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be oij] 18 an< i those of geometry : on the one hand, he shouldCeiy e . e *° write down all the movements that he can con-to p ef ® s Pace, in one analytical method ; and on the other,of ivlij | Uate upon his memory, the object moving in space,

j’fe each of the analytical operations is the expression.»>ini nff e n ° w return to our subject, viz., the mode of deter-8 Urf a „p ^ le projections of the intersections of curved

%ht P re sent the exposition of this method in a clearer‘/which : ! ha11 not at first disclose it with all the eleganceAri(} 8us ceptible, but proceed towards it by degrees.

8<sne ra i ' *“ may oe premisea, mao me exposuioii wm ue*hat th’ ari ^ applicable to any two surfaces whatever; and•bo*, letters used refer to the Figure 26, which

hasg s Particular case of two conic surfaces, with circularv ertical axes, the reader should keep in mind, thattlc alar a ° es Un ^ er consideration may be, each one in par-“ otber than conic (surfaces.

C| trv e ,j 1 9 e neral problem .—The generating lines of two&ij e | 1 SUr * aces being known, and all the given lines whichtl °ns ; t 8 en erators being determined on the plane of projec-Cllrv utur° construct the projections of the curve of doublee, *cli e ’ according to which the two surfaces intersect«

lon —Figure 26. Conceive a series of indefinitee } !!Un I l 'le COnVeniun tly disposed in space ; such planes, for1 'fifn : > ®>ay be all horizontal, as in fact we shall supposel first instance.

tt may be premised, that the exposition will be

”' 1 of tt, — v Here the vertical projection of

5 1 ? w e era will be an indefinite horizontal right line ; and^hlticeg n °*' res t r ained from drawing them at arbitraryao,, >Zont„\^. e ®hall suppose in the vertical projection as many! ‘ r ’ es of tt, 11168 ’, e e ’’ e 6 "’ e e '> & c. as we please, and that the

. * ce, e e y occ. as we pieuse, anu uiat me

Cese lines is the vertical projection of the series of, ®t conceived. This done, work successively,,, p ro - 8u ch planes, and relatively to the line e e', whichle on e ai ^ ct,on > the operation we are about to lay down forc " d'h e ri ,° n S them, that is projected in e e'.

Htvo P ane e e' will intersect the first surface in a certain

% ach of

a >a’.- Whic fi

c '"'cteii '• w 'h be easily constructed, when we are0 f ^.W'th the generation of the surface ; it being theh|, e .8ene rat - e P°* n ts in which the plane ee' is intersected by8i l, ' a and I," 1 ? bne ' Q all its positions. This curve beingt 0 :. 0ri . zon tal, will have its horizontal projection equal,

fof„ " io itgpip "j’ .-I—>

8t, P°ssihl U ’ an “ placed in the same manner ; it is there-?P°8e it , e construct such projection, and here we shallf a /Hhel tabe .the curve egh IK.

p/® in an j®® Plane, e e', will likewise cut the second sur-n > ct i°n if, er Plaiu horizontal curve, whose horizontalMao ^ay h^ a ' S ° be cons tructed, as by the curve f o g.otiji H u' , aa Ppen, that the two curves, wherein the samebo^ P ’ ° r th e tersects the two surfaces, may intersect eachM« v <;r not: if they do not intersect each other,

the s e ® e'j Produced, it proves that, at the height of the^tt/'^rves a t 'T° SUr faces have no common point ; but if

tkw Fi,_ lntppoanf ftnoh /\4V,+1 iott tirill oa in n

a^

a 'a niiQ^ U0 intersect each other, they will do so in a

• C0l 'Se Qllo er 1 °f Points common to the two surfaces, which» 'mentis _ /. .... a • *_*• _

% lre d: and * ^ so many of the points of intersectionar according as the intersecting points of theif ^ Hie fi rst e u Pon the first or second of them, so are theyfile t«/ npo° r Secon d the surfaces proposed ; therefore,■i a ° surf 11 the two curves at once, they are also upon\ tile ho 6 - 8,

i n < f' IZ0nta l projections of the points wherein ther, _ ‘°n of eacb other) should be found both on the

l N j ‘ •. 0 f t , e nrst, and on that of the second ; the points°Wiiq / c °incidence of the two curves pghik,“e horizontal projections of as many points3 i

of the required intersection of the two curved surfaces.

To obtain the vertical projections of the same points, observethat they are all comprised in the horizontal plane e e', andthat their projections must fall upon the line e e'. There-fore, by projecting the points, f, g . . . . upon E e', to/, g. . . .we shall have their vertical projections.

“ By pursuing the same operation for all the other hori-zontal lines, ee', ee', we shall obtain for each of them, in the

horizontal projection, a series of new points, f, g .. &c.,

and in the vertical projection, another new series f g . . . .

&c. Then if the branch of a curve be passed through all

the points f ..., another branch through all the points G..

and so of the rest, the concourse of all these branches whichmay possibly meet one in another, will be the horizontalprojection of the two surfaces; in like manner, if throughall the points^. ... a branch of a curve be passed, through (

all the points g ... . another branch, and so of the others, ;

the concourse of all these branches, which may likewisepossibly meet one in another, will be the vertical projectionof the intersection required.

“ This method is general, even supposing a system of planesto be chosen which intersect a series of horizontal planes.But we shall see, presently, that in certain cases, the choiceof the system of intersecting planes is not indifferent, thatit may sometimes be so made as to be productive of systemsmore easy and more elegant; and that it may even be moreadvantageous, instead of a system of planes, to adopt a seriesof curved surfaces, which vary from each other only in one oftheir dimensions.

“ To construct the intersection of two revolving surfaces,whose axes are vertical, the most advantageous system ofplanes, is a series of horizontal planes; for each of such planesintersects the two surfaces in the circumferences of circleswhose centres are upon the respective axes, whose radii areequal to the ordinates of the generating curves, taken at theheight of the intersecting plane, and whose horizontal projec-tions are circles of known size and position. Here all thepoints of the horizontal projection of the two surfaces arefound by the intersections of the arcs of a circle; and weare aware, that if all the revolving surfaces had their axesrelatively parallel, but not vertical, it would be necessary tochange the planes of projection, and so to choose them as thatone of them should be perpendicular to the axes.

“ Were it required to construct the intersection of twoconic surfaces, with whatever bases, whose traces on thehorizontal plane were given or constructed, the system ofhorizonal planes would demand operations too tedious for theexample: for each of the horizontal planes would intersectthe two surfaces in curves, which, though very nearly resem-bling the traces of the respective surfaces, would not be equalto them; they must be constructed by points, each by itself;whereas, if after having drawn a right line through the givenapices of the two cones, the system of planes passing alongsuch right line be adopted, each of the planes will cut thetwo conic surfaces in four right lines; and these right lines,which will be in the same plane, will cut each other, inde-pendently of the apices, in four points upon the intersectionof the two surfaces. In this case, each of the points of thehorizontal projection of the intersection will be constructedby the intersection of two right lines.

“ For two cylindric surfaces, ,of whatever bases, whosegenerating lines are diversely inclined, the system of hori-zontal planes would not be the most eligible that might beadopted. They would, indeed, each cut the two surfaces incurves similar and equal to their respective traces ; but thecurves that did not correspond vertically to the traces wouldhave for their projections, curves, that would be distant from