a suitableness or relation of agreement between things.

The terms congruity and propriety are not applicable to any single object; they imply a plurality, and obviously signify a particular relation between different objects. Thus we currently say, that a decent garb is suitable or proper for a judge; modest behaviour for a young woman; and a lofty style for an epic poem; and, on the other hand, that it is unsuitable or incongruous to see a little woman sunk in an overgrown fardingale, a coat richly embroidered covering coarse and dirty linen, a mean subject in an elevated style, an elevated subject in a mean style, a first minister darning his wife's stocking, or a reverend prelate in lawn sleeves dancing a hornpipe.

The perception we have of this relation, which seems peculiar to man, cannot proceed from any other cause, but from a sense of congruity or propriety; for, supposing us destitute of that sense, the terms would be to us unintelligible.

It is a matter of experience, that congruity or propriety, wherever perceived, is agreeable; and that incongruity or impropriety, wherever perceived, is disagreeable. The only difficulty is, to ascertain what are the particular objects that in conjunction suggest these relations; for there are many objects that do not: the sea, for example, viewed in conjunction with a picture, or a man viewed in conjunction with a mountain, suggest not either congruity or incongruity. It seems natural to infer, what will be found true by induction, that we never perceive congruity nor incongruity but among things that are connected together by some relation; such as a man and his actions, a principal and his accessories, a subject and its ornaments. We are indeed so framed by nature, as, among things so connected, to require a certain suitableness or correspondence, termed congruity or propriety; and to be displeased when we find the opposite relation of incongruity or impropriety.

If things connected be the subject of congruity, it is reasonable before-hand to expect, that a degree of congruity should be required proportioned to the degree of the connection. And upon examination we find this to hold in fact: where the relation is intimate, as between a cause and its effect, a whole and its parts, we require the strictest congruity; but where the relation is slight, or accidental, as among things jumbled together in the same place, we require little or no congruity: the strictest propriety is required. Congruity quired in behaviour and manner of living; because a man is connected with these by the relation of cause and effect; the relation between an edifice and the ground it stands upon, is of the most intimate kind; and therefore the situation of a great house ought to be lofty; its relation to neighbouring hills, rivers, plains, being that of propinquity only, demands but a small share of congruity: among members of the same club, the congruity ought to be considerable, as well as among things placed for show in the same niche: among passengers in a stage-coach, we require very little congruity; and less still at a public spectacle.

Congruity is so nearly allied to beauty, as commonly to be held a species of it; and yet they differ so essentially as never to coincide: beauty, like colour, is placed upon a single subject; congruity upon a plurality: further, a thing beautiful in itself, may, with relation to other things, produce the strongest sense of incongruity.

Congruity and propriety are commonly reckoned synonymous terms; but they are distinguishable; and the precise meaning of each must be ascertained. Congruity is the genus of which propriety is a species; for we call nothing propriety, but that congruity or suitability which ought to subsist between sensible beings and their thoughts, words, and actions.

In order to give a full view of these secondary relations, we shall trace them through some of the most considerable primary relations. The relation of a part to the whole, being extremely intimate, demands the utmost degree of congruity; even the slightest deviation is disagreeful.

Examples of congruity and incongruity are furnished in plenty by the relation between a subject and its ornaments. A literary performance intended merely for amusement, is susceptible of much ornament, as well as a music-room or a play-house; for in gaiety, the mind hath a peculiar relish for show and decoration. The most gorgeous apparel, however improper in tragedy, is not unfitting to opera-actors; the truth is, an opera, in its present form, is a mighty fine thing; but as it deviates from nature in its capital circumstances, we look not for nature nor propriety in those which are accessory. On the other hand, a serious and important subject admits not much ornament; nor a subject that of itself is extremely beautiful; and a subject that fills the mind with its loftiness and grandeur, appears best in a dress altogether plain.

To a person of a mean appearance, gorgeous apparel is unsuitable; which, besides the incongruity, has a bad effect; for by contrast it shows the meanness of appearance in the strongest light. Sweetness of look and manner, requires simplicity of dress, joined with the greatest elegance. A stately and majestic air requires sumptuous apparel, which ought not to be gaudy, nor crowded with little ornaments. A woman of consummate beauty can bear to be highly adorned, and yet shows best in a plain dress:

For loveliness, Needs not the foreign aid of ornament, But is when unadorn'd, adorn'd the most.

Tasso's Autumn, 203.

Congruity regulates not only the quantity of ornament, but also the kind. The ornaments that enliven a dancing-room ought to be all of them gay. No picture is proper for a church, but what has religion for its subject. All the ornaments upon a shield ought to relate to war; and Virgil, with great judgment, confines the carvings upon the shield of Ajax to the military history of the Romans: but this beauty is overlooked by Homer; for the bulk of the sculpture upon the shield of Achilles, is of the arts of peace in general, and of joy and festivity in particular: the author of Telemachus betrays the same inattention, in describing the shield of that young hero.

