The History of Mathematics: A Brief Course

4. APOLLONIUS^307

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FIGURE 20. Apollonius' construction of the ellipse.

In one sense, this locus definition for an ellipse is not far removed from what

we now think of as the equation of the ellipse, but that small gap was unbridgeable

in Apollonius' time. If we write LM = y and EM = χ in Fig. 20 (so that we are

essentially taking rectangular coordinates with origin at E), we see that Apollonius

is claiming that y^2 = χ • EO. Now, however, EO — EH - OH, and EH is constant,

while OH is directly proportional to EM, that is, to x. Specifically, the ratio

of OH to EM is the same as the ratio of EH to the axis. Thus, if we write

OH = kx—the one, crucial step that Apollonius could not take, since he did not

have the concept of a dimensionless constant of proportionality —and denote the

latus rectum EH by C, we find that Apollonius' locus condition can be stated as

the equation y^2 = Cx - kx^2. By completing the square on x, transposing terms,

and dividing by the constant term, we can bring this equation into what we now

call the standard form for an ellipse with center at (a, 0):

(2) + ^ =

where a = C/(2k) and b = C\fk. In these terms the latus rectum C is 2b^2 /a.

Apollonius, however, did not have the concept of an equation nor the symbolic

algebraic notation we now use, and if he did have, he would still have needed the

letter k used above as a constant of proportionality. These "missing" pieces gave

his work on conies a ponderous character with which most mathematicians today

have little patience.

Apollonius' constructions of the parabola and hyperbola also depend on the

latus rectum. He was the first to take account of the fact that a plane that produces

a hyperbola must cut both nappes of the cone. He regarded the two branches as

two hyperbolas, referring to them as "opposites" and reserving the term hyperbola

for either branch. For the hyperbola Apollonius proves the existence of asymptotes,

that is, a pair of lines through the center that never meet the hyperbola but such

that any line through the center passing into the region containing the hyperbola

does meet the hyperbola. The word asymptote means literally not falling together,

that is, not intersecting.

Books 1 and 2 of the Conies are occupied with finding the proportions among

line segments cut off by chords and tangents on conic sections, the analogs of

results on circles in Books 3 and 4 of Euclid. These constructions involve finding

the tangents to the curves satisfying various supplementary conditions such as being

parallel to a given line.