PRO

343

PRO

in

C, d, e ; draw B p> C a, d r, E s, F t t parallel to

i 1 ^ of i ’ ’ 5

a kcu 3 j n 1 cylinder, cutting the under edge, im, of the^ti 0 to Pi r, s t t; set the proportional compasses in the

i

tl|f* j.

’WU |* ' > » » o, v f SCI. me wui^aooeo m me

S ”Ppose le distance to which I b will throw the shadow, and,l “ 1B to be greater than the length of its shadow ; take

■...y e anCe ’ 1 B > with the longer legs ; make p b equal to theH® t i c °ntained between the points of the opposite ends ;^Ual Stance, jc, iH e , t0 «»e ‘

equal -' e dista

_ _, with the longer legs, and make acHe ( 1 ,^ *' le distance contained between the shorter legs;

1 to " tance > M d, with the longer legs, and make rdi H th e '- 6 ^’ stan ce contained between the opposite points

P°int^°* ntS ’ c> " dl be obtained. In the same manner,n „ Sean dy’willbe found ; through the points, b,c,d, e,f

% Curv e,

which will be the shadow of the lower edge of

7*' n <w us; tlie sun ’ s rays will be in a tangent plane to theX ® llr face at r f: and the part of the edge of thefalls upon the representation of the cylindricT_ tom j t 0 ^ w jp |j e straight.

Jf tlle

n ®o»n , 11 a cylindric abacus, having the same axis with

Pped . Sarne manner, if the representation of a cylinder

p. , "ith q _i_ i_:_ti,« „„™^

o n ,, er > be given, and the representation of the sun’sHn a s t f ° 1 the la -’ SU PP° sin ? the axis of the cylinder perpen-

°tl,

i r - o **

plane of projection, the shadow of the abacus

,° u nd upon the cylindric surface ; and thus for everyic object.

v. .

p Pfismat

JHs XVI —To describe the logarithmic spiral by aJ e po( n t °* nts found in ^ ie curve > ^ ie centre, and two oppo-Jite n tn a straight line passing through the centre, being

ii:

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‘le ,y e 17.

V**,

Let z, No. 1, be the centre; let the straight

Pass through z ; let A and e be two opposite points.

of an the one side, and the other, e, on the other

k 6 ’ Li S y^ t Centre 5 z ; through z draw GC at right angles to,,Ht t | )e tl>e angle A z c, by the straight line B f ; alsopr prop an »^ e E z C, by the straight line D It; find z c, aa v^ 0rt ion°i rtl ° na * between z A and z E ; also find zb, a mean

s' 1 behVlW to 2 B '

tnJ° r leo.’ s tak ® tlle distance, z a

i v ,e s i and

* 6 equal

een z A and z C ; draw any straight line,> and set the proportional compasses in the ratio ofake the distance, z A, with the points of theset that distance from x to a, No. 2, andtli^’ c °ntr^ '* to l' le distance contained between the shorterket Sb ° r fer tbe P 0 ’ n, s of the longer legs to x b, and withl 11 th e e ® S - set the distance x c: contract the distance»i al to .

"ort,

■er i

P°ints of the longer legs toxc, and make x de distance contained between the points of the

-"■'vcaiivjc cuilltlilicu uciwccii tuc puuno ui two

V be • an( l t l |e parts, a b, be, c d, of the straight line,t), e P°itit s e S e °metrical progression. In the same manner,8 , er ies 0 f’ 9' i, &c. may be found, which will continuelr,' > eqn a j Parts as far as there may be occasion ; make z A,%[■ Sa >«e ° Xa ’ ^°’ 2 > 2 B e( l ua l to x i, z c equal to x c.e d th manner > the points, d, e, f, g, h, i, &c.,may be■S’ / d.r c a< , JU 8l’ any number of revolutions ; and a curveHcl * be P° ints W 'H g' ve the spiral required,

by on t | le -A-S the tracing of the curve depends veryOf e foll 0 ®7 e °f the person who performs this operation,f ap;?J ri Pa 3 Se9 ln o ®ethod a curve may be drawn with a pair* 6t it>n.7 to tlig ! )lov ’ded the points do not approach veryc ®ntre. To describe any quadrant, take the

