Page:Cyclopaedia, Chambers - Volume 1.djvu/119

 ALT

(7?)

ALT

ALTERNATIVE, is particularly ufcd for the Choice of two things propofed. — In this Senfe, we fay, To take the Alternative of two Propositions. See Alternate.

ALT1METRY, Altimetria, the Art of taking or meafuring Altitudes, or Heights; whether accessible or inac- cessible. Sec Altitude.

Altimetria makes the firSt Part of Geometry; including the Doctrine and Practice of meafuring both perpendicular and oblique Lines; whether in refpect of Height, or Depth. See Geometry; fee alfo Height, Iz/C.

The Word is compounded of the Latin Alius, high, and ft£Tfs<y, rnetior, I mealure.

ALTITUDE, Altitudo, in Geometry, the third Di- mension of Body; called alfo Height, or Depth. See Height; fee alio Body, Dimension, f$G.

Altitoe, in Opticks, is ufually confider'd as the Angle fubtended between a Line drawn thro' the Eye, parallel to the Horizon, and a Vifual Ray emitted from an Object to the Eye.

For the Laws of the Vifion of Altitude, fee Vision. If thro' the two Extremes of an. Object, S and T, (Tab. Opticks, Fig. 13.) two Parallels, TV and S Q_ be drawn; •the Angle TVS, intercepted between a Ray palling thro' the Vertex S, and terminating the Shadow thereof in V, makes, with the right Line T V, what is called, by fome Writers, the Altitude of the Luminary.

Altitude, in Cofmography, is the perpendicular Height of a Body, or Object; or its Diitance from the Horizon, upwards. Sec Height; fee alfo Horizon.

Altitudes are divided into accejjille and inacceffible. Sec Accessible, and Inaccessible.

There are three Ways of Meafuring Altitudes, viz. Geo- metrically, Trigonometricallv, and Optically. — The firit is fomewhat indirect and unartful; the Second, perform 'd by means of Instruments for the Purpofe; and the third by Sha- dows.

The Instruments chiefly ufed in meafuring of Altitudes, are the Quadrant, theodolite, Geomerrick Quadrat, or Line of Shadows, Sic. the fDeJcriftions, Applications, &c. whereof, fee under their refpeBive Articles, Quadrant, Tiiedolite, Quadrat.

To take Accccfjible Altitudes.

To raeafure an acceffible Altitude, geometrically. — .—Sup- pofe it required to find the Altitude A B, (Tab. Geometry, Pig. 88.) plant a Staff DE perpendicularly in the Ground, Of fuch 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 fame right Line with the Eye C; the length CA is equal to the Altitude A B. If fome other lower Point, as F, prove in the Line with E and the Eye; you mult remove the Staff, £S?G. nearer to the Object: On the contrary, if the Line continued from the Eye over E, mark out feme Point above the Altitude requir'd; the Staff, &c. are to be remov'd farther off, till the Line C E raze the very Point requir'd. — Thus, meafuring the Distance of the Eye C from the Foot of the Object A; the Altitude is had; Since C A = A B.

Or thus : At the Distance of 30, 4.0, or more Feet,

plant a Staff D E, (Fig. 89.) and at a diitance herefrom, in C, another Shorter one, fo as that the Eye being in F; E and B may be in the fame right Line therewith. Mealure the Distance between the two Staves, G F; and between the Shorter Staff and the Object, H F; as alfo, rhe diffe- rence of the Heights of the Staves, G E. To GF, GE

and H F; find a fourth proportional B H. — To this add the Altitude of the fhorter Staff, F C. The Sum is the Altitude requited, AB.

To meafure an acceffible Altitude, trigonometrically. — Sup- pofe it required to find the Altitude A B, (Tab. trigonom. Fig. 25.) chufe a Station in E; and with a Quadrant, Theo- dolite, or other graduared Inffrumcnt duly placed, rind the Quantity of the Angle of Altitude ADC. See Angle.

Meafure the Shortest Diitance of the Station from the Object, viz. D C, which of confequencc is perpendicular to AC. See Distance.

Now, C being a right Angle, 'tis eafy to find the Line AC; Since, in the Triangle A C D, we have two Angles, viz. C and D, and a Side oppofite to one of them, C D, 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 oppofite thereto, DC; fo is the Sine of the other Angle D, to the Side required C A. See Triangle.

To the fide thus found, adding B C, the Sum is the per- pendicular Altitude requir'd.

The Operation is beSt perform'd by Logarithms. See Logarithm.

If there happen an Error in taking the Quantity of the Angle A, (Fig. 24.) the true Altitude B D will be to the falfe one B C; as the Tangent of the true Angle D A B, to the Tangent of the erroneous Angle CAB.

