PRO

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PRO

** *> of ax * s P ara dlel to the picture ,) «»<7 vanishing line,°hy £ tfle plane of that end; through a given point, a, in%•_ f, r P e ndicular, a" a, to uz, to draw the representation of

Let tT " en< ^ °f ^ le c y^ n( ^ er -

ii Jf a cut the vanishing line, u z, at u ; draw h" h, c" c,

, ^ }/ e >f"f parallel to a" a, cutting uz respectively at v,

d'

Jo

> >V y y " > r »**v%**^* W W W, x,i«vv*».£ w Jj.ovmuij u.b f

to % a, , Set ^ le proportional compasses in the ratio of u a® 4" ' ien m a" being greater tlian u a, take the distance,

,; » 0 t a j n j l Le longer legs, and make v b equal to the distance^aini k etween the points of the shorter ends; find thel he p 0 : n " Points, c, d, e,f in the same manner, and through^Present “> h, c > d, e, f draw a curve, which will be theOr, jp tat * on °f the other-, or lower, end of the cylinder.

“6 fo'un i^? e Lase, a, b, c, d, e,f were given, the top would8 Vt er | ln fit® same manner, by making the ratio of the' v ith th e ** S *° ^ le longer legs, as u a to ua", and taking v bLric e 0 p neai 'er extremities, and making v b" equal to the dis-Pointa i/^ le rena ote extremities, and finding the remainingin the same way. In like manner, any• fihi 8 ’ °^’ c ’’ ®’ f’ ma y be found.

lri gly a Pplication of the proportional compasses is exceed-f r ® U 1 in the representation of bows, when required ini^ibit 3 , buildings, as is shown by Figure 9, which**er e t i tae elementary lines of the perspective of a house.

horizontal terminations of the apertures are readilyn ( h e e< L by only drawing perpendicular lines, which may’ er efo 1 . Ve /' t ' ca ' terminations of such apertures, and wouldNtcls J? hp required at all events. The same description ofrefkj es to this figure, as in Figure 8, to which the pro-

n^ EJI ^ — Given the representation of any point

y . u fe, and the intersecting and vanishing lines of the° r hinal J n ’ t0 fi n d the height of a representative line, the. Set * which is parallel to the picture.

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a' V( :en 6 P r °P or tional compasses in the ratio of the distancet di§+ l h e intersecting and the vanishing lines, and^ a ce of the point from the vanishing line, then8t| i th e ^ en gth of the original line with the longer legs,^‘fed °PP os * te points will give the height of the line

^ inte^^ e —ySuppose the distance between the vanishing1 6t 'v eeri _ ecting lines to be three inches, and the distanceih l ' v ° in T re P resen tation of the point or foot of the line to‘ re C ■ a> an< ^ the height of the original line to be six. Huired the Dersneetive height of the same. Set. the

Pti

“es,

°Po t

rtj *—*cu the perspective height of the same. Set thehT ' e §s of com P asses ' n the ratio of 3 to 2; extend the lon-t the t ' le com passes to six inches, and the extent be-°PP os ‘te shorter legs will be the height of the lineH^bich, ' n this case, will be four inches.

t -**»lO VUOO) II lit MW IVUi lUUUVOa

'tatiou fids method all the heights of the repre-

ss; but - a s °hd may be found without any additionalPi^ Ct is of'a- orc * er t° expedite the work, when the originalJure, 0r 'afferent heights in the same plane parallel to thelij dvo ^bere the same height is required to be foundt ' > ro u °, r ® ore points, it will be eligible to draw a straightOf ^bich ?/ ° ne ° ' t * le points, cutting the one or more linesuom'l heights are to be found ; and the same settingu Porf a f Se8 w 'h do for as many heights as are to bei s ^V-rnpie * ~ h ne so drawn.

Cyj.■" eliipjr~® u Ppose a cylinder to be represented, its baset„ lri de r . P? IS > to find any two points in the height of thePof'* 1 'he / avv a straight line, parallel to the vanishing line,* U ° Up o e P res f n tation of the base of the cylinder in two

tii^ to Porti n ea °h point of section draw a perpendicular; setV ah . 1 Uter se „?. nt h oompasses in the ratio of the distance between' 8 hin» i; >n 8 an d vanishing lines, and either point and thene ; then take the height of the cylinder between

the points of the long legs, and the distance between the oppo-site points will give the perspective height upon each perpen-dicular from its respective foot.

In the application hitherto made of the proportional com-passes, the sides of the instrument have not required anygradations of parts; but as these are necessary to the perfec-tion of the instrument, the construction of the scales for thispurpose is as follows.

Problem IX.— To construct the scales of sines for theproportional compasses.

Figure 10.—Let a c be a scale of sines found in the usualmanner; and let c a be produced to R, making a r equal tothe radius; draw R d at any convenient angle with R C; makeR D equal to the length of the proportional compasses, andjoin ci); draw a 9 parallel to £D, and as 9 at the point C isthe chord of 90°, the point 9, in R d, will be the point for theindex of the slider to be set to; then if the legs of the com-passes be set to any angle, the distance between the pointscorresponding to d will be to the distance between the pointscorresponding to R, as the sine of 90° is to radius. In likemanner, join D 8, the next point, 8, in the line AC, and drawA 8 parallel to d 8, cutting r d in 8; then the point, 8 in r d,will be the point for the index of the slider to be set to; then,if the ends of the proportional compasses, corresponding to r,be opened to any radius, the extent of the ends correspond-ing to d is the sine of 80 degrees corresponding to thatradius: in like manner, the points 6, 5, 4, 3, 2, 1, will befound in rd, so that 1, 2, 3, 4, 5, 6, 7, 8, 9, in r d, are thegradations on the side of the proportional compasses corres-ponding to 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90”; hence,if the index be set to any of these numbers, and the pointscorresponding to R be extended to the radius, the distancebetween the opposite points will be the sine of that numberof degrees. The cosines have the same divisions as thesines, but are numbered the contrary way.

In the same manner, by placing the radius in the samestraight line with a common scale of sines or tangents fromzero on A, so as to lengthen the line a c, of tangents, byadding the said radius, A r, and forming the extension, r c,drawing R D at any convenient angle with r c, and dividingRD as in the problem ; then rd will contain the gradationsof a scale of tangents, to be inserted on the side of the pro-portional compasses ; so that when the index, or mark, uponthe slider is brought to any of these numbers, the end cor-responding to R being opened to the radius, the end corre-sponding to d will be the sine or tangent of the anglecorresponding to that number. See the Figure.

Figure 11 shows the application of the method to a scaleof tangents.

Example 1. Figure 13.—To make an angle of 30° at agiven point, A, in the straight line a b.

With any convenient radius, ab, describe an arc, bc; setthe index to 30° on the line of chords; then extend thepoints of the instrument corresponding to r, to the radiusA b, and with the opposite points set off the distance b c, onthe arc, and join c A ; then c A b is the angle required.

Example 2.—To find the tangent of 40° to any given radiuswithin the limits of the points corresponding to r.

Set the index to 40° on the line of tangents, and extendthe points R to the radius ; then the extent between theopposite points will be the tangent of 40°, as required.

Example 3.—To find the sine of 20° to any given radiuswithin the limits of the points corresponding to b.

Set the index to 20° on the line of sines, then extend thepoints corresponding to R to the radius given, and the extentbetween the opposite points will be the sine of the angle, asrequired.

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