DE S

247

DE S

f‘° n m

less o a y require less labour, and bo attended with“g*P en se.

pla Ce » ou ^ there be two heights in the environs of thePlane' l0D1 v ' dd(dl the fortification should defile, the defilingtin e ". tnust be tangent to the surfaces of both, at the sameSol ) ‘i nd t0 determine its position, there is but one'a tli e a | c °ndition ; and it is to be disposed of, by choosing™‘ lee ’ a point, through which the plane may pass, asin ti^'fonnaUe as possible to the conditions prescribed« Tl nr st case.

‘‘ 'J'l* e Second example we shall take, is from painting.WiUiJ> su . r l" aces of bodies, especially when polished, presentPoints, whose lustre may be compared to that of theass ‘^ody by which they are enlightened. The bright-

'ant

’■lOUj

SI

w,

these

m

points is greater, and their extent more con-

of the

. Proportion as the surfaces are more polished,''‘"ch 1 * e SUr * a ees are unpolished, the brilliant points have8Ul f UCe ess lustre, and occupy a greater portion

In *

^ter In j^y® r y surface, the position of the bright point isf ay of )jj| ^ the following condition : that the incidental8 Pi-.ctiit 0r “ J 1 ’ and tlie reflected ray, directed to the eye of the•’ ' n l * le same plane, perpendicular to the plane

‘‘“'shi,

Uln<r

this point, and make equal angles with it; for8 point of the surface acts as a mirror, and reflects

apo,, 11,

a portion of the image of the luminous object.tls ion . "filiation of this point demands the utmost pre-t0|| t(, Ul .„ “ r be the design never so correct, or the apparenti"' n, Hitt 0 i tl ’i iCed with(T P'^lue - m fixin c

jf the 8b ‘ IVe the most palpable errors in the appearanceir, We will give a single, but very striking

1'he

mathematical nicety, the least mistakethe position of the brilliant point wouldmostwill

h-f

Poi,

er «d , v SlJ , lduce of the ball of the eye is polished, and

1 With • —7 - . J ......

. . ect. -.y, a thin moisture, which renders the gloss more

v hen we look upon an open eye, we see a bright

ftit, ,JJ lts surface, of great lustre, but of very limited

"b.

tl,

'vl,

i lv «r a '°i ae P 08 ’ 1 ’ 00 depends upon the situation ofit* 6 “"rfac"^ “ le d ‘ rec t' on of the illuminating object. WereOf V<il ' t 'cal °‘. lde e y e perfectly spherical, it might turn ontli “ l(! brill ,aX ' s w *thout in the least affecting the positioni s e( lir e la, 't point ; but the surface being lengthened inC a “8ed n e l the axis of vision, the position of the pointOb.' 8 ' J.orir^ t ' me that the eye moves upon its verticalo Ur g - ex P er * ei >ce having made us familiar with thistl, “•'ably , J U( lgment, as to the direction of the eye, is con-V ‘“'ght la f sed by it. By the difference of position in] (j oli l(i ,j P^uts upon the balls of the two eyes of a person,' ' tu W;i r ^'' ud S e whether he squint, or not; whether he,, "ioi, j. Us ! and when he does not, to which side hislrected -

pretend to infer from this example, that it isiti, d , u Poii tj’ ‘ n a Picture, that the position of the brillianti s * e °f the eye be geometrically defined ; ourP%iti o merel y to demonstrate how trifling errors as to

the

*tt,

th, ,‘ttit f%, ]Cf 8 for m

i« °f

its

uiay produce considerable distortion in theof the object, though in other respects the

^P r °ceed to the determination of planes tangent* ’!? tin “

apparent outline may remain the same.

lination of planespoints taken on the out-

<i U,,. tn aces , drawn through

j. i'e

of the sphere is one of the most simple that

, e *atb

a ° Ur consideration j it has common generationsii'» number of different surfaces ; we may, forjt among revolving surfaces, and say nothinglVe to it. But its regularity is productive ofsome of which are curious from their

novelty, and with them, in the first instance, we are nowabout to be occupied, not so much on their own account, as toacquire, by the observation of the three dimensions, a habit,of which we shall stand in need, for more general and usefulsubjects.

“ First Question. —Through a given right line to draw atangent plane to the surface of a given sphere.

“ Solution. —First method. Figure 16. Let A, a, be thetwo projections of the centre of the sphere ;BCB, the pro-jection of the great horizontal circle ; E F, e f the twoindefinite projections of the given right line. Through thecentre of the sphere, imagine a plane perpendicular to theright line, and construct, by the methods given under Figure6, the projections G,g, of the point of coincidence of the rightline with the plane.

“From this position, it is evident, that from the givenright line two tangent planes may be drawn to the sphere,the first on one side, the second on the other, and, conse- •quently, that the sphere will be placed between them : thisindicates two different points of contact, whose projectionswe must now construct.

“ If from the centre of the sphere, a perpendicular be con-ceived to fall upon both the tangent planes, they will each hebounded, at the point of contact with the surface of thesphere, by the corresponding planes ; and will both be in theplane perpendicular to the given right line : therefore thetwo points of contact will be in the section of the sphere bythe perpendicular plane ; a section which must be the cir-cumference of one of the great circles of the sphere, and towhich the two sections made in the tangent planes by thesame plane will be tangent.

“ If in the perpendicular plane, and through the centre ofthe sphere, an horizontal line be imagined, whose verticalprojection may be obtained by drawing the horizontal line a h,and its other projection by letting the perpendicular a h fallupon e f ; and if the perpendicular plane be conceived toturn upon this horizontal line, like a hinge, till it becomehorizontal itself ; it is evident that its section with.the sur-face of the sphere would be lost in the circumference bcd,that the two points of contact would then be upon this cir-cumference, and that were the point J constructed, in which,by this movement, the perpendicular plane would meet thegiven right line ; the tangents jc,jd, drawn to the circlebcd, would determine these two points of contact to theposition in which they then appear. It is easy to constructthe point j, or, which is tantamount, to find its distance fromthe point H ; for the horizontal projection of this distance isG H, and the difference of the vertical heights of its extremi-ties is g g ; therefore by transferring the distance G it uponthe horizontal line a h, from g to h, the hypothenuse A g willbe the amount of this distance ; and by transferring g h uponE F, from h to J, and drawing the two tangents j c, J d, thetwo points of contact, c, d, will be determined to the positionthey assumed, whilst the perpendicular plane was laid uponthe horizontal one.

“ Now, to find their projections in the position which theyought naturally to have, we must suppose the perpendicularplane to be restored to its original position, by turning itback upon the horizontal line, or hinge, A h, and it will carrywith it the point J, the two tangents J C, JD, produced tillthey cut ah in the points k k', and the chord c d, which willlikewise cut A h, in the point N. In this movement, it isevident, that Jhe points k, k', n, which are upon the hinge,will be fixed, and that the two points of contact c, d, willdescribe arcs of circles, which will be in planes perpendicularto the hinge, and whose horizontal projections will be obtainedby dropping from the points C, D, the indefinite perpendiculars