of union that prevented the separation of mankind into distinct nations; and some have imagined that the tower of Babel was erected as a kind of fortress, by which the people intended to defend themselves against that separation which Noah had projected.

CONGE D'ELIRE. The king's permission royal to a dean and chapter, in time of a vacation of the see, to choose or elect a bishop. See BISHOP.

CONGELATION, may be defined the transition of a liquid into a solid state, in consequence of an abstraction of heat: thus metals, oil, water, &c. are said to congeal when they pass from a fluid into a solid state. With regard to fluids, congelation and freezing mean the same thing. Water congeals at 32°, and there are few liquids that will not congeal, if the temperature be brought sufficiently low. The only difficulty is, to obtain a temperature equal to the effect; hence it has been inferred that fluidity is the consequence of caloric. See FLUIDITY. Every particular kind of substance requires a different degree of temperature for its congelation, which affords an obvious reason why particular substances remain always fluid, while others remain always solid, in the common temperature of the atmosphere; and why others are sometimes fluid, and at others solid, according to the vicissitudes of the seasons, and the variety of climates. See COLD, FREEZ

ING.

CONGREGATION, an assembly of several ecclesiastics united, so as to constitute one body; as an assembly of cardinals, in the constitution of the pope's court, met for the dispatch of some particular business.

CONGREGATION, is likewise used for assemblies of pious persons, in manner of fraternities.

CONGREGATIONALISTS, in church history, a sect of protestants who reject all church government, except that of a single congregation. In other matters, they agree with the presbyterians. See PRESBYTERIANS.

CONGRESS, in political affairs, an assembly of commissioners, envoys, deputies, &c. from several courts, meeting to concert matters for their common good.

CONGRUITY, in geometry, is applied to figures, lines, &c. which, being laid upon each other, exactly agree in all their parts, as having the very same dimensions.

CONIC sections, as the name imports, are such curve lines as are produced by the mutual intersection of a plane and

the surface of a solid cone. The nature and properties of these figures were the subject of an extensive branch of the ancient geometry, and formed a speculation well suited to the subtle genius of the Greeks. In modern times the conic geometry is intimately connected with every part of the higher mathematics and natural philosophy. A knowledge of those discoveries, that do the greatest honour to the last and the present centuries, cannot be attained without a familiar acquaintance with the figures that are now to engage our attention.

We are chiefly indebted to the preservation of the writings of Apollonius for a knowledge of the theory of the ancient geometricians concerning the conic sections. Apollonius was born at Perga, a town of Phamphylia, and he is said to have lived under Ptolemy Philopater, about forty years posterior to Archimedes. Besides his great work on the conic sections, he published many smaller treatises, relating chiefly to the geometrical analysis, which have all perished. The treatise of Apollonius on the conic sections is written in eight books, and it was esteemed a work of so much merit by his contemporaries, as to procure for its author the title of the great geometrician. Only the four first books have come down to us in the original Greek. On the revival of learning, the lovers of the mathematics had long to regret the original of the four last books 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 conics of Apollonius: but on examination, it was found to contain the first seven books only. Two other Arabic translations of the conics of Apollonius have been discovered by the industry of learned men and as they all agree in the want of the eighth book, we may now regard that part of the treatise as irrecoverably lost. The work of Apollonius contains a very extensive, if not a complete, theory of ti e conic sections. The best edition of it is that published by Dr. Halley, in 1710: to which the learned author has added a restoration of the eighth book, executed with so much ability as to leave little room to regret the original.

Since the revival of learning, the theory of the conic sections has been much cultivated, and is the subject of a great variety of ingenious writings. Dr. Walls, in his treatise "De Sectionibus Conicis,' published at Oxford, in 1655, dcduced

the properties of the curves from a description of them on a plane. Since this time authors have been much divided as to the best way of defining the curves, and demonstrating their elementary properties; many, in imitation of the ancient geometricians, making the cone the groundwork of their theories; while others have followed the example of Dr. Wallis.

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points, D and E, and in these two points only then, having drawn DV and EV to the vertex of the cone, these lines will be both in the conic surface (Cor. Def. 1.) and also in the plane surface; and there are no points, excepting in these lines indefinitely produced, which are common to both the surfaces. Therefore the figure D V E, which is the common intersection of the cone and a plane through the vertex, is a rectilineal triangle.

OF THE CONE AND ITS SECTIONS.

Definitions.

Let ADB be a circle (Fig. 1, Plate I. Conic Sections) and V a fixed point without the plane of the circle; then if a right line passing continually through the point V, be carried round the whole periphery of the circle ADB, that right line being extended indefinitely on the same side of V as the circle, will describe a conic surface; and if it be likewise extended indefinitely on the other side of V, it will describe two opposite conic surfaces.