In judging of propriety with regard to ornaments, we must attend, not only to the nature of the subject that is to be adorned, but also to the circumstances in which it is placed: the ornaments that are proper for a ball, will appear not altogether so decent at public worship; and the same person ought to dress differently for a marriage-feast and for a burial.

Nothing is more intimately related to a man, than his sentiments, words, and actions; and therefore we require here the strictest conformity. When we find what we thus require, we have a lively sense of propriety: when we find the contrary, our sense of impropriety is not less lively. Hence the universal distaste of affectation, which consits in making a show of greater delicacy and refinement than is suited either to the character or circumstance of the person.

Congruity and propriety, wherever perceived, appear agreeable; and every agreeable object produceth in the mind a pleasant emotion: incongruity and impropriety, on the other hand, are disagreeable; and of course produce painful emotions. These emotions, whether pleasant or painful, sometimes vanish without any consequence; but more frequently occasion other emotions, which we proceed to exemplify.

When any slight incongruity is perceived in an accidental combination of persons or things, as of passengers in a stage-coach, or of individuals dining at an ordinary; the painful emotion of incongruity, after a momentary existence, vanishes without producing any effect. But this is not the case of propriety and impropriety: voluntary acts, whether words or deeds, are imputed to the author; when proper, we reward him with our esteem; when improper, we punish him with our contempt. Let us suppose, for example, a generous action suited to the character of the author, which raises him and in every spectator the pleasant emotion of propriety: this emotion generates in the author both self-esteem and joy; the former when he considers the relation to the action; and the latter when he considers the good opinion that others will entertain of him: the same emotion of propriety produceth in the spectators esteem for the author of the action; and when they think of themselves, it also produceth, by means of contrast, an emotion of humility. To discover the effects of an unuitable action, we must invert each of these circumstances: the painful emotion of impropriety generates in the author of the action both humility and shame; the former when he considers his relation to the action, and the latter when he considers what others will think of him: the same emotion of impropriety produceth in the spectators contempt for the author of the action; and it also produceth, by means of contrast, when they think of them. Congruity themselves, an emotion of self-esteem. Here then are many different emotions, derived from the same action, considered in different views by different persons; a machine provided with many springs, and not a little complicated. Propriety of action, it would seem, is a chief favourite of nature, when such care and solicitude is bestowed upon it. It is not left to our own choice; but, like justice, is required at our hands; and, like justice, is enforced by natural rewards and punishments: a man cannot, with impunity, do anything unbecoming or improper; he suffers the chastisement of contempt inflicted by others, and of shame inflicted by himself. An apparatus so complicated, and so singular, ought to rouse our attention: for nature doth nothing in vain; and we may conclude with great certainty, that this curious branch of the human constitution is intended for some valuable purpose.

A gross impropriety is punished with contempt and indignation, which are vented against the offender by corresponding external expressions: nor is even the slightest impropriety suffered to pass without some degree of contempt. But there are improprieties, of the slighter kind, that provoke laughter; of which we have examples without end, in the blunders and absurdities of our own species: such improprieties receive a different punishment, as will appear by what follows. The emotions of contempt and of laughter occasioned by an impropriety of this kind, uniting intimately in the mind of the spectator, are expressed externally by a peculiar sort of laugh, termed a laugh of derision or scorn. An impropriety that thus moves not only contempt, but laughter, is distinguished by the epithet of ridiculous; and a laugh of derision or scorn is the punishment provided for it by nature. Nor ought it to escape observation, that we are so fond of inflicting this punishment, as sometimes to exert it even against creatures of an inferior species: witness a turkeycock swelling with pride, and strutting with displayed feathers; a ridiculous object, which in a gay mood is apt to provoke a laugh of derision.

We must not expect, that these different improprieties are separated by distinct boundaries: far of improprieties, from the slightest to the most gross, from the most risible to the most serious, there are degrees without end. Hence it is, that in viewing some unbecoming actions, too risible for anger, and too serious for derision, the spectator feels a sort of mixed emotion, partaking both of derision and of anger; which accounts for an expression, common with respect to the impropriety of some actions, that we know not whether to laugh or be angry.

It cannot fail to be observed, that in the case of a risible impropriety, which is always flight, the contempt we have for the offender is extremely faint; tho' derisive, its gratification, is extremely pleasant. This disproportion between a passion and its gratification, seems not conformable to the analogy of nature. In looking about for a solution, we must reflect upon what is laid down above, that an improper action not only moves our contempt for the author, but also, by means of contrast, swells the good opinion we have of ourselves. This contributes, more than any other article, to the pleasure we have in ridiculing follies and absurdities; and accordingly, it is well known, that they who put the greatest value upon themselves are the most prone to laugh at others. Pride, which is congruity: a vivid passion, pleasant in itself, and not less so in its gratification, would singly be sufficient to account for the pleasure of ridicule, without borrowing any aid from contempt. Hence appears the reason of a noted observation, That we are the most disposed to ridicule the blunders and absurdities of others, when we are in high spirits; for in high spirits, self-conceit displays itself with more than ordinary vigour.

