PRO

341

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j-^&es, —None of the lines on the proportional com-

the i e Cept tbe tan gents, require the slit to be less than> tho n ® tb °f the shank, deducting that part of the slidel' 6 i 4la e ? U ' e of f'e pin'i'l , Urnb d u 8 ller than do

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PRO

the tangents, tlierefore, cannotand, indeed, the higher tangen-

‘‘IDftric, ' — ' o- -o —

.• te Qed k. S Wou ^ *>e of little use, as the radius would be-di great ^ . evei 7 such increase : and it would be attendedl ' 1 ' v ev er "'accuracy and more labour.

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e com

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aa

-. If the gradations,

0 not exceed 45°, the distance of the extremitiest 0 ij p ' ISses may still be extended to six inches: but, in4o, ^ d tbe tangent of any greater number of degrees9o e * lav e only to subtract that number of degreesv" ^iiu 16 remainder is the complement of the angle ;

' 8 l’nft:. be as h>e tangent of the complement is to radius,V No7 t0 U *e tangent itself. _

ti^^l'ino l6S ® observations it will be very easy to find the1) 6 lfl< lial f° mt °f any line, whatever number of degreese > s <’-c n - that line forms with the primary radial, as may

Vi ,!0 'n". 5 t 4 f '° llowir, S P robl,!rn -

f e r a( ^ (Q *’Nil—7’ 0 find the vanishing ‘point of a line,tigl than w ^ieh makes a greater angle with the primary* (i * 1 ^5°, the vanishing line and its centre being given,Cn^hact P ^ mar y radial.

^Passeg , t le given angle from 90°; set the index of thev "'^nient ^ tbe tangent of the remaining angle; take aU^KlS!3 a ° t part of the distance of the eye from theoff 11 a ny j lnes > anc i set the legs marked it, to that distance;e b ‘e di,t lmaginar y straight line, A e, with the ends K, setv s, n j a nce a

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tslij

b ; take the distance a b, with the oppositej. Set °ff the distance A d ; then let vl be the'in !' e ’ an d c its centre; and, according as the origi-

S - b e sb , 1 *’ 8 r adial, is on the right or left, the distance, A d,

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<< ! et to the

right or left of the centre of the picture,

Joints ne r adial is supposed to contain the distance oftlie le S s marked R.

fttij- appose the radial of the line to make an angle

p 0 |'al to jj 16 primary radial, and the length of the primaryS b 'on/ti Uee beet ’ t0 ^d ,be distance of the vanishingit),] Ub tra et cenl|, e of the picture.

to t v °° b 'om 90°, and the remainder is 40°; set thev ry »«dii tangent of 40° ; take the ninth part of the"ticj* rn ai - k e <3 ’ ° r ^ mches, and extend the ends opposite to•Ht> p e Peat t i R ’ !° d inches, or the ninth part of the distance,lne > ni n 16 ^‘stance of the legs marked r, on the vanisli-, e tlr nes from the centre to the right or left handc > *s ti! an d the extremity of the distance so repeated

- teal 6 Mishin. - ’ *

n

#f C 8ent of

18

-"son of c- ° point re( l uired -! an p r 01 this operation is evident; since the radius

gent „?? rt ’°nal between the tangent of the angle and

’tlj c °mpi lts cora plement, and, therefore, as the tangentIf 6 Rental angle is to the radius, so is the radius

C tl, e ^t "self.

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gle.

the

be d rawn be a rectangle, making unequalintersecting line, the angles made by the

„ , s, des and the *

it»'’ ° r a ]■ “y the•tej-g *; hn e ^

eCt ’ n g linB aWn trough the outer corner parallel to the

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primary radial, being equal to thesides of the rectangle and the intersecting

>ne ;

Pit) m e j - se t the proportional compasses to the tan-■'al ® p i °p f Q ast angle; extend the legs marked r to the\ ^.aiit of. an y Part of the distance that the compassesS -stance’ ^ t * le distance of the opposite legs, or repeatS'f'enC” 1 }?” as the primary radial contains parts,6 >tt er)(i ^' v e th t0 4 ^. e '’ght or left, and the point of exten-^°tti 0 . die e nc j e Va nishing point of that side of the rectangle ;°f it b e f °PP°site to R to the primary radial, or thatVo, tli e ext ° re .’ men tinned, that the compasses will admitt. nsion between the extremities of the legs4 s

marked r, being set from the centre on the other side of thevanishing line, or repeated according to the number of partsinto which the primary radial is divided, will give the vanish-ing point of the other side of the rectangle, viz. of that sidewhich makes the greatest angle with a line perpendicular tothe intersection ; and therefore the vanishing point so foundmust be upon the same side of the centre of the picture thatthe angle itself is upon, being nothing more than the tangentof that angle.

i Scholium .—The vanishing points of a rectangular buildingmay be found arithmetically upon the foregoing principles,viz., as the tangent of the angle made by the radial of oneside and the primary radial is to the primary radial itself, sois the primary radial to the tangent of the angle made by theradial of the other side and the primary radial. Now,admitting the shortest tangent to be to the primary radialas 2 to 3, and the length of the primary radial 3 feet 6 inches,or 42 inches; to find the vanishing points, it will be

3 : 2 : : 422

3 ) 84

28 inches, the distance ofthe vanishing point of the least angle.

And again, 2 : 3 : : 423

2 ) 126

28 ) 1764 ( 63 inches, the tangent168 of the greater angle,- as before.

When the distance of the vanishing points is required,the square root of the sum of the squares of each tangent,and the primary radial, will give the distance from eachrespective vanisliing point ; but if the vanishing points beinaccessible, subtract each tangent from the respective distanceso found, and set the remainder on each side of the centre ofthe picture.

This method of finding the distance would be thought bymany artists very troublesome ; but the following, by theproportional compasses, is very easy : Suppose one radial tomake an angle of 30°, the other will be 60° with the primaryradial; add 30° to 90°, the sum will be 120°; take the halfof 120, which is 60 ; then 30 from 60, there remains 30;the tangent of 30°, set upon the other side of the primaryradial, will give the distance of the vanishing point. Again,add 60 to 90, the sum is 150, the half is 75° ; subtract 60from 75, and the remainder is 15°; then the tangent of 15°,set from the centre of the picture on the other side of theangle of 60°, will give the distance of the vanishing point ofthe line that makes 30° with the intersection.

Again, suppose one of the angles to be 40°, the other will

63 inches, the distance ofthe vanishing point of the greater angle.

Or, because the distance is a mean proportional betweenthe two tangents, it will be

28 : 42 ; : 4242