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H, EM shall also be

>ise of the circle, E L

. M ; and taking their

» CA :CF::CD:

18.) the same things bei V the extremity of the &t to the point G, there ne A H on the axis prodistance of the point O ii'-st to the said extreof the perpendicular G D ic excessof the rectangle i under the segments be. \tiemity of the right line foci, above the rectangle I under the segments cut off rpendicular and the extre^ \i*.

ie right line C H is any how

i* squares of CA and C H

I be equal to twice the rect

I 1, and the square of A H,

i..cause CA, C F, CD, CH

liouals, to twice the rectangle

1 to the square of A H or G F j

F twice the rectangle FCD and

of F D and D G, that is, to the

t FC, CD, and DG, (7. 2.)

the two squares of C A and C H

'to the three squares of FC, CD,

, and taking away the squares of

I C F from both sides, the remaining

-I.; E H F, will be equal to the re

1 m rectangle A D o, and to the square

■ «» {6. t.)

'op IV, (fig. 7 and 8.) Iffrom any point

:i.o hyperbola, there is drawn a right

, ..r.illel to the second axis B b, meeting

iisvcrse axis A a in D; the square of

.averse axis shall be to the square

second axis, as the rectangle con

I under the segments of the tranvcrse

'"'tween the parallel and its extremes,

• »■ square of the parallel.

'top, V. (fig. 8.) If from any point

>t the In perbola there is drawn a right

< parallel to the transverse axis A a,

■ .in:; the second axis in N; the square

>i-rond axis shall be, to the square

transverse, as the sum of the squares

I half of the second axis and its seg

, between the centre and the right line,

■ • square of the line itself; that is,

C A1 : : C B' + G D" : C A2 + the

m-le A Da; that is, as C B' -f C N'

< U'orGN'.

i i|i. W. (fig. 9.) It is another pro

i. of the hyperbola, that the asymptotes,

D d, E c, do never absolutely meet with the curve. See Asymptote.

Prop. VII. It' through any point F (fig. 9.) of the hyperbola, there is drawn a right line I F L parallel to the second axis, and meeting the asymptotes in I aud L -. the rectangle contained under the right lines which are intercepted between the asymptotes and the hyperbola, is equal to the square of the half of the second axis, that is, CB'=IFL=IHL.

Prop. VIII. (fig. 10.) If from any point F of the hyperbola, there is drawn to the transverse diameter, A B, a right line ordinately applied to it F G ; and from the extremity of the diameter there is drawn A H perpendicular to it, and equal to the latut rectum; the square of the ordinate shall be equal to the rectangle applied to the latut rectum, being of the breadth of the abscissa between the ordinate and the vertex, and which exceeds it by a figure like and alike situated to that which is contained under the diameter and the latus rectum.

For join B H, and from the point G let there be drawn G M parallel to A H, and meeting B H in M, and through M let there be drawn M N parallel to A B meeting A H in N, and let the rectangles M N H O, B A H P, be completed. Then since the rectangle A G B, is to the square of G F, as A B is to A H, t. e. as G B is to G M, t. c. as the rectangle A G B is to the rectangle ACM; A G B shall be to the square of G F, as the same A G B to the rectangle A G M : wherefore the square of G F is equal to the rectangle A G M, which is applied to the latvs rectum A H, having the breadth A G, and exceeds the rectangle HA GO, by the rectangle HNHO, like to B A H P; from which excess the name of hyperbola was given to this curve by Apollonins.

Prob. 1. An easy method to describe the hyperbola, fig. 11. having the transverse diameter, D E, and the foci N n given. From N, at any distance, as N F, strike an arch; and with the same opening of the compasses with one foot in E, the vertex, set off E G equal to N F in the axis continued; then with the distance G D, and one foot in n, the other focus, cross the former arch in F. So F is a point in the hyperbola : and by this method repeated may be found any other point, /, further on, and as many more as you please.

