0 LECTURE ON THE HISTORY OF MATHEMATICS BY FRANCIS H. SMITH, A. M. SUPERINTENDENT AND PROFESSOR OF MATHEMATICS OF THE VIRGINIA MILITARY INSTITUTE. [Published at the request of the Cadets.] "GAZETTE OFFICE," LEXINGTON, VA. A. Waddill, Printer. 1841. QA 21 5644 1841 CORRESPONDENCE. V. M. INSTITUTE, JUNE 22, 1841. SIR: We have been appointed a Committee on the part of the Corps of Cadets, to communicate to you the following resolution: 66 Resolved, That we return our thanks to Major FRANCIS H. SMITH, for his able and satisfactory Lecture on the History of Mathematics, and that a Committee be appointed to request a copy for publication." We beg leave to express the hope that you will comply with the request of the Cadets, and to assure you of the satisfaction which the publication of your Lecture would afford. We have the honor to be, Maj. F. H. SMITH. Your obedient servants, J. W. BELL, J. L. BRYAN, Committee. V. M. INSTITUTE, JUNE 23, 1841. GENTLEMEN: I beg leave to acknowledge the receipt of your polite note, communicating a resolution of the Corps of Cadets, requesting, for publication, a copy of my Lecture on the History of Mathematics. I am deeply sensible of the flattering terms in which the Cadets have been pleased to express themselves in their Resolution, and it gives me great pleasure to comply with their request. You must be aware, that my Lecture has been prepared under an unusual press of engagements, which has prevented me from entering as fully into detail as the nature and importance of the subject demanded; while I have not had it in my power to refer to many books, which would have added to the usefulness and accuracy of the Lecture. I present it, however, as it is, in the humble hope that it may in some measure promote the cause of Science in our beloved old Commonwealth, With sentiments of high regard, I am your friend and servant, Cadets L. A. Garnett, J. W. BELL, J. L. BRYAN, FRANCIS H. SMITH, Committee. LECTURE. No occupation can be more agreeable to the student, than tracing the rise and progress of those subjects, which have claimed his time and attention. The inquisitive character of the human mind does not permit him to rest satisfied with the results, merely, of science or of art, but leads him to seek the causes of these results, and the chain of circumstances connected with their developement. The admirer of the beauties of Shakspeare, finds his delight increased, if any reminiscence can be discovered, throwing the least light on the subject or composition of a single play; and a new interest invests the recovery of a simple relict, illustrative of the character or habits of the Bard of Avon. The original copy-right of "Paradise Lost" was with difficulty sold for a few shillings, while the Antiquarian now readily gives a hundred pounds for the manuscript of the Blind Poet. With what increased pleasure do we take up the works of Sir Walter Scott, after reading the interesting biography of Lockhart? Each novel attracts a new feeling, each character a new interest, and we recommence their perusal with a relish never before experienced. Nor are these enquiries without instruction, especially when investigating subjects of a scientific character. The knowledge of the means by which certain results have been obtained, will always facilitate us in the progress of our studies, while we may be also led to the most important discoveries. It was a geometrical problem, solved in some particular cases by the ancients, which led Descartes to the discovery of the science of Analytical Geometry; and the Differential Calculus was suggested to Newton, by the consideration of a geometrical property, which was known as far back as the time of Euclid. The importance given by our Laws to the study of the Mathematics, arising from their acknowledged value in every pursuit of life, has suggested to me the propriety of presenting to you a brief history of their rise and progress. This step is rendered the more necessary from the fact, that so little has been published on the subject; while the works that have appeared are for the most part inaccessible to you. These embarrassments have retarded in no small degree my own investigations; but I am unwilling to withhold from you, on this account, whatever may aid or interest you in the progress of your studies. My object being to present to you a history of the Mathematics, I shall not aim at any originality, but shall simply collect such facts, mentioned by writers on this subject, as may be useful for future reference. My authorities are Montuela-Histoire des Mathematiques, Encyclopœdia Metropolitana, Playfairs History of the physical sciences, Hallam's Literature of Europe and Professor Leslie's Arithmetic. The Science of Arithmetic is so nearly co-existent with the exercise of our mental faculties, that it is impossible to trace with accuracy the steps of its early introduction. A child at the earliest age acquires the habit of comparing quantities with one another, and the result of this comparison will gradually lead him to the idea of number. Beyond this simple conception, his progress will be slow; for until words are formed, he will be unable to separate the idea of any number, from the qualities of the objects with which it is associated. He will have a distinct idea of four cows, as distinguished from five cows, but the idea of these numbers, will not necessarily be the same, as those which represent a like number of sheep. If words, however, be used to represent these numbers, which shall be independent of any qualities of the objects with which they were at first associated, he will soon become accustomed to the words, without any reference to such associations. This process for the formation of abstract numbers might be thus completely effected, by attaching names to the series of natural numbers. Our progress in numeration would still be exceedingly limited, if the names thus assigned were arbitrary, and entirely independent of each other. The memory could not retain such a multitude of disconnected words, while the operations of Arithmetic would be more difficult than could be mastered in the infancy of society. Upon this hypothesis, we might readily conceive the mode of formation of the decimal system of numeration. A person in the habit of counting on his fingers would be led to divide numbers into classes, the units in each class increasing in a tenfold proportion. Names being given to the first nine digits, and also to the units in each ascending class, he might, by combining the names of the digits with those of the units of local value, express any number by the simplest composition of words. This classification leads to a nomenclature which is simple and comprehensive. An examination of the systems of numerical language of various nations shows that it has been established upon such a principle as is here supposed. The decimal system of numeration will be found to have met with very general adoption, a fact which can only be accounted for from the natural practice of numbering by the fingers on the two hands. The Decimal scale will not, however, be found to be the only scale used by the ancients. The habit of counting upon the toes as well as the fingers, would naturally suggest a vicenary scale, which is to be traced among some of the nations of antiquity. Again, a person may stop after counting the fingers on one hand; or he may |
