The aptitude of manking to maths likely progressively appeared since the era of Homo sapiens as it became really obvious in the Middle East, at the time of the city-states and Egypt only, after having, maybe, inspired the peoples of the megaliths. For the purpose of that swift overview, we won't distinguish between arithmetic -the art of using figures- and geometry -the art of reasoning about forms and space- as those two categories are used for the liberal arts. Both, in any case, for the periods with which we concern ourselves, are not distinguished between them

It looks like that mathematics appear in Sumer, that first city-state in the Middle East- in the same time than writing, and for the same utilitarian reasons. Sumerians, then the Babylonians, began developping the first even numeral system for the purpose of trade, agriculture or the computation of fields' surfaces. Such a system is a base-60 one, meaning it works with 6 numerals only. It's from that faraway origin that the use of dividing the hour in sixty minutes, the minute in sixty seconds, or to divide a circle into 360 degrees, takes it source. Those ancient worlds did know the 4 operations, the square and cubic roots, and the quadratic equation (which allow for the drawing of curves). Sumer and Babylon evolved later towards the use of mathematics mainly for astronomy. Babylonians knew a form of trigonometry far more advanced than the modern-day version as they figured it out 1,000 years before the Greeks. That was revealed by a 3,700-year-old clay tablet which holds a series of lines representing the ratios for a series of right-angled triangles ranging from almost a square to almost a flat line. Babylonian trigonometry likely helped Babylonians in terms of building their cities and monuments, and surveying agricultural fields. Babylonian trigonometry differs from the one based upon Pythagoras' theorem as the latter is based on angles, instead of ratios. The same logics, the one of a first, utilitarian use of the mathematics is found too in the ancient Egypt, albeit with less documentary source. The Egyptians, further, were distinguished by applicating geometry to their works of construction, of them, of course, the pyramids. What is important, for those ancient mathematics, is that no uniformity exists, as each of the civilizations created its own systems and concepts. And that is true too for the other civilizations outside the Fertile Crescent, where a history of mathematics may be found also, like in China, the pre-Colombian civilizations of America, or India, for example

It's the Greeks who will have the mathematics turn from their utilitarian use into a knowledge-based approach of it, as they made it a domain of the philosophy. It's the Greeks, thus, who create the concept of the deductive reasoning (like the famed theorems of Thales, or Pythagoras), and it's those who begin to give a real, theoretical structure to the maths! Only the Indian, Jaina mathematicians of between 400 B.C. and 200 A.D. are known to have endeavoured to that. The Greek knowledges into mathematics are taken partly from the Egyptian and the other Middle Eastern civilization ones, and maybe even from India, as they are attaining to their apogee during the hellenistic period, between 300 B.C. and the year 0. The Greeks however, do too keep to have their maths specific in their methods and concepts, with no ability to an universal use. Neither Rome, nor the Byzantine Empire, or even the Early Middle Ages will show particular aptitudes to mathematics. The great Greek knowledges thus, will mainly pass to the Arabs. Those, beginning in about 800 A.D., as they put to profit the pillage of the Greek libraries along with their trade relationship with India and China, will initiate a 700-year period during which they will be the main contributors to the mathematical science. In Baghdad, under the reign of the Caliph Al-Mamun, the "House of Wisdom" is created, a center for the study of mathematics and scholarship generally. The famed Al-Khwarizmi is part of that, writing fundamental treaties in maths, and founding the algebra! Algebra will keep being developed by Abu-Kamil, one of his successors. Al-Battani (855-923), as far as he is concerned, is an astronomer and a mathematician. As an astronomer, his work is of importance, as he corrects the computations of Ptolemy, he draws new tables for the Sun and the Moon, writes about the division of the celestial sphere, he manages to compute the value of the precession of the equinoxes and of the tilt of the Earth's axis. As a mathematician, Al-Battani founded the modern trigonometry, using in his astronomical works the sine and the tangent. Like another Baghdadian astronomer, Al-Battani was originating from the pagan tribe of the 'Sabians', which dwelled in Harran, in the current southeastern part of Turkey, who were worshipers of the stars and other celestial bodies, and close to the Greek culture. The Arab authors, moreover, signaled themselves further, generally, by translating Greek and Indian treaties. It's the Arab mathematicians who allowed for the transmission of the numeral suits of 9 figures (1 to 9), which, thus, began to be a most fundamental base of the mathematics. That numeral suite, in its farthest origins, came from the Buddhist world. It then passed into India, whence it eventually reached to the court of the Caliph Al-Mansur, in 776 A.D. From there, the Arab mathematicians will have it spreading in the whole Middle East. The "zero", as far as it is concerned, originates in India too and is transmitted to the rest of the world trough the Arab mathematicians too, at that same period. The 1 to 9 series, and the '0' will reach the West by the 12th century only

The only mathematical author of the Early Middle Ages, in the West, is Boethius (480-524). Boethius, a 6th century philosopher -with an immense Greek culture- originating from a great Roman -and Christian- family, and as he was a political man of transition in Italy, under the reign of the Ostrogoths, endeavoured, like many other authors of the time, to preserve the antique knowledge amidst the troubles of that time, and especially, in his case, the philosophical works. He wrote too the 'De institutione arithmetica', a translation of a Greek work by Nicomachus (who was a 1st century A.D. neo-Pythagorean) and a series of abstracts from the 'Geometry' by Euclid, the famed Greek scientist from the hellenistic period. The works of Boethius remained at the basis of the mathematical studies until the 12th century. The continuation of the use of the Roman numerals only during the Carolingian era was not prone to a great use of the mathematics

Website Manager: G. Guichard, site Learning and Knowledge In the Carolingian Times / Erudition et savoir à l'époque carolingienne, http://schoolsempire.6te.net. Page Editor: G. Guichard. last edited: 8/28/2017. contact us at geguicha@outlook.com