Subject: [HM] differential triangle
From: Alejandro Rivero (arivero@free4all.co.uk)
Date: Wed Jan 05 2000 - 16:20:58 EST
Thanks very much Julio.
> ...
> 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.
> ...
To me, this point seems original, is it? I mean, the use of a
finite, known, scale MP to which the small triangle is "scaled"
before to apply the rules. In usual calculus only the small
triangle is employed, and then one must suffer the pain of
solving a limit 0/0
Aside: my current ("modern", not "historical") work could be
retold as discussing how one can choose between MP (and PT)
and NQ (and QT) to make the construction. So if anyone happens
to run over some discussion of it (and of course its obvious n
dimensional generalization), I should be interested in knowing,
if only to quote accurately the primary sources.
Thanks again,
Alejandro Rivero
Julio Gonzalez Cabillon wrote:
>
> |/
> M
> /|
> / |
> / |
> / |
> / |
> / |
> N__/______R
> | / |
> |/ |
> | |
> /| |
> / | |
> / | |
> / | |
> / | |
> / | |
> / | |
> / | |
> / | |
> / | |
> / | |
> __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. ...
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