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Zacharias Janssen was born in 1580 and died in 1638, he is known for the first to invent the compound Microscope.

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Molecular Expressions: Science, Optics and You - Timeline - Zacharias Janssen

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Zacharias Janssen is generally believed to be the first investigator to invent the compound microscope.

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William Harvey became the first person to accurately describe the function of the heart and the circulation of blood around the body
https://www.youtube.com/watch?v=7NOU4McjtXs

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Born in 1571 on december 27 in weil, swabia. And realized that figures have a definite ratio between sizes.

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He was born on February 19, 1473 in Poland. He studied law and medicine in Italy. He then began to work as mathematician and astronomer. He proposed that the sun was the center of the world, and that the earth revolved around it. This is called the Copernican heliocentric theory. Copernicus is also described to be the initiator of the Scientific Revolution.

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1. Life and Works

Nicolaus Copernicus was born on 19 February 1473, the youngest of four children of Nicolaus Copernicus, Sr., a well-to-do merchant who had moved to Torun from Cracow, and Barbara Watzenrode, the daughter of a leading merchant family in Torun. The city, on the Vistula River, had been an important inland port in the Hanseatic League. However, fighting between the Order of the Teutonic Knights and the Prussian Union in alliance with the Kingdom of Poland ended in 1466, and West Prussia, which included Torun, was ceded to Poland, and Torun was declared a free city of the Polish kingdom. Thus the child of a German family was a subject of the Polish crown.

The father died in 1483, and the children's maternal uncle, Lucas Watzenrode (1447-1512), took them under his protection. Watzenrode was a very successful cleric - he was to become bishop of Warmia (Ermland in German) in 1489 - and he both facilitated his nephew's advancement in the church and directed his education. In 1491 Copernicus enrolled in the University of Cracow. There is no record of his having obtained a degree, which was not unusual at the time as he did not need a bachelor's degree for his ecclesiastical career or even to study for a higher degree. But the University of Cracow offered courses in mathematics, astronomy, and astrology (see Goddu 25-33 on all the university offerings), and Copernicus's interest was sparked, which is attested to by his acquisition of books in these subjects while at Cracow.

In 1495 Watzenrode arranged Copernicus's election as canon of the chapter of Frombork (Frauenberg in German) of the Cathedral Chapter of Warmia, an administrative position just below that of bishop. He assumed the post two years later, and his financial situation was secure for life. In the meantime, following in his uncle's footsteps, Copernicus went to the University of Bologna in 1496 to study canon law (see Goddu part 2 on what Copernicus may have encountered in Italy). While at Bologna he lived with the astronomy professor Domenico Maria Novara and made his first astronomical observations. In addition, as Rosen (1971, 323) noted, "In establishing close contact with Novara, Copernicus met, perhaps for the first time in his life, a mind that dared to challenge the authority of [Ptolemy] the most eminent ancient writer in his chosen fields of study." Copernicus also gave a lecture on mathematics in Rome, which may have focused on astronomy.

Copernicus's studies at Bologna provided an advantage he did not have at Cracow - a teacher of Greek. Humanism began to infiltrate the Italian universities in the fifteenth century. As Grendler (510) remarked, "By the last quarter of the century, practically all universities had one or several humanists, many of them major scholars." Antonio Cortesi Urceo, called Codro, became professor at Bologna in 1482 and added Greek several years later. Copernicus may have studied with him, for Copernicus translated into Latin the letters of the seventh-century Byzantine author Theophylactus Simocatta ( MW 27-71) from the 1499 edition of a collection of Greek letters produced by the Venetian humanist printer Aldus Manutius. Aldus had dedicated his edition to Urceo. Copernicus had his translation printed in 1509, his only publication prior to the On the Revolutions ( De revolutionibus). It is important to note that Copernicus's acquisition of a good reading knowledge of Greek was critical for his studies in astronomy because major works by Greek astronomers, including Ptolemy, had not yet been translated into Latin, the language of the universities at the time.

Copernicus left Bologna for Frombork in 1501 without having obtained his degree. The chapter then approved another leave of absence for Copernicus to study medicine at the University of Padua. The medical curriculum did not just include medicine, anatomy, and the like when Copernicus studied it. Siraisi (1990, 16) noted that "the reception in twefth-century western Europe of Greek and Islamic technical astronomy and astrology fostered the development of medical astrology...the actual practice of medical astrology was greatest in the West between the fourteenth and the sixteenth centuries." Astrology was taught in the medical schools of Italy. "The importance attached to the study of the stars in medieval medical education derived from a general and widely held belief that the heavenly bodies play an intermediary role in the creation of things here below and continue to influence them throughout their existence. The actual uses of astrology in medical diagnosis and treatment by learned physicians were many and various. 'Astrological medicine' is a vague and unsatisfactory term that can embrace any or all of the following: first, to pay attention to the supposed effect of astrological birth signs or signs at conception on the constitution and character of one's patients; second, to vary treatment according to various celestial conditions...third, to connect the doctrine of critical days in illness with astrological features, usually phases of the moon; and fourth, to predict or explain epidemics with reference to planetary conjunctions, the appearance of comets, or weather conditions" (Siraisi, 1981, 141-42). It is true that astrology required that medical students acquire some grounding in astronomy; nevertheless, it is likely that Copernicus studied astrology while at the University of Padua.

Copernicus did not receive his medical degree from Padua; the degree would have taken three years, and Copernicus had only been granted a two-year leave of absence by his chapter. Instead he matriculated in the University of Ferrara, from which he obtained a doctorate in canon law. But he did not return to his chapter in Frombork; rather he went to live with his uncle in the episcopal palace in Lidzbark-Warminski (Heilsberg in German). Although he made some astronomical observations, he was immersed in church politics, and after his elderly uncle became ill in 1507, Copernicus was his attending physician. Rosen (1971, 334-35) reasonably conjectured that the bishop may have hoped that his nephew would be his successor, but Copernicus left his uncle because his duties in Lidzbark-Warminski interfered with his continuing pursuit of his studies in astronomy. He took up residence in his chapter of Frombork in 1510 and stayed there the rest of his life.

Not that leaving his uncle and moving to Frombork exempted Copernicus from continued involvement in administrative and political duties. He was responsible for the administration of various holdings, which involved heading the provisioning fund, adjudicating disputes, attending meetings, and keeping accounts and records. In response to the problem he found with the local currency, he drafted an essay on coinage ( MW 176-215) in which he deplored the debasement of the currency and made recommendations for reform. His manuscripts were consulted by the leaders of both Prussia and Poland in their attempts to stabilize the currency. He was a leader for West Prussia in the war against the Teutonic Knights, which lasted from 1520-1525. He was physician for the bishop (his uncle had died in 1512) and members of the chapter, and he was consulting physician for notables in East and West Prussia.

Nevertheless, Copernicus began to work on astronomy on his own. Sometime between 1510 and 1514 he wrote an essay that has come to be known as the Commentariolus ( MW 75-126) that introduced his new cosmological idea, the heliocentric universe, and he sent copies to various astronomers. He continued making astronomical observations whenever he could, hampered by the poor position for observations in Frombork and his many pressing responsibilities as canon. Nevertheless, he kept working on his manuscript of On the Revolutions. He also wrote what is known as Letter against Werner ( MW 145-65) in 1524, a critique of Johann Werner's "Letter concerning the Motion of the Eighth Sphere" ( De motu octavae sphaerae tractatus primus). Copernicus claimed that Werner erred in his calculation of time and his belief that before Ptolemy the movement of the fixed stars was uniform, but Copernicus's letter did not refer to his cosmological ideas.

In 1539 a young mathematician named Georg Joachim Rheticus (1514-1574) from the University of Wittenberg came to study with Copernicus. Rheticus brought Copernicus books in mathematics, in part to show Copernicus the quality of printing that was available in the German-speaking cities. He published an introduction to Copernicus's ideas, the Narratio prima (First Report). Most importantly, he convinced Copernicus to publish On the Revolutions. Rheticus oversaw most of the printing of the book, and on 24 May 1543 Copernicus held a copy of the finished work on his deathbed.

2. Astronomical Ideas and Writings

2.1 Pre-Copernican Astronomy

Classical astronomy followed principles established by Aristotle. Aristotle accepted the idea that there were four physical elements - earth, water, air, and fire. He put the earth in the center of the universe and contended that these elements were below the moon, which was the closest celestial body. There were seven planets, or wandering stars, because they had a course through the zodiac in addition to traveling around the earth: the moon, Mercury, Venus, the sun, Mars, Jupiter. Beyond that were the fixed stars. The physical elements, according to Aristotle moved vertically, depending on their 'heaviness' or 'gravity'; the celestial bodies were not physical but a 'fifth element' or 'quintessence' whose nature was to move in perfect circles around the earth, making a daily rotation. Aristotle envisioned the earth as the true center of all the circles or 'orbs' carrying the heavenly bodies around it and all motion as 'uniform,' that is, unchanging.

But observers realized that the heavenly bodies did not move as Aristotle postulated. The earth was not the true center of the orbits and the motion was not uniform. The most obvious problem was that the outer planets seemed to stop, move backwards in 'retrograde' motion for a while, and then continue forwards. By the second century, when Ptolemy compiled his Almagest (this common name of Ptolemy's Syntaxis was derived from its Arabic title), astronomers had developed the concept that the orbit moves in 'epicycles' around a 'deferrent,' that is, they move like a flat heliacal coil around a circle around the earth. The earth was also off-center, on an 'eccentric,' as the heavenly bodies moved around a central point. Ptolemy added a point on a straight line opposite the eccentric, which is called the 'equalizing point' or the 'equant,' and around this point the heavenly bodies moved uniformly. Moreover, unlike the Aristotelian model, Ptolemy's Almagest did not describe a unified universe. The ancient astronomers who followed Ptolemy, however, were not concerned if his system did not describe the 'true' motions of the heavenly bodies; their concern was to 'save the phenomena,' that is, give a close approximation of where the heavenly bodies would be at a given point in time. And in an age without professional astronomers, let alone the telescope, Ptolemy did a good job plotting the courses of the heavenly bodies.

Not all Greek astronomical ideas followed this geocentric system. Pythagoreans suggested that the earth moved around a central fire (not the sun). Archimedes wrote that Aristarchus of Samos actually proposed that the earth rotated daily and revolved around the sun.

During the European Middle Ages, the Islamic world was the center of astronomical thought and activity. During the ninth century several aspects of Ptolemy's solar theory were recalculated. Ibn al-Haytham in the tenth-eleventh century wrote a scathing critique of Ptolemy's work: "Ptolemy assumed an arrangement that cannot exist, and the fact that this arrangement produces in his imagination the motions that belong to the planets does not free him from the error he committed in his assumed arrangement, for the existing motions of the planets cannot be the result of an arrangement that is impossible to exist" (quoted in Rosen 1984, 174). Swerdlow and Neugebauer (46-48) stressed that the thirteenth-century Maragha school was also important in finding errors and correcting Ptolemy: "The method of the Maragha planetary models was to break up the equant motion in Ptolemy's models into two or more components of uniform circular motion, physically the uniform rotation of spheres, that together control the direction and distance of the center of the epicycle, so that it comes to lie in nearly the same position it would have in Ptolemy's model, and always moves uniformly with respect to the equant." They found that Copernicus used devices that had been developed by the Maragha astronomers Nasir al-Din Tusi (1201-1274), Muayyad al-Din al-Urdi (d. 1266), Qutb al-Din al-Shirazi (1236-1311), and Ibn al-Shatir (1304-1375). In addition, Ragep, 2005, has shown that a theory for the inner planets presented by Regiomontanus that enabled Copernicus to convert the planets to eccentric models had been developed by the fifteenth-century, Samarqand-trained astronomer ali Qushji (1403-1474).

Renaissance humanism did not necessarily promote natural philosophy, but its emphasis on mastery of classical languages and texts had the side effect of promoting the sciences. Georg Peurbach (1423-1461) and (Johannes Müller) Regiomontanus (1436-1476) studied Greek for the purpose of producing an outline of Ptolemaic astronomy. By the time Regiomontanus finished the work in 1463, it was an important commentary on the Almagest as well, pointing out, for example, that Ptolemy's lunar theory did not accord with observations. He noted that Ptolemy showed the moon to be at various times twice as far from the earth as at other times, which should make the moon appear twice as big. At the time, moreover, there was active debate over Ptolemy's deviations from Aristotle's requirement of uniform circular motion.

2.2 The Commentariolus

It is impossible to date when Copernicus first began to espouse the heliocentric theory. Had he done so during his lecture in Rome, such a radical theory would have occasioned comment, but there was none, so it is likely that he adopted this theory after 1500. Further, Corvinus, who helped him print his Latin translation in 1508-09, expressed admiration for his knowledge of astronomy, so that Copernicus's concept may have still been traditional at this point. His first heliocentric writing was his Commentariolus. It was a small manuscript that was circulated but never printed. We do not know when he wrote this, but a professor in Cracow cataloged his books in 1514 and made reference to a "manuscript of six leaves expounding the theory of an author who asserts that the earth moves while the sun stands still" (Rosen, 1971, 343; MW 75). Thus, Copernicus probably adopted the heliocentric theory sometime between 1508 and 1514. Rosen (1971, 345) suggested that Copernicus's "interest in determining planetary positions in 1512-1514 may reasonably be linked with his decisions to leave his uncle's episcopal palace in 1510 and to build his own outdoor observatory in 1513." In other words, it was the result of a period of intense concentration on cosmology that was facilitated by his leaving his uncle and the attendant focus on church politics and medicine.

It is impossible to know exactly why Copernicus began to espouse the heliocentric cosmology. Despite his importance in the history of philosophy, there is a paucity of primary sources on Copernicus. His only astronomical writings were the Commentariolus, the Letter against Werner, and On the Revolutions; he published his translation of Theophylactus's letters and wrote the various versions of his treatise on coinage; other writings relate to diocesan business, as do most of the few letters that survive. Sadly, the biography by Rheticus, which should have provided scholars with an enormous amount of information, has been lost. Therefore, many of the answers to the most interesting questions about Copernicus's ideas and works have been the result of conjecture and inference, and we can only guess why Copernicus adopted the heliocentric system.

Most scholars believe that the reason Copernicus rejected Ptolemaic cosmology was because of Ptolemy's equant. They assume this because of what Copernicus wrote in the Commentariolus:

Yet the widespread [planetary theories], advanced by Ptolemy and most other [astronomers], although consistent with the numerical [data], seemed likewise to present no small difficulty. For these theories were not adequate unless they also conceived certain equalizing circles, which made the planet appear to move at all times with uniform velocity neither on its deferent sphere nor about its own [epicycle's] center...Therefore, having become aware of these [defects], I often considered whether there could perhaps be found a more reasonable arrangement of circles, from which every apparent irregularity would be derived while everything in itself would move uniformly, as is required by the rule of perfect motion. (MW 81).

Goddu (381-84) has plausibly maintained that while the initial motivation for Copernicus was dissatisfaction with the equant, that dissatisfaction may have impelled him to observe other violations of uniform circular motion, and those observations, not the rejection of the equant by itself, led to the heliocentric theory. Blumenberg (254) has pointed out that the mobility of the earth may have been reinforced by the similarity of its spherical shape to those of the heavenly bodies.

As the rejection of the equant suggests a return to the Aristotelian demand for true uniform circular motion of the heavenly bodies, it is unlikely that Copernicus adopted the heliocentric model because philosophies popular among Renaissance humanists like Neoplatonism and Hermetism compelled him in that direction. Nor should we attribute Copernicus's desire for uniform circular motions to an aesthetic need, for this idea was philosophical not aesthetic, and Copernicus's replacing the equant with epicyclets made his system more complex than Ptolemy's. Most importantly, we should bear in mind what Swerdlow and Neugebauer (59) asserted:

Copernicus arrived at the heliocentric theory by a careful analysis of planetary models - and as far as is known, he was the only person of his age to do so - and if he chose to adopt it, he did so on the basis of an equally careful analysis.

In the Commentariolus Copernicus listed assumptions that he believed solved the problems of ancient astronomy. He stated that the earth is only the center of gravity and center of the moon's orbit; that all the spheres encircle the sun, which is close to the center of the universe; that the universe is much larger than previously assumed, and the earth's distance to the sun is a small fraction of the size of the universe; that the apparent motion of the heavens and the sun is created by the motion of the earth; and that the apparent retrograde motion of the planets is created by the earth's motion. Although the Copernican model maintained epicycles moving along the deferrent, which explained retrograde motion in the Ptolemaic model, Copernicus correctly explained that the retrograde motion of the planets was only apparent not real, and its appearance was due to the fact that the observers were not at rest in the center. The work dealt very briefly with the order of the planets (Mercury, Venus, earth, Mars, Jupiter, and Saturn, the only planets that could be observed with the naked eye), the triple motion of the earth (the daily rotation, the annual revolution of its center, and the annual revolution of its inclination) that causes the sun to seem to be in motion, the motions of the equinoxes, the revolution of the moon around the earth, and the revolution of the five planets around the sun.