Aq asymptote being taken for a diameter; divided into equal parts, and through all the divisions, which form so many abscissa fixed in the point E, is fixed in the point F, and the end of the thread is fixed in the point E, and the same things performed al before, there will be described another line opposite to the former, which is likewise called an hyperbola ; and both together are called oppo'site hyperbolas. These lines may be extended to any greater distance from the points E F, tag. if a thread is taken of a length greater than that distance. 2. The points E and F are called the foci. 3. And the point C, which bisects the right line between the two focuses, is called the centre of the hyperbola, or of the opposite hyperbola*.' 4. Any right line passing through the centre, and meeting the hyperbolas, is called a transverse diameter; and the points in which it meets them, their vertices: but the right line, which passes through the centre, and bisects any right line terminated by the opposite hyperbolas*, but not passing through the centre, is called a right diameter. 5. The diameter which passes through the foci, is called the transverse axis. 6 If from A or a, the extremities of the transverse axis, there is put a right line A D equal to the distance of the centre C from either focus, and with A, as a centre, and the distance A D, there is a circle described, meeting the right line, which is drawn through the centre of the hyperbola at right angles to the transverse axis, in B b; the line B o, is called the second axis. 7. Two diameters, either of which bisects all the right lines parallel to the other, and which are terminated both ways by the hyperbola, or opposite hyperbolas, are called conjugate diameters. 8. Any right line not passing through the centre, but terminated both ways by the hyperbola, or opposite hyperbolas, and bisected by a diameter, is called an ordinate applied, or simply an ordinate to that diameter: the diameter likewise, which is parallel to that other right line ordinately applied to the other diameter, is said to be ordinately applied to it. 9. The right line which meets the hyperbola in one point only, but produced both ways falls without the opposite hyperbolas, is said to touch it in that point, or is a tangent to it. 10 If through the vertex of the transverse axis a right line is drawn equal and parallel to the second axis, and is bisected by the transverse axis, the right lines drawn through the centre and the extremities of the parallel line are called asymptotes. It. The right line drawn through the centre of the hyperbola, parallel to the tangent, and

[graphic]

equal to the segment of the tangent between the asymptotes, and which is bisected in the centre, is called the second diameter of that which is drawn through the point of contact. 12. A third proportional to two diameters, one of which is the transverse, the other second to it, is called the iattu rectum, or parameter of that diameter, which is the first of the three proportional*. And, 13. Lastly, fig. 9. If upon two right lines A a, B b, mutually bisecting each other at right angles, the opposite hyperbolas A G, a g, are described; and if upon the same right lines there are described two other opposite hyperbolas, B K, b fc, of which the transverse axis, B 6, is the second axis of the two first; and the second axis of the two last, An, is the transverse axis of the two first ; these four are called conjagated hyperbolas, and their asymptotes shall be common.

Prop. I. (fig. 6.) The square of the half of the second axis is equal to the rectangle contained by the right lines between the foci and the vertexes of the transverse

axis.

Let A a be the transverse axis, C the centre, E and F the foci, and B 6 the second axis, which is evidently bisected is the centre C, from the definition; let A B be joined: then since (by def. 6) A B and C F are equal; the squares of A C and C B together, will be equal to the square of C F, that is, (6. 2.) to the square of A C and the rectangle A F n together; wherefore taking away the square of A C which is common, the square of C B will be equal to the rectangle A F a.

Prop. II. If from any point G (fig. 7 and 8.) of the hyperbola, a right line G D is drawn at right angles to the transverse axis A a, and if from the same point there is drawn the right line G F to the focus nearest to that point; the half of the transverse axis C A will be to the distance of the focus from the centre, eta. C F, as the distance of the perpendicular C D, is to the sum of the half of the transverse axis, and the right line drawn to the focus.

Let G E be drawn to the other focus, and on the axis a A produced, let there be set off A H equal G F; then with the centre G, and the distance G F, describe a circle cutting the axis re A in K and F, and the right line E G in the points L and M: then since EF is double CF, and FK double FD, EK shall be also double C D; and since E L or A a, is double C A, and

L M double G E or A H, E M shall also be double C H ; but because of the circle, EL orAa:EF::EK:EM; and taking their halves, it will be as C A : C F : : C D: C H.

