The answer can be obtained without any explicit reference to proportion, but the most instructive mode of regarding the question is as an example of proportion: ten miles bear the same proportion to four miles as two hours and a half bear to one hour. The notion of proportion is suggested by innumerable circumstances of ordinary life, as well as by the questions proposed in books on Årithmetic. For example take a map of England; the distance between London and Cambridge on the map bears the same proportion to the distance between London and Manchester on the map, as the real distance between London and Cambridge bears to the real distance between London and Manchester. Similarly in the plan of a building the lengths of the straight lines on the plan will be in the same proportion as the lengths of the corresponding straight lines of the building.

11. We pass now to some of the rudiments of Geometry. The meaning of most of the common terms is probably known to the reader, but we will draw attention to them.

12. An angle is the inclination of two straight lines to one another which meet together, but are not in the same straight line.

Thus the two straight lines AO, BO, which meet at O form an angle there. The angle is not altered by altering the lengths of the straight lines which form it; thus CO and DO form the same angle as AO and BO. The angle may be denoted

D

in various ways, as the angle AOB, or the angle AOD, or the angle COB, or the angle COD: all mean the same angle.

13. When one straight line is upright to another the angle which the straight lines form is called a right angle, and each straight line is said to be perpendicular to the other. This is put into a more precise form in the following manner: 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.

Thus in the figure if the angle ABC is equal to the angle ABD each of them is a right angle, and AB is perpendicular to DC.

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

15. A triangle is a figure formed by three straight lines. If one of the angles of the triangle is a right angle, the triangle is called a right-angled triangle, and the side opposite to the right angle is called the hypotenuse.

16. A parallelogram is a four-sided figure which has its opposite sides parallel.

Thus AB and CD are parallel, and AC and BD are parallel in the parallelogram ABDC.

It is a property of such a

figure which may be verified

B

by measurement that the opposite sides are equal; thus AB is equal to CD, and AC is equal to BD.

A straight line joining two opposite corners of a parallelogram is called a diagonal. Thus if AD and BC are drawn each of them is a diagonal.

17. A rectangle is a parallelogram with all its angles right angles.

18. A square is a rectangle with all its sides equal.

19. A circle is a plane figure bounded 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: this point is called the centre of the circle. A radius of a circle is a straight line drawn from the centre

to the circumference. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference. An arc of a circle is any part of the circumference.

20. In Arts. 15...19 we have spoken of certain plane figures which present themselves very frequently to our notice. We are now about to mention some solid figures which are also of great importance.

21. A cube is a solid bounded by six equal squares of which every opposite two are in parallel planes. A cube is not an object which comes often under our observation; but an idea of it may be readily obtained. A common brick is usually 83 inches long, 4 inches broad, and 2 inches thick; now it is easy to imagine a brick in which the length, the breadth, and the depth should all be equal: the brick would then be a cube.

22. A sphere is a solid having every point of its surface equally distant from a certain point called the centre of the sphere. A radius of a sphere is a straight line drawn from the centre to the surface. A diameter of a sphere is a straight line drawn through the centre and terminated both ways by the surface. A sphere is sometimes called a globe: marbles and billiard balls are familiar examples of spheres.

23. A right circular cylinder is an upright column standing on a circular base; it is frequently called briefly a cylinder. An uncut lead pencil is an example of this solid. The straight line which joins the centres of the circular ends is called the axis of the cylinder. This is the geometrical axis of the cylinder: in practice the word often

means not a straight line, but a slender cylinder having the same geometrical axis as the other, but projecting beyond it at the ends.

24. A pyramid is a solid bounded by three or more triangles which meet at a point, and by another rectilineal figure. The point is called the vertex of the pyramid, and the rectilineal figure opposite to the vertex is called the base of the pyramid. When three triangles meet at the vertex the base of the pyramid is a triangle; when four triangles meet at the vertex the base is a four-sided figure; and so on. The bases of the famous pyramids of Egypt are

squares.

25. A right circular cone is a solid having a circle for its base, and its vertex on a straight line at right angles to the base through the centre: for a strict definition the reader should consult the Elements of Euclid, or the Mensuration. It is frequently called briefly a cone. straight line which joins the vertex to the centre of the base is called the axis of the cone.

The

26. The centre of a circle or of a sphere is a well known point in connexion with them. It is found convenient to extend the use of the word centre. In some plane figures. a point can be found such that every straight line drawn through it and terminated by the figure is bisected at that point. Thus for a parallelogram the intersection of the diagonals is such a point; and it may be called the centre of the figure. Likewise a cube and a cylinder have each a centre in such a sense.

27. A vast body of important knowledge has been formed in the course of more than two thousand years out of these and a few other definitions and notions. We shall refer the reader for an elementary account of them to the Mensuration, and for fuller information to the Elements of Euclid. Here it will be sufficient to notice a few facts.

28. We often require to find the length of the circumference of a circle when the length of the diameter is known; and this we can do, though not with perfect accuracy, yet with sufficient exactness for any practical pur

pose. The following Rule may be used: multiply the diameter by 34. This Rule makes the circumference a little greater than it ought to be, about a foot too great in a circumference of half a mile. Another Rule which is more accurate is the following: multiply the diameter by 31416. This Rule also makes the circumference a little greater than it ought to be; but the error is very small, being less than a foot in a circumference of 75 miles.

29. To find the area of a circle we must take half the product of the radius of the circle into the circumference, or we may multiply the square of the radius by 3.1416.

30. The reader is supposed to be familiar with the general principle which applies to every measurable thing, namely that it must be measured by a unit of its own kind. For example when we wish to measure lengths we fix on some length for a standard or unit; thus we may take a foot as the unit, and then any length is measured by finding how many times it contains the unit. So also when we wish to measure the areas or sizes of plane figures we fix on some area as the standard or unit; thus we may take a square of which the side is one inch as the unit; such an area is called a square inch. Or we may take a square foot, that is a square of which the side is one foot. In like manner when we wish to measure the bulk of solid figures we fix on some solid as the standard or unit; thus we may take a cubic inch, that is a cube of which the edge is one inch long; or we may take a cubic foot, that is a cube of which the edge is one foot long.

31. The following are the rules for finding the bulk or volume in the case of some solid bodies.

Cylinder. Multiply the area of the base by the perpendicular distance between the two ends.

Pyramid or Cone. Multiply the area of the base by one third of the perpendicular from the vertex on the base. Sphere. Multiply the cube of the diameter by '5236.

32. The following proposition in Geometry is probably the most important fact in the whole range of human