A Source Book in Mathematics, 1200-1800 by D. J. Struik

By D. J. Struik

From the Preface

This resource e-book includes decisions from mathematical writings of authors within the Latin
world, authors who lived within the interval among the 13th and the tip of the eighteenth
century. through Latin international I suggest that there aren't any decisions taken from Arabic or other
Oriental authors, except, as with regards to Al-Khwarizmi, a much-used Latin translation
was to be had. the alternative was once made of books and from shorter writings. often in simple terms a
significant a part of the record has been taken, even supposing sometimes it used to be attainable to include
a whole textual content. All decisions are awarded in English translation. Reproductions
of the unique textual content, fascinating from a systematic standpoint, might have both increased
the measurement of the publication a long way an excessive amount of, or made it essential to decide upon fewer records in a
field the place on the other hand there has been an embarras du choix. i've got indicated in all instances the place the
original textual content may be consulted, and often this is often performed in variants of collected
works on hand in lots of college libraries and in a few public libraries as well.
It has infrequently been effortless to come to a decision to which choices choice might be given. Some
are rather noticeable; elements of Cardan's ArB magna, Descartes's Geometrie, Euler's MethodUB inveniendi,
and a number of the seminal paintings of Newton and Leibniz. within the choice of other
material the editor's selection no matter if to take or to not take was once in part guided through his personal
understanding or emotions, partially through the recommendation of his colleagues. It stands to reason
that there'll be readers who pass over a few favorites or who doubt the knowledge of a particular
choice. in spite of the fact that, i am hoping that the ultimate trend does provide a reasonably sincere photograph of the mathematics
typical of that interval within which the principles have been laid for the speculation of numbers,
analytic geometry, and the calculus.
The choice has been limited to natural arithmetic or to these fields of utilized mathematics
that had an instantaneous referring to the advance of natural arithmetic, equivalent to the
theory of the vibrating string. The works of scholastic authors are passed over, other than where,
as in relation to Oresme, they've got an immediate reference to writings of the interval of our
survey. Laplace is represented within the resource booklet on nineteenth-century calculus.
Some wisdom of Greek arithmetic should be important for a greater understanding1 of
the decisions: Diophantus for Chapters I and II, Euclid for bankruptcy III, and Archimedes
for bankruptcy IV. enough reference fabric for this objective is located in M. R. Cohen and
I. E. Drabkin, A Bource publication in Greek Bcience (Harvard collage Press, Cambridge, Massachusetts,
1948). a few of the classical authors also are simply to be had in English editions,
such as these of Thomas Little Heath.

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Extra info for A Source Book in Mathematics, 1200-1800

Example text

The proof can be given in the same way in all the other cases, since this proof is founded only on the fact that this proportion is true in the preceding base, and that every cell is equal to its preceding one plus the one above it, which is true in all cases. 6 There follow more " consequences," numbered 13-19. 7 The article ends with a " Pro blem": Given the indices of the perpendicular and of the parallel rank of a cell, to find the number of the cell, without using the arithmetic triangle. 4 This seems to be the first satisfactory statement of the principle of complete induction.

It had already been studied by Indian mathematicians, and even in the Cattle Problem, attributed to Archimedes, which leads to a "Pell" equation with A = 4729494 = 2·3·7 · ll ·29·353; see T. L. Heath, A manual of Greek mathematics (Clarendon Press, Oxford, 1931), 337. Fermat, after observing that "Arithmetic has a domain of its own, the theory of integral numbers," defines his problem as follows: Given any number not a square, then there are an infinite number of squares which, when multiplied by the given number, make a square when unity is added.

Corollary 3. By means of the same proof we can conclude that no numbers p and q exist such that p, 2q and p 2 - q2 are squares; if such numbers existed then there would be values for a and b, which would render a 4 + b4 square; for then a = v'p 2 - q2 and b = V2Jjq. Corollary 4. Suppose therefore p = x 2 and 2q = 4y 2 , then p 2 - q2 = x4 - 4y 4 • Then it could not at all happen, that x4 - 4y 4 were a square. Nor could 4x 4 - y 4 be a square; for then 16x4 - 4y 4 would be a square, which reduces it to the former case, because 16x4 is a biquadratic number.

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