With regard to the final causes of congruity and impropriety; one, regarding congruity, is pretty obvious, that the sense of congruity, as one principle of the fine arts, contributes in a remarkable degree to our entertainment. Congruity, indeed, with respect to quantity, coincides with proportion: when the parts of a building are nicely adjusted to each other, it may be said indifferently, that it is agreeable by the congruity of its parts, or by the proportion of its parts. But propriety, which regards voluntary agents only, can never be the same with proportion: a very long nose is disproportional, but cannot be termed improper. In some instances, it is true, impropriety coincides with disproportion in the same subject, but never in the same respect; for example, a very little man buckled to a long toledo: considering the man and the sword with respect to size, we perceive a disproportion; considering the sword as the choice of the man, we perceive an impropriety.

The sense of impropriety with respect to mistakes, blunders, and absurdities, is happily contrived for the good of mankind. In the spectators, it is productive of mirth and laughter, excellent recreation in an interval from business. But this is a trifle in respect of what follows. It is painful to be the subject of ridicule; and to punish with ridicule the man who is guilty of an absurdity, tends to put him more upon his guard in time coming. Thus even the most innocent blunder is not committed with impunity; because, were errors licensed where they do no hurt, inattention would grow into a habit, and be the occasion of much hurt.

The final cause of propriety as to moral duties, is of all the most illustrious. To have a just notion of it, the moral duties that respect others must be distinguished from those that respect ourselves. Fidelity, gratitude, and the forbearing injury, are examples of the first sort; temperance, modesty, firmness of mind, are examples of the other: the former are made duties by the sense of justice; the latter by the sense of propriety. Here is a final cause of the sense of propriety, that must rouse our attention. It is undoubtedly the interest of every man, to suit his behaviour to the dignity of his nature, and to the station allotted him by Providence; for such rational conduct contributes in every respect to happiness, by preserving health, by procuring plenty, by gaining the esteem of others, and, which of all is the greatest blessing, by gaining a justly-founded self-esteem. But in a matter so essential to our well-being, even self-interest is not relied on: the powerful authority of duty is superadded to the motive of interest. The God of nature, in all things essential to our happiness, hath observed one uniform method: to keep us steady in our conduct, he hath fortified us with natural laws and principles, which prevent many aberrations, that would daily happen were we totally surrendered to fallible guides as human CONIC SECTIONS

A RE 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 CXLVI., 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 the 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 stopped 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 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 re- maining part coinciding with the side of the rule being flopped 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 in- tercepted 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 ru- ler 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 pa- rabola 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 perpen- dicular to the directrix, will meet it again, but only once.

Every line passing through the centre of an ellipse is bisected by it; the transverse axis is the greatest of all these lines; the lesser axis the least; and these near- er 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 ordi- nate 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 shown 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 rec- tangle under the abscissas and parameters of their re- spective 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, LK⁴ fig. 5. is to FKF as EF⁴ to GH⁴.

In the hyperbola, the square of the ordinate is to the rectangle contained under the segments of the dia- meters betwixt its vertices, as the square of the dia- meter parallel to the ordinate to the square of the dia- meter to which it is drawn, or as the first diameter to its latus rectum; that is, SX⁴ is to EXK as MN⁴ to KE⁴.

Or if an ordinate be drawn to a second diameter, its square will be to the sum of the squares of the se- cond diameter, and of the line intercepted betwixt the ordinate and centre, in the same proportion: that is, RZ⁴ fig. 6. is to ZG⁴ added to GM⁴, as KE⁴ to MN⁴. These are the most important properties of the conic sections; and, by means of these, it is de- monstrated, that the figures are the same described on a plane as cut from the cone; which we have demon- strated in the case of the parabola.

Sect. II. Equations of the Conic Sections

Are derived from the above properties. The equa- tion of any curve, is an algebraic expression, which denotes the relation betwixt the ordinate and abscissa; 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 \( y^2 = aa + xx : : p : a \); and \( y^2 = \frac{p}{a} \times aa + xx \); 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 thro' 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 \( SX^4 \) as \( GE^4 \) to \( GM^4 \) or \( EP^4 \); that is, as \( GX^4 \) to \( SX^4 \); 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 \( ZG \) 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, \( d \) and \( S \), in a hyperbola, there be drawn lines parallel to the asymptotes \( da \) \( SQ \) and \( Sb \) \( dc \), the rectangle under \( da \) and \( dc \) will be equal to the rectangle under \( QS \) and \( Sb \); also the parallelograms \( da \), \( Gc \), 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 \( GSV \) \( cdW \), are equiangular, \( Wd \) is to \( SV \) as \( cd \) to \( Sb \), and \( RS \) is to \( DY \) as \( SQ \) to \( da \); therefore, \( dc \) 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 \( QS \).

9. 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 \( da \), \( dc \), parallel to the asymptotes; and as \( X \) is to \( da \), so let \( aG \) be to \( GQ \). Draw \( QS \) parallel to \( da \), 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 \( \frac{2}{3} \) 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.

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