Cor. A straight line drawn from the vertex to any point in a conic surface, being produced indefinitely, is wholly in the opposite surfaces.

For a line so drawn will coincide with the line that generates the conic surfaces, when this line, by being carried round the circumference of the base, comes to the proposed point.

11. The solid figure, contained by the conic surface and the circle ADB, is called a core. The point Vis named the vertex of the cone; the line, CV, drawn to the centre of the circle, the axis of the cone; and the circle ADB, the base of the cone.

III. A right cone is when the axis is perpendicular to the plane of the base; otherwise it is a scalene, or oblique

cone.

IV. A right line that meets a conic surface in one point only, and is every where else without that surface, is called a tangent.

PROP. I.

Fig. 1. The common intersection of a conic surface and a plane, VDE, that passes through the vertex, and cuts the base of the cone, is a rectilineal triangle.

For the common section of the plane of the base, and the plane drawn through the vertex (which is a right line 3. 11. E) will cut the periphery of the base in two

PROP. II.

Fig. 2. If a point E, be assumed in a conic surface, and a line, PQ, be drawn through it so as to be parallel to a right line, VB, passing through the vertex, and contained in the conic surfaces; then the right line P Q, will not meet either of the opposite surfaces in another point, but it will fall within the surface in which the assumed point E is, on the one side, and it will be wholly without both surfaces on the other side.

For if a plane be conceived to be drawn through the line VB and the point E, the line P Q, parallel to V B, will be wholly in that plane, 7. 11. E; and the common sections of the plane and the conic surfaces will be the line V B and the line V E C, drawn through the vertex and the point E, Pr. I. Now the line, QP, does not meet either of the lines VB or V C in another point different from E. Also QE, the part of the line that is contained in the angle BV C, is within the cone; and P E, the part of it that is contained in the angle CVN, is without both the opposite surfaces.

PROP. III.

Fig. 3. If a plane be drawn through the vertex of a cone and a tangent of the conic surface GH, it will meet the conic surface only in the line V D, drawn through the vertex of the cone and the point of contact of the tangent.

For, because the point D and the vertex V are common both to the plane surface and to the conic surface, therefore the line VD, indefinitely produced, is likewise common to both surfaces. And because G H meets the conic surface only in the point D, and is every where else without the surface, therefore any line (different from V D) as VF, drawn in one of the conic surfaces, is contained on one side of the plane; and the same line continued in the oppo

site conic surface, as V K, is contained on the other side of the plane.

Cor. 1. Any straight line drawn in the plane VGH, so as to meet the line VD, is a tangent of the conic surfaces.

Cor. 2. No other plane, besides the plane VGH, can be drawn so as to touch the conic surfaces in the line VD, without cutting them.

For RS, the common section of the plane VGH, and the plane of the base, is a tangent to the periphery of the base, Cor. 1. And if there were two such planes, there would likewise be two tangents of a circle drawn through the same point of the periphery, which is absurd.

PROP. IV.

Fig. 4. A right line drawn through a point of a conic surface, so as neither to be a tangent, nor to be parallel to a right line contained in the conic surface, will meet either the same, or the opposite, conic surface again in another point.

Let a plane be drawn through the vertex of the cone and the right line (DB or DC) then that plane will cut the cone; for if it did not, the right line (DB or DC) would be a tangent contrary to the bypothesis. Let VG and VH be the common sections of the plane and the conic surface; then the right line (DB or DC) will not be parallel to VH contained in the conic surface (hyp), therefore it will meet VH either in the same conic surface (as DB), or when produced in the opposite conic surface (as DC.)

PROP. V.

Fig. 5. If either of two opposite conic surfaces be cut by a plane parallel to the base of the cone, the section is a circle, having its centre in the axis of the

cone.

Through V C, the axis of the cone, let two planes be drawn, cutting the base in the lines CD and CE, and the plane parallel to the base in the lines GH and GL, and the conic surfaces in the lines V HD and VLE: then because the base is parallel to the cutting plane, therefore C D is parallel to G H, and CE to GL, 16. 11. E. Therefore, on account of equiangular triangles, 4.

5. E.

DC CV HG: GV CV:CE: GV: GL Ex æquo DC: CE :: HG : GL

But DC = CE, therefore HG GL. And in like manner it may be shown that any right line drawn from G to a point in the intersection of the plane, and the conic surface, is equal to GH; therefore the section is a circle.

