ALTITUDE

ALTITUDE, Altitudo, in Geometry, the third Dimension of a Body; called also Height, or Depth. See HEIGHT; see also BODY, DIMENSION, etc.Altitude, in Optics, is usually considered as the Angle subtended between a Line drawn through the Eye, parallel to the Horizon, and a Visual Ray emitted from an Object to the Eye.For the Laws of the Vision of Altitude, see Vision.If through the two Extremes of an Object, S and T, (Tab. Optics, Fig. 13.) two Parallels, TV and SQ, be drawn; the Angle TVS, intercepted between a Ray passing through the Vertex S, and terminating the Shadow thereof in V, makes, with the right Line TV, what is called, by some Writers, the Altitude of the Luminary. Altitude, in Cosmography, is the perpendicular Height of a Body, or Object; or its Distance from the Horizon, upwards. See HEIGHT; see also HORIZON. Altitudes are divided into accessible and inaccessible. See ACCESSIBLE, and INACCESSIBLE. There are three Ways of Measuring Altitudes, viz. Geometrically, Trigonometrically, and Optically —The first is somewhat indirect and unartful; the second, performed by means of Instruments for the Purpose; and the third by Shadows. The Instruments chiefly used in measuring of Altitudes, are the Quadrant, Theodolite, Geometrical Quadrat, or Line of Shadows, &c. the Descriptions, Applications, &c. whereof, see under their respective Articles, QUADRANT, Theodolite, Quadrat.



To take Accessible ALTITUDES.

To measure an accessible Altitude, geometrically.—
Suppose it required to find the Altitude AB, (Tab. Geometry, Fig. 88.) plant a Staff DE perpendicularly in the Ground, of such height as may equal the height of the Eye. Then, laying prostrate on the Ground, with your Feet to the Staff; if E and B prove in the same right Line with the Eye C; the length CA is equal to the Altitude AB. If some other lower Point, as F, prove in the Line with E and the Eye; you must remove the Staff, &c. nearer to the Object: On the contrary, if the Line continued from the Eye over E, mark out some Point above the Altitude required; the Staff, &c. are to be removed farther off, till the Line CE raze the very Point required.—Thus, measuring the Distance of the Eye C from the Foot of the Object A; the Altitude is had; since CA=AB.Or thus: At the Distance of 30, 40, or more Feet, plant a Staff DE, (Fig. 89.) and at a distance herefrom, in C, another shorter one, so as that the Eye being in F; E and B may be in the same right Line therewith. Measure the Distance between the two Staves, GF; and between the shorter Staff and the Object, HF; as also, the difference of the Heights of the Staves, GE. To GF, GE and HF; find a fourth proportional BH.—To this add the Altitude of the shorter Staff, FC. The Sum is the Altitude required, AB.

To measure an accessible Altitude, trigonometrically.—
Suppose it required to find the Altitude AB, (Tab. Trigonometry, Fig. 23.) choose a Station in E; and with a Quadrant, Theodolite, or other graduated Instrument duly placed, find the Quantity of the Angle of Altitude ADC. See ANGLE. Measure the shortest Distance of the Station from the Object, viz. DC, which of consequence is perpendicular to AC. See DISTANCE.Now, C being a right Angle, 'tis easy to find the Line AC; since, in the Triangle ACD, we have two Angles, viz. C and D, and a side opposite to one of them, CD, to find the Side opposite to the other: for which we have this Canon.—As the Sine of the Angle A, is to the given side opposite thereto, DC; so is the Sine of the other Angle D, to the Side required CA. See TRIANGLE.To the side thus found, adding BC, the Sum is the perpendicular Altitude required.The Operation is best performed by Logarithms, See LOGARITHM.

If there happen an Error in taking the Quantity of the Angle A,(Fig. 24.) the true Altitude BD will be to the false one BC; as the Tangent of the true Angle DAB, to the Tangent of the erroneous Angle CAB.