2.3 On the Revolutions

The Commentariolus was only intended as an introduction to Copernicus's ideas, and he wrote "the mathematical demonstrations intended for my larger work should be omitted for brevity's sake..." ( MW 82). In a sense it was an announcement of the greater work that Copernicus had begun. The Commentariolus was never published during Copernicus's lifetime, but he sent manuscript copies to various astronomers and philosophers. He received some discouragement because the heliocentric system seemed to disagree with the Bible, but mostly he was encouraged. Although Copernicus's involvement with official attempts to reform the calendar was limited to a no longer extant letter, that endeavor made a new, serious astronomical theory welcome. Fear of the reaction of ecclesiastical authorities was probably the least of the reasons why he delayed publishing his book. The most important reasons for the delay was that the larger work required both astronomical observations and intricate mathematical proofs. His administrative duties certainly interfered with both the research and the writing. He was unable to make the regular observations that he needed and Frombork, which was often fogged in, was not a good place for those observations. Moreover, as Gingerich (1993, 37) pointed out,

[Copernicus] was far from the major international centers of printing that could profitably handle a book as large and technical as De revolutionibus. On the other [hand], his manuscript was still full of numerical inconsistencies, and he knew very well that he had not taken complete advantage of the opportunities that the heliocentric viewpoint offered...Furthermore, Copernicus was far from academic centers, thereby lacking the stimulation of technically trained colleagues with whom he could discuss his work.

The manuscript of On the Revolutions was basically complete when Rheticus came to visit him in 1539. The work comprised six books. The first book, the best known, discussed what came to be known as the Copernican theory and what is Copernicus's most important contribution to astronomy, the heliocentric universe (although in Copernicus's model, the sun is not truly in the center). Book 1 set out the order of the heavenly bodies about the sun: "[The sphere of the fixed stars] is followed by the first of the planets, Saturn, which completes its circuit in 30 years. After Saturn, Jupiter accomplishes its revolution in 12 years. The Mars revolves in 2 years. The annual revolution takes the series' fourth place, which contains the earth...together with the lunar sphere as an epicycle. In the fifth place Venus returns in 9 months. Lastly, the sixth place is held by Mercury, which revolves in a period of 80 days" ( Revolutions, 21-22). This established a relationship between the order of the planets and their periods, and it made a unified system. This may be the most important argument in favor of the heliocentric model as Copernicus described it. It was far superior to Ptolemy's model, which had the planets revolving around the earth so that the sun, Mercury, and Venus all had the same annual revolution. In book 1 Copernicus also insisted that the movements of all bodies must be circular and uniform, and noted that the reason they may appear nonuniform to us is "either that their circles have poles different [from the earth's] or that the earth is not at the center of the circles on which they revolve" ( Revolutions, 11). Particularly notable for Copernicus was that in Ptolemy's model the sun, the moon, and the five planets seemed ironically to have different motions from the other heavenly bodies and it made more sense for the small earth to move than the immense heavens. But the fact that Copernicus turned the earth into a planet did not cause him to reject Aristotelian physics, for he maintained that "land and water together press upon a single center of gravity; that the earth has no other center of magnitude; that, since earth is heavier, its gaps are filled with water..." ( Revolutions, 10). As Aristotle had asserted, the earth was the center toward which the physical elements gravitate. This was a problem for Copernicus's model, because if the earth was no longer the center, why should elements gravitate toward it?

The second book of On the Revolutions elaborated the concepts in the first book; book 3 dealt with the precession of the equinoxes and solar theory; book 4 dealt with the moon's motions; book 5 dealt with the planetary longitude and book 6 with latitude. Copernicus depended very much on Ptolemy's observations, and there was little new in his mathematics. He was most successful in his work on planetary longitude, which, as Swerdlow and Neugebauer (77) commented, was "Copernicus's most admirable, and most demanding, accomplishment...It was above all the decision to derive new elements for the planets that delayed for nearly half a lifetime Copernicus's continuation of his work - nearly twenty years devoted to observation and then several more to the most tedious kind of computation - and the result was recognized by his contemporaries as the equal of Ptolemy's accomplishment, which was surely the highest praise for an astronomer." Surprisingly, given that the elimination of the equant was so important in the Commentariolus, Copernicus did not mention it in book 1, but he sought to replace it with an epicyclet throughout On the Revolutions. Nevertheless, he did write in book 5 when describing the motion of Mercury:

...the ancients allowed the epicycle to move uniformly only around the equant's center. This procedure was in gross conflict with the true center [of the epicycle's motion], its relative [distances], and the prior centers of both [other circles]...However, in order that this last planet too may be rescued from the affronts and pretenses of its detractors, and that its uniform motion, no less than that of the other aforementioned planets, may be revealed in relation to the earth's motion, I shall attribute to it too, [as the circle mounted] on its eccentric, an eccentric instead of the epicycle accepted in antiquity (Revolutions, 278-79).

2.4 Rheticus and the Narratio prima

Although Copernicus received encouragement to publish his book from his close friend, the bishop of Chelmo Tiedemann Giese (1480-1550), and from the cardinal of Capua Nicholas Schönberg (1472-1537), it was the arrival of Georg Joachim Rheticus in Frombork that solved his needs for a supportive and stimulating colleague in mathematics and astronomy and for access to an appropriate printer. Rheticus was a professor of mathematics at the University of Wittenberg, a major center for the student of mathematics as well as for Lutheran theology. In 1538 Rheticus took a leave of absence to visit several famous scholars in the fields of astronomy and mathematics. It is not known how Rheticus learned about Copernicus's theory; he may have been convinced to visit Copernicus by one of the earlier scholars he had visited, Johann Schöner, though, as Swerdlow and Neugebauer (16) noted, by "the early 1530's knowledge of Copernicus's new theory was circulating in Europe, even reaching the high and learned circles of the Vatican." Rheticus brought with him some mathematical and astronomical volumes, which both provided Copernicus with some important material and showed him the quality of the mathematical printing available in the German centers of publishing. Rheticus's present of the 1533 edition of Regiomontanus's On all Kinds of Triangles ( De triangulis omnimodis), for example, convinced Copernicus to revise his section on trigonometry. But Rheticus was particularly interested in showing Copernicus the work of the Nuremberg publisher Johann Petreius as a possible publisher of Copernicus's volume. Swerdlow and Neugebauer (25) plausibly suggested that "Petreius was offering to publish Copernicus's work, if not advertising by this notice that he was already committed to do so." Rheticus wrote the Narratio prima in 1540, an introduction to the theories of Copernicus, which was published and circulated. This further encouraged Copernicus to publish his Revolutions, which he had been working on since he published the Commentariolus.

The Narratio prima was written in 1539 and took the form of a letter to Johann Schöner announcing Copernicus's findings and describing the contents of the Revolutions. He dealt with such topics as the motions of the fixed stars, the tropical year, the obliquity of the ecliptic, the problems resulting from the motion of the sun, the motions of the earth and the other planets, librations, longitude in the other five planets, and the apparent deviation of the planets from the ecliptic. He asserted that the heliocentric universe should have been adopted because it better accounted for such phenomena as the precession of the equinoxes and the change in the obliquity of the ecliptic; it resulted in a diminution of the eccentricity of the sun; the sun was the center of the deferents of the planets; it allowed the circles in the universe to revolve uniformly and regularly; it satisfied appearances more readily with fewer explanations necessary; it united all the spheres into one system. Rheticus added astrological predictions and number mysticism, which were absent from Copernicus's work.

The Narratio prima was printed in 1540 in Gdansk (then Danzig); thus, it was the first printed description of the Copernican thesis. Rheticus sent a copy to Achilles Pirmin Gasser of Feldkirch, his hometown in modern-day Austria, and Gasser wrote a foreword that was published with a second edition that was produced in 1541 in Basel. It was published again in 1596 as an appendix to the first edition of Johannes Kepler's Mysterium cosmographicum (Secret of the Universe), the first completely Copernican work by an adherent since the publications by Copernicus and Rheticus.

2.5 Printing On the Revolutions and Osiander's Preface

The publication of Rheticus's Narratio prima did not create a big stir against the heliocentric thesis, and so Copernicus decided to publish On the Revolutions. He added a dedication to Pope Paul III (r. 1534-1549), probably for political reasons, in which he expressed his hesitancy about publishing the work and the reasons he finally decided to publish it. He gave credit to Schönberg and Giese for encouraging him to publish and omitted mention of Rheticus, but it would have been insulting to the pope during the tense period of the Reformation to give credit to a Protestant minister. He dismissed critics who might have claimed that it was against the Bible by giving the example of the fourth-century Christian apologist Lactantius, who had rejected the spherical shape of the earth, and by asserting, "Astronomy is written for astronomers" ( Revolutions, 5). In other words, theologians should not meddle with it. He pointed to the difficulty of calendar reform because the motions of the heavenly bodies were inadequately known. And he called attention to the fact that "if the motions of the other planets are correlated with the orbiting of the earth, and are computed for the revolution of each planet, not only do their phenomena follow therefrom but also the order and size of all the planets and spheres, and heaven itself is so linked together that in no portion of it can anything be shifted without disrupting the remaining parts and the universe as a whole" ( Revolutions, 5).

Rheticus returned to Wittenberg in 1541 and the following year received another leave of absence, at which time he took the manuscript of the Revolutions to Petreius for publishing in Nuremberg. Rheticus oversaw the printing of most of the text. However, Rheticus was forced to leave Nuremberg later that year because he was appointed professor of mathematics at the University of Leipzig. He left the rest of the management of printing the Revolutions to Andrew Osiander (1498-1552), a Lutheran minister who was also interested in mathematics and astronomy. Though he saw the project through, Osiander appended an anonymous preface to the work. In it he claimed that Copernicus was offering a hypothesis, not a true account of the working of the heavens: "Since he [the astronomer] cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past ...these hypotheses need not be true nor even probable" ( Revolutions, xvi). This clearly contradicted the body of the work. Both Rheticus and Giese protested, and Rheticus crossed it out in his copy.

2.6 Sixteenth Century Reactions to On the Revolutions

Copernicus's fame and book made its way across Europe over the next fifty years, and a second edition was brought out in 1566. As Gingerich's census of the extant copies showed, the book was read and commented on by astronomers. (For a fuller discussion of reactions, see Omodeo.) Gingerich (2004, 55) noted "the majority of sixteenth-century astronomers thought eliminating the equant was Copernicus' big achievement."

While Martin Luther may have made negative comments about Copernicus because the idea of the heliocentric universe seemed to contradict the Bible, Philip Melanchthon (1497-1560), who presided over the curriculum at the University of Wittenberg, eventually accepted the importance of teaching Copernicus's ideas, perhaps because Osiander's preface made the work more palatable. His son-in-law Caspar Peucer (1525-1602) taught astronomy there and began teaching Copernicus's work. As a result, the University of Wittenberg became a center where Copernicus's work was studied. But Rheticus was the only Wittenberg scholar who accepted the heliocentric idea. Robert Westman (1975a, 166-67; 2011, chap. 5) suggested that there was a 'Wittenberg Interpretation': astronomers appreciated and adopted some of Copernicus's mathematical models but rejected his cosmology, and some were pleased with his replacement of the equant by epicyclets. One of these was Erasmus Reinhold (1511-1553), a leading astronomer at Wittenberg who became dean and rector. He produced a new set of planetary tables from Copernicus's work, the Prutenic Tables. Although, as Gingerich (1993, 232) pointed out, "there was relatively little to distinguish between the accuracy of the Alfonsine Tables and the Prutenic Tables," the latter were more widely adopted; Gingerich plausibly suggested that the fact that the Prutenic Tables more accurately predicted a conjunction between Jupiter and Saturn in 1563 made the difference. Reinhold did not accept the heliocentric theory, but he admired the elimination of the equant. The Prutenic Tables excited interest in Copernicus's work.

Tycho Brahe (1546-1601) was the greatest astronomical observer before the invention of the telescope. He called Copernicus a 'second Ptolemy' (quoted in Westman 1975, 307) and appreciated both the elimination of the equant and the creation of a planetary system. But Tycho could not adopt the Copernican system, partly for the religious reason that it went against what the Bible seemed to preach. He, therefore, adopted a compromise, the 'geoheliostatic' system in which the two inner planets revolved around the sun and that system along with the rest of the planets revolved around the earth.

Among Catholics, Christoph Clavius (1537-1612) was the leading astronomer in the sixteenth century. A Jesuit himself, he incorporated astronomy into the Jesuit curriculum and was the principal scholar behind the creation of the Gregorian calendar. Like the Wittenberg astronomers, Clavius adopted Copernican mathematical models when he felt them superior, but he believed that Ptolemy's cosmology - both his ordering of the planets and his use of the equant - was correct.

Pope Clement VII (r. 1523-1534) had reacted favorably to a talk about Copernicus's theories, rewarding the speaker with a rare manuscript. There is no indication of how Pope Paul III, to whom On the Revolutions was dedicated reacted; however, a trusted advisor, Bartolomeo Spina of Pisa (1474-1546) intended to condemn it but fell ill and died before his plan was carried out (see Rosen, 1975). Thus, in 1600 there was no official Catholic position on the Copernican system, and it was certainly not a heresy. When Giordano Bruno (1548-1600) was burned at the stake as a heretic, it had nothing to do with his writings in support of Copernican cosmology, and this is clearly shown in Finocchiaro's reconstruction of the accusations against Bruno (see also Blumenberg's part 3, chapter 5, titled "Not a Martyr for Copernicanism: Giordano Bruno").

Michael Maestlin (1550-1631) of the University of Tübingen was the earliest astronomer after Rheticus to adopt Copernicus's heliocentricism. Although he wrote a popular textbook that was geocentric, he taught his students that the heliocentric system was superior. He also rejected Osiander's preface. Maestlin's pupil Johannes Kepler wrote the first book since the publication of On the Revolutions that was openly heliocentric in its orientation, the Mysterium cosmographicum (Secret of the Universe). And, of course, Kepler eventually built on Copernicus's work to create a much more accurate description of the solar system.

Bibliography

A. Complete Works of Copernicus

In 1972 the Polish Academy of Sciences under the direction of J. Dobrzycki published critical editions of the Complete Works of Copernicus in six languages: Latin, English, French, German, Polish, and Russian. The first volume was a facsimile edition. The annotations in the English translations are more comprehensive than the others. The English edition was reissued as follows:

Minor Works, 1992, trans. E. Rosen, Baltimore: The Johns Hopkins University Press (originally published as volume 3 ofNicholas Copernicus: Complete Works, Warsaw: Polish Scientific Publishers, 1985). Referred to herein asMW.

On the Revolutions, 1992, trans. E. Rosen, Baltimore: The Johns Hopkins University Press (originally published as volume 2 ofNicholas Copernicus: Complete Works, Warsaw: Polish Scientific Publishers, 1978). Referred to herein asRevolutions.

B. Other Translations of Copernicus's Works

On the Revolutions of the Heavenly Spheres, 1955, trans. C.G. Wallis, vol. 16 ofGreat Books of the Western World, Chicago: Encyclopedia Britannica; 1995, reprint, Amherst: Prometheus Books.

On the Revolutions of the Heavenly Spheres, 1976, trans. and ed. A.M. Duncan, Newton Abbot: David & Charles.

"The Derivation and First Draft of Copernicus's Planetary Theory: A Translation of the Commentariolus with Commentary," 1973, trans. N.M. Swerdlow, Proceedings of the American Philosophical Society, 117: 423-512.

C. Translations of Other Primary Sources

Bruno, G., 1977, The Ash Wednesday Supper, trans. E.A. Gosselin and L.S. Lerner, Hamden: Archon Books, 1995; reprint, Toronto: University of Toronto Press.

Rheticus, G.J., Narratio prima, in E. Rosen, 1971, 107-96.

D. Secondary Sources

Blåsjö, V., 2014, "A Critique of the Arguments for Maragha Influence on Copernicus," Journal for the History of Astronomy, 45: 183-195.

Blumenberg, H., 1987, The Genesis of the Copernican World, trans. R.M. Wallace, Cambridge, MA: MIT Press.

Cohen, I.B., 1960, The Birth of a New Physics, Garden City: Anchor Books; rev. ed., New York: W.W. Norton, 1985.

---, 1985, Revolutions in Science, Cambridge, MA: Harvard University Press.

Crowe, M.J., 1990, Theories of the World from Antiquity to the Copernican Revolution, New York: Dover Publications.

Feldhay, R. and F.J. Ragep (eds.), 2017, Before Copernicus: The Cultures and Contexts of Scientific Learning in the Fifteenth Century, Montreal: McGill-Queens University Press.

Finocchiaro, M.A., 2002, "Philosophy versus Religion and Science versus Religion: the Trials of Bruno and Galileo," in H. Gatti (ed.), 51-96.

Gatti, H. (ed.), 2002, Giordano Bruno: Philosopher of the Renaissance, Aldershot: Ashgate.

Gillespie, C.C. (ed.), 1970-80, Dictionary of Scientific Biography, New York: Scribner's.

Gingerich, O., 1993, The Eye of Heaven: Ptolemy, Copernicus, Kepler, New York: American Institute of Physics.

---, 2002, An Annotated Census of Copernicus' De revolutionibus, Leiden: Brill Academic Publishers; Nuremberg, 1543 and Basel, 1566.

---, 2004, The Book Nobody Read: Chasing the Revolutions of Nicolaus Copernicus, New York: Walker & Company.

Goldstein, B., 2002, "Copernicus and the Origin of His Heliocentric System," Journal for the History of Astronomy, 33: 219-235.

Goddu, A., 2010, Copernicus and the Aristotelian Tradition: Education, Reading, and Philosophy in Copernicus's Path to Heliocentrism, Leiden: Brill.

Grendler, P., 2002, The Universities of the Italian Renaissance, Baltimore: The Johns Hopkins University Press.

Hallyn, F., 1990, The Poetic Structure of the World: Copernicus and Kepler, trans. D. Leslie, New York: Zone Books.

Koestler, A., 1989, The Sleepwalkers, London: Penguin, reprint of 1959 edition.

Koyré, A., 1957, From the Closed World to the Infinite Universe, Baltimore: The Johns Hopkins University Press.

