Mathematics

For this first edition of Ptolemy’s collected works, Johann Honter drew constellation figures after the manner of Albrecht Dürer. The figures appear in contemporary dress rather than in a classical style.

Featuring Galileo's Handwriting. When Galileo heard news of telescopes invented in the Netherlands he worked out the underlying geometry and crafted one of his own design. In this work, Galileo published the first observations of the heavens made with the telescope.

A “camera obscura” (“dark room”) consists of a box or container in which light enters via a small hole and projects an image on an opposite wall. The image will be reversed and upside-down, but its proportions will be preserved.

This anthology of the works of Alberti, a humanist scholar, contains the printed edition of his treatise, “On Painting,” a work he originally dedicated to Brunelleschi.

The Perspectiva of Peckham (13th century) became the established university textbook on perspective. It was the text Galileo likely used in his early studies of optics. Renaissance artists were well-versed in the classic works on perspective.

This landmark work by Albrecht Dürer presents several variations on the technique of “Alberti’s window.” Here the artist is creating a drawing of a lute with true perspective by means of a string drawn from the object, through the canvas window, to the vanishing point on the wall.

While visiting Florence, Niceron was shown a unique perspective drawing tool devised by the painter Cigoli, one of Galileo’s friends. He viewed examples of anamorphic drawing techniques and Alberti’s perspective boxes. All of these make an appearance in this treatise.

Consider this geometrical drawing, portrayed with true perspective and a mastery of light and shadow. It comes from a treatise on art and mathematics by Luca Pacioli, yet it was not drawn by Pacioli.

This beautiful work by Sirigatti, published in 1596, brings the tradition of perspective drawing up to Galileo’s time. Sirigatti was a member of the Academy of Drawing (Accademia del Disegno), a school for artists and engineers where Galileo studied as a young man.

The frontispiece depicts a variety of optical phenomena: Reflection. Refraction. Perspective. The rainbow. Burning mirrors.

Borgi’s book on the abacus was the most important commercial arithmetic manual in Renaissance Italy. Around 1200, Leonardo of Pisa, also known as Fibonnacci, wrote an earlier manual for the abacus which introduced a sign for zero, Hindu-Arabic numerals, and a base-10 place value system.

Featuring Galileo's Handwritting. Galileo dedicated the manual for his engineering compass to young Cosimo II de Medici, whom he had tutored in mathematics the previous summer.

Ptolemy (Claudius Ptolemaios) lived in Alexandria, Egypt, in the second century. Ptolemy’s technical work on astronomy, originally written in Greek, was titled Almagest (“The Greatest”) by its Arabic translators.

This book recounts the establishment of the Jesuit mission in China in the late 1500s led by Matteo Ricci. When Ricci predicted a solar eclipse in 1592 with greater accuracy than the astronomers of the Chinese court, Emperor Wan-li invited Ricci to Beijing.

Six chief world systems were debated in Galileo’s world: • Ptolemaic: All planets revolve around the central Earth. Geocentric. • Platonic: Like the Ptolemaic, except switches the positions of Venus and Mercury. Geocentric. • Cappellan or Egyptian: Venus and Mercury revolve around the Sun.

This is a replica of an original polyhedral sundial created by Stefano Buonsignori in Florence in 1587, held in the Museo Galileo in Florence.

Euclid was the starting point for a mathematical approach to physics. This is the 1st printed edition. The beautiful woodcuts are hand-colored in this copy. The text of the first page was printed in both black and red ink. The geometrical diagrams were quite difficult to prepare.

This Arabic text of Euclid came from the circle of the Persian astronomer al-Tusi (13th century). Al-Tusi worked in Baghdad and in the observatory of Maragha, in modern northwestern Iran. Printing Arabic with moveable type was a technological challenge.

To provide entertainment at a dinner held by the Grand Duke of Tuscany, Galileo debated the Aristotelian physicist Lodovico delle Columbe on the topic of floating bodies. Galileo employed Archimedes’ mathematical analysis.

In a discussion of optical effects of the atmosphere, Aristotle here addresses the formation of a halo around the Moon. This is one of the most interesting uses of mathematics in all of Aristotle’s writings.

Schreck helped Galileo show the telescope to the Medici family and others in Rome. Once he arrived in China, he wrote this work on engineering in Chinese.

In a work entitled “On the Universe,” Aristotle argued that a 5th element, called ether or the quintessence, composes the celestial spheres that naturally rotate in place above the region where the four lower elements mix together beneath the Moon.

In Natural Magick, della Porta described an optical tube he designed to make far things appear as though they were near. The field of optics was often associated with magical tricks and illusions, and for that reason sometimes held suspect among non-mathematicians.

