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Def. A. Any combination of lines drawn upon a plane is called a plane figure.
Def. XIV. A plane bounded figure is a plane surface bounded by a line or lines.
Def. XX. A plane rectilineal figure is a figure composed of straight lines, drawn upon the same plane.
Def. B. The distance between two points is the length of a straight line drawn from one to the other. A straight line drawn from one point to another is said to join those two points.
Def. C. The area of a bounded figure is the extent of surface enclosed within its boundary. Def. VIII.
A rectilinear angle is the opening formed between two straight lines which meet. It is also defined sometimes to be the inclination of one straight line to another.
Beginners often find difficulty in properly understanding the nature of angular magnitude. The size of an angle has nothing to do with the length of the lines by the meeting of which it is formed, nor has it anything to do with the quantity of space which is (at least partially) enclosed by these lines. Perhaps the best way of explaining the matter to a beginner is the following :-Take a pair of compasses ; when the legs are shut close together, no angle is formed between them ; begin to open the compasses, and one leg forms an angle at the joint with the other leg. The more the compasses are opened the larger this angle becomes -the greater is the slope or inclination which one leg has to the other. It is evident ‘at once that this increasing angle does not depend for size upon the length of the lines that form it, for the legs of the compasses remain of the same length, however widely they may be opened ; and any angle formed by them would remain just the same in size if half the legs were broken off.
Def. D. Two angles are equal when they admit of being placed one upon the other, so that the point of the one angle coincides with the point of the other, and the lines that form the one angle coincide in direction with the lines that form the other,
Def. E. Of two angles one is larger than the other, when a line may be drawn between the lines forming the first angle, from the point where they meet, so as to form with either of them an angle equal to the second angle. Def. X. When one straight line meets another, and
makes the adjacent angles equal to each other, each of those angles is called a right angle, and the one line is said to be perpendicular to the other.*
Def. XI. An obtuse angle is an angle which is larger than a right angle.
Def. XII. An acute angle is an angle which is less than a right angle.
Def. XV. A circle is a plane bounded figure, contained by one line, which is called the circumference, and is such, that every point in it is at the same distance from a certain point within, which is called the centre.
Def. F. A straight line drawn from the centre to a point in the circumference, is called a radius of the circle.
It is another mode of stating the definition of a circle to say, that all the radii of the same circle are equal to one another.
* Beginners are extremely apt to confound perpendicular with straight; and also to suppose that if one line be perpendicular to another, the latter must be horizontal. This is not at all necessary: the lines may be in any conceivable position, so long as their inclination to each other is such as is stated in the definition.
Def. XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
It is obvious that the half of a diameter is a radius.
Def. XXI. A triangle is a plane bounded rectilineal figure, formed by three straight lines.
Def. XXII. A quadrilateral is a plane bounded rectilineal figure, formed by four straight lines.
Def. XXIII. A polygon is a plane bounded rectilineal figure, formed by more than four straight lines.
Def. XXIV. An equilateral triangle is a triangle which has its three sides equal to one another.
Def. XXV. An isosceles triangle is a triangle which has two sides equal. (An equilateral triangle may be regarded as an isosceles triangle if only two of the sides are considered.)
Def. XXVI. A scalene triangle is a triangle no two sides of which are equal.
Def. XXVII. A right-angled triangle is a triangle one of whose angles is a right angle. That side of a right-angled triangle which is opposite to the right angle, is called the hypoten use.
Def. XXVIII. An obtuse-angled triangle is a triangle one of whose angles is obtuse.
Def. XXIX. An acute-angled triangle is a triangle which has all its angles acute.
Def. XXXV. Parallel right lines are lines drawn in the same plane, which, however far they may be produced both ways, do not meet.
Def. G. A parallelogram is a four-sided figure, the opposite sides of which are parallel.
Def. H. A rectangle is a parallelogram which has all its angles right angles.
Def. XXXI. A square is a rectangle which has all its sides equal. (Or, a square is a four-sided figure, with all its sides equal, and all its angles right angles.)
Def. XXXIII. A rhombus is a parallelogram which has all its sides equal, but its angles not right angles.
Def. XXXIV. A trapezium is a four-sided figure, which has two of its sides parallel.
The figures reasoned about in geometry are mental figures-figures which we imagine to ourselves. Diagrams drawn on paper are used only to assist the imagination and memory. With practice the mind may be trained to go through demonstrations of considerable length without any diagram before the eyes. There would therefore be nothing absurd or inconsistent in assuming ourselves at once to be able to construct any required figure, such as a parallelogram, or an equilateral triangle, or to line perpendicular to another, without the aid of any simpler processes; that is to say, we might at once imagine any figure to be before us, and then reason about its properties. But part of the science of geometry consists in showing how the simplest figures that we can conceive may be combined so as to form such as are more complicated. We must, however, assume, that we are able at once to make those elementary figures, by combinations of which all other figures may be produced. Accordingly, in geometry, the following postulates (or “ things demanded") are laid down.
First postulate. Let it be granted, that a right line may be drawn from any given point to any other given point.
Second postulate. Let it be granted, that a given finite right line may be produced (or continued) both ways to any length that we please.
Third postulate. Let it be granted, that a circle may be described with any given point as centre, and at any distance from that centre.
It is obvious, that no combination of simpler figures is either requisite or possible, in order to produce the “construction "'* specified in these postulates.
The most important aspect of the postulates is, however, that they are of a restrictive force. It is implied in them, that no construction is allowable in geometry which cannot be effected by combinations of these three primary constructions.
AXIOMS. The reasonings of geometry are based upon certain primary truths with respect to magnitudes and lines, which are self-evident, and do not admit of being demonstrated by the application of truths of a simpler kind. These primary propositions are called axioms. They are the following:
I. Magnitudes which are equal to the same, are equal to one another.
II. If equals be added to equals, or if equals be added to the same, the sums will be equal.
III. If equals be taken from equals, or if equals be taken from the same, the remainders will be equal.
IV. If equals be added to unequals, the sum of the one of the equals and the greater of the unequals will be greater than the sum of the other equal and the smaller of the two unequals.
V. If equals be taken from unequals, the remainder left, when one of the equals has been taken from the greater of the unequals, will be greater than the remainder left when the other equal has been taken from the smaller of the unequals.
A. If the larger of two unequal magnitudes be added to the larger of two other unequal magnitudes, and the less to the less, the sum of the two larger magnitudes will be larger than the sum of the two less.
* The process by which any figure is produced, is called the construction of the figure.