the questions which follow these definitions, but no explanation is offered of the special treatment which must be accorded them. Eighteen questions exemplify simple equations. The treatment of the problems is made unnecessarily hard by the insistence on the use of letters in the first statement of the equation and its later working out, with the substitution of the numbers given in the problem at the end. There are many purely literal equations among the sets of problems. No attempt is made to place the problems of a geometric nature in a special section. The most advanced work is that under the topic "The Resolution of Adfected Equations by the universal method of Converging Series." The solution of the ninety-fourth question illustrates this method and shows also the kind of work in algebra that was being undertaken in the University of Pennsylvania in 1788. Professor Patterson, who compiled the material for this notebook, must have believed in problems more than in mechanical work. This section. on algebra, covering 71 pages, contains 110 problems. The solutions of many of them are very long, unnecessarily so in some instances. But it is significant that so much space should have been devoted to the application of the algebra to problems rather than to tedious work in operations. Robert Patterson.-This account would be incomplete without some record of the man who worked out such a good course in mathematics for his students. Robert Patterson was a Scotchman, although he was born in Ireland in 1743. Robert early showed promise of his mathematical ability. When he had completed his first formal schooling, he gave evidence of his adventurous and ambitious spirit by setting sail for the New World. He came to the Colonies with the firm intention of becoming a schoolmaster and soon succeeded in impressing his qualifications on the proper authorities in Buckingham, near Philadelphia. He later removed to Philadelphia and offered instruction to navigators in methods of calculating longitude from lunar observations, and related matters. During the American Revolution, Patterson devoted himself to the cause of liberty, and acted for some time as military instructor. He entered the University of Pennsylvania in 1:79, receiving an appointment first as professor and later as vice provost, and remained there for 35 years. The college honored him with the degree of LL. D. in 1816. After holding various offices in the American Philosophical Society, Patterson became its president in 1819, which office he filled until his death in 1824. Professor Patterson was a great teacher, but he seems to have written little for publication. In 1818 there appeared a work en 2 Account taken from Transactions of the American Philosophical Society, Vol. II, New Series, Philadelphia, 1825. Obituary notice of Robert Patterson, LL. D. titled: "A Treatise of Practical Arithmetic, intended for the use of Schools; in two parts." The " Explanatory Notes " of Part I have some interest in connection with algebra. Patterson edited American Solution of a cubic equation from the notebook of a University of Pennsylvania student in 1788. Doctor Pell introduced the "method of registering the steps" shown here editions of several English works on arithmetic, natural and experimental philosophy and astronomy, and in some cases added an ap pendix to the edited work. Besides the Mathematica Compendia, he prepared a work on navigation for the use of his students which is also extant in notebook form. The title-page of this book carries the superscription: "Note-book of Samuel Hayes. Philadelphia, 1789. Navigation by Robert Patterson, A. M.”4 If Robert Patterson had lived in a day when paper and printing were relatively inexpensive, he would have published both the "Mathematica Compendia" and the "Navigation." The character of these works called for their publication. In his long years of service as professor in the University of Pennsylvania, he might have accomplished greater things by the use of textbooks than by the more tedious dictation of lectures. Professor Cajori states that Patterson wrote a small astronomy entitled "The Newtonian System," which was published in 1808. F. Cajori. The Teaching and History of Mathematics in the United States, p. 66, Washington, 1890. * Manuscript Department, Historical Society of Pennsylvania. Chapter V MANUSCRIPT MATERIAL FROM MISCELLANEOUS SOURCES Manuscripts by Robert Brooke.