THE following pages are intended to explain certain mathematical truths, which were discovered by the author while engaged in a series of investigations made during the hours of rest from the labors of the college and the counting room. They consist chiefly of new methods employed in the solution of problems which have heretofore been regarded by mathematicians as impossible; and, although the author's mind has been employed with the subject for a number of years, the result of the investigations are now published for the first time. If the discoveries should not come up to that standard of brilliancy which commands attention, it is hoped that they may be found worthy of a fair and impartial consideration. The author scarcely dares to hope, with the many examples of failure before him, that at the outset the entire mathematical world will bow in submission to his decree, or submit unconditionally to the power of his reason or the force of his logic; nor does he desire that the glorious fabric, which the mathematical genius of the world combined has reared as a monument to the memory of departed greatness, should crumble into dust by a single touch. Ah, no! Rather let the ivy of remembrance forever remain green upon their mausoleums, and the vines of gladness encircle their remains. But if Genius, while pursuing her walks amid these temples of departed greatness, should suddenly be inspired by Wisdom, and conceive Truth, who would be so poor as to refuse a garland with which to crown her brow, where truth sits enthroned ? The discoveries are as follows: 1. The Quadrature of the Circle. 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 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, 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 " come and see." )) و 9 66 who Let us By PROPOSITION 2, PART 1, it is proved that, if from the circumference of the given polygon the syth be deducted, the remaining 18 will be exactly equal to the circle itself; and if the square root of the ' 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 zoth 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 zlth 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 to. 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 gy; by Case 3, 15601; by Case 4, 768378401, 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. HISTORY OF THE QUADRATURE OF THE CIRCLE, TRANSLATED FROM THE FRENCH OF MONTUCLA, By J. BABIN, A. B., LOUISVILLE, Ky. 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 caò 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 |