Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

of twice the cylinder AE, that is, to two thirds of the circumscribing cylinder. Therefore, every sphere &c. Q. E. D.* Ed.

COR. 1. A cone, a hemisphere, and a cylinder, of the same base and altitude, are to one another as the numbers 1, 2, 3. For it has been proved that a cone is one third (6. 12), and a hemisphere two thirds of a cylinder of the same base and altitude.

Cor. 2. Spheres are to one another as the cubes of their di

ameters.

For cylinders of the same altitude are to one another as the cubes of their diameters (1 Cor. 10. 12); and a sphere is two thirds of a cylinder of which the diameter and altitude are both equal to the diameter of the sphere. Therefore spheres &c.

* Similar demonstrations of this proposition may be seen in West's Elements of Mathematics, Saunderson's Algebra, and Keith's Euclid.

THE PRINCIPAL THEOREMS IN BOOK XII.

Prisms and cylinders of equal bases and altitudes are all equal to one another.

Prisms and cylinders of equal altitudes are to one another as their bases; and prisms and cylinders of equal bases are to one another as their altitudes.

The bases and altitudes of equal prisms, or of equal cylinders, are reciprocally proportional.

Similar prisms are to one another as the cubes of their like sides; and similar cylinders are to one another as the cubes of their diameters.

Similar prisms, and similar cylinders, are to one another as the cubes of their altitudes respectively.

If a cone, or a pyramid, be cut by a plane parallel to its base, the section will be similar to the base; and the section and the base will be to each other as the squares of their distances from the vertex.

Pyramids and cones of equal bases and altitudes are all equal to one another.

A pyramid is the third part of a prism, and a cone is the third part of a cylinder, of the same base and altitude.

Cones, and also pyramids, of equal altitudes are to one another as their bases.

Cones, and also pyramids, of equal bases are to one another as their altitudes.

Every sphere is two thirds of its circumscribing cylinder.

Spheres are to one another as the cubes of their di

ameters.

NOTES ON BOOK I.

The first Book of Euclid's Elements contains the principles of all the following Books: it demonstrates some of the most general properties of straight lines, angles, triangles, parallel lines, parallelograms, and other rectilineal figures: it shows the method of constructing certain figures, and of performing different operations.

Some propositions are of little or no use, and may be omitted. Some of these are auxiliary propositions, and were introduced into the Elements merely for the purpose of facilitating the demonstrations of others. Thus, the 16th proposition is implied in the 32nd, and is useless after the 32nd proposition is demonstrated.

The propositions in the first book are not arranged according to the nature of the subjects, but in such order as is adapted to facilitate their demonstrations. By a different arrangement some useless propositions might have been excluded, and perhaps better demonstrations of others might have been given.

DEFINITIONS.

Def. 3. Line signifies a stroke, and, in reference to the operation of writing, expresses the boundary or contour of a figure. A straight line has two radical properties, which are distinctly marked in different languages. Firstly, it holds the same undeviating course, and secondly, it traces the shortest distance between its extreme points. The first property is expressed by the word recta in Latin, and droite in French; and the second property seems to be intimated by the English term straight, which is evidently derived from the verb to stretch. Accordingly Proclus defines a straight line as stretched between its extremities, and consequently must be the shortest distance between them. LESLIE.

Cor. Def. 7. The equality of all right angles is an obvious inference from the formation and definition of a right angle. M. Legendre makes this corollary a proposition, and demonstrates it by a difficult process, which is scarcely intelligible to learners,

AXIOMS.

"ON the principle of congruency Euclid lays down a few simple truths, from which he demonstrates the more complex truths which depend on this principle. Those obvious truths are as follows.

"1. All points coincide.

"2. Straight lines which are equal to one another coincide; and, conversely, straight lines whose extremities coincide are equal.

"3. In two equal angles if the vertexes coincide, and one side of one angle coincide with one side of the other, then the remaining side of the first angle will coincide with the remaining side of the second. Likewise all angles whose sides coincide are equal.

"Though Euclid has not separately enounced those particular axioms subordinate to the general axiom, yet he applies them, as we shall find by analyzing several of his demonstrations." FENN'S EUCLID.

"That which is here numbered the eighth axiom is not properly an axiom, but a definition. It is the definition of geometrical equality; the fundamental principle on which the comparison of all geometrical magnitudes will be found ultimately to depend.

"The geometrical notions of equality and coincidence are the same; and even in comparing together spaces of different figures all our conclusions ultimately rest on the imaginary application of one triangle to another; the object of which imaginary application is merely to identify the two triangles together in every circumstance connected both with magnitude and figure." STEWART'S PHILOSOPHY.

PROPOSITIONS.

Of two propositions one is contrary, or contradictory to the other, when we deny in one what is affirmed in the other. Thus, two lines cannot be equal and unequal at the same time.

Of two propositions one is the converse of the other, when the order of either of them is inverted. Thus, if two sides of a plane triangle be equal, it can be proved that the angles opposite to those sides are equal. Now if we invert the order of this proposition, we shall obtain another proposition which is the converse of it. Thus, if two angles of a plane triangle be equal, it can be proved that the sides opposite to those angles are equal. This proposition is said to be the converse of the former. In the first poposition two sides of a triangle are supposed to be equal, and the equality of the two opposite angles is thence inferred; in the second proposition two angles of

[graphic][subsumed][subsumed]

By a given finite line Euclid means a line given both in position and in magnitude. A line may be given in position, but not in magnitude; or it may be given in magnitude (and therefore is finite), but not in position.

In all the propositions in plane geometry Euclid supposes all lines and all parts of figures to be in the same plane.

Euclid's construction and demonstration of the first proposition are tediously minute, and may be expressed as clearly and more concisely as follows. From the centres A and B, with the radius AB, describe two circles cutting each other in some point C. From C draw CA, CB; then ABC is the triangle required. For AC is equal to AB, because they are radii of the same circle (20 Def), and BC is equal to AB. Therefore AC is equal to BC (1 Ax.). Therefore the three sides AB, AC, BC are equal to one another; therefore ABC is an equilateral triangle.

Modern geometricians generally assume the second and third propositions as postulates, or evident principles, which require no proof. Euclid's demonstration of the second proposition is artificial, and somewhat obscure to students. The demonstration of the third prop. depends on the second, and therefore is equally exceptionable. Euclid's solutions of those problems are so like pedantic trifling that I have retained them with reluctance, and have given others in the Notes, which are practical and modern.

The following demonstrations involve the idea of motion, or the translation of a magnitude from one place to another. Euclid employs motion in some of his demonstrations; and modern geometricians admit it in all parts of mathematics.

« ΠροηγούμενηΣυνέχεια »