# 1795 FASCINATING ORIGINAL HANDWRITTEN STUDY NOTEBOOK OF GEOMETRY, TRIGONOMETRY, SURVEYING AND MORE COMPLEX MATHEMATICS FROM A YOUNG MAN STUDYING AT AMERICA’S MOST RESPECTED UNIVERSITY

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On offer is an exceptional and fascinating handwritten notebook of many different and varied aspects of geometry, arithmetic, trigonometry, finance, and surveying, written by a young man studying at Harvard University in 1795. The book is a terrific example of mathematical and surveying studies at the end of the 18th century, especially as it happens at one of the greatest institutes of learning in America. As such these books are incredibly rare to find outside of university archives. The front cover page reveals the man’s name, William Hardy, written in beautiful gothic hand lettering. “Definitions” which given the definitions of a point, line, chord, curve, tangent, angle, and many other mathematical terms. Going from that base, most probably learned in the first day or so of classes, the book goes into an exceptional number of intermediate mathematical topics. The study book then begins its main sections under the title: “ELEMENTS or First Rudiments of PLAIN Geometry” which give three “POSTULATES” as well as a number of “Terms Used in Geometry.” From these postulates and terms, the book gives 34 “Geometrical Problems” spanning nine pages. A problem is given such as “To describe any pentagon in any given circle” and then a diagram is drawn next to it. The next part continues the study of planar geometry with a number of pages on the “Construction of the Plain Scale.” This section ends with a full page diagram of “THE PLAIN SCALE” that is certainly a thing of mathematical beauty. There follow a few pages of “Another Method of constructing P. Scale” and ending with another full page “PLAIN SCALE.” The notebook then dives into “Plain Trigonometry” followed by “A Solution of Cases of R. Ang. Triangles” and a section “Of Oblique Ang. Trigonometry.” After all this background mathematics, the notebook then begins on what this class was most probably based around. The section is entitled “Surveying” and a definition is given: “Surveying in the largest sense of the word is the measuring, laying out, dividing, and leveling land. Mensuration consists of three parts. 1. Measuring the fields. 2. Protracting it. 3. Casting up the contents.” Each of these parts is explained in detail with surveying “rules” and accompanied alongside a number of case problems in which the rule is used to solve practical problems that would arise in this profession. The section contains a number of charts, diagrams of shapes and angles, and long-form mathematical calculations to solve the problem at hand. This section is 18 pages long and makes up one of the longest sections of the book. Though the next section, “Mensuration of Superficies and solids,” is a bit of a mathematical addendum to the surveying section. This chapter has 35 ‘Articles’ explaining different problems and aspects of Euclidian geometry, such as: “Art. 6. To measure a triangle.”; “Art. 11. To measure any regular polygon.”; “Art. 22. To find the area of a semicircle, the diameter being given.”; “Art. 28. To measure a cube” and many more. Each of these articles comes with an explanation of the article at hand as well as diagrams, charts, and examples needed to understand the problem. This section is followed by a number of sections on mathematical finance. The first sections are: “Simple Interest, “Interest by Decimals,” “Discount,” and “Discount by Decimals.” They again include rules on how to calculate interest/discounts alongside questions and examples to use the knowledge practically. For example: (Rule 2 Exm. Calculate the interest of £327.10 at 6 percent per annum for 210 days.”). It is of interest to note that Hardy uses pounds, shillings, and pence for the financial equations here. Why he doesn’t use the American Dollar (which had been officially set up in 1792) is very interesting. It could be that the pound was still the de facto currency base for educated young men to know, but I do not have a definitive answer to this question. There are also a number of pages under the title, “Barter” which is explained as “The exchanging of one commodity for another and teaches traders to proportion their quantities without cost.” It is essentially just mathematical equivalencies. More finance follows with sections about “Loss and Gain,” “Equation of Payments,”“Discount by Compound Interest,” “Annuities or Payments in of Compound Interest,” and “Annuities, Leases &c. Taken in Reversion at Compound Interest.” The study book then returns back to equivalencies with two sections: “Alligation Medial” (which is used to find the quantity of a mixture given the quantities of its ingredients) and “Alligation Alternate” (which is used to find the amount of each ingredient needed to make a mixture of a given quantity). Finally the last section deals with mathematical “Permutations and Combinations.” The cover of the book is a brown burlap which is faded and discolored. The both the front and back cover boards are warped. The book no longer lies flat as the top and bottom of the book both curl upward noticeably. Many of the pages have some discoloration or wear from use and age, especially on the edges. Many of the pages are only precariously bound to the spine. The book is roughly 125 pages in length of which there is writing on all but a few. All the writing in the book is in very good condition. Some of the blank ink used shows some fading, but for the most part, the ink used was probably very good quality as much of it still looks very good. This book is over 220 years old and as such is quite fragile. It should be treated with care. (For reference: The Harvard University Archives holds similar late 18th century mathematical notebooks by students Ephraim Eliot, William Emerson Faulkner, Samuel Griffin, Ebenezer Hill, and Thomas Noyes.); Manuscript; Folio - over 12" - 15" tall; KEYWORDS: HISTORY OF, WILLIAM HARDY, MATHEMATICAL FORMULA, EUCLIDIAN GEOMETRY, TRIGONOMETRY, FINANCE, FINANCIAL CALCULATIONS, HARVARD UNIVERSITY, BOSTON, MASSACHUSETTS, SURVEYING SCIENCE, GEOMETRICAL PROBLEMS, PLANAR GEOMETRY, PLANE SCALE, MENSURATION, ALLIGATION, POST REVOLUTIONARY WAR AMERICA, HIGHER EDUCATION THE UNITED STATES, STUDYING AND TEACHING, CYPHER, CYPHERING, CIPHER, 18TH CENTURY AMERICA, UNITED STATES AFTER INDEPENDENCE, COMPLEX MATHEMATICS, ARITHMATIC, ALGEBRA, AMERICANA, HANDWRITTEN, MANUSCRIPT, DOCUMENT, LETTER, AUTOGRAPH, WRITER, HAND WRITTEN, DOCUMENTS, SIGNED, LETTERS, MANUSCRIPTS, HISTORICAL, HOLOGRAPH, WRITERS, AUTOGRAPHS, PERSONAL, MEMOIR, MEMORIAL, ANTIQUITÉ, CONTRAT, VÉLIN, DOCUMENT, MANUSCRIT, PAPIER ANTIKE, BRIEF, PERGAMENT, DOKUMENT, MANUSKRIPT, PAPIER OGGETTO D'ANTIQUARIATO, ATTO, VELINA, DOCUMENTO, MANUSCRITTO, CARTA ANTIGÜEDAD, HECHO, VITELA, DOCUMENTO, MANUSCRITO, PAPEL