DE S

239

DE S

recti , a »gle S (projections which are reducible to the system ofany ' p 6: ( r angles) it will be easy to ascertain the length ofp ara H lts dimensions : for this dimension will either be* 1 . 1 to one of the two planes of projection, or it will be at

® moment oblique to them both. In the first case,

• 1 Ulr ed length of the dimension will be equal to its pro-

fitfile

^ c tio n

3ecti 0 " ’ ln t ' le second, it may be reduced from the two pro-^ t * le method just described.

jectio 6 corae now to describe the mode by which the pro-are „ S s °lids, terminated by planes and rectilinear anglesop er . < ? nstru eted ; though there is no general rule for thisbe o' 011 : indeed, the construction of these projections willposit; ^ ° r ' ess easy, according to the method in which thethe t i ,e apices of the angles of the solid is defined ;

definif re the operation being governed by that of thethere • The case ' 3 precisely here as in algebra, in which

tio n „ 1S T no general method of reducing a problem to equa-

tes

titi,

°iO(le

general ineuiou oi reuucing a promein to eqitn every particular instance, the process depends

'es i

in which the relation between the given quan-v ari et ,, n r lose sought for, is expressed : and it is only by ahols nf i exam ples that young students can learn how to lay

lit, th ~ -

So

e 'vi s - eSe a ®nities, and to express them in equations.^atn D i e ’ In descriptive geometry, it is only by a multitude of

6c boofg CS ’, an< t by the use of the rule and compasses in ourhotig ’ la ' we can acquire the habit of forming construc-*boiiii a ° r accustom ourselves to make choice of the most•hay ^and elegant methods in each particular case. Weller observe, that, as in analyses, when a problem is*HHatio t0 ec l ua tions, there are methods of treating those' ) »kno w S ’ ar ® °f deducting from them the value of each; so also, in descriptive geometry, whencon4 Ctl0 . ns are made, there are certain general methodstes Pecti r ' ,ct ' n g whatever may result from the terms, and” Nor'® Potions of bodies.

^ aWpv, 18 '■bis comparison between descriptive geometr

*“ 3 V/Umjjtuiauii uctvvccii i

? 0, inent e / a a J'°S e ther useless; for these sciences are intimatelya,, t ‘ -There is no construction of descriptive geometry,

te 9uire 111 a T be reduced to an analysis ; and when questions*bay ij e than three unknown quantities, each analysis

‘Iti

upon as the record of a spectacle in geometry.Ir ’ u °b to be wished, that, these two sciences were

j'l'lieq to" UCl1 to b e wished, that .......—.... .......

8| ice lyi Seiner : descriptive geometry would carry that evi-rated a | 18 * ts peculiar characteristic, into the most com-J^ical an ‘'vfical operations ; while on the other hand alge-'•di j, al yses would give to geometry, that generality“.’The !, a . nd ? in need of.

JS s j* In ciple upon which we ground the theory of pro-5 Cft , o r e ° n venient for describing the position of a point inC'| S!ef lUenti' it; °f an indefinite or terminated right line, and,Ot ^ band ^ or representing the form and position of a^.'be ap; c nated either with plane faces, rectilinear arretes,an i 'be n <iS °f s °bd angles ; for when once we are acquaintedbn? 8s > thg < ) Sll: ’ on °f ail its arretes, and of the apices of all its

Q0| J | v,li tlil ItO ClllCtCOj Orliu \Ji

p 0 j ded, e'tt ^ ''self is entirely known. But were all bodies■ btg ^ ler by an uniformly curved surface, all whose

li3 - *

^^ 0Verne ^ by the same law, as in spheres, or by

cjvV'^ suvp Cte ^ ass emblage of several parts of differently

^ouid ’

a s in a body turned on a lathe ;

this

pnri-

bnjbhte 0 y ot °nly be inconvenient, impracticable, and

°uld.

advantage of forming an idea of the shape,

y .' y r jp st 80 be insufficient through want of variety.Mq^lf, ^ an ? e : it is easy to perceive that this principleb to d e ° . be inconvenient and impracticable, if we%]j°hl(l jj e Sc nbe all the points of a curved surface ; because\ U its ^ n ecessary not only to indicate each of them, asa e t'Vo° 1 ' 1Z ? nta b as it 3 vertical projection, but also toProjections of the same point united together, in

order to avoid a combination of the horizontal projection of !one point with the vertical projection of another; and the most jready mode of thus uniting these projections, being to join ithem by one perpendicular right line to the line of intersec- ition of the two planes of projection, the draught would !become surcharged with a prodigious number of lines, and !cause a confusion, which would increase in proportion as wewould aim at accuracy and precision. !

“ We shall now prove this method to be insufficient, and \destitute of the requisite copiousness. I

“ Amongst the vast variety of differently-curved surfaces, jthere are some which extend only through a finite and cir-cumscribed portion of space, and whose projections arelimited, as to extent, in every direction; as in the case of asphere, the extent of whose projection on a plane is reducedto that of a circle, having its circumference equal to that ofthe sphere ; and we must allow the plane, on which theprojection falls, to be of dimensions sufficient to receive it.But all cylindric surfaces are as indefinite in a certain direc-tion, as the right line by which they are generated ; and theplane itself, the most simple of all surfaces, is indefinite intwo ways. There are likewise a great number of surfaces,whose protuberant particles ( nappes ) shoot at once into allthe regions of space. Now, as the planes on which projec-tions are received, are unavoidably of a limited extent, thismode of describing the nature of a curved surface, had we noother than that of the two projections of each of the points :by which it passes, could be only applicable to those of which jthe points of the surface correspond to the size of the planes jof projection ; all beyond this, could neither be expressed jnor known: consequently, this mode would be insufficient. |Lastly, it would want variety, because we could not deducefrom it anything relative either to tangent planes to the sur-face, nor to its normals, nor to its two curvatures in eachpoint, nor to its lines of inflection, nor to its returning arretes,nor to its multiplied lines and points, nor, in a word, to anyof the affections necessary to be considered in operating ona curved surface.

“ It is therefore necessary, that we should have recourse jto some new principle, not only compatible with the former,but also capable of supplying its place, whenever it becomesin itself insufficient for our purpose. It is this new principlethat we are now about to lay down.

“ Every curved surface may be considered as generated bythe movement of a curved line, either inflexible in formwhen its position is changed, or variable both in form andsituation. As the universality of this proposition may renderit difficult of comprehension, we shall explain it by somefamiliar examples.

“ Cylindric surfaces may be generated in two principalways, viz., either by the movement of a right line, whichkeeps constantly parallel to a given line whilst in motion, yetinclining always towards a given curve; or by the movementof the curve itself, taken in the foregoing instance as the con-ductor, which so moves, as that while from one point itinclines towards a given line, all its other points may describeparallels to this line. In both these modes of origin, thegenerating line, which is a right line in the first case, and acurve, of whatever description, in the second, invariablyretains its form ; only changing its position in space. '

“ Conical surfaces have, in like manner, two principalmodes of being generated. First, they may be consideredas generated by an indefinite right line, which being forced to !pass always upon a given point, moves so as to lean constantly ^towards a given curve, that directs its movement. The only 1point through which it always passes the right line, is thecentre of the surface, improperly called its apex or head. In