the altitude sought; the decimals being rejected as unimportant. ALT1t Ude, in Astronomy, is the arch measuring the height of a celestial object above the horizon. It is either true or apparent. The apparent altitude is that which appears by observations made on the surface of the earth. And the true is that which results by correcting the apparent, on account of refraction and parallax. The quantity of the refraction is different at different allitudes; and the quantity of the parai lax is different according to the distance of the different luminaries: in the fixed stars this is too small to be observed ; in the sun it is only about 83 seconds; but in the moon it is at a mean about 58 minutes. The altitude of a celestial object may be very accurately determined, by measuring the arc of an oblique great circle intercepted between the star and the horizon, and the inclination of the same great circle to the horizon. Meridian ALT1t upk of any celestial object, is an arch of the meri dian intercepted between the horizon and the centre of the object upon the meridian. The altitude of a celestial boy is greatest when it comes to the meridian of any place (the poles of the earth excepted, for there the altitude of a fixed body is subject to no variation;) and the altitude of any star which sets not, is least, and the depression of any star which does set, is greatest when in the opposite part of the meridian. Altitude of the Pole, is an arch of the meridian intercepted between the horizon and the pole: it is equal to the latitude. Alt it UD E of the Equinorial, is the elevation of that circle above the horizon, and is equal to the complement of the latitude. Refraction of A Ltitude, is an arch of a verticle circle, whereby the altitude of an heavenly body is increased by refraction. And Parallar of Altitude, is an arch of a verticle circle, whereby the altitude is decreased by parallax. ALT1t UD E Instrument, or Equal Altitude Instrument, one used to observe a celestial object when it has the same altitude on the east and west sides of the meridian. Observations of this kind are made for the purpose of obtaining tne true time of the sun’s passing the meridian : various modes of calculation have been recommended at different times; but we know of none preferable to the following, invented by Dr. Rittenhouse, the American astronomer. Suppose there are four sets of altitudes obtained on two successive days, (viz. one set in the morning, and one in the afternoon, of each day,) the instrument being kept at exactly the same height both days; then take the difference in , time between the forenoon observations of the two days, and also between the afternoon observations. Call half the diff. of the two differences, X; And half the sum of the two dif. ferences, Y; Let the half interval between the two observations of the same day, be Z. Tuen, if the times of the alti tudes observed on the second day be both nearer 12, or both farther from 12 per clock than on the first day,+X will be the daily variation of the clock, from apparent time, and Y will be the daily difference in time, of the sun’s coming to the same altitude, arising from the change of declination. And the proposition will be, 24” : Y = Z: E, the equation sought ; which will be found the same (without any sensible difference) as the equation obtained from the tables. But if one of the observations on the second day be nearer 12, and the other more remote from 12, than on the first day,...then Y will become the daily variation of the clock from apparent time, and X will be the daily difference in time of the sun’s being at the same altitude. And the proportion will be....24" : X = Z. : E. The equation E, thus obtained, is to be subtracted from the mean noon, if the sun’s meridian altitude be daily increasing ; but to be added, if it be daily decreasing. AMBIGUOUS Case, in Trigonometry, is that which arises in the solution of a problem, in which an acute angle and its opposite side are two of the given parts, and one of the sides about the given angle is the third part. With these data, the angle opposite the unknown side may be either obtuse or acute, because every sine answers to two angles which are supplements to each other. AM big Uous Sign, in Algebra, is that in which both plus and minus enter; being written thus, --, and is read plus or minus. AMPLITUDE, in Astronomy, is an arch of the horizon, intercepted between the true east and west points, and the centre of the sun or a star at its rising or setting; so the amplitude is of two kinds; eastern and western. These are also called northern or southern, as they fall in the northern or southern quarters of the horizon; and the compliment of the amplitude, or the distance of the point of rising or setting from the N. or S. point of *; horizon, is called the azimuth. And this is of the same name with the declination : viz. north, when the declination is north; and south, when that is south. AM P Litude, in Projectiles, the right line upon the ground, subtending the curvelinear path in which a projectile moves. AMPLITU De, Magnetical, is an arch of the horizon, contained between the sun or a star at its rising or setting, and the magnetical east or west point of the horizon indicated by the magnetical compass, or the amplitude or azimuth compass; or it is the difference of the rising or setting of the sun from the east or west points of the compass. In order to ascertain this amplitude, place the compass on a steady place