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XIII. A CIRCLE is a plane figure contained by one line, which is called the CIRCUMFERENCE, and is such, that all straight lines drawn to the circumference from a certain point (called the CENTRE) within the figure are equal to one another.
XIV. Any straight line drawn from the centre of a circle to the circumference is called a RADIUS.
XV. A DIAMETER of a circle is a straight line drawn through the centre and terminated both ways by the circumference.
Thus, in the diagram, O is the centre of the circle ABCD, OA, OB, OC, OD are Radii of the circle, and the straight line AOD is a Diameter. Hence the radius of a circle is half the
XVI. A SEMICIRCLE is the figure contained by a diameter and the part of the circumference cut off by the diameter.
XVII. RECTILINEAR figures are those which are contained by straight lines.
The PERIMETER (or Periphery) of a rectilinear figure is the sum of its sides.
XVIII. A TRIANGLE is a plane figure contained by three straight lines.
XIX. A QUADRILATERAL is a plane figure contained by four straight lines.
XX. A POLYGON is a plane figure contained by more than four straight lines.
When a polygon has all its sides equal and all its angles equal it is called a regular polygon.
XXI. AN EQUILATERAL Triangle is one which has all its sides equal.
XXII. An IsoSCELES Triangle is one which has two sides equal.
The third side is often called the base of the triangle.
The term base is applied to any one of the sides of a triangle to distinguish it from the other two, especially when they have been previously mentioned.
A RIGHT-ANGLED Triangle is one in which one of the angles is a right angle.
The side subtending, that is, which is opposite the right angle, is called the Hypotenuse.
XXIV. An OBTUSE-ANGLED Triangle is
one in which one of the angles is obtuse.
It will be shewn hereafter that a triangle can have only one of its angles either equal to, or greater than, a right angle.
XXV. An ACUTE-ANGLED Triangle is one in which ALL the angles are acute.
XXVI. PARALLEL STRAIGHT LINES are such as, being in the same plane, never meet when continually produced in both directions.
Euclid proceeds to put forward Six Postulates, or Requests, that he may be allowed to make certain assumptions on the construction of figures and the properties of geometrical magnitudes.
Let it be granted
I. That a straight line may be drawn from any one point to any other point.
II. That a terminated straight line may be produced to any length in a straight line.
III. That a circle may be described from any centre at any distance from that centre.
IV. That all right angles are equal to one another.
VI. That if a straight line meet two other straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced shall at length meet upon that side, on which are the angles, which are together less than two right angles.
The word rendered
"Postulates" is in the original
In the first three Postulates Euclid states the use, under certain restrictions, which he desires to make of certain instruments for the construction of lines and circles.
In Post. 1. and II. he asks for the use of the straight ruler, wherewith to draw straight lines. The restriction is, that the ruler is not supposed to be marked with divisions so as to measure lines.
In Post. III. he asks for the use of a pair of compasses, wherewith to describe a circle, whose centre is at one extremity of a given line, and whose circumference passes through the other extremity of that line. The restriction is, that the compasses are not supposed to be capable of conveying distances.
Post. IV. and v. refer to simple geometrical facts, which Euclid desires to take for granted.
Post. VI. may, as we shall shew hereafter, be deduced from a more simple Postulate. The student must defer the consideration of this Postulate, till he has reached the 17th Proposition of Book I.
Euclid next enumerates, as statements of fact, nine Axioms
or, as he calls them, Common Notions, applicable (with the exception of the eighth) to all kinds of magnitudes, and not necessarily restricted, as are the Postulates, to geometrical magnitudes.
I. Things which are equal to the same thing are equal to one another.
II. If equals be added to equals, the wholes are equal.
III. If equals be taken from equals, the remainders are equal.
IV. If equals and unequals be added together, the wholes are unequal.
V. If equals be taken from unequals, or unequals from equals, the remainders are unequal.
VI. Things which are double of the same thing, or of equal things, are equal to one another.
VII. Things which are halves of the same thing, or of equal things, are equal to one another.
VIII. Magnitudes which coincide with one another are equal to one another.
IX. The whole is greater than its part.
With his Common Notions Euclid takes the ground of authority, saying in effect, "To my Postulates I request, to my Common Notions I claim, your assent."
Euclid develops the science of Geometry in a series of Propositions, some of which are called Theorems and the rest Problems, though Euclid himself makes no such distinction.
By the name Theorem we understand a truth, capable of demonstration or proof by deduction from truths previously admitted or proved.
By the name Problem we understand a construction, capable of being effected by the employment of principles of construction previously admitted or proved.
A Corollary is a Theorem or Problem easily deduced from, or effected by means of, a Proposition to which it is attached. We shall divide the First Book of the Elements into three sections. The reason for this division will appear in the course of the work.
It is well known that one of the chief difficulties with learners of Euclid is to distinguish between what is assumed, or given, and what has to be proved in some of the Propositions. To make the distinction clearer we shall put in italics the statements of what has to be done in a Problem, and what has to be proved in a Theorem. The last line in the proof of every Proposition states, that what had to be done or proved has been done or proved.
The letters Q. E. F. at the end of a Problem stand for Quod erat faciendum.
The letters Q. E. D. at the end of a Theorem stand for Quod erat demonstrandum.
Hyp. stands for Hypothesis, supposition, and refers to
something granted, or assumed to be true.