Euclid. # Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

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Wherefore, if two triangles, &c. Q.E.D.

BOOK VI. PROP. XXXUI. 291

PROPOSITION XXXIII. THEOREM.

In equal circleSy angles, whether at the centers or circumferences, have

the same ratio which the circumferences on which they stand have to one

another : so also have the sectors.

Let ABC, DEFhe equal circles; and at their centers the angles

BGC, EIIF, and the angles BAC, EDF, at their circumferences.

As the circumference BC to the circumference FF, so shall the

angle BGC be to the angle EIIF, and the angle BAG to the

angle ^Di^;

and also the sector BGC to the sector EIIF,

Take any number of circumferences CK, KL, each equal to BC,

and any number whatever FM, MN, each equal to EF:

and join GK, GL, HM, UN.

Because the circumferences BC, CK, KL are all equal,

the angles BGC CGK, KGL are also all equal : (ill. 27.)

therefore what multiple soever the circumference BL is of the cir-

cumference BC, the same multiple is the angle BGL of the angle

BGC:

for the same reason, whatever multiple the circumference EN is of

the circumference EF^ the same multiple is the angle EHN of the

angle EHF:

and if the circumference BL be equal to the circumference EN,

the angle BGL is also equal to the angle EHN; (ill. 27.)

and if the circumference BL be greater than EN,

likewise the angle BGL is greater than EHN-, and if less, less:

therefore, since there are four magnitudes, the two circumferences

BC, EF, and the two angles BGC, EHF; and that of the circum-

ference BC, and of the angle BGC, have been taken any equimultiples

whatever, viz. the circumference BL, and the angle BGL : and of the

circumference EF, and of the angle EHF, any equimultiples what-

ever, viz. the circumference EN, and the angle EHN:

and since it has been proved, that if the circumference BL be greater

than EN)

the angle BGL is greater than EHN;

and if equal, equal ; and if less, less ;

therefore as the circumference BC to the circumference EF, so is the

angle BGC to the angle EHF: (v. def. 5.)

but as the angle BGC is to the angle EHFy so is the angle B AC to

the angle EDF: (v. 15.)

for each is double of each ; (ill. 20.)

therefore, as the circumference BCh to EF, so is the angle BGC to

the angle EHF, and the angle B AC to the angle EDF.

02

^92 Euclid's elements.

Also, as the circumference ^C to JEJF, so shall the sector BGChe

to the sector EHF.

Join BC, CK, and in the circumferences, BC, CK, take any points

X, O, and join BX, XC, CO, OX,

Then, because in the triangles GBC, GCK,

the two sides BG, GC are equal to the two CG, GX each to each,

and that they contain equal angles ;

the base ^Cis equal to the base CX, (l. 4.)

and the triangle GBC to the triangle GCX:

and because the circumference BC is equal to the circumference CX,

the remaining part of the whole circumference of the circle ABC, is

equal to the remaining part of the whole circumference of the same

circle : (ax. 3.)

therefore the angle BXC is equal to the angle COX; (ill. 27.)

and the segment BXC is therefore similar to the segment COX;

(III. def. 11.)

and they are upon equal straight lines, BC, CX:

but similar segments of circles upon equal straight lines, are equal

to one another: (lil. 24.)

therefore the segment BXC is equal to the segment COX :

and the triangle BGCwas proved to be equal to the triangle CGX;

therefore the whole, the sector BGC, is equal to the whole, the

sector CGX:

for the same reason, the sector XGZ is equal to each of the sectors

BGC, CGX:

in the same manner, the sectors EHF, FH3Â£, MHN may be

proved equal to one another :

therefore, what multiple soever the circumference BL is of the circum-

ference BC, the same multiple is the sector BGL of the sector BGC;

and for the same reason, whatever multiple the circumference EN

is of EF, the same multiple is the sector EHN of the sector

EHF:

and if the circumference BL be equal to EX, the sector BGL is

equal to the sector EHN;

and it the circumference BL be greater than EN, the sector BGL

is greater than the sector EHN;

and if less, less ;

since, then, there are four magnitudes, the two circumferences BC,

EF, and the two sectors BGC, EHF, and that of the circumference

BC, and sector BGC, the circumference BL and sector BGL are any

equimultiples whatever; and of the circumference EF, and sector

EHF, the circumference EN, and sector EHN are any equimultiples

whatever ;

BOOK VI. PROP. B, C. 293

and since it has been proved, that if the circumference BL be greater

than EN, the sector BGL is greater than the sector EIIN ;

and if equal, equal ; and if less, less :

therefore, as the circumference ^Cis to the circumference JEF, so

is the sector BGC to the sector EHF. (v. def. 5.)

