tion as the hair lengthens or shortens, it causes the cylinder to turn in one or the other direction, and, by a necessary consequence, the little index turns likewise, the motions of which are measured on the circumference of a graduated circle, about which the index performs its revolution as in common clocks. In this manner a very small variation in the length of the hair becomes perceptible, by the much more considerable motion that it occasions in the extremity of the index; and it will be easily conceived, that equal 'degrees of expansion, or of contraction in the hair, answer to equal arcs described by the extremity of the index. To give to the scale such a basis, as may establish a relation between all the hygrometers that are constructed upon the same principles, Saussure assumes two fixed terms, one of which is the extreme of humidity, and the other that of dryness: he determines the first by placing the hygrometer under a glass receiver, the whole interior surface of which he had completely moistened with water; the air being saturated by this water acts by its humidity upon the hair to lengthen it. He moistened anew the interior of the receiver, as often as it was necessary; and he knew that the term of extreme humidity was attained, when, by a longer continuance under the receiver, the hair ceased to extend itself. To obtain the contrary limit of extreme dryness, the same philosopher made use of a hot and well dried receiver, under which he included the hygrometer with a piece of iron plate, likewise heated and covered with a fixed alkali. This salt, by exercising its absorbent faculty upon the remaining humidity in the surrounding air, causes the hair to contract itself, until it has attained the ultimate limit of its contraction. The scale of the instrument is divided into a hundred degrees. The zero indicates the limit of extreme dryness, and the number one hundred that of extreme humidity. The effects of moisture and of dryness upon the hair are modified by those of heat, which act upon it, sometimes in the same sense, and sometimes in a contrary one; so that, if it be supposed, for example, that the air is heated about the hygrometer on one part, this air, whose dissolving faculty with regard to the water will be augmented, will take away from the hair a portion of the water, which it had imbibed, thus tending to shorten the hair; while, on the other part, the heat, by penetrating it, will tend, though much more feebly, to lengthen it; and hence the total effect will be found to consist of two partial and contrary effects, the one hygrometric, the other pyrometric. In observations which require a certain precision, it is therefore necessary to consult the thermometer at the same time with the hygrometer; and, on this account, the inventor has constructed, from observation, a table of correction, which will put it in the power of philosophers always to ascertain the degree of humidity of the air, from the ef. fect produced by the heat. De Luc, who devoted his attention to the same object has followed a different method. This philosopher employed for the construction of his hygrometers, a very thin slip of whale-bone, which performs the same office as the hair in the hygrometer of Saussure. He kept this whale-bone bent by means of a spring, the action of which he preferred to that of a weight: he determined the degree of extreme humidity, by immersing the slip of whale-bone entirely under water; and to fix the opposite limit, which is that of extreme dryness, he made use of calcined lime, which he inclosed with the hygrometer under a glass bell. The choice of lime is founded on this, that the calcination having produced a higher degree of dryness, if it be afterward left to cool, so far that it may be placed without inconvenience under the glass bel; destined for the experiment, it will be still found, as to sense, in the same state of dryness, since it is very slow in acquiring humidity; and thus all its absorbent faculty will be employed to dry up, by little and little, the air contained under the receiver, and to make the hygrometer itself pass to a state which approaches the nearest possible to extreme dryness. The hygrometer has been long neglected in meteorological observations; it is necessary to associate with it the thermometer and the barometer, to be in a state to unravel the complication of different causes which influence the variations of the atmosphere ; and it is only by the aid of a long series of observations, made by these various instruments, together with all the indications which are deduced from the state of the heavens, that we can obtain such data as will enable us to prognosticate, with great probability, the temporary changes, and to arrive at a plausible theory upon this subject, so interesting, and so naturally calculated to excite our curiosity. Mr. Marshall says, that a simple instrument of this sort may be formed by means of “a flaxen line (large well manufactured whip cord) five feet long, and having a graduated scale fixed to an index, moving on a fulcrum. The length of the index, from the fulcrum to the point should be ten inches; that of the lever, from the fulcrum to the middle of the eye, to which the cord is fixed, two and a half.” He adds, that “the principle on which this hygrometer acts is obvious. The air becoming moist, the cord imbibes its moisture; the line, in consequence, is shortened, and the index rises. On the contrary, the air becoming dry, the cord discharges its moisture, — lengthens, -and the index falls. It may be true,” he says, “that no two hygrometers will keep pace with each other sufficiently to satisfy the curious. He will venture to say, however, from seven months close attention, that two hygrometers, on this simple construction, have coincided sufficiently for the uses of agriculture. It is true,” he adds, “they diminished in the degree of action; but as the scale may be readily diminished in extent, and as a fresh line may be so cheaply and so readiiy supplied, this is not a valid objection.” It is remarked, that “this diminution in the degree of action depends considerably on the construction; the propriety, or rather delicacy, of which rests, almost