the following:-"My son, get money, honestly if you can-but get money." Thus the youth is trained to the art of profit; the majority have no other idea of Education than that it shall advance the pupils in the world of trade or commerce. Thus the great principle in Education has been the utilitarian principle, without any regard to higher relationships with which we have to deal. Education has had exclusive reference to the life of the body; we ought not, most assuredly, to condemn the life of the body, but to tie it again to the life of the soul. To search for the profoundness of science, to propagate the reign of truth, which is the reign of God, is, then, what is demanded for superior instruction. In the higher studies, we learn to devote ourselves to thought for the sake of thought itself, to render adoration to truth because it is truth. Thus the soul enlarges itself by scientific disinterestedness: the views become enlarged and universalitized by a general acquaintance with things. And by our very philosophical speculations the character becomes purer, and is strengthened by its attachment to truth, and by a submission to the only eternal principles. Man born in relation to infinity, the love of science, as it bears him to the most elevated heights, brings the fallen nature again towards heaven. In these higher regions of the spirit, religious instruction becomes the sanctuary of the great thoughts of every age, of the glorious inspirations of every country, of noble characters venerated throughout all time. There we find united all that has given to reason the strength of conscience, all that has fired the generous heart, all that has ever been exalted upon earth. Classical antiquity in this respect is one of the most powerful means of this Education. We often praise the study of Greek and Latin in its relation to the logical construction of these languages, and we ought to praise them; for what truly strikes us in these admirable idioms, is the energetic expressions of a people breathing the free atmosphere of liberty; of a liberty secured and sheltered from the hollow hypocrisies of the Haut Ton. There we find the vigorous impressions of the soul, which are no less developed in the language, than in the deeds of heroes, the sublime monuments of art, the breathing picture, or the living marble. Nothing is more stirring to the bosom than the enthusiasm of the grand and the beautiful, expressed in the narrow and harmonious language of the Greek historians, philosophers, and finally by their poets, who are all that and something more.

What influence, what spontaneous magic in the literature of those times, where genius is mixed with public and familiar life; where a book appears as the action of a man, and not as the product of industry; where the conscience is the literary law of the world, and where the degree does not dignify nor degrade talent. By what great images, by what noble concep

tions the great men of antiquity excite our admiration. A rigid study of the classics is then, in our opinion, the first and almost indispensable condition of a really superior Education. If they be not useful on the utilitarian principle, they are in that higher walk and range of thought, which will tend to make a people what they would be in the eternal annals.

From a

Let the higher kinds of study be more common. thousand institutions let men go forth ripened by profound meditation, and ennobled by the contemplation of the eternal emblems, rendered energetic by the adoration of truth, and habituated to consider everything in the most elevated light. These men, instead of teaching men to live by dark and deep expedients, will show them how to walk with singleness of eye in the perfect day. They will embrace in their vast intelligence the history of the past, the wants of the present, and the development of the future; and will judge of events after the laws of that eternal world, of whom Christ is both God and King. Capable of every sacrifice except the convictions of conscience, such apostles of truth, not animated by a selfish ambition, which seeks to support itself upon false greatness in the low passions of a debased population, will go forth burning with the ambition to elevate their fellow-creatures even to themselves, and to survive in the noble thoughts that they have popularized in the community. Rich in the treasure of the souls they have stored with wisdom and religion, and fired with the sublimity of patriotic virtue; scatter through a country men of this cast, and like a light, which though little, enlightens a large space; or as the heart, which circulates the principle of life to the extremities of the arteries, so will such men disseminate truth and justice, religion and piety, through the deepest ramifications of our social confederation to the remotest periods of time-nay more, into eternity itself.

ON THE ARITHMETIC OF SQUARES, OR QUADRATES.

The great improvements that have been made in the science of numbers, of late years, appear to have nearly exhausted that subject, and arithmetic is considered as one of those sciences, to which little can now be added; and which, in its nature, seems capable of little further improvement. The noble invention of logarithms has extended the limits of mathematical science to a great and useful extent, not only by bringing forward a new and compendious method of computation; but also in the discovery of some of the most useful and important subjects depending thereon. The following is the subject of a mode of computation by means of square numbers, which,

I believe may be usefully applied in many branches of mathematical science, with nearly the same compendiousness as logarithms; and, in cases where considerable multiplications are required, it may be more usefully applied than ever logarithms can.

A table of logarithms from 1 to 10,000, can only multiply two equal numbers not exceeding 100, without the trouble of finding new numbers for the product, but by a table of squares from 1 to 10,000, the product of any two numbers under 10,000 may be readily found, or of any two numbers whose sum does not exceed 20,000, integral or fractional, as will appear in the course of this essay, with nearly the same advantages in division.

This mode of computation depends on a numerical property, which is very simple; it is as follows:

The product of the sum, and difference of any two numbers, are equal to the difference of their squares. Demonstration.-Let

number; then,

2x+a their sum,

the lesser, and r+a the greater

a their difference,

2x xa+x=2 a x+a' their product,

x+a2 + x2=2a x+a2 the difference of their squares. Or if we take x and y for the two numbers, the product of x+y and y3 — y2. Let 4 and 6 be the two numbers; then 4+6=10, and 6-4-2; their product: 10+2=20, and 62-4' 20, which was to be proved.

