DE S

262

DE S

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of the sphere required will be found, whose position will bedetermined by their common intersection.

“ In order to simplify the construction, we shall supposethe planes of projection to be so chosen as that the one whichwe consider as horizontal may be the same with one of thefaces of the pyramid.

“ Figure 39.— Plate 4. With this view, let A, b, c, d, bethe given horizontal projections of the apices of four solidangles of the pyramid, and a, b, c, d', their vertical projec-tions ; conceive through the apex of the pyramid, planesperpendicular to the three sides of the base ; such planeswill be vertical, and their horizontal projections will be theright lines d e, d f, d g, falling perpendicularly from the pointd on the sides a c, C b, b A, of the base. Each of these planeswill cut the base of the pyramid and the face that passesthrough the arete in two right lines, comprehending betweenthem an angle equal to that which the face forms with thebase. Then, by measuring on l m the right lines d e, d f, d g,beginning from the vertical line D d d', from d to e,f g, anddrawing from the apex, d', the right lines d' e, d, f d! g, thelatter will form with l m angles equal to those formed by thecorresponding faces of the pyramid with the base ; and if eachof these three angles be divided into two equal parts by theright lines ee’,f gg', the angles formed by these last lineswith i, m, will be equal to those formed by the base with the| faces of a second pyramid, having the same base with thegiven pyramid, and whose apex will be in the centre of the! sphere required.

“ To find the apex of this second pyramid, let it be cut bya horizontal plane, drawn at an arbitrary height, whosevertical projection may be obtained by drawing any horizontalline whatever, as p n. This line will cut e ef ff, g g' in thepoints, h ', i', k', from which let the vertical lines h' h, i’ i, h' h,fall upon l m ; and by measuring the three distances h e, if,k g , on the respective perpendiculars, from E to h, from r toj, and from G to k, we shall have in h, j, k, the horizontalprojections of points taken in the three faces of the secondpyramid, and which will be found upon the arbitrary hori-zontal plane. Then, by drawing through the points h, j, k,to the respective sides of the base, parallel lines, as? N, n o, op, they will be the projections of the sections of',he three faces of the second pyramid, upon the same hori-sontal plane ; they will also intersect each other in points, asN, o, p, which will be projections of as many points of thethree aretes of the second pyramid ; and by drawing fromthese points to the apices of the respective angles of the base,indefinite right lines, as a p, b o, c n, such line will be theprojections of the aretes ; lastly, the single point Q, whereinthey all three meet, will be the horizontal projection of theapex of the second pyramid, and consequently of the centreof the sphere required.

“ To obtain the vertical projection of this centre, draw, first,the indefinite right line of projection Q qq', on which it willbe found ; then project the three points, N, o, p, on the hori-zontal line n, p, in n, o, p; draw through the projections,a, 6, c, of the apices of the respective angles of the base, theright lines ap, bo,cn, and they will be the vertical projec-tions of the three aretes ; and the single point, q',, whereinthe three last lines cut each other, and which will be at thesame time on the right line Q q q', will be the vertical pro-jection of the centre of the sphere.

“ Lastly, the vertical line q <f will be evidently equal to theradius of the inscribed sphere, and the points Q, q, will be theprojections of the point of contact of the surface of the spherewith the plane of the base.

“ We have shown by what considerations we are enabledto determine the position of a point, after having ascertained

. • .we ^

its distances from three points of known position ;

now proceed to the actual construction of this quest*® ' ; n t,“ Third Question .—To construct the projection o ® ^ j 3whose distances from three other points given 1,1 "known. . ,j on b®

“ Solution.—Figure 40. Let the planes of p)'°J e ® r passso chosen as that the one considered as horizontal & ^jcU'along the three given points, and let the other be p el P . j n e<b

are J°points :

fii) u

lar to the right line by which two of the pointsThen let a, b, c, represent the three given 'join

a', b', c', their given distances from the point req u ’ r ® r jy t®two of the points by the right line a b, perpendm® 0 f tb®

position

0

which draw L M, and it will determine the p 1vertical plane of projection. Take the points <JiS'

centres, and describe with radii equal to the respe®^ e» e ^tances a', b', c', three arcs of circles, which ® h tb®

other by two’s, in the points D, E, r, J, p,

on 3

intersecting points of these arcs taken by two s, : eC tklines d e, f j, p Q, and they will be the horizontal P ^ g plier eSof the circumferences of circles wherein the tb® 3 ^

intersect each other ; and the single point N, wn - eC tiori 0three lines meet, will evidently be the horizontal p r Jthe point required. _ „ s #rI1

“ In order to obtain the vertical projection 0 ^ . tb®/^point, draw the indefinite line of projection N n -^ & \ to 1 ’observing that the circle projected in d E is P^ ra g rn ust ®vertical plane, and that its croieetion on this P ia j

it)

tb®

vertical plane, and that its projection ona circle of equal radius, project the line a b upon n »> ‘ qU 3 Jpomt r, from which, as a centre, and with an interval hto d b, or the half of d e, describe the circle d n e n', a " ncircumference will cut the right line N n ii in two P® 11 ?* p 0 ifl {which will be indifferently the vertical projection of f)required. r J

According to the other circumstances of the questi^may determine whether the two points n, n', ought 10 ,. jj ofused ; and in case of one only being necessary, "

Prefer, and which to reject. . Jt^

“ The reader may propose to himself the constructs J inprojections of a point whose distances from three hues 9space are known. &

“Fourth Question.— -An engineer, when surveying a t b®whether for the purpose of stu ) ^rb 3

tainous country, whether for the purpose < J ‘ ~ jjjc "b ca lform of the earth, or for making a draught * or Zpogi' 9 ? Ltsdependent on such form, is furnished with a ^g eT evt P° o(chart, whereon not-only the projections of the a ititu“ e3of the earth are exactly laid down, but also > ^ b}’ yiS'-all these points above a level surface, are ' nt j 1C ca iied ^j l9 rt,placed on their sides respectively, commonly t ]ie 0 pehe meets with a remarkable point, not place ^ j )aS b®either in consequence of an omission, or bee®”, erl gin e6 > e r,remarkable since the chart was drawn. ^ grapb oC1no instrument of observation about him, excep j urB b-li Iie .' t |j ( )U tused for measuring angles, furnished with a P j re d, of“ Placed in such circumstances, he is i 19

quitting the station, to construct on the _ u ota, v

the point in question, and to find the suita c rib®^

height above the level surface. _ nrrectly “ e ,„t tb®

“ Mode of Solution .—Among the points c . ( j era tioD, tb®

on the chart, nearest to the one under c °® sl g j a re n® ^ ftfiengineer select three, of which two at le® . ^0 1®.same altitude with that on which he stan .,j 1 ’^eobserve the angles formed by the zenith. W1 - n gle °b® edirected to these three points, and by this p 0 ' 11 j

he may resolve the question. . t i, e tbrf® f, nK

' “ For instance : Let a, b, c, represen ^ t p e cb^ 0observed, whose horizontal projections are py a 3whose vertical projections he may cons >