from which the horizon may be clearly seen, and looking through the sight vanes of the compass, turn the instrument round till the centre of the sun or star may be seen through the narrow slit, which is one of the sight vanes, exactly in the thread which bisects the aperture in the other sight vane ; and when the centre of the celestial object, whether rising or setting, is just in the horizon, push the stop in the side of the box, so as to stop the card, and then read the degree of the card which stands just against the siducial line in the box; and this gives the amplitude required. Then subtracting from this amplitude the known or true amplitude, and the difference will be the variation of the magnetic needle. ANALEMMA, a projection of the sphere, on the plane of time meridi an, orthographically made by perpendiculars from every point of that plane; the eye being supposed to be at an infinite distance, and in the E. or W. point of the horizon. ANALYSIS is, ol. the method of resolving mathematical problems, by reducing them to equations; and may be divided into ancient and modern. The ancient analysis is the me. thod of proceeding from the thing sought taken for granted, through its consequences, to something that is, really granted or known; in which sense it is opposed to synthesis, or composition, which commences with the last step of the analysis, and traces the several steps backwards, making that in this case antecedent, which in the other was consequent, till we arrive at the thing sought, which was assumed in the first step of the analysis. Montucla, in his Histoire des Math. has given an example illustrating the method pursued by the ancients in their analysis, as follows: Pro B. From the extremities of the base A and B, of a given segment of a circle, it is required to draw two lines AC, BC, meeting at the point C, in the circumfer. ence, so that they shall have a given ratio to each other, suppose that of F to G. A NALYsis. Suppose the thing done, viz. that (Plate I. fig. 8.) AC : BC = F: G, and let there be drawn RH, making the angle ABH equal to the angle ACB, and meeting AC produced in H. Then the angle A being also common, the two triangles ABC, ABH, are equiangular; and, therefore, AC: BC = AB : BH in the given ratio; also AB bein given, BH is given in position . magnitude. SY not Bi Esis, Construction. Draw BH, making the angle ABH, equal to that which may be contained in the given segment, and take AB to BH, in the given ratio of F to G. Draw ACH and BC. Demonstration. The triangles abo, ABH, are equi-angular, therefore AC : CB = AB : BH difference between any angle and a semicircle. A Right ANGLE is equal to its supplement; an Obtuse ANGLE is greater than its supplement; and an Acute ANGLE is less. Hence, all right angles are equal, and each is measured by a quadrant, or the fourth part of a circle. Wertically opposite ANGLEs are formed by two lines which cross each other. At each crossing there are two pairs of such angles; those of each pair are equal, and the supplements of the other pair. Alternate ANG Les, are those made on the opposite sides of a line cutting two other lines; and if these two lines are parallel, the alternate angles are equal. Exterior ANG LEs, are those formed by the sides of any rightlined figure, and the adjacent sides produced ; they are the supplements of the interior angles. Hence the exterior and interior angles are equal to twice as many rightangles as the figure has sides; and it can be shewn that all the exterior angles of any figure are equal to four right-angles. Hence, again, if any figure have its angles ali equal, the measure of one of its angles, calling n the number, and R, a right-angle, will be of any tight-lined figure, is equal to twice as many right angles, wanting four, as the figure has sides. d An Angle at the Centre of a Circle, is that whose angular point is at the centre. An ANGLE at the Circumference, is that whose angular point is in any part of the circumference. An angle at the centre is double an angle at the circumference, when both stand on the same arc. An ANGLE in a Semi-Circle, is an angle at the circumference contained in a semi-circle, or standing upon a semi-circle or diameter. An angle in a semi-circle, is a right-angle. An angle in a segment, greater than a semi-circle, is less than a right-angle. An angle in a segment, less than a semi-circle, is greater than a right-angle. ANGULAR, something relating to, or having angles. ANGULAR Motion, is that which is performed by an oscillating or vibrating body, as referred to the angle which it describes or passes over in a given time, the vertex of which is the point of suspension, or centre of notion. Hence all points in a pendulum have the same angular motion, although their absolute motions are different from each other, being greater or less, according to their distance from the centre of suspension. ANGULAR Motion is also sometimes used to denote a motion which is partly curvilinear, and partly rectilinear; as the notion of a coach-wheel on a plane. ANGULAR Sections, is a term used by Vieta to denote a species of analytical trigonometry, relating to the law of increase and decrease of the sines and chords of multiple arcs. Vieta demonstrated that if a semi-circle be divided into equal arcs, and the radius be taken equal to unity, and the first chord = r ; then will the chords drawn from the one extremity, beginning with the diameter respectively, be Dia = 2 1st = a 2nd a’ – 2 3rd = a 8 – 3a. |