Mathematics

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.

Despite a lack of publications, Leonardo’s fame grew as word of his notebooks spread. The first book by Leonardo to be printed was his Treatise on Painting, published a century after his death.

Niccolò Tartaglia argued for the use of mathematics in physics, engineering and art. Tartaglia’s frontispiece shows Euclid guarding the gate of knowledge. Just inside, Perspectiva stands among the sciences that open the way to Philosophia.

Manuscript copy by Hajime Terai from original written in 1765 by Ise; illustrations copied by Odani.

This is the first English translation of Newton’s masterwork in physics. The Copernican idea that the Earth moves as a planet required a thorough revision of physics. Galileo undertook this task in his Discourse on Two New Sciences, published 80 years after Copernicus.

In a 1611 book published by the Academy of the Lynx, the Jesuit astronomer Christoph Scheiner argued that sunspots are little planets circling the Sun like Venus. Galileo answered Scheiner with this book.

A sundial consists of a gnomon, which casts the Sun’s shadow, and a dial on which the shadow indicates the time. This simple portable sundial features a gnomon that can be adjusted according to one’s latitude. Sundials have ranged in size from pocket-dials to monumental architecture.

In his dialog entitled The Timaeus, Plato taught that the cosmos is constructed from regular geometrical figures known as the Pythagorean solids. Wherever one finds an emphasis upon mathematical demonstrations in science, one may credit Plato and the Pythagoreans.

Capella described the seven liberal arts. The first three are grammar, logic or dialectic, and rhetoric. Then come the mathematical sciences, geometry and arithmetic. Geometrical circles in motion make astronomy. Numbers in motion make music.

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.

Schickard, a friend of Kepler’s, designed this planisphere or “astroscopium” to calculate the positions of the stars for any day and hour of the year. Schickard also devised a calculating machine to produce astronomical tables according to Kepler’s laws.

Galileo’s engineering compass employed scales of his own innovative design, useful for an astonishing variety of calculations in the field.

Erasmus Reinhold, a professor at Wittenberg who was sympathetic to Copernicus, published the first Greek edition of Ptolemy’s Almagest.

This Copernican cosmic section, the first published in Italy, appears in a sympathetic account, known to Tycho and to Kepler, which may have influenced Galileo. Naibod was a professor of mathematics at Padua who likely studied with Erasmus Reinhold in Wittenberg.

In this poster-sized work, the first publication of observations made with a microscope, Cesi and Stelluti studied the anatomy of the bee. The text includes classical references to bees as well as new knowledge, integrated in a tabular outline.

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.

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.

Galileo employed his scientific acumen to engage in the literary debates of the day. Here he considered the merits of Tasso and Ariosto, comparing both with Dante. Using his new physics of tensile strength, Galileo refuted Ariosto’s indiscriminate descriptions of giants.

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 this book, Napier presented logarithmic methods of calculation in more than 50 pages of explanation, followed by 90 pages of numerical tables. “Logarithm” derives from “logos” (proportion) and “arithmos” (number).

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.

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 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?

The crest of the Barberini family, showing three busy bees, appears at the top of the frontispiece. Galileo’s supporter, Cardinal Maffeo Barberini, had become Pope Urban VIII. The election of Barberini seemed to assure Galileo of support at the highest level in the Church.

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.

For two decades, Tycho and his assistants at Uraniborg produced thousands of astronomical observations of unprecedented quality. Tycho’s large-scale observing instruments, together with sophisticated new error correction techniques, increased observational precision by a factor of twenty.

From his new astronomy, using Tycho’s observations, Kepler calculated these tables of the positions of the Sun, Moon and planets. Kepler adopted John Napier’s recently invented computational method of logarithms, published in 1614.

Although Galileo eloquently championed mathematical methods in science, the main target of his wit and sarcasm in The Assayer was Grassi, a fellow astronomer, whose mathematical methods proved that comets move above the Moon.

In this work, Tycho explained two problems posed for Copernicus by the absence of stellar parallax: 1. Due to the annual movement of the Earth around the Sun, one would expect to see stars appear to shift in position. This parallax evaded detection, even at Uraniborg. 2.

Archimedes (d. 212 B.C.) developed the law of the lever with his Treatise on the Balance. He contributed to arithmetic by devising methods for expressing extremely large numbers. He deduced many new geometrical theorems on spheres, cylinders, circles and spirals.

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.

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...

This work was attributed to Proclus (5th century), one of the most important Neoplatonic philosophers of late antiquity. It became one of the most popular introductions to astronomy during the Italian Renaissance, appearing in more than 70 16th-century editions.

In De mundi aetherei (1588), Tycho reported that the comet of 1577 displayed no detectable parallax and thus moved, contrary to Aristotle, in the regions of the heavens beyond the Moon, passing through multiple celestial spheres. The ancient solid spheres melted.

The Rudolpine Tables were not a best seller. Three years later, Kepler and his son-in-law Jacob Bartsch published this little extract to stir up interest in the Rudolphine Tables and boost sales. It contains predictions of the transits of Mercury and Venus across the disk of the Sun in 1631.

Danti was a cosmographer in the court of Cosimo I de Medici. Visitors to Florence today may view his stunning maps of the world in the Hall of Maps of the Palazzo Vecchio, as well as armillary spheres and a quadrant he mounted on the facade of the church of Santa Maria Novella.

In De revolutionibus, Copernicus placed the Sun in the center of the universe and set the Earth in annual motion around the Sun. This is the 3d edition, printed in 1617, the year after the Inquisition stimulated fresh interest in the work by placing it on the Index of Prohibited Books.

Archimedes died defending the ancient city of Syracuse, on the island of Sicily, from the Carthaginian navy. Reports attributed the defense of the city to his ingenuity, including giant mirrors capable of setting attacking ships in the harbor on fire.

Astronomers use astrolabes for dozens of astronomical operations including telling time by the Sun or stars and determining the positions of planets.

This medieval introduction to astrology was frequently translated from Arabic into Latin. Al-Qabisi lived in the 10th century in Syria.

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.

In University study from the 13th through 16th centuries, the most common introduction to the geocentric cosmos was the medieval work, On the Sphere, by Sacrobosco.

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.

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.

Kepler, Galileo and Guidobaldo were the leading optical theorists of their generation. Galileo studied with Guidobaldo while he was composing this treatise.

The frontispiece of Riccioli’s treatise depicts not two, but three major systems of the world. The Ptolemaic system rests discarded (lower right corner) because of the phases of Venus and Mercury (upper left corner). All-seeing Argus looks on, holding a telescope.

As a young scientist-engineer, Galileo wrote two manuscripts on motion. The first, Delle macchine, written ca. 1592; reflected the tradition of Aristotle’s Mechanics. It was never printed. The second, revised study, Le mechaniche, written ca.

The Copernican idea that the Earth moves as a planet required a thorough revision of physics. Galileo undertook this task in his Discourse on Two New Sciences, published 80 years after Copernicus.

Organization of nursing as a profession: Florence Nightingale championed social reform and the organization of nursing as a profession.

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.

Scientific theories may be accepted on the basis of a weighing of many complex factors rather than a single determinative observation or crucial experiment. From antiquity, Copernicanism had been rejected due to a failure to observe stellar parallax.

The Foucault pendulum swings in a constant plane or direction, and thus reveals the rotation of the Earth turning underneath.

The Foucault pendulum swings in a constant plane or direction, and thus reveals the rotation of the Earth turning underneath.