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[It will be seen hereafter that a triangle can only have one right angle, or one obtuse angle.]
29. An acute-angled triangle is that which has three acute angles.
[In any triangle the word base is often applied to any one of the sides to distinguish it from the other two, and the angular point opposite to that side is called the vertex.]
30. Of four-sided figures :
A square is that which has all its sides equal, and all its angles right angles.
31. An oblong is that which has all its angles right angles, but not all its sides equal.
32. A rhombus has all its sides equal, but its angles are not right angles.
33. A rhomboid has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.
34. All other four-sided figures besides these are called trapeziums.
[Four-sided figures which have two opposite sides parallel are often called trapezoids.]
35. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet.
36. A parallelogram is a four-sided figure, of which the opposite sides are parallel.
37. The diameter (or diagonal) of a parallelogram is the straight line joining two of its opposite angles.
[At the close of the definitions Euclid inserts three Postulates, that is, three things which he asks to be allowed to do (from postulare, to ask), and twelve axioms, that is, statements which he claims to be taken as true without proof, from a Greek word meaning to claim. These will now be given, and some remarks will be made on their nature afterwards.]
Let it be granted,
1. That a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line. 3. That a. circle
be described from any centre, at any distance from that centre.
1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals the wholes are equal.
3. If equals be taken from equals the remainders are equal.
4. If equals be added to unequals the wholes are unequal.
5. If equals be taken from unequals the remainders are unequal.
6. Things which are double of the same thing are equal to one another.
7. Things which are halves of the same thing are equal to one another.
8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
9. The whole is greater than its part.
12. If a straight line meet two straight lines, so as to make the 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 on that side on which are the angles which are less than two right angles.
[Geometrical propositions (things proposed) are of two kinds, problems and theorems. In a problem some geometrical construction is required to be made, as to describe an equilateral triangle on a given finite straight line.' In a theorem, some asserted geometrical property is to be demonstrated (or shown true), as the angles at the base of an isosceles triangle are equal to one another.' Thus in a problem there are data (things given) and quæsita (things required) and a solution is sought; in a theorem there are the hypothesis (supposition or things admitted), and the conclusion (things to be shown true), and a demonstration or proof is required. In the example of a problem just given, the thing given is the straight line, the thing required (the construction to be made) is an equilateral triangle upon it. In the example of a theorem the hypothesis is that the triangle is isosceles, the conclusion (what has to be shown true) is the statement that the base angles are equal.
The postulates are problems of which the possibility is admitted without proof, as self-evident ; similarly the axioms are theorems or statements which cannot be made more evident by demonstration.
Every proposition consists of the following parts: first, there is the general enunciation; then the particular enunciation, that is, the previous general enunciation repeated, but with reference to and explained by the diagram. Next the construction, in which certain lines, which are necessary to do what has to be done, or to prove what has to be proved, are drawn.* Then follows the demonstration, showing the truth or falsehood of the theorem, or the possibility or impossibility of the problem. The demonstration is said to be direct when the conclusion is obtained directly from the hypothesis. It is indirect when it is proved that the introduction of any other supposition leads to an absurdity: in other words, when the conclusion is obtained by showing that some absurdity follows from supposing the required result to be untrue. This method is called the reductio ad absurdum.
* In some cases no construction is required ; see, for instance, note to Prop. 15.
The beginner will understand these distinctions and divisions more clearly when he has learnt a few propositions and has separated each into its different parts. When they are once mastered, his task will become very much easier.
I now proceed to give the abbreviations which will be allowed in the first book. for point
for angle rt.
triangle st. straight
is (or are) equal to, sq. square
or, equal to int. interior
li parallel ext. exterior
circle rectil. rectilinear ce circumference equilat. equilateral
» perpendicular rem. » remaining,
» parallelogram or remainder
because adj. adjacent
It is usual to place the letters Q.E.F. (an abbreviation for quod erat faciendum, that is, which was to be done) at the end of the discussion of a problem, and the letters Q.E.D. (an abbreviation for quod erat demonstrandum, that is, which was to be demonstrated, or proved), at the end of the discussion of a theorem. Hyp. stands for hypothesis, constr. for construction, dem. for demonstration.]