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43. On a given straight line describe an equilateral and equi-

angular octagon.

VI.

44. Inscribe a circle in a rhombus.

45. Having given the distances of the centers of two equal circles

which cut one another, inscribe a square in the space included between

the two circumferences.

46. The square inscribed in a circle is equal to half the square

described about the same circle.

47. The square is greater than any oblong inscribed in the same

circle.

48. A circle having a square inscribed in it being given, to find a

circle in which a regular octagon of a perimeter equal to that of the

square, may be inscribed.

49. Describe a circle about a figure formed by constructing an

equilateral triangle upon the base of an isosceles triangle, the vertical

angle of which is four times the angle at the base.

oO. A regular octagon inscribed in a circle is equal to the rectangle

k5

202 GEOMETRICAL EXERCISES

contained by the sides of the squares inscribed in, and circumscribed

about the circle.

51. If in any circle the side of an inscribed hexagon be produced

till it becomes equal to the side of an inscribed square, a tangent

drawn from the extremity, without the circle, shall be equal to the

side of an inscribed octagon.

VII.

52. To describe a circle which shall touch a given circle in a given

point, and also a given straight line.

53. Describe a circle touching a given straight line, and also two

given circles.

54. Describe a circle which shall touch a given circle, and each of

two given straight lines.

66, Tm'o points are given, one in each of two given circles ; describe

a circle passing through both points and touching one of the circles.

6Q. Describe a circle touching a straight line in a given point, and

also touching a given circle. AVhen the line cuts the given circle,

shew that your construction will enable you to obtain six circles

touching the given cii'cle and the given line, but not necessarily in the

given point.

57. Describe a circle which shall touch two sides and pass through

one angle of a given square.

58. If two circles touch each other externally, describe a circle

which shall touch one of them in a given point, and also touch the

other. In what case does this become impossible ?

59. Describe three circles touching each other and having their

centers at three given points. In how many different ways may this

be done ?

VIII.

60. Let two straight lines be drawn from any point within a circle

to the circumference: describe a circle, which shall touch them both,

and the arc between them.

61. In a given triangle having inscribed a circle, inscribe another

circle in the space thus intercepted at one of the angles.

62. Let AB, A C, be the bounding radii of a quadrant; complete

the square A JB DC and draw the diagonal AD-, then the part of the

diagonal without the quadrant will be equal to the radius of a circle

inscribed in the quadi*ant.

63. If on one of the bounding radii of a quadrant, a semicircle be

described, and on the other, another semicircle be described, so as to

touch the former and the quadrantal arc; find the center of the circle

inscribed in the figure bounded by the three curves.

64. In a given segment of a circle inscribe an isosceles triangle,

such that its vertex may be in the middle of the chord, and the base

and perpendicular together equal to a given line.

65. Inscribe three circles in an isosceles triangle touching each

other, and each of them touching two of the three sides of the triangle.

IX.

66. In the fig. Prop. 10, Book iv, shew that the base BD\& the

ON BOOK IV. 203

side of a regular decagon inscribed in the larger circle, and the side of

a regular pentagon inscribed in the smaller circle.

67. In the fig. Prop. 10, Book IV, produce DC to meet the circle

in F, and draw l^F; then the angle ABF shall be equal to thi'ee times

the angle BFD.

68. If the alternate angles of a regular pentagon be joined, the

figure formed by the intersection of the joining lines will itself be a

regular pentagon.

69. 1^ ABODE be any pentagon inscribed in a circle, and AC,

BD, CE, DA, EB be joined, then are the angles ABE, BCA, CDB,

DEC, EAD, together equal to two right angles.

70. A watch-ribbon is folded up into a flat knot of five edges, shew

that the sides of the knot form an equilateral pentagon.

71. If from the extremities of the side of a regular pentagon

inscribed in a circle, straight lines be drawn to the middle of the arc

subtended by the adjacent side, their diff'erence is equal to the radius ;

the sum of their squares to three times the square of the radius ; and

the rectangle contained by them is equal to the square of the radius.

72. Inscribe a regular pentagon in a given square so that four

angles of the pentagon may touch respectively the four sides of the

square.

73. Inscribe a regular decagon in a given circle.

74. The square described upon the side of a regular pentagon in

a circle, is equal to the square on the side of a regular hexagon, together

with the square upon the side of a regular decagon in the same circle.

X.

75. In a given circle inscribe three equal circles touching each

other and the given circle.

76. Shew that if two circles be inscribed in a third to touch one

another, the tangents of the points of contact will all meet in the same

point.

77. If there be three concentric circles, whose radii are 1, 2, 3 ;

determine how many circles may be described round the interior one,

having their centers in the circumference of the circle, whose radius is

2, and touching the interior and exterior circles, and each other.

78. Shew that nine equal circles may be placed in contact, so that

a square whose side is three times the diameter of one of them will

circumscribe them.

XL

79. Produce the sides of a given heptagon both ways, till they

meet, forming seven triangles; required the sum of their vertical

angles.

80. To convert a given regular polygon into another which shall

have the same perimeter, but double the number of sides.

81. In any polygon of an even number of sides, inscribed in a

circle, the sum of the 1st, 3rd, 5th, &c. angles is equal to the sum of

the 2nd, 4th, 6th, &c.

82. Of all polygons having equal perimeters, and the same number

of sides, the equilateral polygon has the greatest area.

BOOK V.

DEFINITIONS.

I.

A LESS magnitude is said to be apart of a greater magnitude, when

the less measures the greater; that is, 'when the less is contained a

certain number of times exactly in the greater/

11.