< haf the

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esn e , tadiu8 that bisects it from one extremity of t etXt, ' be an arc, and with the same radius from the'O’ty describe another arc cutting the former ■ ,P°mt of intersection, with the same distance,W 0u 8li ,v arc between the two extremities, and it will passL tn ’ddle point, very nearly ; thus, take the dts-C>b, as a radius; from the extremity A, of the arc A b C6 r adi,.„ ar ® n ear the centre ; from the point C, wit is > describe another arc, cutting the former near

the centre z; then, from the point of intersection, describe thearc A c, which will pass through the point b, very nearly. Inthe same manner the successive arcs, c e, e g, gi, &c.,maybe described ; and the curve thus formed will be so near asnot to be detected by the eye.

Problem XVII.— To draw the representation of the meri-dians of a solid of revolution upon a plane parallel to theaxis of the solid; given an axal section, that is, a section of •the solid, passing along the axis, upon a plane parallel to the jsaid axis.

Figure 18—Let a b c d e v kj i h gf be the axal section, g vthe axis itself, and a f the base perpendicular to q v ; in q vtake any number of points, r, s, t, u; through these drawb h, c i, dj, and e k, perpendicular to q v, meeting the curveon the one side at b, c, d, e, and on the other at h, i,j, k ; thenit is obvious that the lines b h,c i, dj, eh, will be bisected ;now, supposing the meridians to be formed on the surface ofthe solid by the intersections of five planes at equal anglesround the axis, and that one of these planes is parallel to theplane of projection, and let a vf a be the representation ofthat plane ; draw A b parallel to a f; produce v q to meet a bat w ; from w as a centre, with the radius qa, or qf describethe semicircle A 1 23 4b ; divide the semicircle into fiveequal parts by the points of section 1, 2, 3, 4 ; draw 3 x and4 y perpendicular to A b, cutting A b at x and y ; make q lequal towy; set the proportional compasses in the ratio ofq a to q l; take the distance rb, with the longer legs, andwith the opposite legs set off the distance r m ; take the dis-tance sc, with the longer legs, and with the opposite pointsset off the distance sn ; take the distance t d, with the longerlegs, and with the shorter legs set off the distance t o ; lastly,take the distance u e, with the longer legs, and with theshorter legs set off the distance up ; then draw the curveImnopv, which will be one of the meridians, as required.

In the same manner all the others may be found.

Besides the uses of the proportional compasses, whichhave already been shown, they may he applied to trigo-nometry in finding the sides and angles of triangles ; suppose,in a right-angled triangle, that the two legs were given, tofind the angles; the analogy is, as the one side is to theother, so is radius to the tangent of the angle opposite thelatter side ; set the proportional compasses in the ratio of thetwo sides containing the right angle ; then the index willshow the tangent of the angle on the line of tangents.

Again, suppose the hypothenuse and one of the legs weregiven, to find the angles ; the analogy in this will be, asthe hypothenuse is to the given leg, so is radius to the sineof the angle opposite to that leg ; set the proportional com-passes in the ratio of the hypothenuse to the given leg, andthe index will be against the sine of the angle.

Lastly, suppose the angles and the hypothenuse to be given :the method of proceeding in this case is exactly the reverseof the last; thus, set the proportional compasses to the sine ofthe angle, then extend the longer legs to the hypothenuseof the angle, and the shorter legs will contain the length ofthe leg required. |

In the same manner, if the angles and one of the legs weregiven, to find the other leg ; set the proportional compasses Ito the tangent of the leg required, then take the length ofthe given leg with the longer legs, and the distance betweenthe points of the shorter legs will be the leg of the triangle,as required.

PROPYL2EA, (Greek irpo, before or in front of, and irvXova gate) the entrance to a temple or other large building, con-sisting mostly of a gateway flanked by towers or other erec-tions. The Egyptian temples were universally adorned withmagnificent propylsea, which consisted of lofty truncated