Chap. VIII.

GEOMETRY.

753

The CoJumtK G has a circular plane at top and bottomand a cylindrical surface; the irregular figure P is boundedby numerous planes running through QRS, &c.

To draw accurately such figures, it is necessary to intersectthem by planes in a vertical as well as in a horizontal direc-tion : the form P indicates the taste which prevailed in thetime of Louis XIV. , and which took precedence over thesimple .shaft previously in use; to ascertain the quantity orweight of such irregular figures the greatest care in theirmeasurement is required.

The difference between the diameters at O and N in thecolumn H I is termed its diminution, and the proportionswhich govern this in architecture should always be drawn froma study of nature. Smeaton , in preferring the trunk of theoak for his model .of the Eddystone lighthouse to the column,showed that he had thought well on the subject, and by adoptingthe curve line for its section, he produced less resistance to thewaves as they rose up its sides: where columns carry weightsthey must be proportioned to the load, and many writers haveurged that eight or nine diameters is as much as should at anytime be depended upon for stone or timber; when used as poststhe ancients varied their marble columns from four to tendiameters in height, but on no occasion do they appear to loadthem, when applied to temples, beyond their own weight.

K is an irregular cylinder, hollowed in the extent of itsheight, and formed of different horizontal planes.

The contents of such forms are not very easily obtained, toacquire which a variety of dimensions are necessary : to modelthem the turning lathe is employed : their curvature can thusbe fashioned to the purposes for which they are intended.

A Cylinder has three superficies, formed by a rectangularparallelogram ; the solid A is generated by turning the paral-lelogram Z B C on its axis B C.

The cylinder is a solid figure, the surface of which is partlyplane and partly curved, the plane portions being two equaland parallel circles, and the curved portion such that any pointbeing taken in the circumference of either circle, the straightline which is drawn through it, parallel to the line joining theircentres, lies wholly in the surface.

The base of a cylinder is the circular ends R and S, that atQ is shown in perspective.

Wherever a cylinder is made use of, either for support or asa gudgeon attached to machinery, we must recollect that itsstiffness, when compared to that of its circumscribing prism, isas three times its mass to four times that of the prism. Whena cylinder is compared with a prism of the same length andweight, its vibrations, according to Dr. Young, will be lessfrequent, in the ratio of 300 to 307, or nearly of 43 to 44 : thenit may be said that the stiffness of a cylinder is to that of itscircumscribing prism, as three times the bulk of the cylinderto four times that of the prism ; the authority before citedalso observes that the force of each stratum of the cylinder maybe considered as acting on a lever, of which the length is equalto its distance x from the axis, for though there is no fixedfulcrum at the axis, yet the whole force is exactly the same asif such a fulcrum were placed there, since the opposite actionsof the opposite parts would remove all pressure from thefulcrums; the tension of each stratum being also as thedistance x, and the breadth being called 2 y, the fiuxion of theforce on either side of the axis will be 2x*yx, while that ofthe force of the prism is 2:r 2 .zand its fluent ^ x s . But the

fluent of 2 x*yx, or 2v^(l— xx) x*x, calling the radius unity,is ^ (z— y*x), z being the area of the portion of the section in-cluded between the stratum and the axis, of which the fluxionis y x, for the fluxion of z - y* x \s y x — y* x — 3 y* x y — y x*x —

3 C

Fig. 736.

Fig. 737.

Fig. 739.

c

Fig. 740.

Fig. 74!.