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Def. I. And it may be added, or of which the circumferences are

equal. And conversely : if two circles be equal, their diameters and

radii are equal ; as also their circumferences.

Def. I. states the criterion of equal circles. Simson calls it a theorem ;

and Euclid seems to have considered it as one of those theorems, or

axioms, which might be admitted as a basis for reasoning on the equality

of circles.

Def. II. There seems to be tacitly assumed in this definition, that a

straight line, when it meets a circle and does not touch it, must necessarily,

when produced, cut the circle.

A straight line which touches a circle, is called a tangent to the circle ;

and a straight line which cuts a circle is called a secant.

Def. rv. The distance of a straight line from the center of a circle

is the distance of a point from a straight line, which has been already

explained in note to Prop.xi. page 53.

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 drawm from the center to the circumference, divide

the circle into two unequal sectors, which become equal when the two

radii are in the same straight line. As Euclid, however, does not notice

re-entering angles, a sector of the circle seems necessarily restricted

to the figure which is less than a semicircle. A quadrant is a sector

whose radii are perpendicular to one another, and which contains a fourth

part of the circle.

Def. VII. No use is made of this definition in the Elements.

Def. XI. The definition of similar segments of circles as employed in

the Third Book is restricted to such segments as are also equal. Props.

XXIII. and xxiv. are the only two instances, in which reference is made

to similar segments of circles.

Prop. I. â€¢â€¢ Lines drawn in a circle," always mean in Euclid, such

lines only as are terminated at their extremities by the circumference.

If the point G be in the diameter CÂ£, 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,

H 5

154

Indirect demonstrations are more frequently employed in the Third

Book than in the First Book of the Elements. Of the demonstrations

of the forty- eight propositions of the First Book, nine are indirect : but

of the thirty-seven of the Third Book, no less than fifteen are indirect

demonstrations. The indirect is, in general, less readily appreciated

by the learner, than the direct form of demonstration. The indirect form,

however, is equally satisfactory, as it excludes every assumed hypothesis

as false, except that which is made in the enunciation of the proposition.

It may be here remarked that Euclid employs three methods of de-

monstrating converse propositions. First, by indirect demonstrations as

in Euc. I. 6: iii. 1, &c. Secondly, by shewing that neither side of a

possible alternative can be true, and thence inferring the truth of the

proposition, as in Euc. i. 19, 25. Thirdly, by means of a construction,

thereby avoiding the indirect mode of demonstration, as in Euc. i. 47 :

III. 37.

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. It excludes the idea of the

circumference of a circle being flexible, or capable under any circum-

stances, of admitting the possibility of the line falling outside the circle.

If the line could fall partly within and partly without the circle, the

circumference of the circle would intersect the line at some point between

its extremities, and any part without the circle has been shewn to be

impossible, and the part within the circle is in accordance with the

enunciation of the Proposition. If the line could fall upon the cir-

cumference and coincide with it, it would follow that a straight line

coincides with a curved line.

From this proposition 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. ii. depends on the foU

lowing axiom, '* If a point be taken nearer to the center of a circle than

the circumference, that point falls within the circle."

Take any point E in AB, and join DA, DE, DB. (fig. Euc. iir. 2.)

Then because DA is equal to DB in the triangle DAB',

therefore the angle DAB is equal to the angle DBA ; (i. 5.)

but since the side AE oi the triangle DAE is produced to B,

therefore the exterior angle DEB is greater than the interior and opposite

^ng\Q DAE\ (i. 16.)

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 Z)S is greater than the side DE ; ^

but DB from the center 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. viii. An arc of a circle is said to be convex or concave with

respect to a point, according as the straight lines drawn from the point

NOTES TO BOOK III. 155

meet the outside or inside of the circular arc : and the two points found

in the circumference of a circle by two straight lines drawn from a given

point to touch the circle, divide the circumference into two portions, one

of which is convex and the other concave, with respect to the given point.

Prop. IX. This appears to follow as a Corollary from Euc. iii. 7.

Prop. XT. and Prop. xii. In the enunciation 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 to touch each other in any

point, the straight line which joins their centers being produced, shall

pass through that point in which the circles touch each other. In

Prop. XIII. 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. XIII, The following is Euclid's demonstration of the case, in

which one circle touches another on the inside.

If possible, let the circle EBF touch the circle j4BC on the inside,

in more points than in one point, namelj' in the points By D. (fig. Euc.

III. 13.) Let P be the center of the circle ABC, and Q the center of EBF.

Join P, Q ; then PQ produced shall pass through the points of contact B, D.

