Every right line drawn perpendicular to a chord at its middle, passes through the centre of the circle and through the middle of the arc subtended by the chord.--Division of an arc into two equal parts.--To pass the circumference of a circle through three points not in the same right line. The tangent at any point of a circumference is perpendicular to the radius passing through that point. The arcs intercepted in the same circle between two parallel chords, or between a tangent and a parallel chord, are equal. _Measure of angles._ If from the summits of two angles two arcs of circles be described with the same radius, the ratio of the arcs included between the sides of each angle will be the same as that of these angles.--Division of the circumference into degrees, minutes, and seconds.--Use of the protractor. An angle having its summit placed, 1º at the centre of a circle; 2º on the circumference of that circle; 3º within the circle between the centre and the circumference; 4º without the circle, but so that its sides cut the circumference; to determine the ratio of that angle to the right angle, by the consideration of the arc included between its sides. From a given point without a circle, to draw a tangent to that circle. To describe, on a given line, a segment of a circle capable of containing a given angle. _To make surveys for plans._ (_Lever des plans._) Tracing a straight line on the ground.--Measuring that line with the chain. Measuring angles with the graphometer.--Description of it. Drawing the plan on paper.--Scale of reduction.--Use of the rule, the triangle, and the protractor. To determine the distance of an inaccessible object, with or without the graphometer. Three points, A, B, C, being situated on a smooth surface and represented on a map, to find thereon the point P from which the distances AB and AC have been seen under given angles. “The problem of the three points.” “The _Trilinear_ problem.” _Of the contact and of the intersection of circles._ Two circles which pass through the same point of the right line which joins their centres have in common only that point in which they touch; and reciprocally, if two circles touch, their centres and the point of contact lie in the same right line. Conditions which must exist in order that two circles may intersect. _Properties of the secants of the circle._ Two secants which start from the same point without the circle, being prolonged to the most distant part of the circumference, are reciprocally proportional to their exterior segments.--The tangent is a mean proportional between the secant and its exterior segment. Two chords intersecting within a circle divide each other into parts reciprocally proportional.--The line perpendicular to a diameter and terminated by the circumference, is a mean proportional between the two segments of the diameter.