Prop. III. (fig. 7 and 8.) the same things being supposed, if from A the extremity of the transverse axil nearest to the point G, there is set off a right line A H on the axis produced, equal to the distance of the point G from the focus F, nearest to the said extremity ; the square of the perpendicular G D shall be equal to the excessof the rectangle E H F, contained under the segments between^ H (the extremity of the right line A H) and the foci, above the rectangle A D a contained under the segments cut off between the perpendicular and the extremities of the axis.

For since the right line C H is any how cut in A, the squares of CA and C H together will be equal to twice the rectangle A C H, and the square of A H, tr. *.) i. e. because C A, C F, C D, C H axe proportionals, to twice the rectangle F C D, and to the square of A H or G F; that is, to twice the rectangle FCU and the squares of F D and D G, that is, to the squares of F C, C D, and D G, (7. 2.) wherefore the two squares of C A and C H are equal to the three squares of FC, C D, and D G; and taking away the squares of C A and C F from both sides, the remaining rectangle E H F, will be equal to the remaining rectangle A D a, and to the square ofDG(6. *.)

Prop. IV. (fig. 7 and 8.) If from any point G of the hyperbola, there is drawn a right line parallel to the second axis B b, meeting the transverse axis A a in D ; the square of the transverse axis shall be to the square of the second axis, as the rectangle contained under the segments of the tranverse axis between the parallel and its extremes, to the square of the parallel.

Prop. V. (fig. 8.) If from any point G of the h> perbola there is drawn a right line parallel to the transverse axis A a, meeting the second axis in N; the square of the second axis shall be, to the square of the transverse, as the sum of the squares of the halt of the second axis and its segment, between the centre and the right line, to the square of the line itself; that is, CB1: C A' : : C B* + G D' : C A' + the rectangle A D a ; that is, as C B' -J- C Na istoCD'orGN'.

Prop. VI. (fig. 9.) It is another property of the hyperbola, that the asymptotes,

D d, E e, do never absolutely meet with the curve. See Asymptote.

Prop. VII. If through any point F (fig. 9.) of the hyperbola, there is drawn a right line I F L parallel to the second axis, and meeting the asymptotes in I and L; the rectangle contained under the right lines which are intercepted between the asymptotes and the hyperbola, is equal to the square of the half of the second axis, that is, CB' = IFL = IHL.

Prop. VIII. (fig. 10.) If from any point F of the hyperbola, there is drawn to the transverse diameter, A B, a right line ordinately applied to it F G ; and from the extremity of the diameter there is drawn A H perpendicular to it, and equal to the latui rectum; the square of the ordinate shall be equal to the rectangle applied to the lutut rectum, being of the breadth of the abscissa between the ordinate and the vertex, and which exceeds it by a figure like and alike situated to that which is contained under the diameter and the latus rectum.

For join B H, and from the point G let there be drawn G M parallel to A H, and meeting B H in M, and through M let there be drawn M N parallel rb A B meeting A H in N, and let the rectangles M N H O, B A H P, be completed. Then, since the rectangle A G B, is to the'square of GF, as A B is to A H, i. e. as G B is to G M, i. e. as the rectangle A G B is to the rectangle ACM; A G B shall be to the square of G F, as the same A G B to the rectangle A G M : wherefore the square of GF is equal to the rectangle A G M, which is applied to the latui rectum A H, having the breadth A G, and exceeds the rectangle HA GO, by the rectangle M N H O, like to B A H P; from which excess the name of hyperbola was given to this curve by Apollouius.

Prob. 1. An easy method to describe the hyperbola, fig. 11. having the transverse diameter, D E, and the foci N n given. From N, at any distance, as N F, strike an arch; and with the same, opening of the compasses with one foot in E, the vertex, set off E G equal to N F in the axis continued; then with the distance G D, and one foot in n, the other focus, cross the former arch in F. So F is a point in the hyperbola: and by this method repeated may be found any other point, /, further on, and as many more as you please.