Hence, such Error will be greater in a greater Altitude, than in a less: and hence also, the Error is greater if the Angle be lesser, than if greater—To avoid the Inconveniences of both which, the Station is to be pitched at a moderate Distance; so as the Angle of Altitude, DEB, May be nearly half right.Again, if the Instrument were not horizontally placed but inclined, e. g. to the Horizon in any Angle: The true Altitude will be to the erroneous one, as the Tangent of the true Angle, to that of the erroneous one.

To measure an accessible Altitude optically, by the Shadow of the Body. See SHADOW.

To measure an accessible Altitude by the geometrical Quadrat.
Suppose it required to find the Altitude A B, (Tab. Geom. Fig. 90.) choosing a Station at pleasure in D, and measuring the Distance thereof from the Object, DB; turn the Quadrat this and that way, till the Top of the Tower A, appear through the Sights.If, then, the Thread cut the right Shadows, say, As the Part of the right Shadow cut off, is to the side of the Quadrat; so is the Distance of the Station DB, to the Part of the Altitude A E,—If the Thread cut the versed Shadows, say, As the Side of the Quadrat is to the Part of the versed Shadow cut off; so is the Distance of the Station DB, to the Part of the Altitude AE.AE, therefore, being found in either Case, by the Rule of Three; and the Part of the Altitude BE added thereto, the Sum is the Altitude required.

To take Inaccessible Altitudes.

To measure an inaccessible Altitude, geometrically —.
Suppose AB, (Fig.89.) an inaccessible Altitude, so that you cannot measure to the Foot thereof. Find the Distance CA, or FH, as taught under the Article Distance: proceed with the rest as in the Article for accessible Distances.

To measure an inaccessible Altitude, trigonometrically.— Choose two Stations, G and E, (Tab. Trigonon, Fig. 25.) in the same right Line with the required Altitude AB, and at such distance from each other, DB, as that neither the Angle F AD, be too small, nor the other Station G too near the Object, A B.—With a proper Instrument, take the Quantity of the Angles ADC; AFC, and CFB. See ANGLE.—And also measure the Interval FD, Then, in the Triangle AFD, we have the Angle D, given by Observation; and the Angle A FD, by subtracting -the observed Altitude AEC, from two right Angles; and consequently the third Angle DAF, by subtracting the other two from two right ones: and also the Side FD: From whence the Side A F is found by the Canon above laid down, in the Problem of accessible Altitudes: And again, in the Triangle A CF, having a right Angle C, and one oblique Angle F, and a side AF; the Side AC, and the other C R, are found by the same Canon. Lastly, in the Triangle FCB, having a right Angle C, observed Angle CFB, and a Side CF; the other side CB, is found by the same Canon.Adding, therefore, AC and CB the Sum is the Altitude required, AB.

To find an inaccessible Altitude, by the Shadow, or the geometrical Quadrat. Choose two Stations in D and H, (Tab. Geom. Fig. 90.) and find the Distance DH or CG: observe what part of either the right or versed Shadow is cut by the Thread. If the right Shadows be cut in both Stations, say, As the Difference of the right Shadows in the two Stations, is to the Side of the Square; so is the Distance of the Stations GC to the Altitude E A. If the Thread cut the versed Shadow at both Stations, say, As the Difference of the versed Shadows marked at the two Stations, is to the lesser versed Shadow; so is the Distance of the Stations GC, to the Interval GE. Which being had; the Altitude EB is also found by means of the versed Shadow in G; as in the Problem for accessible Altitudes. Lastly, if the Thread in the first Station G, cut the right Shadows; and in the latter, the versed Shadows: say, As the Difference of the Product of the right Shadow into the versed, subtracted from the Square of the Side of the Quadrat, is to the Product of the Side of the Quadrat into the versed Shadow: so is the Distance of the Stations GC, to the Altitude a AE.The utmost Distance at which an Object may be seen being given; to find its Altitude.——Suppose the Distance DB, (Tab. Geography, Fig. 9.) turn this into Degrees; by which means, you will have the Quantity of the Angle C.From the Secant of this Angle subtract the whole Sine BC;the Remainder will be AB, in such Parts, whereof BC is 10000000.—Then say, as 10000000 is to the Value of ABin such Parts; so is the Semidiameter of the Earth BC 19695359, to the Value of the Altitude AB in Paris Feet.The Sun’s Altitude may also be found without a Quadrant, or any the like Instrument, by erecting a Pin or Wire perpendicularly, as in the Point C, (Tab. Astronomy, Fig. 62.) from which Point you had described the Quadrantal Arch AF. Make CE equal to the Height of the Pin or Wire, and through E draw ED parallel to CA, and make it equal to CG, the length of the Shadow; then will a Ruler, laid from C to D, intersect the Quadrant in B; and BA is the Arch of the Sun’s altitude, when measured on the Line of Chords. See CHORD.