For since P is the center of the circle ABC, PB is equal to PD, but PB

is greater than QD, much more then is QB greater than QD. Again,

since the point Q is the center of the circle EBF, QB is equal to QD ; but

QB has been shewn to be greater than QD, which is impossible. One circle

therefore cannot touch anotheron theinsideinmore points than in one point.

Prop. XVI. may be demonstrated directly by assuming the following

axiom ; ** If a point be taken further from the center of a circle than the

circumference, that point falls without the circle."

If one circle touch another, either internally or externally, the two

circles can have, at the point of contact, only one common tangent.

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 ^rop. VIII.

The best practical method of drawing a tangent to a circle from a given

point without the circumference, is the following : join the given point

and the center of the circle, upon this line describe a semicircle cutting

the given circle, then the line drawn from the given point to the inter-

section will be the tangent required.

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

Prop, xviii. appears to be nothing more than the converse to Prop.

XVI., because a tangent to any point of a circumference of a circle is a

straight line at right angles at the extremity of the diameter which meets

the circumference in that point.

Prop. XX. This proposition is proved by Euclid only in the case in

which the angle at the circumference is less than a right angle, and the

demonstration is free from objection. If, however, the angle at the cir-

cumference be a right angle, the angle at the center disappears, by the

two straight lines from the center to the extremities of the arc becoming

one straight line. And, if the angle at the circumference be an obtuse

angle, the angle formed by the two lines from the center, 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 center is less than two

1

166

EUCLID S ELEME^'TS.

right angles. If, however, the defect of an angle from four right angles

may be regarded as an anf^le, the proposition is universally true, as may

be proved by drawing a line from the angle in the circumference through

the center, and thus forming two angles at the center, 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. v. 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. xxiT. The converse of this Proposition, namely : If the oppo-

site angles of a quadrilateral figure be equal to two right angles, a circle

can be described about it, is not proved by Euclid.

It is obvious from the demonstration of this proposition, that if any

side of the inscribed figure be produced, the exterior angle is equal

to the opposite angle of the figure.

Prop. XXIII. It is obvious from this proposition that of two circular

segments upon the same base, the larger is that which contains the

smaller angle.

Prop. XXV. TTie three cases of this proposition may be reduced to one,

by drawing any two contiguous chords to the given arc, bisecting them,

and from the points of bisection drawing perpendiculars. The point in

which they meet will be the center of the circle. This problem is equi-

valent to that of finding a point equally distant from three given points.

Props. XXVI â€” XXIX. The properties predicated in these four proposi-

tions with respect to equal circles, are also true when predicated of

the same circle.

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. And

that if the diameter of a circle be one of the equal sides of an isosceles

triangle, the base is bisected by the circumference.

Prop. XXXV. The most general case of this Proposition might have

been first demonstrated, and the other more simple cases deduced Irom it.

But this is not Euclid's method. He always commences with the more

simple case and proceeds to the more difficult afterwards. The following

process is the reverse of Euclid's method.

Assuming the construction in the last fig. to Euc. iii. 35. Join FA, FD,

and draw FK perpendicular to AC, and FL perpendicular to BD.

Then (Euc. ii. 5. ) the rectangle AE, EC with square on EK is equal to

the square on AK: add to these equals the square on FK: therefore the

rectangle AE, EC, with the squares on EK, FK, is equal to the squares

on AK, FK. But the squares on EK, FK are equal to the square on EF,

and the squares on AK, FK-axe equal to the square on AF. Hence the

rectangle AE, EC, with the square on EF is equal to the square on AF.

In a similar waj' may be shewn, that the rectangle BE, ED with the

square on EF is equal to the square on FD. And the square on FD is

equal to the square on JD. Wherefore the rectangle AE, EC with the

square on EPis equal to the rectangle BE. ED with the square on EF.

Take from these equals the square on EF, and the rectangle AE, EC

U equal to the rectangle BE, ED.

QUESTIONS ON BOOK ITT. 157

The other more simple cases may easily be deduced from this genera-

case.

The converse is not proved by Euclid ; namely, â€” If two straight lines

intersect one another, so that the rectangle contained by the parts of

one is equal to the rectangle contained by the parts of the other ; then

a circle may be described passing through the extremities of the two

lines. Or, in other words :â€” If the diagonals of a quadrilateral figure

intersect one another, so that the rectangle contained by the segments

of one of them is equal to the rectangle contained by the segments of the

other ; then a circle may be described about the quadrilateral.

Prop. XXXVI. The converse of the corollary to this proposition may

be thus stated:â€” If there be two straight lines, such that, when pro-

duced to meet, the rectangle contained by one of the lines produced, and

the part produced, be equal to the rectangle contained by the other

line produced and the part produced; then a circle can be described

passing through the extremities of the two straight lines. Or, If two

opposite sides of a quadrilateral figure be produced to meet, and the

rectangle contained by one of the sides produced and the part produced,

be equal to the rectangle contained by the other side produced and the

part produced ; then a circle may be described about the quadrilateral

figure.

Prop, xxxvii. The demonstration of this theorem may be made

shorter by a reference to the note on Euclid iii. Def. 2 : for if DB meet

the circle in B and do not touch it at that point, the line must, when

produced, cut the circle in two points.

It is a circumstance worthy of notice, that in this proposition, as well

as in Prop, xlviii. Book i. Euclid departs from the ordinary ex absurdo

mode of proof of converse propositions.

QUESTIONS ON BOOK III.

1. Define accurately the terms radius^ arc, circumference, chord, secant.

2. How does a sector differ in form from a segment of a circle ? Are

they in any case coincident ?

3. What is Euclid's criterion of the equality of two circles? What

is meant by a given circle ? How many points are necessary to deter-

mine the magnitude and position of a circle?

4. When are segments of circles said to be similar? Enunciate the

propositions of the Third Book of Euclid, in which this definition is em-

ployed. Is it employed in a restricted or general form ?

5. In how many points can a circle be cut by a straight line and by

another circle ?

6. When are straight lines equally distant from the center of a circle?

7. Shew the necessity of an indirect demonstration in Euc. iii. 1 .

8. Find the centre of a given circle without bisecting any straight

line.

9. Shew that if the circumference of one of two equal circles pass

through the center of the other, the portions of the two circles, each of

which lies without the circumference of the other circle, are equal.

10. If a straight line passing through the center of a circle bisect a

straight line in it, it shall cut it at right angles. Point out the excep-

tion ; and shew that if a straight line bisect the arc and base of a segment

of a circle, it will, when produced, pass through the center.

158 etjclid's elemenets.

11. If any point be taken within a circle, and a right line be drawn

from it to the circumference, how many lines can generally be drawn

equal to it ? Draw them.

12. Find the shortest distance between a circle and a given straight

line without it.

13. Shew that a circle can only have one center, stating the axioms

upon which your proof depends.

14. Why would not the demonstration of Euc. in. 9, hold good, if

there were only two such equal straight lines ?

15. Two parallel chords in a circle are respectively six and eight inches

in length, and one inch apart ; how many inches is the diameter in length ?

16. Which is the greater chord in a circle whose diameter is 1 inches ;

that whose length is 5 inches, or that whose distance from the center is

4 inches ?

17. What is the locus of the middle points of all equal straight lines

in a circle ?

18. The radius of a circle BCDGF, (fig. Euc. in. 15.) whose center

is Â£, is equal to five inches. The distance of the line FG from the center

is four inches, and the distance of the line BC from the center is three

inches, required the lengths of the lines FG, BC.

19. If the chord of an arc be twelve inches long, and be divided into

two segments of eight and four inches by another chord : what is the

length of the latter chord, if one of its segments be two inches ?

20. What is the radius of that circle of which the chords of an arc

and of double the arc are five and eight inches respectively ?

21. If the chord of an arc of a circle whose diameter is 8^ inches,

be five inches, what is the length of the chord of double the arc of the

same circle ?

22. State when a straight line is said to touch a circle, and shew

from your definition that a straight line cannot be drawn to touch a circle

from a point within it.

23. Can more circles than one touch a straight line in the same

point ?

24. Shew from the construction. Euc. in. 17, that two equal straight

lines, and only two, can be drawn touching a given circle from a given

point without it : and one, and only one, from a point in the cir-

cumference.

25. What is the locus of the centers of all the circles which touch

a straight line in a given point ?

26. How may a tangent be drawn at a given point in the circum-

ference of a circle, without knowing the centej* ?

27. In a circle place two chords of given length at right angles to.

each other.

28. From Euc. in. 19, shew how many circles equal to a given

circle may be drawn to touch a straight line in the same point.

29. Enunciate Euc. in. 20. Is this true, when the base is greater

than a semicircle ? If so, why has Euclid omitted this case ?

30. The angle at the center of a circle is double of that at the circum-

ference. How will it appear hence that the angle in a semicircle is aright

angle ?

31. What conditions are essential to the possibility of the inscription

and circumscription of a circle in and about a quadrilateral figure ?

32. What conditions are requisite in order that a parallelogram may

be inscribed in a circle ? Are there any analogous conditions requisite

that a parallelogram may be described about a circle ?

33. Define the angle in a segment of a circle, and the angle on a seg-

QUESTIONS ON BOOK III. 159

raent ; and shew that in the same circle, they are together equal to two

right angles.