A greater magnitude is said to be a multiple of a less, when the

greater is measured by the less, that is, * when the greater contains the

less a certain number of times exactly.'

III.

" Ratio is a mutual relation of two magnitudes of the same kind to

one another, in respect of quantity."

IV.

Magnitudes are said to have a ratio to one another, when the less

can be multiplied so as to exceed the other.

V.

The first of four magnitudes is said to have the same ratio to the

second, which the third has to the fourth, when any equimultiples

whatsoever of the first and third being taken, and any equimultiples

whatsoever of the second and fourth ; if the multiple of the first be less

than that of the second, the multiple of the third is also less than that

of the fourth : or, if the multiple of the first be equal to that of the

second, the multiple of the third is also equal to that of the fourth: or,

if the multiple of the first be greater than that of the second, the mul-

tiple of the third is also greater than that of the fourth.

VI.

Magnitudes which have the same ratio are called proportionals.

N.B. 'When four magnitudes are proportionals, it is usually ex-

pressed by saying, the first is to the second, as the third to the fourth.'

VII.

ViTien of the equimultiples of four magnitudes (taken as in the

fifth definition), the multiple of the first is greater than that of the

second, but the multiple of the third is not greater than the multiple

of the fourth ; then the first is said to have to the second a greater

ratio than the third magnitude has to the fourth : and, on the contrary,

the third is said to have to the fourth a less ratio than the first has to

the second.

VIII.

" Analogy, or proportion, is the similitude of ratios."

DEFINITIONS.

205

IX.

Proportion consists in three terms at least.

X.

When three magnitudes are proportionals, the first is said to have

to the third, the duplicate ratio of that which it has to the second.

XT.

When four magnitudes are continual proportionals, the first is said

to have to the fourth, the triplicate ratio of that which it has to the

second, and so on, quadruplicate, &c., increasing the denomination

still by unity, in any number of proportionals.

Definition A, to wit, of compound ratio.

When there are any number of magnitudes of the same kind, the

first is said to have to the last of them the ratio compounded of the

ratio which the first has to the second, and of the ratio w^hich the

second has to the third, and of the ratio which the third has to the

fourth, and so on unto the last magnitude.

For example, if A, B, C, D be four magnitudes of the same kind, the first

A is said to have to the last D, the ratio compounded of the ratio of ^ to _B,

and of the ratio of B to C, and of the ratio of C to D ; or, the ratio of A to

D is said to be compounded of the ratios of A to B, B to C, and C to D.

And if A has to B the same ratio which i' has to F; and B to C the

same ratio that G has to H; and C to D the same that iThas to i; then,

by this definition, A is said to have to I) the ratio compounded of ratios

which are the same with the ratios of S to F, O to H, and /l to X. And the

same thing is to be understood when it is more briefly expressed by saying,

A has to jb the ratio compounded of the ratios of JEtoF,G to S, and Kto L.

In like manner, the same things being supposed, if M has to iV the same

ratio which A has to D ; then, for shortness' sake, M is said to have to JV

the ratio compounded of the ratios of B to F, Q to JT, and K to L.

XII.

In proportionals, the antecedent terms are called homologous to

one another, as also the consequents to one another.

* Geometers make use of the following technical words, to signify certain

ways of changing either the order or magnitude of proportionals, so that

they continue still to be proportionals.'

XIII.

Permutando, or altemando by permutation, or alternately. This

word is used when there are four proportionals, and it is inferred that

the first has the same ratio to the third which the second has to the

fourth ; or that the first is to the third as the second to the fourth :

as is shewn in Prop. xvi. of this Fifth Book.

XIV.

Invertendo, by inversion ; when there are four proportionals, and

it is inferred, that the second is to the first, as the fourth to the third.

Prop. B. Book v.

206 Euclid's elements.

^ XV.

Componendo, by composition ; when there are four proportionals,

and it is inferred that the first together with the second, is to the

second, as the third together with the fourth, is to the fourth. Prop.

18, Book V.

XVI.

Dividendo, by division ; when there are four proportionals, and it is

inferred, that the excess of the first above the second, is to the second,

as the excess of the third above the fourth, is to the fourth. Prop. 17,

Book V.

xvn.

Convertendo, by conversion ; when there are four proportionals, and

it is inferred, that the first is to its excess above the second, as the

third to its excess above the fourth. Prop. E. Book v.

XVIII.

Ex sequali (sc. distantia), or ex aequo, from equality of distance :

when there is any number of magnitudes more than two, and as many

others such that they are proportionals when taken two and two of

each rank, and it is inferred, that the first is to the last of the first rank

of magnitudes, as the first is to the last of the others : * Of this there

are the two following kinds, which arise from the diflferent order in

which the magnitudes are taken, two and two.'

XIX.

Ex sequali, from equality. This term is used simply by itself, when

the first magnitude is to the second of the first rank, as the first to the

second of the other rank ; and as the second is to the third of the first

rank, so is the second to the third of the other ; and so on in order : and

the inference is as mentioned in the preceding definition ; whence this

is called ordinate proportion. It is demonstrated in Prop. 22, Book v.

XX.

Ex sequali in proportione perturbata seu inordinate, from equality

in perturbate or disorderly proportion*. This term is used when the

first magnitude is to the second of the first rank, as the last but one is

to the last of the second rank ; and as the second is to the third of the

first rank, so is the last but two to the last but one of the second rank:

and as the third is to the fourth of the first rank, so is the third from

the last to the last but two of the second rank ; and so on in a cross

order: and the inference is as in the 18th definition. It is demon-

strated in Prop. 23, Book v.

