elevation above the ground: and is otherwise called its height or depth; the former, when measured from bottom to top, the latter when measured from top to bottom.

Altitude of a figure, is the distance of its vertex from the base, or the length of a perpendicular let fall from its vertex to the base. The altitudes of figures are useful in computing their areas or solidities.

Altitude, or height of any point of a terrestrial object, is the perpendicular let fall from that point to the plane of the horizon. Altitudes are distinguished into accessible and inaccessible.

Accessible Altitude of an object, is that to whose base there is access, to measure the nearest distance to it on the ground, from any place.

Inaccessible Altitude, of an object, is that to whose base there is not free access, by which a distance may be measured to it, by reason of some impediment, such as water, wood, or the like.

To measure or take Altitudes. If an altitude cannot be measured by stretching a string from top to bottom, which is the direct and most accurate way, then some indirect way is used, by actually measuring some other line or distance which may serve as a basis, in conjunction with some angles, or other proportional lines, either to compute, or geometrically determine, the altitude of the object sought.

There are various ways of measuring altitudes, or depths, by means of different instruments, and by shadows or reflected images, on optical principles. There are also various ways of computing the altitude in numbers, from the measurements taken as above, either by geometrical construction, or trigonometrical calculation, or by simple numeral computation from the property of parallel lines, &c.

The instruments mostly used in measuring altitudes, are the quadrant, theodolite, geometrical square, line of shadows, &c.; the descriptions of each of which may be seen under their respective names.

To measure an Accessible Altitude Geometrically. Thus, suppose the height of the accessible tower AB be required. First, by means of two rods, the one longer than the other: plant the longer upright at C; then move the shorter back from it, till by trials you find such a place, D, that the eye placed at the top of it at E, may see the top of the other, F, and the top of the object B straight in a line: next measure the distances DA or EG and DC or EH, also HF the difference between the heights of the rods: then, by similar triangles, as EH: EG:: HF: the 4th proportional GB; to which add AG or DE, and the sum will be the whole altitude AB sought. Fig. 1. pl. 12.

Or, with one rod CF only: plant it at such a place C, that the eye at the ground, or near it, at I, may see the tops F and B in a right line: then, having measured IC, IA, CF, the 4th proportional to these will be the altitude AB sought,

Or thus, by means of shadows. Plant a rod ab at a, and measure its shadow ac, as also the shadow AC of the object AB; then the 4th proportional to ac, ab, AC will be the altitude AB sought. Fig. 6.

Or thus, by means of optical reflection. Place a vessel of water, or a mirror or other. reflecting surface, horizontal at C; and move' off from it to such a distance, D, that the eye E may see the image of the top of the object in the mirror at C: then, by similar figures, CD: DE :: CA: AB the altitude sought. Fig. 7.

Or thus, by the geometrical square. At any place, C, fix the stand, and turn the square about the centre of motion, D, till the eye there see the top of the object through the sights or telescope on the side DE of the quadrant, and note the number of divisions cut off the other side by the plumb-line EG: then as EF: FG:: DH: HB; to which add AH or CD, for the whole height AB. Fig. 2.

To measure an Accessible Altitude Frigonometrically. At any convenient station, C, with a quadrant, theodolite, or other graduated instrument, observe the angle of elevation ACB above the horizontal line AC; and measure the distance AC. Then, A being a right angle, it will be, as radius is to the tangent of the angle A, so is AC to AB sought. Fig. 4.

If AC be not horizontal, but an inclined plane; then the angle above it must be observed at two stations C and D in a right line, and the distances AC, CB, both measured. Then, from the angle C, take the angle D,' and there remains the angle CBD: hence in the triangle BCD, are given the angles and the side DC, to find the side CB; and then in the triangle ABC, are given the two sides CA, CB, with the included angle C, to find the third side AB. Fig. 5.

Or thus, measure only the distance AC, and the angles A and C: then, in the triangle ABC, are given all the angles and the side AC, to find the side AB.

To measure an Inaccessible Altitude, as a hill, cloud, or other object. This is commonly done, by observing the angle of its altitude at two stations, and measuring the distance between them. Thus, for the height AB of a hill, measure the distance CD at the foot of it, and observe the quantity of the two angles C and D. Then, from the angle C taking the angle D, leaves the angle CBD); hence As sine CBD · sine D :: CD: CB; and As radius: sine ACB:: CB: AB the altitude. Fig. 8.

