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PRO

340

PRO

'Example 4.—To find the cosine of 50°, the radius beinggiven within the limits of the instrument.

Set the index to 50° on the line of cosines ; extendthe points corresponding to E to the radius, and the oppositepoints will be the cosine of the angle required.

Example 5 .—The chord, sine, cosine, or tangent, of anynumber of degrees being given, to find the radius.

Extend the points corresponding to d to the chord, sine,cosine, or tangent, and the opposite points corresponding toE will be the radius of the circle required.

Peoblem: X.— To draw the representation qf a house, bymaking the centre of the picture the dividing point, havingthe dimensions of the building given.

Figure 12.—Let vi be the vanishing line, h It, the inter-secting line, and C, the centre of the picture ; and let thebuilding touch the intersecting line in a ; also, let the vanish-ing points of the sides be supposed to be given.

Set the proportional compasses to the cosine of the anglewhich the one side of the building (viz., say that on theright) makes with the intersecting line by Example 4, Pro-blem IX., say 30° ; then, with the ends of the instrumentcorresponding to e, take the original measure from a plan ;or, if the measures are known in feet, &c., take them from ascale, without being at the trouble of drawing a plan ; takethe distance of the corner of the building, and the nearestside (say 3 feet 6 inches) from the scale, with the remotepoints, and with the opposite points set off the distance a bon the intersecting line ; take the breadth of the windowfrom the scale (say 3 feet) with the remote points, and withthe opposite points set off the distance b c, on the intersectingline ; take the breadth of the next pier from the scale (say 4feet) with the remote points, and with the opposite points setoff the distance c d : proceed in this manner to h, so that thewhole extent, a h, will represent a distance of 24 feet. Nextproceed to the end of the building, which is supposed to beon the left hand; here, as the angle of the building is supposedto be a right angle, and as the right-hand side was supposedto make an angle of 30° with the intersecting line, the left-hand side will therefore make an angle of 60°; set the indexof the proportional compasses to 60° in the line of cosines ;then, from the scale, take the breadth of the end (say 20 feet)with the remote ends, and with the opposite ends set off thedistance, A k ; set the breadth of the piers from k to j, androm a to i, in the same manner, draw the lines A h and a kto the vanishing points of the sides.

From the points, b, c, d, e, f, g, h, as also from i, j, k, drawlines to C, cutting a h at b, c, d, e,f g, h, and a k at i,j, k :from the points of section draw perpendiculars tovL; set theheights of the apertures, and the height of the building itself,upon the corner A a': say that the sills of the lower windowsare three feet high, the windows six feet, the space betweenthe lower windows and the upper, four feet, the upper win-dows four feet also, and the distance to the parapet three feet;therefore make a l equal to three feet, l m equal to six feet,m n equal to four feet, n o equal to four feet, and o A' equalto three feet, exactly to the scale or natural measures of thebuilding : from these points of section draw lines to thevanishing points, and complete the whole representation. Insetting up the heights, common dividers may be used. Bythe last setting of the proportional compasses, if the thicknessof the walls are taken from the scale by the remote ends,marked e, and the distance of the two opposite points be setupon the intersecting line from the points, c, e, g, towards A;and if, from the points of section, lines be drawn to C, thecentre of the picture, to cut the representative base line, a h,of the building, and perpendiculars be drawn from the pointsof section in a h, these perpendiculars will give the reveals

its

of the windows ; that is, they will show the thick * 16walls on the sides of the windows. j afi e,

Peoblem XI. — Given the vanishing line of n P 0’centre and distance, and the inclination of a “ 1ie ^ of ^plane to the intersection ; to find the vanishing P° . ;l gr d erline without drawing any lines in the vanishing p‘ an ’ 1,1

to find the vanishing point of the original line, as "position to the intersecting line. _ . ,j ne n'itj’

Figure 14.—Let the angle made by the origins ^ , v iUthe intersection be 30°; subtract 30° from 90°, an sge 3 to^remain ; bring the index of the proportional comps c otO'on the line of tangents ; extend the extremities o the

passes, marked e, to the radius ; then set the dista^ yj,,

side of the rectangle will be 40°. Now, su PPf^ t0 sts**^

siae oi tne rectangle win De w. iw™, t tor to *of the picture to be 12 inches, and the spec a„ n j s liiog

2 feet, or 24 inches, from the centre of the

ind

aal P

iartSi

slide

the

and suppose the proportional compasseswhich is the usual length : divide 24 into ®^ bes }of which, in this instance, will contain 4 an gen ts l 0 at tindex of the instrument to 50° on the line o sixth P^g

with the legs b, take the extent of 4 inches, p0 ints ° n1 i }

of 2 feet, and repeat the distance of the °PP 0 p ; c ture> t 0 ."i)ingvanishing line, from the centre, C, of >p ie .P g t he van* 6 : » 9the right-hand, six times, to v, and v will 8 ’ _ ban d s ’ d jjal,point. To find the vanishing-point of the prima* - ? T . t 9 n'the radial of this side makes 40° with * fine o . 4

item

inches, or the sixth of 2 feet; repeal 0 f tne

the points of the opposite ends from the cen ^ , v jUtowards the left hand, six times to v > . an .„„ tfln g' e '

LUWU1-U9 U1C ICll IIUUU, BIA -- (O^ 0

vanishing point on the left-hand side ot tn ^y b e fin£The vanishing points of all lines wha e intf s flll glein this manner, having the angle forme ^ to r>. s !

line and the original : for this is alwaysformed by the radial and the para

is aiw»j“ * . a iH* - r

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an d

t} ! e distance“". u

andl ) 6 18 the tan gent l of S !h ng point from the <x -“" vra®th P 16 ra <h'al 0 f f ke angle made by the primary ,. uSi

trn P r °P 0 r £i°j}a] comn lUe ’ primary radial being _e vanishing point Passe8> being set as above, will g lV

luainvu av, iw iuv< ittuiuo f mvii avv »*— t

opposite extremities from c to v, on the vanishing ^and v will be the vanishing point of the line, as re ^ 3(an ce 0The proportional compasses will answer to any <* ta ) ; iri£the eye from the vanishing line, however great, ; tb*one-lialf, one-third, one fourth, &c. of the distanc^^jfiesvanishing line, as most convenient, with the e p 0 iifi smarked b ; then repeating the distance between ^ yicoe 3 ’of the opposite extremities, twice, three times, 0&c., accordingly. Vin»l ^ ne '

By this method, the vanishing point of every poi rlt

of any plane figure, may be found without j ; n6 fro ** 1

of sight placed in a perpendicular to the vanishml e nie nt ’the centre of the- picture. This will be very con a nthe extension is entirely confined to the vanisbin^Ldtb the 0thus it requires the paper to be of no greater brwhat is sufficient for the picture intended. . ..„ poi nt *Example. —Let it be required to find the van be place® #of a rectangle, supposing the right-hand side to r40° with the intersecting line. . 3 ; n a r ’®.' v

Subtract 40° from 90°, the number containe pjadeangle, and the remainder, 50°, will give the an eL t j, ethe primary, or shortest radial, and the radia 0 ^ t j,e i eC ,hand side, or the angle made by the original si 6 0 rig' n ^tangle on the right-hand, and a perpendicular ‘1 contendplane to the intersecting line ; then the a » ^between the primary radial and the radial t j ie le^o

atlinei

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to be e ach