An asymptote being taken for a diameter; divided into equal parts, and through all the divisions, which form so many abscisses continually increasing equally, ordinates to the curve being drawn parallel to the other Asymptote; the abscisses will represent an infinite series of natural numbers, and the corresponding hyperbolic, or asymptotic spaces, will represent the series of logarithms of the same number. Hence different hyperbolas will furnish different series of logarithms; so that to determine any particular series of logarithms, choice must be made of some particular hyperbola. Now the most simple of all hyperbolas is the equilateral one, t. e. that whose asymptotes make a right angle between themselves.

Equilateral hyperbola is that wherein the conjugate axes are equal.

Apollonian hyperbola is the common hyperbola, or the hyperbola of the first kind: thus called in contradistinction to the hyperbolas of the higher kinds, or infinite hyperbolas: for the hyperbola of the first kind, or order, has two asymptotes; that of the second order has three; that of the third, four, &c

HYPERBOLE, in rhetoric, a figure, whereby the truth and reality of things are excessively either enlarged, or diminished. See Rhetoric.

HYPERBOLIC, or hyperbolical, something relating either to an hyperbole, or an hyperbola.

Hyperbolic cylindroid, is a solid figure, whose generation is given by Sir Christopher Wren, in the " Philosophical Transactions." Thus, two opposite hyperbolas being joined by the transverse axis, and through the centre a right line being drawn at right angles to that axis; and about that, as an axis, the hyperbolas being supposed to revolve; by such revolution, a body will be generated, which is called the hyperbolic cylindroid, whose bases, and all sections parallel to them, will be circles. In a subsequent transaction, the same author applies it to the grinding of hyperbolical glasses: affirming, that they must be formed this way or not at all. Hyperbolic leg of a curve, is that which approaches infinitely near to some asymptote. Sir Isaac 'Newton, reduces all curves, both of the first and higher kinds, into those with hyperbolic legs, and those with parabolic ones.

Hyperbolic line is used by some authors for what we call the hyperbola itself. In this sense, the plane surface, terminated by the curve line, is called the hyperbola, or hyperbolic space; and the

curve line that terminates it the hyperbolic line.

HYPERICUM, in botany, St. Johm't icort, a genus of the Polyadelphia Polyaundria class and order. Natural order of Rotacex. Hype.rica, Jussieiu Essential character: calyx five parted; petals five; filaments many, connected at the base in five bundles. There are fifty-seven species. These are principally shrubs or under shrubs, with cylindrical, ancipital, or quadrangular stems; leaves frequently with pellucid dots; flowers sometimes in cymes, frequently in corymbs, with the peduncle* often trichotomous and three flowered.

HYPHYDRA, in botany, a genus of the Monoecia Gy nandria class and order. Essential character: male, calyx one-leafed, three parted; corolla none; stamens six, inserted above the germ: female, calyx and corolla none; style triangular, with three stigmas; capsule one celled, three valved; seed single. There is but one species, riz. H. Ilmiatilis, a little plant which grows three or four feet under water; it is a native of Guiana.

HYPNUM, in botany, a genus of the Cryptogamia Musci class and order. Natural order of Musci or Mosses. Generic character: capsule oblong; peristomiiim double, outer with sixteen broadish teeth, inner membranaceous, equally laciniated , segments broadish with capillary ones interposed. Males germaceous on different plants. Botanists differ greatly as to the number of species, some reckon forty, others fifty, and Dr. Withering enumerates seventy, and to facilitate the investigation of the species he has thrown them into seven divisions.

HYPOCHOERIS, in botany, a genus of the Syngenesis Polygamia /EqtiaUs class and order. Natural order of Composite Si'iniflosculosi. Cichoraceae, Jussieu. Essential character: calyx subimbricate ; down feathered; receptacle chaffy. There are five species.

HYPOTHECATE, in law, to hypothecate a ship, is to pawn the same for necessaries; and a master may hypothecate either ship or goods for relief, when in distress at sea; for he represents the traders as well as owners; and in whose hands soever a ship or goods hypothecated come, they are liable. But it has been recently held in the Court of King's Bench, that if the master pay for the repairs himself, and do not hypothecate the ship, he has no lien upon the ship for his debt.