=

From this it also appears evident, that the difference of any two numbers, is equal to the product of the sum and difference of their square roots.

When I first observed this property, I wrote it down as nothing more than a curiosity; and a long time afterwards, as I was accidentally looking over my papers, on sight of the one that contained this proposition, I was struck with the thought, that by it, the multiplication of any two numbers could be effected with less trouble than by the common process of multiplication; which has caused me to write this short

account.

OF FINDING THE PRODUCTS AND QUOTIENTS OF NUMBERS BY THE AFORESAID PROPERTY.

From what has been said, it will be very easy to apply the before-mentioned property to the finding of products, or multiplication of numbers. In order to this, it must be observed, that the sum and difference of two numbers being given, the numbers may be found as follows; half the sum of the given sum and difference is the greatest number; and half the difference, of the given sum and difference is the least: for if

and y represent the two numbers, and s and d their sum and difference, then x+y=s, or x=s—y, and X- -y=d, or x=d+y; hence s-y=d+y, whence y=s-d; and the value of x is s+d.

Now to find the product of any two numbers, call the greater factor the sum, and the less the difference of two numbers, which may be found by the before-mentioned rule: and the difference of the square of these two numbers will be the required product. Thus, let the two factors to be multiplied be 6 and 12; then 6+12-9 the greater number, and 12-62

3 the less; now 92-32-81-9-72, the product required. Any number multiplied by the reciprocal of any other number, will be the quotient of the said number divided by the other number and by having the reciprocals of numbers in the form of decimals, they may be annexed to the whole numbers, and be used with them with great ease; and division may be performed in the same manner as multiplication, by taking the reciprocal of the divisor for a factor, instead of taking as a divisor the whole number: thus, let the number 15 be to be divided by 5, whose reciprocal is . 2, then 15 . 2 =7.6, and 15.27.4, and 7.62-7.4-57.76-54.76

2

2

3 the quotient; the same thing may be more easily performed by using the decimal as a whole number, and cutting off a proper number of figures and decimals at the last, which may also be done in multiplying whole numbers with decimals.

From what has been said it will easily appear, that by having a table of squares and reciprocals of all the numbers between 1 and 10,000, that the product or quotient arising from any two numbers under that sum may be found by the simple operation of taking the difference of the squares of the half sum, and difference of the numbers to be multiplied and divided; and the easy method of constructing such a table, and of finding the square of any number by it, that is not included in the table, as also, in connecting the squares in case of error, must render this method of computation very easy. These things can be done much readier than in logarithms, and a table of squares may be carried to a much greater extent than a table of logarithms with much less trouble; but a table of squares of all numbers from 1 to 10,000 may be sufficient for most purposes.

Mixed numbers or fractions may be reduced to decimals, and then be worked as whole numbers, cutting off a proper number of figures for decimals, at the end of the operation; and it must be observed, that there will be always double the number of decimals in the square of any number as the

that number.

are in

THE NATURE OF SQUARES FARTHER EXPLAINED BY THE PARABOLIC

CURVE.

The equation of the common parabola is a x=y2, and when the invariable quantity a is equal to 1, then the equation is ry. From this it appears that the abscissa of the curve is every where equal to the square of the corresponding ordinate; hence, a table of squares would be no other than parabolic abscissas, and their corresponding ordinates are the square roots, or numbers of which they are squares.

By the application of the before mentioned property of numbers, it appears, that in any parabola whose equation is x=y", the difference of any two abscissas, is equal to the product, or rectangle, of the sum and difference of their corresponding ordinates. Let y=any ordinate and y' will be the corresponding abscissa; and let y+a be another ordinate, then the corresponding abscissa will be y' +2 a y+a2, and their dif ference is y+2 ay+a2— y2=2a ya; the sum and difference of the ordinates are 2 y + a and a; hence, their product is 2 a y+a2.

If a parabola of the kind above-mentioned were properly divided, calculations might be made with it after the manner of the logarithmic scale.

OF THE METHOD OF INCREASING THE SQUARES IN CASE OF FRACTIONS OR GREAT NUMBERS.

When a table of squares is made, the square of any number is readily found by as far as it extends; but sometimes fractions, which should be converted to decimals, enter into the operations, and then it will be necessary to increase the squares so as to answer our design, as also in case the square of a greater number should be required. The square of any number may be readily found by multiplying the number by itself: but when we have a table of squares, this operation may be performed with much less trouble.

If the side of a square be increased by a given quantity, the square of the increased line will exceed the former square by twice the rectangle made by that quantity and the side of the first square, together with the square of the aforesaid part.

This appears by squaring any binomial, one part of which is the root or side, and the other the increased quantity; thus

2

a+b2 = a2 + 2 ab+b. If it were required from the square of 10, which is 100, to find the square of 10.4, or 104; then 10 X.4 x 2+.16+100=108.16, the square of 10.4 or 10816 the square of 104; and the square of n +.5= n2 + n + 1, which might be often applied very usefully.