2. A Common Measure of the Side and Diagonal of the Square. 3. An Infinite Series of Right-angled Triangles, with a Rule for their Solution.

For information concerning the History of the Quadrature of the Circle, the reader is referred to the Introduction, which begins on page 9, of this book. But before we proceed too far in our investigation of the subject, it seems proper to inquire first what is the circle. If a draughtsman or mechanic take an ordinary pair of dividers, and with one foot as a center, and the other starting at a certain point, cause it to describe a curve which is constantly receding upon itself, this point will return to the point from whence it started, when it is said to be an inclosed curve; and the curve, which is described by one point rotating around the other point within, is said to be the circumference of the circle, every point of which is equally distant from the point within; and this point within is called the center of the circle; and the plane figure which is inclosed by the circumference is said to be the circle itself. But a mathematical circle is more difficult to comprehend. If we say that to make a dot with a pencil that it is a point, the definition is sufficient for mechanical purposes; but a mathematical point has position only, and no magnitude, because it has no size. So, also, a mathematical circumference is a curved line constantly receding upon itself; but, like a mathematical straight line, it has length only, without either breadth or thickness. A mathematical circle, then, is a plane figure, which is inclosed by a curved line so finely defined as to be invisible, not only to the naked eye but by the means of the most powerful microscope which it is likely ever will be made, yet its existence can be as certainly determined, mathematically, as if it were drawn mechanically upon wood or paper, and not only its figure, but its dimensions, and consequently the ratio or pro

portion of its different parts one to another. But it may be objected that it has no real existence, because we can not see it. It may be answered that it has a mathematical existence, which can be so plainly demonstrated by sensible figures that the human mind finds it impossible to doubt it. Just as the Deity has an eternal existence, for neither can we see Him "and live," but it can be shown that He is, for "the invisible things of Him, from the creation of the world, are clearly seen, being understood by the things that are made, His eternal power also, and divinity;" and it is a wonderful truth that space, which is also said to be infinite, can only even be partially measured by the aid of the power which we obtain from the science of numbers, because, by their aid, we can reason mathematically and truly far beyond what we can see. It may be said with truth that the circle, the square, and even the triangle are emanations from the divine intelligence, as well as the science of numbers, by the aid of which they are measured, because they are a contrivance; and if there is contrivance there must have been design; but there is contrivance, therefore there was design; and if there was design, there must have been a designer; and it is very evident this designer was not man, who has expended all the talent, energy, and ability he ever had in trying to find out what the circle is, and has ignominiously failed. And after all the struggle, by simply turning our eyes to Holy Writ, there we find it, simple and beautiful as truth itself, of which it is a fit emblem, for it has no fault-it is perfect. "Verily the foolishness of God is wiser than men." The Designer, then, must have been one of infinite power and wisdom, "higher than the heavens," "who dwelleth in light inaccessible; and if so, He must have used such numbers as in His infinite wisdom would best accomplish that result, and these numbers must be such as is commonly found in all His works. Let us "come and see."

By PROPOSITION 2, PART 1, it is proved that, if from the circumference of the given polygon the 'th be deducted, the remaining will be exactly equal to the circle itself; and if the square root of the 8 be extracted, the result will be the number 7, which is the base of the system.

Again, by PROPOSITION 3, PART 1, it is proved that, if from the sums of the squares of the two sides of any square the th be deducted, the square root of the remaining can be extracted exactly, which will be 7, the base of the system or the generating number; and if the square root of the 6th be extracted, it will give the number 1, which is another generating number.

COINCIDENCE FROM HOLY WRIT.

Levit. xxiii., xxiv., and xxv., see the numbers 7, 49, and 50. Genesis, Exodus, Leviticus, Numbers, Deuteronomy, and Apocalypse for the number 7.

By Case 2, PART 1, it is shown that the sine No. 1 was divided into 14 parts, and each part was 76.

COINCIDENCE FROM HOLY WRIT.

See Genesis, Exodus, Leviticus, Numbers, and St. Matthew for the number 14.

See Genesis and Daniel for the number 70.

By Case 2, the inscribed double triangle for double the number of sides was proved to be ; by Case 3, 19801; by Case 4, 768398401, and so on ad infinitum.

COINCIDENCE FROM HOLY WRIT.

See Genesis, Exodus, Leviticus, Numbers, and Apocalypse for the number 2.

See Job, chapters 1 and 2, for the circumference and diameter. See Genesis for the circle.

See Apocalypse for the square.

"And the stone which the builders rejected was composed of three triangles."

IT has seemed to us fit to treat here separately this Great Question, on account of its great celebrity, and we shall not hesitate to give some of the silly notions to which this problem has given rise in ill-balanced and enthusiastic minds.

To square the circle is to assign the geometrical dimensions of a square equal to the circle. The quadrature of the circle has been attempted in several ways, by endeavoring to find a square or any other rectilineal figure equal to the circle. As it was soon found that the rectangle of the radius, by half the circumference, is equal to the area of the circle, the problem was soon reduced to finding the length of the circumference in terms of the radius. We can not believe that Archimedes was the first who made known this truth, for it is a necessary consequence of what was already known on the measure of regular polygons of which the circle is the extreme limit, the last of all.

The circle being after rectilineal figures, the most simple in appearance, geometricians very naturally soon began to seek for its measure. Thus we find that the philosopher Anaxagoras occupied himself with this question in his prison. Then Hippocrates of Chios tried the same problem, and it led him to the discovery of what is called his lune, a sur

face in the shape of a crescent, bounded by two arcs and exactly equal to a given square. He also found two unequal lines which were together equal to a rectilineal figure, so that if their relation could have been found the solution of the problem would have been obtained. But this no one has yet been able to do, nor is it likely ever to be done. We are also indebted to Simplicius for the history of a disciple of Pythagoras, named Sextus, who claimed to have solved the problem, but his reasoning has not been transmitted to us. Finally, this inquiry at that early day became so famous that Aristophanes in ridiculing Meton makes him appear on the stage, in his "Comedy of the Clouds," as promising to square the circle about 430 years before Christ. This was all the more amusing from the fact that people generally suppose that to attempt to square the circle is the same thing as trying to make a circle square, which implies a manifest contradiction. Yet this was that Meton, so renowned for his discovery of the cycle of nineteen years, of whom, together with Socrates, the comedian made a public laughing stock.

Aristotle mentions two of his contemporaries, Bryson and Antiphon who worked at the quadrature of the circle. Nothing could have been more grossly inaccurate than Bryson's pretended quadrature; for he made the circumference of a circle equal to 3 times the diameter. But Antiphon stated that having inscribed a square in a circle, if an isoceles triangle having the chord for the base be inscribed in each of the remaining segments, and similar triangles in the remaining eight segments, and so on, the sum of all these rectilineal figures would be equal to the circle; nothing could be more true than this, and Aristotle was undoubtedly wrong in calling Antiphon a paralogist, for one of Archimedes' two quadratures of the parabola depends upon the same operation; but this method has not yet succeeded with the circle.

It might be supposed that Archimedes applied himself to the solution of this problem, and that he gave his approximate measure of the circumference of a circle only for want of the long-sought-for vigorously exact measure. His discoveries on the spiral, if they have preceded his book on the dimensions of the circle, might well inspire him with the hope of finding the length of the circumference. However, that may be, Archimedes showed, about the year 250 before Christ, that if the diameter of a circle is 1, its circumference is less than 348 or 34, and more than 310; by taking 34 the error is less than the 7 of the diameter. The calculation of Archimedes is singularly skillful, and