Wherefore, in equal circles, &c. Q.E.D.

PROPOSITION B. THEOREM.

If an angle of a triangle be bisected by a straight line which likeioise cuts

the base ; the rectangle contavied by the sides of the triangle is equal to the

rectangle contained by the segments of the base, together with the square on

the straight line which bisects the angle.

Let ABC he a triangle, and let the angle BA C be bisected by the

straight line AD.

The rectangle BA, AC shall be equal to the rectangle BD, DC^

together with the square on AD.

//

E

Describe the circle A CB about the triangle, (iv. 5.)

and produce AD to the circumference in E, and join EC.

Then because the angle BAD is equal to the angle CAE, (hyp.)

and the angle ABD to the angle AEC, (ill. 21.)

for they are in the same segment ;

the triangles ABD, AEC are equiangular to one another : (l. 32.)

therefore as BA to AD, so is EA to AC-, (vi. 4.)

and consequently the rectangle BA, ^ C is equal to the rectangle EA,

AD, (VI. 16.)

that is, to the rectangle ED, DA, together with the square on ^D ;

(II. 3.)

but the rectangle ED, DA is equal to the rectangle BD, DC\ (III. 35.)

therefore the rectangle BA, AC'ys, equal to the rectangle BD, DC,

together with the square on AD.

Wherefore, if an angle, &c. Q. E. D.

PROPOSITION C. THEOREM.

If from any angle of a triangle, a straight line be drawn 'perpendicular to

the base ; the rectangle contained by the sides of the triangle is equal to the

rectangle contained by the perpendicular and the diameter of the circle de-

scribed about the triangle.

Let ABC ha a triangle, and AD the perpendicular from the angle

A to the base B C.

The rectangle i?^, ^Cshall be equal to the rectangle containedby

AD and the diameter of the circle described about the triangle.

294:

EUCLID S ELEMENTS.

Describe the circle jiCJB about the triangle, (iv. 5.) and draw its

diameter AE, and join JEC.

Because the right angle J5DA is equal to the angle JECA in a

semicircle, (iii. 31.)

and the angle ABD equal to the angle -4^C in the same segment;

(III. 21.) the triangles ABD, AjECare equiangular:

therefore as BA to AI), so is UA to AC; (vi. 4.)

and consequently the rectangle BA, AC is equal to the rectangle ^A,

AD. (vi. 16.) If therefore from any angle, &c. Q.E.D.

PROPOSITION D. THEOREM.

The rectangle contained by the diagonals of a quadrilateral figure inscribed

in a circle, is equal to both the rectangles contained by its opposite sides.

'Lei AB CD he any quadrilateral figure inscribed in a circle, and

'^om AC, BD.

The rectangle contained by A C, BD shall be equal to the two

rectangles contained by AB, CD, and by AD, BC.

Make the angle ABE equal to the angle DBC: (I. 23.)

add to each of these equals the common angle EBB,

then the angle ABD is equal to the angle EBC:

and the angle BDA is equal to the angle B CE, because they are

in the same segment : (IIL 21.)

therefore the triangle ABD is equiangular to the triangle BCE:

wherefore, ;as ^Cis to CE, so is BD to DA ; (vi. 4.)

and consequently the rectangle BC, AD is equal to the rectangle

BD, CE : (VL 16.)

again, because the angle ABE Is equal to the angle DBC, and the

angle BAE to the angle BDC, (in. 21.)

the triangle ABE is equiangular to the triangle BCD :

therefore as BA to AE, so is BD to DC;

wherefore the rectangle BA, DC is equal to the rectangle BD, AE:

but the rectangle BC, AD has been shewn to be equal

to the rectangle BD, CE ;

therefore the whole rectangle AC, BD is equal to the rectangle

AB, DC, together with the rectangle AD, BC. (il. 1.)

Therefore the rectangle, &c. Q. E. D.

This is a Lemma of CI. Ptolemseus, in page 9 of his MeydXtj Syvra^is.

