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A Pplic a

°f c U^J Ca ^ 0n °f the method of constructing the intersections“l n p. SUr f a ces to the solution of various questions.

'he r n J UjUre 26, we have defined the mode of constructing. ons of the intersection of two curved surfaces,"'e 'heir form and position ; which we have done in

that is, without attending to the nature of thewhose

'I'lie exrf , w * 10se solutions would require such operations''hi b B ,. 0s ’''°n of this method, even in this abstract manner,

h,„ ue tnn„j _

as 0n

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'°nnd sufficient for most of the arts ; for instance, m

... d carpentry, the curved surfaces there considered,fk tler allv°r ^ Struct ' on °h whose intersections may be required,j ■aiselv ° rrn 'ho principal object of attention, and presentJay b ef .„l a “a'urally. T1 ■ - ' • ••-*■-—

. ^ -j. But as descriptive geometry will one

^th 0(]a e a principal part of the national education, itst l ' n 8) no less necessary to artists than reading,

0 poijjj *' arithmetic ; we conceive it must prove usefulJalyji ( ? ut > by a few examples, how it may furnish thetb' hi at ° r l ' le solution of a

great number of questions,

n uviutll/u VI U givuil J.1UUJVV1 Vi <^UVUI/lViiW)

* 8 aianj) U ' St s '»ht, seem not to admit of being treated in’/ tb e We shall begin with such examples as require

' V ^ cl >'tbe nterSection:

of planes, and then proceed to those in1 Tfi e o 1 , n ' ersec tions of curved surfaces are necessary.>ni n „ question that forcibly occurs to those who are6 ccatr P . le P elements ordinary geometry, is the finding of

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h' e arbit. a c * lc ' e whose circumference passes throughj. ls points on the plane. The determination

of

the intersection of two right lines, upon eachty>t s ge npr la necessaril y found, surprises the pupil as well

■ -a gene r vcaHulll y iouhu, surprises uie pupu as wei^ all n p a as because it yields a mode of execution,tld EQ ? tr y treated in the same manner, which it may

W ^ ' v ould . . *

^ c Ulti Vat j a much greater number of geniuses, wouldtli e 0 ;V an< l practised by a far more numerous class ofJanced .! ‘ nar y instruction of the nation would be moreto .Ju tlj',.' . r ! a " le science itself carried to a greater extent,a °rie i lee dimensions, there exists a question analogous^ rst o'* < l?°' e< l> with which we shall begin.

%- 6re > wii est * on -—To find the centre and radius of aairily ;° 8e su rface passes through four points, giventon ^' o/tt tion Space -

' r om Uriel y ,^l le four points being given by their hori-0n e of J, Itlca l projections, conceive right lines drawn

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. — vjvvuviic, wuiigi v c Aigim lines uitttr u

. ft nd y ,. m to each o' the others ; and trace the hori-ti, _ e jtical projections of such lines. Then, in con-of these right lines, it will appear evident,

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»Hcb 0 extrenp ■ centre must be at equal distances fromVdj '‘ght li n Itles >'t will be on the plane perpendicular to'’ill '•^' n gthe > an ? *' ra wn through its middle. Therefore'lie jjpve tl le P r ?j ec t*°us of the line into equal parts, whichSl,e>of u P r qiecti° ns of its middle, and by constructing' r ;iA,, a U'e, as j® P'ane drawn through the point perpendicular^ a Plan 38 ^ 6en t ,e f" ore described, we shall obtain the'J 1 C0 llS j® u Pon which the centre required will be found., ‘l ) t; l i 1| lS|v fc]y f 0 6rin S the two other right lines, and working' 'e c 'l>e tra C e ei ' C '* °f them, a like operation, we shall|\t be fe s°u4°V hree severa l pl anes ) upon each of which

l‘t for will be found. Now, as the centre

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l' " =re but”.** t ^ le f lrst ail d second of these planes, it cant l* v etb t ^* e 3' ne °f their intersection ; therefore,

• 16 -,J >0 h each i P r °j ec t>ons of this intersection, we shallane of projection, a line containing that of'°n of t | tlle same reason, if the projections of thebl' 3 ‘lie ,e °n f , *, e ® rst and third planes be constructed, weof lr °'i 6c tio n ' ?' ane °f projection another line contain-' fttl iti P r °jecti ° 'l )e centre. Hence we have upon eache the D ° r n . tw ? right lines, whose intersection will" °J e ction of the required centre of the

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“ By using the intersection of the second and third planes,we shall obtain a third right line, that passes through thecentre, and whose projections also pass through those required,which furnishes a means of verification.

“ As to the radius, it is evident, that a right line drawnthrough the projection of the centre and that of one of thegiven points, will be its projection ; whence we may obtainboth the horizontal and vertical projections of the radius,and consequently its size.

“ When the position of the planes of projection can bechosen at pleasure, the preceding method may be considerablysimplified. Thus, suppose the plane that we have consideredas horizontal ( Figure 38) to pass through three given points,so as that of the given projections a, b, c, d, of the fourpoints, the three first may be blended with their respectivepoints ; then, having drawn the three right lines a b, a c,a D, suppose the vertical plane to be parallel to a d, that isto say, that the right lines l m and a d are parallel to eachother ; the vertical projections of the three first points willbe upon l m, in such points as a, b, c, and that of the fourthwill be given somewhere in a point, d, of the right line d d,perpendicular to L M. This done, the line drawn from thepoint A to B, being horizontal, any plane perpendicular to itwill be vertical, having for its horizontal projection a lineperpendicular to ab. It is the same with respect to theright line drawn from A to C. Therefore, by drawingthrough the middle of a b, the indefinite perpendicular e e,we shall have the horizontal projection of a vertical plane,that passes through the centre of the sphere ; consequently,the horizontal projection of the centre will be somewhere onthe line e e. So, likewise, by drawing through the middleof a c, the indefinite perpendicular r f we shall have theprojection of a second vertical plane, that passes throughthe centre of the sphere, and the horizontal projection of thiscentre will be in some point of the right line f f: therefore,the point, G, of the intersection of the two lines e e, vf, willbe the horizontal projection of the centre of the sphere, andits vertical projection will consequently be upon the indefiniteright line of projection G g g.

“ The line drawn from the point A to the fourth point beingparallel to its vertical projection a d, any plane perpendicularto it, will also be perpendicular to the vertical plane of pro-jection, and will have its vertical projection in a right lineperpendicular to a d. Hence, by drawing through themiddle of a d, the indefinite perpendicular H h, we shallobtain the projection of a third plane, that passes throughthe centre of the sphere ; therefore, the vertical projectionof the centre being at the same time upon g g and h h, mustbe in the point k, of the intersection of these two lines.

“ Lastly, by drawing the two right lines a g, ok, we shallevidently have the two projections of the same radius of thesphere ; therefore, by measuring A G upon L M, from g to J,we shall have, in the right line J k, the size of the radiusrequired.

“ Second Question. —To inscribe a sphere in a giventriangular pyramid; that is to say, to find the position of thecentre of the sphere and the size of its radius.

“ Solution. —As the surface of the inscribed sphere musttouch the four faces of the pyramid, it is evident that a planepassing through the centre of the sphere and through eachof six aretes, would divide the angle formed by the two facesthat pass through the same arfete, into two equal parts. If,then, three of the six aretes be chosen, which do not all passthrough the same apex of a solid angle, and if through eachof these argtes a plane be made to pass, dividing in twoequal parts the angle formed by the two corresponding faces,we shall obtain three planes, upon each of whieh the centre

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