ALTITUDE

, in Geometry is the third dimension of body, considered with respect to its elevation above the ground: and is otherwise called its height when measured from bottom to top, or its depth 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 sigures are useful in computing their areas or solidities.

, 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 whose base there is access to, to measure the nearest distance to it on the ground, from any place.

Inaccessible Altitude, of an object, is that whose base there is not free access to, 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 instrumcnts 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 rode, 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.

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.

Or thus, by means of Optical Reflection. Place a vessel of water, or a mirror or other reflecting smface, 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.

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.

To measure an Accessible Altitude Trigonometrically. 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.

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, CD 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 sind the side CB; and then in the triangle ABC, are given the | two sides CA and CB, with the included angle C, to find the third side AB.

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 augle CBD; hence As sine [angle]CBD: sine [angle]D :: CD : CB; and As rad.: sine [angle]ACB :: CB : AB the altitude.

And for a balloon, or cloud, or other moveable object C, let two observers at A and B, in a plane with C, take at the same time the angles A and B, and measure the distance between them AB; then calculate the altitude CD exactly as in the last example.

To find the height of an object, by knowing the utmost distance at which its top can be just seen in the horizon. As suppose the top H of a tower FH can be just seen from E when the distance EF is 25 miles, supposing the circumference of the earth to be 25000 miles, or the radius 3979 miles or 21009120 feet. First, as 25000 : 25 :: 360° : 21′ 36″ equal to the angle G; then as radius : sec. [angle]G :: EG : GH, which will be found to be 21009536 feet; from which take EG or GF, and there remains 416 feet, for FH the height of the tower sought.— Or rather thus, as 10000000 radius: 198=sec. [angle]G—radius :: 21009120=EG : 416 = FH, as before.

Or the same may be found easier thus: The horizon dips nearly 8 inches or 2/3 of a foot at the distance of 1 mile, and according to the square of the distance for other distances; therefore as 1² or 1 : 25² or 625 :: 2/3 : 2/3 of 625 or 416 feet, the same as before.

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 barometer observed at the bottom and top of the same. Which see under the article Barometer.

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

, is the arch of a vertical 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 but about 8 3/4 seconds; but in the moon it is about 52 minutes.

Altitudes are observed by a quadrant, or sextant, or by the shadow of a gnomon or high pole, and by various other ways, as may be seen in most books of astronomy.

Meridian Altitude, is an arch of the meridian intercepted between any point in it and the horizon. So if HO be the horizon, and HEZO the meridian; then the arch HE, or the angle HCE, is the meridian altitude of an object in the meridian at the point E.

, or elevation, of the Pole, is the angle OCP, or arch OP of the meridian, intercepted between the horizon and pole P.

This is equal to the latitude of the place; and it may be found by observing the meridian altitude of the pole star, when it is both above and below the pole, and taking half the sum, when corrected on account of refraction. Or the same may be found by the declination and meridian altitude of the sun.

, or elevation, of the equator, is the angle HCE, or arch HE of the meridian, between the horizon and the equator at E; and it is equal to ZP the colatitude of the place.

Altitude of the Tropics, the same as what is otherwise called the solstitial altitude of the sun, or his meridian altitude when in the solstitial points.

, or height, of the horizon, or of stars &c seen in it, is the quantity by which it is raised by refraction.

Refraction of Altitude, is an arch of a vertical circle, by which the true altitude of the moon, or a star, or other object, is increased by means of the refraction; | and is different at different altitudes, being nothing in the zenith, and greatest at the horizon, where it is about 33′.

Parallax of Altitude, is an arch of a vertical circle, by which the true altitude, observed at the centre of the earth, exceeds that which is observed on the surface; or the difference between the angles LM and IK of altitude there; and is equal to the angle I L formed at the moon or other body, and subtended by the radius IL of the earth.

It is evident that this angle is less, as the luminary is farther distant from the earth; and also less, for any one luminary, as it is higher above the horizon; being greatest there, and nothing in the zenith.

Altitude of the Nonagesimal, is the altitude of the 90th degree of the ecliptic, counted upon it from where it cuts the horizon, or of the middle or highest point of it which is above the horizon, at any time; and is equal to the angle made by the ecliptic and horizon where they intersect at that time.

Altitude of the cone of the earth's or moon's shadow, the height of the shadow of the body, made by the sun, and measured from the centre of the body. To find it, say, As the tangent of the angle of the sun's apparent semidiameter is to radius, so is 1 to a 4th proportional, which will be the height of the shadow, in semidiameters of the body.

So, the greatest height of the earth's shadow, is 217.8 semidiameters of the earth, when the sun is at his greatest distance, or his semidiameter subtends an angle of about 15′ 47″; and the height of the same is 210.7 semidiameters of the earth, when the sun is nearest the earth, or when his semidiameter is about 16′ 19″: And proportionally between these limits for the intermediate distances or semidiameters of the sun.

The altitudes of the shadows of the earth and moon, are nearly as 11 to 3, the proportion of their diameters.

, or exaltation, in astrology, denotes the second of the five essential dignities, which the planets acquire by virtue of the signs they are found in.

Altitude of motion, is a term used by Dr. Wallis, for the measure of any motion, estimated in the line of direction of the moving force.

, in speaking of fluids, is more frequently expressed by the term depth. The pressure of fluids, in every direction, is in proportion to their altitude or depth.

Altitude of the mercury, in the barometer and thermometer, is marked by degrees, or equal divisions, placed by the side of the tube of those instruments.

The altitude of the barometer, or of the mercury in its tube, at London, is usually comprised between the limits of 28 and 31 inches; and the mean height, for every day in several years, is nearly 29.87 inches.

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.

, circles of, parallels of, quadrant of, &c. See the respective words.

Equal Altitude Instrument, is an instrument used to observe a celestial object, when it has the same or an equal altitude, on both sides of the meridian, or before and after it passes the meridian: an instrument very useful in adjusting clocks &c, and for comparing equal and apparent time.