-Another set of notebooks which were probably prepared, at least in part, under the direction of Professor Patterson is that by Robert Brooke. He is given as a nongraduate of the University of Pennsylvania, presumably of the class of 1793.1 The first 2 of these books bears the superscription, "Robert Brooke, His Book-March 10, 1783. Arithmetick." It contains 180 pages 20.5 cm. by 33 cm., covering the subject of arithmetic in a complete fashion. A second book, dated 1792, is on "Practical Geometry." While Brooke was given to dating his work rather freely, he failed to set down any indication of the time at which he began to prepare the material on algebra. This part of the notebook consists of 41 pages, 19 cm. by 33 cm., and has a complete set of algebra topics. They bear a resemblance to the Mathematica Compendia by Robert Patterson, and the work was probably done under him, although this is a briefer course. The topics are as follows: Definitions, Praxis on the Signs, Addition, Subtraction, Multiplication, Division, Involution, Binomials, Multinomials, Evolution, Fractions, Simple Equations, Equations with Two Unknowns, Quadratic Equations, Adfected Equations. The survival of earlier symbols and forms may be noted in this work. The signs for inequality are the Oughtred symbols found in the Harvard manuscripts, but in the opposite order, and hence they resemble the present-day symbols. Still another undated book contains no name, but is unmistakably in the handwriting of Robert Brooke. It consists of 36 pages and deals with the "Application of Algebra to the Invention of Theorems." These applications are made to Arithmetic Progression, Geometric Progression, Annuities, Compound Interest, Extraction of Square Root, Mensuration of Superficies, Mensuration of Solids, and Spheric Trigonometry. 1 University of Pennsylvania. Biographical Catalogue of the Matriculates of the College 1749-1893, XIII, Philadelphia, 1894. 2 This notebook and all the others by Robert Brooke herein described are in one package in the manuscript department, Historical Society of Pennsylvania, hereafter referred to as Hist. Soc. Pa. Other notebooks by the same man are devoted to: "Applications of Plain Trigonometry to Astronomy," "Spheric Geometry," "Applications of Astronomy," "Dialing," and "Fluxions." 3 There is sufficient resemblance between the notebook on algebra by Robert Brooke and one by Andrew Porter, jr., to warrant the inclusion of the Porter book at this point. The latter was "Andrew Porter jun.'.'s Book 1792." Thirty-nine pages, 19 cm. by 31 cm., show a fair knowledge of elementary algebra and constitute an abbreviated edition of the algebra work by Brooke. The order of topics agrees with this same book as far as "Quadratic Equations." Only the heading for that topic appears, and the work stops there. Symbols and forms and, in some cases, exercises are the same. 4 "Practical Mathematics."-Brief mention will now be made of such other notebooks containing algebraic material as have been found during a careful search in many libraries of the East. The number of these books extant does not compare with the number containing arithmetic, and yet a bit of algebraic material crops up in the most unlooked for places. A set of five volumes beautifully printed by hand, by one Thomas Sullivan, in 1796, is entitled "Practical Mathematics." Virtually every topic of interest in applied mathematics at that time is included in these volumes, as well as some purely theoretical work. A collection of algebraic problems in such a series could not prove to be other than an unexpected treasure. Algebra is used here for its applications under the heading: "The Application of Algebra to the Solution of Problems." These 24 problems are drawn from geometry, but are not of a really useful nature. In a good piece of work on trigonometry by one Joseph King,5 which contains the date 1740, although in another hand than that of King, familiarity with the use of letters in algebraic expressions is shown. The proportion xy::y:2 appears, followed by "Then xz=y2 and √xz=y." Ten principles of trigonometry are given, and the tenth one reads: "By Algebra or Analytical Investigation." Somewhat more extended is the algebra in "William Winthrop's Book, began June 6, 1769," and carried over into the year 1770. This book has 20 problems and a few statements, such as "The Sum and Difference of any two Numbers multiply'd to-gether produces the Difference of their squares." Later in the book there are several pages on algebra in Latin. Some of these notebooks give us side lights on the attitudes of the college boys who were compelled to prepare them. One such book 3 Hist. Soc. Pa. 4 Manuscript Department, Library of Congress. 5 Hist. Soc. Pa. 6 Harvard University Library. 7 American Antiquarian Society. 7 |