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volumes and surfaces of prisms or pyramids raised upon those polygons as bases, differ from the volume and surface of the cylinder or the cone raised upon the circle as a base, by volumes and surfaces less than the least that can be assigned: the required equivalence, or the required relation, is then at once inferred from a General Theorem in the tract on proportion. The reference is, however, made merely to avoid the necessity of stating the reductio ad absurdum in each proposition. The like process is employed for determining the volume and superficies of a sphere: the first is found from the cylinders formed by the revolution of rectangles inscribed within, and described about the circle by whose revolution the sphere is formed, and the other, from the surfaces produced by the revolution of a polygon inscribed within, and of one described about the circle.
Now Euclid, in proving (Prop. 2. B. 12.) by means of polygons inscribed in the circles, that circles are to one another as the squares of their diameters, does not state expressly that the sides of the inscribed polygons are less than the least assignable lines, but that the series of inscribed polygons is continued till there is obtained, in one of the circles, a polygon whose area shall differ from that of the circle by a magnitude less than that by which an assumed magnitude differs from the circle; this assumed magnitude may, however, differ from the circle by the least assignable magnitude; and therefore the last inscribed polygon may differ from the circle by a surface less than the least that can be assigned. But it is easy to show that a polygon inscribed in a circle cannot differ from the circle by a surface less than any that may be assigned, unless, at the same time, the sides of the polygon are, each, less than the least line that can be assigned. For, suppose AB to be a side of a polygon inscribed in the circle whose centre is c, and let CE be drawn perpendicularly to AB; consequently bisecting it in D. Then, if AB be an assignable line, its half, AD, will be assignable, and AD2 will be an assignable surface: but AD2 AC2 — CD2 = (AC+CD). (AC-CD)=(AC+ CD). DE;
therefore DE is an assignable line: but the segment AEB is greater than the triangle AEB, that is, greater than the rectangle AD. DE, which, being contained by lines not less than the least that can be assigned, is not less than the least surface that can be assigned; consequently, while AB is an assignable line, the polygon inscribed in the circle will differ from the circle by an area which is not less than the least
that can be assigned. Hence, in the demonstrations of Euclid, relating to the magnitudes of figures of revolution, it must be understood that the sides of the polygons inscribed in the circles are less than the least lines that can be assigned; and, if it can be shown that there may be inscribed in the circle a polygon of that kind, it will follow that the method employed in this work is in accordance with the spirit of the ancient geometry. Now it is a necessary corollary from the first proposition of Euclid's tenth book, that by continual bisections of the circumference of a circle, the whole circumference may, at length, be divided into parts, each of which has magnitude, and is less than the least line that can be assigned. Each part thus found is a curve line, for, otherwise, it must either be a point or a straight line: it cannot be a point, because a point has no magnitude, neither can it be a straight line, because, then, the circle would be a rectilineal polygon, which is absurd; therefore, granting that each part is a curve line, however small, it may be conceived that a straight line joining any two of the successive points of bisection, will cut off from the circle some segment, however small, and will be less than any assignable line: also, if a perpendicular be let fall from the centre of the circle upon such line, it will be less than a semidiameter of the circle. Thus, there may be conceived to be inscribed in a circle, a polygon whose sides are less than any lines that can be assigned, which, however, will possess the properties of any similar rectilineal figure whose sides are of assignable length.
The determination of the ratios which the surfaces, or the volumes, of solids of revolution bear to one another, might be investigated at length by processes corresponding to that by which the ratio of circles to one another is found; but this is scarcely necessary, after it has been shown that those surfaces, or volumes, are equivalent (the bases and altitudes of the figures being equal) to the areas of plane rectangles, or to the volumes of prisms. For it is easy then to infer that, whatever be the nature of the surfaces, or solids of revolution, they have the same ratios to one another as exist between the equivalent rectangles or prisms: in the present edition, these relations are, therefore, put in the form of corollaries to the propositions in which the surfaces, or volumes of the figures of revolution, are determined.
A POINT is that which has no parts, or which has no magnitude.
A line is that which has length without breadth.
The extremities of a line are points.
A straight line is that which lies evenly between its extreme points.
A superficies is that which has only length and breadth.
The extremities of a superficies are lines.
A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
"A plane angle is the inclination of two lines to one another "in a plane: the lines meeting, but not lying in the same "direction."
A plane rectilineal angle is the inclination of two straight lines to one another; the lines meeting, but not lying in the same direction.
N. B. • When several angles are at one point B, any one of them is expressed by three letters, of which the letter at 'the vertex of the angle, that
' is, at the point in which the straight lines containing the angle meet one another, is 'put between the two other 'letters; one of these two is
'somewhere upon one of those straight lines, and the other upon the other line: Thus the angle which is contained by 'the straight lines AB, CB, is named the angle ABC, or C BA; 'that which is contained by AB, DB is named the angle ABD, or DBA; and that which is contained by DB, CB is called 'the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E.'
When a straight line standing on another straight line makes the adjacent angles equal to one
another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
An obtuse angle is that which is greater than a right angle.
An acute angle is that which is less than a right angle.
"A term or boundary is the extremity of any thing."
A figure is that which is inclosed by one or more boundaries.
A circle is a plane figure contained by one line which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.
This point is called the centre of the circle.
A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. A semidiameter of a circle is now usually called the radius.
A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.
"A segment of a circle is the figure contained by a straight "line, and the circumference it cuts off."
Any portion of the circumference of a circle is now usually designated
Rectilineal figures are those which are contained by straight lines.
Trilateral figures, or triangles, by three straight lines.
Quadrilateral, by four straight lines.
Multilateral figures, or polygons, by more than four straight lines.
Of three-sided figures, an equilateral triangle is that which has three equal sides.
An isosceles triangle is that which has only two sides equal.