DE S

243

DE S

if d —--

to sunk e ,p arr >ed from d to d, the right line d b will be equal

.<-{;[! distance.

pqiinj t ® re fore, taking b as a centre, and, at an interval•t win B describing the arc of a circle, the point e, whereSec ond°Vj the first, will be the point of coincidence of thehii e A g 81 . w 'th the horizontal plane ; consequently, the rightana]p V1 ^ he the horizontal projection of such side, and'• Tf 6 B . A E that of the angle observed.

*he j lne preceding questions barely convey an idea ofan t/ 01 * projections ; they are inadequate to a display

;! Sen,

a >l th eeral

resources : but in proportion as we rise to more

°t>erati 0 * Ons 'derations, we shall take care to introduce such

Of / 111 ue most conducive to tms object.

• " Tli * >lQnes ta ngent to curved surfaces, and of normals.

"i s ev ® 18 n o curved surface but what may be generated°re, iffr Wa y s > by the movement of curved lines; there-? l| pp 0Sef j orn any point of a surface, two generating lines bel!* Passi t0 s P r ’ n o in the position they would naturally havesiinn ® ea . c * 1 ot i*er through such point, and if the tangentsKNe ,]g e< \ ' n this point, to each of the two generators, thethe poinT ' 11 ^ 6 ^ suc h two tangents is the tangent plane.° t her i „ , the surface in which the two generators cut eachatl d to tk 18 a t the same time common to the two tangentse tangent plane, is the point of contact between the“ an d the plane. J

fr-Mi '• ’

>■(';t (:e , -

v tli 1S P e, 'pendicular to the ground of the surface,it* 1 * 1 that* 6 1 ’ rect ' on of such ground coincides, in every part,tS hrolo'n ° tbe tangent plane, which may be considered as. “a , Sation.faces . kn °wled g

r ‘8ht line drawn through the point of contact, per-r jy to the tangeut plane, is said to be normal to the

s, i s "' ,,ie uge of tangents and of normals to curved sur-J® hulisp er y use ful in a great number of arts ; in many, iteacl, c nSa ^ e ' We shall here adduce only a single example‘‘ ’fli e Se ’ ^ected in architecture and painting,ti e > are S6Vei a ^ portions which compose vaults of hewnw?"°Us v Ca ^ e< ^ voussoirs, and the faces on which two con-'eth 6l . t? Uss °irs touch each other, are denominated joints,be c 16 v ° u ssoirs form but a single course, or whether. *Th e m P lase d in two successive courses,iv^'tions ° Slt '? n the joints of vaults is subject to several• yhich must necessarily be complied with, andmu -^all de - - -

>0 tk Se bUra G . ,

but at present we must confine our attention

'** necessarily ue euuipneu wiwi, turn

dis C()Ui- 8la h demonstrate in succession, in the sequel of

,;' e °bi P ,,t e

is of , rnoie immediately before us.

f >at th e \' e c °bditions required in the position of joints,C aea of 4 be a11 Perpendicul ar to each other, and to the

iiw

of u nc «, ,i

^ ok. Va Ult.°"l of the joints be made oblique to the surface>'t !l 0bt ase a ’ ° ne °f two contiguous voussoirs will formoth^'op ant ^ the other, an acute one ; and in the

fvk f > the t C1 these voussoirs would exert against eachj n 0 an gles would present an unequal resistance,angle ° llse< l uen ce of the fragility of the materials, thev 0ii» a8 onda ° U ^ hilge, and spoil the shape of the vault, as8 °irs, t j|® er the edifice. The reduction of vaults into,a ng ei I? re ^ 0l 'e, absolutely requires a knowledge ofSt Us , , an ^ bormals to the curved surface of the arch,‘o tl,'’ e 'v, sc 6 anot her example, from an art, which, at‘b , r|) le. etlls to require a much less rigid attentionNs, '•'tin* ;

• tx ■ "TIuTq 8 generally considered as consisting of twoin ip 6 ’ s > P r operly speaking, the art; its object is

-j q - Any material deviation from this rule,

iitni ri j s l r °y s the general symmetry of the structure, but' Ce i if * eS tbe hrmness and durability of the vault. Foi

1 giv eil '? s P ect nt°r a determinate emotion, to createmea, or to place him in a situation the most

favourable for receiving a certain impression ; it supposesin the artist a great knowledge of philosophy ; exacts, onhis part, a most intimate acquaintance with the nature ofthings, the mode in which they affect us, and the moverments, even involuntary, by which such affection manifestsitself. This can only be the result of a very refined educa-tion, such as no one receives, and such as we are far fromgiving to young artists : it is governed by no general rule,but is subject solely to genius.

“ The other part of painting may be properly called thetrade: its object is a correct execution of the conceptions ofthe former. Here nothing is arbitrary ;• all is foreseen, bythe help of sound reasoning, as the necessary result of deter-minate subjects and given circumstances. When the formand position of an object are ascertained; when its nature,and the number and positions of all the bodies by which itmay be illumined, whether by direct light or reflected rays,are understood ; when the position of the eye of the spectatoris determined ; and when, in a word, every circumstancethat can influence the vision, is well established and known,the tint of each of the points of the visible surface is abso-lutely determined. Whatever relates to the colour of thistint, or its brightness, depends on the position of the planetangent in this point, with respect to the illuminating bodiesand to the eye of the spectator. This may be discoveredby mere reasoning, and when so determined should be appliedwith accuracy. Every diminution, or exaggeration, willchange the appearances, alter the forms, and produce aneffect quite different from that intended by the artist.

“ I am aware that the rapidity of execution, which is oftennecessary, rarely admits of the use of a method whichdeprives the genius of all corporeal succours, and leaves itto the exercise of its faculties alone, as well as that it is muchmore easy for the painter to look at objects, to observe theirtints, and to imitate them : but were he accustomed to con-sider the positions of tangent planes, and the two curvaturesof surfaces in each of theif points (curvatures which willform the subject of subsequent lessons) he would not failto derive from this material method, a more advantageousresult: it would enable him to supply effects which theomission of some circumstances had prevented him fromproducing, and to suppress others which had arisen fromextraneous incidents.

“In conclusion, we may remark, that the vague expres-sions, such as flats, which painters are in the constanthabit of using, are standing evidences of the need inwhich they are of more accurate knowledge, and of deeperreflection.

“Besides its utility in the arts, the knowledge of planestangent and normals to curved surfaces, is one of the mostfertile methods employed in descriptive geometry for thesolution of questions, which it would be very difficult toresolve by any other process, as will appear from the follow-ing examples.

“The general mode of determining the plane tangent toa curved surface, consists, as we have already remarked, inconceiving at the point of contact, the tangents with twodifferent generating curves, which would pass through thispoint, and in constructing the plane that would pass throughthese two right lines. In some particular cases, in order toshorten the construction, the strict letter of this mode isdeparted from, but ah equivalent is always adopted.

“ As to the construction of the normal, we shall not dwellupon it particularly, as it reduces itself merely to that of aright line perpendicular to the tangent plane, which is suffi-ciently understood.

“ First Question .—From a supposed point on a cylindric