NOTES TO BOOK YL

In this Book, tlie theory of proportion exhibited in the Fifth Book, is

applied to the comparison of the sides and areas of plane rectilineal figures,

both of those which are similar, and of those which are not similar.

Def. I. In defining similar triangles, one condition is sufficient, namely,

that similar triangles are those which have their three angles respectively

equal ; as in Prop. 4, Book vi, it is proved that the sides about the equal

angles of equiangular triangles are proportionals. But in defining similar

figures of more than three sides, both of the conditions stated in Def. i,

are requisite, as it is obvious, for instance, in the case of a square and a

rectangle, which have their angles respectively equal, but have not their

sides about their equal angles proportionals.

The following definition has been proposed : ** Similar rectilineal

figures of more than three sides, are those which may be divided into the

same number of similar triangles," This definition would, if adopted,

require the omission of a part of Prop. 20, Book vi.

Def. III. To this definition may be added the following :

A straight line is said to be divided harnioyiically, when it is divided

into three parts, such that the whole line is to one of the extreme segments,

as the other extreme segment is to the middle part. Three lines are in

harmonical proportion, when the first is to the third, as the diflerence be-

tween the first and second, is to the difference between the second and

third ; and the second is called a harmonic mean between the first and third.

The expression â€¢ harmonical proportion' is derived from the following

fact in the Science of Acoustics, that three musical strings of the same

material, thickness and tension, when divided in the manner stated in the

definition, or numerically as 6, 4, and 3, produce a certain musical note,

its fifth, and its octave.

Def. IV. The term altitude, as applied to the same triangles and paral-

lelograms, will be different according to the sides which may be assumed

as the base, unless they are equilateral.

Prop. I. In the same manner may be proved, that triangles and paral-

lelograms upon equal bases, are to one another as their altitudes.

Prop. A. When the triangle ABC is isosceles, the line which bisects

the exterior angle at the vertex is parallel to the base. In all other cases,

if the line which bisects the angle BAC cut the base BCin the point G,

then the straight line BD is harmonically divided in the points Gr, C.

For EG is to GC as BA is to AC ; (vi. 3.)

and BD is to DC as BA is to AC, (vi. a.)

therefore BD is to DC as BG is to GC,

but BG = BD - DG, and GC = GD - DC.

Wherefore BD is to DC as BD - DG is to GD - DC.

Hence BD, DG, DC, are in harmonical proportion.

Prop. IV is the first case of similar triangles, and corresponds to the

third case of equal triangles, Prop. 26, Book i.

1

EUCLID S ELEMENTS.

Sometimes the sides opposite to the equal angles in two equiangular

triangles, are called the correspondmg sides, and these are said to be pro-

portional, which is simply taking the proportion in Euclid alternately.

The term homologous (o^aoXoyos), has reference to the places the sides

of the triangles have in the ratios, and in one sense, homologous sides may

be considered as corresponding sides. The homologous sides of any two

similar rectilineal figures will be found to be those which are adjacent to

two equal angles in each figure.

Prop. V, the converse of Prop, iv, is the second case of similar triangles,

and corresponds to Prop. 8, Book r, the second case of equal triangles.

Prop. VI is the third case of similar triangles, and corresponds to Prop.

4, Book I, the first case of equal triangles.

The property of similar triangles, and that contained in Prop. 47, Book

I, are the most important theorems in Geometry.

Prop. VII is the fourth case of similar triangles, and corresponds to the

fourth case of equal triangles demonstrated in the note to Prop. 26, Book i.

Prop. IX. The learner here must not forget the diiFerent meanings of

the word part, as employed in the Elements. The word here has the

same meaning as in Euc. v. def. 1.

It may be remarked, that this proposition is a more simple case of the

next, namely. Prop. x.

Prop. XI. This proposition is that particular case of Prop, xii, in which

the second and third terms of the proportion are equal. These two

problems exhibit the same results by a Geometrical construction, as are

obtained by numerical multiplication and division.

Prop. XIII. The difierence in the two propositions Euc. ii. 14, and

Euc. VI. 13, is this : in the Second Book, the problem is, to make a rect-

angular figure or square equal in area to an irregular rectilinear figure,

in which the idea of ratio is not introduced. In the Prop, in the Sixth

Book, the problem relates to ratios only, and it requires to divide a line

into two parts, so that the ratio of the whole line to the greater segment

may be the same as the ratio of the greater segment to the less.