Cor. If through a point situated within or without a conic surface, two straight lines, both parallel to the plane of the base of the cone, (that is parallel to straight lines in that plane,) be drawn to cut or touch the conic surface: then the

rectangle contained by the two segments (between the point and the conic surface,) of one of the lines when it cuts, or the square of its segments when it touches the conic surface, is equal to the rectan gle contained by the two segments of the other line when it cuts, or to the square of its segment when it touches the conic surface.

For a plane drawn through the two lines will be parallel to the plane of the base, 15. 16. E; and it will intersect the conic surface of the periphery of a circle; whence the corollory is manifest, 35 and 36. 3. E.

When a straight line drawn through a point, situated within or without a cone, meets one or both of the conic surfaces in two points, it is called a secant; and the two parts of such a line, between the point through which it is drawn, and the conic surface or surfaces, are called the segments of the secant. And when a line, drawn from a point without a cone, touches one of the conic surfaces, that part of it between the point from which it is drawn and the conic surface is denoted by the word tangent, in the following propositions.

PROP. VI.

Fig. 6,7, and 8. If a straight line be drawn from the vertex of a cone, to a point, as B, in the plane of the base, but not in the periphery of the base; and, through any point, as P, situated without or within the cone, another straight line, parallel to the former, be drawn to cut or touch the conic surface or opposite surfaces; then the square of the line drawn from the vertex of the cone to the point B is to the rectangle under the segments of the secant, or to the square of the tangent, drawn from the point P, as the rectangle under the segments of any line drawn from B, to cut the base of the cone, is to the rectangle under the segments of any line parallel to the base

of the cone, drawn through the point P, to cut the conic surface.

Fig. 6. Let the point B be without the base of the cone, and let QR, drawn through P, without or within the conic surface, be parallel to VB, and let it cut the conic surface in Q and R: through P and the line VB draw a plane cutting the conic surface in the lines V G and V H, and the plane of the base in the line BGH; and through P draw LK parallel to G H. Because V B and PRQ are parallel, therefore the line PRQ is contained in the plane B V P, 7. 11. E.; and the points Q and R are in the lines VH and V G, the common sections of the plane and the conic surface. Because Q P is parallel to V B, and L K to GH, therefore the triangle QPL is equiangular to the triangle VBH, and the triangle PKR to the triangle V GB: therefore 4. 6. E.

VB PR: BG: PK VB: PQ BH : PL

Consequently, VB: PR X PQ :: BG X BH: PK X PL, 23. 6. E. But the rectangle BG X BH is equal to the rectangle under the segments of any other line drawn from B to cut the base of the cone, 35, and 36. 3. E; and the rectangle PK KL is equal to the rectangle under the segments of any other line, parallel to the plane of the base, drawn from P to cut the conic surface, Cor. Pr. 5; and hence the proportion is manifest in this case.

Fig. 7. And if the point B be within the base of the cone, and a straight line as (PQR), parallel to the line V B that joins the point B and the vertex of the cone, be drawn to cut the opposite sur faces through a point P, situated without or within the cone: the proposition may be demonstrated, in this case, in the very same words as in the former case.

And if the point P (fig 8.) be without the cone as well as the line V B, and PS, parallel to V B, be drawn to touch the conic surface, instead of cutting it; then the plane PVB will meet the conic surface in a line V S M; and B M will touch the base of the cone, and PN, parallel to B M, will touch the conic surface. And because the two triangles SPN and V B M are equiangular, therefore,

VB: PS: : BM: PN And VB: PS2:: BM2: PN

But BM is equal to the rectangle under the segments of any line drawn from B to cut the base of the cone; and PN is equal to the rectangle under the segments of any line, parallel to the base of the cone, drawn from P to cut the conic surface; and hence the proposition is manifest in this case also.

PROP. VII.

Fig. 9. If a point be assumed without or within a cone, and two lines be drawn through it to meet a conic surface, or opposite surfaces, and so as to be parallel to two straight lines given by position; then the rectangle under the segments of the secant, or the square of the tangent, parallel to one of the lines given by position, has to the rectangle under the segments of the secant, or to the square of the tangent parallel to the other line given by position, a ratio that is constantly the same, wherever the point (from which the lines are drawn) is assumed, without or within the cone.

Let VB and VC be two straight lines, (fig. 9.) drawn from the vertex of a cone to the plane of the base, and given by po. sition (or parallel to lines given by position) and let P Q and M N be two straight lines drawn through any assumed point, as R, to cut the conic surface, and so as to be respectively parallel to C V and V B and as C V is to the rectangle CK X CL (contained by the segments of any line drawn from C to cut the base of the cone,) so let D, any assumed line or magnitude be to E; and as VB is to BGXBH (the rectangle contained by the segments of any line drawn from B to cut the base of the cone,) so let F be to E and draw S T parallel to the base of the cone through the point R; then, Pr. 6.