---, 1973, The Astronomical Revolution: Copernicus, Kepler, Borelli, trans. R.E.W. Maddison, Ithaca: Cornell University Press.

Kuhn, T., 1957, The Copernican Revolution, Cambridge, MA: Harvard University Press.

Morrison, R., 2014, "A Scholarly Intermediary between the Ottoman Empire and Renaissance Europe," Isis, 105: 32-57.

---, 2017, "Jews as Scientific Intermediaries in the European Renaissance," in Feldhay and Ragep (eds.), 198-214-97.

Omodeo, P.D., 2014, Copernicus in the Cultural Debates of the Renaissance: Reception, Legacy, Transformation, Leiden: Brill.

Ragep, F.J., 2005, "Ali Qushji and Regiomontanus," Journal for the History of Astronomy, 36: 359-71.

---, 2007, "Copernicus and His Islamic Predecessors," History of Science, 45: 65-81.

---, 2016, "Ibn al-Shatir and Copernicus: The Uppsala Notes Revisited," Journal for the History of Astronomy, 47: 395-415.

---, 2017, "From Tun to Turin: The Twists and Turns of the Tusi Couple," in Feldhay and Ragep (eds.), 161-97.

Rosen, E.,1970a, "Copernicus," in Gillespie (ed.), 3: 401-11.

---, 1970b, "Rheticus," Gillespie (ed.), 11: 395-97.

---, 1971, Three Copernican Treatises, 3d ed., New York: Octagon Books.

---, 1975, "Was Copernicus' Revolutions Approved by the Pope?"Journal of the History of Ideas, 36: 531-42.

---, 1984, Copernicus and the Scientific Revolution, Malabar, FL: Krieger Publishing Co.

Saliba, G., 2007, Islamic Science and the Making of the European Renaissance, Cambridge, MA: MIT Press.

Shumaker, W., 1979, The Occult Sciences in the Renaissance: A Study in Intellectual Patterns, Berkeley: University of California Press, reprint of 1972 edition.

Siraisi, N., 1981, Taddeo Alderotti and His Pupils: Two Generations of Italian Medical Learning, Princeton: Princeton University Press.

---, 1990, Medieval and Early Renaissance Medicine: An Introduction to Knowledge and Practice, Chicago: University of Chicago Press.

Swerdlow, N., 2000, "Copernicus, Nicolaus (1473-1543)," in Encyclopedia of the Scientific Revolution, W. Applebaum (ed.), New York: Garland Publishing, 162-68.

---, 2017, "Copernicus's Derivation of the Heliocentric Theory from Regiomontanus's Eccentric Models of the Second Inequality of the Superior and Inferior Planets" Journal for the History of Astronomy, 48: 33-61.

Swerdlow, N. and O. Neugebauer, 1984, Mathematical Astronomy in Copernicus's De Revolutionibus, 2 vols., New York: Springer-Verlag.

Westman, R., 1975a, "The Melanchthon Circle, Rheticus, and the Wittenberg Interpretation of the Copernican Theory," Isis, 66: 165-93.

---, 1975b, "Three Responses to the Copernican Theory: Johannes Praetorius, Tycho Brahe, and Michael Maestlin," in Westman (ed.), 1975c.

---, (ed), 1975c, The Copernican Achievement, Berkeley: University of California Press.

---, 2011, The Copernican Question: Prognostication, Skepticism, and Celestial Order, Berkeley: University of California Press.

Yates, F., 1979, Giordano Bruno and the Hermetic Tradition, Chicago: University of Chicago Press, reprint of 1964 edition.

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Nicolaus Copernicus was born on 19 February 1473, the youngest of four children of Nicolaus Copernicus, Sr., a well-to-do merchant who had moved to Torun from Cracow, and Barbara Watzenrode, the daughter of a leading merchant family in Torun. The city, on the Vistula River, had been an important inland port in the Hanseatic League.

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Who is Andreas Vesalius? What did he do?

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Who is Andreas Vesalius? What did he do?

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Andreas Vesalius started his career at the University of Paris. He was a teacher who dissected human corpse for himself, instead of reading the book while someone else dissected the corpse for him. He used executed criminal bodies. He also proved that Galen was very wrong about the human body. Vesalius proved that he was wrong by showing his students everything he did while in the classes he taught. He also wrote a book about the human body. No he didn't steal corpses. He just studied human anatomy

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Galileo Galileo was born in Italy and lived from 1564 to 1642. He is best known for his additions to physics and the theory of inertia and motion. His philosophy that everything can be written with math helped develop the idea of experiments and the scientific method. Finally his discoveries with the telescope revolutionized astronomy idea of a heliocentric solar system, but in the process resulted in a inquisition against himself.

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What Gabriel Daniel Fahrenheit did was make the first reliable thermometers in 1686-1736

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By the early 1700s, at least 35 different temperature scales had been proposed. At that time, a Dutch instrument maker by the name of Daniel Gabriel Fahrenheit became famous for his mercury thermometers. The Fahrenheit scale he developed is still the most widely used temperature scale in the United States.

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Anders Celcius was a Swedish astronomer, know for inventing the celcius tempature scale.

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Marie and Antoine Lavoiser discovered oxygen the basis of combustion during the french revolution. Lavoiser was beheaded for the murder of charlotte corday

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marie and antoine lavoisier - Google Search

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Isaac Newton Biography - Newton's Life, Career, Work - Dr Robert A. Hatch

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Life & Character - Isaac Newton was born prematurely on Christmas day 1642 (4 January 1643, New Style) in Woolsthorpe, a hamlet near Grantham in Lincolnshire. The posthumous son of an illiterate yeoman (also named Isaac), the fatherless infant was small enough at birth to fit 'into a quartpot.'

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1608-1647

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He was the single greatest experimental scientist , he knew everything from astomatics to chemistry to biology and excetra , he and some other scientist discovered the universal joint . He was born on July 18 1635 , And died on March 3 1703.

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He found ratios for shapes mostly triangles and circles

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a Swiss physician, alchemist, and astrologer of the German Renaissance. He was a pioneer in several aspects of the "medical revolution" of the Renaissance, emphasizing the value of observation in combination with received wisdom. he is known for being a medical genius.
http://knarf.english.upenn.edu/People/paracels.html

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Joseph Priestley was an English scientist who authored more than 150 publications. He is noted for his groundbreaking contributions to chemistry of gases

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he made a vaccination to smallpox and was known for his contribution to immunization.
lived 1749-1823
came from Berkeley, United Kingdom

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Johannes Kepler was born on December 27, 1571, a premature child. To be precise, according to his own records, the pregnancy lasted 224 days, 9 hours and 53 minutes. (This rather odd piece of information, and the quotes from Kepler's horoscopes for his family given below, I found in Koestler's book, reference 1.)

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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1200696/

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He was called "the Father of Modern Science" for empirical research and was the first Englishman to write a poetry of beauty. He was born in October 28, 1909 and died in April 28, 1992.
https://www.theartstory.org/artist-bacon-francis.htm
https://www.bnl.gov/bera/activities/globe/sir_francis_bacon.htm

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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1200696/

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In full:

Galileo Galilei

(Show more)

Galileo (born February 15, 1564, Pisa [Italy]-died January 8, 1642, Arcetri, near Florence) Italian natural philosopher, astronomer, and mathematician who made fundamental contributions to the sciences of motion, astronomy, and strength of materials and to the development of the scientific method. His formulation of (circular) inertia, the law of falling bodies, and parabolic trajectories marked the beginning of a fundamental change in the study of motion. His insistence that the book of nature was written in the language of mathematics changed natural philosophy from a verbal, qualitative account to a mathematical one in which experimentation became a recognized method for discovering the facts of nature. Finally, his discoveries with the telescope revolutionized astronomy and paved the way for the acceptance of the Copernican heliocentric system, but his advocacy of that system eventually resulted in an Inquisition process against him.

Early life and career

Galileo was born in Pisa, Tuscany, on February 15, 1564, the oldest son of Vincenzo Galilei, a musician who made important contributions to the theory and practice of music and who may have performed some experiments with Galileo in 1588-89 on the relationship between pitch and the tension of strings. The family moved to Florence in the early 1570s, where the Galilei family had lived for generations. In his middle teens Galileo attended the monastery school at Vallombrosa, near Florence, and then in 1581 matriculated at the University of Pisa, where he was to study medicine. However, he became enamoured with mathematics and decided to make the mathematical subjects and philosophy his profession, against the protests of his father. Galileo then began to prepare himself to teach Aristotelian philosophy and mathematics, and several of his lectures have survived. In 1585 Galileo left the university without having obtained a degree, and for several years he gave private lessons in the mathematical subjects in Florence and Siena. During this period he designed a new form of hydrostatic balance for weighing small quantities and wrote a short treatise, La bilancetta ("The Little Balance"), that circulated in manuscript form. He also began his studies on motion, which he pursued steadily for the next two decades.

In 1588 Galileo applied for the chair of mathematics at the University of Bologna but was unsuccessful. His reputation was, however, increasing, and later that year he was asked to deliver two lectures to the Florentine Academy, a prestigious literary group, on the arrangement of the world in Dante's Inferno. He also found some ingenious theorems on centres of gravity (again, circulated in manuscript) that brought him recognition among mathematicians and the patronage of Guidobaldo del Monte (1545-1607), a nobleman and author of several important works on mechanics. As a result, he obtained the chair of mathematics at the University of Pisa in 1589. There, according to his first biographer, Vincenzo Viviani (1622-1703), Galileo demonstrated, by dropping bodies of different weights from the top of the famous Leaning Tower, that the speed of fall of a heavy object is not proportional to its weight, as Aristotle had claimed. The manuscript tract De motu ( On Motion), finished during this period, shows that Galileo was abandoning Aristotelian notions about motion and was instead taking an Archimedean approach to the problem. But his attacks on Aristotle made him unpopular with his colleagues, and in 1592 his contract was not renewed. His patrons, however, secured him the chair of mathematics at the University of Padua, where he taught from 1592 until 1610.

Although Galileo's salary was considerably higher there, his responsibilities as the head of the family (his father had died in 1591) meant that he was chronically pressed for money. His university salary could not cover all his expenses, and he therefore took in well-to-do boarding students whom he tutored privately in such subjects as fortification. He also sold a proportional compass, or sector, of his own devising, made by an artisan whom he employed in his house. Perhaps because of these financial problems, he did not marry, but he did have an arrangement with a Venetian woman, Marina Gamba, who bore him two daughters and a son. In the midst of his busy life he continued his research on motion, and by 1609 he had determined that the distance fallen by a body is proportional to the square of the elapsed time ( the law of falling bodies) and that the trajectory of a projectile is a parabola, both conclusions that contradicted Aristotelian physics.

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Galileo | Biography, Discoveries, Inventions, & Facts

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Galileo, the brilliant Italian polymath, revolutionized our understanding of the universe through his groundbreaking discoveries in astronomy and contributions to scientific methodology.

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Gerardus Mercator was a German-Flemish cartographer, Geographer, and cosmographer , he was known for creating the 1569 world map.

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Paracelsus - The Renaissance Pt. 1 - Secret Teachings

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Paracelsus: The Renaissance Pt. 1 - Secret Teachings Documentary Paracelsus - born Philippus Aureolus Theophrastus Bombastus von Hohenheim, was a Swiss German philosopher, physician, botanist, astrologer, and general occultist. He is credited as the founder of toxicology. He is also a famous revolutionary for utilizing observations of nature, rather than referring to ancient texts, something of radical defiance during his time.

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A tool to measure atmospheric pressure, mostly used for forecasting the weather.

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Robert Boyle - Man of Science, Man of Faith

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Robert Boyle has been called the "Father of Modern Chemistry". He was one of a handful of thinkers who pioneered the scientific method and helped displace alchemy and replace it with chemistry. But few are aware of his spiritual pursuits, both as a supporter of Bible translation and a defender of Christianity against Atheism.

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The Mystery of Matter: “OUT OF THIN AIR” (Documentary)

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Lived 1596 - 1650.
René Descartes invented analytical geometry and introduced skepticism as an essential part of the scientific method. He is regarded as one of the greatest philosophers in history.

His analytical geometry was a tremendous conceptual breakthrough, linking the previously separate fields of geometry and algebra. Descartes showed that he could solve previously unsolvable problems in geometry by converting them into simpler problems in algebra. He represented the horizontal direction as x and the vertical direction as y. This concept is now indispensable in mathematics and other sciences.

Beginnings

René Descartes was born into a well-educated, upper-class family on March 31, 1596 in the French village of La Haye en Touraine. The village is now called Descartes, Indre-et-Loire in his honor.

René's father was Joachim Descartes, a lawyer at Brittany's Court of Justice. His mother was Jeanne Brochard, daughter of the Lieutenant General of Poitiers. René was their third child.

A year after René's birth his mother and her fourth child died during childbirth.

René's father's worked six months of the year in the Court of Justice in Rennes, about 200 miles (300 km) from their home. René was raised by his grandmother and his great-uncle. His father remarried when René was four and began living in Rennes permanently. Despite this, there was always affection between René and his father.

From birth René suffered poor health and had a permanent cough. Local doctors thought he would not survive infancy. His father employed a nurse who devoted herself to René's care. As an adult he believed his nurse saved his life - he paid her a permanent pension.

Education

At the age of about ten or eleven René was finally considered healthy enough to begin school. He boarded at the Jesuit School at La Flèche in Anjou. In a concession to his delicate health he was allowed to rise later in the morning than other students.

René spent seven or eight years at La Flèche learning logic, theology, philosophy, Latin and Greek. In his final two years he also learned mathematics and physics. The physics was that of Aristotle - almost entirely wrong.

He was a boy of prodigious curiosity, asking questions endlessly.

René learned something of Galileo's work including his recent amazing discovery of Jupiter's moons. At this time, Galileo had still not published his greatest works overturning Aristotle's physics; his trouble with the Catholic Church lay in the future.

At the age of 18, in 1614, René Descartes left La Flèche. Later he recorded his thoughts about his education as follows:

On Mathematics

On Philosophy

On the Sciences

The Law, the Army, and Mathematics

His father encouraged René Descartes to follow in his footsteps and study law. He did this, graduating from the University of Poitiers in 1616, aged 20, with a diploma and license in church and civil law.

Rather than becoming a lawyer, however, Descartes went traveling for about two years, including spending some time in Paris. He joined the Dutch States Army in Breda in 1618 and began studying engineering in a military academy. He soon met the Dutch scientist and philosopher Isaac Beeckman. In 1619 Descartes wrote Beeckman:

Descartes, aged 23, was now profoundly serious in his desire to make new discoveries in mathematics.

René Descartes' Contributions to Science

The Miracle

On November 10, 1619 Descartes was dozing in a warm, stove-heated room in the German town of Neuburg an der Donau.

There he had a series of dreams that would ultimately change the way scientists work. He believed a spirit sent by God gave him new ideas about:

The Scientific Method

Analytical Geometry

Philosophy

18 years later, in 1637, he published his ideas in Discours de la mèthode (Discussion of the Method), La Gèomètrie (Geometry), Les Mètèores (Meteorology), and La Dioptrique (Optics). The first two of these works contain his most significant contributions.

The Method

In Discussion of the Method Descartes shared his framework for doing science.

One of his main lines of thought was skepticism - that everything should be doubted until it could be proved.

His four main ideas for scientific progress were:

1. Never accept anything as true until all reasons for doubt can be ruled out.

2. Divide problems into as many parts as possible and necessary to provide an adequate solution.

3. Thoughts should be ordered, starting with the simplest and easiest to know, ascending little by little, and, step by step, to more complex knowledge.

4. Make enumerations so complete, and reviews so general, that nothing is omitted.

It is ironic that Descartes' own method might lead us to doubt that a dream 18 years earlier could have been the true source of his ideas!

Analytical Geometry

Descartes made the revolutionary discovery that he could solve problems in geometry by converting them into problems in algebra.

In La Gèomètrie he showed that curves could be expressed in terms of x and y on a two-dimensional plane and hence as equations in algebra.

The Cartesian coordinate system used in the image below is named in his honor. (Descartes' name in Latin is Cartesius.)

Descartes never actually drew an x- or y-axis in his work. These were assumed in his diagrams. The axes were formally introduced by the mathematician Frans van Schooten and other mathematicians in Leiden who translated La Gèomètrie from French into Latin, while developing it further. Latin editions of La Gèomètrie were released in 1649, 1659 and 1661.

Descartes also introduced the modern notation for exponents. For example, rather than writing a.a.a he would write a ³.

By unleashing the mathematical power of algebra to tackle problems in geometry, Descartes surpassed the expertise of Ancient Greece's brilliant geometers: he could now solve problems that had defeated them.

Analytical geometry was independently invented earlier by Pierre de Fermat, who lived in France at the same time as Descartes. Fermat worked on mathematics for his own pleasure and often kept his results private. He did, however, enjoy issuing challenges to other mathematicians to solve problems.

In 1638 Fermat sent a work entitled Introduction to Plane and Solid Loci to the mathematician, Marin Mersenne, to show how problems he had posed at an earlier date could be solved. Fermat's approach was different from Descartes'. Descartes showed how geometry could be expressed as algebra; Fermat showed how algebra could be expressed as geometry.

Influencing Isaac Newton and the Invention of Calculus

Calculus has been crucial to the progress of mathematics and the sciences. It was developed in the 1660s by Isaac Newton, and developed independently in the 1670s by Gottfried Leibniz.

