PREFACE.

A FEW remarks of a personal and prefatory character, it may be proper in this place to address to the reader. Some thirty years ago, while in college, I had paid some little attention to Geometry, having gone with my class through three or four of the fifteen books of Euclid's Elements. But the knowledge obtained, even of the few books read, was somewhat superficial; and pursuits in after-life not requiring exercise in the science, thirty years disuse had suffered every demonstration and almost every principle derived from Euclid to fade from the mind. About two years and a half ago, JOHN A. PARKER, Esq., of New York, a gentleman whose life had mainly been passed in mercantile and commercial pursuits, applied to me, as an old acquaintance and friend, to examine some original papers, in which he claimed to have solved the most celebrated problem in mathematics, the quadrature of the circle. He had several years before discovered what he believed to be the true ratio of the circumference of a circle to its diameter, and had, during the interval, made repeated endeavors to have his papers examined, and his positions acknowledged by mathematicians. But he had found very few to give them even a slight examination, and none to concede the truth of his conclusions.

I took up his papers and read them with great care. I was at once much impressed with the boldness, strength, and originality of his reasoning, and finally convinced of the truth of his solution of that remarkable problem, which had long since been pronounced by mathematicians and learned societies as an impossibility. I became strongly interested

in the whole subject of Geometry. I took down my old Euclid, and brushed off the dust of thirty years; I went to the bookstalls and bookstores and searched for different works on Geometry, till I had picked up fifteen or twenty, which I examined, some partially, some thoroughly, but all with a zest. Mr. PARKER's reasonings and demonstrations led to the conclusion that the circumference of a circle was not a line coinciding with the perimeter of the circle, as geometers had hitherto considered it, but a line wholly and perfectly outside of the circle, and consequently that it must be a magnitude entirely distinct from the circle, and must have breadth. In addition to these reasonings of Mr. PARKER, which were entirely original with him, I found upon research that the learned and acute mathematician, Dr. BARROW, had come to the decided conclusion that mathematical number always expressed magnitude. And I also found some remarks of Aristotle, which seemed to lead to the same conclusion.

Here a great question intensely pressed upon my mind,—if mathematical number always represents magnitude, mathematical lines represented by numbers are magnitudes, and must have breadth; and if they have breadth, is it not possible by some geometrical demonstration to prove what that breadth is? The thought pursued me day and night, for it would not leave me even in my sleeping hours. I set myself down steadily to the task for a year and a half, and the present volume is the result. The breadth of mathematical lines is not only perfectly established, but the whole subject of Geometry is simplified, cleared from obscurities and difficulties, and placed, as it were, on a new foundation. But let it not be supposed that the new laws and principles of Geometry, developed and demonstrated in this work, have been derived from hypothesis and theoretical reasoning. They rest not upon so unsafe a basis. They were reached by the pure methods of the inductive philosophy of Bacon. I went to work upon original diagrams with the Greek rule and compasses in my hand, and spent a long and laborious year in digging out my facts. I examined such varieties of geometrical forms

as the imagination could suggest, and as patient thought and labor were able to investigate. I measured, computed and compared diameters, areas, and circumferences of plane figures, and diameters, solidities, and surfaces of solid figures, at the same time examining and comparing the roots of all these various quantities; and from the facts thus gradually collected and arranged, the general laws and principles of the science presented themselves clearly to view, and demanded the acknowledgment of their high prerogatives.

It is proper here also to remark, that a work made up almost entirely of new geometrical principles, and embracing a great variety of original arithmetical calculations, all prepared by one individual, without being revised by others, cannot reasonably be expected to be entirely free from errors. Some slip of the pen, some oversight of the eye, some figure missed, or some typographical error unperceived or uncorrected, may very probably be found to mar, in some degree, the work. But if nothing shall be discovered to invalidate the principles laid down, as they are intended to be explained, the Author trusts that minor errors, should such appear, will be charitably and cheerfully overlooked by the reader.

The work of Mr. PARKER on the Quadrature of the Circle is in preparation for the press, and is expected soon to be published. It is therefore unnecessary, and would be hardly appropriate here for me to enter into any elaborate consideration of it. I have already expressed my conviction of the truth of his ratio of the circumference of a circle to its diameter. That ratio is 20612 for circumference, and 6561 for diameter, which is the smallest expression of the perfect ratio that can be given in whole numbers. This ratio, in a circle whose diameter is 1, gives for circumference 3.141594+. The approximate ratio obtained by geometers, and generally received as correct, is 3.141592+. Mr. PARKER, it is seen, differs from this in the sixth decimal figure. And he shows conclusively that the method of geometers in obtaining this approximate ratio, which is by means of inscribed and circumscribed

polygons, necessarily leads to an error in the sixth decimal place. To test the truth of his perfect ratio, Mr. PARKER has, with a bold concep tion and singular originality, applied it to some of the astronomical cir cles, and obtained remarkable and startling results, indicating that in the motions and periods of the heavenly bodies there are perfect mathematical relations much more wonderful and extensive than have yet been understood.

Hippocrates squared a portion of a circle more than two thousand years ago, in the figure called "the lune of Hippocrates ;" and I have myself squared other portions of a circle by similar methods. And I think when the reader has seen, in the demonstrations and principles exhibited in the following pages, what perfect harmony prevails between the circle and all rectilineal figures, and how the circle controls all rectineal figures by one simple and uniform law, he will have no doubt that the whole circle may be perfectly squared.

SEBA SMITH.

NEW YORK, July 4, 1850.