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1. A point is that which is without parts. 2. A line is length without breadth. 3. The extremities of a line are points. 4. A right line, is that which lies evenly between its extremities.
5. A superficies, is that which has only length and breadth.
6. The boundings of a superficies are lines.
7. A plane superficies, is that which lies evenly between its extreme right lines.
8. A rectilineal angle, is the inclination
of two right lines to each other, which touch, but do not form one straight line.
An angle is designated either by one letter at the vertex; or three, of which the middle one is at the vertex, the remaining two any place on the legs.
9. The legs of an angle, are the lines which make thé angle.
10. The vertex of an angle is the point in which the legs mutually touch each other.
11. When a right line standing on a right line makes the adjacent angles equal, which are ABC and ABD, each of these angles is equal to a right angle, and the right line AB standing on the other, is called the perpendicular.
12. The angle ABC, which is greater
than a right angle, is called obtuse.
13. The angle ABD, which is less C than a right angle, is called acute.
14. A plane figure is a plane superficies, bounded on every side by one or more lines.
15. A circle is a plane figure, contained by one line, which is called the circumference; to which from a certain point within the figure, all right lines drawn are equal.
16. That point is called the centre of the circle.
17. A diameter of a circle is a right line drawn through the centre, and both of its extremities terminate in the circumference.
18. A radius is a right line drawn from the centre to the circumference.
19. A semicircle, is the figure which is contained by the diameter and the part of the circumference which the diameter cuts off.
20. A rectilineal figure, is a plane superficies contained by right lines.
21. A triangle is a plane superficies, which is contained by three right lines.
22. An equilateral triangle is that which has three sides equal.
23. An isosceles triangle (or æquicrurum) is that which hath two equal sides.
24. A scalene triangle, is that which hath three unequal sides.
25. A right angled triangle is that which has a right angle.
26. An obtuse angled triangleis that which hath an obtuse angle.
27. An acute angled triangle is that which hath three acute angles.
28. Parallel right lines, are those which are in the same plane, and although produced, never meet.
29. A quadrilateral figure,
is a rectilineal figure which is contained by four right lines.
30. A parallelogram, is a quadrilateral figure, whose opposite sides are parallel.
31. A square is a quadrilateral figure, which is equilateral and equiangular.
32. Rectilineal figures which have more than four sides, are called polygons.
1. Let it be granted that a right line can be drawn from any point to another.
2. That a terminated right line can be produced to any distance.
3. That a circle can be described from any centre, with any radius.
COMMON NOTIONS, OR AXIOMS.
1. Things which are equal to the same are equal to one another.
2. If equals be added to equals, the wholes will be equal.
3. If from equals, equals be taken, the remainders are equal.
4. If to unequals, equals be added, the wholes are unequal.
5. If from unequals, equals be taken, the remainders are unequal.
6. Things which are double of the same, or of equals, are equal to one another.
7. Things which are halves of the same, or of equals, are equal to one another.
8. Magnitudes which coincide with one another are equal to one another.
9. The whole is greater than its part.
10. Two right lines cannot enclose a space:
11. All right angles are equal to one another.
12. If two right lines, meeting a right line, make the internal angles on the same side less than two right angles; these two right lines, being produced, will meet one another on that side at which the angles are less than two right angles.
On a given finite right line (AB) to describe an equilateral triangle.
From the centre A, and with the interval AB, describe the circle BCD, (by Post. 3). From the centre B and with the interval BA, describe the circle ACE. From the intersection C, draw the right lines CA, and CB, to the extremities of the given right line AB. (by Post. 1).
It is manifest that ABC is a triangle, constructed upon the given right line AB; but it is equilateral; for AC is equal to AB, because they are the radii of the same circle, DCB (by Def. 15). But BC is equal to BA, because they are the radii of the same circle ACE. Therefore, since both AC and BC are equal to the same, AB, they are equal to one another (by Ax. 1); and therefore the triangle ACB is equilateral.
SCHOL.-Draw AF and FB, and it can be similarly demonstrated that the triangle AFB is equilateral.
From a given point (A) to draw a right line equal to a given finite right line (BC).
From the given point A draw a right line AB to either extremity B of the given right line (by Post. 1); upon AB construct an equilateral triangle ADB (by Prop. 1). From the centre B, with the interval BC, describe a circle GCF (by Post. 3), producing DB, till it meets its circum
ference in G. From D as a centre, with the interval DG, describe a circle GLO. The circumference of it