COL

181

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of ma y diminished by a curve, according to any

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following methods :

—Figure 1. Take the semi-diameter, A b, atdi 4tn P °f the shaft, and apply it from c to d on the semi-ajj a ® er > c e, at the bottom ; with the radius, c e, describetj • > E p. Draw d f perpendicular to e c, divide the arcE n 0 an y number of equal parts (as four, in this example)

into tjT ” n ’ n F » a ^ so divide the representative axis, c b,ttitoiiat? , arae number of equal parts, c m, m m, m m, m B ;'h'awp P°' n * s m i m > & c - and also through the points n,cut the' 1168 m t and v n ’ P ar;l d e ^ to e c, of which the lines v nt° n jj representative axis at v ; make all the lines m t equalv n, beginning next to the base in succession,Hicij . A B 5 then through all the points, t, draw a curve,ditnini p cont our of the column. The edge of the'he r lln g rule, h, shown upon the other side, is justc°lu mn e £ s e !’ n that is to say, it is concave, the contour of the

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as

‘"to.

JVlET s convex,in jv or> If -—Figure 2. The points d and f, being found,

- ‘Sure 1

e - instead of dividing the arc, e f, as in Figure 1,Partg^ 8 P arts » divide the straight line d f into the equalthe L *» * *» xx, x f, and the representative axis C b, into°Perfiti 0 6 num i )er > an d complete the other parts of the

etlCe be - "’ 88 ’ n Fi S ure i- The same letters of refer-a >naij Ur ln S fixed to the like parts, show the process to be

8 haft, and^ — Figure 3. b C being the altitude of the

fion B A > at right angles to it, the quantity of dirninu-Pciritg E ° ne s ide of it: divide C b into equal parts, at thePartg ) a ’ E ’ E ; also divide a b into the same number of equalj^allel t t le P°' nts d, e, f; draw the lines d g, e h, f i,jj 0 AB > again from the points d, e,f, to the point C,^ l > C, (j a S> ^ G c > e H C, fl C ; and through the points A, G,' cun 'e, which will give the contour required.

lt> Pinyj, D rV— Figure 4. A B and b C, being the same assuppose it were required to give less<i any j ° t ^ le contour of the column ; between a and bate . rme diate point, g, nearer towards A or b, as thea Is 'n tbe 8 1Bten ded to be flatter or quicker (in this examplef y tm r „i mi ddle of a b) ; draw the line g c, divide a g intob , Ur ); divil ° f e( l ual parts, Ad, de,effg, (as here intoe n p > f o f B C ' nto the same number of equal parts, b d,1 c dra w d g, e h, f i, parallel to b a, draw d G c,i s A en through the points A, g, h, i, c, draw aghic,4* T5i o£ e £ Urve required.

0 E 8 to it ~~~figure 5. Join a c, and draw A L at right

'he an d c E rQee t> n g c b produced at l ; draw A d parallel toe a , 8 atn e riu P® r P e ndicular to it; divide h C and A d, each intolatt e e< l Uid parts, the former at the points e, f, g,a) SQ °f eq Ua [ r at a, x, K ; also, divide a b into the same num-C 11 c, f £ a !? s ’ A «. effg, g B ; join emh, fni, gok ;if r ^qui re( j C) 9 o c, and draw amnoc, which is the

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"icse‘^icula ,. 1 —Figure 6. Join ago, and bisect it by a'I'aty 6 tli e ^ F G ’ on the centre, c, with the radius, C A,bp 0tl ®Ec, a*'? AD 5 divide ad into two equal parts in e ;"ad l( i ed» e P a rallel to g a draw f i ; make an angle, Cfi,bile •’ an d with & ^ oart * or ru l e > P u t in pins at the points c'hg ! 8 m 0Ve( j „ a Pencil, upon the angular point f, while theNil" 1 at E, and r ° m F to C ’ keeping the side F i of it uponP°>nt P t ^ I . e same side f c, upon the pin at C, theKil 8 h j , will describe the contour of the columnnr and C. i n

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like manner, by removing the

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’ n d putting it in a, the part f a may

be

l0l > fio ® Curv e might have been found by one continuedA to c, as follows : suppose the line d C to have

3 a

been produced to a point, k ; the supposed line, c k, to have \

been equal to CD ; and an angle, having been made upon the I jedge of a thin board, equal to A C K ; then the contour, AFC, j

would have been described in the same manner, between the j

points A and c, as each of the former parts shown by the i

figure. It is obvious, that by this last method, it would be j

requisite to have the machine twice the length of that in thefirst method ; from which it would become more unmanage-able in the formation of the curve, and inconvenient in manysituations, for want of space to extend it to the necessarydistance.

Method VII. —Figure 7. Let a b be the representativeaxis of the column ; ad, bc, the semi-diameters of the topand bottom of the shaft ; produce C b to f ; on the point D,with a radius, c b, describe, an arc, cutting A B at e ; drawD e f ; in A B, take any number of points, m, and draw thelines F mn; make each line in n, equal to b c ; then drawD##...C| which will be the curve required.

Method VIII.— Figure 8. The points e and f being foundas in Method VII. place a rule with a canal, or groove, onthe axis A b, and put a pin in f ; take another rule, d f,having a groove on the under side, and lay this groove onthe pin at f ; put another pin through the rod at e, into thegroove AB; then, with a pencil, through D, and moving j

the ruler d f, while the pin e slides in the groove a b, and \

the groove on the under side of d f on the pin f, the contourd c, will be described with one movement.

Method IX.— Figure 9. A b being the axis of the column;

A D and B C the lower and upper diameters; draw D e parallelto a b ; find the point f, as in Methods VII. and VIII., anddraw e c f ; divide A f into any number of equal parts, bythe points g; also divide de into the same number of equalparts, by the points h; join each corresponding g h ; makeevery g i equal to a d, and the curve drawn through all thepoints i, will be that required. This mode is practised whenroom cannot be found for Figures 7 and 8, with which it isthe same in principle. j

Observations on the several methods. —By the First Method,the curvature of the shaft becomes continually less towardsthe superior diameter, and if the contour were extendedbeyond the shaft, it would meet in a point, of which itsdistance from b would be a fourth proportional to the lengthof the arcs ef,fg, Figure 1, and the axis C b ; the semi-axalsection of the whole thus produced, is a figure of the samenature as the figure of the sines.

By the Second Method, the curvature of the shaft is con-tinually increased towards the superior diameter, and if thecontour were extended above A b, Figure 2, it would termi-nate at a distance from c, which would be a fourth propor-tional to d f, C io, c b ; the contour thus produced, would be !an elongated semi-ellipsis. I

By the Third Method, the curvature of the shaft is con- jtinually less towards the superior diameter, and if the contourwere extended, the two sides would meet in a point, whosedistance from the lower end of the shaft would be the secondroot of a fourth proportional to the quantity of diminution,the semi-diameter at the bottom, and the square of the altitudeof the shaft. The shaft and the part thus produced, forms .the half of a parabolic spindle ; and the axal section is twoequal semi-parabolas, joined together by a common ordinate,which forms the axis of the column.

By the Fourth Method, the curvature is likewise parabolic;but the axal section is two equal portions of a parabola, lessthan semi-parabolas.

By Methods V. and VI., the curvature is everywhere thesame, and is consequently the arc of a circle : this contourwould therefore meet in a point, which would be distant from