1. ARITHMETIC METHOD FOR SOLVING EQUATIONS And applications

2. Heron, Diophantus, Brahmagupta and Al-Khwarizmi are remembered together because they all found (or however tried to discover) an arithmetic method for solving equations. In fact they only came up with formulas formed by sequential numerical calculus and they didn’t use incognita.

3. WHEN: it’s not so clear but he probably was born on the 10 AD and he died around the 70 AD;
WHERE: he lived in Egypt and spent lots of his time in the library of the university of Alexandria;
WHAT: he found some important formulas, like the formula of the area of a triangle, the one for the area of a orbitary quadrilateral and the one for the area of a cyclic quadrilateral; he invented the first steam engine and some instruments for daily life;
HOW: his most famous book is “ Metrica”, but he wrote many others; he also taught maths, mechanics and physical science;

4. Nothing is really known about Hero’s life in Egypt
Nothing is really known about Hero’s life in Egypt. As a college student Hero loved to be in the library of the University of Alexandria, because he particularly enjoyed the series of gardens and the thousands books. Hero was strongly influenced by the writings of Ctesibius of Alexandria. He may have been a student of Ctesibius. When he became older he taught at the University of Alexandria. Hero taught mathematics, mechanics, and physical science. He wrote many books and he used them as text for his students, and manuals for technicians.

5. He found the formula of the area of a triangle
He found the formula of the area of a triangle. This formula may be Archimedes’s but its presentation is credited to Heron. The area can be computed if you know the length of one side of the triangle (a, b, c) with “s” as the semi perimeter:
The second formula that Hero found is useful for determining the area of an cyclic quadrilateral (which means a quadrilateral inscribed in a circle):

6. At that time it was used as a toy. 
A hollow ball was supported on two brackets on the lid of a basin of boiling water. 
One bracket was hollow and conducted steam. 
The steam escaped from two bent pipes on the top, creating a force that made it spin around. 
The movement of the ball was used to make puppets dance. 
Although it was very simple, Hero's aeolipile illustrated Sir Isaac Newton's third law of motion which tells that for every action there is an equal and opposite reaction. 
Hero's steam engine has been the first step for the development of the jet engine.

7. To open the doors, the priest lit a fire on the altar, heating the air within and expanding it.
This expansion in volume forced water out of the sphere and into the bucket, which moved downwards under the extra weight. This bucket was connected to a rope coiled around a spindle and, as the bucket moved downwards, this spindle revolved, making the doors open.
Once the fire died down, the air contracted and, to avoid leaving a vacuum, the water siphoned from the bucket back into the sphere, causing the bucket to rise with the aid of a counterweight. As a result, the doors swung shut.
While it’s not sure that he actually built this particular device, one can imagine the wonder of a congregation seeing this machine in action; they would surely have believed that it was magic!

8. The Dioptra was a practical invention of Heron that became fundamental for building sprawling cites and erecting temples and monuments. It also became a mainstay of the Greek astronomers, allowing them to judge the position and elevation of celestial bodies.
Maybe Heron hasn’t been his creator and maybe the Dioptra have existed before, but he wrote an extensive treatise about the use of this device. The device consisted of a circular table fixed to a sturdy stand, and this was calibrated and inscribed with angles.
The surveyor levelled the device, using small water levels for accuracy, and used the disc to measure the angle between two distant objects with the aid of a rotating bar fitted with sights.
Using triangulation, the surveyor could also measure the distance between two objects, especially if the terrain as too difficult for Heron’s other surveying device, the Odometer.
Cleverly, the rotating disc could be tilted, allowing surveyors and astronomers to make vertical measurements.

9. Heron’s organ included a small wind-wheel, which powered a piston and forced air through the organ pipes, creating sounds and tweets, ‘like the sound of a flute’.
This device is believed to be the first example of wind powering a machine.

10. One of Heron’s lasting contribute to science is the syringe, a device he used to control the delivery of air or fluid with precision.
This device, as modern syringes, used suction to keep air or liquid and, when the plunger was depressed, it forced the liquid out at a controllable rate.
This device is obviously our syringes’ ancestor.

11. That fountain seemed to power itself, and used some very sophisticated pneumatic and hydraulic principles.
The fountain contained two reservoirs, one of which was filled with water.
As water was poured into the upper tray, it flowed down to the first reservoir, where it compressed the air.
This compressed air was forced into the second reservoir, where it forced the water out and created a powerful jet.
This device operated until the bottom reservoir became filled with water, when it had to be reset.

12. WHEN: he was born around 200-214 and he died between 284 and 298;
WHERE: he studied at the University of Alexandria, in Egypt;
WHAT: he recognized fractions as numbers and he improved algebraic notation;
HOW: his major contribute is a collection of 13 books called Arithmetica, he wrote books with a series of about 150 problems.

13. We only know a little about his life, but we can count how many years he lived thanks for an algebraic puzzle rhyme: “Here lies Diophantus: God gave him his boyhood one-sixth of his life, One-twelfth more as youth while whiskers grew rife; And then yet one-seventh eve marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage Met fate at just half his dad's final age. Four years yet his studies gave solace from grief; Then leaving scenes earthly he, too found relief." Did you solve the puzzle? The answer is 84 years old.

