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As an example, let it be required to divide the product of 76.4 and 35.84, by the product of 473.9 and 4.76





Quotient 1.214

Ar. Co. 7.32431

Ar. Co. 9.32239

log. 1.55437

log. 1.88309




1. GEOMETRY is that science wherein the properties of magnitude are considered.

- 2. A point is that which has position, but not magnitude.

3. A line has length but not breadth.

4. A straight, or right line, is the shortest line that can be drawn between any two points.

5. A superficies or surface has length and breadth, but pot thickness.

6. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

7. 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, as A, Fig. 1.

Note. When several angles are formed about the same point, as at B, Fig. 2, each particular angle is expressed by three letters, whereof the middle letter shows the angular point, and the other two, the lines that form the angle; thus, CBD or DBC signifies the angle formed by the lines CB and DB.


8. The magnitude of an angle depends on the inclination that the lines which form it have to each other, and not on the length of those lines. Thus the angle DBE is greater than the angle ABC, Fig. 3.

9. When a straight line CD stands on another straight line AB, so as to incline to neither side, but makes the angles on each side equal, then those angles ADC and BDE are called right angles, and the line CD is said to be perpendicular to AB, Fig. 4.

10. An acute angle is that which is less than a right angle, as BDE, Fig. 4.

11. An obtuse angle is that which is greater than a right angle, as ADE, Fig. 4.

12. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet, as AB, CD, Fig. 5.

13. A figure is a space bounded by one or more lines.

14. A plane triangle is a figure bounded by three straight lines, as ABC, Fig. 6.

15. An equilateral triangle has its three sides equal to each other, as A, Fig. 7.

16. An isosceles triangle has only two of its sides equal, as B, Fig. 8.

17. A scalene triangle has three unequal sides, as ABC, Fig. 6.

18. A right angled triangle has one right angle, as ABC, Fig. 9: in which the side AC opposite to the 19. An obtuse angled triangle has one obtuse angle, as C, Fig. 10.

20. An acute angled triangle has all its angles acute, as ABC, Fig. 6.

21. Acute and obtuse angled triangles are called oblique angled triangles.

22. Any plane figure bounded by four right lines, is called a quadrilateral.

23. Any quadrilateral, whose opposite sides are parallel, is called a parallelogram, as D, Fig. 11.

24. A parallelogram, whose angles are all right, is called a rectangle, as E, Fig. 12.

25. A parallelogram whose sides are all equal, and angles right, is called a square, as F, Fig. 13.

26. A rhomboides is a parallelogram, whose opposite sides are equal and angles oblique, as D, Fig. 11.

27. A rhombus is a parallelogram, whose sides are all equal and angles oblique, as G, Fig. 14.

28. Any quadrilateral figure that is not a parallelogram, is called a trapezium.

29. A right line joining any two opposite angles of a quadrilateral figure, is called a diagonal.

30. That side AB upon which any parallelogram. ABEC, or triangle ABC is supposed to stand, is called the base; and the perpendicular CD falling thereon from the opposite angle C, is called the altitude of the parallelogram, or triangle, Fig. 15.

31. All plane figures contained under more than four sides, are called polygons; of which those having five sides, are called pentagons; those having six sides, hexagons, and so on.

32. A regular polygon is one whose angles, as well as sides, are all equal.

33. A circle is a plane figure, bounded by one curve line ADEB, called the circumference or periphery, every part of which is equally distant from a certain point C within the circle, and this point is called the centre, Fig. 16.

34. The radius of a circle is a straight line drawn from the centre to the circumference, as CB, Fig. 17.

35. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the cir cumference, as AE, Fig. 17. It divides the circle into two equal parts, called semicircles.

36. A quadrant is one quarter of a circle, as ACB, Fig. 17.

Note. The fourth part of the circumference of a circle, is also called a quadrant.

37. A segment of a circle is the figure contained by a right line, and the part of the circumference it cuts off: thus AEBA and AEDA are segments of the circle ABED, Fig. 16.


38. An arc of a circle is any part of the circumference,

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