« PreviousContinue »
Thus, the angle contained by the straight lines AB and BC is ex pressed either by ABC or CBA, and the angle contained by AB and
BD is expressed either by ABD or DBA. When there is only one angle at any given point, it may be expressed by the letter at that point, as the angle E.
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 anything. 14. A figure is that which is enclosed by one or more
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. 16. And this point is called the centre of the circle, [and
straight line drawn from the centre to the circumference "led a radius of the circle).
17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
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 part of the circumference which it 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.
27. A right-angled triangle is that which has a right angle.
28. An obtuse-angled triangle is that which has an obtuse angle:
29. An acute-angled triangle is that which has three acute angles.
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 is that which has all its sides equal, but its angles are not right angles.
33. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.
34. Parallel straight lines are such as are in the same plane, and which being produced
ever so far both ways do not meet. 35. A parallelogram is a four-sided figure of which the opposite sides are parallel ; and the diagonal is the straight line joining two of its opposite angles. All other four-sided figures are called trapeziums.
Postulates. 1. Let it be granted 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. And that a circle may be described from any centre, at any distance from that centre.
Axioms. 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 are equal to one another.
7. Things which are halves of the same 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. 10. Two straight lines cannot inclose a space. 11. All right angles are equal to one another. 12. If a straight line meet two 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 on that side on which are the angles which are less than two right angles.
Explanation of Terms and Abbreviations.
An Axiom is a truth admitted without demonstration.
A Theorem is a truth which is capable of being de monstrated from previously demonstrated or admitted truths.
A Postulate states a geometrical process, the power of effecting which is required to be admitted.
A Problem proposes to effect something by means of admitted processes, or by means of processes or constructions, the power of effecting which has been previously demonstrated.
A Corollary to a proposition is an inference which may be easily deduced from that proposition. The sign = is used to express equality.
means angle, and a signifies triangle.
The sign > signifies “is greater than," and < "is less
than." + expresses addition ; thus AB + BC is the line
whose length is the sum of the lengths of
AB and BC. expresses subtraction ; thus AB - BC is
the excess of the length of the line AB above
that of BC.
Proposition 1.- Problem,
tres A and B, and ra
To describe an equilateral triangle on a given finite straight line.
Let AB be the given straight line.
It is required to describe an equilateral triangle on AB. From cen
CONSTRUCTION.- From thecentre
A, at the distance AB, describe the dius=AB,
circle BCD (Post. 3). circles.
From the centre B, at the distance BA, describe the circle ACE (Post. 3).
From the point C, in which the
circles cut one another, draw the straight lines CA, CB to the points A and B (Post. 1).
Then ABC shall be an equilateral triangle. AC=AB. Proof.—Because the point A is the centre of the circle
DCD, AC is equal to AB (Def. 15). BC=AB. Because the point B is the centre of the circle ACE, BC
is equal to BA (Def. 15).
Therefore AC and BC are each of them equal to AB.
But things which are equal to the same thing are equal to
one another. Therefore AC is equal to BC (Ax. 1). AB=BC
Therefore AB, BC, and CA are equal to one another.
Therefore the triangle ABC is equilateral, and it is de scribed on the given straight line AB. Which was to be done.
AC and BC each=AB, AC=BC.