When this method is used to find the altitude of a cloud, balloon, or other moveable object: the angles of its altitude must be taken at the same moment by two observers who are situated in a vertical plane which will pass through the object.

When an object can neither be approached to, nor receded from, in the line of direction of that object, its altitude may still be found without having recourse to horizontal angles, thus: Let any two distances CD=D, DE=&

rd

altitude = from √ ( 1⁄2 cot 2 6-f § col 2a — cot ? b)' which we have this logarithmic rule:-Double the log. cotangents of the angles of elevation at the extreme stations, find the natural numbers answering thereto, and take half their sum; from which subtract the natural number answering to twice the log. cotangent of the middle angle of elevation: then, half the log. of this remainder subtracted from the log of the measured distance CD or DE, will be the log. af the height of the object.

It is manifest that in these, or any other methods of ascertaining altitudes trigonometrically; the accuracy of the result will depend upon the accuracy and care with which the angles are taken. Some mathematicians (particularly Mr. Cotes, in his excellent little treatise De æstimatione errorum in mixta mathesi,") have enquired in what manner the result would be effected by any supposed error in the angles. This matter may be exemplified by referring to the most simple case in taking altitudes, namely, when the object is accessible, and perpendicular to the horizon, as AB, fig. 4. It has been found by a fluxional process, that the error of the height AB, is to the measure of the error of the angle C, as the double of this height is to the sine of double the observed angle. The error, therefore, which can prevail in the determination of the said height, will be the least possible when the sine of double the observed angle is the greatest possible, i. e. when this angle is 45°. Hence appears the propriety of moving to and fro on the line AC, until the observed angle is as near as possible to 45o, If a mistake of one minute of a degree be committed in the determination of the angle C: then, if radius be = 10000000, the are of a minute which measures the supposed error, will be 2009; the double of which 15818: but this quantity is to the sine of double the angle 459, i. e. to radius, as 1 to 1719; therefore, on this supposition the error in the height will be 1-1719th part of the height itself. If the error of the angle C be increased or diminished, the error of the height will be increased or diminished in the same ratio. And, if the angle observed be greater or less than 450, the error of the height will be increased in the ratio of radius to the sine of double the said angle,

Other methods of taking altitudes, &c. may be seen in Dr. Hutton's comprehensive treatise on Mensuration, and in Mr. Bonnycastle's and most other works on Trigonometry.

There is a very easy method of taking great

terrestrial altitudes, such as mountains, &c. by means of the difference between the heights of the baronieter observed at the bottom and top of the same; which see under the article BA

ROMETER.

Altitude of the Eye, in perspective, is a right line let fall from the eye, perpendicular to the geometrical plane.

Altitude of the Pyramids in Egypt, was measured so long since as the time of Thales, which he effected by means of their shadow, and that of a pole set upright beside them, making the altitudes of the pole and pyramid proportional to the lengths of their shadows. Plutarch has given an account of the manner of this operation, which is one of the first geometrical observations we have an exact account of.

Altitude, in astronomy, is the arch of a ver tical circle, measuring the height of the sun, moon, star, or other celestial object, above the horizon.

This altitude may be either true or apparent. The apparent altitude is that which appears by sensible observations made at any place on the surface of the earth. And the true altitude 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 84 seconds; but in the moon it is about 58 minutes, at a mean.

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. This may be put in practice by means of the equatoreal, thus: let the sine of the estimated altitude of the object be s, elevate the equatoreal circle above the horizon to an angle, the sine of which = =s, rad. being = 1. The declination circle being set to 0, direct the line of collimation to the star, by the equatoreal and azimuth circles moved in their own planes; observe the arc of the equatoreal circle intercepted between the index and VI; if the sine of this arc p, the sine of the observed altitude will be equal to ps, radius being 1 This indirect method has many advantages; and is, in general, less exposed to errors than the direct method in about the proportion of 1 to 7. Atwood's Lectures, p. 198, 227.

Meridian Altitude of the Sun, or any celes tial object, is an arch of the meridian intercepted between the horizon and the centre of the object upon the meridian. The altitude of a celestial body 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 me ridian,

Altitude of the Pole, is an arch of the meridian intercepted between the horizon and the pole: it is equal to the latitude of the place.

Altitude of the Equinoctial, is the elevation of that circle above the horizon, and is always equal to the complement of the latitude.

