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264

DES

of land, whose situations are accurately marked on the chart,and of such kind as not to be in the same plane with the pointof the station, let the engineer measure the three angles whichthe visual rays directed towards those three points formwith each other; and by means of' this simple observation hewill be able to resolve the question.

“ Thus, by letting A, b, c, represent the three points ofobservation, and supposing them to be joined by the rightlines A B, b C, C A , the engineer will be in possession of thehorizontal projections of such lines, traced on the chart; andby means of the quotas of the three points, he may obtainthe differences of the altitudes of the extremities of the lines;he may, therefore, ascertain the size of each of them.

“ This done, if in any plane whatever, drawn through A b,a rectangular triangle, bad (Figure 42) be conceived asconstructed, with A B for its base, whose angle in b is thecomplement of the angle under which the side ab has beenobserved, the angle in d will be equal to the angle of obser-vation, and the circumference of a circle described throughthe points A, b, d, will possess the property that if from anypoint whatever, e, of the arc ad b, two right lines be drawnto the points A, b, the angle, in e, which they include, willbe equal to the angle of observation. If, then, the plane ofthe circle be conceived to turn upon a b, as on a hinge, thearc AD b will generate a revolving surface, all whose pointswill be endowed with the same property, viz. that if from anypoint of the surface two right lines be drawn to the pointsA, B, they will include an angle equal to the angle of observa-tion. Now, it is evident that the points of such revolvingsurface alone possess this property ; therefore the surfacepasses through the point of the station.

“ By operating in a similar manner upon the two otherlines b c, C a, two other revolving surfaces will be obtained,on each of which the point of the station will be found ; thispoint will therefore be at the same time upon three differentrevolving surfaces, determined as to their form and position ;consequently it must be a point of their common intersection.Thus, in the construction of the horizontal and vertical pro-jections of tlie intersections of these three surfaces, taken inpairs, the points in which the three projections cut each otherwill be the projections of the point answering the question.Tlie horizontal projection will be its position on the chart,and tlie vertical projection its elevation above or below thepoints of observation.

“ Were this question to be treated by analysis, it wouldgenerally lead to an equation of the sixty-fourtli degree ; for| eacli of the revolving surfaces has four distinct faces, two ofwhich are generated by tlie arc adb, and the other two bytlie arc A f b. As each of the faces of the' first surface maybe cut by all those of the second, sixteen branches may be theresult in tlie curve of intersection ; and as these may beagain cut by tlie four faces of the third surface, they mayproduce sixty-four points of intersection in tlie three surfaces ;though they would not all apply to the solution of the ques-tion. Thus, if from any point, as F, of the arc A F b, linesbe drawn to the extremities of Ab, the angle afb, includedby them, would not be equal to the angle of observation, butwould be its supplement. The faces generated by the arcafb, and the analogous faces in the other revolving surfaces,cannot therefore serve towards the solution of the question ;and all the intersecting points belonging to any of these facesare foreign to the problem.

“ In descriptive geometry, we may, and indeed shouldexclude tlie arc A v b, and those analogous to it, in the twoother surfaces 5 each of which will then have but two faces,i and the number of their possible points of intersection will bej reduced to eight. Of these, four will be on one side of the

on

plane passing through the three revolving axes, and 0 nthe other. The observer, being always aware of the ® n otwhich he is placed relatively to this plane, will,construct the intersections of the opposite side, sonumber of points will in fact be reduced to fo 01, jiiy

among these four points, if they all exist, he a

know which is placed with respect to the points a, b > ’ | lf eemanner similar to that of the station relatively topoints of the country that he has observed. j an es

“ Construction .—Select the position of. the two P passprojection so as that the one taken as horizontalthrough the three points observed, and that the o p^ggebe perpendicular to the right line drawn through two .^glepoints. Let then abo (Figure 42) represent. * a 11 ®

formed by the points observed, considered in it® P pra' v 'a', b', c', the three angles given by the observation^ ^jcl 1perpendicularly to the sides a b, the right line ^ ’ t j 0 n iwill indicate the position of the vertical plane of P 1 give 0 ’and construct, according to the directions al r/eVoVfthe generating arcs A e d b, b g c, C f a, of three ^ q’liei'i

surfaces, whose sides, ab,bc,ac, are the axe®; c ird eS '

taking the point A as a centre, describe arcs_ jnR ra- torS '

as eof, at pleasure, and they will cut the

points as E, F, .e indefinite perpendicaeach

e ene v ihci>’

otK

whose axes meet in A, in such

drop upon the respective axes, the „ (

f.e', ff' ; these perpendiculars will intersect hori^lsomewhere in a point, as h, which will be tn gU rla cprojection of a point of intersection of the ' ^ <b' a ".p

whose axes are ab, a c; and the curve, A ^ h 0 *through all the points h .. . thus found, will ^ et przontal projection of this intersection. ^ eX ’ c entr e ’ 9 rjecting the axis A B in a, take the point a as a ^ { j ie p e ^describe from it, with radii successively e 1 ua ^ 0 fpendiculars ee', arcs of circles, as ee’h, °“_f 8 0 ; n t in

Wi

11

r ----

by piojecting the point h, to its corresponding p°Jshall have the vertical projection of a point of the iof the two same revolving surfaces ; and the curv

drawn through all the points, h _so constructed.

* P ro j ec, '°n of this intersection.

lhe same method may be pursued for the t" r ° ;„trevolving about the axes ab, bc ; vie. taking t 0

coincidence, b, of the two axes as a centre, describe g t1f0circles, as dkg, at pleasure ; which arcs will cu TeS pe c

from which drop ° / nU lheS eP er

generators in points, as d, g,

tive axes the indefinite perpendiculars d, D , d t iie G ^ 0 crpendiculars will cut each other in a point, as J > ,1- ‘ > 1

BJF drawn through all the points J... will ^ t l^zontal projection of the intersection of the first an ^ kievolving surfaces. Taking the point a as a ce “ V r ‘$radii successively equal to the perpendiculars vX, ’ s poflf:

the co ;/yi t)' 0

rs

: thethroes 1 *, e cti

gh 81 So-

ares of circles, as rfd'i, on which project in ^

ing points J; and the curve, ,aip, d rawnpoints i, will be the vertical projection of the sa . . n -w’ 1 11 r o-“ Here we shall find that all the points, P • ‘j’ or i 2 on til „fl<l

two curves ahp,bjp, cut each other, are ^in-jections of so many points applicable to tn g a Jifthat all the points, p .... in which the c _ u ^ s 0 f th eintersect each other, are the vertical project 10

tio°

Ml

Points.

] t,!e f >osi ^on'’of t tTer,! > hU t S f r Un<] ’ wiil aot giV t ’rt

height, because the £?!?* ° f stat ’ 0M on the ch r the ^but it will be * horizontal plane is not that of j p,

be eas ? t0 hnd them on the true P I*» es ° ,

on ^ 01 i ; f ‘ t l,ef;J ;

jection.

gets

Jtion. «rnY

“Sixth Question .—The general i3 '^oh

an enemy, but not having a chart ot tne yhow to form his plan of attack. But being