u y l

22 3

CYL

By logarithms.

Then as radius 10.000000

is to the sine of 50” 9.884254so is 3.5 544068

10.428322

10 .

to the shorter axis .428322 = nat. num 2.68

2.68

3.5

2)6.18

3.09 mean diameter

3*

9.27

.44

9.71 elliptic primeter.22

1942

1942

.The

213.62 the area required.

^'ffieter y. V ? met hod °f finding the extension of the elliptic^tical n 61n ^ t ^ le most expeditious, is the most useful for4c «ur acv P ur P°ses, but those who wish to work with greaterAcyj^y.eonsult the word Ellipsis.

?. !| S t >itu ( j e 8r is said > to he given in position, when its base and®'^n. ’ an( i the inclination and length of its axis, are

it s n *j-^ Sa ’fi to be given on the surface of a cyclinder,o; 1 * 6 p:»nii,.i ance from that point to the base, measured in a

trt +Vio i — I, „ „ J xl xi

3.-

, x i J. - * ---

10 t he axis is known, and the point on the^Ur e o Ceof the base given.

tt i ' , ' iCe , to iT? Ven a right cylinder and three points on itsi\ 0u 9h thefh ^ l€ sect i° n °S cylinder, by a plane passingtlj 8tar, Ces 0 f f ee P°ints. Let the straight lines, A, b, c, be thepc/ c }’liti(} e tae t hree points, the circle d e f it, the base oftho* 1 * 8 o and ’ 1 t ^ le centre °f the base. Through theto ret ° draw F ^ raw the straight line p o ; perpendicular(jj 4 ’ ®, c, 6ac * 1 °f the lines »g,ee,pi, respectively equal| heeti n J' aw D . E Q an< f ghq, intersecting at Q; also drawin ol ° °’ t h en Q °> which is the intersection of

° ta ke an ne ' • Through l draw kist, cuttingQ o at t.e Utf Hrn to m T. Point m, draw m u perpendicular to g o, pro-Q o a I In 0 p > an( i draw st u perpendicular to p o,an u - From the point o, with the distance o r,Perp 14p and 1 °’ Cutt ' n g mu at u; perpendicular from r t,l o aT 1 ^* c ul ar . N ’ cu tting p o at p and N. Draw p p and n n'•'d J ^ r <iw t r° p °> cu tting og produced at p, and n. ParallelN i r eq Ual ( e hnes p r and fnlh; make o t equal to oi^, a nd lj t ° p E > and draw r 1st; make l s equal to l r,^ti'oh a will ec l ua i to R l, the semi-diameter of the>x TkJ r and If be the semi-axis of the elliptic

%>^ ation _ .

w 6 > o ’ l ^ at if ’ 13 s h° wn under the article Inclined

j t>o 1 the heicyn’ F ’ h e an y three points, and the lines.N: a w J: l '> thnt 1 th ree points in the space above the^ *fti the or - ^ ° he the intersection of the plane in. ® S^ 'Smal plane p o q or its continuation: undergraphy and Stereotomx it is shown, that

if p o q and p op be any two plane angles, to be placed at aright angle with each other, that the plane angle go m, will bethe hypothenusal plane, and consequently when the three planes,POQ,POp, and you are turned into their position; the straightline o k will coincide with o u; therefore if a plane be perpen-dicular to the intersection of two planes, it will be at rightangles to each of these planes ; and that the section of anyvertical line, whose seat is given, will be found by drawing aline on the original plane, parallel to the intersection fromthe seat, to the intersecting line of the vertical plane; thencea line perpendicular to this intersection to the common inter-section of the vertical and cutting planes, and thence a linefrom this point parallel to the cointersection; then, by makingthis line equal to the line drawn from the seat to the commonintersection of the original and inclined planes, and thusthe point r being the seat of r, and l, the seat of l, and theplane passing through rlst, being perpendicular toQO, thecommon intersection of the inclined and original planes, r 1stwill be the intersection of a plane passing through the axisand at right angles to the intersection o Q, and therefore per-pendicular to Ow; consequently r s and / h are at rightangles to, and bisect each other, and are therefore the axes ofthe ellipsis.

The segment of a cylinder is any portion of the cylindermade by a plane parallel to the axis.

The plane parallel to the axis is called the chord-plane.

The two lines of concourse formed by the cylindric surfaceand the chord-plane, are termed the sides of the chord-plane,and the other two lines of concourse made by the chord-plane and the two ends of the solid, are termed the bases orends of the chord-plane.

A line of position is said to be given, when the said line isdrawn through a given point on each side of the chord-plane,or to make a given angle at a given point in the base of thechord-plane, or in the base continued.

The position of the cutting plane is said to be given, whenthe line of position through which the cutting plane passes,and the angle which the cutting plane and chord plane makewith each other, are known.

Figures 4 and 5.— To find the section of the segment ofa cylinder; given its base and the position of the cuttingplane. —Let ABC be the base of the solid, a d and c ethe sides of the chord plane perpendicularly situated inrespect of A c, the chord of the base, and let d e be the lineof position of the cutting plane. Through any point/in d edraw hf g, at right angles to de ; make the angle gfk equalto the inclination which the planes of the chord plane and sec-tion make with each other; take any point, m, in/ g, Figure 4,ere fh, Figure 5, and draw m k parallel to d e, cutting/^ in k,and m n parallel to c a, cutting d e at n; make J h equal to fk,and join h n. Parallel to e C draw m q, cutting a c at p andn r, meeting it at r: make p q equal to m k, and join r q. Tofind any point in the curve of the section, take any point, a,in the arc ABC; draw a b parallel to q r, cutting A C at b ;draw b c parallel to C E, cutting D e at c, and draw c dparallel to n h; make c d equal to b a, and the point d is inthe curve.

In the same manner, as many points may be found as willbe sufficient to draw the curve with accuracy.

This method is an improvement upon that published inThe Carpenter's Guide, in the year 1792; but as thatmethod has been very generally employed, it will be well toinsert it likewise in the plate, as the connection of the prin-ciple may be clearly deduced from the one to the other.

Figures 6 and 7 .— To find the section of the segment ofa cylinder, by another and older method, supposing everythinggiven as before. —Let ABC, No. 2, be the inclinations of the