In La Gèomètrie, Descartes showed how he could find tangents to curves. This process is a vital part of differential calculus. His mathematical competitor Fermat was also able to find tangents to curves; his methods were actually simpler than Descartes'. Both Descartes and Fermat helped guide Newton and Leibniz's development of calculus.

Silenced by Fear of the Church

Four years before he released his 1637 works, Descartes had intended publishing The World.

In 1633, however, he learned that the Catholic Church had tried Galileo for heresy and sentenced him to life in prison. This was reduced to permanent house arrest because Galileo was rather elderly. The Church also prohibited Galileo's works.

Descartes, like Galileo, believed the sun sits at the center of the solar system. He decided not to risk the Church's wrath and did not publish The World. (Galileo could have been burned at the stake if his trial had gone worse than it did.)

Philosophy

Descartes is regarded as one of the greatest philosophers of all time. Here we are concerned with science rather than philosophy, so we will restrict ourselves to noting his most famous declaration:

"I think therefore I am."

This could also be expressed as:

"I can think, therefore I exist."

Descartes regarded this statement as the unshakeable foundation that all other philosophy could be built upon.

His most famous philosophical work is Meditations on First Philosophy, published in 1641.

Deducing the Laws of Nature from First Principles

Descartes' most comprehensive work, Principles of Philosophy, was published in 1644. In it he tried to deduce all of nature's laws from first principles. Although the book had much to commend it to philosophers, its science was incorrect.

He argued that action at a distance is impossible and agreed with the Ancient Greek philosopher Aristotle that there could be no vacuum. Soon, however, all his power as a philosopher would be defeated by scientific experiments.

An experiment is a question which science poses to Nature, and a measurement is the recording of Nature's answer.

Max Planck, Theoretical Physicist

In 1654 Otto von Guericke constructed the first vacuum pump. In 1662 Robert Boyle demonstrated that the magnetic force can travel through a vacuum, establishing that action at a distance is possible.

Unfortunately Descartes did not live long enough to learn of these developments.

Some Personal Details and the End

Descartes was wealthy enough to pursue his own interests. His father gifted him a number of properties which Descartes, at the age of 24, sold. This raised enough money for him to live on comfortably for the rest of his life.

Descartes spent much of his life on the move. He lived for 20 of his later years in various locations in Holland. He also studied and taught mathematics there. He found he could work better in Holland, with fewer distractions than in France.

Although he never married, in 1635, aged 39, he became a father. His partner was an Amsterdam servant, Helena Jans van der Strom. Their daughter was named Francine. Mother and daughter lived with Descartes in his house - he told people Francine was his niece. He planned to educate his daughter in France, but sadly, at the age of five, she died of scarlet fever. Francine's mother later got married, with Descartes playing a fatherly role by paying the wedding dowry.

In 1649 Descartes was invited to Stockholm by Queen Christina of Sweden. She wanted him to set up a new academy of science.

René Descartes died, aged 53, of pneumonia in Stockholm on February, 11 1650. He was buried at the Adolf Fredriks Church in Stockholm.

In death, as in life, Descartes was mobile. 16 years after his first burial his remains were moved and buried in the Saint-Ètienne-du-Mont church in Paris, France. In 1819 his remains minus skull and finger were moved again, this time to the Abbey of Saint-Germain-des-Près in Paris, where he now rests.

In 1663, despite his efforts to avoid such a fate - he regarded himself as a devout Catholic - a number of Descartes' works joined Galileo's on the index of books prohibited by the Catholic Church. Over 300 years later, in 1966, this index was finally discontinued.

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Further Reading
René Descartes
The Method Meditations and Philosophy
M. Walter Dunne, Washington and London, 1901

Elizabeth S. Haldane
Descartes: His Life and Times
John Murray, London, 1905

René Descarte, translators David E. Smith and Marcia L. Latham
The Geometry
Dover Publications Inc. 1954

Tom Sorell
Descartes (Past Masters)
Oxford University Press, 1987

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Introduction

One of the most important threads in seventeenth century mathematics was the introduction of infinitesimal methods for solving area and volume problems. Foremost among these newfound techniques was Cavalieri's Principle, a creation of Bonaventura Cavalieri (1598-1647) that used Eudoxus' theory of proportions (as found in Book V and Book VI of Euclid's Elements) to study infinitesimal slices of objects and derive a proportion between the objects themselves. Ironically, when mathematicians in the seventeenth century wanted to understand Cavalieri's method, often they didn't turn to the work of Cavalieri himself. Instead, they turned to the work of his contemporary Evangelista Torricelli (1608-1647). Torricelli's sole published work, hisOpera Geometrica (1644), was a lengthy and wide-ranging tome, but one of the most important sections for understanding Cavalieri's method was de Dimensione Parabolae.

De Dimensione Parabolae consisted of two parts--both of which were devoted to alternate proofs of the classical result fromQuadrature of the Parabola by Archimedes of Syracuse (287-212 BCE). The first part of de Dimensione Parabolae was Quadratura Parabolae Pluribus modis per duplicem positionem, more antiquorum, absolutae ("The Quadrature of the Parabola solved with many methods through a two-fold placing in the manner of the ancients"). As the name implies, this section was devoted to finding the area of a segment of the parabola using Archimedean techniques--i.e., the method of exhaustion. However, in the second section, Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta ("The Quadrature of the Parabola solved with many methods through the new geometry of indivisibles"), Torricelli presented 11 Propositions which gave unique and wide-ranging approaches to solving the quadrature of the parabola, all of which involve Cavalieri's method or similar infinitesimal methods. As Torricelli himself noted, in the process he demonstrated that

it is certain that this wonderful geometry is a shortcut for invention, and that it confirms countless almost inscrutable theorems with brief, direct, and affirming demonstrations, which is certainly not able to be done easily through the ancient teaching. To be sure, this is truly the Royal Road in the mathematical thorn hedges, that Cavalieri, creator of these wonderful inventions, first among everyone opened up and made public for the common good [Torricelli, p. 56].

The purpose of this paper is to give a translation of Torricelli's Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta. To put the work in context, we will first look briefly at the 17th century mathematical figures who provided a milieu for its development. Then, following an overview of the results in Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta, we will present a slightly modernized introduction to Archimedes' Quadrature of the Parabola--the work that was the focus of Torricelli's de Dimensione. With an eye toward understanding how Torricelli's work might be used in a classroom today, we also look closely at an example from the translation which, while not strictly an application of Cavalieri's method, demonstrates just how conceptually different these new infinitesimal methods were from the traditional approach presented in Quadrature of the Parabola that had been the gold standard for mathematical rigor for centuries.

Some historical background

It's impossible to talk about Torricelli without introducing some other figures who were undoubtedly his most important mathematical influences: Galileo Galilei, Bonaventura Cavalieri, and the relatively unknown figure of Benedetto Castelli. Galileo (1563-1642) was, of course, the famous Italian scientist whose trouble with the Roman Inquisition is discussed even to this day. Cavalieri (1598-1647) was the 17th century mathematician whose work formalizing indivisibles would be the basis for Torricelli's work in de Dimensione Parabolae. However, it was Castelli (1578-1643) who would undoubtedly have the most important social and mathematical impact on Torricelli's life.

Benedetto Castelli

Born Antonio Castelli to relatively wealthy parents in Brescia, Castelli took the name Benedetto when he entered the Benedictine monastic order in 1595. Eventually, he was transferred to a monastery in Padua, where Galileo had been teaching since 1592. Castelli became Galileo's student, and eventually the two became close friends, as evidenced by their frequent correspondence after Castelli was transferred to another monastery near Naples in 1607. In 1610, when the publication of Sidereus Nuncius enabled Galileo to obtain a position as Chief Mathematician to the Medicis with a nominal position in mathematics at the University of Pisa, Castelli quickly obtained a transfer to a Benedictine Abbey in Florence. In 1613, with Galileo's help, Castelli too obtained a professorship in mathematics at the University of Pisa, where he remained until 1626, when he took a position teaching mathematics and solving problems in hydraulics for the Pope in Rome. Significantly, there was no Benedictine monastery in Pisa, so Castelli resided in a Jesuati monastery which from 1616 to 1620 was also the home of a young Jesuati named Bonaventura Cavalieri. Castelli was widely regarded as an outstanding teacher, and it was from Castelli that Cavalieri learned much of his mathematics. Castelli also introduced Cavalieri to Galileo, with whom Cavalieri exchanged more than 100 letters.

One of the interests these men shared was a mathematical technique called the method of indivisibles, which determined ratios between plane or solid figures by dividing them up into parallel, indivisible slices and studying the ratios of these slices. They weren't the first to stumble on this approach. (Unbeknownst to seventeenth century mathematicians, Archimedes had discovered a similar technique in his lost work on The Method of Mechanical Theorems. Johannes Kepler (1571-1630), who discovered the laws of planetary motion and was a Galileo correspondent, had also utilized similar ideas in his Nova stereometria doliorium vinariorum.) But they were probably the ones most responsible for promoting it. In his Dialogues Concerning Two New Sciences (1638) Galileo demonstrated the equality between the volume of a "soup bowl" AFBED of revolution to a cone DCE of the same height by demonstrating that on each planar slice perpendicular to the shared axis CF of the figures, the annulus on ON had the same area as the circle on radius PL.

(See [Baron, p. 119] and [Katz, p. 515].) Cavalieri's Geometria indivisibilibus continuorum nova quadam ratione promota was intended to be the most thorough exposition of the theory of indivisibles that was available at the time. However, as Andersen notes, it was "so difficult that [nineteenth century mathematical historian] Maximilien Marie suggested that if a prize existed for the most unreadable book, it should be awarded to Cavalieri for Geometria" [Andersen, p. 294].

Evangelista Torricelli

Torricelli was a latecomer to this group of mathematicians. Born in 1608 in Faenza to a family with limited means, his education was overseen by his uncle Jacopo, a Benedictine monk. He entered a Jesuit college in 1624 and, when Castelli moved to Rome in 1626, Torricelli spent the next six years as both Castelli's student and his secretary. From 1632 to 1641, Torricelli was secretary to Giovanni Ciampoli, another close friend of Galileo, and in 1641, three month's before Galileo's death, Torricelli joined Galileo as an assistant in Arcetri (across the Arno from Florence), where Galileo had been living under house arrest since his run-in with the Inquisition in 1633. After Galileo's death, Torricelli would go on to hold his position as court mathematician to the Medicis. Though he is now most famous for his discovery of barometric pressure, his only published work, the Opera Geometrica of 1644, which contained de Dimensione Parabolae and several other works, was very well regarded at the time. In particular, because Cavalieri's work on indivisibles was so difficult to understand, Torricelli's Opera Geometrica, and in particular Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta, was often the source from which mathematicians learned about the method of indivisibles--hence its importance in the history of mathematics.

Overview of the Translation

As noted above, the Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta is the second half of de Dimensione Parabolae. The original Latin version (see here or here) runs for 29 pages and consists of an introduction and Propositions 11 through 21 of de Dimensione Parabolae, interspersed with an additional 14 lemmas and several scholia and "Aliter" sections. Torricelli notes his focus at the outset of this section:

Until now, the matter about the measurement of the parabola has been related in the manner of the ancients. It remains that we should approach the same measurement of the parabola with a certain new but marvelous system--namely, by the aid of the Geometry of Indivisibles and with diverse methods in this manner.

There is no formal subdivision of the section, but the "diverse methods" that Torricelli cites can be roughly subdivided as follows:

Propositions 11-13: Solving the quadrature of the parabola through proportions from classical geometry. For Proposion 11, Torricelli uses Euclid's proportion (Euclid XII.10) between the cone and the cylinder. For Proposition 12, he uses Euclid's result (Euclid XII.2) that two circles are in the same ratio as the squares of the diameters (see Lemma 20 of Torricelli). For Proposition 13, he uses Archimedes' proportion (see [Heath, p. 43]) between the sphere and the cylinder.

Proposition 14: Solving the quadrature of the parabola using Archimedes' Equilibrium of Planes I, which determines the center of gravity of a triangle (See [Heath, p. 201]). He also makes use of a more recent result on the location of the center of equilibrium of a cone, which had been established by Federico Commandino (1506-1575) in his Liber de Centro Gravitatis Solidorum (1565).

Propositions 15-16: Solving the quadrature of the parabola by several approaches which generalize Archimedes' geometric summation method found in Quadrature of the Parabola to an infinite geometric series. This will be discussed in the section below.

Proposition 17: Solving the quadrature of the parabola using Archimedes' proportion between the first turn of a spiral and the circumscribing cylinder. Logically, this belongs with Propositions 11-13, but it requires a more general version of Cavalieri's Principle that Torricelli develops in Lemma 29.

Proposition 18-21: Solving the quadrature of the parabola using centers of gravity. Determining centers of gravity was a major area of research in the sixteenth and seventeenth centuries, and there were several major results extending the results of Archimedes (in Plane Equilibrium I and II), who determined the center of gravity of the triangle and the parabola. Proposition 18 finds the quadrature of the parabola with the center of gravity of a triangle and Proposition 19 finds the quadrature with the center of gravity of a parabola. It makes use of a very brief proof (Lemma 30) which shows how infinitesimals can demonstrate quite easily Archimedes' result on the center of gravity of a parabola found in On the Equilibrium of Planes II. Lemma 31 is the first published proof of the Pappus-Guldin Theorem, and it is used in Proposition 20 to establish a proof of the quadrature of the parabola using the Pappus-Guldin Theorem. Proposition 21 presents a quadrature of the parabola very similar in spirit to Proposition 1 of Archimedes' On The Method (which was unknown at the time), and Proposition 22 presents a quadrature proof using the center of gravity of a hemisphere, which Luca Valerio (1552-1618) had established in his work de Centro Gravitatis Solidorum libri tres (1603).

Some Prerequisites for Reading the Translation

De Dimensione Parabolae builds on the geometrical tradition that mathematicians in the seventeenth century inherited from Euclid and Archimedes. The primary goal of the work is to establish equivalences between many classical results in geometry and Archimedes' well-known result from the Quadrature of the Parabola, so it's necessary to have some familiarity with these results. These, in turn, rely on results in Book V and Book VI of Euclid's Elements. However, our translation begins in the middle of a larger work, and Torricelli sometimes refers to earlier results in this work. For the sake of completeness, we summarize these here. More details on the mathematical background of this work can be found in [Leahy, pp. 175-177].

Lemma 3: If a parabola were to have three tangents, two at the base and a third through the vertex, the triangle comprised by the tangents will be eight times the triangle which arises from a fourth tangent having been drawn through the vertex of either semiparabola.

Lemma 3 is referenced in Proposition 15, where Torricelli generalizes Archimedes' result expressing the quadrature of the parabola in terms of a triangle inscribed inside of a parabola to the triangle comprised of the three given tangent lines to the parabola.

Lemma 7: If a triangle having the same base and the same altitude as the parabola is inscribed in a parabola, two other triangles are also inscribed in like manner in the remaining portions. The triangle inscribed first will be eight times either of the triangles inscribed afterwards.

As Torricelli notes, "This Lemma is demonstrated by Archimedes in Proposition 21 of Quadrature of the Parabola". It will also be demonstrated by us below.

Lemma 11: An entire semiparabola hangs at equilibrium from a point on the base in which it is thus divided so that the part terminated at the curve is to the remaining part as five is to three.

Lemma 11 generalizes Archimedes' result on the center of gravity of a segment of the parabola found in Book II of On the Equilibrium of Planes.

Lemma 18: If a first magnitude were to a second thus as a third is to a fourth, and thus however many times it is pleasing, and if all of the firsts and also all of the thirds are equal among themselves, then all of the first magnitudes together are to all of seconds together, as all of the thirds together are to all of the fourths.

Lemma 18 is Torricelli's version of Cavalieri's Principle. It is generalized in Lemma 29, which is stated and proved in the translation below.

Torricelli also cites, e.g., "the explanation of Proposition 9" at several points, referring to the proof of this Proposition. This proof establishes a result (also established in Propositions 16 and 17 of Archimedes' Quadrature of the Parabola [Heath, pp. 244-246]) which is equivalent to Archimedes' result on the area of a parabola: Construct the "tangent triangle" to the segment of the parabola, consisting of the tangent line \(DQ\) at one end of the base \(Qq\) of the segment of the parabola, the base itself, and a line \(Dq\) parallel to the diameter \(PV\) through the other endpoint \(q\) of the base.

Then \( {1 \over 4} \Delta DqQ = \Delta QPq\). Why? Proposition I.35 of the Conics of Apollonius of Perga (262-190 BCE) [Apollonius, p. 62] shows that \(TP = PV \). Thus, triangles \( \Delta TQV\) and \( \Delta QPq\) have the same area. A similarity argument shows that \( \Delta DqQ\) has four times the area of \( \Delta TVQ\). Thus, \( {1 \over 4} \Delta DqQ = \Delta QPq\), and so to show that \( area(par(QPq)) = {4 \over 3} \Delta QPq\) it is necessary and sufficient to show that \( area(par(QPq)) = {1 \over 3} \Delta QDq\).

Archimedes' Quadrature of the Parabola

Torricelli's de Dimensione establishes the amazing result that Archimedes' computation of the area of a segment of a parabola is logically equivalent to most of the major geometrical results known at the time. Given its centrality to Torricelli's work, it's useful to review in modern terms the key result in Archimedes'Quadrature of the Parabola, which determines the area of a segment of a parabola in terms of an inscribed triangle:

Proposition 24 of QP: Every segment bounded by a parabola and a chord \(Qq\) is equal to four-thirds of the triangle which has the same base as the segment and equal height [Heath, p. 251].

