We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length
$t{{R}^{1/3}}$
on a circle of radius
$R$
, for any given
$t\,>\,0$
. In particular we prove that any arc of length
${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$
on a circle of radius
$R$
, with
$R\,>\,\sqrt{65}$
, contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points,
$\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$
, each of which lies on an arc of length
${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$
on a circle of radius
${{R}_{n}}$
.