Commentary on Proposition IV.1

Back to Proposition IV.1.

Of this problem as it stands there are of course an infinite number of solutions; and, if a particular point be chosen as one extremity of the chord to be “fitted in,” there are two solutions. More difficult cases of “fitting into” a circle a chord of given length are arrived at by adding some further condition, e.g. (1) that the chord is to be parallel to a given straight line, or (2) that the chord, produced if necessary, shall pass through a given point. The former problem is solved by Pappus (III. p. 132); instead of drawing the chord as a tangent to a circle concentric with the given circle and having as radius a straight line the square on which is equal to the difference between the squares on the radius of the given circle and on half the given length, he merely draws the diameter of the circle which is parallel to the given direction, measures from the centre along it in each direction a length equal to half the given length, and then draws, on one side of the diameter, perpendiculars to it through the two points so determined.

The second problem of drawing a chord of given length, being less than the diameter of the circle, and passing through a given point, is more important as having been one of the problems discussed by Apollonius in his work entitled νεύσεις, now lost. Pappus states the problem thus (VII. p. 670): “A circle being given in position, to fit into it a straight line given in magnitude and verging (νεύουσαν) towards a given (point).” To do this we have only to place any chord HK in the given circle (with centre O) equal to the given length, take L the middle point of it, with O as centre and OL as radius describe a circle, and lastly through the given point C draw a tangent to this circle meeting the given circle in A, B. AB is then one of two chords which can be drawn satisfying the given conditions, if C is outside the inner circle; if C is on the inner circle, there is one solution only; and, if C is within the inner circle, there is no solution.

H L K O A C B

Thus, if C is within the outer (given) circle, besides the condition that the given length must not be greater than the diameter of the circle, there is another necessary condition of the possibility of a solution, viz. that the given length must not be less than double of the straight line the square on which is equal to the difference between the squares (1) on the radius of the given circle and (2) on the distance between its centre and the given point.