AXIOMS.

I.

Equimultiples of the same, or of equal magnitudes, are equal to

one another.

n.

Those magnitudes, of which the same or equal magnitudes are

equimultiples, are equal to one another.

â€¢ Prop. 4. Lib. ii. Archimedis de sphaera et cylindro.

BOOK V. PROP. I, II. 201

III

A multiple of a greater magnitude is greater than the same mul-

tiple of a less.

IV.

That magnitude, of which a multiple is greater than the same

multiple of another, is greater than that other magnitude.

PROPOSITION I. THEOHEM.

If any number of magnitudes be equimultiples of as many, each of each : what

multiple soever any one of them is of its part, the same multiple shall all the

first magnitudes be of all the other.

Let any number of magnitudes AB, CD be equimultiples of as

. many others ^E, F, each of each.

Then whatsoever multiple AB is of F,

the same multiple shall AB and CD together be of ^ and i^ together.

A G B C H D

Because AB is the same multiple of U that CD is of F,

as many magnitudes as there are in AB equal to F, so many are

there in C.D equal to F.

Divide AB into magnitudes equal to F, viz. AG, GB ;

and CD into CH, HD, equal each of them to F;

therefore the number of the magnitudes CH, HD shall be equal to

the number of the others A G, GB ;

and because AG is equal to F, and CJTto F,

therefore A G and CH together are equal to F and i^ together : (l. ax. 2.)

for the same reason, because GB is equal to F, and HD to F-,

GB and HD together are equal to F and F together :

wherefore as many magnitudes as there are in AB equal to F,

so many are there in AB, CD together, equal to F and F together :

therefore, whatsoever multiple u4.B is of F,

the same multiple is AB and CD together, of F and F together.

Therefore, if any magnitudes, how many soever, be equimultiples

of as many, each of each ; whatsoever multiple any one of them is

of its part, the same multiple shall all the first magnitudes be of all

the others : ' For the same demonstration holds in any number of

magnitudes, which was here applied to two.' Q. E. D.

PROPOSITION II. THEOREM.

Jf the first magnitude be the same multiple of the secojid that the third is of

the fourthf and the fifth the same multiple of the second that the sixth is of the

fourth; then shall lite first together with the fifth be the same viultiple of the

second, that the third together with the sixth is of the fourth.

Let AB the first be the same multiple of C the second, that DF

the third is of F the fourth :

208

and BG the fifth the same multiple of C the second, that EH the

sixth is of F the fourth.

Then shall A G, the first together with the fifth, be the same mul-

tiple of C the second, that DH, the third together with the sixth, is

of F the fourth.

A B Q D E H

Because AB is the same multiple of Cthat DE is of JP;

there are as many magnitudes in AB equal to C, as there are in DE

equal to F.

in like manner, as many as there are in BG equal to C, so many are

there in EH equal to F:

therefore as many as there are in the whole A G equal to C,

so many are there in the whole DH equal to F:

therefore AG is the same multiple of C that DH is of F;

that is, AG, the first and fifth together, is the same multiple of the

second C,

that DH, the third and sixth together, is of the fourth F.

If therefore, the first be the same multiple, &c. q.e.d.

Cor. From this it is plain, that if any number of magnitudes AB,

BG, GHhe multiples of another C;

and as many DE, EX, KL be the same multiples of F, each of each:

then the whole of the first, viz. AH, is the same multiple of C,

that the whole of the last, viz. DL, is of F.

PROPOSITION III. THEOREM.

Jf the first he the same multiple of the second, tohich the third is of the fourth:

and if of the first and third there be taken equimultiples; these shall be equi-

multiples, the one of the second, and the other of the fourth.

Let A the first be the same multiple of B the second, that C the

third is of D the fourth :

and of ^, Clet equimultiples EF, GHlae taken.

Then EF shall be the same multiple of B, that GH is of D.

E K F G L H

A-

B D

Because JE^i^is the same multiple of ^, that 6r^is of C,

there are as many magnitudes in EF equal to ^, as there are in GH

equal to C :

let J^jPbe divided into the magnitudes EK, KF, each equal to A ;

and GH into GL, LH, each equal to C:

therefore the number of the magnitudes EK, XF shall be equal to

the number of the others GL, LH\

BOOK V. PROP. IV. 209

and because A is the same multiple of B, that C is of D,

and that EK is equal to A, and GL equal to C:

therefore EK is the same multiple of B, that GL is of Z) ;

for the same reason, XFis the same multiple of B, that LIT is of D :

and so, if there be more parts in EF, GH, equal to ^, C:

therefore, because the first EK is the same multiple of the second B,

which the third GL is of the fourth E,

and that the fifth ^-Pis the same multiple of the second B, which the

sixth LH is of the fourth E ;

EF the first, together with the fifth, is the same multiple of the second

B, (V. 2.)

which (?-Â£r the third, together with the sixth, is of the fourth E.

If, therefore, the first, &c. q.e.d.

PROPOSITION IV. THEOEEM.

If the first of four magnitudes has the same ratio to the second which the

third has to the fourth ; then any equimultiples whatever of the first and third

shall have the same ratio to any equimultiples of the second and fourth, viz, ' the

equimultiple of the first shall have the same ratio to that of the second, which the

equimultiple of the third has to that of the fourth.'

Let A the first have to B the second, the same ratio which the third

C has to the fourth E ;

and of A and C'let there be taken any equimultiples whatever E, F;

and of B and E any equimultiples whatever G, H.

Then E shall have the same ratio to G, which F has to H.

K M-

E G-

A B-

C D-

F H-

L N-

Take of E and F any equimultiples whatever K, L,

and of G, JTany equimultiples whatever M, N\

then because E is the same multiple of ^, that Fh of (7;

and of E and F have been taken equimultiples K, L ;

therefore K is the same multiple of A, that i is of C: (v. 3.)

for the same reason, M is the same multiple of J5, that N is of E.

And because, as A is to B, so is C to E, (hyp.)

and of A and C have been taken certain equimultiples K, L,

and of B and E have been taken certain equimultiples M, N',

therefore if K be greater than M, L is greater than N\

and if equal, equal ; if less, less : (v. def. 5.)

but K, L are any equimultiples whate\er of E, F, (constr.)

and M, N any whatever of G, E[\

therefore as E is to G, so is F to H. (v. def. 5.)

Therefore, if the first, &c. Q. E. d.

Cor. Likewise, if the first has the same ratio to the second, which

the third has to the fourth, then also any equimultiples whatever of

210 EUCLID'S ELEMENTS.

the first and third shall have the same ratio to the second and fourth;

and in like manner, the first and the third shall have the same ratio to

any equimultiples whatever of the second and fourth.

Let A the first have to B the second the same ratio which the

third C has to the fourth D.

and of A and C let E and jPbe any equimultiples whatever.

Then E shall be to ^ as i^ to D.

Take of J?, i^any equimultiples whatever, K, L,

and of B, D any equimultiples whatever G, II:

then it may be demonstrated, as before, that K is the same multiple

of A, that X is of C:

and because A is to B, as C is to D, {hyp.)

and of ^ and (7 certain equimultiples have been taken viz., ^and X;

and of B and I) certain equimultiples G, H;

therefore, if Khe greater than G, L is greater than II\

and if equal, equal ; if less, less : (v. def 5.)

but K^ L are any equimultiples whatever of E, F, (constr.)

and G, H any whatever of ^, Z> ;

therefore as E is to B, so is F to D. (v. def. 5.)

And in the same way the other case is demonstrated.

PROPOSITION V. THEOREM.

Jf one magvitude he the same multiple of another, which a magnitude talcen

from the first is of a magnitude taken from the other; the remainder shall be the

same multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the same multiple of CE, that ^X^ taken

from the first, is of CF taken from the other.

The remainder EB shall be the same multiple of the remainder

FE, that the whole AB is of the whole CD,

F D

Take A G the same multiple of FB, that AE is of CF:

therefore AE is the same multiple of CF, that EG is of CD : (v. 1.)

but AE, by the hypothesis, is the same multiple of CF, that AB is

of CD;

therefore EG is the same multiple of CD that AB is of CD;

wherefore EG is equal to AB : (v. ax. 1.)

take from each of them the common magnitude AE;

and the remainder ^6^ is equal to the remainder EB.

Wherefore, since AE is the same multiple of CF, that ^ 6^ is of FD,

(constr.)

and that A G has been proved equal to EB ;

therefore AE is the same multiple of CF, that EB is of FD :

but AE is the same multiple of CF that AB is of CD : (hyp.)

therefore EB is the same multiple of FD, that AB is of CD.

Therefore, if one magnitude, &c. Q. E. D.

BOOK V. PROP. VI. 211

PROPOSITION VI. THEOREM.

Jf two magnitudes he equimultiples of two others, and if equimultiples of

the'ie he taken from the first two ; the remainders are either equal to these others,

or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the two E, F,

and let AG, CZT taken from the first two be equimultiples of the

same E, F.

Then the remainders GB, HD shall be either equal to E, F, or

equimultiples of them.

AGE E

K C H D r â€”

First, let GB be equal to E\

HD shall be equal to F.

Make CK equal to F:

and because AGh the same multiple of E, that CJI'is of F: (hyp.)

and that GB is equal to E, and CK to F;

therefore AB is the same multiple of E, that XII is of F:

but AB, by the hypothesis, is the same multiple of^, that CD is otF;

therefore KH is the same multiple of i^, that CD is of i^:

wherefore KH is equal to CD: (v. ax. 1.)

take away the common magnitude CH,

then the remainder KC is equal to the remainder HD :

but KCis equal to F: (constr.)

therefore HD is equal to F.

Next let GB be a multiple of E.

Then HD shall be the same multiple of F.

K c H ^D r â€”

Make CKthe same multiple of i^, that GB iso^ E:

and because A G is the same multiple of E, that CH is of F: (hyp.)

and GB the same multiple of E, that CK is of F;

therefore AB is the same multiple of E, that KHis of F: (y. 2.)

but AB is the same multiple of E, that CD is of i^; (hyp.)

therefore KH Is the same multiple of F, that CD is of F]

wherefore KHis equal to CD : (v. ax. 1.)

take away CH from both;

therefore the remainder KC is equal to the remainder HD :

and because GB is the same multiple of E, that KC is of F, (constr.)

and that KC is equal to HD ;

therefore HD is the same multiple of F, that GB is of E.

If, therefore, two magnitudes, &c. q.e.d.

21^

PROPOSITION A. THEOREM.

7/ the first of four magnitudes has the same ratio to the second, which the

third has to the fourth ; then, if the first be greater than the second, the third

is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each :

then, by def. 5th of this book, if the double of the first be greater

than the double of the second, the double of the third is greater than

angular octagon.

