Early Modern Period (Renaissance) (1450 A.D. – 1700 A.D.)

A. Trigonometry and Logarithms• Publication of precise trigonometry tables, improvement of

surveying methods using trigonometry, and mathematical analysis of trigonometric relationships. (approx. 1530 – 1600)

• Logarithms introduced by Napier in 1614 as a calculation aid. This advances science in a manner similar to the introduction of the computer.

B. Symbolic Algebra and Analytic Geometry • Development of symbolic algebra, principally by the French

mathematicians Viete and Descartes • The Cartesian coordinate system and analytic geometry deve-

loped by Rene Descartes and Pierre Fermat (1630-1640).C. Creation of the Calculus

• Calculus co-invented by Isaac Newton and Gottfried Leibniz. Major ideas of the calculus expanded and refined by others, especially the Bernoulli family and Leonhard Euler. (approx. 1660 – 1750).

• A powerful tool to solve scientific and engineering problems, it opened the door to a scientific and mathematical revolution.

The Renaissance (reborn") was a cultural movement that spanned the period roughly from the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The changes of the Renaissance were not uniformly experienced across

Europe .

Although the Renaissance saw revolutions in many intellectual pursuits, as well as social and political upheaval, it is perhaps best known for its artistic developments and the contributions of such polymaths as Leonardo da Vinci and Michelangelo, who inspired the term "Renaissance man".

A New World ViewA new "world view" was created - the way one looked at and answered basic questions:What are humans?What is the purpose of life?Why am I here?

Italy was the birthplace of the Renaissance for many reasons. First of all, since Italy was the center of ancient Roman history, it was natural for the Renaissance to start there.

Some things such as architectural remains, antique statues, coins and inscriptions reminded Italians of the glory of the Roman Empire.

Secondly, Italy was different from the rest of Europe in another way. Italian cities had survived the Middle Ages and cities like Florence, Milan, Venice, and Genoa all grew into wealthy, prosperous cities of manufacturing and trade. Rome and Naples also contributed to the Renaissance cultural recovery.

Wealthy Italian merchants encouraged the cultural rebirth by applying political and economic leadership. They also had great attitudes that helped shape the Italian Renaissance. These merchants suggested education and individual achievements and they spent quite a bit of money on the arts.

A New Conception of the WorldSkepticism (Şüphecilik). Humans began to have a questioning attitude and challenged tradition and authority. Secularism (Laiklik). They believed life on earth was more important than the afterlife. There was greater emphasis on this life and less on the afterlife and on God - the world is to be enjoyed.

During the early Renaissance, Florence was a very important city. It came to symbolize the Italian Renaissance. Florence was very much like ancient Athens because it produced a great number of talented poets, artists, architects, scholars, and scientists in a short period of time. Humanism was an intellectual movement that was at the heart of the Italian Renaissance. It put an emphasis on man, his intellect, and his life on earth. It also stated that the Church shouldn't rule certain matters and it rediscovered ancient items, thoughts, and beliefs. Humanist scholars hoped to use their wisdom to increase their understanding of their own times.

Galileo Galilei(1564-1642)

Galileo was born in Pisa, Italy on February 15, 1564.

His father, was a musician. Galileo was the first of six (though some people believe seven) children.

His family belonged to the nobility (asalet) but was not rich. In the early 1570's, he and his family moved to Florence.

In 1581, Galileo began studying at the University of Pisa .

At the University of Pisa, Galileo learned the physics of the Ancient Greek scientist, Aristotle. However, Galileo questioned the Aristotelian approach to physics. Aristotelians believed that heavier objects fall faster through a medium than lighter ones.

Galileo eventually disproved this idea by asserting that all objects, regardless of their density, fall at the same rate in a vacuum. To determine this, Galileo performed various experiments in which he dropped objects from a certain height. In one of his early experiments, he rolled balls down gently sloping inclined plane and then determined their positions after equal time intervals. He wrote down his discoveries about motion in his book, De Motu, which means "On Motion."

In 1592, Galileo was appointed professor of mathematics at the University of Padua. While teaching there, he frequently visited a place called the Arsenal, where Venetian ships were docked and loaded. During his visits to the Arsenal, he became fascinated by nautical technologies, such as the sector and shipbuilding. A year later, he patented a model for a pump. His pump was a device that raised water by using only one horse. Galileo's belief in the Copernican System (Güneş merkezli evren) eventually got him into trouble with the Catholic Church (dünya merkezli evren).

A committee of consultants declared to the Inquisition (soruşturma) that the Copernican proposition that the Sun is the center of the universe was a heresy (sapkınlık). Because Galileo supported the Copernican system, he was warned that he should not defend Copernican theories. In 1633, Galileo was found guilty of heresy for his Dialogue, and was sent to his home near Florence where he was to be under house arrest for the remainder of his life. In 1638, the Inquisition allowed Galileo to move to his home in Florence, so that he could be closer to his doctors.

By that time he was totally blind. In 1642, Galileo died at his home outside Florence. SEE VIDEO 1

Napier’s Logarithms

In his Mirifici Logarithmorum Canonis descriptio (1614) (A Description of the Wonderful Canon of Logarithms) the Scottish John Napier introduced the concept of logarithms as an aid to calculation.

John Napier1550-1617

Henry Briggs and the Development of Logarithms

Napier’s concept of a logarithm is not the one used today. Soon after Napier’s book was published the English mathematician Henry Briggs collaborated with him to develop the modern base 10 logarithm.

Tables of this logarithm and instructions for their use were given in Briggs’ book Arithmetica Logarithmica (1624).

A page from this work is shown on the left.

Leibniz’s Calculus

Leibniz and Newton independently developed the calculus during the same time period. Although Newton’s version of the calculus led him to his great discoveries, Leibniz’s concepts and his style of notation form the basis of modern calculus.

Gottfried Leibniz

1646 - 1716

A diagram from Leibniz's famous 1684 article in the journal Acta eruditorum.

Newton and Leibniz

Newton: 1643 – 1727

Leibniz: 1646 - 1716