Commentary on Proposition III.8

Back to Proposition III.8.

[Folio 0026 | p. 20]

As De Morgan points out, there are here two assumptions similar to that tacitly made in the proof of III. 7, namely that K falls within the triangle DLM and E outside the triangle DFM. These facts can be proved in the same way as the assumption in III. 7. Let DE meet FM in Y and LM in Z. Then, as before, MZ is less than ML and therefore than MK. Therefore K lies further than Z from the foot of the perpendicular from M on DE. Similarly E lies further than Y from the foot of the same perpendicular.

Heron deals with lines on opposite sides of the diameter through the external point in a manner similar to that adopted in his previous note. For the case where E, F are the second points in which AE, AF meet the circle the method answers well enough. If AE is nearer the centre D than AF is,

sqs. on DG, GE $=$ sqs. on DH, HF

and

sqs. on DG, GA $=$ sqs. on DH, HA,

whence, since DG $<$ DH, it follows that GE $>$ HF, and AG $>$ AH, so that, by addition, AE $>$ AF.

But, if K, L be the points in which AE, AF first meet the circle, the method fails, and Heron is reduced to proving, in the first instance, the property usually deduced from III. 36. He argues thus: AKD being an obtuse angle,

[II. 12]sq. on AD $=$ sum of sqs. on AK, KD and twice rect. AK, KG.

ALD is also an obtuse angle, and it follows that

sum of sqs. on AK, KD and twice rect. AK, KG is equal to

sum of sqs. on AL, LD and twice rect. AL, LH.

Therefore, the squares on KD, LD being equal,

sq. on AK and twice rect. AK, KG $=$ sq. on AL and twice rect. AL, LH,

or

sq. on AK and rect. AK, KE $=$ sq. on AL and rect. AL, LF,

i.e.

rect. AK, AE $=$ rect. AL, AF.

But, by the first part, AE $>$ AF. Therefore AK $<$ AL.

3. 7, 8 deal with the lengths of the several lines drawn to the circumference of a circle (1) from a point within it, (2) from a point outside it; but a similar proposition is true of straight lines drawn from a point on the circumference itself: If any point be taken on the circumference of a circle, then, of all the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is; of any others that which is nearer to the straight line which passes through the centre is greater than one more remote; and from the same point there can be drawn to the circumference two straight lines, and only two, which are equal to one another, one on each side of the greatest line.

[Folio 0027 | p. 21]

The converses of III. 7, 8 and of the proposition just given are also true and can easily be proved by reductio ad absurdum. They could be employed to throw light on such questions as that of internal contact, and the relative position of the centres of circles so touching. This is clear when part of the converses is stated: thus (1) if from any point in the plane of a circle a number of straight lines be drawn to the circumference of the circle, and one of these is greater than any other, the centre of the circle must lie on that one, (2) if one of them is less than any other, then, ($a$) if the point is within the circle, the centre is on the minimum straight line produced beyond the point, ($b$) if the point is outside the circle, the centre is on the minimum straight line produced beyond the point in which it meets the circle.