Archimedes' proof of this result (see also, e.g., [Katz, pp. 108-109] or [Boyer, pp. 52-53]) is an application of the method of exhaustion, which was the traditional approach among Greek mathematicians for solving area and volume problems. The proof also hinges on what is essentially a summation formula for a finite geometric series and an ingenious observation about how triangles can be inscribed inside a segment of a parabola. This result about inscribed triangles in turn depends on two more fundamental properties about the nature of the parabola which can be found in [Heath, p. 248]:

1. Given any chord \(Qq\) in a parabola and the point \(P\) on the parabola where the tangent at \(P\) is parallel to \(Qq\), the line \(PV\) parallel to the axis of the parabola bisects \(Qq\).

2. Given \(Qq\) and \(PV\) as above, if the points \(W\) on \(PV\) and \(R\) on the parabola are such that \(RW\) is parallel to \(Qq\), then \[ {PV \over PW} = {QV^2 \over RW^2}. \]

With these results in hand, Archimedes constructs a sequence of successively smaller and non-overlapping triangles inscribed inside of the segment of the parabola. Starting with the original triangle \(\Delta PQq\) from the proposition, another triangle \(\Delta PRQ\) with one-eighth the size of \(\Delta PQq\) and inscribed in the parabola along the side of \(PQ\) of \(\Delta PQq\) is constructed as follows: With \(Qq\) bisected at \(V,\) consider the point \(M\) bisecting \(QV\) and the point \(R\) on the parabola such that \(RM\) is parallel to \(PV\) (and hence parallel to the axis of the parabola itself by definition of \(PV\)). Label the intersection of \(RM\) and \(PQ\) by \(Y\) and let \(RW\) be parallel to \(Qq\).

It follows that \(RM\) is 3/4 of \(PV\). Why? From property (2) above, \[ {PV \over PW} = {QV^2 \over RW^2} = {(2 MV)^2 \over RW^2} = 4 {RW^2 \over RW^2} = 4, \] because \(RMVW\) is a parallelogram (and thus \(MW = RV\)). This implies that \( {1 \over 4} PV = PW\) and so \( RM = WV = PV - PW = {3 \over 4} PV\). Now note that the triangles \(\Delta QYM\) and \(\Delta QPV\) are similar, because \( RM\) and \( PV\) are parallel. Since \( QM = {1 \over 2} QV\), this similarity implies that \( YM = {1 \over 2} PV\) as well. Hence \(YR = RM - YM = {1 \over 4} PV\) and so \(YM = 2 YR\). Thus, \(\Delta PRQ\) and \(\Delta PMQ\) have the same base (\( PQ\)), but the height of \(\Delta PMQ\) is twice the height of \(\Delta PRQ\), and it follows that in terms of area \(\Delta PMQ = 2 \Delta PRQ\).

Likewise, \(\Delta PQM\) and \(\Delta PQV\) have the same height but \(\Delta PQV\) has twice the base of \(\Delta PQM\), so we also have \(\Delta PQV = 2 \Delta PQM\) in terms of area. A similar argument shows that \(\Delta PQq = 2 \Delta PQV\). It follows that \( \Delta PRQ = {1 \over 8} \Delta QPq\), because \[ 8 \Delta PRQ = 4 \Delta QPM = 2 \Delta PQV = \Delta QPq. \] A triangle \(\Delta Pqr\) with the same ratio to \(\Delta PQq\) can likewise be constructed on the side \(Pq\), so altogether the original triangle has four times the total area of the two newly-constructed triangles.

Also, note the geometrically obvious fact that the area of the three triangles together make an even better approximation to the area of the segment of the parabola (which we will denote by \( par(PQq) \) in what follows) than the first triangle alone.

But why stop there? Each line \(PQ\) and \(Pq\) is itself the base of a segment of the parabola and \(rm\) and \(RM\) are parallel to the axis of the parabola (just like \(PV\)), so the same construction can be applied to the sides of \(\Delta PRQ\) and \(\Delta Prq\), resulting in four new triangles with combined area \((1/4)^2 \Delta PQq\). Obviously, the sum of these seven triangular areas is an even better approximation to the area of the segment of the parabola. And so on. As Archimedes will show eventually (see the discussion of the method of exhaustion below), the difference between the sums of these triangular areas and the area of the sector of the parabola is going to zero as \(n\) gets larger. At the \(n\)th step of the approximation another \( (1/4)^n \Delta PQq\) is added to the approximation, so the total area of the approximation after \(n\) stages is the finite geometric series \[ \Delta PQq + (1/4) \Delta PQq + (1/4)^2 \Delta PQq + \cdots + (1/4)^n \Delta PQq. \]

The next observation in Archimedes' proof is essentially (from our perspective today) an algebraic identity that was designed to understand how this finite geometric series determined by the triangular areas grows as more triangles are added. Note first that for any quantity \(A\), simple arithmetic shows that \[ {1 \over 4} A + {1 \over 3}\cdot {1 \over 4} A= {1 \over 4} A + {1 \over 12} A = {1 \over 3} A. \] Next, consider a finite sequence \(A_1, A_2, \ldots, A_n\) such that \(A_i = {1 \over 4} A_{i-1}\). Since \({1 \over 4} A_i +{1 \over 12} A_i = {1 \over 3} A_i\) for each \(i\), \[ {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + {1 \over 12} A_1 + \cdots + { 1\over 12} A_n = {1 \over 3} A_1 + \cdots + {1 \over 3} A_n. \] But since \(4 A_i = A_{i-1}\) for \(i = 2, 3, \ldots n\), we have \({1 \over 12} A_{i-1} = {1 \over 3} A_i\) and this equation can be rewritten as \[ {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + {1 \over 3} A_2 + \cdots + { 1\over 3} A_{n+1} = {1 \over 3} A_1 + \cdots + {1 \over 3} A_n. \] By canceling, this becomes \[ {1 \over 4} A_1 +\cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} = {1 \over 3} A_1. \] Adding \(A_1\) to both sides, we have Archimedes' geometric summation formula: \[ A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} = {4 \over 3} A_1. \]

If \(A_1 = area(\Delta PQq)\), then \((1/4) A_1\) is the area of the triangles added at the second step, \((1/4)A_2\) is the area of the triangles added at the third step, and so on. So from a modern perspective, the identity \[ A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} = {4 \over 3} A_1 \] makes Archimedes' area formula geometrically obvious. The right-hand side never changes as \(n \rightarrow \infty\). On the left-hand side, since \(A_{n+1} = (1/4)^n \Delta PQq\), \({1 \over 3} A_{n+1} \rightarrow 0\) as \(n \rightarrow \infty\). The remaining terms are the sum of the inscribed, non-overlapping triangular areas, which converge to the area of the parabola. Thus, Archimedes' area formula \( par(PQq) = {4 \over 3} \Delta PQq\) follows.

However, Archimedes would never take that step. Greek mathematicians, perhaps stung by the criticisms of Zeno of Elea (490-430BCE) in his paradoxes (or perhaps not--see [Baron, p. 22-25]), did not use infinite processes in formal proofs and never developed the concept of the limit. Instead, Archimedes employed the method of exhaustion, a technique found first in Antiphon, later perfected by Eudoxus, and adopted by Archimedes and others [Boyer, p. 32]. This was the standard method of solving area and volume problems for Greek mathematicians and, unlike our modern approach, doesn't require passage to the limit at any step. But it pays for this conceptual clarity with a lengthy double reductio ad absurdum proof, and that's exactly the sort of proof that Archimedes submits in the Quadrature of the Parabola.

In particular, to show that the area equation \(par(QPq) = {4 \over 3} \Delta QPq\) is true, Archimedes supposes that the equation isn't true. Then there are two possible relations between the areas:

1. \(par(QPq) > {4 \over 3} \Delta QPq\)

2. \(par(QPq) < {4 \over 3} \Delta QPq\).

Both of these will be shown to lead to a contradiction.

To show that \(par(QPq) > {4 \over 3} \Delta QPq\) yields a contradiction, Archimedes notes that since each inscribed triangle is greater than half of the parabolic segment in which it lies (see [Heath, p. 248]), the sum of the triangular areas \(A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n\) can be made as close to \(par(QPq)\) as we want for large enough \(n\). In particular, since \({4 \over 3} \Delta QPq < par(PQq)\) by assumption, we must also have \[ {4 \over 3} \Delta QPq < A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n \] for some finite number \(n\). However, \[ A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n < A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} \] and his algebraic identity for the geometrical series shows: \[ A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} = {4 \over 3} A_1 = {4 \over 3} \Delta PQq \] and so \({4 \over 3} \Delta QPq < {4 \over 3} \Delta QPq\). Hence a contradiction.

To show that \(par(QPq) < {4 \over 3} \Delta QPq\) yields a contradiction, note that since \(A_{n+1} = \left({1 \over 4}\right)^{n} A_1\), it follows that \(A_{n+1} \) can be made arbitrarily small for some finite \(n\). Since (by assumption) \(par(QPq) < {4 \over 3} A_1\), we have \[ A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n + { 1\over 3} A_{n+1} = {4 \over 3} A_1 > par(QPq). \] Because \(A_{n+1}\) can be made arbitrarily small, this shows we must eventually have \( A_1 + {1 \over 4} A_1 + \cdots + {1 \over 4} A_n > par(QPq)\) for some finite \(n\). Hence another contradiction--this time of the fact that each of the non-overlapping regions \(A_1, \ldots, A_n\) is inscribed in the segment of the parabola \(par(QPq)\), so their sum must be smaller than the area of \(par(QPq)\).

By trichotomy, \(par(QPq) = {4 \over 3} \Delta QPq\) is the only possibility.

Torricelli's Quadrature of the Parabola (One Version)

The method of exhaustion was a logically sound approach to solving area and volume problems. However, as Boyer notes, "the cumbersomeness of its application led later mathematicians to seek a more direct approach to problems in which the application of some such procedure would have been indicated" [Boyer, p. 35]. In the seventeenth century, Cavalieri's method was the premier example of this "more direct approach", and in the section of de Dimensione Parabolae, given in translation below, Torricelli presented eleven different examples of how to find the quadrature of the parabola using Cavalieri's method and other infinitesimal methods. Modern interpretations of some of these Cavalierian approaches have already been given in [Anderson, p. 356] and [Leahy, pp.178-183]. As a specific example we will discuss another approach found in Proposition 15 of the work. (See also [Baron, p. 183-185] for another discussion of this result.)

What was noteworthy about the approach in Proposition 15 was that it directly extended Archimedes' own technique for finding the quadrature of the parabola. More precisely, whereas Archimedes stopped after finitely many terms of his geometric series and let the method of exhaustion take care of the rest, Torricelli used a geometrical argument to demonstrate how to add the infinitely many terms of the same geometric series and arrive at the same area formula.

To justify this infinite summation Torricelli introduced a geometrical figure he called a flexilineum, which was constructed as follows: Consider two lines \(B_1D\) and \(C_1D\) that intersect at \(D\), and a sequence of points \(B_1, B_2, \ldots\) on \(B_1D\), and \(C_1, C_2, \ldots\) on \(C_1D\) such that for \(i = 1, 2\ldots \) the segments \(B_iC_i\) are mutually parallel, as are the segments \(B_{i+1}C_i\). The flexilineum consisted of the infinitely many conjoined segments \(B_1 C_1 B_2 C_2 \ldots\)

Torricelli made several observations about the flexilineum that were key to his proof. First, he noted that the terms of the sequence of lengths \(B_iC_i\) are in a continuous geometric proportion. That is, \[ {B_1 C_1 \over B_2 C_2} = {B_2 C_2 \over B_3 C_3} = {B_3 C_3 \over B_4 C_4} = \cdots \] To see why, note that since both families of line segments \(B_i C_i\) and \(B_{i+1} C_i\) are parallel among themselves, Euclid VI.2 implies that for each \(i\) \(\Delta B_i C_i D\) is similar to \( \Delta B_{i+1} C_{i+1} D\), and \(\Delta B_{i+1} C_i D \) is similar to \(\Delta B_{i+2} C_{i + 1} D\). So by similar triangles we have \[ \eqalign{ {B_i C_i \over B_{i +1} C_{i +1}} &= { C_i D \over C_{i+1} D} \cr &= {B_{i+1} D \over B_{i+2} D} \cr &= {B_{i+1} C_{i+1} \over B_{i+2} C_{i+2}} \cr} \] and the result follows. A similar argument shows that the line segments \(B_{i+1} C_i\) are also in a continuous geometric proportion.

Torricelli's second and most important result about the flexilineum was the derivation of a geometrical representation for the sum of the lengths of the infinitely many line segments \(B_i C_i\). To find this sum geometrically, he extended each of the segments \(B_{i+1} C_i\) so that they intersected the extension of \(B_1 C_1\) in a point we will call \(D_{i+1}\). (Note: \(D_1 = B_1\) and \(D_2 = C_1\) by definition as well.) Similarly, he let the line parallel to all the \(B_{i+1} C_i\) and passing through \(D\) intersect the extension in a point \(L\).

All of the lines \(B_i D_i\) are parallel to each other. By assumption, the lines \(B_i C_i\) are parallel to each other as well. Consequently the quadrilateral \(D_i B_i C_i D_{i+1}\) is a parallelogram, so \(D_i D_{i+1} = B_i C_i\) and the length of the segment \(B_1 L\) (that is, the infinite sum of the lengths of all of the \(D_i D_{i+1}\) segments) is clearly the same as the infinite sum of the lengths of all of the segments \(B_i C_i\).

Torricelli was so concerned to get this proof right that he spent several pages filling in details. He noted, for instance, that even though the proof deals with ratios of lengths, it is still valid for ratios of any other type of magnitude [Torricelli, p. 67]. It's also an easy exercise to show that given any ratio \(a/b\) it's possible to construct a flexilineum such that \( B_i C_i / B_{i+1} C_{i+1} = a/b\).

Returning to the original problem of the quadrature of the parabola, let \( B_i C_i\) represent \(A_i\), the area of the triangles added at the \(i^{th}\) stage of Archimedes' construction. Then by Archimedes' construction the segments \(B_i C_i\), \(i = 1,2,\ldots\), are in a continuous geometrical proportion in a ratio of 1 to 4. That is, \[ {1 \over 4 } = {B_1 C_1 \over B_2 C_2} = {B_2 C_2 \over B_3 C_3} = \cdots \] With this interpretation, Torricelli's flexilineum showed that the sum of the areas \(A_1, A_2, \ldots\) (i.e., the area of the segment of the parabola) is the same as the length \(B_1 L\).

The final step in Torricelli's proof was to compute \( B_1 L\). To find the length of \(B_1 L\), he established a proportion between:

1. the first length \(B_1 C_1\) (which is the area of the first triangle \(\Delta QPq\) in Archimedes' construction)

2. the extended length \(B_1 L\) (which is the area of the segment \(par(QPq)\) of the parabola)

3. the difference \(B_1 C_1 - B_2 C_2\) (which is the difference between the first triangle and the triangles added at the second step--that is, \( {3 \over 4} \Delta PQq\)).

To see this proportional relation, note that if \(V\) is placed on \(B_1 L\) so that \(V B_2\) is parallel to \(C_1 D\), then \(V C_1 = B_2 C_2\), since \(V C_1 C_2 B_2\) is a parallelogram.

Likewise, we have \(\Delta B_1 V B_2\) similar to \( \Delta B_1 C_1 D\) and \(\Delta B_1 C_1 B_2\) similar to \(\Delta B_1 L D\). With some work, the similarity of these triangles can be used to show: \[ {B_1 C_1 - B_2 C_2 \over B_1 C_1} = {B_1 V \over B_1 C_1} = {B_1 B_2 \over B_1 D} = {B_1 C_1 \over B_1 L}, \] which we can solve for \[ B_1 L = { (B_1 C_1)^2 \over (B_1 C_1 - B_2 C_2)}. \] In terms of the quadrature of the parabola, this means \[ par(PQq) = {(\Delta PQq)^2 \over {3 \over 4} \Delta PQq} = {4 \over 3} \Delta PQq, \] and thus Torricelli derived Archimedes' quadrature formula for a segment of a parabola.

In the Classroom and Conclusion

In the Classroom

Solutions to area problems are some of the most fundamental and easy-to-grasp results in calculus. The Fundamental Theorem of Calculus is an amazing tool for solving these problems, but it isn't the only tool. Historically, the quest to solve general area problems can be traced to Archimedes' particular solution to the area problem for a parabola, and there is a rich history of diverse methods for solving this specific area problem which stretches from ancient Greece to the modern calculus. If a detailed discussion of Archimedes' solution to the area problem for the parabola is included in a history class, the Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta can also provide examples of the many different approaches to solving this particular problem, and so is an important original source in the history of calculus. It is also accessible to any student who is comfortable with Books V and VI of Euclid's Elements and has experience reading original sources up to the level of Archimedes. Here are some specific examples of how it might be used as an original source in an upper level history of mathematics class:

For those students who have read Archimedes' Quadrature of the Parabola, in particular the proof of the area formula given in Propositions 21-14, Torricelli's Lemmas 24-27 and Proposition 15 present a proof identical in spirit but without resort to the method of exhaustion. What is the purpose of Torricelli's painstaking attempt to justify his infinite summation in geometrical terms? How does this contrast with the method of exhaustion that Archimedes utilized? Does it lack anything in rigor? (See the section "Torricelli's Quadrature of the Parabola" above for a modern explanation of how Torricelli was attempting to wrestle with infinite summations.)

As an extension of the previous exercise, Torricelli's Lemma 28 (with the accompanying "In Another Way" discussion) and Proposition 16 give a nearly identical proof of Archimedes' result using triangles that are tangent to the parabola instead of inscribed in it. What would an Archimedean proof using the method of exhaustion for this result look like?

