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to discriminate the minor differences. In time, indeed, the new resident forgets the brown colour and does not notice it at all.

Similarly do we appreciate the facial merits of our loved relatives, who may be homely or even repulsive to others. To add another example; it has often been remarked that a shepherd recognizes by their features the members of his flock, which look alike to ordinary persons.

Most people have not developed a sense of the relations between numbers, and have not practised thinking about them—hence their inability to recognize and remember them. When this faculty of the mind has been developed by practice of number arguments, the numbers will become familiar realities with strong features of their own, and will be remembered with comparative ease.

Let us suppose that you want to remember your new telephone number, which is 8715. Write the number down, look at it, and do all the reasoning that you can about it, on the following lines: the first number is even and it is the biggest; the other three are odd, and of those three the biggest comes first and is one less than the even number; the middle odd number is the smallest possible; if you add the last two you have a descending series from 8; the addition of the two middle numbers equals the first—and so on.

It is a great help in the remembering of long numbers to divide them into groups, in much the same way as long words are divided into syllables. The present number conveniently breaks into 8 and 715.

Looking over the balcony where I am writing this paragraph I see a motor car standing in the road—number 208457. This easily splits into two parts, 2084 and 57. The first part has only even numbers, if we may consider 0 in the even series; the last part has two odd numbers, which are ascending and successive, and follow in order (5 after 4) from the first part. The first part begins with the smallest positive even number, ascends after o to the highest and then goes on to half that or double the first—and so on.

The following happens to be the number on a certain passport: 062246. It presents the peculiarity of being composed only of even numbers. It splits comfortably into three, 06, 22 and 46. The middle pair is easily remembered, and the other two may be compared. Both end in 6; the first number of the last pair is the sum of the middle pair, and the second number follows it successively; the sum of the last pair is equal to the sum of all the rest—and so on.

There is no group of numbers that cannot be discussed in this way. After considering for half a minute any telephone or other number you will find it pleasantly reclining in your mind whenever you want to remember it. The arguments will disappear, but the number will remain, and you will probably soon find also that your observation and memory for numbers have been greatly improved, so that you can remember them far better than before, even without special intention and without resort to these number arguments.

Let us now turn to a method of remembering numbers Which I have called "Number Diagrams."

Look for a little while at the first diagram above, which is nothing more than a square containing nine dots in the centres of the nine equal divisions into which it is easily broken up in the imagination.

Then look at the second diagram, and imagine that the divisions of the square have the values of 1 to 9, as shown.

In the first diagram the middle dot can be supposed to

stand for the number 5, the dot in the lower left-hand corner for the number 7, that in the upper right-hand corner for 3, and so on. Thus, an imaginary square containing a dot or a little dash, as below, will constitute a diagram for the number 6.

Two Digits. To form a diagram for a number having two digits, simply draw a line from the one position to the other, straight if the smaller comes first, curved if the bigger comes first, as in the following, representing 34, 95 and 28.    Three Digits. If the number contains three or more digits, always begin with a straight line and end with a curved one; thus we may express 458, 242, 6138, 5736, 24691 and 759523 by. If the three numbers happen to lie in a straight line, a break in the line should be made, as will be seen in the following diagrams of 258 and 1598:  A little complication is introduced if two similar digits happen to come together, but the difficulty is overcome by the device of making a little tick across the line to indicate the second similar digit; thus, for 553, 227 and 445599 we form—   A further complication arises in connexion with the cipher. In this case insert a little circle into the series; thus, for 20, 202 and 22005550 we have—   If the cipher comes first in the number, detach it at the beginning if there are only two digits, but attach it if there are more, as in the following, representing 02, 026 and 073. A decimal point may be indicated by a dot placed in that one of the nine divisions of the square which corresponds to the position of the number before which it is to be placed. Thus if the point is to be placed before the first digit, it will be put in the first division, and so on, as in the following examples, showing 423, 4 23 and 42 3. It is a help to make the number diagrams of a generous size in the imagination—as big as an average picture or even a window frame.

The two practices in this chapter lend themselves to immediate employment in practical affairs, so no special exercises need be prescribed.

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