An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

LIMITS

5

C, of the circle, at the same instant. It is clear that the perpendicular from P1 intersects the radius from P in a point of
the curve, and that the vertical intercept is the ordinate y, of
that point; that OP x and =, the measuring arc to the
angle PCO. Also - =- Let the student show that
x.' 2(
7rr
y-= ( — ) )tan
2(a
is the equation of the curve.
(a(a — x\
(a -, x) _ a - ( r-) _2 a 2\ a )
Now 3= -
7TX f7 rr 7T a,-\
cot tan - - - tan - ---
2 \2 2 a\ a
7r '(a - (a
and t a becomes indefinitely near to unity when x is
tan ----
2 a
taken indefinitely near to a; therefore
2a
YI =,-'= a 7r
Hence, if yo is the ordinate for x = a,
2a Diameter of circle
Jo Maximum ordinate
We have then a mechanical means of determining an approximate value of 7r, by simply dividing the diameter of the circle by
the maximum ordinate as found by construction. An approximate value of wr (one of the earliest determinations) was found
in some such manner by Dinostratus, about 400 B.C. The
curve is frequently called the Qzadratrix of Dinostratls, but
was invented about half a century before the time of his quad1atlurc by Hippias, supposed of Elis. The curve is also employed in the trisection (transceiZdcntal) of the arc, and, indeed,
may be used to find any fraction of an arc. An irregular
draughting curve is of great assistance in the construction.