MEN
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M E N
th
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® height of the tower to be known, to find the distance of any
'J<-Ct on the plane below ; it is only the converse of the formercase. You may also, having the base and angles, easily find the'ypothenuse AC , or how far it is from the toil of the tower to the‘ration, which is useful in many cases. See Trigonometry.
, •. To measure an inaccessible height by the geometrical qua-Ur ant, &c. at two stations—The limb 15C, tig. 3, of the common* u rveying quadrant, ABC, is divided into 90 “, and each of these®fther divided into as many equal parts as the space will allow,cither diagonally or otherwise. On one of the semi-diametersj 7 C, are fitted two moveable sights; and to the centre is some-wiies also fixed a label, or moveable index, AD, bearing twoether sights; but in lieu of these last sights there is sometimes■Ued a telescope: also from the centre there is lmng a thread with* plummet; and on the under-side or face of the instrument is fil-ed a ball and socket, by means of which it may be put into anyPosition. The general use of it is for taking angles in a verticalP'une, comprehended under right lines going from the centre of' e instrument, one of which is horizontal, and the other is direct-th t0 . Some visible point. But besides the parts already described,^dure is frequently added on the face near the centre, a kind ofcompartment, EF, called the quadrant, or geometrical square; anddus it comprises the last mentioned instrument. This quadrantmay be used in different situations: for observing heights or depths,/‘plane must be disposed perpendicularly to the horizon ; but toa ke horizontal distances, its plane is disposed parallel thereto:Sain, heights and distances may be taken two ways, viz. by means1 the fixed sights and plummet, or by the label. As to the mail-er of measuring angles by this quadrant: let there be an angle in.vertical plane, comprehended between a line parallel to the ho-'2cm HK, and the right line 11A, fig. 4, coming from the sun,l °on, a star, or any remarkable point of a tower or hill: now todeasure this angle RAH by the quadrant, let the instrument L>e1 <aced in the vertical plane, so as that its centre A may be in thedgular point, and let the sights on the side CA be directed to-1 ,Orel's the object at R; then the degrees and minutes in the archcut off by the plummet or perpendicular, AD, will measuret .' e angle RAH : for, from the make of the quadrant, BAG is a, 8 pt angle ; therefore BAR is likewise a right angle, being equalBut, because HK is horizontal, and AD perpendicular, tot, e , horizon, HAD will be a right angle ; and, therefore, BAR =p* A D, and BAR—HAB = HAD—IIAB, or RAH = BAD:Ad the arch BD is the measure of the angle BAD, consequentlyis likewise the measure of RAH. The remaining arch on the :quadrant, DC , is the measure of tiie angle RAZ, comprehendedr t. » and AZ which points to the
es; or is equal to, the zenith-
II. To find the height of an inaccessible object.
W" }}y Quadrant. From any convenient station B, mea-aiirf t )e distance BA in a direct line with the foot of the object,IW? 1 b°th stations A, U, take the angles of elevation I)AC ,Tji The difference of these angles will give the angle ADB.ha v ?n m . *be triangle ABD, from the principles of trigonometry we]j[, e this proportion; as sine of ADB to sine of DAB, so is AB tot 0 T}’ f , ncxt > ' n the triangle BDC, as radius to sine of B, so is BD^ > tiie height of the object as required. If the line measur-d e , e pot horizontal, as CD, lig. 12, then the angle ACD must bejY'nined.
th e ’% the Square. At the station A, find by the squareCft^port-wn pf AC to CD, and at tiie station 1? find tiie ratio ofe 0fls 0 QE; hence the ratio of AC to CB will lie given, andevidently that of AB to BC, from which BC, and consequently’ bray be found. Let AB =r d, and AC : CD : : m : n ; and
CD
■P
PlummetCD— n q
If at both stations
the
„ R-
9—» 100
: q, then CD = ~ . l dn q—n p
cut the side of the square remote from the eye,d
If the side contiguous to the eye,
CD:
If tiie plummet cut the opposite angles of the
vot,,
140.
square at the first station 15, then CD :
« d
100 —n
If live height of
tiie tower, AF, fig. 12 , be wanted, tiie angle BCF mav be foundwith the quadrant, which being taken from the angle AQB alreadyknown, tiie angle ACF will remain; but the angle FAC waaknown before; therefore the remaining angle AFC will be know n.But the side AC was supposed found by the last problem; there-fore in tiie triangle AFC, all the angles, and one of the sides AC being known, AF, the height of the tower above the bill, may hefound by trigonometry.
