I. INTRODUCTION.

1. THE late Dr Whewell, congratulating a friend famous for his knowledge and ability on the birth of a son, said, "Young as he is he will learn more than you in the next twelve months." The remark may appear simple but it is striking from its truth; for it is curious to notice how soon a child placed under reasonably favourable circumstances gains the rudiments of all the science which the wisest men can teach. At a very early age the child begins to arrange and classify; he sees that some of the objects around him can move themselves, and that others cannot, suggesting the broad distinction between things which have life and things which have not life. Again, further subdivisions soon become clear; thus for example among living things he learns to bring together in his thoughts all such as fly, and to call them by the name of birds. Even if he does not use a common name for a class of things which in some respects are like each other, he can hardly fail to notice the fact of likeness. Thus the water in which he is bathed, the milk he drinks, the ink he is forbidden to touch, must seem to him in some respects like each other, and different from the chairs and tables and toys of his nursery; though he has not learned to call the former fluids and the latter solids.

T. P.

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2. One of the most important words to be found in our language is Law. The original sense of the word is that of a rule or command which must be obeyed. Thus it is the duty of all people to obey the Law of the land; and it is the duty of children to obey the Law of their parents. In another sense the word Law is used to denote the unwavering constancy with which certain results will follow when the circumstances are the same. Thus, for example, we say it is a law that a stone will fall down again if it be thrown up into the air; by this we mean that from repeated observation we are certain this result will happen. There are many such facts, which are called Laws of Nature; and from our infancy we begin to learn these laws, and to shape our conduct according to them. These are the Laws with which we shall be occupied in the present work; Laws not in the sense of duties but of facts.

3. That Laws in the sense here adopted extend throughout his little world soon becomes obvious to a child. He discovers and remembers that if he falls against a hard body he hurts himself, that if he pulls his toy cart with a string it follows him. He observes too the prevalence of Law in one of the most important fields of his early education, namely language; and at first he even exaggerates the range of this principle, and assumes that there are no exceptions to it. Thus he learns by habit that the past time of an English verb is usually made by adding d or ed to the present; and accordingly he constructs such forms for himself. For example a child says, “he fighted me”; thus conjugating the verb according to the general law, before he has learned by experience that to fight is an irregular verb, the past time of which is fought.

4. Those who wish to have a profound knowledge of Natural Philosophy must acquire considerable familiarity with Mathematics; but it will be possible to understand the elementary principles of the subject with the aid of a little skill in the operations of Arithmetic, and some acquaintance with the figures of Geometry. It will be convenient to mention the most important particulars which we shall assume to be known.

5. There are certain signs which are used as very convenient abbreviations in Arithmetic; among these are = - for. equal to, for added to, and for diminished by. The use of these signs is exemplified by such statements as 12−5=7=4+3. We may say that = is the sign of equality, + the sign of addition and the sign of subtraction.

6. The sign x denotes multiplication; thus 6x7=42.

7. If a number be multiplied by itself the product is called the square of the number; thus 6×6=36, so that 36 is the square of 6. The product of the square of a number into the number is called the cube of the number; thus 36×6=216, so that 216 is the cube of 6.

8. A fraction means a part or parts of some whole or unit. Thus is a fraction; it means that some whole or unit is to be divided into eight equal parts and five of them taken. If the whole or unit is a weight of one pound, that is of sixteen ounces, an eighth part is two ounces, and five such parts are ten ounces. If the whole or unit is a shilling, that is twenty-four halfpence, an eighth part is three halfpence, and five such parts are sevenpence-halfpenny. In the fraction the 5 is called the numerator, and the 8 the denominator.

9. The product of two fractions is obtained by multiplying the two numerators for a new numerator, and the two denominators for a new denominator. Thus §×=1. It is explained in books on Arithmetic that the term product is conveniently and naturally used in this case, although the meaning of the term may seem somewhat different from that which it has in the multiplication of whole numbers. Thus 2×3=6, that is 6 is the product of 2 and 3; so that the product is greater than either of the factors 2 and 3. But, that is is the product of and; and in this case the product is less than either of the factors and 3.

10. The notion of proportion is one of the most important of those which are illustrated in Arithmetic. Suppose that a man walks four miles in one hour, and we have to find how far he can walk in two hours and a half.

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 Arithmetic. 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

B

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.

A

B

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.