Johann Kepler's Uralten Messekunst Archimedes

Johannes Kepler ( 1571—1630 ) was concerned that Austrian wine merchants were cheating their customers by gauging the volume of their barrels incorrectly. To correct the situation, he undertook a study of the volume of wine barrels. He published his findings, Nova Stereometria Doliorum vinarorum, in 1615. Forsaking classical techniques of volume calculation, Kepler produced solids of revolution, dissected them into an infinite number of circular laminae and obtained a volume summation. He applied this technique to consider solids other than wine barrels; in total studying the volumes of 92 different solids. Written in Latin this work was scholarly and had a limited audience. In order to increase his financial returns in 1616, he published a popular German language version of his work, Ausszag aus der Uralten Messekunst Archimedes. The page images are from the Messekunst. Page 27 contains a discussion on the volume of a torus. Page 28 returns to a consideration of the volume of wine barrels.

Johann Kepler's Uralten Messekunst Archimedes

Boethius's Liber Circuli

Illustration from the 1503 edition of Boethius’s Liber Circuli. Folio 86 discusses the quadrature of a circle.

Boethius's Liber Circuli

Billingsley Euclid

In 1570 Sir Henry Billingsley (d.1606) published the first English-language edition of Euclid's Elements: The elements of geometrie of the most ancient philosopher Euclide of Megara [sic].

The image is Folio 314 of Billingsley's Elements. This page contains three pop up models of pyramids. These pop-up models occur throughout Book XI on solid geometry and were hand-glued into each copy of the work.

Billingsley Euclid

Maria Agnesi's Analytical Institutions

Pages from the original Italian version of the Instituzioni analitche ad uso della gioventu italiana (Foundations of Analysis for the Use of Italian Youth) of Maria Agnesi (1718-1799). The text was one of the earliest treatments of calculus written on the European continent. Because Agnesi originally wrote this to instruct her younger brothers in analysis, she explained concepts very clearly and gave numerous examples.

Among Agnesi's examples was a description of a curve which she called la Versiera. We give here (pp. 380-381) her geometric description of the curve and her derivation of its analytic formula. The figure is given below.

Maria Agnesi's Analytical Institutions

Leonhard Euler's Integral Calculus

Leonard Euler’s discussion of volume 1 of his Integral Calculus concerning the integration of logarithmic and exponential functions on page 121 of Integral Calculus, vol. 1 (1768). Note that Euler used lx to represent what we write as ln(x).

The complete Integral Calculus in three volumes appeared in the interval 1768 – 1770. This was the first complete textbook published on the integral calculus.

Leonhard Euler's Integral Calculus