Hence, fuch Error will be greater in a greater Altitude than in a lefs : and hence alfo, the Error is greater if t ne Angle be leffer, rhan if greater. — To avoid the Inconvenien. ces of both which, the Station is to be pitch 'd on at a mo-' derate Distance; fo as the Angle of Altitude, DEB, may- be nearly half right. J

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 meafure an acceffible Altitude optically, by the Sha- dow of the Body. See Shadow.

To meafure an acceffible Altitude by the geometrical 3i fa.

drat. Suppofe it required to find the Altitude A B, (Tab.

Geom. Fig. 90.) chufing a Station at pleafure in D, and meafuring the Diitance thereof from the Object, D B; turn rhe Quadrat this and that way, till the Top of the Tower A, appear thro' the Sights.

If, then, the Thread cut the right Shadows, fay, As the Part of the right Shadow cut off, is ro the Side of the Qua- drat; fo is the Distance of the Station D B, to the Part of the Altitude A E. — If the Thread cut the verfed Shadows fay, As the Side of the Quadrat is to the Part of the verfed Shadow cut off; fo is the Distance of the Station D B, to the Parr of the Altitude A E.

A E, therefore, being found in cither Cafe, by the Rule of Three; and the Part of the Altitude B E added thereto ■ the Sum is the Altitude required.

To take Inacceffiblc Altitudes.

To meafure an inacceffiblc Altitude, geometrically. — ,. Suppofe A B, (Fig.89.) an inaccessible Altitude, fo that you cannot meafure to the Foot thereof. Find the Distance CA, or FH, as taught under the Article Distance : pro- ceed with the reft as in the Article for accessible Diltances.

To meafure an inacceffible Altitude, trigonometrically.— Chufe two Stations, G and E, (Tab. Trigonom. Fig. 2;.) in the fame right Line with the required Altitude AB, and at fuch distance from each other, D F, as that neither the Angle FAD, be too fmall, nor the other Station G too near the Object, A B. — With a proper Instrument, take the Quantity of the Angles AD C; AFC, andCFB. See Angle.— And alfo meafure the Interval FD.

Then, in the Triangle A F D, we have the Angle D, given by Obfervation; and the Angle A F D, by Subtracting the obferved Altitude A F C, from two righr Angles; and consequently the third Angle DAF, by fubtraaing rhe other two from two right ones: and alfo the Side F D : From whence the Side A F is found by the Canon above laid down, in the Problem of acceffible Altitudes. And again, in the Triangle A C F, having a right Angle C, an obferved Angle, F, and a fide A F; the Side A C, and the other C F, are found by the fame Canon. Laflly, in the Triangle FCB, having a righr Angle C, obferved Angle C FB, and a Side CF; the other fide C B, is found by the fame Canon.

Adding, therefore, AC and CB; the Sum is the Alti- tude required, A B. To find an inacceffible Altitude, by the Shadow, or till

geometrical Quadrat. Chufe two Stations in D and H,

(Tab. Geom. Fig. 90.) and find the Distance DH or CG : obferve what part of either the right or verfed Shadow is cut by the Thread. .

If the right Shadows be cut in both Stations, fay, As the Difference of the right Shadows in the two Stations, is to the Side of the Square; fo is the Distance of the Stations

G C to the Altitude E A. If the Thread cut the verfed

Shadow at borh Stations, fay, As the Difference of the ver- fed Shadows mark'd ar the two Stations, is to the leffer verfed Shadow; fo is the Distance of the Stations GC, to the Interval G E. Which being had; the Altitude E B is alfo found by means of the verled Shadow in G; as in the Problem for acceffible Altitudes.

Laftly, if the Thread in the firSt Station G, cut the right Shadows; and in the latter, the verfed Shadows : fay, As the Difference of the Product of the right Shadow into the verfed, fubtracted from rhe Square of rhe Side of the Qua- drar, is to the Product of the Side of the Quadrat into the verfed Shadow : fo is the Distance of the Stations G C, to the Altitude requir'd, AE.

the utmofl \Difiance at which an ObjeEl may be Jin,

being given; to fmd.its Altitudes Suppofe the Distance

D B, (Tab. Geography, Fig. 9.) rum this into Degrees; by which means, you will have the Quantity of the Angle C : From the Secant of this Angle fubtraft the whole Sine B C; the Remainder will be A B, in fuch Parts, whereof BC is 10000000. — Then fay, as ioooqooo is to the Value of AB in fuch Parts; fo is the Semidiameter of the Earth B C 19S953 59, to the Value of the Altitude A B in 'Paris Feet. The Sun's Altitude may alfo be found without a Qua- drant, or any the like Instrument, by erecting a Pin or Wire perpendicularly, as in the Point C, (Tab, AJlronowy, H-