solely, on this point: the weight of the index should be so proportioned to the weight of the lever and cord, that the cord may be kept perfectly straight, without being too much stretched. He made one with a long heavy index, and in order to gain a more extensive scale with a short lever; but, even when it was first put up, it could barely act; and in a few weeks it flagged, and was not able to raise the index, though the air was uncommonly moist. He therefore made another with the same length, both of index and lever, but with a lighter index, and a heavier lever, so as to gain the proportion above mentioned, and it has acted exceedingly well.” He thinks that no farmer, “who wishes to profit by the hygrometer, should have less than two. Three or four would be more advisable. They would then assist in correcting each other; and, in case of renewal or alteration, there would be no danger of losing the state of the atmosphere; which, if only one is kept, must necessarily be the case. The principle on which this hygrometer is formed is not, he says, confined to a small cord, and an index of ten inches long : it may be extended to a rope, of any length or thickness, and to an index and scale of almost any dimensions and extent.” But one, or more, on a portable construction, might, he thinks, be found useful. An axe is the form he has thought of; the edge, graduated, will constitute the scale ; and the handle will receive the cord: this may be hung up in the shade, exposed to the action of the air; or, by means of a spike in the end of the handle, it may be placed in the open field. By placing it on fallow ground, it may be actuated by the perspiration of the earth; among vegetables by vegetable perspiration; by the means of one, or, more probably, by the means of several, placed at varied heights, the different degrees of moisture at different altitudes may be ascertained, &c. In fact, he considers the hygrometer, whether it is a prognostic of the weather or not, as a most valuable oracle to the farmer. See WEATHER. HYMEN, in anatomy, a thin membrane stretched transversely across the vagina, at a small distance from its entrance. It is sometimes found entire, completely intercepting the passage—sometimes it exists but partially—and often not at all. HYMENAE, in botany, a genus of the Decandria Monogynia class and order. Natural order of Lomentaceae. Leguminosae, Jussieu. Essential character: calyx five-parted ; petals five, almost equal; style twisted inwards; legume filled with farinaceous pulp. There is only one species, viz. H. courbaril, locust tree. The wild bees are fond of building their nests in this tree, which grows to a considerable size in the West Indies, and is looked upon as excellent timber; but it must be very old before it is cut, otherwise the heart will be but small. It is in great request for wheel-work in the sugar mills, particularly for cogs to the wheels, being remarkably hard and tough. Professor Jacquin says, that a cubic foot weighs about a hundred pounds, and that it will take a fine polish. HYMENOPTERA, in natural history, the fifth order of insects according to the Linnaean system. The insects of this order are furnished with four membranaceous wings, and also with a sting, or a process resembling one. The wasp and the bee are insects of this order. It con- HYOBANCHE, in botany, a genus of the Didynamia Angiospermia class and order. Natural order of Personatae. Pediculares, Jussieu. Essential character: calyx, seven-leaved; corolla ringent, without any lower lip; capsule two-celled, many-seedcq. There is but one species, viz. H. sanguinea, a native of the Cape of Good Hope, and is parasitical at the roots of shrubs. HYOSCYAMUS, in botany, henbane, a genus of the Pentandria Monogynia class and order. Natural order of Luridae. Solaneze, Jussieu. Essential character: corolla funnel-form, obtuse; stamina inclined; capsule two-celled, covered with a lid. There are eight species. HYOSERIS, in botany, swine's lettuce or succory, a genus of the Syngenesia Polygamia Equalis class and order. Natural order of Compositae Semiflosculosae. Cichoracer, Jussieu. Essential character: calyx almost equal ; down hairy and calycled; receptacle naked. There are ten species. HYPECOUM, in botany, a genus of the Tetrandria Ingynia class and order. Natural order of Corydales. Papaveraceae, Jussieu. Essential character: calyx twoleaved ; petals four, the two outer broader, and trifid; fruit a silique. There are three species. HY PELATE, in botany, a genus of the Polygamia Monoecia class and order. Essential character: calyx five-leaved; corolla five-petalled; stigma bent down, three-cornered: drupe one seeded. There is but one species, viz. H. trifoliata, a native of Jamaica, where it is common in the low lands. HYPER BOLA, in geometry, the section, GEH, (Plate VII. Miscel. fig. 5.) of a cone, ABC, made by a plane, so that the axis, EF, of the section inclines to the opposite leg of the cone, BC, which, in the parabola, is parallel to it, and in the ellipsis intersects it. The axis of the hyperbolical section will meet also with the opposite side of the cone, when produced above the vertex at D. Definitions. 1. If at the point E (fig. 6.) in any plane, the end of the rule EH be so fixed, that it may be freely carried round, as about a centre; and at the other • end of the rule H there is fixed the end of a thread shorter than the rule, and let the other end of the thread be fixed at the point F, in the same plane; but the distance of the points EF must be greater than the excess of the rule above the length of the thread; then let the thread be applied to the side of the rule EH, by the help of a pin G, and be stretch ed along it; afterwards let the rule be carried round, and in the mean time let the thread, kept stretched by the pin, be constantly applied to the rule: a certain line will be described by the motion of the pin, which is called the hyperbola. But if the extremity of the same rule, which was 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 as before, there will be described another line opposite to the former, which is likewise called an hyperbola; and both together are called opposite hyperbolas. These lines may be extended to any greater distance from the points EF, viz. 