VI.

44. Inscribe a circle in a rhombus.

45. Having given the distances of the centers of two equal circles

which cut one another, inscribe a square in the space included between

the two circumferences.

46. The square inscribed in a circle is equal to half the square

described about the same circle.

47. The square is greater than any oblong inscribed in the same

circle.

48. A circle having a square inscribed in it being given, to find a

circle in which a regular octagon of a perimeter equal to that of the

square, may be inscribed.

49. Describe a circle about a figure formed by constructing an

equilateral triangle upon the base of an isosceles triangle, the vertical

angle of which is four times the angle at the base.

oO. A regular octagon inscribed in a circle is equal to the rectangle

k5

202 GEOMETRICAL EXERCISES

contained by the sides of the squares inscribed in, and circumscribed

about the circle.

51. If in any circle the side of an inscribed hexagon be produced

till it becomes equal to the side of an inscribed square, a tangent

drawn from the extremity, without the circle, shall be equal to the

side of an inscribed octagon.

VII.

52. To describe a circle which shall touch a given circle in a given

point, and also a given straight line.

53. Describe a circle touching a given straight line, and also two

given circles.

54. Describe a circle which shall touch a given circle, and each of

two given straight lines.

66, Tm'o points are given, one in each of two given circles ; describe

a circle passing through both points and touching one of the circles.

6Q. Describe a circle touching a straight line in a given point, and

also touching a given circle. AVhen the line cuts the given circle,

shew that your construction will enable you to obtain six circles

touching the given cii'cle and the given line, but not necessarily in the

given point.

57. Describe a circle which shall touch two sides and pass through

one angle of a given square.

58. If two circles touch each other externally, describe a circle

which shall touch one of them in a given point, and also touch the

other. In what case does this become impossible ?

59. Describe three circles touching each other and having their

centers at three given points. In how many different ways may this

be done ?

VIII.

60. Let two straight lines be drawn from any point within a circle

to the circumference: describe a circle, which shall touch them both,

and the arc between them.

61. In a given triangle having inscribed a circle, inscribe another

circle in the space thus intercepted at one of the angles.

62. Let AB, A C, be the bounding radii of a quadrant; complete

the square A JB DC and draw the diagonal AD-, then the part of the

diagonal without the quadrant will be equal to the radius of a circle

inscribed in the quadi*ant.

63. If on one of the bounding radii of a quadrant, a semicircle be

described, and on the other, another semicircle be described, so as to

touch the former and the quadrantal arc; find the center of the circle

inscribed in the figure bounded by the three curves.

64. In a given segment of a circle inscribe an isosceles triangle,

such that its vertex may be in the middle of the chord, and the base

and perpendicular together equal to a given line.

65. Inscribe three circles in an isosceles triangle touching each

other, and each of them touching two of the three sides of the triangle.

IX.

66. In the fig. Prop. 10, Book iv, shew that the base BD\& the

ON BOOK IV. 203

side of a regular decagon inscribed in the larger circle, and the side of

a regular pentagon inscribed in the smaller circle.

67. In the fig. Prop. 10, Book IV, produce DC to meet the circle

in F, and draw l^F; then the angle ABF shall be equal to thi'ee times

the angle BFD.

68. If the alternate angles of a regular pentagon be joined, the

figure formed by the intersection of the joining lines will itself be a

regular pentagon.

69. 1^ ABODE be any pentagon inscribed in a circle, and AC,

BD, CE, DA, EB be joined, then are the angles ABE, BCA, CDB,

DEC, EAD, together equal to two right angles.

70. A watch-ribbon is folded up into a flat knot of five edges, shew

that the sides of the knot form an equilateral pentagon.

71. If from the extremities of the side of a regular pentagon

inscribed in a circle, straight lines be drawn to the middle of the arc

subtended by the adjacent side, their diff'erence is equal to the radius ;

the sum of their squares to three times the square of the radius ; and

the rectangle contained by them is equal to the square of the radius.

72. Inscribe a regular pentagon in a given square so that four

angles of the pentagon may touch respectively the four sides of the

square.

73. Inscribe a regular decagon in a given circle.

74. The square described upon the side of a regular pentagon in

a circle, is equal to the square on the side of a regular hexagon, together

with the square upon the side of a regular decagon in the same circle.

X.

75. In a given circle inscribe three equal circles touching each

other and the given circle.

76. Shew that if two circles be inscribed in a third to touch one

another, the tangents of the points of contact will all meet in the same

point.

77. If there be three concentric circles, whose radii are 1, 2, 3 ;

determine how many circles may be described round the interior one,

having their centers in the circumference of the circle, whose radius is

2, and touching the interior and exterior circles, and each other.

78. Shew that nine equal circles may be placed in contact, so that

a square whose side is three times the diameter of one of them will

circumscribe them.

XL

79. Produce the sides of a given heptagon both ways, till they

meet, forming seven triangles; required the sum of their vertical

angles.

80. To convert a given regular polygon into another which shall

have the same perimeter, but double the number of sides.

81. In any polygon of an even number of sides, inscribed in a

circle, the sum of the 1st, 3rd, 5th, &c. angles is equal to the sum of

the 2nd, 4th, 6th, &c.

82. Of all polygons having equal perimeters, and the same number

of sides, the equilateral polygon has the greatest area.

BOOK V.

DEFINITIONS.

I.

A LESS magnitude is said to be apart of a greater magnitude, when

the less measures the greater; that is, 'when the less is contained a

certain number of times exactly in the greater/

11.

