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 extre ie right line C H is any how i* squares of CA and C H
i..cause CA, C F, CD, CH liouals, to twice the rectangle 1 to the square of A H or G F j
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 ■ «» {6. t.)
: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.
>t the In perbola there is drawn a right < ■ .in:; the second axis in N; the square >i-rond axis transverse, as the sum of the squares
, ■ • 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. i. D d, E c, do never absolutely meet with the curve. See Asymptote. Prop. VII. 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. Aq asymptote being taken for a diameter; divided into equal parts, and 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 Prop. I. (fig. 6.) The square of the half of the second axis is equal to the rectangle contained by the right lines axis. Let A a be the 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 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 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, 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 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 Prop. VI. (fig. 9.) It is another property of the hyperbola, that the asymptotes, D d, E 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. 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 An asymptote being taken for a diameter; divided into equal parts, and through all the divisions, which 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 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 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 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 HYPTIS, in botany, a genus of the Didynamia Gyrunospcrmia class and order. Natural order of Verticillatas. Lubiatae, Jussieu. Essential character 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 |