In this mathematics textbook, Schott explained the rod-based calculating machine designed by his fellow Jesuit, Athanasius Kircher. The philosopher Leibniz also created a calculating machine, described in 1666, which he offered to the emperor of China.

Pannochieschi, head of the University of Pisa, defended Columbe, widening the debate over floating bodies and exemplifying the Aristotelian physicists’ reaction to Galileo’s use of Archimedean methods. In response, Galileo published a 2d ed.

This book is Schall’s account of the Jesuit mission in China after Ricci. Working closely with Chinese collaborators, Schall oversaw the publication of more than 30 scientific works in Chinese which drew upon Galileo, Copernicus, Tycho, Kepler and John Napier.

This work of sports medicine analyzes the physics of bones and muscles. Borelli, a practicing mathematician and engineer as well as a physician, analyzed the musculoskeletal system in terms of the mechanics of the lever and other simple machines.

Members of the Academy of the Lynx preferred to communicate with each other in code. Della Porta was the most accomplished cryptographer of the Renaissance. This work includes a set of movable cipher disks to code and decode messages.

Ptolemy’s influential music theory was related to his astronomy. Through sight, we apprehend beauty through astronomy. Through hearing, we apprehend beauty through harmony.

Some of Galileo’s most avid opponents were Aristotelian physicists who, lacking training in mathematics, were unable to refute Galileo’s arguments.

In this masterwork of physics, Galileo studied the two sciences of tensile strength and motion. The science of tensile strength considers how larger objects must bear more and more weight to perform the same action.

Not all versions of Euclid’s Elements were created equal. Clavius prepared this edition for his students at the Rome College (Collegio Romano). If these editions of Euclid were used in different courses, which course would you take?

Although best known for his astronomy, Ptolemy (2nd century) brought the same mathematical methods to bear on various topics, including optics, geography, and astrology. This is the first printed edition of his geography, which established mathematical methods in cartography.

Nocturnal dials tell time by the stars. This instrument replicates an original nocturnal dial created by Girolano della Volpaia in Florence in 1569, held in the Museo Galileo in Florence.

Kepler’s contributions reached far beyond the realm of astronomy, to meteorology, mathematics, geology, mineralogy and crystallography. Kepler published this 24-page pamphlet, a study of the snowflake, as a New Year’s greeting for a friend.

In a work entitled “On the Universe,” Aristotle argued that a 5th element, called ether or the quintessence, composes the celestial spheres that naturally rotate in place above the region where the four lower elements mix together beneath the Moon.

Munster’s Cosmography was one of the most popular books of the 16th century. In addition to the map of the world, it includes separate maps for America, Africa, Asia and Europe. First published in 1540, at least 24 editions were published in the following century.

The slide rule is based on logarithms. With a slide rule, one may quickly and reliably calculate to a precision of about 3 digits. Until the pocket calculator became available in the mid-1970‘s, slide rules were in constant use by scientists and engineers.

Aristotle’s Mechanics contained an analysis of the principles of motion and simple machines. While no longer accepted as an authentic work by Aristotle, its influence among Renaissance scientist-engineers was profound, as illustrated in this commentary by Baldi.

These notes comprise one of the most important papers in the history of computing. Lovelace explained how Babbage’s “analytical engine,” if constructed, would amount to a programmable computer rather than merely a calculator.

Tartaglia’s compass (also known as a “sector”) incorporated the functions of a quadrant and a caliper measuring device. His “new science” investigated the ballistics of cannonballs, laying a foundation for Galileo’s studies of projectile motion and free fall.

Aristarchus, the Copernicus of antiquity, proposed in the 3rd century B.C.E. that the Sun lies at the center of the universe and that the Earth and other planets revolve around the Sun.

By far the best known 16th-century defender of Copernicus was Johann Kepler. In this work he demonstrated that vast empty regions lying between the planetary spheres, which were required by Copernicus, were not wasted space.

In this work, Viviani reconstructed an ancient study of conic sections by Aristaeus the Elder (4th century B.C.E.). Viviani became Galileo’s student and assistant in 1639.

Apollonius (3rd century B.C.E.) examined the properties of conic sections; namely, the: • circle (cuts a cone horizontally, perpendicularly to the axis of the cone) • ellipse (cuts a cone to make a closed curve) • parabola (cuts a cone parallel to a side of the cone) • hyperbola (cuts a cone in...

This book is Orsini’s manual for using a measuring stick instrument which he designed and called a “radio latino.” With its changing angles, multiple sight lines, and various scales, it was useful for making astronomical measurements, surveying uneven topography, measuring a cannon’s bore or...