A greater magnitude is said to be a multiple of a less, when the

greater is measured by the less, that is, * when the greater contains the

less a certain number of times exactly.'

III.

" Ratio is a mutual relation of two magnitudes of the same kind to

one another, in respect of quantity."

IV.

Magnitudes are said to have a ratio to one another, when the less

can be multiplied so as to exceed the other.

V.

The first of four magnitudes is said to have the same ratio to the

second, which the third has to the fourth, when any equimultiples

whatsoever of the first and third being taken, and any equimultiples

whatsoever of the second and fourth ; if the multiple of the first be less

than that of the second, the multiple of the third is also less than that

of the fourth : or, if the multiple of the first be equal to that of the

second, the multiple of the third is also equal to that of the fourth: or,

if the multiple of the first be greater than that of the second, the mul-

tiple of the third is also greater than that of the fourth.

VI.

Magnitudes which have the same ratio are called proportionals.

N.B. 'When four magnitudes are proportionals, it is usually ex-

pressed by saying, the first is to the second, as the third to the fourth.'

VII.

ViTien of the equimultiples of four magnitudes (taken as in the

fifth definition), the multiple of the first is greater than that of the

second, but the multiple of the third is not greater than the multiple

of the fourth ; then the first is said to have to the second a greater

ratio than the third magnitude has to the fourth : and, on the contrary,

the third is said to have to the fourth a less ratio than the first has to

the second.

VIII.

" Analogy, or proportion, is the similitude of ratios."

DEFINITIONS.

205

IX.

Proportion consists in three terms at least.

X.

When three magnitudes are proportionals, the first is said to have

to the third, the duplicate ratio of that which it has to the second.

XT.

When four magnitudes are continual proportionals, the first is said

to have to the fourth, the triplicate ratio of that which it has to the

second, and so on, quadruplicate, &c., increasing the denomination

still by unity, in any number of proportionals.

Definition A, to wit, of compound ratio.

When there are any number of magnitudes of the same kind, the

first is said to have to the last of them the ratio compounded of the

ratio which the first has to the second, and of the ratio w^hich the

second has to the third, and of the ratio which the third has to the

fourth, and so on unto the last magnitude.

For example, if A, B, C, D be four magnitudes of the same kind, the first

A is said to have to the last D, the ratio compounded of the ratio of ^ to _B,

and of the ratio of B to C, and of the ratio of C to D ; or, the ratio of A to

D is said to be compounded of the ratios of A to B, B to C, and C to D.

And if A has to B the same ratio which i' has to F; and B to C the

same ratio that G has to H; and C to D the same that iThas to i; then,

by this definition, A is said to have to I) the ratio compounded of ratios

which are the same with the ratios of S to F, O to H, and /l to X. And the

same thing is to be understood when it is more briefly expressed by saying,

A has to jb the ratio compounded of the ratios of JEtoF,G to S, and Kto L.

In like manner, the same things being supposed, if M has to iV the same

ratio which A has to D ; then, for shortness' sake, M is said to have to JV

the ratio compounded of the ratios of B to F, Q to JT, and K to L.

XII.

In proportionals, the antecedent terms are called homologous to

one another, as also the consequents to one another.

* Geometers make use of the following technical words, to signify certain

ways of changing either the order or magnitude of proportionals, so that

they continue still to be proportionals.'

XIII.

Permutando, or altemando by permutation, or alternately. This

word is used when there are four proportionals, and it is inferred that

the first has the same ratio to the third which the second has to the

fourth ; or that the first is to the third as the second to the fourth :

as is shewn in Prop. xvi. of this Fifth Book.

XIV.

Invertendo, by inversion ; when there are four proportionals, and

it is inferred, that the second is to the first, as the fourth to the third.

Prop. B. Book v.

206 Euclid's elements.

^ XV.

Componendo, by composition ; when there are four proportionals,

and it is inferred that the first together with the second, is to the

second, as the third together with the fourth, is to the fourth. Prop.

18, Book V.

XVI.

Dividendo, by division ; when there are four proportionals, and it is

inferred, that the excess of the first above the second, is to the second,

as the excess of the third above the fourth, is to the fourth. Prop. 17,

Book V.

xvn.

Convertendo, by conversion ; when there are four proportionals, and

it is inferred, that the first is to its excess above the second, as the

third to its excess above the fourth. Prop. E. Book v.

XVIII.

Ex sequali (sc. distantia), or ex aequo, from equality of distance :

when there is any number of magnitudes more than two, and as many

others such that they are proportionals when taken two and two of

each rank, and it is inferred, that the first is to the last of the first rank

of magnitudes, as the first is to the last of the others : * Of this there

are the two following kinds, which arise from the diflferent order in

which the magnitudes are taken, two and two.'

XIX.

Ex sequali, from equality. This term is used simply by itself, when

the first magnitude is to the second of the first rank, as the first to the

second of the other rank ; and as the second is to the third of the first

rank, so is the second to the third of the other ; and so on in order : and

the inference is as mentioned in the preceding definition ; whence this

is called ordinate proportion. It is demonstrated in Prop. 22, Book v.

XX.

Ex sequali in proportione perturbata seu inordinate, from equality

in perturbate or disorderly proportion*. This term is used when the

first magnitude is to the second of the first rank, as the last but one is

to the last of the second rank ; and as the second is to the third of the

first rank, so is the last but two to the last but one of the second rank:

and as the third is to the fourth of the first rank, so is the third from

the last to the last but two of the second rank ; and so on in a cross

order: and the inference is as in the 18th definition. It is demon-

strated in Prop. 23, Book v.

