Italian Pietro Mengoli was born in 1625 in Bologna. Mengoli received two Ph.D.s in the University of Bologna, Italy: one in philosophy in 1650, and the second in civil and canon law in 1653. After Cavalieri’s death, Pietro’s mathematics professor, Mengoli became chair of the University of Bologna, as well as a parish priest in Bologna. Although Mengoli had many disciplines, including astronomy, he studied mathematics with Cavalieri, and his studies resulted in his recognition in mathematics for discovering proofs on infinite series forty years before Johann Bernoulli.

Mengoli begins with the summation of geometric series in his book *Novae quadraturae arithmeticae, seu de additione fractionum *published in 1650. He then showed that “the harmonic series does not converge. In doing so he became the first person to prove that it was possible for a series whose terms tend to zero to be made larger than any given number” (JJ O’Connor & EF Robertson. Para. 4). In other words, the series diverges to an infinite amount. Another one of his investigations resulted in proving that the summation of the alternating harmonic series is equal to, or better said, “converges to” the natural logarithm of 2. So, even though Mengoli wasn’t greatly acknowledged by the public, he was still the first one to prove that the harmonic series diverges; and he proved a few more infinite series.

Pietro also published a second book in 1659 called *Geometriae Speciosae Elementa. *This book was based on a theory of limits in geometrical figures, and “it contained a definition of a definite integral in terms of the area of a plane figure rigorously given by constructing and inscribing parallelograms which sum to equal limits”; three decades before Newton and Leibniz, “Mengoli was setting up the basic rules for calculus” (O’Connor & Robertson, para. 9). Both Newton and Leibniz were influenced by Mengoli’s work, either directly or indirectly, which is the reason why Mengoli is acknowledged.

His few books that Mengoli published were only known to Collins, Leibniz, and Wallis. Pietro’s recognition “lies in the transitional position of his mathematics, midway between Cavalier’s method of indivisbles and Newton’s fluxions and Leibniz’ differentials” (O’Connor & Robertson, para. 8). In other words, Mengoli’s studies in calculus and infinite series established some groundwork for Newton and Leibniz to advance in their insightful ideas in calculus. Pietro Mengoli continues to be investigated, and he also contributed in other fields of science, such as astronomy. His Italian written books may also have contributed to the lack of acknowledgment that the math community had given Megoli; either way, we still know him as one of the first mathematicians that proof that the harmonic series diverges to infinity.

The mathematician and astronomer Eratosthenes is well-known for his technique in finding prime numbers called the “Sieve of Eratosthenes” and he is also known for closely approximating the circumference of the Sun.

Eratosthenes was born in Cyrene, Greece in 276 BC, which is now known Libya in North Africa. He studied in Athens, but then Ptolemy III, the ruler of Cyrene at the time, appointed Eratosthenes head librarian in the Great Library at Alexandria University (now in Egypt). Eratosthenes method of the prime number sieve is used as an important tool in number theory research, and he was the first person to successfully measure the circumference of the Earth quite accurately. When he was about 80 years old, he went blind and because of this blindness he decided to starve himself to death.

Eratosthenes was born in Cyrene, Greece in 276 BC, which is now known Libya in North Africa. He studied in Athens, but then Ptolemy III, the ruler of Cyrene at the time, appointed Eratosthenes head librarian in the Great Library at Alexandria University (now in Egypt). Eratosthenes method of the prime number sieve is used as an important tool in number theory research, and he was the first person to successfully measure the circumference of the Earth quite accurately. When he was about 80 years old, he went blind and because of this blindness he decided to starve himself to death.

While head librarian in the Great Library at Alexandria, Eratosthenes presented a step-by-step procedure for finding prime numbers up to a given limit that he called the Sieve of Eratosthenes. Let us remember that prime numbers are natural numbers that are only divisible by one and itself. To find these so called primes, Eratosthenes propose to write down all known natural numbers starting from the first prime, which is two, and above and then “sieving out” every multiple of that prime number, and then moving on to the next prime, three, and “sieve out” all the multiples of the new prime which in result cancels out the composite numbers and we are left with the primes; this process could continue on forever. The sieve is a great way to find single digit primes, but this process becomes very tedious to find double or triple digit primes. The sieve is still a great ancient form used to find prime numbers.

Another of Eratosthenes great works is his geometrical method for finding the circumference of the Earth. Details of his work were lost in his treatise *On the Measurement of the Earth*. Authors J J O’Connor and E F Robertson wrote a great article about Eratosthenes of Cyrene; they mention that details of Eratosthenes “calculations appear in works by other authors such as Cleomedes, Theon of Symyrna, and Strabo” (para.12). Anyways on the summer solstice, Eratosthenes noticed that the sunlight reached down to the bottom of a well in Syene, which is near the city of Aswan in Egypt by the Nile. Then, after he calculated the distance between the well in Syene and the library of Alexandria, he measured the angle of a shadow cast in Alexandria which measured seven degrees. And since sunlight rays are parallel lines, Eratosthenes figured that the shadow cast by the building must a transversal line cutting through the center of the Earth.

