Subject: Re: [HM] Barrow
From: Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Date: Tue Jan 04 2000 - 10:18:53 EST
On 2 Jan 00, Alejandro Rivero asked:
> What is exactly the small triangle of Barrow?
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N__/______R
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__A_______T___________Q_________P_____________
The "differential triangle" (or Barrow's small triangle) is
the triangle MRN. See Barrow's description below according
to Child's translation:
"Let AP, PM be two straight lines given in position, of
which PM cuts a given curve in M, and let MT be supposed
to touch the curve at M, and to cut the straight line a T
In order to find the quantity of the straight line PT,
I set off an indefinitely small arc, MN, of the curve;
then I draw NQ, NR parallel to MP, AP; I call MP = m,
PT = t, MR = a, NR = e, and other straight lines,
determined by the special nature of the curve, useful
for the matter in hand, I also designate by name; also I
compare MR, NR (and through them, MP, PT) with one
another by means of an equation obtained by calculation;
meanwhile observing the following rules.
RULE 1. In the calculation, I omit all terms containing
a power of $a$ or $e$, or products of these (for these
terms have no value).
RULE 2. After the equation has been formed, I reject all
terms consisting of letters denoting known or determined
quantities, or terms which do not contain $a$ or $e$ (for
these terms brought over to one side of the equations,
will always be equal to zero).
RULE 3. I substitute $m$ (or MP) for $a$, and $t$ (or PT)
for $e$. Hence at length the quantity of PT is found.
Moreover, if any indefinitely small arc of the curves
enters the calculation, an indefinitely small part of the
tangent, or of any straight line equivalent to it (on
account of the indefinitely small size of the arc) is
substituted for the arc. But these points will be made
clearer by the following examples. ...
All the best,
Julio Gonzalez Cabillon
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