34. State and prove the converse of Euc. iii. 22.

35. All circles which pass through two given points have their centers

in a certain straight line.

36. Describe the circle of which a given segment is a part. Give

Euclid's more simple method of solving the same problem independently

of the magnitude of the given segment.

37. In the same circle equal straight lines cut off equal circumfer-

ences. If these straight lines have any point common to one another, it

must not be in the circumference. Is the enunciation given complete ?

38. Enunciate Euc. iii. 31, and deduce the proof of it from Euc. iii. 20.

39 . What is the locus of the vertices of all right-angled triangles which

can be described upon the same hypotenuse ?

40. How may a perpendicular be drawn to a given straight line from

one of its extremities without producing the line f

41. If the angle in a semicircle be a right angle; what is the angle

in a quadrant ?

42. The sum of the squares of any two lines drawn from any point

in a semicircle to the extremity of the diameter is constant. Express

that constant in terms of the radius.

43. In the demonstration of Euc. iii. 30, it is stated that â€¢* equal

straight lines cut off equal circumferences, the greater equal to the greater,

and the less to the less :" explain by reference to the diagram the meaning

of this statement.

44. How many circles may be described so as to pass through one,

two, and three given points ? In what case is it impossible for a circle

to pass through three given points ?

45. Compare the circumference of the segment (Euc. iir. 33.) with

the whole circumference when the angle contained in it is a right angle

and a half.

46. Include the four cases of Euc. iii. 35, in one general proof.

47. Enunciate the propositions which are cqnverse to Props. 32, 35

of Book III.

48. If the position of the center of a circle be known with respect

to a given point outside a circle, and the distance of the circumference to

the point be ten inches : what is the length of the diameter of the circle,

if a tangent drawn from the given point be fifteen inches ?

49. If two straight lines be drawn from a point without a circle, and

be both terminated by the concave part of the circumference, and if

one of the lines pass through the center, and a portion of the other

line intercepted by the circle, be equal to the radius : find the diameter

of the circle, if the two lines meet the convex part of the circumference,

a, 6, units respectively from the given point.

50. Upon what propositions depends the demonstration of Euc. iil

35? Is any extension made of this proposition in the Third Book ?

51. What conditions must be fidfilled that a circle may pass through

four given points ?

52. Why is it considered necessary to demonstrate all the separate

cases of Euc. iii. 35, 36, geometrically, which are comprehended in one

formula, when expressed by Algebraic symbols ?

53. Enunciate the converse propositions of the Third Book of Euclid

which are not demonstrated ex absurdo : and state the three methods

which Euclid employs in the demonstration of converse propositions in

the First and Third Books of the Elements.

GEOMETEICAL EXEECISES ON BOOK III.

PROPOSITION I. THEOREM.

If AB, CD be chords of a circle at right angles to each other, prove that the

sum of the arcs AC, BD is equal to the sum of the arcs AD, BC.

Draw the diameter FGIT parallel to AB, and cutting CD in H.

D

Then the arcs FDG and FCG are each half the circumference.

Also since CD is bisected in the point H,

the arc FD is equal to the arc FC,

and the arc FD is equal to the arcs FA, AD, of which, AF is

equal to J5G,

therefore the arcs AD, BG are equal to the arc FC;

add to each CG,

therefore the arcs AD, BC are equal to the arcsi^C, C(r, which make

up the half circumference.

Hence also the arcs A C, DB are equal to half the circumference.

Wherefore the arcs AD, BC are equal to the arcs A C, DB.

PROPOSITION II. PROBLEM.

The diameter of a circle having been produced to a given point, it is required

to find in the part produced a point, from which if a tangent be drawn to the

circle, it shall be equal to the segment of the part produced, that is, between the

given point and the point found.

Analysis. Let AFB be a circle whose center is C, and whose dia-

meter AB is produced to the given point D.

Suppose that G is the point required, such that the segment GD

is equal to the tangent GF di-awn from G to touch the circle in F.

F

E

Join DF and produce it to meet the circumference again in JP;

join also CF and CF.

Then in the triangle GDF, because GD is equal to GF,

therefore the angle GFD is equal to the angle GDF-,

I

GEOMETRICAL EXERCISES ON BOOK 111. 161

and because CE is equal to CF,

the angle CJEF is equal to the angle CFF;

therefore the angles CFF, GED are equal to the angles CFE

*GDE:

but since GE is a tangent at E,

therefore the angle CEG is a right angle, (iii. 18.)

hence the angles CEF, GEF are equal to a right angle,

and consequently, the angles CFE, EDG are also equal to a right

angle,

wherefore the remaining angle FCD of the triangle CFD is a right

equal. And conversely : if two circles be equal, their diameters and

radii are equal ; as also their circumferences.

