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straight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle, are equal to one another, viz. the rectangle BA, A E, to the rectangle CA, AF: for each of them is equal to the square of the straight line AD, which touches the circle.

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If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meels shall touch the circle.

Let any point D be taken without the circle ABC, and from it let two straight lines DCA and DB be drawn, of which

DCA cuts the circle, and DB meets it. If the rectangle AD,DC be equal to the square of DB; then DB shall touch the circle.

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Draw the straight line DE, touching the circle ABC, in the point B; (111. 17.)

find F, the centre of the circle, (111. 1.)

and join FE, FB, FD. Then FED is a right angle: (111. 18.) and because DE touches the circle ABC, and DCA cuts it, therefore the rectangle AD, DC is equal to the square of DE: (111. 36.) but the rectangle AD, DC is, by hypothesis, equal to the square of DB; therefore the square of DE is equal the square of DB; (ax. 1.) and the straight line DE equal to the straight line DB:

and FE is equal to FB; (1. def. 15.) wherefore DE, EF are equal to DB, BF, each to each; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal to the angle DBF: (1. 8.)

but DEF was shewn to be a right angle ; therefore also DBF is a right angle: (ax. 1.)

and BF, if produced, is a diameter ;
and the straight line which is drawn at right angles to a diameter,
from the extremity of it, touches the circle; (u. 16. Cor.)

therefore DB touches the circle ABC.
Wherefore, it from a point, &c. Q.E.D.

NOTES TO BOOK III.

In the third Book of the Elements are demonstrated the properties of the circle, assuming all the properties of figures demonstrated in the first and second books.

A new conception is introduced in the third book, namely, that of similarity, and applied to the proof of properties connected with similar segments of circles.

It may be worthy of remark, that the word circle will be found sometimes taken to mean the surface included within the circumference, and sometimes the circumference itself. A circle is said to be given in position when the position of its centre is known, and in magnitude when its radius is known.

Def. 1 states the criterion of equal circles. Simson calls it a theorem; and Euclid seems to have considered it as one of those theorems, or definitions involving an axiom, which might be admitted as a basis for reasoning on the equality of circles.

Def. vi, X. An arc of a circle is any portion of the circumference; and a chord is the straight line joining the extremities of an arc. Every chord except a diameter divides a circle into two unequal segments, one greater than, and the other less than a semicircle. And in the same manner, two radii drawn from the centre to the circumference, divide the circle into two unequal sectors, which become equal when the two radii are in the same straight line. A quadrant is a sector whose radii are perpendicular to one another, and contains a fourth part of the circle.

Prop. 1. If the point G be in the diameter CE, but not coinciding with the point F, the demonstration given in the text does not hold good. At the same time, it is obvious that G cannot be the centre of the circle, because GC is not equal to GE.

Prop. II. In this proposition, the circumference of a circle is proved to be essentially different from a straight line, by shewing that every straight line joining any two points in the arc falls entirely within the circle, and can neither coincide with any part of the circumference, nor meet it except in the two assumed points. From which follows the corollary, that, “a straight line cannot cut the circumference of a circle in more points than two.

Commandine's direct demonstration of Prop. 11 depends on the following axiom, “If a point be taken nearer to the centre of a circle than the circumference, that point falls within the circle.”

Take any point E in AB, and join DA, DE, DB.
Then because DA is equal to DB in the triangle DAB;
therefore the angle DAB is equal to the angle DEB; (v. 1.)

but since the side A E of the triangle DAE is produced to D, therefore the exterior angle DEB is greater than the interior and opposite angle DAE;

but the angle DAE is equal to the angle DBE,

therefore the angle DEB is greater than the angle DBE.
And in every triangle the greater side is subtended by the greater angle;

therefore the side DB is greater than the side DE;
but DB from the centre meets the circumference of the circle,

therefore DE does not meet it.
Wherefore the point E falls within the circle.

And E is any point in the straight line AB, therefore the straight line AB falls within the circle.

Prop. vii and Prop. viii exhibit the same property ; in the former, the point is taken in the diameter, and in the latter, in the diameter produced.

Prop. XI and Prop. XII. In the enunciations it is not asserted that the contact of two circles is confined to a single point. The meaning appears to be, that supposing two circles touch each other in any point, the straight line which joins their centres being produced shall pass through that point in which the circles touch each other.

