curve that we desire to trace, is referred to two right lines, of which one is called the line, or axis of the abscissa; the other, the line, or axis of the ordinates. We then determine the ratio which obtains between the abscissæ and ordinates, and the analytical expression of this ratio gives the equation of the curve.
For example, let the curve AMBM be a circle, AB its diameter, C the center, and AD an indefinite line, drawn from A perpendicular to AB.
Suppose now any point M of the circumference to be referred to the two lines AB, AD, by the lines MP, ME parallel to AD, AB. Then will MP be an ordinate; and ME, or its equal AP, an abscissa to the point M. The lines AD, AB, are the axes of the ordinates and abscissæ.
Call the diameter AB, 2a; AP, x; PM, y. Then by (Euclid 8.6) PM2 AP PB, or in symbols y2= (2a-x) — 2 ax—xx. this relation is the same for any point M whatever, this is the equation of the circle.
3. Any expression in which a quantity enters, is for the sake of conciseness, called a function of that quantity. Thus, for example, we say, that the equation of the circle expresses the constant equality of a function of each ordinate (its square), to a function of the corresponding abscissa (its product by the remainder of the diameter).
The abscissæ, and corresponding ordinates of a curve, are called the co-ordinates; and as the length of these lines varies at each instant, they are called variable, or indeterminate quantities, in opposition to the constant or determinate quantities.
The point from which we begin to compute the abscissæ, is called the origin of the abscissa. We may fix this point at pleasure before we commence investigating the equation of the co-ordinates; but this position once determined, it must always remain the same during the details of the same calculation.
Usually the origin of the abscissæ is placed at either the summit or center of the curve.
And as at setting out from their origin, we may take the coordinates on two opposite sides, it is customary to designate those on one side by the sign+, and those on the other by the sign-; so that an abscissa is considered as positive, when it is on that part of the axis which is considered as positive. The choice of this part is altogether arbitrary; but when once made, it must be adhered to.
4. The ordinates may be either perpendicular or oblique to the line of the abscissæ, provided that they are parallel to one another. Generally they are supposed to be perpendicular, and are distinguished as positive or negative, according as they are on the one or the other side of the axis of the abscissæ. Sometimes they proceed from a fixed point, as we shall hereafter explain.
These things premised, let us describe the curve whose equation is y2=2ax-xx. We already know (2) that it is the circumference of a circle whose diameter is 2 a; but if we were ignorant of this, the construction of this equation would soon inform us of it.
Let then a denote a constant quantity which I suppose=5, and let there be drawn an indefinite right line BD, on which I take AD-10=2 a; and which I divide into ten equal parts AP, PP, &c. Let A be B the origin of the abscissæ, BD their axis, AD the direction of the positive abscissæ, then AB will be that of the negative, if the curve required has any. To the point A let there be
drawn the indefinite perpendicular EF which I take as the axis of the ordinates, and of which I consider AE as the positive part. Lastly, let AP=x, and PM=y.
By the equation itself y=±√2 ax-xx, from which it is clear that when x=0, we have y=0; therefore the curve has the point A in common with the line of the abscissæ. If we make x=1, y becomes + 3; if x=2, becomes 4, &c.; so that the corresponding values of x and y are x=0, 1, 2, 6, 7, 8, 9, 10 y=0,3,4,±√21,±√24, ±5,±√24,±√21, ±4, ±3, ±0 These values of y determine the lengths of so many ordinates PM, whose extremities M are points of the curve required; and since these values are at once both positive and negative, it is evident that drawing through the point A two equal branches, one of which passes through the points M, above the axis of the abscissæ, and the other through the corresponding points M', M' below, we shall have the curve required.
With respect to the description, it will be the more exact in proportion as we multiply the divisions of the line AD. In this manner we may describe any curve by referring each of its points M to two lines BD, EF, given in position; for if we compleat the parellelogram APMN, of which we know the two sides AP or NM, and I'M, the intersection of these last two lines will determine the point M of the curve. This parallelogram is called the parallelogram of the co-ordinates.