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(CV1: CK X C L, or) (D: E:: PR X RQ SR X RT, and BV2: BG' X BH, or) F E MRX RN:SR XR T. Therefore, invertendo and ex æquo.

::

D: F:: PR X RQ: MR X RN.

And as the same reasoning applies whereever the point R is assumed, therefore the ratio of the rectangles P R R Q, and MRX RN, is the same with, or equal to the constant ratio of D to F, wherever the point R is assumed.

And in like manner may the proposition be demonstrated in all other cases,or

in all positions of the lines P Q, and M N, whether they cut, or touch, the same or opposite surfaces.

PROP. VIII.

Fig. 10. If a right line, as P T, drawn through a point P in the surface of a cone, so as to be parallel to a right line V B contained in the conic surface meet two parallel lines (in the points R and S) that cut or touch the conic surface or opposite surfaces: then PR is to P S as the rectangle under the segments of the secant, or the square of the tangent, drawn through the point R, is to the rectangle under the segments of the secant, or to the square of the tangent drawn through the point S.

Through the two parallels PT and V B (fig. 10.) draw a plane cutting the conic surface again in the line VA, and the plane of the base in the line BA; and, through Rand S, draw M N and H G parallel to A B. Because P T is parallel to V B, and RN to S G, therefore RNGS is a parallelogram; and RN is GS. It is obvious that the triangles PMR and PHS are equiangular: therefore P R is to PS as MR is to H S, 4. 6. E, or as MRX RN is to HS XS G, 1. 6. E. But MRX RN and HSXSG are respecttively equal to the rectangles contained by the segments of any two lines, parallel to the base of the cone, drawn through Rand S to cut the conic surface, Cor. Pr; 5, and hence the proposition is manifest, when P T meets two lines parallel to the plane of the base.

And if P T meet two parallel lines DE and I K, not parallel to the plane of the base; then let the same construction be made as before: and because D E is parallel to I K, and M N to G H; therefore,

DRX RE: MRXRN:: ISXSK:

HSXSG;

Alternando, D RXRE: ISXSK:: MRXRN: HSXS G. Therefore, as is obvious from what has already been shewn,

PR: PS: DRXRE: ISX S K.

And if S be without the cone, and the line drawn through it touch the conic surface instead of cutting it, the reasoning is still the same, when the square of the tangent is taken in place of the rectangle under the segments of the se

cant.

PROP. IX.

Fig. 11. Let a scalene cone be cut by a plane drawn through the axis perpendi. culiar to the plane of the base, making the triangular section VA B; and let V D, cutting A B produced in D, be drawn so as to make the angle B V D equal to the angle V A B, and draw M N in the plane of the base, perpendicular to AD; then every section of the cone, as PS Q, made by a plane parallel to the plane V MN (called a subcontrary section) is a circle; and every circular section of the cone, which is not parallel to the base, is a subcontrary section.

Draw T S in the plane of the section parallel to M N, which is plainly possible, because the two planes P Q and V M N are parallel because T S is parallel to MN, a line in the plane of the base, therefore every plane drawn through S T will cut the base in a line parallel to S T (16. 11. E.:) therefore L O K, the common section of the base, and a plane drawn through V and S T is parallel to S T and MN (9. 11. E.): therefore K O L is perpendicular to A B, and it is bisected in O; therefore S T is bisected in R. Again, the line P Q is parallel to V D, therefore V D': PRXRQ:ADXDB: TRX RS (6.) But if a circle be described about the triangle A V B, D V will be a tangent of that circle (32. 3. E): therefore V D'AD X D B, and consequently, PRXRQ=TR× RS, or R T3 (36. 3. E.) Because the plane A V D is perpendicular to the base (hyp) and MN is perpendicular to AD: therefore M N is perpendicular to the plane A V D: therefore, T R, parallel to M N, is perpendicu lar to the same plane, and to PQ. And hence, from what has already been shewn, the section P Q is a circle.

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Next, let P Q be a circular section, not parallel to the base of the cone: draw a plane through the vertex, parallel to the plane P Q, and let it cut the base in the line M N draw A D through the centre of the base perpendicular to M N, and let a plane drawn through V and A D cut the parallel planes in the lines P Q and V D, and the conic surface in the lines A V and V B: draw the plane V TLK S through ST parallel to MN, as before. It is shewn, as above, that T S is bisected in R: and, in like manner, it may be proved that any other line, as G H, parallel to M N, is bisected. Because PQ, a line in a circle, bisects two or more parallels, it is a diameter of the circle, and it cuts all the