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

Altitude of the Earth's or Moon's Shadow, in eclipses. See ECLIPSE.

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 (independent of tables) that is preferable to the following me, thod of deducing the true time of the sun's passing the meridian, by the clock, from a comparison of four equal altitudes, observed on two succeeding days. The rule was invented by the celebrated Dr. Rittenhouse, the Ameri

can 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 the exact time of the sun's passing the meridian per clock, may be readily obtained by the following

Rule. Take the difference in time between the forenoon observations of the two days, and also between the afternoon observations. Call half the difference of the two differences, X; And half the sum of the two differences, Y; Let the half interval between the two observations of the same day, be Z.

Then, if the times of the altitudes 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 ... 24h: Y Z E, the equation sought: which will be found the same (without any sensible difference) as the equation obtained

:

from the tables.

But of 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 tame altitude. And the proportion will be ... 24h: 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. The reason of all this is very plain; and its mode of application so obvious that it is needless to give examples in this place: several, however, may be seen in the first volume of the American Transactions, whence the rule was extracted. Altitude, circles of, paralleis of, quadrant of, &c. See the respective words.

ALTMORE, a town of Ireland, in the county of Tyrone. Lat. 54. 35 N. Lon. 7. 18 W

ALTO, in music, the highest natural tenor voice.

ALTO ET BASSO, or in Alto and in Basso, in law, signifies the absolute reference of all differences, small and great, high and low, to some arbitrator or indifferent person.

ALTO RIPIENO, in music. The tenor of the great chorus, which sings or plays in the full part of the concert, occasionally,

ALTOGETHER. ad. (from all and toge ther.) Completely; without restriction; with out exception (Swift).

ALTON, a town in Hampshire, with a market on Saturday. Lat. 51. 22 N. Lon. 0.56 W.

ALTORF, or ALTDORF, a district of Nuremberg, in Germany. Here is a university, a library, and a physic garden. Lat. 49. 20 N. Lon. 11. 22 È.

ALTORF, a town of Suabia, Germany. Lat. 47. 50 N. Lon. 9. 30 E.

ALTRINGHAM, a town of Cheshire, with a market on Tuesdays. Lat. 53. 24 N. Lon. 2. 34 W.

ALVA (Ferdinand Alvarez, duke of), was born in 1508. He distinguished himself by his valour and military skill when young, and in 1538 was made general by Charles V. whom he served against the German protestants, the French, and the pope. As Alva, however, was a bigoted catholic, the last service displeased. him, and he asked the forgiveness of the pontiff whom he had vanquished. Philip II. sent him into the Low Countries in 1567, to reduce. them to the Spanish yoke, from which they were about to revolt. Here he established a council which was called the Bloody Tribunal. He filled the United Provinces with terror and scenes of carnage, for which his memory is held in detestation to this day. After obtaining great advantages over the malcontents, the tide of success turned in their favour so rapidly, that Alva, in a fit of dejection, solicited his recal, in 1573, which was granted. He enjoyed considerable marks of distinction from his sovereign for some time, but at last fell into disgrace through the misconduct of one of his sons. He was afterwards employed against Portugal, where he greatly added to his military renown, by driving don Antonio from the throne, in 1581. He died the next year, aged 74. Watkins.

ALUCETA, a name given by Fabricius to various species (constituting a tribe or family) of the lepidopterous inscet phalana or moth, corresponding with the tinca of Gmelin. See PHALENA,

ALUDELS, certain pots or vessels made of earthen-ware or glass, open at both ends, and may be inserted and applied above each other, so that the whole shall form a pipe or tube more or less long according to the number of aludels composing it. The aludel which terminates this tube above, ought to be closed in its upper part, or to have but a very small opening. The tube composed of these aludels is nothing but a kind of capital or head, which may be enlarged or lengthened at pleasure, and adapted to a cucurbit. This apparatus is intended to collect and retain dry and volatile matters, which may be reduced into flowers by sublimation. It was employed for the preparation of flowers of sulphur, of arsenic, of antimony, of benjamin, &c.; but is now generally discarded, since the shops are supplied from wholesale laboratories, where more convenient apparatus are employed. See pl. 11. fig. 7., where A is the cucurbit, B a series of aludels, and C the capital.