Using Euclid’s proposition that alternate-interior-angles are equal, Eratosthenes then knew that the Earth’s arc between Syene and Alexandria is seven degrees.Also, he hired some people to measure the distance between the Syene and Alexandria, and the distance they measure about 800 km. Now using ratios and simple geometric calculations, he concluded that the circumference of the Earth was approximately 41, 000 km (about 25,000 miles).

** **__ 7°__ = __ 800 km__

**360°** C (**circumference of the Earth)**

** C = **__(800 km * 360°)__ = 41142.8571429 km

**7°**Modern circumference of the Earth is 24,901 miles. So, Eratosthenes came real close to actual measurements of the circumference of the Earth which is why he is still remembered as a great astronomer as well.

Eratosthenes is revered for his two great contributions in mathematics and astronomy: the Sieve of Eratosthenes and measuring the circumference of the Earth. The sieve is still used in modern number theory research. His scientific calculations for measuring the circumference of the Earth have awarded him with recognition beyond he and his peers could ever dream of. His early studies in Geometry helped him see the connections necessary in order for him to do what many people behind his time have failed to do.

O’Coonor, J J and Robertson, E F. “Eratosthenes of Cyrene”

Rosenberg, Matt. “Eratosthenes”

http://geography.about.com/od/historyofgeography/a/eratosthenes.htm>

http://www.math.wichita.edu/history/men/eratosthenes.html

Eudoxus of Cnidus was well known for his two major contributions in mathematics: the theory of proportions and his method of exhaustion. Eudoxus was born around 395-390 BC in Cnidus, Asia Minor, which is now known as Turkey. Despite the Eudoxus’s work being lost, many mathematicians throughout history have referred to his work; including, the famous Euclid in his book V Elements dealing with the theory of proportions and book X dealing with the method of exhaustion. Eudoxus’s contributions are fundamental tools for the composition of further important mathematical theories and proofs.

During Eudox’s time period, the Pythagoreans had come up with the idea of commensurability. Commensurability means that the ratio of lengths of two line segments can be express as rational numbers. This idea was later rejected by a counter example in where the ratios could not be express as whole numbers; in other words, the ratios could only be expressed with irrational numbers, as the illustration on the left shows. Thus Eudoxus came up with his theory of proportions, which resolved many of the problems that were doubtfully proven by commensurability. His theory is now found in Book V Euclid’s Elements.

Eudoxu’s second major contribution is the method of exhaustion, now known as a limit. Eudoxus calculated the curvilinear areas and the volumes of regular polygons. With this method, “the successive inscribed figures ‘exhaust’ the original figure” (Wise, David). This technique lasted for 2 millenniums, until calculus was invented in the 17th century. With this technique, the great Archimedes was able to approximate the circumference of a circle.

Both of Eudoxus’ contributions set the stage for Euclid and Archimedes. He is said to be the second greatest mathematician of antiquity, Archimedes being the first. Thanks to Eudoxu’s theory of proportions we have irrational numbers, such as pi and square root of two, and thanks to his method of exhaustion mathematicians were able to approximate the curvilinear areas and volumes of curve shapes and figures for almost 2000 years. Eudoxus and his two major advances in mathematics are worthy of recognition.

Mendell, Henry Ross. “Eudoxus of Cnidus”

Jan. 2014. http://www.britannica.com/EBchecked/topic/195005/Eudoxus-of-Cnidus

Wise, David. “Eudoxus’ Influence on Euclid’s Elements with a close look at The Method of

Exhaustion”

Krimson. “The Method of Exhaustion is not a method, and exhaustive in a very particular

sence.”

The Moscow Papyrus is one of the oldest written mathematical texts from ancient Egypt around 2000 B.C to 1800 B.C. The Moscow Papyrus contains 25 mathematical problems. Some of the problems are unreadable or too damaged to translate. Problem 14 from the Moscow Papyrus shows an illustration with an example to find the volume of a truncated pyramid. The modern formula for a similar figure, called frustum, is V= (a^2 + a*b + B^2)*(h/3), where a,b, and h are shown in the diagram below.

The text is now located in the largest museum in Europe, the Pushkin Museum of Fine Arts in Russia. This ancient piece of mathematical history inspires awe and wonder in those appreciative of mathematics because the Egyptians never explained “how” their example in problem 14 worked, nor did they show any deductive reasoning behind this problem.

Finding the volume of a truncated pyramid is very challenging if doing so by experiments alone. One cannot just stumble to this conclusion just by trial and error. No one knows how the Egyptians derived the formula, and considering that the author of the text remains unknown, we might never know. However, many historians of mathematics have their theories, but even their theories are imminent. Mathematical historians have little to no evidence to prove their theories, since the text was written over 4000 years ago.

Amazingly, considering the age of the Moscow Papyrus, it is still being carefully examine by Egyptologists. This is a valuable piece of history and is worthy of being recognized in a great museum like the Pushkin Museum. One can only wonder of the brilliance or good fortune of the Egyptians.

Vetter, Quido. “Problem 14 of the Moscow Mathematical Papyrus”

Carlesdorce. “Moscow Mathematical Papyrus”

Mastin, Luke. “Egyptian Mathematics”

Dunham, William. “ Journey Through Genius The Great Theorems of Mathematics” New York: