PROPOSITION 6.

If two circles touch one another, they will not have the

same centre.

For let the two circles ABC, CDE touch one another at the point C;

I say that they will not have the

therefore FE is also equal to FB, the less to the greater: which is impossible.

Therefore F is not the centre of the circles ABC, CDE. Therefore etc.

Q. E. D.

The

The English editions enunciate this proposition of circles touching internally, but the word (evrós) is a mere interpolation, which was no doubt made because Euclid's figure showed only the case of internal contact. fact is that, in his usual manner, he chose for demonstration the more difficult case, and left the other case (that of external contact) to the intelligence of the reader. It is indeed sufficiently self-evident that circles touching externally cannot have the same centre; but Euclid's proof can really be used for this

case too.

Camerer remarks that the proof of 111. 6 seems to assume tacitly that the points E and B cannot coincide, or that circles which touch internally at C cannot meet in any other point, whereas this fact is not proved by Euclid till III. 13. But no such general assumption is necessary here; it is only necessary that one line drawn from the assumed common centre should meet the circles in different points; and the very notion of internal contact requires that, before one circle meets the other on its inner side, it must have passed through points within the latter circle.

PROPOSITION 7.

If on the diameter of a circle a point be taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest 5 the nearer to the straight line through the centre is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of the least straight line.

Let ABCD be a circle, and let AD be a diameter of it; 10 on AD let a point F be taken which is not the centre of the circle, let E be the centre of the circle,

and from Flet straight lines FB, FC, FG fall upon the circle ABCD;

I say that FA is greatest, FD is least, and of the rest FB is 15 greater than FC, and FC than FG.

25 the two sides BE, EF are equal to the two sides CE, EF. But the angle BEF is also greater than the angle CEF; therefore the base BF is greater than the base CF.

30

For the same reason

CF is also greater than FG.

Again, since GF, FE are greater than EG,

and EG is equal to ED,

35 FD.

GF, FE are greater than ED.

Let EF be subtracted from each;

[I. 24]

therefore the remainder GF is greater than the remainder

Therefore FA is greatest, FD is least, and FB is greater than FC, and FC than FG.

I say also that from the point F only two equal straight lines will fall on the circle ABCD, one on each side of the 40 least FD.

For on the straight line EF, and at the point E on it, let the angle FEH be constructed equal to the angle GEF [1. 23], and let FH be joined.

Then, since GE is equal to EH,

45 and EF is common,

the two sides GE, EF are equal to the two sides HE, EF; and the angle GEF is equal to the angle HEF;

therefore the base FG is equal to the base FH.

[1.4]

I say again that another straight line equal to FG will not

50 fall on the circle from the point F.

For, if possible, let FK so fall.

Then, since FK is equal to FG, and FH to FG,
FK is also equal to FH,

the nearer to the straight line through the centre being 55 thus equal to the more remote: which is impossible.

Therefore another straight line equal to GF will not fall from the point F upon the circle;

therefore only one straight line will so fall. Therefore etc.

Q. E. D.

4. of the same diameter. I have inserted these words for clearness' sake. The text has simply eλaxiorη dè ʼn λaný, “and the remaining (straight line) least." 7, 39. one on each side. The word is not in the Greek, but is necessary to give the force of ἐφ' ἑκάτερα τῆς ἐλαχίστης, literally on both sides, sides, of the least."

"one

or" on each of the two

A

B

E F

De Morgan points out that there is an unproved assumption in this demonstration. We draw straight lines from F, as FB, FC, such that the angle DFB is greater than the angle DFC and then assume, with respect to the straight lines drawn from the centre E to B, C, that the angle DEB is greater than the angle DEC. This is most easily proved, I think, by means of the converse of part of the theorem about the lengths of different straight lines drawn to a given straight line from an external point which was mentioned above in the note on III. 2. This converse would be to the effect that, If two unequal straight lines be drawn from a point to a given straight line which are not perpendicular to the straight line, the greater of the two is the further from the perpendicular from the point to the given straight line. This can either be proved from its converse by reductio ad absurdum, or established directly by means of 1. 47. Thus, in the accompanying figure, FB must cut EC in some point M, since the angle BFE is less than the angle CFE.

Therefore EM is less than EC, and therefore than EB.

