The words usually given as a part of the corollary "and that a straight line touches a circle at one point only, since in fact the straight line meeting it in two points was proved to fall within it" are omitted by Heiberg as being an undoubted addition of Theon's. It was Simson who added the further remark that "it is evident that there can be but one straight line which touches the circle at the same point."

PROPOSITION 17.

From a given point to draw a straight line touching a given circle.

Let A be the given point, and BCD the given circle; thus it is required to draw from the point A a straight line touching the circle BCD.

For let the centre E of the circle

be taken :

[III. 1] let AE be joined, and with centre E and distance EA let the circle AFG be described;

from D let DF be drawn at right angles to EA,

and let EF, AB be joined ;

I say that AB has been drawn from

the point A touching the circle BCD.

For, since E is the centre of the circles BCD, AFG,

EA is equal to EF, and ED to EB;

therefore the two sides AE, EB are equal to the two sides FE, ED;

and they contain a common angle, the angle at E;

therefore the base DF is equal to the base AB, and the triangle DEF is equal to the triangle BEA, and the remaining angles to the remaining angles; [1.4] therefore the angle EDF is equal to the angle EBA. But the angle EDF is right;

therefore the angle EBA is also right.

Now EB is a radius ;

and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [1. 16, Por.] therefore AB touches the circle BCD.

Therefore from the given point A the straight line AB has been drawn touching the circle BCD.

Q. E. F.

The construction shows, of course, that two straight lines can be drawn from a given external point to touch a given circle; and it is equally obvious that these two straight lines are equal in length and equally inclined to the straight line joining the external point to the centre of the given circle. These facts are given by Heron (an-Nairīzī, p. 130).

It is true that Euclid leaves out the case where the given point lies on the circumference of the circle, doubtless because the construction is so directly indicated by III. 16, Por. as to be scarcely worth a separate statement.

An easier solution is of course possible as soon as we know (111. 31) that the angle in a semicircle is a right angle; for we have only to describe a circle on AE as diameter, and this circle cuts the given circle in the two points of contact.

PROPOSITION 18.

If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.

For let a straight line DE touch the circle ABC at the point C, let the centre F of the circle ABC be taken, and let FC be joined from F to C;

I say that FC is perpendicular to
DE.

For, if not, let FG be drawn
from F perpendicular to DE.
Then, since the angle FGC is
right,

the angle FCG is acute; [1. 17] and the greater angle is subtended by the greater side;

therefore FC is greater than FG. But FC is equal to FB;

A

therefore FB is also greater than FG,

E

the less than the greater: which is impossible.

Therefore FG is not perpendicular to DE.

B

G

[1. 19]

Similarly we can prove that neither is any other straight

line except FC ;

therefore FC is perpendicular to DE.

Therefore etc.

Q. E. D.

3. the tangent, ἡ ἐφαπτομένη.

Just as III. 3 contains two partial converses of the Porism to III. 1, so the present proposition and the next give two partial converses of the corollary to III. 16. We may show their relation thus: suppose three things, (1) a tangent at a point of a circle, (2) a straight line drawn from the centre to the point of contact, (3) right angles made at the point of contact [with (1) or (2) as the case may be]. Then the corollary to III. 16 asserts that (2) and (3) together give (1), III. 18 that (1) and (2) give (3), and III. 19 that (1) and (3) give (2), i.e. that the straight line drawn from the point of contact at right angles to the tangent passes through the centre.

PROPOSITION 19.

If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn. For let a straight line DE touch the circle ABC at the point C, and from C let CA be

drawn at right angles to DE;

I say that the centre of the circle is on AC.

For suppose it is not, but, if possible, let F be the centre, and let CF be joined.

Since a straight line DE touches the circle ABC,

and FC has been joined from the centre to the point of contact,

FC is perpendicular to DE; therefore the angle FCE is right.

But the angle ACE is also right;

therefore the angle FCE is equal to the angle ACE,

the less to the greater: which is impossible.

Therefore F is not the centre of the circle ABC. Similarly we can prove that neither is any other point except a point on AC.

Therefore etc.

Q. E. D.

We may also regard 111. 19 as a partial converse of III. 18. Thus suppose (1) a straight line through the centre, (2) a straight line through the point of contact, and suppose (3) to mean perpendicular to the tangent; then III. 18 asserts that (1) and (2) combined produce (3), and III. 19 that (2) and (3)

produce (1); while again we may enunciate a second partial converse of 111. 18, corresponding to the statement that (1) and (3) produce (2), to the effect that a straight line drawn through the centre perpendicular to the tangent passes through the point of contact.

We may add at this point, or even after the Porism to 11. 16, the theorem that two circles which touch one another internally or externally have a common tangent at their point of contact. For the line joining their centres, produced if necessary, passes through their point of contact, and a straight line drawn through that point at right angles to the line of centres is a tangent to both circles.

PROPOSITION 20.

In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.

Let ABC be a circle, let the angle BEC be an angle 5 at its centre, and the angle BAC an angle at the circumference, and let them have the same circumference BC as base;

I say that the angle BEC is double of 10 the angle BAC.

15

For let AE be joined and drawn through to F.

Then, since EA is equal to EB, the angle EAB is also equal to the

angle EBA;

A

E

therefore the angles EAB, EBA are double of the angle EAB.

But the angle BEF is equal to the angles EAB, EBA;

[1. 32] therefore the angle BEF is also double of the angle 20 EAB

25

For the same reason

the angle FEC is also double of the angle EAC. Therefore the whole angle BEC is double of the whole angle BAC.

Again let another straight line be inflected, and let there be another angle BDC; let DE be joined and produced to G.

Similarly then we can prove that the angle GEC is double of the angle EDC,

30 of which the angle GEB is double of the angle EDB; therefore the angle BEC which remains is double of the angle BDC.

Therefore etc.

Q. E. D.

25. let another straight line be inflected, kekλáo0w dǹ wád (without eveîa). The verb kλáw (to break off) was the regular technical term for drawing from a point a (broken) straight line which first meets another straight line or curve and is then bent back from it to another point, or (in other words) for drawing straight lines from two points meeting at a point on a curve or another straight line. KEKλáσbai is one of the geometrical terms the definition of which must according to Aristotle be assumed (Anal. Post. 1. 10, 76 b 9).

The early editors, Tartaglia, Commandinus, Peletarius, Clavius and others, gave the extension of this proposition to the case where the segment is less than a semicircle, and where accordingly the "angle" corresponding to Euclid's "angle at the centre" is greater than two right angles. The convenience of the extension is obvious, and the proof of it is the same as the first part of Euclid's proof. By means of the extension III. 21 is demonstrated without making two cases; III. 22 will follow immediately from the fact that the sum of the "angles at the centre" for two segments making up a whole circle is equal to four right angles; also 11. 31 follows immediately from the extended proposition.

But all the editors referred to were forestalled in this matter by Heron, as we now learn from the commentary of an-Nairizi (ed. Curtze, p. 131 sqq.). Heron gives the extension of Euclid's proposition which, he says, it had been left for him to make, but which is necessary in order that the caviller may not be able to say that the next proposition (about the equality of the angles in any segment) is not established generally, i.e. in the case of a segment less than a semicircle as well as in the case of a segment greater than a semicircle, inasmuch as III. 20, as given by Euclid, only enables us to prove it in the latter case. Heron's enunciation is important as showing how he describes what we should now call an "angle" greater than two right angles. (The language of Gherard's translation is, in other respects, a little obscure; but the meaning is made clear by what follows.)

"The angle," Heron says, "which is at the centre of any circle is double of the angle which is at the circumference of it when one arc is the base of both angles; and the remaining angles which are at the centre, and fill up the four right angles, are double of the angle at the circumference of the arc which is subtended by the [original] angle which is at the centre."

Thus the "angle greater than two right angles" is for Heron the sum of certain "angles" in the Euclidean sense of angles less than two right angles. The particular method of splitting up which Heron adopts will be seen from his proof, which is in substance as follows.

Let CDB be an angle at the centre, CAB that at the circumference.
Produce BD, CD to F, G;

take any point E on BC, and join BE, EC, ED.

Then any angle in the segment BAC is half of the angle BDC; and the sum of the angles BDG, GDF, FDC is double of any angle in the segment BEC.