Copyright 2009 by Jeff Suzuki
The web is a great place to do research...provided you know what you're doing. I've written a guide to URLs that you might find useful. This page is mainly for my own sanity, so it's obviously failed...
Many libraries (and Google) have decided that digitizing their collections, particularly their rare books collections, is a good idea. And I'm very happy with that, since it allows me to go through dusty old archives without having to get to those dusty old archives. (Not that I'm against travel...but there are logistical issues). There's just one problem: sometimes, you read a book, and then...lose it. That's what this page is about. Others who do research in the history of mathematics might find it useful, so here goes...
Ancient Egypt
Mesopotamia
Cuneiform texts from the Nippur expedition
Digitized collection of cuneiform texts and translations
Islamic mathematics
Algebra Al Khwarismi (Karpinski translation of Robert of Chester)
Al Khwarismi (Rosen translation)
Hindu mathematics
Bhaskara's Vija Ganita
Aryabhata's Aryabhattiya
Varahamihira's Panchasiddhantika (English translation at end)
Greek mathematics
Euclid's Elements (Heath translation) Volume 1 (Books I and II) Volume 2 (Books III through IX) Volume 3 (Books X through XV)
Medieval mathematics
The Earliest Arithmetics in English
Leonardo of Pisa's Liber Abaci
Renaissance mathematics
Albrecht Durer's Four Books on Measurement (German, and Gothic font besides). Many constructions based on the Geometria Deutsch by Matthias Roritzer.
Kepler's Harmonia Mundi (English translation).
I'm semiliterate in Spanish, Italian, French, Latin, and German (with the proviso that anything that's written in Gothic type is essentially unreadable for me). Fortunately, I know enough math to be able to interpolate between the bits I do understand. The following references are mainly for my own purposes, but anyone who's interested in the deep history of mathematics (and who has some ability to read in a foreign language) may be want to use them as a starting point. In no particular order:
Fermat's coordinate geometry (Latin and French). Descartes gets the credit, but Fermat's work is a lot closer to what we teach: perpendicular axes, algebraic equations generating curves, and lines having slope.
Descartes use of a line to find the tangent. In The Geometry, Descartes found the center of a tangent circle. In this letter to Claude Hardy, written in 1638, he used a straight line.
Fermat's method of extrema (Latin original and French translation). Said to be a precursor to calculus, though it is more likely based on algebraic considerations, as he goes on to discuss Viete's theory of equations. This seems to be one of the earliest appearances of the difference quotient in the form (f(x + h) - f(x))/h, and Fermat's rationale is the best I've seen (given that most calculus books don't give any rationale for it!). Fermat used this to find the tangent to a cycloid (which is a transcendental curve and is not amenable to the analysis by means of repeated roots!) Fermat used his method to find the inflection point of a curve by identifying it with an extremum of the slope of the tangent line). In 1730 Claude Rabuel modified Descartes's method to find inflection points.
Descartes tries to show Fermat's method is only usable on the simplest curves. He claims Fermat cannot apply his method to the curve now called the folium of Descartes. Fermat solves the problem in a later letter.
The first appearance of the Fermat primes. A good example of how mathematics is interconnected: the Fermat primes appear in connection with a geometric construction problem, and are relevant to another geometric construction problem...but the two construction problems are otherwise unrelated.
The first (?) appearance of the quadratic formula. Viète was the first to be able to write a general equation, so the quadratic formula (which expresses the roots of a quadratic in terms of its coefficients) could not have appeared much earlier.
The problem of Adrianus Romanus. Romanus challenged all the world to solve a 45th degree equation. Viète recognized its connection to the expansion of sin(45x), and solved it overnight.
Viète on the relationship between the roots and coefficients of a quadratic equation. This is one of the earliest results on the theory of equations, recognizing that in a quadratic equation, the product of the roots equals the constant and the sum equals the negative coefficient of the linear term.
Viète on the correspondence between arithmetic operations and geometric construction. Descartes does something similar at the beginning of his Geometry, which leads to accusations by Parisian mathematicians that Descartes plagiarized Viète.
Gauss discusses constructing the 17-sided regular polygon (Latin).
Argand's identification of multiplication by i as a 90 degree rotation (English translation).
Hamilton's introduction of the product i^2 = -1
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