Def. I. states the criterion of equal circles. Simson calls it a theorem ;

and Euclid seems to have considered it as one of those theorems, or

axioms, which might be admitted as a basis for reasoning on the equality

of circles.

Def. II. There seems to be tacitly assumed in this definition, that a

straight line, when it meets a circle and does not touch it, must necessarily,

when produced, cut the circle.

A straight line which touches a circle, is called a tangent to the circle ;

and a straight line which cuts a circle is called a secant.

Def. rv. The distance of a straight line from the center of a circle

is the distance of a point from a straight line, which has been already

explained in note to Prop.xi. page 53.

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 drawm from the center to the circumference, divide

the circle into two unequal sectors, which become equal when the two

radii are in the same straight line. As Euclid, however, does not notice

re-entering angles, a sector of the circle seems necessarily restricted

to the figure which is less than a semicircle. A quadrant is a sector

whose radii are perpendicular to one another, and which contains a fourth

part of the circle.

Def. VII. No use is made of this definition in the Elements.

Def. XI. The definition of similar segments of circles as employed in

the Third Book is restricted to such segments as are also equal. Props.

XXIII. and xxiv. are the only two instances, in which reference is made

to similar segments of circles.

Prop. I. â€¢â€¢ Lines drawn in a circle," always mean in Euclid, such

lines only as are terminated at their extremities by the circumference.

If the point G be in the diameter CÂ£, 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,

H 5

154

Indirect demonstrations are more frequently employed in the Third

Book than in the First Book of the Elements. Of the demonstrations

of the forty- eight propositions of the First Book, nine are indirect : but

of the thirty-seven of the Third Book, no less than fifteen are indirect

demonstrations. The indirect is, in general, less readily appreciated

by the learner, than the direct form of demonstration. The indirect form,

however, is equally satisfactory, as it excludes every assumed hypothesis

as false, except that which is made in the enunciation of the proposition.

It may be here remarked that Euclid employs three methods of de-

monstrating converse propositions. First, by indirect demonstrations as

in Euc. I. 6: iii. 1, &c. Secondly, by shewing that neither side of a

possible alternative can be true, and thence inferring the truth of the

proposition, as in Euc. i. 19, 25. Thirdly, by means of a construction,

thereby avoiding the indirect mode of demonstration, as in Euc. i. 47 :

III. 37.

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. It excludes the idea of the

circumference of a circle being flexible, or capable under any circum-

stances, of admitting the possibility of the line falling outside the circle.

If the line could fall partly within and partly without the circle, the

circumference of the circle would intersect the line at some point between

its extremities, and any part without the circle has been shewn to be

impossible, and the part within the circle is in accordance with the

enunciation of the Proposition. If the line could fall upon the cir-

cumference and coincide with it, it would follow that a straight line

coincides with a curved line.

From this proposition 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. ii. depends on the foU

lowing axiom, '* If a point be taken nearer to the center of a circle than

the circumference, that point falls within the circle."

Take any point E in AB, and join DA, DE, DB. (fig. Euc. iir. 2.)

Then because DA is equal to DB in the triangle DAB',

therefore the angle DAB is equal to the angle DBA ; (i. 5.)

but since the side AE oi the triangle DAE is produced to B,

therefore the exterior angle DEB is greater than the interior and opposite

^ng\Q DAE\ (i. 16.)

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 Z)S is greater than the side DE ; ^

but DB from the center 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. viii. An arc of a circle is said to be convex or concave with

respect to a point, according as the straight lines drawn from the point

NOTES TO BOOK III. 155

meet the outside or inside of the circular arc : and the two points found

in the circumference of a circle by two straight lines drawn from a given

point to touch the circle, divide the circumference into two portions, one

of which is convex and the other concave, with respect to the given point.

Prop. IX. This appears to follow as a Corollary from Euc. iii. 7.

Prop. XT. and Prop. xii. In the enunciation 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 to touch each other in any

point, the straight line which joins their centers being produced, shall

pass through that point in which the circles touch each other. In

Prop. XIII. 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. XIII, The following is Euclid's demonstration of the case, in

which one circle touches another on the inside.

If possible, let the circle EBF touch the circle j4BC on the inside,

in more points than in one point, namelj' in the points By D. (fig. Euc.

III. 13.) Let P be the center of the circle ABC, and Q the center of EBF.

Join P, Q ; then PQ produced shall pass through the points of contact B, D.

