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to each of these eqnals add the square of GF; therefore (Ax. 2.)

3. The rectangle AE, EC, together with the squares of EG, GF, is equal to the squares of AG, GF; but the squares of EG, GF, are equal (I. 47.) to the square of EF; and the squares of AG, GF, are equal to the square of AF; therefore

4. The rectangle AE, EC, together with the square of EF, is equal to the square of AF; that is, to the square of FB: but (II. 5.)

5. The square of FB is equal to the rectangle BE, ED, together with the square of EF; therefore (Ax. 1.)

6. The rectangle AE, EC, together with the square of EF, is equal to the rectangle BE, ED, together with the square of EF: take away the common square of EF, and therefore (Ax. 3.)

7. The remaining rectangle AE, EC, is equal to the remaining rectangle BE, ED. Lastly, let neither of the straight lines AC, BD, pass through the centre.

Take (III. 1.) the centre F, and through E, the intersection of the straight lines AC, DB, draw the diameter GEFH. And because the rectangle AE, EC, is equal, as has been shown, to the rectangle GE, EH; and for the same reason, the rectangle BE, ED, is equal to the same rectangle GE, EH; therefore (Ax. 1.)

The rectangle AE, EC, is equal to the rectangle BE, ED. Wherefore, if two straight lines, &c. Q.E.D.

PROP. XXXVI.-THEOREM.

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

Let D be any point without the circle ABC, and let DCA, DB, be two straight lines drawn from it, of which DCA cuts the circle, and DB touches the same: the rectangle AD, DC, is equal to the square of DB.

Either DCA passes through the centre, or it does not : first, let it pass through the centre E.

Join EB; therefore (III. 18.)

1. The angle EBD is a right ungle: and because the straight line AC is bisected in E, and produced to the point D, (II. 6.)

2. The rectangle AD, DC, together with the square of EC, is equal to the square of ED; and CE is equal to EB; therefore

3. The rectangle AD, DC, together with the square of EB, is equal to the square of ED: but the square of ED is equal (I. 47.) to the squares of EB, BD, because EBD is a right angle; therefore (Ax. 1.)

4. The rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: take away the common square of EB; therefore (Ax. 3.)

5. The remaining rectangle AD, DC, is equal to the square of the tangent DB. But if DCA does not pass through the centre of the circle ABC, take (III. 1.) the centre E, and draw EF perpendicular (I. 12.) to AC, and join EB, EC, ED.

And because the straight line EF, which passes through the centre, cuts the straight line AC which does not pass through the centre, at right angles, it shall likewise bisect it (III. 3.); therefore

1. AF is equal to FC: and because the straight line AC is bisected in F, and produced to D, (II. 6.)

2. The rectangle AD, DC, together with the square of FC, is equal to the square of FD:

to each of these equals add the square of FE; therefore (Ax. 2.)

3. The rectangle AD, DC, together with the squares of CF, FE, is equal to the squares of , FE: . but the square of ED is equal (I. 47.) to the squares of DF; FE, because EFD is a right angle; and the square of EC is likewise equal to the squares of CF, FE; therefore (Ax. 1.)

4. The rectangle AD, DC, together with the square of EC, is equal to the square of ED: and CE is equal to EB; therefore

5. The rectangle AD, DC, together with the square of EB, is equal to the square of ED: but the squares of EB, BD, are equal (I. 47.) to the square of ED, because EBD is a right angle; therefore

6. The rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: take away the common square of EB; therefore (Ax. 3.)

7. The remaining rectangle AD, DC, is equal to the square of DB. Wherefore, if from any point, &c. Q.E.D.

COR.-If from any point without a circle there be drawn two 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.

Thus :

The rectangle BA, AE, is equal to the rectangle CA, AF: for each of them is equal to the square of the straight line AD which touches the circle.

PROP. XXXVII.-THEOREM.

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 meets shall touch the circle.

Let any point D be taken without the circle ABC, and from it let two

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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, DB touches the circle.

Draw (IlI. 17.) the straight line DE, touching the circle ABC; find its centre F, and join FE, FB, FD. Then (III. 18.)

1. FED is a right angle: and because DE touches the circle ABC, and DCA cuts it, (III. 36.)

2. The rectangle AD, DC, is equal to the square of DE: but the rectangle AD, DC, is, by hypothesis, equal to the square of DB: therefore (Ax. 1.)

3. The square of DE is equal to the square of DB, and the straight line DE equal to the straight line DB: and FE is equal (I. Def. 15.) to FB; wherefore DE, EF, are equal to DB, BF, each to each; and the base FD is common to the two triangles DEF, DBF; therefore (1. 8.)

4. The angle DEF is equal to the angle DBF: but DEF was shown to be a right angle, therefore also

5. DBF is a right angle: and FB, 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 (III. 16.); therefore

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

EUCLID'S ELEMENTS OF GEOMETRY.

Book IV.

DEFINITIONS.

A rectilineal figure is said to be inscribed in another rectilineal figure,

when all the angular points of the inscribed figure are upon the sides of the figure upon which it is inscribed, each upon each.

II.

In like manner, a 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 angular

points 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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