AXIOMS.

I.

Equimultiples of the same, or of equal magnitudes, are equal to

one another.

n.

Those magnitudes, of which the same or equal magnitudes are

equimultiples, are equal to one another.

â€¢ Prop. 4. Lib. ii. Archimedis de sphaera et cylindro.

BOOK V. PROP. I, II. 201

III

A multiple of a greater magnitude is greater than the same mul-

tiple of a less.

IV.

That magnitude, of which a multiple is greater than the same

multiple of another, is greater than that other magnitude.

PROPOSITION I. THEOHEM.

If any number of magnitudes be equimultiples of as many, each of each : what

multiple soever any one of them is of its part, the same multiple shall all the

first magnitudes be of all the other.

Let any number of magnitudes AB, CD be equimultiples of as

. many others ^E, F, each of each.

Then whatsoever multiple AB is of F,

the same multiple shall AB and CD together be of ^ and i^ together.

A G B C H D

Because AB is the same multiple of U that CD is of F,

as many magnitudes as there are in AB equal to F, so many are

there in C.D equal to F.

Divide AB into magnitudes equal to F, viz. AG, GB ;

and CD into CH, HD, equal each of them to F;

therefore the number of the magnitudes CH, HD shall be equal to

the number of the others A G, GB ;

and because AG is equal to F, and CJTto F,

therefore A G and CH together are equal to F and i^ together : (l. ax. 2.)

for the same reason, because GB is equal to F, and HD to F-,

GB and HD together are equal to F and F together :

wherefore as many magnitudes as there are in AB equal to F,

so many are there in AB, CD together, equal to F and F together :

therefore, whatsoever multiple u4.B is of F,

the same multiple is AB and CD together, of F and F together.

Therefore, if any magnitudes, how many soever, be equimultiples

of as many, each of each ; whatsoever multiple any one of them is

of its part, the same multiple shall all the first magnitudes be of all

the others : ' For the same demonstration holds in any number of

magnitudes, which was here applied to two.' Q. E. D.

PROPOSITION II. THEOREM.

Jf the first magnitude be the same multiple of the secojid that the third is of

the fourthf and the fifth the same multiple of the second that the sixth is of the

fourth; then shall lite first together with the fifth be the same viultiple of the

second, that the third together with the sixth is of the fourth.

Let AB the first be the same multiple of C the second, that DF

the third is of F the fourth :

208

and BG the fifth the same multiple of C the second, that EH the

sixth is of F the fourth.

Then shall A G, the first together with the fifth, be the same mul-

tiple of C the second, that DH, the third together with the sixth, is

of F the fourth.

A B Q D E H

Because AB is the same multiple of Cthat DE is of JP;

there are as many magnitudes in AB equal to C, as there are in DE

equal to F.

in like manner, as many as there are in BG equal to C, so many are

there in EH equal to F:

therefore as many as there are in the whole A G equal to C,

so many are there in the whole DH equal to F:

therefore AG is the same multiple of C that DH is of F;

that is, AG, the first and fifth together, is the same multiple of the

second C,

that DH, the third and sixth together, is of the fourth F.

If therefore, the first be the same multiple, &c. q.e.d.

Cor. From this it is plain, that if any number of magnitudes AB,

BG, GHhe multiples of another C;

and as many DE, EX, KL be the same multiples of F, each of each:

then the whole of the first, viz. AH, is the same multiple of C,

that the whole of the last, viz. DL, is of F.

PROPOSITION III. THEOREM.

Jf the first he the same multiple of the second, tohich the third is of the fourth:

and if of the first and third there be taken equimultiples; these shall be equi-

multiples, the one of the second, and the other of the fourth.

Let A the first be the same multiple of B the second, that C the

third is of D the fourth :

and of ^, Clet equimultiples EF, GHlae taken.

Then EF shall be the same multiple of B, that GH is of D.

E K F G L H

A-

B D

Because JE^i^is the same multiple of ^, that 6r^is of C,

there are as many magnitudes in EF equal to ^, as there are in GH

equal to C :

let J^jPbe divided into the magnitudes EK, KF, each equal to A ;

and GH into GL, LH, each equal to C:

therefore the number of the magnitudes EK, XF shall be equal to

the number of the others GL, LH\

BOOK V. PROP. IV. 209

and because A is the same multiple of B, that C is of D,

and that EK is equal to A, and GL equal to C:

therefore EK is the same multiple of B, that GL is of Z) ;

for the same reason, XFis the same multiple of B, that LIT is of D :

and so, if there be more parts in EF, GH, equal to ^, C:

therefore, because the first EK is the same multiple of the second B,

which the third GL is of the fourth E,

and that the fifth ^-Pis the same multiple of the second B, which the

sixth LH is of the fourth E ;

EF the first, together with the fifth, is the same multiple of the second

B, (V. 2.)

which (?-Â£r the third, together with the sixth, is of the fourth E.

If, therefore, the first, &c. q.e.d.

PROPOSITION IV. THEOEEM.

If the first of four magnitudes has the same ratio to the second which the

third has to the fourth ; then any equimultiples whatever of the first and third

shall have the same ratio to any equimultiples of the second and fourth, viz, ' the

equimultiple of the first shall have the same ratio to that of the second, which the

equimultiple of the third has to that of the fourth.'

