XXVIII. A polygon of four sides is called a quadrilateral.

XXIX. A quadrilateral whose four sides are equal is called a lozenge.

xxx. A lozenge which has a right angle is called a square.

XXXI. A polygon which has five sides is called a pentagon; one which has six sides, a hexagon, and so on.

THE CIRCLE.

XXXII. A circle is a plane figure formed by a curved line called the circumference, and is

such that all right lines drawn
from a certain point within the
figure to the circumference are A
equal to one another. This point
is called the centre.

XXXIII. A radius of a circle is

any right line drawn from the centre to the circumference, such as CD.

XXXIV. A diameter of a circle is a right line drawn through the centre and terminated both ways by the circumference, such as AB.

From the definition of a circle it follows at once that the path of a movable point in a plane which remains at a constant distance from a fixed point is a circle; also that any point P in the plane is inside, outside, or on the circumference of a circle according as its distance from the centre is less than, greater than, or equal to, the radius.

POSTULATES.

Let it be granted that—

1. A right line may be drawn from any one point to any other point.

When we consider a straight line contained between two fixed points which are its ends, such a portion is called a finite straight line.

II. A terminated right line may be produced to any length in a right line.

Every right line may extend without limit in either direction or in both. It is in these cases called an indefinite line. By this postulate a finite right line may be supposed to be produced, whenever we please, into an indefinite right line.

III. A circle may be described from any centre, and with any distance from that centre as radius.

B

If there be two points A and B, and if with any instruments, such as a ruler and pen, we draw a line from A to B, this will evidently have some A irregularities, and also some breadth and thickness. Hence it will not be a geometrical line no matter how nearly it may approach to one. This is the reason that Euclid postulates the drawing of a right line from one point to another. For if it could be accurately done there would be no need for his asking us to let it be granted. Similar observations apply to the other postulates. It is also worthy of remark that Euclid never takes for granted the doing of anything for which a geometrical construction, founded on other problems or on the foregoing postulates, can be given.

AXIOMS.

1. Things which are equal to the same, or to equals, are equal to each other.

Thus, if there be three things, and if the first, and the second, be each equal to the third, we infer by this axiom that the first is equal to the second. This axiom relates to all kinds of magnitude. The same is true of Axioms II., III., IV., V., VI., VII., IX.; but VIII., X., XI., XII., are strictly geometrical.

II. If equals be added to equals the sums will be equal.

III. If equals be taken from equals the remainders will be equal.

IV. If equals be added to unequals the sums will be unequal.

v. If equals be taken from unequals the remainders will be unequal.

VI. The doubles of equal magnitudes are equal. VII. The halves of equal magnitudes are equal. VIII. Magnitudes that can be made to coincide are equal.

The placing of one geometrical magnitude on another, such as a line on a line, a triangle on a triangle, or a circle on a circle, &c., is called superposition. The superposition employed in Geometry is only mental, that is, we conceive one magnitude placed on the other; and then, if we can prove that they coincide, we infer, by the present axiom, that they are equal. Superposition involves the following principle, of which, without explicitly stating it, Euclid makes frequent use:-"Any figure may be transferred from one position to another without change of form or size."

IX. The whole is greater than its part.

This axiom is included in the following, which is a fuller statement:

IX'. The whole is equal to the sum of all its parts.

x. Two right lines cannot enclose a space.

This is equivalent to the statement, "If two right lines have two points common to both, they coincide in direction," that is, they form but one line, and this holds true even when one of the points is at infinity.

XI. All right angles are equal to one another.

This can be proved as follows:-Let there be two right lines AB, CD, and two perpendiculars to them, namely, EF, GH, then if AB, CD be made to coincide by superposition, so that the point E will coincide with G; then since a right angle is equal to its supplement, the line EF must coincide with GH. Hence the angle AEF is equal to CGH.

C

D

XII. If two right lines (AB, CD) meet a third line (AC), so as to make the sum of the two interior angles (BAC, ACD) on the same side less than two right angles, these lines being produced shall meet at some finite distance.

A

This axiom is the converse of Prop. xvII., Book I.

B

EXPLANATION OF TERMS.

Axioms. "Elements of human reason," according to DUGALD STEWART, are certain general propositions, the truths of which are self-evident, and which are so fundamental, that they cannot be inferred from any propositions which are more elementary; in other words, they are incapable of demonstration. "That two sides of a triangle are greater than the third" is, perhaps, self-evident; but it is not an axiom, inasmuch as it can be inferred by demonstration from other propositions ; but we can give no proof of the proposition that things which are equal to the same are equal to one another," and, being self-evident, it is an axiom.

Propositions which are not axioms are properties of figures obtained by processes of reasoning. They are divided into theorems and problems.

A Theorem is the formal statement of a property that may be demonstrated from known propositions. These propositions may themselves be theorems or axioms. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. Thus, in the typical theorem,

If X is Y, then Z is W,

(1.)

the hypothesis is that X is Y, and the conclusion is that Z is W.

Converse Theorems.-Two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. Thus the converse of the theorem (1.) is—

If Z is W, then X is Y.

(II.)

From the two theorems (1.) and (II.) we may infer

two others, called their contrapositives. Thus the contrapositive

of (1.) is, If Z is not W, then X is not Y; (m.)

of (I.) is, If X is not Y, then Z is not W. (IV.) The theorem (IV.) is called the obverse of (1.), and (I.) the obverse of (II.).

A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.

The Solution of a problem is the method of construction which accomplishes the required end.

The Demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed.

The Enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done.

Postulates are the elements of geometrical construction, and occupy the same relation with respect to problems as axioms do to theorems.

A Corollary is an inference or deduction from a proposition.

A Lemma is an auxiliary proposition required in the demonstration of a principal proposition.

A Secant or Transversal is a line which cuts a system of lines, a circle, or any other geometrical figure.

Congruent figures are those that can be made to coincide by superposition. They agree in shape and size, but differ in position. Hence it follows, by Axiom vIII., that corresponding parts or portions of congruent figures are congruent, and that congruent figures are equal in every respect.

Rule of Identity.-Under this name the following principle will be sometimes referred to:-" If there is but one X and one Y, then, from the fact that X is Y, it necessarily follows that Y is X."-SYLLABUS.