HYPOTHENUSE, in geometry, the longest side ol'a right angled triangle; or it is that «ii!c of which subtends the right angle. Euclid, lib. i. proposition 47, demonstrates, that, in every rectilinear right angled triangle, the square of the. hypothenuse is equal to the squares of both the other sides. This celebrated problem was discovered by Pythairoras, who is said to have sacrificed a hecatomb to the Muses, in gratitude for the discovery.

HYPOTHESIS, in general, denotes something supposed to be true, or taken for granted, in order to prove or illustrate a point in question. An hypothesis is either probable or improbable, according ai it accounts rationally or not for any phenomenon; of the former kind we may reckon the Copernican system and I lay • gens's hypothesis concerning the ring of Saturn , and the Ptolemaic system may be esteemed an instance of the latter.

HYPOXIS, in botany, a genus of the Hexandria Monogynia class and order. Natural order of Coronarite. Narcissi, Jussieu. Essential character: calyx a two valved glume; corolla, six-parted, permanent, superior; capsule narrower at the base. There are fourteen species.

HYRAX, in natural history, a genus of Mammalia, of the order Glires. Generic character: front teeth in the upper jaw two broad and somewhat distant; in the lower jaw four, broad, flat contiguous and notched; grinders large, four on each side in both jaws; fare-feet, four-toed, hind-feet three-toed; no tail; no clavicles. What distinguishes this genus from the whole class of Glires, besides, is the circumstance of having four teeth instead of two in the lower jaw, and indeed, the teeth in general are differently formed. There are two species.

H. capensis, or the Cape hyrax, is about as large as a rabbit, and abounds in the mountainous districts near the Cape of Good Hope, leaping from rock to rock with extreme agility, feeding by day, and retreating at night to the clefts and holes of the mountains. It has no power of borrowing any recess for itself. Its sound is a reiterated squeak. It subsists entirely on vegetable food, and prepares a bed for its repose and comfort in its favourite recess. It may be easily familiarized, and in a state of domestication is extremely cleanly and alert.

H. tyriacus or the bristly hyrax, is to be met with particularly in Ethiopia and

Abyssinia, and particularly under the rocks of the Mountains of the Sun. Its full length is about seventeen inches. These animals are called by the natives of these countries Ashkokos. They are gregarious, and, occasionally, seen in companies of several scores basking before the clefts of the rocks in the open sunshine. They are gentle, weak and fearful, but if handled with roughness will bite with great severity. They are supposed to live on grain, fruits, and roots, and when kept in confinement, they will live upon bread and milk. They feed without any voracity, and even the pangs of hunger could not impel them to attack chickens or smaller birds which have been thrown to them in that state, in the way of experiment. Their motion is not firm upon their legs, but rather by stealing along, by a few paces at a time, upon their bellies, in the manner of the bat in approaching its prey. For the Hyrax, see Mammalia, Plate XII. fig. 5.

HYPTIS, in botany, a genus of the Didynamia Gyrunospcrmia class and order. Natural order of Verticillatas. Lubiatae, Jussieu. Essential character 's calyx turbinate; corolla with a very spreading border; lower lip semibifid; anthers hanging down. There are two species.

HYSSOPUS, in botany, hyssop, a genus of the Didynamia Gymnospermia class and order. Natural order of Vcrticillatae, Jussieu. Essential character: corolla, lower lip with a small middle crenate segment;* stamens straight, distant. There are three species.

HYSTERICS. See Medicine.

HVSTR1X, porcupine, in natural history, a genus of quadrupeds of the order Glires. Generic character: two fore-teeth in the upper and the under jaw, cut obliquely, eight grinders; body with spines and hair; toes four or five on the fore feet. There are five species. H. cristata, or the common porcupine, is about two feet in length, exclusively of the tail. It is found in Africa and India, and is seen not unfrequently in the warmer climates of Europe, particularly in Italy and Sicily. It is covered on the upper part of its body with variegated spines, or quills, which are long and sharp, and which, when irritated, it erects with particular intenseness, and a rustling and alarming noise, giving the idea of formidable hostility. It was supposed by the ancients to possess the power of darting these with unerring, and sometimes fatal, aim against its adversaries; but it is ascertained to cm

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