The result in this proposition obtained by a Geometrical construction,

is analogous to that which is obtained by the multiplication of two

numbers, and the extraction of the square root of the fjroduct.

It may be observed, that half the sum of ^i^ and BC is called the

Arithmetic mean between these lines ; also that BD is called the Geo-

metric mean between the same lines.

To find two mean proportionals between two given lines is impossible

by the straight line and circle. Pappus has given several solutions of

this problem in Book iii, of his Mathematical Collections ; and Eutocius

has given, in his Commentary on the Sphere and Cylinder of Archimedes,

ten different methods of solving this problem.

Prop. XIV depends on the same principle as Prop, xv, and both may

easily be demonstrated from one diagram. Join DF, FE, EG in the fig.

to Prop. XIV, and the figure to Prop, xv is formed. We may add, that

there docs not appear any reason why the pro]3erties of the triangle and

parallelogram should be here separated, and not in the first proposition of

the Si:ith Book.

Prop. XV holds goou wnen one angle of one triangle is equal to the

defect from what the corresponding angle in the other wants of two right

angles.

This theorem will perhaps be more distinctly comprehended by the

learner, if he will bear in mind, that four magnitudes are reciprocally

NOTES TO BOOK VI. 297

proportional, when the ratio compounded of these ratios is a ratio of

equality.

Prop. XVII is only a particular case of Prop, xvi, and more properly,

might appear as a corollary: and both are cases of Prop. xiv.

Algebraically, Let AB, CD, E, F, contain a, b, c, d units respectively.

Then, since a, i, c, d are proportionals, .â€¢-=-.

b (i

Multiply these equals by bd, .'. ad = be,

or, the product of the extremes is equal to the product of the means.

And conversely. If the product of the extremes be equal to the pro-

duct of the means,

or ad = be,

then, dividing these equals by bd, /. t = -, ,

or the ratio of the first to the second number, is equal to the ratio of the

third to the fourth.

Similarly may be shewn, that if - = - ; then ad=b*.

d

And conversely, if adâ€”b^\ then t = -, â€¢

o a

Prop. XVIII. Similar figures are said to be similarly situated, when

their homologous sides are parallel, as when the figures are situated on

the same straight line, or on parallel lines ; but when similar figures are

situated on the sides of a triangle, the similar figures are said to be similarly

situated when the homologous sides of each figure have the same re-

lative position with respect to one another ; that is if the bases on which

the similar figures stand, were placed parallel to one another, the re-

maining sides of the figures, if similarly situated, would also be parallel

to one another.

Prop. XX. It may easily be shewn, that the perimeters of similar

polygons, are proportional to their homologous sides.

Prop. XXI. This proposition must be so understood as to include all

rectilineal figures whatsoever, which require for the conditions of simila-

rity another condition than is required for the similarity of triangles.

See note on Euc. vi. Def. i.

Prop. XXIII. The doctrine of compound ratio, including duplicate and

triplicate ratio, in the form in which it was propounded and practised by

the ancient Geometers, has been almost wholly superseded. However

satisfactory for the purposes of exact reasoning the method of expressing

the ratio of two surfaces, or of two solids by two straight lines, may be in

itself, it has not been found to be the form best suited for the direct ap-

plication of the results of Geometry. Almost all modern writers on Geo-

metry and its applications to every branch of the Mathematical Sciences,

have adopted the algebraical notation of a quotient -4 i5 : BC ; or of a

AB

fraction â€” - ; for expressing the ratio of two lines AB, BC : as well as that

x> G

of a product AB x BC, or AB .BC, for the expression of a rectangle.

The want of a concise and expressive method of notation to indicate the

proportion of Geometrical Magnitudes in a form suited for the direct ap-

plication of the results, has doubtless favoured the introduction of Alge-

braical symbols into the language of Geometry. It must be admitted,

Jiowever, that such notations in the language of pure Geometry are liable

o5

298

to very serious objections, chiefly on the ground that pure Geometry does

not admit the Arithmetical or Algebraical idea of a product or a quotient

into its reasonings. On the other hand, it may be urged, that it is not

the employment of symbols which renders a process of reasoning pecu-

liarly Geometrical or Algebraical, but the ideas which are expressed by

them. If symbols be employed in Geometrical reasonings, and be under-

stood to express the magnitudes themselves and the conception of their Geo-

metrical ratio, and not any measures, or numerical values of them, there

would not appear to be any very great objections to their use, provided

that the notations employed were such as are not likely to lead to mis-

conception. It is, however, desirable, for the sake of avoiding confusion

of ideas in reasoning on the properties of number and of magnitude, that

the language and notations employed both in Geometry and Algebra

should be rigidly defined and strictly adhered to, in all cases. At the

commencement of his Geometrical studies, the student is recommended

not to employ the symbols of Algebra in Geometrical demonstrations.