ALVEARIUM, in anatomy, (from alveare, a bee-hive,) that part of the meatus auditorius, or conch of the external ear, which contains its wax.

ALVEHEZIT, among Arabian writers, denotes what we ordinarily call falling-stars, or star-shot.

ALVEOLATE. (Alveolatum s. favosum) receptacle. Divided into open cells, like an honey-comb, with a seed fodged in each: as in onopordum.

ALVEOLUS, in anatomy, the socket-like cavity in the jaws, wherein each of the teeth

is fixed.

ALVEOLUS, in the history of fossils, a marine body, not known at present in its recent state, but frequently found fossile. The alveoli are of a conic shape, and composed of a number of cells, like so many beehives jointed into one another, with a pipe of communication, like that of the nautilus. They are sometimes met with entire, but more frequently truncated, or with their smaller ends broken off.

ALVEUS, in antiquity, a small boat made of the trunk of a single tree.

ALVEUS, (from alvus, a paunch, being as it were the reservoir whence a fluid is carried.) The tube or channel of a fluid; in medicine, chiefly that of the chyle.

ALVIDUCA, (Alviduca sc. medicamenta.) Medicines which open the bowels.

ALVIFLUXUS. A purging.

ALUMEN, (alum, Arab.) Alum. A neutral salt formed by a combination of the earth called alumine, alumina or pure clay, with sulphuric acid. The commercial and medicinal alum is the same, and is called in the new chemical nomenclature sulphas alumina acidulus cum potassa, (sulphat of alumine with potass or potash) ;-by Bergman, argilla vitriolata. It is afforded by the ores which are dug out of the earth for this purpose, and manufactured by first decomposing the ore, then lixiviating it, evaporating the lixivium, and then crystallizing the alum, which exhibits the form of tetraheral pyramids, applied to each other, base to

base; the angles being occasionally truncated. The following are the chief species of ore.

Alum native, or fossil: alum found crystallized by nature without the assistance of art. Alum plumore, or plume alum: alum natorally crystallized in the form of threads or fibres resembling feathers.

Alum prepared, or purified: alum dissolved in pure hot water, (rain or distilled,) and crystallizing after a sufficient evaporation.

Alum rock, or ice-alum: so named from Rocca, now called Edessa in Syria, where it is found in large transparent masses of native crystallization, but not very pure. It was in this place that the earliest alum manufactories were establised of which we have any account.

Alum Roman: prepared in the territory of Civita-Vecchia, from native masses not unlike the rock alum. It is imported in lumps of the size of eggs, covered with a reddish tincture.

Alum saccharine: a composition of common alum dissolved in rose water, clarified by the whites of eggs, and formed, after boiling to a due consistency, in the shape of a sugar-loaf; whence its name. It is used as a cosmetic.

Alum, on its first taste, imparts a sweetness; but is soon felt to be strongly astringent; on account of which virtue it is of extensive use in medicine and surgery. Internally it is given in hæmeptoe or blood-spitting,colica pictonum, chronic pains of the bowels and enuresis. Externally it is applied as a styptic to bleeding vessels and to phagadenic ulcers.

Exposed to the fire, alum at first becomes liquefied; much aqueous vapour or water of crystallization exhales from it, and it swells into a large white mass, rough and full of cavities all over its surface. This production is termed alumen ustum or burnt alum; and is sometimes employed in surgery to destroy fungous flesh, as well as for particular kinds of ophthalmies. Besides this preparation alum enters the aqua aluminis composita, and the coagulum aluminis of the pharmacopoeias.

Alum, or alumen, dissolves in from ten to fifteen times its weight of cold water, according to its purity; but boiling water will dissolve more than its weight of alum. It crystallizes by evaporation and cooling, the figure of its crystals varying with circumstances: half its weight of water is retained in crystallizing. The crystals dissolve in about seventeen times their weight of cold water. Alum swells when heated, loses its regular form and the water which it contained, and becomes a light white substance called burnt, or calcined alum. In a more violent degree of heat it loses part of its acid, and becomes tasteless; is no longer susceptible of crystallization, but precipitates from its solution, in a very fine adhesive pow der. Magnesia, baryt, and the alkalies, precipitate it from this solution; but the alkalies, added in excess, re-dissolve it.

By the addition of more alumine the glass selenite of Baumé is formed, which is almost tasteless and insoluble, and exhibits cubic crystals.

If three parts of alum, and one of flour or