Let A the first have to B the second, the same ratio which the third

C has to the fourth E ;

and of A and C'let there be taken any equimultiples whatever E, F;

and of B and E any equimultiples whatever G, H.

Then E shall have the same ratio to G, which F has to H.

K M-

E G-

A B-

C D-

F H-

L N-

Take of E and F any equimultiples whatever K, L,

and of G, JTany equimultiples whatever M, N\

then because E is the same multiple of ^, that Fh of (7;

and of E and F have been taken equimultiples K, L ;

therefore K is the same multiple of A, that i is of C: (v. 3.)

for the same reason, M is the same multiple of J5, that N is of E.

And because, as A is to B, so is C to E, (hyp.)

and of A and C have been taken certain equimultiples K, L,

and of B and E have been taken certain equimultiples M, N',

therefore if K be greater than M, L is greater than N\

and if equal, equal ; if less, less : (v. def. 5.)

but K, L are any equimultiples whate\er of E, F, (constr.)

and M, N any whatever of G, E[\

therefore as E is to G, so is F to H. (v. def. 5.)

Therefore, if the first, &c. Q. E. d.

Cor. Likewise, if the first has the same ratio to the second, which

the third has to the fourth, then also any equimultiples whatever of

210 EUCLID'S ELEMENTS.

the first and third shall have the same ratio to the second and fourth;

and in like manner, the first and the third shall have the same ratio to

any equimultiples whatever of the second and fourth.

Let A the first have to B the second the same ratio which the

third C has to the fourth D.

and of A and C let E and jPbe any equimultiples whatever.

Then E shall be to ^ as i^ to D.

Take of J?, i^any equimultiples whatever, K, L,

and of B, D any equimultiples whatever G, II:

then it may be demonstrated, as before, that K is the same multiple

of A, that X is of C:

and because A is to B, as C is to D, {hyp.)

and of ^ and (7 certain equimultiples have been taken viz., ^and X;

and of B and I) certain equimultiples G, H;

therefore, if Khe greater than G, L is greater than II\

and if equal, equal ; if less, less : (v. def 5.)

but K^ L are any equimultiples whatever of E, F, (constr.)

and G, H any whatever of ^, Z> ;

therefore as E is to B, so is F to D. (v. def. 5.)

And in the same way the other case is demonstrated.

PROPOSITION V. THEOREM.

Jf one magvitude he the same multiple of another, which a magnitude talcen

from the first is of a magnitude taken from the other; the remainder shall be the

same multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the same multiple of CE, that ^X^ taken

from the first, is of CF taken from the other.

The remainder EB shall be the same multiple of the remainder

FE, that the whole AB is of the whole CD,

F D

Take A G the same multiple of FB, that AE is of CF:

therefore AE is the same multiple of CF, that EG is of CD : (v. 1.)

but AE, by the hypothesis, is the same multiple of CF, that AB is

of CD;

therefore EG is the same multiple of CD that AB is of CD;

wherefore EG is equal to AB : (v. ax. 1.)

take from each of them the common magnitude AE;

and the remainder ^6^ is equal to the remainder EB.

Wherefore, since AE is the same multiple of CF, that ^ 6^ is of FD,

(constr.)

and that A G has been proved equal to EB ;

therefore AE is the same multiple of CF, that EB is of FD :

but AE is the same multiple of CF that AB is of CD : (hyp.)

therefore EB is the same multiple of FD, that AB is of CD.

Therefore, if one magnitude, &c. Q. E. D.

BOOK V. PROP. VI. 211

PROPOSITION VI. THEOREM.

Jf two magnitudes he equimultiples of two others, and if equimultiples of

the'ie he taken from the first two ; the remainders are either equal to these others,

or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the two E, F,

and let AG, CZT taken from the first two be equimultiples of the

same E, F.

Then the remainders GB, HD shall be either equal to E, F, or

equimultiples of them.

AGE E

K C H D r â€”

First, let GB be equal to E\

HD shall be equal to F.

Make CK equal to F:

and because AGh the same multiple of E, that CJI'is of F: (hyp.)

and that GB is equal to E, and CK to F;

therefore AB is the same multiple of E, that XII is of F:

but AB, by the hypothesis, is the same multiple of^, that CD is otF;

therefore KH is the same multiple of i^, that CD is of i^:

wherefore KH is equal to CD: (v. ax. 1.)

take away the common magnitude CH,

then the remainder KC is equal to the remainder HD :

but KCis equal to F: (constr.)

therefore HD is equal to F.

Next let GB be a multiple of E.

Then HD shall be the same multiple of F.

K c H ^D r â€”

Make CKthe same multiple of i^, that GB iso^ E:

and because A G is the same multiple of E, that CH is of F: (hyp.)

and GB the same multiple of E, that CK is of F;

therefore AB is the same multiple of E, that KHis of F: (y. 2.)

but AB is the same multiple of E, that CD is of i^; (hyp.)

therefore KH Is the same multiple of F, that CD is of F]

wherefore KHis equal to CD : (v. ax. 1.)

take away CH from both;

therefore the remainder KC is equal to the remainder HD :

and because GB is the same multiple of E, that KC is of F, (constr.)

and that KC is equal to HD ;

therefore HD is the same multiple of F, that GB is of E.

If, therefore, two magnitudes, &c. q.e.d.

21^

PROPOSITION A. THEOREM.

7/ the first of four magnitudes has the same ratio to the second, which the

third has to the fourth ; then, if the first be greater than the second, the third

is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each :

then, by def. 5th of this book, if the double of the first be greater

than the double of the second, the double of the third is greater than

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