For since P is the center of the circle ABC, PB is equal to PD, but PB

is greater than QD, much more then is QB greater than QD. Again,

since the point Q is the center of the circle EBF, QB is equal to QD ; but

QB has been shewn to be greater than QD, which is impossible. One circle

therefore cannot touch anotheron theinsideinmore points than in one point.

Prop. XVI. may be demonstrated directly by assuming the following

axiom ; ** If a point be taken further from the center of a circle than the

circumference, that point falls without the circle."

If one circle touch another, either internally or externally, the two

circles can have, at the point of contact, only one common tangent.

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 ^rop. VIII.

The best practical method of drawing a tangent to a circle from a given

point without the circumference, is the following : join the given point

and the center of the circle, upon this line describe a semicircle cutting

the given circle, then the line drawn from the given point to the inter-

section will be the tangent required.

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

Prop, xviii. appears to be nothing more than the converse to Prop.

XVI., because a tangent to any point of a circumference of a circle is a

straight line at right angles at the extremity of the diameter which meets

the circumference in that point.

Prop. XX. This proposition is proved by Euclid only in the case in

which the angle at the circumference is less than a right angle, and the

demonstration is free from objection. If, however, the angle at the cir-

cumference be a right angle, the angle at the center disappears, by the

two straight lines from the center to the extremities of the arc becoming

one straight line. And, if the angle at the circumference be an obtuse

angle, the angle formed by the two lines from the center, 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 center is less than two

1

166

EUCLID S ELEME^'TS.

right angles. If, however, the defect of an angle from four right angles

may be regarded as an anf^le, the proposition is universally true, as may

be proved by drawing a line from the angle in the circumference through

the center, and thus forming two angles at the center, 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. v. 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. xxiT. The converse of this Proposition, namely : If the oppo-

site angles of a quadrilateral figure be equal to two right angles, a circle

can be described about it, is not proved by Euclid.

It is obvious from the demonstration of this proposition, that if any

side of the inscribed figure be produced, the exterior angle is equal

to the opposite angle of the figure.

Prop. XXIII. It is obvious from this proposition that of two circular

segments upon the same base, the larger is that which contains the

smaller angle.

Prop. XXV. TTie three cases of this proposition may be reduced to one,

by drawing any two contiguous chords to the given arc, bisecting them,

and from the points of bisection drawing perpendiculars. The point in

which they meet will be the center of the circle. This problem is equi-

valent to that of finding a point equally distant from three given points.

Props. XXVI â€” XXIX. The properties predicated in these four proposi-

tions with respect to equal circles, are also true when predicated of

the same circle.

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. And

that if the diameter of a circle be one of the equal sides of an isosceles

triangle, the base is bisected by the circumference.

Prop. XXXV. The most general case of this Proposition might have

been first demonstrated, and the other more simple cases deduced Irom it.

But this is not Euclid's method. He always commences with the more

simple case and proceeds to the more difficult afterwards. The following

process is the reverse of Euclid's method.

Assuming the construction in the last fig. to Euc. iii. 35. Join FA, FD,

and draw FK perpendicular to AC, and FL perpendicular to BD.

Then (Euc. ii. 5. ) the rectangle AE, EC with square on EK is equal to

the square on AK: add to these equals the square on FK: therefore the

rectangle AE, EC, with the squares on EK, FK, is equal to the squares

on AK, FK. But the squares on EK, FK are equal to the square on EF,

and the squares on AK, FK-axe equal to the square on AF. Hence the

rectangle AE, EC, with the square on EF is equal to the square on AF.

In a similar waj' may be shewn, that the rectangle BE, ED with the

square on EF is equal to the square on FD. And the square on FD is

equal to the square on JD. Wherefore the rectangle AE, EC with the

square on EPis equal to the rectangle BE. ED with the square on EF.

Take from these equals the square on EF, and the rectangle AE, EC

U equal to the rectangle BE, ED.

QUESTIONS ON BOOK ITT. 157

The other more simple cases may easily be deduced from this genera-

case.

The converse is not proved by Euclid ; namely, â€” If two straight lines

intersect one another, so that the rectangle contained by the parts of

one is equal to the rectangle contained by the parts of the other ; then

a circle may be described passing through the extremities of the two

lines. Or, in other words :â€” If the diagonals of a quadrilateral figure

intersect one another, so that the rectangle contained by the segments

of one of them is equal to the rectangle contained by the segments of the

other ; then a circle may be described about the quadrilateral.

Prop. XXXVI. The converse of the corollary to this proposition may

be thus stated:â€” If there be two straight lines, such that, when pro-

duced to meet, the rectangle contained by one of the lines produced, and

the part produced, be equal to the rectangle contained by the other

line produced and the part produced; then a circle can be described

passing through the extremities of the two straight lines. Or, If two

opposite sides of a quadrilateral figure be produced to meet, and the

rectangle contained by one of the sides produced and the part produced,

be equal to the rectangle contained by the other side produced and the

part produced ; then a circle may be described about the quadrilateral

figure.

