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at the lower station, and the sign when it is lowest at that station. Exam.-Find the height of a mountain, from the following observations taken at the foot and summit:

... 71° 55°

|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 ex

Bar. At. ther. De. th.cepted, for there the altitude of a
fixed body is subject to.no varia-
tion ;) and the altitude of any star
which sets not, is least, and the
depression of any star which does

Low. stat. 29.862 ... 68°
Hig. stat. 26:137 ... 63°
Here we have, d= 5°, diff. detach.

thermometer.

...

And............ƒ = 63° mean of de-set, is greatest when in the oppo

tach. thermometer.

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the altitude sought; the decimals being rejected as unimportant.

ALTITUDE, 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 altitudes; and the quantity of the parallax 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 8 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 ALTITUDE of any celestial object, is an arch of the meri

site 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.

ALTITUDE of the Equinoxial, is the elevation of that circle above the horizon, and is equal to the complement of the latitude.

Refraction of ALTITUDE, is an arch of a verticle circle, whereby the altitude of an heavenly body is increased by refraction. And Parallax of Altitude, is an arch of a verticle circle, whereby the altitude is decreased by parallax.

ALTITUDE 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 the 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.

Then, 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.

AMBIGUOUS Sign, in Algebra, is that in which both plus and minus enter; being written thus, t, and is read plus or minus.

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As cosine 51° 32. 9.7938317 is to radius 10-0000000 9.6001181

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so is sin. dec. 23° 28
To amplitude 39° 48′ 9-8062364

And this is of the same name with the declination: viz. north, when the declination is north; and south, when that is south.

AMPLITUDE, in Projectiles, the right line upon the ground, subtending the curvelinear path in which a projectile moves.

AMPLITUDE, 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 indi cated 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 horiWith these data, the angle op- zon may be clearly seen, and lookposite the unknown side may being through the sight vanes of the either obtuse or acute, because compass, turn the instrument every sine answers to two angles round till the centre of the sun or which are supplements to each star may be seen through the other. 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.

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 ANALEMMA, a projection of the the horizon, is called the azimuth. | sphere, on the plane of the 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, generally, the method of resolving mathematical problems, by reducing them to equations; and may be divided into ancient and modern.

The ancient analysis is the method 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:

therefore

AC: CB AB: BH which is the given ratio by con struction.

Modern ANALYSIS comprehends algebra, arithmetic of infinites, infinite series, increments, fluxions, &c.; for an account and illustra tion of each of which see the respective articles.

ANEMOMETER, in Mechanics, a machine for measuring the force of the wind.

ANEMOSCOPE, a term sometimes given to a wind-dial, which points out the course of the wind by an index that is connected by the spindle on which the vane works.

ANGLE, in Geometry, the opening between two lines which meet at a point, or the inclination of two lines which would meet if produced. When an angle is spoken of without reference to the lines by which it is formed, it is named by a single letter at the point; and when the lines are referred to by a letter in the one line, a letter at the point, and a letter on the other line.

PROB. From the extremities of An Angle is measured by an arch the base A and B, of a given seg-of a circle, the point being the cenment of a circle, it is required to tre, and the containing lines the draw two lines AC, BC, meeting extremities. 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.

ANALYSIS.

ANGLES are of several kinds or denominations, as rectilinear, curvilinear, spherical, mixed, solid, &c. A Rectilinear ANGLE is formed by the meeting of two right lines.

A Curvilinear ANGLE is formed by the meeting of two curve lines.

A Mixtilinear ANGLE is formed by the meeting of a right line and curve.

Suppose the thing done, viz. that (Plate I. fig. 8.) AC: BC F: G, and let there be drawn BH, making the angle ABH equal to the angle ACB, and meeting AC produced in H. Then the angle A A Spherical ANGLE is that which being also common, the two trian-is formed on the surface of a gles ABC, ABH, are equiangular; sphere, by the intersection of two and, therefore, great circles. See SPHERE, and SPHERICAL Trigonometry.

