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1. All right angles are equal to one another.
2. ABC and DEF are right angles.
3. Therefore ABC is equal to DEF.
The two first statements are called the premisses, and the last the conclusion; and it will be noticed that the principle which underlies and justifies the reasoning is this, that whatever is true of an entire class must be true of every member of that class. The first premiss makes an assertion which is true of all right angles; the second asserts that particular angles ABC, DEF belong to the class of right angles; and hence what is asserted of all right angles may be asserted concerning them, viz. that they are equal to one another. Now as all geometrical proofs may be reduced to a series of processes similar to the above in which the truth of the conclusion drawn is evident from the very form in which the argument is stated, we may always be certain that the results arrived at by this method are true so long as the premisses are true. But in Euclid the assertions contained in the premisses are either, 1: truths contained in the definitions and axioms, or 2: truths which are given in the supposition of the proposition under discussion, or 3: truths which follow from the construction, or 4: truths which have been previously proved. And at each step the references to the definitions ('Def:'), axioms (Ax:'), constructions ('Const:'), &c., show whence the statements to which they are attached have been derived. Thus starting with the elementary truths in the definitions and axioms Euclid proceeds to prove other and more complex truths which are stated in propositions. The constructions employed to aid in demonstrating these truths are such as are
demanded in the postulates or have been already effected. The statements advanced in the course of the demonstration are drawn from certain recognised and specified sources ; and the arguments are cast in such a form as to ensure that the conclusion shall be correctly drawn. Every new result thus established is felt to be as true as those from which it has been deduced, and becomes in its turn a sure stepping-stone to still further results. So we are conducted steadily and surely, step by step, lo the most remote and important consequences.
From this it will be seen that geometry is an exact science, and that our knowledge regarding it possesses a certainty and completeness such as we can attain to in but few subjects besides.
The practical applications of geometry, which are many and important, will be pointed out hereafter. But apart from these there are great advantages to be derived from the study of geometry as an abstract science. We see a complete body of truth reared upon the basis of a few elementary truths clearly conceived. The discussion at each step is always conducted in an orderly and systematic manner. The terms employed are used in a precise and definite sense. The line between the known and unknown is always clearly and distinctly marked, and finally the reasoning is always conducted in an exact and rigorous manner, so that we feel the utmost certainty as to the truth of the results obtained. The study of a subject so conducted not only teaches us to argue correctly, but trains to habits of precision in the use of terms and to a proper appreciation of the value of order and method, thus affording mental discipline of a most important kind.
1. A POINT is that which has no parts, or which has no magnitude.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is that which lies evenly between its extreme points.
5. A superficies is that which has only length and breadth.
6. The extremities of a superficies are lines.
7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
10. 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.
11. An obtuse angle is that which is greater than a right angle.
12. An acute angle is that which is less than a right angle.
13. A term or boundary is the extremity of any thing.
14. A figure is that which is enclosed by one or more boundaries.
16. And this point is called the centre of the circle.
17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
15. 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.
18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.
19. A segment of a circle is the figure contained by a straight line and the circumference which its cuts off.
20. Rectilineal figures are those which are contained by straight lines.
21. Trilateral figures, or triangles, by three straight lines.
22. Quadrilateral figures by four straight lines.
23. Multilateral figures, or polygons, by more than four straight lines.
24. Of three-sided figures an equilateral triangle is that which has three equal sides.
25. An isosceles triangle is that which has only two sides equal.
26. A scalene triangle is that which has three unequal sides.