spendthrift to consider the ways of the ant; and does not Esop seem to have borrowed from this idea his fable of the ant and the grasshopper? That great prince in delineating the portrait of true wisdom, paints her in saying, "To understand a parable, and the interpretation, the words of the wise, and their dark sayings."

Mental amusements, then, have been esteemed, in all ages, and by persons of every condition; and the pleasure they excite is the purer as they affect only the more delicate parts of the mind. The human intellect, as is well known, has its peculiar pleasures; every thing that increases knowledge, pleases and exalts it; we are always gratified when we comprehend a difficulty which has checked the progress of others, or have unveiled a mystery, concealed from persons possessed of less penetration than ourselves.

Besides, these amusements, purely intellectual, may be enjoyed at little expence; they do not fatigue the body, on which they make no impression; and, on this account, they ought to be preferred to sensual pleasures, the enjoyment of which creates disgust, injures the health as well as fortune, and almost always deranges the economy of a peaceful and tranquil life.

The class of mathematicians has always arrogated the right of treating of mathematical and philosophical recreations. In compiling the present collection, Ozanam, and those who have written on the same subject have been our guides; and from their works we have selected the greater part of what we now offer to the public; for these amusements are the production neither of one man, nor one age, but of a great number of the learned, of artists, and of many ages of research and of observation.

By the long experience we have had, we are induced to hope that young persons, who are often disgusted with the formality of study, and who, on that account, sometimes conceive an aversion to the most useful branches of science, will find in the greater part of the amusements which are here presented to them, some things suited to their taste, and easy to be comprehended. When the first difficulties are surmounted, they will become so many steps to conduct them gradually, by the most agreeable path, to the solution of problems, which at first may appear too difficult and abstruse for their age.

As it is impossible to understand properly all these amusements without the knowledge of certain principles, the application of which is often necessary, they are preceded by an introduction, calculated to facilitate the solution of the most difficult problems.

The reader is requested to observe, that the figures, inclosed within parenthesis, which occur in the course of the following work, refer to that section in the introduction, where the necessary explanation will be found.

AMUSEMENTS.

INTRODUCTION.

1. DIFFERENT symbols or signs, established by general practice, are sometimes employed in order to simplify calculations, and facilitate the resolution of certain problems. Thus,

Thus, it may be easily conceived that 2 + 3 = 5; 21; that 3 2; that 2 < 3; and that

that 3

123 or 12 = 4.

OF FRACTIONS.

2. Besides the application of the common rules to whole numbers, with which every body is acquainted, it is sometimes indispensably necessary to perform the same operations with fractional numbers.

A Fraction is one or more parts of a whole. Every fraction is expressed by two characters, placed one above the other, with a line between them, in this 3 a &c. The upper character, which is

manner:

4'6

called the numerator, expresses how many parts are taken of the whole; and the other, called the denominator, denotes the quality of these parts. Thus the fraction signifies that the whole is divided into fourths, and that 3 of them are taken.

It hence follows, that a fraction is greater according as the numerator is greater; and, on the other hand, less as the denominator is greater. Thus > ;

<. By a necessary consequence, all fractions, the two characters of which are equal, denote exactly the same value; == //

6 &c.

3. To reduce a whole number to a fraction, which shall have a determinate denominator, we must multiply the whole number by the given denominator, and place the product above the latter. Thus 8, reduced to a fraction, having 3 for its denominator, is 24: and 5 reduced to a fraction having the same denominator as 2, is 35.

4. To reduce two fractions to the same denominator, the numerator of the first must be multiplied by the denominator of the second, and the numerator of the second by the denominator of the first: these two products will be the numerator of two new fractions; and the product of the two denominators will be the common denominator. Thus 2 and 3, reduced to the same denominator, give and. Any number of fractions may, in like manner, be reduced to a common denominator, provided that each numerator be multiplied by the denominators of the other fractions, and that the product of all the denominators be taken for a common denominator. Thus, for example, the three fractions, 4,,, when reduced to the same denominator, 21, 56, 36 give

2

84.

5. To add two fractions, we must first reduce them to a common denominator, and then add their numerators. Thus the sum of the two fractions 3 and 4, is 21 + 20

35

=.

6. To subtract one fraction from another, they must first be reduced to the same denominator, and the nu

merator of the less must then be taken from the numerator of the greater. Hence the difference of the fractions and is 35.

7. To multiply two fractions together, we must make a new fraction, the numerator of which shall be the product of the two numerators, and the denominator. Thus the product of by is

8. To divide one fraction by another, we must make a fraction, the numerator of which shall be equal to the product of the numerator of the first multiplied by the denominator of the second; and the denominator equal to the product of the numerator of the second multiplied by the denominator of the first. The quotient of divided by, will therefore be .

9. Sometimes it is necessary to simplify a fraction, by reducing it to its simplest expression, or what is called its lowest terms: nothing is necessary for this purpose, but to divide the numerator and denominator by the greatest common measure or divisor. Thus the fractions and, reduced to their simplest expression, give and }.

OF POWERS.

10. By the power of a quantity, is understood its product by unity or by itself a certain number of times. Thus, the first power of 2 is 2: its second power or square is 2 x 2; its third power or cube is 2 x 2 x 2, and so on. Hence it is evident, that to obtain any power whatever of a given quantity, it must be multiplied by itself as many times less 1, as are equal to the number which denotes that power.

The power of any quantity is expressed sometimes, algebraically, or numerically, by the figure which denotes its degree, as in the following examples: a', á2, a3, a*, &c. 4', 4a, 43, 44, &c.

11. An algebraic character is sometimes accompa nied by two figures, as 263. The first of these is called the coefficient, and the second the exponent: the former denotes how many times the quantity is added to itself; and the second indicates the power. Thus, the