AC: BC

AB: BH

SYNTHESIS.

in the given ratio; also AB being A Solid ANGLE is formed by the given, BH is given in position and mutual inclination of more than magnitude. two plane angles, the sum of which is less than four right angles. meetConstruction. Draw BH, making in a common point. ing 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 ABC, ABH,

A Salient ANGLE has its point outwards; a Re-entering ANGLE has its point inwards; a Direct ANGLE measures less than half a circle, and a Retroflected ANGLE more. are equi-angular, A Supplemental ANGLE is the

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.

Vertically 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 ANGLES, 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.

of any right-lined figure, is equal to twice as many right angles, wanting four, as the figure has sides.

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

Exterior ANGLES, are those formed by the sides of any right-right-angle.

ANGULAR, something relating to, or having angles.

lined figure, and the adjacent sides An angle in a segment, less than produced; they are the supple- a semi-circle, is greater than a ments of the interior angles. Hence right-angle. 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 all equal, the measure of one of its angles, calling n the number, and R, a right-angle, will be 2n4R (2-2) R

n

n

2

EXAMPLES.

== R

n-2

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2

1. Required the angle of square?

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a

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 motion. 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 2. Hence 90° X partly rectilinear; as the motion of a coach-wheel on a plane.

2

2

90° angle. = 2. Required the angle of a pentagon?

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The sum of all the external angles of any figure is equal to four right angles.

Internal ANGLES, are the angles within a figure, formed by the meeting of each two adjacent sides; as the angles a, b, c, &c.

The sum of all the inward angles

ANGULAR Sections, is a term used by Vieta to denote a species of analytical trigonometry, relat ing to the law of increase and de.

crease 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 = x; then will the chords drawn from the one extremity, beginning with the diameter respectively, be

Dia 2
1st 2

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&c. &c.

ANIMATED Needle, a needle touched with a magnet. ANNUAL, in Astronomy, any thing which relates to the year, or which returns yearly.

ANNUITIES, signify any interest of money, rents, or pensions, payable from time to time, at particular periods.

The most general division of annuities is into annuities certain, and annuities contingent; the payment of the latter depending upon some contingency; such, in particular, as the continuance of a life. ANNUITIES have also been divided into annuities in possession, and annuities in reversion; the former meaning such as have commenced, or are to commence immediately; and the latter, such as will not commence till some particular future event has happened, or till some given period of time has expired.

immediately will amount to £100 in two years; and so on for any number of years or payment; and the sum of the values of all the payments will be the present value of the annuity.

Let the interest be supposed to be 4 per cent. The sum which improved at 4 per cent. interest for the year, will produce £100 at the end of the year, is that sum which has the same proportion to 100 as 100 has to 104; the sum of the interest and principal. Say therefore, as 104: 100

100:

100 X 100
104

value of the first payment.

= 96.15,

And that sum which in two years will amount to £100, at the same rate of interest, is evidently that £96-15. because we have seen that which in one year will amount to this sum will, in a year, amount to £100; we have, therefore, on the same principle as above.

104: 100 96.15;

96.15

104

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=92.45

92.45
104

85.48

85.48
104

82-19

value of 4th payment.

and so on.

ANNUITIES may be farther con- bears a constant ratio to principal, But since the interest of money sidered as being payable yearly, we may represent these proporhalf-yearly, or quarterly. The present value of an Annuitying generally £1 as the yearly tions more conveniently by assumis that sum which being improved at compound interest, will be suf. ficient to pay the annuity.

The present value of an Annuity certain, payable yearly, and the first payment of which is to be made at the end of a year, is computed as follows:

payment, and r as the amount of
£1 for one year; and whatever
multiplied by the yearly payment,
results are thus produced, being
will be the value of the annuity
above proportions become
in such case. By this means the

year r: 1=

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Let the annuity be supposed £100; the present value of ther:1=1: first payment of it, or of a hundred pounds to be received a hence, is that sum in hand, which being put to interest will amount to £100 in a year. In like manner, the present value of the second payment, or of £100 to be received two years hence, is that sum which being put to interest

37

1 1 r:1= ::

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and so on.

The value, therefore, of an annuity of £1 for n years, is equal to the sum of the series.

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