a winged-boat, which conveyed him round by the northern part of the earth back to his place of rising in the east. Milton alludes to this in his Comus :

'Now the gilded car of day

His golden axle doth allay

In the steep Atlantic stream,

And the slope sun his upward beam
Shoots against the dusky pole,

Racing toward the other goal,

Of his chamber in the east.'

The abode of the gods was on the summit of Mount Olympus, in Thessaly. A gate of clouds kept by the goddess named the Seasons, opened to permit the passage of the celestials to earth and to` receive them on their return. The gods had their separate dwellings; but all, when summoned, repaired to the palace of Jupiter, as did also those deities, whose usual abode was the earth, the waters, or the underworld.

It was also in the great hall of the palace of the Olympian King that the gods feasted each day on ambrosia and nectar, their food and drink; the latter being handed round by the lovely goddess Hebe. Here they conversed of the affairs of heaven and earth; and as they quaffed their nectar Appollo, the God of Music, delighted them with the tones of his lyre, to which the Muses sang in responsive strains. When the sun was set the gods retired to sleep in their respective dwellings. The following lines from the Odyssey will show how Homer conceived of Olympus:

'So saying Minerva, goddess azure-eyed,

Rose to Olympus, the reputed seat,

Eternal of the gods, which never storms

Disturb, rains drench, or snow invades; but calm

and cloudless shines with purest day;

There the inhabitants, divine, rejoice

Forever.""

The above fable though very beautiful, was one of those silly notions which the wisest of all the ancients believed as an established fact, and from it may be seen how few, even among those who profess to be wise, really think for themselves-and how the vast majority of mankind allow others to do their thinking for them; furthermore by far the greater number of mathematical truths which are now valued so highly, and referred to with so much confidence, were discovered and first demonstrated by these very Greeks and their contemporaries; and the most plausible method known up to this present time for the solution of the quadrature of the circle was discovered, it is supposed, during the highest cultivation of the arts and sciences, by

these people, whose genius for invention probably exceeded that of any nation which followed them till we come to the American. Mr. Pope, in eulogizing the genius of Homer, says:

"Homer is universally allowed to have had the greatest invention of any writer whatever. The praise of judgment Virgil has justly contested with him,* and others may have their pretensions as to particular excellencies, but his invention remains yet unrivalled. Nor is it a wonder if he has ever been acknowledged the greatest of poets who most excelled in that which is the very foundation of poetry.

It is the invention that in different ages distinguishes all great geniuses. The utmost stretch of human study, learning, and industry, which masters everything besides, can never attain to this. It furnishes art with all her materials, and without it judgment itself can but steal wisely; for art is only like a prudent steward that lives on managing the riches of nature. Whatever praises may be given to works of judgment, there is not even a single beauty in them to which the invention must not contribute; as in the most regular gardens art can only reduce the beauties of nature to more regularity, and such a figure which the common eye may better take in, and is therefore more entertained with. And the reason why common critics are inclined to prefer a judicious and methodical genius to a great and fruitful one, is because they find it easier for themselves to pursue their observations through an uniform and bounded walk of art, than to comprehend the vast extent of nature."

When Copernicus maintained, in opposition to the theory of the ancients, that the earth moved on its own axis, and that it moved around the sun instead of the sun moving around the earth, history teaches us with what difficulty he maintained his position, though in the end he succeeded in establishing his theory in opposition to all.

When Columbus, following in the footsteps of Copernicus, maintained that the earth being round another continent was necessary on the opposite side of the earth to maintain its equilibrium, he was treated as a visionary and madman, until through the charity of "Her Most Catholic Majesty," "Isabella, then Queen of Castile and Arragon," he was provided with means to prosecute to a successful issue his discoveries; and in return for the ingratitude of nations he gave them a "New World." In reference to those cases where it is necessary to give instructions for the demonstration of a theorem or the solution of a problem, or of combining our assumption with results already established, it is to be observed, supposing the problem to be the solution of the quadrature of the circle,

1st. That the cosine must always intercept the sine at right angles, and the radius must always intercept the tangent at right angles.

2d. The tangent is always greater than the arc to which it belongs, and the sine is always less than the arc to which it belongs, for the tangent lies wholly without the circle while the sine lies wholly within the circle, and the arc to which they belong must lie between them; therefore, as far as the sine and the tangent agree with one another, that is as far as they can be expressed by the same quantity, either one of them may be taken for the arc of the circle to which they belong. 3d. The inscribed polygon must not at any time extend outside the given circle nor the circumscribed polygon come within it; and any method which may be adopted for the solution of the quadrature of the circle by the means of regular inscribed and circumscribed polygons, which are made to approach the circle (one from within and the other from without) by doubling the number of sides, will not give a correct solution as long as there is danger of forcing a limit between the two polygons; neither does it follow as a necessary consequence that the circle is the limit of the two polygons.

