Cyrano wrote:without a great degree of understanding of geometry, trigonometry, algebra (from Persian al-jabr which might have been based on Indian works of Brahmagupta etc.
There is little reason for doubt, algebra was fairly well developed in India well before al-Khwarizmi's time:
https://mathshistory.st-andrews.ac.uk/P ... chapter-7/
The Bakhshali manuscript.... L Gurjar discusses its date in detail, and concludes it can be dated no more accurately than 'between 2nd century BC and 2nd century AD'. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried by R Hoernle). His evidence includes the language in which it was written ('died out' around 300 AD), discussion of currency found in several problems, and the absence of techniques known to have been developed by the 5th century. Further support of these dates is provided by several occurrences of terminology found only in the manuscript, (which form the basis of a paper by M Channabasappa).
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The B. Ms. highlights developments in Arithmetic and Algebra. The arithmetic contained within the work is of such a high quality that it has been suggested:
... In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic. [LG, P 53]
This quote 'throws open' the traditional Eurosceptic opinion of the history (and origins) of mathematics. Yet even today histories of mathematics rarely acknowledge this contribution of the Indian sub-continent and the B. Ms. is rarely if ever mentioned.
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By the end of the 2nd century AD mathematics in India had attained a considerable stature, and had become divorced from purely practical and religious requirements, (although it is worth noting that over the next 1000 years the majority of mathematical developments occurred within works on astronomy). The topics of algebra, arithmetic and geometry had developed significantly and it is widely thought that the decimal place value system of notation had been (generally) perfected by 200 AD, the consequence of which was far reaching.
From Aryabhatiya, written by Aryabhatta around 5th century CE:
https://mathshistory.st-andrews.ac.uk/P ... hapter-10/
In the mathematical verses of the Aryabhatiya the following topics are covered:
Arithmetic:
Method of inversion.
Various arithmetical operators, including the cube and cube root are though to have originated in Aryabhata's work. Aryabhata can also reliably be attributed with credit for using the relatively 'new' functions of squaring and square rooting.
Algebra:
Formulas for finding the sum of several types of series.
Rules for finding the number of terms of an arithmetical progression.
Rule of three - improvement on Bakshali Manuscript.
Rules for solving examples on interest - which led to the quadratic equation, it is clear that Aryabhata knew the solution of a quadratic equation.
Trigonometry:
Tables of sines, not copied from Greek work (see Figure 8.2.1).
Gupta comments:
...The Aryabhatiya is the first historical work of the dated type, which definitely uses some of these (trigonometric) functions and contains a table of sines. [RG3, P 72]
Spherical trigonometry (some incorrect).
Geometry:
Area of a triangle, similar triangles, volume rules.
It has been suggested that Aryabhata's geometry was borrowed from the Jaina works, but this seems unlikely as it is generally accepted he would not have been familiar with them.
Also of relevance is the use of 'word numerals' and 'alphabet numerals', which are first found in Aryabhata's work. We can argue that this was not due to the absence of a satisfactory system of numeration but because it was helpful in poetry. C Srinivasiengar quaintly describes it as an:
...Exceedingly queer, if original method of enumeration. [CS, P 43]
However the work of Aryabhata also affords a proof that:
...The decimal system was well in vogue. [CS, P 43]
Of the mathematics contained within the Aryabhatiya the most remarkable is an approximation for π\piπ, which is surprisingly accurate. The value given is: π\piπ = 3.1416
With little doubt this is the most accurate approximation that had been given up to this point in the history of mathematics. Aryabhata found it from the circle with circumference 62832 and diameter 20000. Critics have tried to suggest that this approximation is of Greek origin. However with confidence it can be argued that the Greeks only used π=sqrt10\pi =sqrt{10}π=sqrt10 and π=227\pi =\large\frac{22}{7}π=722 and that no other values can be found in Greek texts.
^^ this entire thesis is well worth a read if one is interested in the history of Indian mathematics. Lots of useful info.
This article has some good references as well, but it does not allow me to quote from it:
https://www.myindiamyglory.com/2019/05/ ... ent-india/
https://www.aramcoworld.com/Articles/Se ... Connection
No less significant to modern mathematics are the works of Muhammad ibn Musa al-Khwarizmi. Born in the late eighth century. In the Khwarazam oasis, in what is now Khiva, Uzbekistan, Al-Khwarizmi moved to Baghdad during the reign of Al-Mamun. There, he served as a teacher and scholar in the famous Bayt al-Hikma (House of Wisdom), where the arts of translation and scholarship reached their zenith. His writings freely reference mathematical computations borrowed from Indus Valley. In his Kitab al-Jabr wa al-Muqabala (The Book of Manipulation and Restoration), he lays out its purpose:
[To teach] What is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.
This composition served as a popular introduction to what became algebra, based on methods acquired from India, which al-Khwarizmi simplified from their original metrical (poetic) forms, writing them out in prose with explanations that have resonated ever since. The Kitab al-Jabr wa al-Muqabala, translated into Latin, made a significant impact in Europe—so much so that part of its title, al-jabr (“restoration”), became synonymous with the equation theory that we know today as algebra.
So there you have it - al-Khwarizmi's work was written as prose, making it easy for people to learn from, whereas the earlier versions used in India were done for more specific purposes. Without taking away credit from al-Khwarizmi for understanding the source material and expanding on it, the origin is clearly Indic. Even his book title contains the word "compendium" (Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala) implying he is collecting and summarizing other works, along with his contributions. Just like the so-called "Arabic numerals", though his works on the latter credit India in the book's title itself:
kitāb al-ḥisāb al-hindī, or
The book of Indian Mathematics.
There is this reference to Diophantus as the "father of Algebra". Nonsense. If one were to consider the formulae laid out in the Shulba Sutras, the Indic knowledge of algebra to express and solve practical problems predates Diophantus (~200-300 CE) by well over 700 years (800 BCE - 400 BCE), even if one were to go by western sources of Indian chronology. al-Khwarizmi was perhaps aware of this hoary history, so didn't claim to have invented the field, and in any case, historical appropriation of others' ideas seems to be a uniquely European trait.