Fig.144. base, is a circle equal to its base. Any plane section through the axis or parallel to the axis, is a rectangle.

The straight line A'C', also perpendicular to CC', generates the circle A'D'B', also equal and parallel to ADB ; this we call the superior base. The line CC' about which the rectangle revolves, is called the axis of the cylinder, and is perpendicular to the bases.

276. If we conceive a circle, as ADB, around Fig.145. which a straight line, as AA' (fig. 145) moves always parallel to itself, but oblique to the plane of the circle; this line will generate the surface of an oblique or inclined cylinder. Or we may conceive this cylinder to be generated by the motion of the circle ADB parallel to itself, along the straight line AA', every point in the circle describing a straight line by this motion. The circumference of the circle will generate the cylindrical surface; the centre of the circle will generate the axis CC'; and as the generating circle is always parallel to itself, every section of this cylinder parallel to the base, will be a circle equal to the base. Any section of this cylinder through the axis or parallel to the axis, is a parallelogram. Sections parallel to the axis and having the same inclination to the base as the axis has, are rectangles.

277. We have here given two methods of generating the inclined cylinder. In the first, we call the straight line AA' which generated the cylindrical surface, the generatrix, and the circle by whose circumference its motion was guided, we call the directrix. In the latter method we call the circle by which the cylinder was generated, the generatrix; and the straight line by which its motion was directed, the directrix. If in either of these two cases the straight line had been perpendicular to the plane of the circle, the body generated would have been a right cylinder.*

278. A slight inspection will satisfy us that the cyl

* These methods of defining a cylinder are preferable to the description given in Art. 274; for they will apply to all cylinders, if, instead of the term circle, we substitute a plane surface embraced by a curve line; as it is not an essential property of a cylinder, that any section of it should be a circle. Nor is it an essential property of a cone, that any section of it should be a circle.

inder belongs to the family of prisms; and therefore to find the area and volume of a cylinder we should proceed as we would with the prism in the same cases. That is, the area of any right cylinder has for its measure the height of the cylinder multiplied by the periphery of the base; and the volume of any cylinder is measured by the the product of its base multiplied by its height. This may be readily demonstrated.

279. If we inscribe a regular polygon in the circular base of the right cylinder (fig. 146) and circumscribe about Fig.146. the circle a similar polygon, and make these two polygons the bases of two prisins the one inscribed in the cylinder and the other circumscribed about it; as the number of sides of these polygons may be so great that the difference between the polygons shall be as little as we please, it is manifest that we may erect upon the polygons two prisms, the one inscribed in the cylinder and the other circumscribed about it, whose difference shall be less than any assignable magnitude.

The area of the cylinder is less than the area of the circumscribed prism by an indefinitely small magnitude m, and greater than that of the inscribed prism, by an indefinitely small quantity m'. The perimeter of the base of the circumscribed prism exceeds the circumference of the base of the cylinder, by a quantity less than any assignable magnitude, which excess we denotede by d. This circumference exceeds the perimeter of the base of the inscribed prism by an indefinitely small excess which we designate by d'. Denoting the area of the cylinder by A and the height by H, we shall have for the area of the exterior prism A+ m = H × (circ. + d); and for the area of the interior prism, A — m' HX (cire. — d' ) ; × that is, if any magnitude, however small, be added to the area of the cylinder, something must be added to the product of its height multiplied by the circumference of the base, to express this area; and if any magnitude however small be subtracted from the area of this cylindrical surface, a corresponding magnitude must be subtracted from the above mentioned product, to express this area; that is―The product of the height of the right cylinder multiplied by the circumference of the base, (being the measure of an area neither greater nor less) is the measure of the area of the cylindrical surface.

280. The learner is requested to give an analogous

Fig.147.

proof that―The volume of a cylinder has for its measure the product of the area of its base multiplied by its height.

Remark. It is manifest that this last proposition applies as well to oblique as to right cylinders, there being nothing in the reasoning which supposes the axis of the cylinder perpendicular to the base.

281. If the semicircle ACB (fig. 147) revolve about its diameter AB, it will generate a sphere; the semicircumference generates the spherical surface. Every point in this semi-circumference, as D, generates the circumference of a circle whose centre is in the diameter AB (here called the axis) and whose plane is perpendicular to this axis. The extremities A, B, of the axis, are, in reference to this revolution, called poles. They are particularly the poles of all the circles whose circumferences are generated by the several points in the arc ACB of the revolving semicircle.