14. This is his major work and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of determinate and indeterminate equations. Nowadays we only have six of the original thirteen books which belonged to Arithmetica. Though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources.

15. Diophantus himself refers to a work which consists of a collection of lemmas called “The Porisms” (or Porismata), but this book is completely lost. Many scholars and researchers believe that The Porisms may have actually been a part included inside Arithmetica or indeed may have been the rest of Arithmetica. However we know three lemmas contained in The Porisms bacause Diophantus refered to them in the Arithmetica. One of them is that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers.

16. WHEN: he probably was born in 598 and he died in the 670;
WHERE: he lived in India;
WHAT: he found solutions for linear equations, he used firstly the zero, he found the formula for the area of a cyclic quadrilateral, he wrote the theorem of rational triangles, he gave important ideas of astronomy;
HOW: his major book is “Brahmasphutasiddhanta”, but he also wrote other three books about astronomy and arithmetic.

17. He mostly studied mathematics and astronomy and his works are really important for the modern studies.
MATHEMATICS:
He was the first to use zero as a number;
He gave the solution of the general linear equation;
He gave rules for dealing with five types of combinations of fractions;
He gave an approximate and an exact formula for the area of a cyclic quadrilateral;
He gave a theorem on rational triangles;
ASTRONOMY:
He rebutted the idea that the Moon is farther from the Earth than the Sun;
He found methods for calculating the position of heavenly bodies over time and for calculating solar and lunar eclipses;
he observed that the Earth and heaven were spherical and that the Earth is moving.

18. For a cyclic quadrilateral with sides of length a, b, c, and d, the area is given by where s is the semiperimeter

19. Given: Draw chord AC. Extend AB and CD so they meet at point P: Angle ADC and Angle ABC subtend the same chord AC from the two arcs of the circle. Therefore they are supplementary. Angle ADP is supplementary to Angle ADC. So Triangle PBC and Triangle PDA are similar. The ratio of similarity is Area of Triangle PDA = * Area of Triangle PBC Area ABCD = Area of Triangle PBC - Area of Triangle PDA

20. Perhaps Brahmagupta's most important innovation was his treatment of the number zero. Brahmagupta was the first who studied the behavior of zero in common arithmetical equations, relating zero to positive and negative numbers (which he called fortunes and debts). He correctly stated that multiplying any number by zero yields a result of zero, but erred, as did many other ancient mathematicians, in attempting to define division by zero. Nevertheless, Brahmagupta is sometimes referred to as the "Father of Zero”.

21. He was the first one who described addition, subtration, multiplication and division.
ADDITION: “[The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero”.
SUBTRACTION: “A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added”.

22. MULTIPLICATION: “The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero”.
DIVISION: “A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.
A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root”.

23. “Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals.
The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle].
Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular]”.

24. Brahmagupta changed the idea that the Moon is farther from the Earth than the Sun. He explained this showing the illumination of the Moon by the Sun. He told that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies.

25. WHEN: he probably was born around 780 AD and he died around 850 AD;
WHERE: he lived in Baghdad and he studied at “the House of Wisdom”;
WHAT: he worked on the first and second degree (linear and quadratic terms), he founded the mathematical concept of the algorithm, he worked on geography and astronomy;
HOW: he wrote two important books: “The book of calculations by completion and balancing” and “The work of Al-Khwarizmi” (which was named after him). His most important contribute is his skill of getting easy something really difficult.

26. Al-Khwarizmi received this name because in his book appeared for the first time the word “algebra”. His book is divided in two parts: in the first one he discussed about linear and quadratic terms, in the second one he focused in the aspect of business and the applications involved. Al-Khwarizmi found six standard forms of equations: 1. squares equal to roots (10x²= 20x) 2. squares equal to numbers (10x² = 25) 3. roots equal to numbers (10x = 20) 4. squares and roots equal to numbers (x² + 10x = 39) 5. squares and numbers equal to roots (x² + 39 = 10x) 6. roots and numbers equal to squares (10x + 39 = x²)

27. This is the method found by Al-Khwarizmi for solving a quadratic equation.
He used geometry in order to explain the equation x²+10x=39.
He starts with a square of side x, which
therefore represents x².
To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square.
Now the figure has area x² + 10x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 x 5/2 = 25/4.
Hence the outside square has area
4x25/4+39 = = 64.
The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

28. He did all his works rhetorically or spoken
He did all his works rhetorically or spoken. In fact notation didn’t appear until the 16th century.
He didn’t use negative numbers, so we can imagine he read Euclid’s “Element” which explain the non-use of negative.
He was the first one who used the number zero: this is because he integrated Greek and Hindu knowledge.
He reviewed Ptolemy’s ideas on geography and he checked them in detail. With other geographers he also produced the first map of the known world changing it into a globe.

29. WEBSITES USED: www.bsu.edu/web/cvjones/AlgBridge/father.htm
www-groups.dcs.st-and.ac.uk/.../Mathematicians/Heron.html
en.wikipedia.org/wiki/Brahmagupta

30. Made by: Bandini Francesca “C
Made by: Bandini Francesca “C.Colombo” classical high school Genoa, Italy