Hence the point B in which FB meets the circle is further from the foot of the perpendicular from E on FB than M is;

therefore the angle BEF is greater than the angle CEF.

Another way of enunciating the first part of the proposition is that of Mr H. M. Taylor, viz. "Of all straight lines drawn to a circle from an internal point not the centre, the one which passes through the centre is the greatest, and the one which when produced passes through the centre is the least; and of any two others the one which subtends the greater angle at the centre is the greater." The substitution of the angle subtended at the centre as the criterion no doubt has the effect of avoiding the necessity of dealing with the unproved assumption in Euclid's proof referred to above, and the similar substitution in the enunciation of the first part of III. 8 has the effect of avoiding the necessity for dealing with like unproved assumptions in Euclid's proof, as well as the complication caused by the distinction in Euclid's enunciation between lines falling from an external point on the convex circumference and on the concave circumference of a circle respectively, terms which are not defined but taken as understood.

Mr Nixon (Euclid Revised) similarly substitutes as the criterion the angle subtended at the centre, but gives as his reason that the words "nearer" and "more remote" in Euclid's enunciation are scarcely clear enough without some definition of the sense in which they are used, Smith and Bryant make

the substitution in 111. 8, but follow Euclid in III. 7.

On the whole, I think that Euclid's plan of taking straight lines drawn from the point which is not the centre direct to the circumference and making greater or less angles at that point with the straight line containing it and the centre is the more instructive and useful of the two, since it is such lines drawn in any manner to the circle from the point which are immediately useful in the proofs of later propositions or in resolving difficulties connected with those proofs.

Heron again (an-Nairizi, ed. Curtze, pp. 114-5) has a note on this proposition which is curious. He first of all says that Euclid proves that lines nearer the centre are greater than those more remote from it. This is a different view of the question from that taken in Euclid's proposition as we have it, in which the lines are not nearer to and more remote from the centre but from the line through the centre. Euclid takes lines inclined to the latter line at a greater or less angle; Heron introduces distance from the centre in the sense of Deff. 4, 5, i.e. in the sense of the length of the perpendicular drawn to the line from the centre, which Euclid does not use till III. 14, 15. Heron then observes that in Euclid's proposition the lines compared are all drawn on one side of the line through the centre, and sets himself to prove the same truth of lines on opposite sides which are more or less distant from the centre. The new point of view necessitates a quite different line of proof, anticipating the methods of later propositions.

The first case taken by Heron is that of two straight lines such that the perpendiculars from the centre on them fall on the lines themselves and not in either case on the line produced.

Let A be the given point, D the centre, and let AE be nearer the centre than AF, so that the perpendicular DG on AE is less than the perpendicular DH on AF

E

A

H

Therefore, by addition, AE> AF.

The other case taken by Heron is that where one perpendicular falls on the line produced, as in the annexed figure. In this case we prove in like manner that

and

GE > HF,

GA> AH.

Thus AE is greater than the sum of HF, AH,

whence, a fortiori, AE is greater than the difference

of HF, AH, i.e. than AF.

E

H

B

Heron does not give the third possible case, that, namely, where both perpendiculars fall on the lines produced, The fact is that, in this case, the foregoing method breaks down. Though AE be nearer to the centre than AF in the sense that DG is less than DH,

AE is not greater but less than AF Moreover this cannot be proved by the same method as before.

For, while we can prove that

GE > HF,

GA > AH,

H

E

B

we cannot make any inference as to the comparative length of AE, AF. To judge by Heron's corresponding note to III. 8, he would, to prove this case, practically prove 111. 35 first, i.e. prove that, if EA be produced to K and FA to L,

rect. FA, AL =rect. EA, AK,

from which he would infer that, since AK> AL by the first case,

AE < AF.

An excellent moral can, I think, be drawn from the note of Heron. Having the appearance of supplementing, or giving an alternative for, Euclid's proposition, it cannot be said to do more than confuse the subject. Nor was it necessary to find a new proof for the case where the two lines which are compared are on opposite sides of the diameter, since Euclid shows that for each line from the point to the circumference on one side of the diameter there is another of the same length equally inclined to it on the other side.

PROPOSITION 8.

If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the centre is greatest, while of the rest

H. E. II.

2