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 focus's, is called the centre of the hyperbola, or of the opposite hyperbolas. 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 pass. ing 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 AD, equal to the distance of the centre C from either focus, and with A, as a centre, and the distance AD, 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 b 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 o 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. 11. The right line drawn through the centre of the hyperbola, parallel to the tangent, 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 transverse, the other second to it, is called the latus rectum, or parameter of that diameter, which is the first of the three proportionals. And, 13. Lastly, fig. 9. If upon two right lines A a, Bob, mutually bisecting each other at right angles, the opposite hyperbolas AG, a g, are described; and if upon the same right lines there are described two other opposite hyperbolas, BK, b k, of which the transverse axis, B 6, is the second axis of the two first and the second axis of the two last, A a, is the transverse axis of the two first; these four are called conjugated hyperbolas, and their asymptotes shall be common. Prop. 1. (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 b the se. cond axis, which is evidently bisected in the centre C, from the definition : let A B be joined: then since (by def. 6.) AB and CF are equal; the squares of AC and CB, together, will be equal to the square of CF, that is, (6. 2.) to the square of AC and the rectangle AF a together; wherefore, taking away the square of AC, which is common, the square of CB will be equal to the rectangle AF a. Prop. II. It from any point G (fig. 7 and 8.) of the hyperbola, a right line GD is drawn at right angles to the transverse axis, A a, and if from the same point there is drawn the right line GF to the focus nearest to that point; the half of the transverse axis CA will be to the distance of the focus from the centre, viz. CF, as the distance of the perpendicular CD is to the sum of the half of the transverse axis, and the right line drawn to the focus. . Let GE be drawn to the other focus, and on the axis a A produced, let there be set off AH equal GF; then, with the centre G, and the distance GF, describe a circle cutting the axis a A in K and F, and the right line EG in the points L and M : then since EF is double CF, and FK double FD, EK shall be also double CD; and since EL or A a is double CA, and LM double GE or A H, EM shall also be double CH but because of the circle, EL or A a EF: : EK: EM ; and taking their halves, it will be as CA: CF: : CD : CH. Prop. III. (fig. 7 and 8.) the same things being supposed, if from A, the extremity of the transverse axis nearest to the point G, there is set of a right line AH on the axis produced, equal to the distance of the point G from the focus io, nearest to the said extremity; the square of the perpendicular Gl) shall be equal to the excess of the rectangle EHF, contained under the segments between H (the extremity of the right line AH) and the foci, above the rectangle AD a, contained under the segments cut off between the perpendicular and the extremi. ties of the axis. For since the right line CH is any how cut in A, the squares of CA and CH together will be equal to twice the rectangle ACH, and the square of AH, (7. 2.) i. e. because CA, CF, CD, CH, are proportionals to twice the rectangle FCD, and to the square of AH or G F : that is, to twice the rectangle of FCD and the squares of FD and D.C., that is, to the squares of FC, CD, and De, (7.2) wherefore the two squares of CA and Cit are equal to three squares of FC, CD and DG and taking away the squares of CA and CF from both sides, the remaining rectangle EHF, will be equal to the remaining rectangle AD a, and to the square of DG (6.2) 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 5, 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 transverse axis, between the parallel and its extremities, to the square of the parallel. Prop. V. (fig. 8). If from any point G of the hyperbola 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 half of the second axis and its segment, between the centre and the right line, to the square of the line itself; that is, CB’ CA* : : CB--G D* : C A*-H the rectangle AD a , that is, as C B+C N* is to C D* or G N*. 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, C B" = 1 FL = I H L. Prop VIII. (fig. 10.) If from any point F of the hyperbola, there is drawn to the transverse diameter, AB, a right line ordinately applied to it FG ; and from the extremity of the diameter there is drawn AH pérpendicular to it, and equal to the latus rectum ; the square of the ordinate shall be equal to the o: applied to the latus 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 BH, and from the point G let there be drawn GM parallel to AH, and meeting BH in M, and through M let there be drawn MN parallel to AB, meeting AH in N, and let the rectangles MNHO, BAHP, be completed. Then since the rectangle AGB is to the square of GF, as AB is to AH, i. e. as GB is to GM, i.e. as the rectangle A G B is to the rectangle AGM; A G B shall be to the square of GF, as the same AGB to the rectangle AGM : wherefore the square of G F is equal to the rectangle A G M, which is applied to the latus rectum, AH, having the breadth AG, and exceeds the rectangle H A G O by the rectangle MNHO, like to BAHP; from which excess the name of hyperbola was given to this curve by Apollonius. 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 EG equal to NF in the axis continued; then with the distance GD, 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 f. 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 symptote; the abscisses will represent an infinite series of natural numbers, and the corresponding hyperbolic or asymptotic spaces will re. present 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, i.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. HYPER Bolic 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. |