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“ (3.) Having ascertained the projections, n, n, of one ofthe points of the required common perpendicular, it will besufficient to obtain the projection of the perpendicular itself,to draw through the points N, n, the right lines N P, np,perpendicular to the respective traces A e, p b; and theparts N p and np of these perpendiculars, comprised betweenthe projections of the two given right lines, will be the pro-jections of the required shortest distance.

“ (4.) In conclusion, if the size of this shortest distancebe desired to be known, it may be constructed by the processof Figure 3.

“ The consideration of a cylindric surface touched by aplane, was not essential to the solution of the precedingquestion. After having supposed a plane parallel to the twogiven right lines, we might through each of these lines havedrawn to such plane, a perpendicular plane ; and the inter-section of these two planes would have been the direction ofj the required shortest distance. We content ourselves withj announcing this second method, and advise the reader to seekI its construction by way of exercise.

“ In the several questions which we have resolved relativeto planes tangent to curved surfaces, we have always sup-posed the point, through which the tangent plane should bedrawn, to be taken on the surface, and to be itself the pointof contact: this condition alone sufficed to determine theposition of the plane. But it is different when the pointthrough which the plane should pass is taken out of the; surface.

“ In order to determine the situation of a plane, it mustsatisfy three several conditions, each equivalent to that of: passing through a given point. Now, in general, the pro-perty of being tangent to a given curved surface, when the! point of contact is not indicated, is only equivalent to one of' these conditions : if, therefore, we propose to determine the! position of a plane by conditions of this nature, we shalli generally have occasion for three. For instance: supposei three curved surfaces to be given, and that a plane be tan-! gent to one of them, in any point whatever; we can conceive’ that such plane would move around the surface, withoutJ ceasing to touch it: it would do so in every direction; onlyI the point of contact would shift its situation on the surface,in proportion as the tangent plane changed its position ; andthe direction of the point of contact would be similar to thatof the motion of the plane. Suppose this movement to be; made in a, certain direction till the plane meet the second: surface, and touch it in a given point: then the plane would! be tangent to the two first surfaces at once, and its position| would not yet be fixed. Indeed, the plane may be supposedto turn about the two surfaces, without ceasing to touch; them both. It will no longer, however, be free to move inevery direction, as before, but will be confined to one only.In proportion as the plane changes its position, the twopoints of contact will move each upon the surface to whichit belongs ; so that if a right line be conceived as passingthrough those points, their movements will be in the samedirection with respect to such line, when the plane touchesthe two surfaces on the same side ; and they will be in acontrary direction, when it touches one surface on one side,and the other on the contrary side. Lastly, imagine thismotion, which is the only one that can now take place, to bej continued till the plane touch the third surface in a certain; point; then its position will become fixed, and it can nolonger move without ceasing to be tangent to one of thethree surfaces.

“ Hence we may perceive, that to determine the position; of a plane by means of indeterminate contacts with givencurved surfaces, we shall generally require three such sur-

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faces. Thus, were it proposed to draw a tangent l* I ’ e

to 1

given curved surface, this condition would be e fi u ' va ',qg ofonly one of the three to which the plane is cap 11

answering: we might, again, take two others, , - vand for example, make the plane pass through twopoints, or, which is the same thing, along a given r, o l [v v 0Were it essential that the plane should be tangent . efesurfaces at once, two conditions would be fulfilled > jfiwould remain but one to be disposed of, and the P' an j j , lS tiy>only be brought to pass through one given point. t j,er®when the plane touches three given surfaces at onC .®’ g p 0 si"remains no longer any condition to be disposed of j 1tion is determined. urf aCeS

“ The preceding observations relate to curvedgenerally; yet we must except from them, whatever t [iecylindric, conic, and developable surfaces ; in w b utcontact with the plane is not reduced to a single P° ^jcl*extends along the whole length of an indefinite l |Iie > q|ieloses itself in the generator, in one of its P 0S ' tI0ll g 3 1 ’ r fiice s >property of a plane touching one only of ^ ese gU bj ectwould be equivalent to two conditions, since it wouit to pass along a right line ; and there would on v ^ugl*one condition to be disposed of, viz., to make it P ass sS toa given point. It were in vain, therefore, to P 1 *q 0 ttf®draw a plane, that should be at one time tangen th eI ' 6of these surfaces, much less to three of them, un ^ t heS ewere some peculiar circumstances which should renconditions compatible. „ r oce e<)

“ It may not be altogether useless, before we P tl> eI ?farther, to illustrate by a few examples, the neP e3Sis for drawing planes tangent to curved surface® ^ t lie= dpoints taken from the outside of them. The , a tion s- .examples is selected from the construction of ioft' it ’j“In treating of the general principles of f° rtl ” t j, e gi’oO 11 ,taken for granted, first, that, in every direction,by which the place is surrounded, at least w * t ’'I^gjjce ^of cannon-shot, is flat, and free from every eH ^ rl „y, i ^might be converted to advantage by a besieging ‘ ^ - shypothesis being settled, the draught of the p ‘ l ^determined, with its half-moons, covered ways,works ; the bearings of the various parts of the ^ ni; 1 /

upon each other are then marked out, so th e * r ^^in'contribute, in the most efficacious manner, to ^ t j, e se P 1 eand reciprocal defence. But, in order to a PP^, ese nts s® ^ciples to cases where the surrounding country P^ an d j 1 (0height, of which besiegers might take advance ^ n>awhich it is requisite that the fortification s ** 0 ,^ t i ie re bfdefile, a new consideration presents itself. I up° n t t0a single eminence, two points should be fixe tai'S^idplace, through which might be conceived a P a ta a » 0 [the height, from which it is desirable to defi■ ^ ie pa 1 e ,

plane is denominated the defiling plane; andthe fortification must receive the same relief a 0 li« ^as they would have had above the horizonta P ^ a^l.^ji-country been quite level: by this means, t iey^ jpe "tli® 1 ”relatively upon each other, and collectively U P w0 y|d ;

bouring height, a command equal to what t ' ; t |, e fo‘ ;

wise have possessed over the fiat country : an first ^ ■tion will possess the same advantages as in . w h' u ,\ e i0 jAs to the choice of the two points, ’ irol iq )n foi ' ,l, t r H> e ^defiling plane ought to pass,' it must bethe two following conditions: 1st, I hat P° sSl ,jc« °

by the plane with the horizon, be the e ^ gt , r v ^jiy,order that, the platforms having less ® °P e ’ ( ,| Ill eii t9 ’defence may be attended with fewer 1U1 P^ tl‘ e 11That the relief of the fortification a ^ ^ j ts c ° lPground, be likewise as little as possible, t u

at pl eilS .

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