For those students who have read the first Proposition of Archimedes' The Method, Torricelli's Lemmas 32-33 and Proposition 20 should be familiar, because (with some minor differences) Torricelli's proof is the same as the proof Archimedes' gives in the method. Has Torricelli indepedently rediscovered Archimedes'Method more than 250 years before it finally came to light again in 1906?

Those students who have read the proof of Proposition 8 of Book II of Archimedes' On the Equilibrium of Planes know that it depends on the formula for the area of a segment of a parabola given inQuadrature of the Parabola. Torricelli's Proposition 30 demonstrates (using the Law of the Lever as found in Book I ofOn the Equilibrium of Planes) that conversely one can find the area formula for a segment of a parabola once one knows the center of gravity of a parabola (a result which Torricelli proves quite easily using infinitesimals in Lemma 30 of his work). Do you think Archimedes was aware that the two results were equivalent?

Perhaps the most important part of Torricelli's work is that it articulates Cavalieri's method of indivisibles better than Cavalieri himself did. Moreover, whereas Cavalieri restricted himself to comparing figures having equal altitudes, Torricelli used indivisibles much more freely. Those students who have read about Cavalieri's Principle at the level of, say, [Katz, p. 514-517], would benefit from reading Torricelli's most general expression of Cavalieri's Principle as it is found in Lemma 29. How does Torricelli's statement of Cavalieri's Principle compare to Cavalieri's own statement?

Immediately following Lemma 29, Torricelli uses this version of Cavalieri's Principle to show (in Proposition 17) that Proposition 24 of Archimedes' On Spirals can be used to find Archimedes' area formula for a parabolic segment. Was Archimedes aware that these two results were also equivalent?

One issue that students may have with our translation is that we have made no attempt to modernize the language of proportionality used in the text. Students may find it easier to read if they understand how to make the translation to modern fractional notation themselves. Thus, the proportion "DF will be to FB as CE is to DF" may be easier to understand when written as "DF/FB = CE/DF". For a model of how to construct modern interpretations such as this of some of Torricelli's arguments, see [Leahy, pp.178-183].

Conclusion

Infinitesimal methods for solving area and volume problems were some of the most important developments in mathematics during the seventeenth century. Cavalieri's method of indivisibles was the most well known of these techniques, but his works were not widely understood. As Andersen notes, Torricelli is "an important link between Cavalieri's method and the general understanding of it", and his sole work, Opera Geometrica, in particular was "influential in spreading knowledge of the method of indivisibles" [Andersen, p. 356]. Among the various parts of this work, Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta stands out as one of the most clear, concise, and compelling presentations of Cavalieri's method that was available at the time. We hope that the translation presented here wil aid in understanding the importance of this work.

Acknowledgments and References

Acknowledgments

The authors of the translation would like to thank Brenda Fineberg for her assistance with this translation. We would also like to acknowledge the indirect support of Google, which made the original Latin version of Torricelli's Opera Geometrica available through books.google.com. The original Latin is also available here. The authors would also like to thank the reviewers, whose meticulous reading of the translation and its preamble found numerous errors and yielded many ideas for improvement.

References

1. Kirsti Andersen, Cavalieri's method of indivisibles, Archive for History of Exact Science 31 (1985) 291-367.

2. Apollonius, Densmore, Dana, ed. Taliaferro, Catesby, trans. Conics Books I-III. Santa Fe, NM: Green Lion Press, 1998.

3. Archimedes, Thomas Little Heath, ed. The Works of Archimedes. Dover, New York, 2002.

4. Margaret E. Baron, The Origins of the Infinitesimal Calculus, Pergamon Press, Oxford, 1969.

5. Boyer, Carl B. The History Of The Calculus And Its Conceptual Development. New York: Dover 1959.

6. Katz, Victor, A History of Mathematics: An Introduction (3rd edition). Boston: Addison-Wesley 2009.

7. Leahy, Andrew, Evangelista Torricelli and the 'Common Bond of Truth' in Greek Mathematics, Mathematics Magazine 87 (2014) 174-184.

8. Evangelista Torricelli, de Dimensione Parabolae, Opera Geometrica, 1644. Translated by Kasandara Sullivan and the author.

Appendix: The Quadrature of the Parabola
solved by many methods through the new geometry of indivisibles

Translated by Andrew Leahy and Kasandara Sullivan

Until now, the matter about the measurement of the parabola has been related in the manner of the ancients. It remains that we should approach the same measurement of the parabola with a certain new but marvelous system--namely, by the aid of the Geometry of Indivisibles and with diverse methods in this manner. In fact, with the principal theorems of the ancients assumed, as much of Euclid as of Archimedes (they may concern very diverse matters themselves), it is amazing that from every single one of these the quadrature of the parabola is able to be elicited with so little trouble and vice versa, as if there were a certain common bond of truth. In fact, from the premise that a cylinder is three times its own inscribed cone, hence it follows that the parabola is 4/3 of its own inscribed triangle. Indeed, if you prefer to assume that the cylinder is 3/2 of its own inscribed sphere, the quadrature of the parabola is immediately inferred. The same result is concluded by supposing the demonstration which shows that the center of gravity of a cone is placed on its axis, so that the part which is near the vertex is three times the remainder. The parabola is no less squared by also supposing that the space bound by a spiral line in the first revolution and by the line which is the beginning of the revolution is 1/3 of the first circle. On the other hand, with the quadrature of the parabola supposed, all the aforementioned theorems are able to be demonstrated easily. Moreover, for my part I would not dare to assert that this Geometry of Indivisibles is a thoroughly new invention. Rather, I would have believed that the old geometers used this one method in the discovery of the most difficult theorems, although they would have produced another way more acceptable in their demonstrations, either for concealing the secret of the art or lest any opportunity for contradiction be proffered to envious detractors. Whatever it is, it is certain that this wonderful geometry is a shortcut for invention, and that it confirms countless almost inscrutable theorems with brief, direct, and affirming demonstrations, which is certainly not able to be done easily through the ancient teaching. To be sure, this is truly the Royal Road in the mathematical thorn hedges, that Cavalieri, creator of these wonderful inventions, first among everyone opened up and made public for the common good.

Proposition 11.

The parabola is 4/3 of a triangle that has the same base and height.1

Let ABC be a parabola with tangent CD and let AD be parallel to the diameter. Let the parallelogram AE be drawn, and let a circle with diameter AD be conceived which is the base of a cone having vertex at point C and likewise is the base of some cylinder ACED of the same height with the aforementioned cone.

Now let some line FG be drawn parallel to AD, and let a plane parallel to the circle on AD be conceived to pass through the line itself. FG will then be to IB as the line DA is to IB--that is, as the square on DC is to the square on CI (because of the parabola). Or as the square on DA is to the square on IG (because of similar triangles)--that is, as the circle on DA is to the circle on IG--namely, as the circle on FG is to the same circle on IG. And it is this always. All the first magnitudes are equal to the line DA and therefore equal among themselves. Also, all the thirds are equal to the circle on DA, and on account of this equal among themselves. Therefore, by Lemma 18, all the firsts together--namely the parallelogram AE--will be to all the seconds together--namely, to the trilineum ABCD--as all the thirds together--namely, the cylinder AE--are to all the fourths together together--that is, to the cone ACD.2 Therefore, the parallelogram AE is three times the trilineum ABCD.3 With the parallelogram AE cut in half, the triangle ACD will be 3/2 of the trilineum ABCD. By conversion of the ratio, the triangle ACD will be three times the parabola itself. On account of this, from the explanation of Proposition 9, the parabola will be 4/3 of its own inscribed triangle. Quod erat &c.

We will also square the parabola by a different computation with the principles of indivisibles demonstrated previously, by which it will be able to be done with brevity. Moreover, we, scraping the earth with less daring, will resolve it by means of the immense ocean of Cavalieri's Geometry. Whoever wishes will be able to see all these things (shall I say in a fountain, or in a sea?) around the middle of the second book of Cavalieri's Geometry of Indivisibles.

Lemma 20.

The squares of all the parts of a straight line together are in a ratio of 1/3 to the same number of squares of the whole line together.

Let AB be a straight line. I say that all the squares of all the parts of the line AB together are 1/3 of the same number of the squares of that same line AB.

Indeed, let ACDB be a square with the diameter AD drawn. Let the figure be revolved around the axis AB until it returns to its starting point. It is clear that a cylinder CH is described by the square and also the cone DAH which has vertex at A is described by the triangle ABD. Now let EF be drawn parallel to CA and let AF--or FG (indeed they are equal)--be one of the infinite parts of the whole line AB.

Now the square on the whole line AB is to the square on the part AF as the square on EF is to square on FG (because of equality)--namely, as the circle on diameter EL is to the circle on diameter GI (Euclid XII.2). And it will always be so. Also, the first magnitudes individually are equal to the square on AB, and the thirds are always equal to the circle on DH. Therefore, by Lemma 18 all the firsts together (that is, as many lines AB squared as the line itself has parts) will be to all the squares of the parts as all the thirds together (that is, as the cylinder CH) will be to all fourths together (namely, to the cone DAH). Therefore, just as many squares of some line as the line itself has parts are to all squares of its very parts as the cylinder CH is to the cone DAH--namely, 3 to 1. By transposing, the proposition stands. Quod demonstrandum fuerat &c.

Lemma 21.

All rectangles which are bounded both by a straight line together with its own individual pieces and by the remainders are in a ratio of 2/3 to the same number of squares of the same straight lines.

With the figure in the preceding lemma assumed, some point F should be taken on the line AB. The rectangle contained by BAF (as one straight line) and by FB will be one of all the aforementioned rectangles. (In other words, one side is composed from the whole line AB together with the part AF; the other side is clearly FB, without a doubt the remaining part.)

Moreover, the aforementioned rectangle contained by BAF as one line and by FB is the same as the rectangle EIL (because of the equality of sides). And this is always true in this manner wherever the point F is. But all the rectangles contained both by the lines intersecting in the trapezoid CAHD (one of which is EI) and by the remainders (one of which is IL) together with all the squares of the intermediate section (one of which is FI) are equal (because of Euclid II.5) to all the squares on the halves (one of which is FL). In fact, all squares of the intermediate sections (one of which is FI) are to all the squares of the halves (one of which is FL) as 1 is to 3 (by the preceding lemma). Therefore, if all squares of the intermediate sections are taken away, all the rectangles will remain (one of which is EIL). That is, all the rectangles contained both by AB together with its parts and by the remainders will remain as 2/3 of all the squares which are made from the halves--that is, the same number of all the squares of the whole AB. Quod fuerat ostendendum &c.

Proposition 12.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose diameter is BE, and around the parabola let there be a parallelogram DC. Let a line FG be drawn parallel to the diameter. FG will be to GI as BE is to GI, or as the rectangle CEA is to CGA--that is, as the square on CE is to the rectangle CGA. And it will always be so. Also, the first magnitudes are always equal to the line BE. Moreover, the thirds are always equal to the square on CE. Therefore (by Lemma 18) all the firsts together (that is, the parallelogram AB) will be in relation to all the seconds together (namely, the semiparabola AIBE) as all the thirds together (clearly just as many squares on the line CE as the line itself has parts) in relation to all the fourths together (namely, all the rectangles contained by CE with its parts and by the remaining parts). Therefore (from the preceding lemma) the parallelogram AB will be 3/2 of its own semiparabola. Also, the whole parallelogram DC will be 3/2 of the whole parabola--indeed, as 6 is to 4. Therefore the parabola will be to its own inscribed triangle (which is in fact 1/2 of the parallelogram DC) as 4 is to 3--that is, 4/3. Quod erat &c.

We are able to square the parabola by the same argument without the trouble of the former lemmas, with yet a different supposition--namely, by supposing the proportion which the cylinder has in relation to its inscribed sphere--a proportion which is indeed 3/2, as shown from Archimedes' On the Sphere and the Cylinder, Book I

Proposition 13.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola, around which is the parallelogram AD. Let there be a semicircle on the diameter AC, around which is the rectangle AE. Then with the axis AC fixed, let it be conceived that the semicircle itself is revolved around the axis in such a way that from its own revolution a sphere is circumscribed. A cylinder likewise arises from the revolution of the rectangle AE.

With some point G now assumed, let a line GF be drawn parallel to the diameter HB, and through the same point G let the plane GL be drawn perpendicular to the axis AC.

The line FG will be to GI as BH is to GI (because of equality). That is, as the rectangle CHA is to the rectangle CGA, or as the square on HN is to the square on GM (because of the circle), or as the square on GL is to the square on GM--namely, as the circle on the semidiameter GL in the cylinder is to the circle on the semidiameter GM in the sphere. And it will always be thus wherever the point G is assumed. Moreover, all the firsts are equal among themselves, as are all the third magnitudes among themselves. Therefore, all the firsts (namely, the parallelogram AD) will be to all the seconds (namely, to the parabola ABC) as all the thirds (that is, the cylinder) are to all the fourths together (clearly to the sphere). But the cylinder to the sphere is 3/2. Therefore, the parallelogram AD will certainly be 3/2 of the parabola, and the parabola itself will be 4/3 of its own inscribed triangle, as was concluded in the preceding lemma. Quod &c.

Lemma 22.

Suppose some number of magnitudes will have been hung from given points on a balance, and the same number of magnitudes of another type, equally proportional with the aforementioned magnitudes, will hang from the same points. The center of equilibrium of each type of the magnitudes will be one and the same.

Let some number of magnitudes C, D, E and F of the first type be hung from whatever points on a balance AB. Let just as many magnitudes G, H, I, and L of a second type hang from the same points and let them be proportional--namely, as C is to D so is G to H, and as C is to E so is G to I, etc. I say that the same point on the balance is the common center of equilibrium of each type of suspended magnitudes.

Indeed, since as C is to D so is G to H, they will counterbalance from the same point, so the two magnitudes C and D will be as the two magnitudes G and H.

Further, since as C is to D so is G to H, by convertendo and componendo DC will be to C as HG is to G. Moreover, C is to E as G is to I. Therefore, CD together will be to E as GH together is to I by equality. Wherefore the magnitudes CD and E will counterbalance from the same point from which the two GH and I counterbalance.

Additionally, since as CD is to E so is GH to I by the things just mentioned, by componendo CDE will be to E as GHI is to I. But E is to C as I is to G, and C is to F as G to L. Wherefore CDE together will be to F as GHI together is to L by equality. Therefore, the two magnitudes CDE and F will have the same point of equilibrium which the two magnitudes GHI and L have. And thus also if there are many magnitudes, all the way to infinity, quod erat propositum &c.

Lemma 23.

If a parabola should have a tangent at the base and indeed a line parallel to the diameter for the other side, the trilineum bound by the parabolic curve, the tangent, and the aforementioned parallel will balance at equilibrium from the point on the tangent where it is divided so that the part nearest to the point of tangency is three times the remaining part.

Let ABC be a parabola whose tangent at the base is CD and let AD be parallel to the diameter. I say that the mixed trilineum ABCD balances at equilibrium from the point on the tangent CD where it is divided so that the part towards the tangent C is three times the remaining part.

Let the figure be imagined in such a way that DA is perpendicular to the horizontal, and let a circle on the diameter DA be conceived which is the base of a cone having vertex at point C.

With some point E already assumed, let EF be drawn parallel to DA itself, and let a plane parallel to the base of the cone pass through the EF.

Therefore the line DA will be to EB as the square on DC is to the square on CE (because it is a parabola), or as the square on DA is to the square on EF--that is, as the circle on DA is to the circle on EF. And it is always so wherever the point E. Therefore, since proportional magnitudes of two types are suspended from the same points, as was required in the preceding lemma, all the magnitudes of the first type together (that is, all the lines of the trilineum ABCD, or the trilineum itself) will have the same point of equilibrium as all the magnitudes of the second type together (that is, all the circles of the cone ACD, or the cone itself) have. Moreover, the cone balances at equilibrium from the point which divides CD in such a way that the part toward C is three times the remaining part, since the line DA is perpendicular to the horizontal.4 Therefore, the trilineum ABCD also balances at equilibrium from the same point. Quod erat propositum, &c.

Proposition 14.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose diameter DE is understood to be perpendicular to the horizontal. Let CF and AD be tangents, and let AF in fact be parallel to the diameter.

Let FH then be assumed as 1/4 of the whole FC. The mixed trilineum ABCF will balance at equilibrium from the point H (by the preceding Lemma). Also, let FI be taken as 1/3 of the whole FC. The whole triangle AFC will balance at equilibrium from I.5 Clearly the parabola will balance at equilibrium from D, since it has a center on the diameter. Therefore, the trilineum ABCF will be reciprocally to the parabola itself as DI is to IH--namely, 2 to 1. (In particular, if FC is 12 of some parts, FD is 6, FI is clearly 4, and FH is 3. Therefore, DI is 2 and IH is 1.) Consequently, by componendo, the whole triangle AFC will be three times the parabola. The remainder of the quadrature is completed as was done in Proposition 9. Quod erat &c.

In another way.

With the same things assumed as above, let FH be supposed 1/4 part of the whole FC. The mixed trilineum ABCF will balance at equilibrium at the point H. Also, let FI be supposed 1/3 part of FD itself. Then indeed the triangle FDA will balance at equilibrium from the point I. Clearly, the mixed trilineum ABCD will balance at equilibrium from the point D. (For the entire triangle ADC balances at equilibrium from the point D, and the removed parabola balances at equilibrium from the same point D. Therefore, it is also necessary that the remaining trilineum ABCD balance at equilibrium from the point D.) And so the triangle FDA will be to the trilineum ABCD reciprocally as DH is to HI--namely, as 3 is to 1. By conversion of the ratio, the triangle ADC will be to the parabola as 3 is to 2, or as 6 is to 4. Wherefore, the parabola will be to the triangle ABC as 4 is to 3--namely, 4/3. Quod erat propositum demonstrare &c.