Prop, xxxvii. The demonstration of this theorem may be made

shorter by a reference to the note on Euclid iii. Def. 2 : for if DB meet

the circle in B and do not touch it at that point, the line must, when

produced, cut the circle in two points.

It is a circumstance worthy of notice, that in this proposition, as well

as in Prop, xlviii. Book i. Euclid departs from the ordinary ex absurdo

mode of proof of converse propositions.

QUESTIONS ON BOOK III.

1. Define accurately the terms radius^ arc, circumference, chord, secant.

2. How does a sector differ in form from a segment of a circle ? Are

they in any case coincident ?

3. What is Euclid's criterion of the equality of two circles? What

is meant by a given circle ? How many points are necessary to deter-

mine the magnitude and position of a circle?

4. When are segments of circles said to be similar? Enunciate the

propositions of the Third Book of Euclid, in which this definition is em-

ployed. Is it employed in a restricted or general form ?

5. In how many points can a circle be cut by a straight line and by

another circle ?

6. When are straight lines equally distant from the center of a circle?

7. Shew the necessity of an indirect demonstration in Euc. iii. 1 .

8. Find the centre of a given circle without bisecting any straight

line.

9. Shew that if the circumference of one of two equal circles pass

through the center of the other, the portions of the two circles, each of

which lies without the circumference of the other circle, are equal.

10. If a straight line passing through the center of a circle bisect a

straight line in it, it shall cut it at right angles. Point out the excep-

tion ; and shew that if a straight line bisect the arc and base of a segment

of a circle, it will, when produced, pass through the center.

158 etjclid's elemenets.

11. If any point be taken within a circle, and a right line be drawn

from it to the circumference, how many lines can generally be drawn

equal to it ? Draw them.

12. Find the shortest distance between a circle and a given straight

line without it.

13. Shew that a circle can only have one center, stating the axioms

upon which your proof depends.

14. Why would not the demonstration of Euc. in. 9, hold good, if

there were only two such equal straight lines ?

15. Two parallel chords in a circle are respectively six and eight inches

in length, and one inch apart ; how many inches is the diameter in length ?

16. Which is the greater chord in a circle whose diameter is 1 inches ;

that whose length is 5 inches, or that whose distance from the center is

4 inches ?

17. What is the locus of the middle points of all equal straight lines

in a circle ?

18. The radius of a circle BCDGF, (fig. Euc. in. 15.) whose center

is Â£, is equal to five inches. The distance of the line FG from the center

is four inches, and the distance of the line BC from the center is three

inches, required the lengths of the lines FG, BC.

19. If the chord of an arc be twelve inches long, and be divided into

two segments of eight and four inches by another chord : what is the

length of the latter chord, if one of its segments be two inches ?

20. What is the radius of that circle of which the chords of an arc

and of double the arc are five and eight inches respectively ?

21. If the chord of an arc of a circle whose diameter is 8^ inches,

be five inches, what is the length of the chord of double the arc of the

same circle ?

22. State when a straight line is said to touch a circle, and shew

from your definition that a straight line cannot be drawn to touch a circle

from a point within it.

23. Can more circles than one touch a straight line in the same

point ?

24. Shew from the construction. Euc. in. 17, that two equal straight

lines, and only two, can be drawn touching a given circle from a given

point without it : and one, and only one, from a point in the cir-

cumference.

25. What is the locus of the centers of all the circles which touch

a straight line in a given point ?

26. How may a tangent be drawn at a given point in the circum-

ference of a circle, without knowing the centej* ?

27. In a circle place two chords of given length at right angles to.

each other.

28. From Euc. in. 19, shew how many circles equal to a given

circle may be drawn to touch a straight line in the same point.

29. Enunciate Euc. in. 20. Is this true, when the base is greater

than a semicircle ? If so, why has Euclid omitted this case ?

30. The angle at the center of a circle is double of that at the circum-

ference. How will it appear hence that the angle in a semicircle is aright

angle ?

31. What conditions are essential to the possibility of the inscription

and circumscription of a circle in and about a quadrilateral figure ?

32. What conditions are requisite in order that a parallelogram may

be inscribed in a circle ? Are there any analogous conditions requisite

that a parallelogram may be described about a circle ?

33. Define the angle in a segment of a circle, and the angle on a seg-

QUESTIONS ON BOOK III. 159

raent ; and shew that in the same circle, they are together equal to two

right angles.

