CTL
222
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which is a general property of every two diameters, formedby the intersection of planes passing along the axis at rightangles to each other. These diameters, it is evident, will beparallel to tangents to the curve, at the extremities of eachother. For, suppose two planes touching the cylindric sur-face, each in a line which passes from the extremities of eachradius at a right angle with each other, these tangent planeswill be respectively parallel to each plane passing along theaxis and through the said radii; the two tangent planes, andthe two planes passing along the axis, will be at right anglesto each other, or each opposite pair will be parallel; therefore,if these be cut by a fifth plane, the intersections of the oppo-site planes with such fifth plane, in any position, will beparallel lines ; and since the touching planes do not cut thesurface of the cylinder, the lines at the extremities of eachdiameter will be parallel to the curve, and are what aretermed conjugate diameters.
A cylinder is a species of prism, because all its parallelsections are equal, and every section parallel to the base, isequal to that base ; therefore the solidity of a cylinder isdetermined by multiplying the area of an end by the paralleldistance between the two ends.
Example I. What is the solidity of a cylinder, whoseheight is 10 feet, and the diameter of the base 2 feet6 inches ?
in.
6 = = . 5 in decimals
2.5 = the diameter2.5
6.25
.7854
2500
3125
5000
4375
4.908750
10
49.087500
Example II. What is the solidity of a cylinder, the circum-ference of the base being 7.85 feet, and the height 15 feet?7.857.85
3925
6280
5495
61.6225
.07958
4929800
3081125
5546025
4313575
4.903918550
15
24519592750
4903918550
73.558778250
The curved surface of every cylinder is equal to a rect fone of whose dimensions is the length of the ax '\
equal to the length of the side of the cylinder, and thedimension equal to the perimeter or girt. This is a S (principle, whether the cylinder have its ends perpior oblique to the axis ; and hence the following— b 0 {
Rule. Multiply the girt of the cylinder by the l eD othe axis, and the product will be the cylindric surface. ^Example. Suppose a cylinder, girt 5 feet 9 inches, an ^length of its axis, or side parallel to the axis, be7 inches, what is the superficial content of the surface ‘
Decimals.
9.583
5.75
5
3
47915
35
67081
81
47915
i 121
55.10225
12
10
1
45
1.22700
12
55
.1.3
2.724
See the first method in this example un( ^ er . ^
cle
Decimal Multiplication. These two methods won)“ ^agreed exactly, but no decimal corresponding to 7 ind 1be precisely found. _ 0 „ld
The method by the girt is what every man in practic®o-irtnaturally prefer, if the object be before him, si nce , eq u^can be as easily measured as the diameter, and wJ fJ1 bo9 saccuracy, by means of a string; nor would any P ers0 . 1 !’ a dd*'mind is crowded with the affairs of business, ta ^ etional trouble of finding the circumference from the 1when it may be so expeditiously obtained an ot &® f f0 edHowever, for the satisfaction of those who wish to be i $3of every mode of operation, we shall add the folI °” 0 t besome cases may occur, in which the perimeter ca ^(itobtained without the diameter ; though the contrary
.___ . 2 the
before to multiply the circumference by the ax’ s >product is the superficial content. This is so easf ^ t0 tnBut where the axis is. inen” jj a rn ef ^
as likely be the case. .
If the cylinder be a right cylinder, find its cir cwhich will then be the same as its ends ; .then^P g)
lina^Vi an sample, nut wnere me ^ -base, the section of the cylinder will be elliptic, theof the circular ends will be the greater axis, the lesser o be
tlms found° n tLe inclination of the cylinder, andAs the radius
is to the sine of inclination,
so is the diameter of the circular ends
to the shorter axis of the ellipsis. ^ (j
From the two axes being now found, find the p er “°e ellipsis, then proceed as in the first rule. f e L
Example. Suppose the length of the axis to D ^ oand the inclination of the same to the plane of req uir edegrees, and the diameter of each end 3 feet 6 inch eS >the area of the curved surface.