With the following observation on geometric progressions said first, we may approach the quadrature of the parabola from yet another supposition.

Lemma 24.

If two straight lines should intersect each other and between these should be described a certain constant flexilineum from alternately parallel lines, all lines which are parallel among themselves will be in continuous proportion.

Let two straight lines AB and CB intersect each other at point B, and between these let a flexilineum CADEFG etc. be described in such a way that CA, DE, FG, etc. are parallel among themselves, and likewise AD, EF, and the remaining in turn are assumed parallel among themselves. I say that AC, ED, and GF are in continuous proportion.

Indeed, since they are parallel, as AC is to ED, so is AB to BE, or DB to BF--that is, ED to GF. (Euclid VI.2 and VI.4.) Constat ergo quod propositum fuerat.

Lemma 25.

With two straight lines intersecting each other as shown above, if the two lines AC and DE are parallel between themselves and with CD joined, let it be conceived that the flexilineum ACDE is continued in infinitum all the way to the point B of intersection. I say that in a flexilineum of this kind are to a hair each and all the terms which are in the progression of the proportion of AC to DE continued in infinitum.6

Let F be assumed equal to AC itself and G equal to DE. Also, let the proportion of F to G be conceived as continued in its own infinitely many terms from F to H.

Now if it is possible, let there be some term or terms in the progression from F to H which are not found in the flexilineum and let I be greatest term of those terms which, though they are in the progression FH, are not in the flexilineum. Therefore, the term L preceding such an I will be in the flexilineum. Let it be MN. Since L is to I as F is to G, or as AC is to DE, or as NM is to PO, following next, and L and NM are equal, I and PO will also be equal. Therefore, the term I, which was assumed to not be in the flexilineum, was found in the same.

We would demonstrate in completely the same method that there is no term in the flexilineum which is not also in the progression FH etc. We would then conclude that all the terms in the flexilineum are precisely the terms of the proportion of AC to DE continued in infinitum, since it was demonstrated that no term which is in the progression FH is missed in the flexilineum nor is any term which would not also be found in the progression FH present in the flexilineum.

Lemma 26.

With infinitely many straight lines of a greater than inequality assumed in continuous proportion, to find a straight line which is equal to all the aforementioned lines together.

Let A and B be the first two lines of the given progression, in which CD is supposed equal to the greater A and EF to the lesser B. Let CD and EF be parallel, and let DF and CE, which intersect by necessity, be joined. Thus let them intersect at point G and, with CF drawn, let GL be parallel to the CF.

I say that the line DL is equal to all the infinite terms of the progression ABM taken together.

Indeed, let it be conceived that the flexilineum DCFE etc. is continued in infinitum all the way to the point G. All the lines will be in the flexilineum itself, or the terms of the given progression ABM.

Now let HE, NI, and the remaining lines parallel to them be extended all the way to DL. EF will be equal to CP itself, HI equal to PQ, and NO equal to QR, and so on one at a time (by Euclid I.34). Indeed, each line which is in the flexilineum will have its own corresponding small part on the line DL equal to itself until the flexilineum should arrive at the ultimate point G. At that time, moreover, there will not be anything from the flexilineum or from the line DL that will be left over, but the flexilineum itself as much as the line DL will also have been completely used up. Indeed, GL itself, which is drawn from the final point G of the flexilineum, is the last of all the parallels which are drawn all the way to DL. Therefore, all the lines of the flexilineum together, of which the first is CD, taken alternately (that is, all the lines of the progression ABM) are equal to all the small parts of the line DL taken together (that is, DL itself). Quod erat ostendendum &c.

Lemma 27.

With infinite magnitudes supposed in a continuous geometric proportion of a greater than inequality, the first magnitude will be the mean proportional between the first difference and the aggregate of all the magnitudes.

Indeed, with the preceding construction assumed, let FV be drawn parallel to the GC. Then DV will be the first difference. But DV is to the first magnitude DC as FD is to DG (Euclid VI.4)--that is, as DC is to the aggregate DL of all of them. Quod erat demonstrandum &c.

Comment.

We will not hesitate to assert that this is also true for numbers and any sort of magnitudes. We will convey a demonstration even more universal, especially since it is quite brief. The conclusion of this truth, when it had been contrived in passing by us for the most celebrated Cavalieri, also itself established the same Theorem with the following demonstration, which has been already supplied by us in the first proof.

This is said in advance. But if there will be however many magnitudes, either finite or infinite in number, the antecedents of which are always larger than the consequents, the first magnitude of all will be equal to all the differences taken together with the smallest magnitude itself.

This was known among geometers and is demonstrated as was done by us in Lemma 15, where we show that the parallelogram AE is equal to all the differences between the following parallelograms and the smallest parallelogram OC.

Now let magnitudes be supposed infinite in number in a continuous geometric proportion of a greater than inequality. It is evident that the smallest magnitude of all either will or will not be a point. Therefore, in this case the first magnitude will be equal to all such differences.

Moreover, since the magnitudes are assumed in a continuous geometric proportion, the differences will also be proportional in the same ratio and therefore (with a conversion having been done) as the first difference is to the first magnitude, so will the second difference be to the second magnitude, and thus always. Consequently, as one is to one, so summarily will all be to all--namely as the first difference is to the first magnitude, so are all the differences together (that is, the first magnitude itself) to all the magnitudes together. Therefore, it stands that the first magnitude is the mean proportion between the first difference and the aggregate of all of them.

Proposition 15.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola in which is inscribed the triangle ABC. I say that the parabola is 4/3 of the triangle ABC.

Indeed, let the two triangles ADB and BEC also be inscribed in the remaining portions ADB and BEC of the parabola. The triangle ABC will be four times the two triangles ADB and BEC together (Lemma 7). Let four inscribed triangles also be conceived in the remaining small parts AD, DB, BE, and EC. Both triangles ADB and BEC together will be four times the aforementioned subsequent triangles together (Lemma 7), and always in this manner. Therefore, the parabola is nothing other than a certain aggregate of magnitudes infinite in number in a 4 to 1 proportion, of which the first is the triangle ABC, and indeed the second consists of the two triangles ADB and BEC. Therefore, the first magnitude ABC will be the mean proportional between the first difference and the aggregate of all of them--namely, the parabola.

Consequently, suppose the triangle ABC is 4. Therefore, the two triangles ADB and BEC together are 1, and the first difference (of course between 4 and 1) is 3. Therefore, the aggregate of all the infinite magnitudes (namely the parabola itself) will be (by lemma 27) to the first magnitude (that is, to the inscribed triangle ABC) as the first magnitude itself is to the first difference--clearly, as 4 is to 3. Namely, 4/3. Quod erat propositum demonstrare &c.

In Another Way.

Let ABC be a parabola whose diameter is DB, let AD and CD be tangents to the base, and also let EF be tangent to the. Moreover, in the remaining trilinei ABE and BCF let the two triangles GEH and IFL be inscribed (as was required for the construction in the third and fourth lemmas). Likewise, let four triangles be conceived in the four remaining mixed trilinei, and always in this way. So the entire trilineum ABCD will be nothing other than a certain aggregate of magnitudes infinite in multitude in a 4 to 1 proportion (Corollary 1 of Lemma 3), of which the first is the triangle EDF, the second in truth consists of the two triangles GEH and IFL, and the third in truth from the following four, etc. Therefore, the aggregate of all (namely, the mixed trilineum ABCD) will be to the first magnitude (namely, to the triangle EDF) as the first magnitude itself is to the first difference (Lemma 27)--clearly, as 4 is to 3.

Thus, since the trilineum ABCD is to the triangle EDF as 4 is to 3, the same trilineum will be to the triangle ADC as 4 is to 12. Therefore, the parabola will be to the triangle ADC as 8 is to 12 and to its own inscribed triangle as 8 to 6--namely, 4/3. Quod erat demonstrandum &c.

Lemma 28.

If straight lines AB, CD, EF, etc. infinite in number are in a continuous geometric proportion of a greater than inequality and moreover another progression BG, DH, FI, etc. is supposed so that just as the first AB is to the first BG, so the second CD is to the second DH, and so the third EF is to the third FI, and thus always, I say that the entire aggregate of the progression AB, CD, EF, etc. is to the aggregate of the progression BG,DH, FI as AB is to BG.

Let all the terms of the two progressions be conceived to be in the flexilinei (in like manner with Lemma 25). With AD and GD joined, let OL be drawn parallel to AD itself and OM parallel to DG itself. BL will be equal to all the infinite terms AB, CD, EF, etc. (Lemma 26), and indeed OM will be equal to all the infinite terms of the remaining progression BG, DH, and FI (Lemma 26).

Now as LB is to BA, so is OB to BD--that is, MB to BG (Euclid VI.4). Therefore, by permutando, the aggregate LB is to the aggregate BM as AB is to BG--namely, as one magnitude is to the other. Quod erat &c.

This theorem had been able to be substituted for the demonstration in Proposition 12 of Book 5 of Euclid. Indeed, it is one and the same with the Theorem in the aforementioned Proposition. But in fact, since almost all are of the opinion that Euclid there supposed a multitude finite in number, we have determined with the help of flexilinei that,

Proposition 16.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose diameter is DE, whose tangents at the base are AD and CD, and whose tangent through the vertex is in fact FBG. Let ABC be the inscribed triangle. I say that the parabola is 4/3 of the triangle ABC.

For since the EB is equal to the BD (because it is a parabola),the line AC is in fact twice the line FG and the inscribed triangle ABC will be twice the triangle FDG contained by the tangents. And this is also always true around the remaining parabolic parts AIB and BOC. (Indeed, AIB is a parabola whose tangents at the base are AF and BF. Therefore, the inscribed triangle AIB will be twice the tangent triangle LFM. The same is also true for the other part. Therefore, the two triangles AIB and BOC together are twice the two triangles LFM and NGP together.) Therefore, since the two progressions are both in a continuous proportion of magnitudes infinite in multitude (namely, the one inside the parabola whose first term is the triangle ABC and second term is the two triangles AIB and BOC, together, etc, and also the other progression outside the parabola whose first term is namely the triangle FDG and second moreover is the triangles LFM and NGP together, etc.) and since the individual terms of the progression inside the parabola are twice the individual terms of the progression outside the parabola, therefore the entire aggregate of the first progression will be twice the entire aggregate of the second progression (Lemma 28). In particular, the parabola itself will be twice the mixed trilineum ABCD. Therefore, by componendo and conversion of the ratio, the triangle ADC will be 3/2 of its own parabola--namely, as 6 is to 4. Therefore, the parabola will be to the triangle ABC as 4 is to 3--clearly, 4/3. Quod erat ostendendum &c.

With the aid of infinitesimals the quadrature of the parabola is able to be gained with yet other results assumed. We suppose Propositions 14 and 25 which Archimedes demonstrated in the book on Spiral lines with a lemma of this kind set out beforehand.

Lemma 29.

If a first magnitude is to a second as a third is to a fourth, and thus however often it will have been pleasing, and if all the firsts and also all the thirds are proportional in the same order, then all the firsts together will be to all the seconds together as all the thirds together are to all the fourths together.

Let the first A be to the second B as the third C is to the fourth D, and E to F as G to H, and thus however often it will have been pleasing. And let all the firsts A, E, I, etc. and all the thirds C, G, M etc. be proportional in order--namely, as A is to E so should C be to G. Moreover, as A is to I, so should C be to M, etc., and thus always. I say that all the firsts A, E, I etc. together are to all the seconds B, F, L etc. together as all the thirds C, G, M etc. together are to all the fourths D, H, N etc. together.

Let O, P, Q etc. be taken as individuals equal to the first of the firsts--that is, to A itself--and let there be just as many as there are all the firsts A, E, I, etc. Likewise, let R, S, T etc. be assumed to be just as many as there are all the thirds, and let the individuals R, S, T be equal to the first of the thirds--namely, C itself.

Now on account of equality, as O is to A so will R be to C. Further, since P is equal to A itself, and S to C itself (because of the supposition) as P is to E so will S be to G. And this always. All O, P, Q are equal, and likewise all R, S, T are equal. Therefore, all O, P, Q, etc. together are to all A, E, I, etc. as all R, S, T together are to all C, G, M together (Lemma 18). Finally, by convertendo, all R, S, T are to all O, P, Q as all C, G, M are to all R, S, T. Remember this.

Since indeed as O is to A so is R to C and as A is to B so is C to D, O will be to B as R is to D (by equality). For completely the same reason, we conclude that as P is to F so is S to H (by equality), and thus concerning the rest. Therefore, all O, P, Q etc. together will be to all B, F, L etc. as all R, S, T etc. together are to all D, H, N etc. (Lemma 18). Wherefore, by equality, all A, E, I etc. will be to all B, F, L etc. as all C, G, M etc. are to all D, H, N etc. Quod erat ostendendum.

Proposition 17.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose tangent is AE. Indeed, let the diameter be parallel to CE, and let some FD be drawn parallel to CE itself. EC will be to FB in length as EA is to AF² (because it is a parabola), or as EC² is to FD. Therefore EC, FD, and FB will be in a continuous proportion.

Then let the intervals AC and AD become two circles with center A. Let the beginning of a spiral be placed on the semidiameter AC, and let the spiral itself be AGC.

So DF will be to FB as CE is to DF, or as CA is to AD--that is, as CA is to AG, or as the entire circumference CLHC is to the arc CLH (Proposition 14 of On Spiral Lines). That is, as the entire circumference DPGD is to the arc DPG. Indeed, it will always be thus wherever the point D is taken. All the first magnitudes and likewise all the thirds, are proportional in the way in which they ought to be (as we will show below). Moreover, all the firsts together (namely, the triangle AEC) will be to all the seconds together (namely, the mixed trilineum ABCD) as all the thirds together (namely, the circle CLH) are to all the fourths together (that is, to the remainder of the circle itself with the region inside the spiral CAGC subtracted) by the preceding lemma. Moreover, the circle CLH is 3/2 of the aforementioned space with the region inside the spiral subtracted (Proposition 25 of On Spiral Lines). Therefore, the triangle ACE will also be 3/2 of the mixed trilineum ABCE. By conversion of the ratio, the triangle ACE will be three times the parabola ABC. The remainder of the quadrature can be completed as in Proposition 9. Propositione factum est.

Moreover, we will now show what was assumed--namely, that all the firsts and all the third magnitudes are proportional in the way that is required in the preceding lemma.

Let some MO be drawn parallel to FD itself as in the previous figure, and let us suppose that the FD is the first of the firsts and that the circumference DPG is itself first of the thirds. Therefore, DF will be to OM as DA is to AO, or as the circumference DPG is to the circumference whose semidiameter is AO etc. And thus always. Quod oportebat &c.

We will also square the parabola by a way as of yet untried--of course, with its center of gravity found with the prior help of indivisibles. Moreover, we suppose a lemma which Archimedes showed in the Book 2 of on the Equilibrium of Planes--that is, that the centers of gravity of parabolas divide their own own diameters in the same proportion.

Lemma 30.

The center of gravity of a parabola divides the diameter so that the part ending at the vertex is 3/2 of the remainder.7

Let ABC be a cone whose base is AMC, axis is BC, and the axial triangle is in fact ABC. Let the cone be cut by the plane EFG as was demanded in Proposition 11 of The Conics, Book I. The section will be that which is called a parabola, and its diameter will be FH. Now let the center of gravity of the parabola EFG be some point, call it I. It must be shown that the line FI is 3/2 of the IH.

Let the line AIL be drawn through the point I, and let the cone be cut by another plane MNO parallel to the EFG. The section on MNO will be a parabola, and its center of gravity will be P. (For since they are parabolas, as FI is to IH so is NP to PR. But I is assumed the center of gravity of the parabola EFG. Therefore, by Proposition 7 of of On the Equilibrium of Planes, Book II, P will be the center of gravity of the parabola MNO.) And thus always, wherever the plane MNO. Therefore, the centers of gravity of all the parabolas which are in the cone ABC are found individually on the line AL. Wherefore, the common center of gravity of all the same aforementioned parabolas together will also be on the line AL. Moreover, all the parabolas are the same as the cone itself. Therefore the center of the cone is on the line AL, but since it is also on the axis BD, the center of the cone will be on the common point of intersection S. Therefore, BS will be three times SD itself.

Let the line DQ be drawn from the center of the base parallel to AL. CQ and QL will be equal (Euclid VI.2). Moreover, since BS is three times SD (because S is the center of the cone), BL will also be three times LQ (Euclid VI.2). Therefore, BL is 3/2 of LC. Wherefore, FI will also be 3/2 of IH. Quod erat propositum &c.

Proposition 18.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose diameter is BD, and indeed let the inscribed triangle be ABC. I say that the parabola is 4/3 of the triangle ABC.