As every point of the arc of the semicircle is at the same distance from its centre, and as this centre does not move during the process of revolution; it follows that every point of the spherical surface is equally distant from the point O, the centre of the generating semicircle and also the centre of the sphere. It is also evident that any radius of the semicircle, in any part of its revolution, is also radius of the sphere; and that these radii are all equal.

282. If we suppose the sphere to be cut by a plane passing through its centre, it is perfectly manifest that the section will be a circle, whose radius will be the radius of the sphere.

But suppose the plane to pass through the sphere on one side of the centre; and let the section be DGFH; from the centre of the sphere, draw OE perpendicular to this section, and draw radii to the several points in the periphery of this section; they will be equal oblique lines, and will therefore meet the plane DGFI, at equal distances from the perpendicular OE (206). Every point in this periphery is therefore at the same distance from E; the section is consequently a circle whose centre is E. And as this reasoning applies to any plane section on either side of the centre, we infer that-Every plane section of a sphere is a circle; and also that-Radius

perpendicular to any plane section of the sphere, passes Fig.147. through its centre.

23. Remark. As DE is always one side of a rightangled triangle whose hypothenuse is radius of the sphere, DE must be less than radius; the plane section whose radius is radius of the sphere, is called a great circle; other plane sections are called less circles.

284. As two great circles are planes, their intersection must be a straight line; and as each passes through the centre of the sphere, the line of intersection must pass through the centre of each of them, that is- Two great circles mutually bisect each other.

285. Any portion of a spherical surface embraced by the arcs of three great circles, is called a spherical triangle. As two circumferences of great circles bisect each other, if each angle of a spherical triangle is salient, each of the arcs must be less than a semi-circumference.

286. Let CMI be a spherical triangle, and draw the radii OC, OM, OI; it is evident that they determine a triedral angle whose vertex is at O, and whose plane angles are measured by the arcs CI, CM, IM. And as the sum of any two of these plane angles is greater than the third (220) the sum of the arcs which measure them must be greater than the third arc (113); that is -- The sum of any two sides of a spherical triangle, is greater than the third. Whence we infer that-The shortest way from one point to another, on the surface of a sphere, is in the arc of a great circle passing through these points. For if we suppose it shorter to pass through any point out of this arc, that point and the two points proposed may be made the vertices of a spherical triangle, one side of which will be the arc first proposed and less than the sum of the other two; the inference is therefore manifestly true.

287. Suppose the two semi-circumferences ACB and AIB which meet cach other in A and B, to be cut by the arc IM of another great circle; we shall have MI← MB +BI; and consequently AM + AI + IM < AMB + AIB; that is-The sum of the three sides of any spherical triangle, is less than the circumference of a great circle.

Remark. We might have deduced this truth from the limit of the magnitude of the plane angles which form the triedral angle at the centre of the sphere (222); this sum being less than four right angles, the sum of the

Fig.147. arcs which measure these angles. must be less than an entire circumference (27).

288. As any straight line drawn through any point in a curve line (in the plane of the curve) and oblique to a right-line tangent at that point, must cut the curve line. (111), the tangent may be considered as representing the direction of the curve at that point. It follows from this, that-The angle which any two curves make with each other at their intersection is neither more nor less than the angle which their respective tangents at this point make with each other.

The angle CIM of the spherical triangle will therefore be the same as the angle which the tangents to the two arcs CI, MI, at the point I, make with each other. These tangents are perpendicular to radius 10 (111), and are in the planes of the two circles respectively; but being in these planes and perpendicular to their intersection, their angle is the inclination of these planes, or the diedral angle made by the two faces of the triedral angle whose edge is IO. We therefore say—A spherical triangle determines a triedral angle whose vertex is at the centre of the sphere, whose plane angles are measured by the sides of the spherical triangle, and whose diedral angles are respectively the angles of the spherical triangle.

289. It will be readily seen that as the sum of the plane angles in a triedral angle, approaches to four rightangles, the value of each of the diedral angles approaches to two right-angles; and their sum of course approaches to six right-angles; and the less the plane angles of the triedral angle the nearer does the sum of the diedral angles approach to two right-angles. Therefore-The sum of the three angles of a spherical triangle is not constant; and the limits of this variation are two right-angles and six right-angles.

Remark. The other properties of spherical triangles analogous to those of plane triangles, discussed in articles 38 to 52 inclusive, the learner is requested to investigate by processes analogous to those adopted in the articles referred to.

290. If through the point E the centre of the less circle DGFH, a straight line be drawn perpendicular to this circle, it will pass through the centre of the sphere (282); it is called the axis of this circle, and is also the axis of