The values of y increase here from 0 to 5, and then decrease in the same proportion to zero again. Hence we may conclude 1o, that there is one ordinate PM greater than any other; this is called the maximum of the ordinates. (The investigation of the maxima and minima is one of the most curious analytical subjects; we shall hereafter give examples of it.)
2ly. We may also conclude, that the curve whose equation is yy=2 ax-xz, is a curve returning into itself, and compleatly including a space. It does not extend beyond the point A ; for there its
abscissæ becoming negative, the values of y would be imaginary, which indicates that none of its branches can extend on the other side of the origin of the abscissæ. Let us now investigate some of its properties.
5. From C, the middle of the line AD, I draw the right lines CM, and thereby form as many right-angled triangles CPM, in which CM2 PM2 + CP2 = y2+a2 — 2 ax + xx; therefore since y'-2 ax-xx, we shall always have CM-a; and consequently all the points M of the curve are at an equal distance from the centre C, a distinguishing property of the circumference of the circle.
Again the equation yy-2 ax-xx, gives xy::y:2a-x, or AP PM:: PM; PD; therefore every perpendicular PM is a mean proportional between the two segments of the diameter AD, another property of the circle.
Draw the chord AM, and we shall have AM-2 ax; therefore :AM::AM: 2 a; this shews that in the required curve any chords drawn from the point A to any point M, are mean proportionals between the diameter AD, and the corresponding segment AP; this also agrees with the circle.
If we draw the chord MD, we shall have AM2+ MD2— 4 a2= AD', a property of the right-angled triangle. Therefore all the angles AMD are right-angles, as they ought to be in the circle.
Inscribing the quadrilateral figure AMDM', we shall find that AMX M'D+AM' × MD — AD × MM'. And in the same manner we may discover the remaining properties.
6. Let it now be required to describe a curve, supposing the equation of its co-ordinates to be y2—ax.
We see at once that this curve must cut the axis of the abscissæ at their origin, since supposing to, we also have y=0. It is also evident that it must have two equal branches, one positive, the other negative. These branches will extend ad infinitum, receding from their axis, as we assign greater values to x. But these values must all be positive, otherwise the ordinates would become imaginary. these reasons, the curve will have the form MAM.
7. Again suppose yy=xx-aa. It is clear that if the curve to which this equation belongs, cuts the axis of the abscissæ, or even only touches it in some points, we shall determine them by supposing yo. On this supposition we have x = ±a. Therefore taking in an indefinite right line BD a point A for
the origin of the abscissæ, and two parts AS, As equal to the given quantity a, the curve must pass through the points S, s, which are called its vertices, or summits.
To find the direction of its branches, let AD be the side of the positive abscissæ; we shall have y±±√ (x2-a2 ) = ± √(x+a) (-a), this gives two branches, the one SM, the other SM', whose course extend indefinitely, provided x is greater than a. If it were less, the values of y would be imaginary; hence the curve does not pass on the other side of the point S, so long as the abscissæ are positive.
Suppose that we take them y=±√(−x+a) (—x—a). values of y will be imaginary. between the points A and s.
negative, the equation will become Here as long as x is less than a, the Hence no part of the curve will lie
If x-a, we find as above, y-o: if xa, then y has two real values, one positive, the other negative; and these values constantly increasing, the curve will have two new branches, opposite to, but equal to the two first. The axis of the abscissæ is BD, that of the ordinates, EF; assigning values to x, we shall determine those of y or PM, and the parallelogram of the co-ordinates will give the points M, m, &c. through which the required curve must pass. We shall soon have occasion to examine the properties of these two curves.
8. Let us now investigate the curve whose equation is bx2+x3
2ly. For each value of x, there are two values of y. are both positive and negative ordinates. It now remains to determine the points at which they cease to be real.
sly. Take a positive, but less than a or AD, and we have for y two values PM, PM', which continue to increase till xa, when b+x
they become infinite; for then y±(+); supposing there
fore as usual, that zero expresses an infinitely small quantity, or
that o=—, y is infinite. That is to say, we must prolong the
line GH, without limit, that it may meet the two branches of the