Let AD and DC be divided in half at the points E and F, and let EG and FH be drawn parallel to the diameter. These will be the diameters of the segments AGB and BHC. Let the centers of gravity of the aforementioned segments be O and N. GO and HN will each be 3/2 of the remainders OI and NL (by the preceding lemma). Let ON be joined. The common center of gravity of the two segments will be on ON itself (by Proposition 8 of On the Equilibrium of Planes, Book I). But it is also on BD. (For as the center of the whole parabola is on BD, so also is the center of the triangle ABC.) Wherefore, the point P will be the center of the segments AGB and BHC. Assume BD is 60 parts. GE will be 45 parts (since it is 3/4 of BD), the IE will be 30 parts, and the EO--that is, DP--36 parts. Let Q be the center of gravity of the triangle ABC; DQ will be 20 parts. Let R be the center of the parabola; RD itself will be 24 parts (by the preceding lemma). Therefore, PR will be 12 parts and RQ 4 parts. But as PR is to RQ so reciprocally is the triangle ABC to the two segments AGB and BHC. Wherefore, the triangle ABC will be to the two portions AGB and BHC as 12 is to 4--namely, as 3 is to 1. By componendo and conversion of the ratio, the parabola ABC will be to the triangle inscribed in it as 4 is to 3--namely, 4/3. Quod erat propositum &c.

We will undertake the quadrature of the parabola by a hitherto new calculation, with the following lemma assumed, which indeed was said to have come forth from the Cavalerian School. Indeed, it was devoted to the measurement of a certain solid arising from the parabola itself having been revolved around its axis. Moreover, the lemma is of a type from the author Ionas Antonio Roccha, a distinguished geometer.8

Lemma 31.

If a plane figure is balanced above some straight line of its own which divides the figure itself, the moments of the segments of the figure are as the solids of revolution described by the segments having been revolved around the dividing line.

Let ACDBFE be a plane figure which the straight line AB divides, and let the figure be imagined to be balanced above the line AB. I say that the moment of the segment ACDB is to the moment of the segment AEFB as the solid of revolution arising from the revolution of the segment ACDB around the axis AB is to the solid of revolution arising from the revolution of the remaining segment around the same axis of revolution.

For with two given points H and I assumed on the line AB, let the lines CE and DF be drawn through H and I perpendicular to the AB, and let the segments DH and HF be cut in half at the points L and M.

Therefore the moment of the line DH will have a ratio to the moment of the line HF compounded from the ratio of the magnitudes DH to HF, and from the ratio of the distances LH to HM--or DH to HF.9 Therefore, the moment of the line DH will be to the moment of the line HF as the square on DH is to the square on HF.

It will be shown in the same manner that the moment of the line CI is to the moment of the line IE as the square on CI is to the square on IE, and thus always.

Further, the moment on DH is to the moment on CI (on account of the same reason as above) as the square on DH is to the square on CI, and in this way always. Therefore, (by Lemma 29) all the first magnitudes together, (namely, all the moments of the figure ACDB) will be to all the seconds together (namely, to all the moments of the remaining figure AEFB) as all the thirds together (namely, all the squares of the figure ACDB) are to all the squares of the remaining figure, or as all the circles of the figure ACDB (namely, the circular solid described by its rotation around the axis AB) are to all the circles of the remaining figure AEFB (namely, to the circular solid arising from its revolution around the same axis AB). Quod erat ostendendum &c.

With this having been set forth (which, as we indeed declared thoroughly above, was undertaken by others, was introduced here as another's, and which I believe was not well known until now), we will square the parabola, with the demonstration, which is proved in many ways, assuming that the Cylinder is twice its own inscribed parabolic conoid.10

Proposition 19.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABCD be a semiparabola, around which is a rectangle DE. Let a point F be assumed such that AF is to FD as 5 is to 3, and let FG be drawn parallel to the diameter. The center of gravity of the semiparabola will be on the FG (by Lemma 11). Let it be some point, call it I, let LIM be drawn parallel to AD through I, and let IN be taken equal to IM itself. Also, let PQ (wherever it may fall) be understood as extended parallel to the diameter CD in such a way that the rectangular parallelogram DP is equal to the semiparabola itself. Then let a rectangle DR be conceived applied to the line CD in such a way that it balances at equilibrium with the semiparabola with the balance above the line CD. Let the center of the aforementioned rectangle be the point S and, with TSX drawn parallel to the AD, let the line IS be joined.

Now it is clear from the lemma above that the cylinder made from the rectangle DR revolved around the axis DC will be equal to the parabolic conoid made from the rotation of the semiparabola ACD around the same axis of revolution CD, since the moments of the plane figures are assumed equal. Therefore, the cylinder made from the rectangle DR will be 1/2 of the cylinder made from the rectangle DE, and consequently the square on TX will be 1/2 of the square on ML. (For cylinders of equal height are among themselves as the squares of the bases.) Remember this.

In fact, MN is to TX as IM is to TS (Euclid VI.4)--for they are halves of the same things), or as IV is to VS--in particular, (because the plane figures balance at equilibrium above the line CD, or from the point V) reciprocally as the rectangle DR is to the semiparabola (or to the rectangle DP equal to the semiparabola itself), or as their bases TX to MO. Therefore, TX is the mean proportional between MN and MO. Wherefore, the rectangle NMO will be 1/2 of the square on LM, since it is equal to the square on TX.

Clearly the ratio of the square on LM to the rectangle NMO is composed from the ratio of LM to MN (which is 4/3 by construction, for we assumed the point F is such that AF is to FD as 5 is to 3) and from the ratio of LM to MO, which indeed was unknown, but now shows itself as 3/2 by necessity. For the ratio of 2 to 1 is composed from 4/3 and 3/2, as also was made known by the Cantorians themselves, as was seen in the three numbers 4,3,2.

Therefore, the rectangle DE to DP itself, or to the semiparabola, will be 3/2, and the semiparabola to the triangle ACD will be 4/3. Quod erat ostendendum &c.

Lemma 32.

Let ABC be a parabola with base AC and tangent CD. Let AD be parallel to the diameter. With some point E assumed, let EF be drawn parallel to the diameter. I say that as FE is to EB, so is CA to AE.

For DA is lengthwise to the FB as DC ² is to CF ², or as DA ² is to FE ² (because it is a parabola). Therefore, DA, FE, and FB are in a continuous ratio. Remember this.

Now as AC is to CE so is AD to EF, or EF to FB. By conversion of the ratio, as CA is to AE so is FE to EB. Quod erat ostendend. &c.

Lemma 33.

Any parabola is equal to two parabolas taken together which indeed have a base equal to that one, a diameter in fact half, and which are equally inclined.

Let ABC be a parabola whose diameter is BH. Let the two other parabolas AEC and AGC be on the same base and indeed let the diameters HE and HG both be 1/2 of the diameter HB, but inclined equally to the base. I say that the parabola ABC is equal to the figure AECG.

For let some point be taken on the base AC, and let it be M. With PMN drawn parallel to the diameter BH, BH will be to NM as the rectangle AHC is to the rectangle AMC, or as the line HE is to the line MO. By permutando, as BH is to HE so will NM be to MO. Wherefore, NM will be twice the MO. It will be shown in completely the same manner that NM is also twice the MP. Therefore, the entire NM is equal to OP itself. And this always. On that account, all lines of the figure ABC together (namely, the parabola ABC itself) will be equal to all the lines of the figure AECG together (namely, to the two parabolas AEC and AGC). Quod erat &c.

Proposition 20.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a parabola whose diameter BE is conceived as perpendicular to the horizontal, and let the parabola itself be inverted. Let CA be extended to D, so that CA and AD are equal, and let DC be a balance whose fulcrum is A. Let CF be drawn tangent to the parabola and AF parallel to the diameter EB. Suppose GH is equal to AC itself, and let GH be divided in half at I. Let IL and IM both be half of the line EB and be inclined equally to the base as EB itself is to AC. Also, let the two parabolas GLH and GMH be constructed, which (by the preceding lemma) together will be equal to the parabola ABC, and let the figure GLHM be suspended from the point D.

Now let the points O and N be taken equally in distance from the points I and E, respectively. With NQ drawn parallel to EB and ROS to LM, NP will be equal to RS itself, as in the preceding lemma.

Now QN is to RS as QN is to NP (because of equality), or reciprocally as DA is to AN (by Lemma 32). Therefore, the lines QN and RS balance at equilibrium, and thus always. Therefore, all the lines of the triangle AFC together (namely, the triangle itself) balance at equilibrium all the lines of the figure GLHM together (namely, the figure GLHM itself).

Let AV be 1/3 of the entire AC. It is clear that if the line from V is sent down parallel to AF itself, the center of gravity of the triangle AFC will be on it, and the line will be perpendicular to the horizontal. Therefore, the triangle AFC will be suspended centrally from the point V, and the triangle AFC will reciprocally be to the region GLHM as DA is to AV--namely, 3 to 1.

Moreover, since the region GLHM is equal to the parabola ABC, the triangle AFC will also be three times the parabola ABC.

The remainder of the quadrature is solved as was done in Proposition 9. Quod &c.

Proposition 21.

The parabola is 4/3 of the triangle that has the same base and height.

Let ABC be a semiparabola whose diameter is CE and ordinate is AE, and let CD be tangent. Let the parallelogram AECD be completed. It is clear that all the lines of the mixed trilineum DABC which are in fact parallel to the diameter are in the same ratio among themselves as all of the circles of a cone which has axis DC and vertex C. Therefore, the center of gravity of all the lines of the trilineum DABC, will be on that line which divides the balance DC just as the center of gravity of the cone divides the same balance--namely, so that the part nearest to C is 3 times the remainder (by Lemma 22). Therefore, let CF be made 3 times FD itself, and let FM be drawn parallel to CE. The center of gravity of the trilineum DABC, wherever it may be, will be on the line FM.

Likewise, all lines which are drawn parallel to the diameter of the semiparabola ABCE are in the same ratio among themselves which are all circles of some hemisphere whose axis is AE and vertex clearly is A (by Lemma 22). Therefore, the center of gravity of all the lines suspended from the balance AE, or from the semiparabola itself, will be on that line which thus divides the balance AE as the center of gravity of the hemisphere divides the balance. Namely, so that the part terminated at A is to the remainder as 5 is to 3.11 Therefore, let AI be to IE as 5 is to 3, and let IH be drawn parallel to CE. The center of the parabola, wherever it is, will be on the line IH. Finally, let GL be drawn which cuts in half the sides AE and DC. The center of gravity of the parallelogram DE, which is O, will be on GL. Suppose that the center of gravity of the semiparabola is some point P, and let PO be extended onto N. N will be the center of gravity of the trilineum DABC. Now the semiparabola is to the trilineum as NO is to OP, or as ML is to LI--namely, as 2 is to 1. (For by construction the whole AE is 8 of some kind of parts, AM is 2 of such parts, ML is 2, LI is 1, and the remaining IE will be 3.) Therefore, the semiparabola will be to the parallelogram as 2 is to 3, or as 4 is to 6, and the semiparabola is to its inscribed triangle as 4 is to 3--namely, 4/3. Quod &c.

FINIS.

Translation Notes

1. A word about typesetting: Torricelli uses a combination of italics and Roman type, depending on the type of claim he is making (e.g., "Lemma" or "Proposition"). We have followed him in our typesetting of the translation, so, e.g., a figure ABCD may sometimes be referred to as ABCD or ABCD depending on the context.

2. A trilineum is a generalization of a triangle in which one or more of the sides is curved instead of rectilinear. The terminology goes back at least to Commandino.

3. Here Torricelli is using Euclid XII.10, which establishes that a cone is 1/3 of the cylinder in the same base with the same height.

4. In Proposition XXII of his Liber de Centro Gravitatis Solidorum (1565), Federico Commandino established that the center of gravity of a cone--i.e., the point at which it balances at equilibrum--lies at the point on its axis which is 1/4 of the way from the base to the vertex. In other words, the distance from the center of gravity to the vertex is three times the distance from the center of gravity to the base.

5. Proposition 14 of Archimedes' On the Equilibrum of Planes I establishes that the center of gravity of a triangle lies at the intersection of any two median lines. It follows from this that the center of gravity lies at a point on a given median line that is 2/3 of the way from the vertex to the opposite side. A well-known property of centers of gravity is that a figure will be suspended at equilibrium from a point on its boundary if and only if the line connecting that point to the center of gravity is vertical, which means parallel to FA in this case. Since the triangle is right, FI must be twice IC by similarity.

6. The Latin version of this sentence reads: Dico in huiusmodi flexilineo esse omnes, & singulos ad unguem terminos qui sunt in progressione proportionis AC ad DE. in infinitum continuatae. "Ad unguem" means literally "to a fingernail" or "exactly".

7. This result is the main conclusion of Archimedes' On the Equilibrium of Planes II (see Proposition 8 of the work), which was established with much effort on Archimedes' part. Torricelli is demonstrating how much easier it is to prove this result using indivisibles.

8. Giannantonio Rocca (1607-1656) was a mathematical correspondent of Cavalieri and Galileo, among others. His demonstration was apparently already established by 1628, two years before the publication of Guldin's result. (See p. 83 of Seventeenth-Century Indivisibles Revisited, ed by Vincent Julien.) He was instrumental in helping Cavalieri formulate a response to Guldin's criticism of his work in Cavalieri'sExercitationes. (See pp.154-155 of Infinitesimal: How a Dangerous Mathematical Theory Shapred the Modern World by Amir Alexander.) Letters between Cavalieri and Rocca can be found inLettere d' uomini illustri del secolo XVII a Giannantonio Rocca filosofo e matematico Reggiano con alcune del Rocca a' medesimi (Moderno 1765).

9. For a discussion of what a "compounded" ratio is, see the discussion of Euclid Definition V.9 on pages 132-133 of Heath, Thomas trans., Euclid: The Thirteen Books of the Elements, Vol. 2. Dover (1956). Also, the definition of amoment stems from Archimedes' Law of the Lever: Given two masses m and M, the masses will balance on a lever when placed distances d and D (respectively) from a fulcrum, where d and D satisfy the reciprocal proportion m/M = D/d. By cross-multiplfying, we have m d = M D. This product is called themoment of the system. For, say, the line segment DH the center of gravity is the midpoint L. Thus, the moment is (algebraically speaking) \( DH \cdot {1 \over 2} DH = {1 \over 2} DH^2\). When taking the ratio of the moment of DH to the moment of HF, the "1/2" terms will cancel. Hence the ratio of the moments of the line segments is the same as the ratios of the squares. Note that the conclusion of this Lemma is a version of the Pappus-Guldin Theorem.

10. This follows from Proposition 21 of Archimedes' On Conoids and Spheroids, which states (in the Heath edition) that "Any segment of a paraboloid of revolution is half as large as the cone or segment of a cone which has the same base and the same axis", after applying Euclid XII.10.

11. In Proposition XXXI of De Centro Gravitatis solidorum libri tres (1603), Luca Valerio established that the center of gravity of a hemisphere was at the point on the axis which is 3/8 of the way from the center to the circumference--i.e., which divides the axis in a ratio of 5 to 3.

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A Translation of Evangelista Torricelli's Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta

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One of the most important threads in seventeenth century mathematics was the introduction of infinitesimal methods for solving area and volume problems. Foremost among these newfound techniques was Cavalieri's Principle, a creation of Bonaventura Cavalieri (1598-1647) that used Eudoxus' theory of proportions (as found in Book V and Book VI of Euclid's Elements) to study infinitesimal slices of objects and derive a proportion between the objects themselves.

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https://www.youtube.com/watch?v=19G8z0c73f4

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https://youtu.be/NrgxvTnoo3A

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https://www.famousscientists.org/anders-celsius/

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The history of the barometer (and how it works) - Asaf Bar-Yosef

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 Gabriel Fahrenheit was a man that made the thermometer and he was born in 1686. He was only fifteen when his parents 

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 Gabriel Fahrenheit was a man that made the thermometer and he was born in 1686. He was only fifteen when his parents 

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And now a page from our "Sunday Morning" Almanac: May 24, 1686, 329 years ago today ... a day for temperate celebration.

For that was the birthday of Daniel Gabriel Fahrenheit, the European physicist who used purified mercury to create the first modern thermometer.

Fahrenheit also invented his namesake temperature scale, where water freezes at 32 degrees, and boils at 212.

For centuries, Fahrenheit's mercury thermometer was the standard in countless fields, including medicine.

A somewhat overheated British industrial film from 1953 followed the thermometer-making process step-by-step:

"The tubing must be perfectly straight, so she's gauging it for possible twists, but few of these thermometer strips fail to measure up to the highest degree of accuracy! ... A high standard of manufacture, great care taken to pack them up, and the result is, we hope, to save many of us from 'packing up.'"

Today, modern digital thermometers have largely replaced mercury devices in doctors' offices and hospitals.

And in most of the world outside the United States, Fahrenheit's temperature scale has largely been replaced by Centigrade, where water freezes at zero degrees, and boils at 100.

Still, there remains good reason to salute Daniel Fahrenheit's contributions:

How hot is hot?
How cold is cold?
A vexing problem in days of old.
Back then it was almost cause for derision,
To claim to take temperatures with precision.
The first to do it,
And do it right,
Was Daniel Gabriel Fahrenheit.

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And now a page from our "Sunday Morning" Almanac: May 24, 1686, 329 years ago today ... a day for temperate celebration. For that was the birthday of Daniel Gabriel Fahrenheit, the European physicist who used purified mercury to create the first modern thermometer.

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pictures of daniel gabriel fahrenheit - Google Search

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The Galileo Project | Science | Tycho Brahe

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Tyge (Latinized as Tycho) Brahe was born on 14 December 1546 in Skane, then in Denmark, now in Sweden. He was the eldest son of Otto Brahe and Beatte Bille, both from families in the high nobility of Denmark. He was brought up by his paternal uncle Jörgen Brahe and became his heir.

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Tycho Brahe is a Danish astronomer, born 1546 died 1601.

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Zacharias Janssen- born in 1580, he wanted to see closer, so he invented the microscope and i think tha tis pretty dope