In Prop. xii, it is proved that a circle cannot touch another in more points than one, by assuming two points of contact, and proving that this is impossible.

Prop. xv and xvi. The converse of these propositions are not proved by Euclid.

Prop. xvi may be demonstrated directly by assuming the following axiom ; “If a point be taken further from the centre of a circle than the circumference, that point falls without the circle.”

Prop. xvii. When the given point is without the circumference of the given circle, it is obvious that two equal tangents may be drawn from the given point to touch the circle, as may be seen from the diagram to Prop. VIII.

Circles are called concentric circles when they have the same centre.

Prop. xviii appears to be nothing more than a corollary to Prop. xvi. Because a tangent to any point of the circumference of a circle is a straight line at right angles at the extremity of the diameter which meets the circumference in that point.

In Prop. XVI, AE is proved to be perpendicular to AB, and in Prop. XVIII, AB is proved to be perpendicular to AE; which is the same thing.

Prop. xx. This proposition is proved by Euclid only in the case in which the argle at the circumference is less than a right angle, and the demonstration is free from objection. If, however, the angle at the circumference be a right angle, the angle at the centre disappears by the two straight lines from the centre to the extremities of the arc becoming one straight line. And, if the angle at the circumference be an ubtuse angle, the angle formed by the two lines from the centre, does not stand on the same arc, but upon the arc which the assumed arc wants of the whole circumference.

If Euclid's definition of an angle be strictly observed, Prop. xx is geometrically true only when the angle at the centre is less than a right angle. If, however, the defect of an angle from four right angles may be regarded as an angle, the proposition is universally true, as may be proved by drawing a line from the angle in the circumference through the centre, and thus forming two angles at the centre in Euclid's strict sense of the term.

In the first case, it is assumed that, if there be four magnitudes, such that the first is double of the second, and the third double of the fourth, then the first and third together shall be double of the second and fourth together : also in the second case, that if one magnitude be double of another, and a part taken from the first be double of a part taken from the second, the remainder of the first shall be double the remainder of the second, which is, in fact, a particular case of Prop. 5, Book v.

Prop. xxi. Hence, the locus of the vertices of all triangles upon the same base, and which have the same vertical angle, is a circular arc.

Prop. xxxI suggests a method of drawing a line at right angles to another when the given point is at the extremity of the given line.

Prop. xxxv. It is possible to prove the most general case of this proposition, and from it to deduce the other cases. The converse of Prop. xxxv is not proved by Euclid.

The properties of the circle demonstrated in the third book may be divided under three heads. ]. Those which relate to the centre. 2. To similar segments.

3. To the equal rectangles contained by the segments of the lines which intersect each other within and without the circle.

BOOK IV.

DEFINITIONS.

I. A RECTILINEAL figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each,

II. In like manner,

figure is said to be described about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each.

III. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.

IV. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle.

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V. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure.

VI. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described.

VII. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

PROPOSITION I. PROBLEM. In a given circle to place a straight line, equal to a given straight line which is not greater than the diameter of the circle.

Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle.

It is required to place in the circle ABC a straight line equal to D.

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Draw BC the diameter of the circle ABC.
Then, if BC is equal to D, the thing required is done ;
for in the circle ABC a straight line BC is placed equal to D.
But, if it is not, FC is greater than D; (hyp.)

make CE equal to D, (1. 3.)
and from the centre C, at the distance CE, describe the circle AEF,
and join CA.

Then CA shall be equal to D. Because C is the centre of the circle AEF, therefore CA is equal to CE: (1. def. 15.)

but CE is equal to D; (constr.)

therefore D is equal to CA. (ax. 1.) Wherefore in the circle ABC, a straight line CA'is placed equal to the given straight line D, which is not greater than the diameter of the circle. Q.E. F.

PROPOSITION II. PROBLEM.

In a given circle to inscribe a triangle equiangular to a given triangle.

Let ABC be the given circle, and DEF the given triangle. It is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.

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Draw the straight line GAH touching the circle in the point A, (111. 17.)

and at the point A, in the straight line AH,
make the angle HAC equal to the angle DEF; (1. 23.)

and at the point A, in the straight line AG,

make the angle GAB equal to the angle DFE;
and join BC: then ABC shall be the triangle required.

Because HAG touches the circle ABC,
and AC is drawn from the point of contact,

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