1.
This is Theon
of Alexandria (AD 335? - 405?), father of Hypatia. He authored
an edition of Euclid's Elements which included these definitions
of the terms analysis and synthesis; they were not written
by Euclid, but may have been included in the Elements by later editors
copying in a portion of some other ancient text dating from about the time
of Euclid.
2. "The
science of correct discovery" alludes to the procedure for solving
for the unknown quantity in a problem. This process involves, first,
the translation of the given information into some algebraic formulation
as an equation or inequality (zetetics), the manipulation of this
equation by the rules of algebra (poristics), and finally, the interpretation
of this manipulation as a solution of the problem (exegetics).
These three terms have not been
retained in standard mathematical terminology.
3. A syllogism is a logical argument that derives a conclusion from a pair of premises (like the famous "All men are animals. Socrates is a man. Therefore, Socrates is an animal."). An enthymeme is a syllogism in which one of the premises is suppressed.
4. The law of homogeneous terms was imposed by those like Viète who used algebraic techniques to analyze geometric problems. For them, the unknown value in an algebraic problem is some length, area or volume. So for instance, an equation like x3 + ax = b would make no sense unless x represented some length, a represented an area and b a volume, because, since the first term of the equation was a cube, it had to stand for some 3-dimensional volume. Consequently, the second term ax had to also stand for some volume, since only homogeneous quantities could be added. Thus, since x was a length, a had to be a 2-dimensional area. Similarly, the sum b had to also be a volume. It would take some time before the majority of mathematicians would be able to divorce algebra from having to refer to geometry and allow it to simply describe the relations between numbers. Still, it was possible, even for Viète, to speak of higher-dimensional quantities (4-, 5-, and 6-dimensional objects, which are called plano-planes, plano-cubes, and cubo-cubes, respectively, belying their geometric nature) even though these concepts could not be visualized in reality except in the abstract. Until then, the law of homogeneity held sway.
5. Here, class refers to geometric dimension, moving from point to line, to plane, etc.
6. The terms of an equation, like x3 + 4ax2 = 3b3, say, are said to be affected by their coefficients; the term 4ax2, for instance, represents 4 copies of the volume found by multiplying length a with area x2. The term on the right is affected in a different way: here we have 3 copies of volume b3.
7. The subscript p denotes that D represents a plane figure.
8. The subscript s denotes that Z represents a solid figure.
9. If Al-Khwarizmi is the father of algebra, then Diophantus of Alexandria (AD 200? - 284?) is its grandfather. Dipohantus' monumental work Arithmetica, a work in eight books, gives a collection of dozens of problems, of which the following is typical, taken from the English translation of Jacques Sesiano of the ninth century Arabic text of Qusta ibn-Luqa (Springer-Verlag, 1982):
Problem iv.10: We wish to find a cubic number such that, when we increase it by an arbitrary multiple of the square having the same side, the sum is a square number.It is important to note that this translation obscures the fact that Dipohantus surely did not originally use notation like x3 in his third century manuscript. He identified the unknown x with a symbol that looks like V and stood for the first two letters of the Greek word ariqmoV (number), the square x2 by the symbol Du, the first two letters of dunamiV (power), and the cube x3 by Ku, the first two letters of kuboV (cube). Nonetheless, this work is known to Viète and strongly influences his geometric predisposition to algebra. Viète talks of how Diophantus "exhibits his method in numbers and not in symbols", by which he means that, as we see in the example above, the algebraic problem is solved not in general (for a number of simplifying assumptions are made during the course of the solution) but in specific, with the ultimate goal being to find some solution, not all solutions.We put x as the side of the cube, so the cube is x3; we put for the multiplicative factor 10, and we add ten times the square of the cube's side, or x2, to x3, thus obtaining x3 + 10x2, and this is equal to a square. We assume its side to be x's [in] such [quantity] that their square is larger than 10x2, thus making the reduction possible. Putting 4x as the side of that [square], the square is 16x2, hence x3 + 10x2 equals 16x2. Let us remove the common quantity 10x2 so that 6x2 is equal to x3. Dividing that by x2, we obtain x equal to 6. [Thus] x3 is 216. The square of the side is 36; ten times that is 360, and adding this to x3 gives 576, which is a square with 24 as its side.
10. Viète has described in this proposition the first six instances of what today we call the Binomial Theorem, that is, the expansion of the powers (A + B)n of the binomial A + B. He provides a "uniform method" for finding these successive expansions.
11. In other words, (A - B)(An + An-1B2 + An-2B3 + ... + Bn) = An+1 - Bn+1.
12. He is deriving the two formulas
13. Now
he states that (A + B)(An-An-1B2
+ An-2B3-
... + Bn) = An+1 +Bn+1
if
n
+
1 is odd, and if n + 1 is even,
(A +B)(An-An-1B2
+ An-2B3-
... -Bn)
= An+1 -
Bn+1.
In both cases the signs are meant to alternate in the big factor on the
left of each equation.
14. This
is the alternate formulation of the previous set of formulas (note 13):
where we use + if n + 1 is odd and - if n + 1 is even.
15. That
is, if (A ± B)n has an even number
of terms (or, if n + 1 is even), then
Return to the text
Return to the calendar