4th. If by any means the inscribed polygon could be made to approach the circle from within, without regard to the circumscribed polygon, by doubling the number of sides, and this result could be reduced to a commensurable quantity, it is very evident that such a method would give the true quadrature as far as it could be carried; for in that case the inscribed polygon could never be forced to extend beyond or become greater than the circle, and if it could be continued till it would reach the circle, that is to infinity, which is impossible, it would give the true quadrature.

5th. A common measure of two straight lines, as for example the sine and tangent, can not be regarded as a measure of the circle, as no part of it, however small, is straight.

6th. Neither is it necessary for the solution of the quadrature of the circle that the polygon should be regular, for it was demonstrated by Gauss, a mathematician of Göttingen, in his disquisitiones, Arithmeticae, published in 1801, that polygons of 17 sides, 257 sides, and in general any number of sides expressed by 2n+1, can be inscribed in a circle when 2+1 is a prime number.

7th. If the square root of two be taken for the radius of the given circle, and a radius be assumed in terms of the inscribed square (the side of which will be 2 and the area 4), and this radius as a variable be made to approach as its constant the true radius, so near that the difference between it and the true radius shall be less than any assignable

quantity, so that it may be taken for the true radius, and at the same time if a circumference be assumed in terms of the inscribed square and of the radius, and this circumference as a variable be made to approach as its constant the true circumference, so near that the difference between it and the true circumference shall be made less than any assignable quantity, so that it may be taken for the true circumference, provided that this radius and circumference are selected in accordance with theorem second, then will the ratio between the assumed diameter and the assumed circumference be the same as the ratio between the true diameter and the true circumference, and this ratio will be the repeating decimal 3.142857 or 34 to infinity; and, consequently, when the assumed radius becomes the true radius and the assumed circumference becomes the true circumference, both of which happen at the same moment, then the circle is infinite, that is, no part, however small, is straight.

The following selection consists of the most approved methods of solving the quadrature of the circle; they are inserted here, just as they are to be found in the works of the most approved authors recently published, for the purpose of giving the student a more general knowledge of the subject.

The first method is taken from "Robinson's Elements of Geometry," which gives the approximate ratio of the circumference to the diameter, by means of inscribed and circumscribed polygons to 6144 sides, commencing with the hexagon; it is as follows:

PROPOSITION III.-THEOREM.

When the radius of a circle is unity, its area and semi-circumference are numerically equal.

Let R represent the radius of any circle, and the Greek letter π, the half circumference of a circle whose radius is unity. Since circumferences are to each other as their radii, when the radius is R, the semi-circumference will be expressed by R.

Let m denote the area of the circle of which R is the radius; then, by Theorem 1, we shall have, for the area of this circle, R2 = m, which, when R = 1, reduces to = m.

This equation is to be interpreted as meaning that the semi-circumference contains its unit, the radius, as many times as the area of the circle contains its unit, the square of the radius.

REMARK.-The celebrated problem of squaring the circle has for its object to find a line, the square on which will be equivalent to the area of a circle of a given diameter; or, in other words it proposes to find the ratio between the area of a circle and the square of its radius.

An approximate solution only of this problem has been as yet discovered, but the approximation is so close that the exact solution is no longer a question of any practical importance.

PROPOSITION IV.-PROBLEM.

Given, the radius of a circle unity, to find the areas of regular inscribed and circumscribed hexagons.

D

d

B

Conceive a circle described with the radius CA, and in this circle inscribe a regular polygon of six sides (Prob. 28, B. IV), and each side will be equal to the radius CA; hence, the whole perimeter of this polygon must be six times the radius of the circle, or three times the diameter. The chord bd is bisected by CA. Produce Cb and Cd, and through the point A, draw BD parallel to bd; BD will then be a side of a regular polygon of six sides, circumscribed about the circle, and we can compute the length of this line, BD, as follows: The two triangles, Cbd and CBD, are equiangular, by construction; therefore,

C

Now, let us assume CA = Cd = the radius of the circle, equal unity; then bd = 1, and the preceding proportion becomes

In the right-angled triangle Cad, we have,

Ca 1:1 BD

(1)

(Th. 39, B. I).

(Ca)2+(ad)=(Cd)2,

(Ca)2+1=1, because Cd=1, and ad=}.

13. This value of Ca, substituted in proportion (1), gives

But the area of the triangle Cbd is equal to bd (= 1,) multiplied by Ca = 13; and the area of the triangle CBD is equal to BD multiplied by CA. Whence,

and

area, Cbd = }

3.

1 area, CBD = 13.

But the area of the inscribed polygon is six times that of the triangle Cbd, and the area of the circumscribed polygon is six times that of the triangle CBD. Let the area of the inscribed polygon be represented by p, and that of the circumscribed polygon by P.

Then p=31/3, and P=