34. State and prove the converse of Euc. iii. 22.

35. All circles which pass through two given points have their centers

in a certain straight line.

36. Describe the circle of which a given segment is a part. Give

Euclid's more simple method of solving the same problem independently

of the magnitude of the given segment.

37. In the same circle equal straight lines cut off equal circumfer-

ences. If these straight lines have any point common to one another, it

must not be in the circumference. Is the enunciation given complete ?

38. Enunciate Euc. iii. 31, and deduce the proof of it from Euc. iii. 20.

39 . What is the locus of the vertices of all right-angled triangles which

can be described upon the same hypotenuse ?

40. How may a perpendicular be drawn to a given straight line from

one of its extremities without producing the line f

41. If the angle in a semicircle be a right angle; what is the angle

in a quadrant ?

42. The sum of the squares of any two lines drawn from any point

in a semicircle to the extremity of the diameter is constant. Express

that constant in terms of the radius.

43. In the demonstration of Euc. iii. 30, it is stated that â€¢* equal

straight lines cut off equal circumferences, the greater equal to the greater,

and the less to the less :" explain by reference to the diagram the meaning

of this statement.

44. How many circles may be described so as to pass through one,

two, and three given points ? In what case is it impossible for a circle

to pass through three given points ?

45. Compare the circumference of the segment (Euc. iir. 33.) with

the whole circumference when the angle contained in it is a right angle

and a half.

46. Include the four cases of Euc. iii. 35, in one general proof.

47. Enunciate the propositions which are cqnverse to Props. 32, 35

of Book III.

48. If the position of the center of a circle be known with respect

to a given point outside a circle, and the distance of the circumference to

the point be ten inches : what is the length of the diameter of the circle,

if a tangent drawn from the given point be fifteen inches ?

49. If two straight lines be drawn from a point without a circle, and

be both terminated by the concave part of the circumference, and if

one of the lines pass through the center, and a portion of the other

line intercepted by the circle, be equal to the radius : find the diameter

of the circle, if the two lines meet the convex part of the circumference,

a, 6, units respectively from the given point.

50. Upon what propositions depends the demonstration of Euc. iil

35? Is any extension made of this proposition in the Third Book ?

51. What conditions must be fidfilled that a circle may pass through

four given points ?

52. Why is it considered necessary to demonstrate all the separate

cases of Euc. iii. 35, 36, geometrically, which are comprehended in one

formula, when expressed by Algebraic symbols ?

53. Enunciate the converse propositions of the Third Book of Euclid

which are not demonstrated ex absurdo : and state the three methods

which Euclid employs in the demonstration of converse propositions in

the First and Third Books of the Elements.

GEOMETEICAL EXEECISES ON BOOK III.

PROPOSITION I. THEOREM.

If AB, CD be chords of a circle at right angles to each other, prove that the

sum of the arcs AC, BD is equal to the sum of the arcs AD, BC.

Draw the diameter FGIT parallel to AB, and cutting CD in H.

D

Then the arcs FDG and FCG are each half the circumference.

Also since CD is bisected in the point H,

the arc FD is equal to the arc FC,

and the arc FD is equal to the arcs FA, AD, of which, AF is

equal to J5G,

therefore the arcs AD, BG are equal to the arc FC;

add to each CG,

therefore the arcs AD, BC are equal to the arcsi^C, C(r, which make

up the half circumference.

Hence also the arcs A C, DB are equal to half the circumference.

Wherefore the arcs AD, BC are equal to the arcs A C, DB.

PROPOSITION II. PROBLEM.

The diameter of a circle having been produced to a given point, it is required

to find in the part produced a point, from which if a tangent be drawn to the

circle, it shall be equal to the segment of the part produced, that is, between the

given point and the point found.

Analysis. Let AFB be a circle whose center is C, and whose dia-

meter AB is produced to the given point D.

Suppose that G is the point required, such that the segment GD

is equal to the tangent GF di-awn from G to touch the circle in F.

F

E

Join DF and produce it to meet the circumference again in JP;

join also CF and CF.

Then in the triangle GDF, because GD is equal to GF,

therefore the angle GFD is equal to the angle GDF-,

I

GEOMETRICAL EXERCISES ON BOOK 111. 161

and because CE is equal to CF,

the angle CJEF is equal to the angle CFF;

therefore the angles CFF, GED are equal to the angles CFE

*GDE:

but since GE is a tangent at E,

therefore the angle CEG is a right angle, (iii. 18.)

hence the angles CEF, GEF are equal to a right angle,

and consequently, the angles CFE, EDG are also equal to a right

angle,

wherefore the remaining angle FCD of the triangle CFD is a right

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