In the same year that Galileo published The Assayer, Grassi delivered these lectures to Jesuit students in the Rome College (Collegio Romano).

This essay on meteorology contains Descartes’ explanation of the optics of the rainbow and his law of refraction. Descartes’ ambitious aim was to produce a new body of writings that would completely displace the Aristotelian corpus.

Prior to Newton, fewer than half a dozen astronomers accepted Kepler’s three laws. Galileo was typical in ignoring everything Kepler did. Yet this beautiful book is an exception: it clearly demonstrated that Kepler’s laws were more accurate than anything that had come before.

Despite a lack of publications, Leonardo’s fame grew as word of his notebooks spread. The first work by Leonardo to be printed was his Treatise on Painting, published a century after his death. That 1651 Italian edition is on display at the Fred Jones Jr. Museum of Art in Spring 2016.

The abacus, an ancient calculating machine, is still in use around the world. Counters consisting of pebbles in the sand, beads on a wire or knots on a string are shifted back and forth on each level to represent different quantities, such as units and tens.

Galileo’s thermoscope, developed in the context of pneumatic engineering, was an ancestor to the thermometer. Galileo pioneered scientific investigations with the thermoscope along with his two Paduan friends, Giovanni Sagredo and Santorio Santorio.

Astronomers use quadrants and sextants to measure angular distances in the night sky, such as the angular divergence between a planet and the nearest bright star. One may also measure the height of the North Star above the horizon, which is equal to one’s latitude on the Earth.

The physics of bones and muscles: Borelli, a practicing mathematician and engineer as well as a physician, analyzed the musculoskeletal system in terms of the mechanics of the lever and other simple machines. Borelli studied under Galileo’s student Castelli, along with Torricelli.

Color-coded, graphical proofs occur in this masterpiece of visual presentation and design. Text is dramatically reduced in favor of a strategy of visual thinking.

Stevin’s work represents that of a scientist-engineer in the Low Countries, whose major works appeared in Dutch. Like the scientist-engineers of Italy, Stevin maintained water systems and improved fortifications. He investigated the mechanics of motion, falling bodies and hydraulics.

This book explains how to use the terrestrial and celestial globes, an armillary sphere (which shows the movements of the sky), and an orrery (which models the motions of the planets). Martin operated an instrument shop in London.

Analyzing the center of gravity of an object was a traditional problem addressed using the methods of Archimedes. Galileo referred to Valerio as “the Archimedes of our age” and recommended him for membership in the Academy of the Lynx.

Gilbert, a physician to Queen Elisabeth I, wrote the first experimental treatise devoted to magnetism. Gilbert discerned analogies between the Earth and magnets, and reasoned that the Earth itself is a magnet.

An orrery, also called a planetarium, shows the choreography of planets as they dance with coordinated precision around the Sun. From antiquity, geocentric (Earth-centered) models of the moving planets were constructed, such as the Antikythera device and large mechanical clocks.

Hero described five simple machines: the lever, pulley, wheel, wedge and screw. In this theoretical investigation of the foundations of mechanics, Guidobaldo demonstrated that all five machines could be deduced from the principle of the lever.

Ferguson’s books, orreries, clocks and mechanical devices were studied with interest by Benjamin Franklin, Thomas Paine and William Herschel, among others.

This is the first separate English edition of Galileo’s Discourse on Two New Sciences, his masterwork in mathematical physics. The “two new sciences” are tensile strength and motion.

Algarotti’s popular introduction to Newtonian science went through many editions and aided in the dissemination of Newtonian ideas on the European continent. It was dedicated to Fontenelle. Like Fontenelle’s Plurality of Worlds, it was written as an entertaining dialogue.

Galileo’s mechanics demonstrated that projectiles follow a parabolic path. This is true whether the projectile is a cannonball, an arrow or a football. This set of four Japanese Samurai manuscripts, drawn on rice paper in the mid 1800’s, was copied by hand from mid-16th-century sources.

Galileo’s mechanics demonstrated that projectiles follow a parabolic path. This is true whether the projectile is a cannonball, an arrow or a football. This set of four Japanese Samurai manuscripts, drawn on rice paper in the mid 1800’s, was copied by hand from mid-16th-century sources.

Galileo’s mechanics demonstrated that projectiles follow a parabolic path. This is true whether the projectile is a cannonball, an arrow or a football. This set of four Japanese Samurai manuscripts, drawn on rice paper in the mid 1800’s, was copied by hand from mid-16th-century sources.