Prob. Ilf. To find tiie distance of a given place from au inoaccessible object. Fig. 13.
Let A be the inaccessible object; it is required to find its dis*tance from the given station B. Measure any convenient distanceBC, as the base of a triangle, whose vtrtex is at A. Then, thetheodolite being placed at B, let the diameter be directed towardsthe station C, and the moveable index towards the object A, andthe intercepted arch shews the number of degrees in tbeangle ABC.
I n like manner let tiie angle BC A he measured, and the angle at Awill be known by subtracting their sum from ISO’. Then in thetriangle CAB, BA may be found by the following proportion. Assine of A to sine of C, so is BC to I5A.
Prob. IV. To find the distance between two inaccessible ob-jects. Fig. 14.
Let a proper distance, CD, be measured as the base of two tri-angles whose vertices are at the objects A, B. Then the angles atC and D being measured by the theodolite, we find as in the lastproblem the sides AD, DB; and as the included angle ADB isgiven, the other angles of tiie triangle DBA may be found by thefollowing proportion. As the sum of AD and BD, to their differ-ence, so is the tangent of half the sum of the angles DBA, DAB,to the tangent of half their difference. Then half the difference ofthese angles added to half the sum gives the greater; and half thedifference subtracted from half the sum Jeayes the less. In thetriangle BD A we now know all tiie angles ; also two of the sides;hence we may find AB by either of these proportions. As sine ofDAB to sine of ADB, so is DB to BA; or, as sine of ABD to sineof ADB, so is DA to AB.
Note. That it is not necessary that the points A, B, C, and D,be in one plane.
To measure any distance, at land or sea, by the quadrant. Inthis operation, the index AH, fig. 2 , is to be applied to tiie instru-ment, and the instrument is to be placed horizontally at the point A,fig. 15, then let it be turned till the remote point, F, whose distanceis to be measured, be seen through the fixed sights: and bringingthe index to be parallel with the other side of tiie instrument, ob-serve through its sights any accessible mark B, at a distance ; thencarrying the instrument to the point B, let the immoveable sightsbe directed to the first station- A, and the sights of the index tothe point F. If the index cut the right side of the square, as inK, the proportion will he 100 : n (BR : RK) :: BA (the distanceof the stations to be measured with a chain) : AF, the distancesought. But if the index cut the reclined side of the square, inthe point L; then the proportion is n : 100 (LS : SB) :: 1!A 1AG, the distance sought; which, accordingly, may be found bytiie rule of three. If these admeasurements be accurately laiddown on paper by means of the plain scale, &c. the results will beobtained w ith great ease instrumental!}'.
Mensuration of Superficies.
A superficies or surface is measured by another superficies,which may be called the standard-measure, or tiie measuring unit,and may be a square inch, foot, yard, &c. Thus if E, fig. 16 , hea square inch, foot, &c. and if, when applied to the surfaceABCD, it be found that ABCD contains it a certain number oftimes, suppose 12 , then the figure ABCD is said to contain somany square inches, feet, &c. and if any part of the figure remainabove an even number of the measuring units, it is to be measuredby a less unit, or estimated in fractional parts. But as such appli-cation of a measuring surface would, in almost every practicalcase, be very inconvenient, the same result may much more easilybe obtained, by a determination of certain linear dimensions, andthe application of the principles derived from geometry. Seeo M Geometry.