Suppose, e.g., the Altitude be required of a Tower AB, whose Top is visible at the Distance of five Miles: Then will DCB 20', from whose Secant 10000168, subtracting the whole Sine 10000000; the Remainder AB is 168, which will be found 331 Paris Feet.

Altitude of the Eye, in Perspective, is a right Line let fall from the Eye, perpendicular to the geometrical Plane.

Altitude, in Astronomy, is the Distance of a Star, or other Point in the Mundane Sphere, from the Horizon. See SPHERE, HORIZON, DISTANCE, &c.

This Altitude may either be true or apparent—If regard be had to the rational, or real Horizon; the Altitude is said to be true or real: If to the apparent, or sensible Horizon; the Altitude is apparent—Or rather, the apparent Altitude is such as it appears to our Observation; and the true, that from which the Refraction has been subtracted. See TRUE, APPARENT, &c.

The Altitude of a Star, or other Point, is properly an Arch of a Vertical Circle, intercepted between the assigned Point and the Horizon. See VERTICAL.—Hence, Meridian Altitude.—The Meridian being a vertical Circle; a Meridian Altitude, that is, the Altitude of a Point in the Meridian, is an Arch of the Meridian intercepted between it and the Horizon. See OBSERVATION. To observe the Meridian Altitude of the Sun, of a Star, or other Phenomenon, by means of the Quadrant, see Meridian Altitude. To observe a Meridian Altitude by means of a Gnomon, see Gnomon, Altitude of the Pole—Since the Meridian passes through the Poles of the World; the Altitude of the Pole, is an Arch of the Meridian, intercepted between the Pole and the Horizon. To observe the Altitude or Elevation of the Pole, see Elevation, and Pole. The Altitude of the Pole coincides with the Latitude of the Place. See LATITUDE.

Altitude of the Equator, is the Complement of the Altitude of the Pole to a Quadrant of a Circle. See ELEVATION OF THE EQUATOR. To find the Altitudes of the Sun, Stars, &c. by the Globe. See GLOBE.

Altitude of the Nonagesimal, is the Altitude of the 90th Degree of the Ecliptic, reckoned from the East Point. See NONAGESIMAL. Refraction of Altitude, is an Arch of a Vertical Circle, as SS, (Tab. Astronomy, Fig. 28.) whereby the Altitude SE, of a Star or other Body, is increased by means of the Refraction. See REFRACTION.

Parallax of Altitude, called also simply Parallax; is the difference CB, (Tab. Astro. Fig. 27.) between the true and apparent Place of a Star; or, the Difference BC, between the true Distance of a Star AB, and the observed Distance AC, from the Zenith. The Parallax diminishes the Altitude of a Star, or increases its Distance from the Zenith. To find the Parallax of Altitude, &c. see Parallax.

Altitude of a Figure, in Geometry, is the Distance of its Vertex, from its Base; or the length of a Perpendicular let fall from the Vertex to the Base. See FIGURE, BASE, and VERTEX. Thus, KM, (Tab. Geometry, Fig.19.) being taken for the Base of the Right-Triangle, KLM: the Perpendicular KM, will be the Altitude of the Triangle. Triangles of equal Bases and Altitudes, are equal; and Parallelograms, whose Bases and Altitudes are equal to those of Triangles, are just double thereof. See TRIANGLE, PARALLELOGRAM, &c.

Altitude of Motion, is a Term used by Dr. Wallis, for the Measure of any Motion, estimated according to the Line of Direction of the moving Force. See MOTION.

ALTITUDE