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The manuscript probably comes from England, but the scribe is unknown. This page contains four propositions from Book I with their diagrams. First is proposition 26 at the top , which is the AAS congruence theorem. Next is proposition 27, stating that if a line falling on two straight lines makes the alternate angles equal, then the two lines are parallel.

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Proposition 28 states that if such a lines an exterior angle equal to the opposite interior angle or makes the interior angles on one side equal to two right angles, then the two lines are parallel. Proposition 46 demonstrates how to construct a square on a give straight line, while proposition 47 is the Pythagorean Theorem. Note that the scribe has two versions, neither very neat, of the famous diagram illustrating Euclid's proof of this theorem. Finally, proposition 9 states that if a straight line is cut into equal and unequal segments, then the sum of the squares on the two unequal segments is equal to twice the square on half the original line plus twice the square on the segment between the points of section.

All of these propositions can be translated into algebraic results and easily checked, but Euclid treats these as geometric propositions and draws diagrams in the first two propositions confirming the proposed equality of areas.

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These pages are from Plimpton MS This is a manuscript from c. Pages from the Boethius are elsewhere in this document. At the bottom of the page are the opening definitions of Book II. The diagrams, however, are not entirely in line with the propositions to which they refer. For example, the last diagram is one accompanying proposition II Note that the circles are all drawn with a compass and the straight lines with straightedges.

The text contains the various definitions of Book V dealing with ratio and proportion. The illustration in the right margin is of definition 5, the famous definition of equal ratio. Note the fancy initial letter desinating the beginning of a book. The two diagrams toward the top illustrate the definitions of similar figures. The text also includes propositions 1 and 2 of the book.

P5: Linee guida per la codifica e l'interscambio del testo elettronico

Proposition 1 states that "triangles and parallelograms which are under the same height are to one another as their bases," while proposition 2 is "if a straight line is drawn parallel to one of the sides of a triangle, then it will cut the sides of the triangle proportionally, and conversely. The proposition, actually a construction, states, "To a given straight line to apply a parallelogram equal to a given rectilinear figure and deficient by a parallelogrammic figure similar to a given one; thus the given rectilineal figure must not be greater than the parallelogram described on the half of the straight line and similar to the defect.

In this form, the problem can be translated and was translated by Islamic mathematicians into the question of solving a quadratic equation. This is not the chain rule you know from calculus, of course! The two images below are of the same manuscript Plimpton MS of Boethius's Arithmetic as that above.

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  • Although highly valued for its content, the manuscript was never printed. It contains several classical problems of the times: grains of wheat on a chessboard, the hare and the hound, jealous husband, and the testament of the dying man. They are given in the right column and were obtained by a technique called Ruisai Shosa-ho. Using color and pictures, Byrne attempted to teach Euclid's geometry by minimizing textual discussion, including labels. This innovative approach stressed visualization. Oliver Byrne was trained as a civil engineer but was also employed as a surveyor and a sometime professor of mathematics.

    The image, a monument for astronomy, pays homage to the great astronomers of the past: Hipparchus, Ptolemy, Copernicus and Tycho Brahe. Copernicus and Brahe appear to be debating an issue while Hipparchus ponders and Ptolemy computes. It contained the positions of over stars and directions for locating the planets within our solar system. Kepler finished the work in and dedicated it to his patron, the Emperor Rudolf II, but actually published it in It is the first scientific work which extensively employs a new concept of logarithms.

    The table's findings support Kepler's laws and the theory of a heliocentric astronomy.

    For additional images from this work, see also Mathematical Treasure: Kepler's Rudolphine Tables and defense of Brahe. Brahe was the foremost astronomer of his time, particularly noted for his accurate celestial measurements. His angular observations were consistently correct within one minute of arc. This level of accuracy was due in large part to the excellence of his self designed instruments, most of which are described in this book.

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    This functioning quadrant was actually painted on the wall of his observatory, Uraniborg at Hven. Petrus Apianus Peter Apian was a German humanist known for his work in mathematics, astronomy, and cartography. This is the title page of his Instrumentum primi mobilis , which discusses the use of measuring instruments and presents new developments in the compiling of trigonometric tables. This book is also noteworthy in that it contains the first European printing of the Latin translation by Gerard of Cremona of the revised edition of Ptolemy's Almagest by Jabir ibn Aflah fl.

    Jabir is commonly known in the West by the Latinized version of his name, "Geber". This page of Instrumentum directly precedes a listing of sine values. In this book, Apian reviewed the theories of Vernerus, or Johannes Werner , a Nuremburg priest and mathematician who devised a method of using lunar observations to find longitude, and explained applications of trigonometry specifically, sines and chords in geography.

    Convergence 's Mathematical Treasures also include images from Apian's Folium populi: Instrumentum and from his Cosmographia , the latter extended from his original edition by his student Gemma Frisius. The complete work in three volumes appeared in the interval — This was the first complete textbook published on the integral calculus. The entire Integral Calculus is available at the Euler Archive. Note that Euler used lx to represent what we write as ln x.

    An excerpt, pp. Christopher Clavius S.

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    Renowned as a teacher and writer of textbooks, Clavius was particularly active in the reform of the Gregorian calendar. Alberti - was an Italian engineer and architect. The instrument shown contains a compass for marking bearings. This is the title page of the Academia Algebrae by Johann Faulhaber - He had earlier exhibited the formulas for smaller values of k.

    Unfortunately, he left little indication as to how he had developed these formulas. On this double page , signature B, f. The final formula appears in the third from the last line of the paragraph with all the algebraic notation. For more information on Faulhaber and sums of powers in general, consult the article in this magazine by Janet Beery, " Sums of Powers of Positive Integers. Recorde explains in this poem the reason for the name of what is essentially an algebra text, one of the earliest in England.

    Note that the owner of this particular copy wrote notes to help him understand the various names and abbreviations for the powers. Recorde explains subtraction of polynomials by use of a poem Sig. On this page Sig. Ff, f. On these pages Sig. Ii, f.

    Mathematical Treasures from the Smith and Plimpton Collections at Columbia University

    Kk, f. This problem is entitled a "question of jorneying" and requires knowledge of the formula for the sum of an arithmetic progression. His textbooks were valued and widely used. The missionary, Matteo Ricci S.

    On page 76, in obtaining the quotient of and , the division by is undertaken with the first four digits of the dividend. A remainder is obtained, , indicating the division process must continue. On page 77, the quotient of and is sought. Although van Heuraet was not the first to accomplish a rectification, a task that Descartes had said could not be done, this is the first publication of a general procedure, a procedure very close to our standard calculus procedure for finding the length of a curve. Note also that van Heuratet uses Descartes's symbol for "equal" rather than our modern equal sign. Finally, he shows that to find the length of a parabola he needs to be able to find the area under a hyperbola. Barozzi - was a Venetian nobleman, mathematician, astronomer and humanist.