How far it may be necessary or advisable to employ them when he fully

understands the nature of the subject, is a question on which some diffe-

rence of opinion exists.

Prop. XXV. There does not appear any sufficient reason why this pro-

position is placed between Prop. xxiv. and Prop. xxvi.

Prop. XXVII. To understand this and the three following proposi-

tions more easily, it is to be observed :

1 . â€¢' That a parallelogram is said to be applied to a straight line, when

it is described upon it as one of its sides. Ex. gr. the parallelogram JC

is said to be applied to the straight line JB.

2. But a parallelogram ^E is said to be applied to a straight line

JB, deficient by a parallelogram, when JD the base of JE is less than

y4B, and therefore ^E is less than the parallelogram ^C described upon

AB in the same angle, and between the same parallels, by the parallelo-

gram DC; and DC is therefore called the defect of JE.

3. And a parallelogram //G is said to be applied to a straight line

JB, exceeding by a parallelogram, when .VF the base of ^G is greater

than AB, and therefore JG exceeds AC the parallelogram described

upon AB in the same angle, and between the same parallels, by the

parallelogram 5G."â€” Simson.

Both among Euclid's Theorems and Problems, cases occur in which

the hypotheses of the one, and the data or qusesita of the other, are

restricted within certain limits as to magnitude and position. The

determination of these limits constitutes the doctrine of Maxima and

Minima, Thus : â€” The theorem Euc. vi. 27 is a case of the maximum

value which a figure fulfilling the other conditions can have ; and the

succeeding proposition is a problem involving this fact among the

conditions as a part of the data, in truth, perfectly analogous to Euc. i.

20, 22 ; wherein the limit of possible diminution of the sum of the two

sides of a triangle described upon a given base, is the magnitude of

the base itself: the limit of the side of a square which shall be equal to

the rectangle of the two parts into which a given line may be divided,

is half the line, as it appears from Euc. ii. 5 : â€” the greatest line that can

be drawn from a given point within a circle, to the circumference,

Euc. III. 7, is the line which passes through the center of the circle ;

and the least line which can be so drawn from the same point, is the part

produced, of the greatest line between the given point and the circum-

ference. Euc. III. 8, also affords another instance of a maximum and a

minimum when the given point is outside the given circle.

QUESTIONS ON BOOK VI. 299

Prop. XXXI. This proposition is the general case of Prop. 47, Book i,

for any similar rectilineal figure described on the sides of a right-angled

triangle. The demonstration, however, here given is wholly independent

ofEuc. I. 47.

Prop. XXXIII. In the demonstration of this important proposition,

angles greater than two right angles are employed, in accordance with

the criterion of proportionality laid down in Euc. v. def. 5.

This proposition forms the basis of the assumption of arcs of circles

for the measures of angles at their centers. One magnitude may be as-

sumed as the measure of another magnitude of a different kind, when the

two are so connected, that any variation in them takes place simultane-

ously, and in the same direct proportion. This being the case with

angles at the center of a circle, and the arcs subtended by them ; the

arcs of circles can be assumed as the measures of the angles they subtend

at the center of the circle.

Prop. B. The converse of this proposition does not hold good when

the triangle is isosceles.

QUESTIONS ON BOOK YI.

1 . Distinguish between similar figures and equal figures.

2. "What is the distinction between homologoy^s sides ^ and equal sides

in Geometrical figures ?

3. What is the number of conditions requisite to determine similarity

of figures ? Is the number of conditions in Euclid's definition of similar

figures greater than what is necessary ? Propose a definition of similar

figures which includes no superfluous condition.

4. Explain how Euclid makes use of the definition of proportion in

Euc. VI. 1.

5. Prove that triangles on the same base are to one another as their

altitudes.

6. If two triangles of the same altitude have their bases unequal,

and if one of them be divided into w equal parts, and if the other contain

n of those parts ; prove that the triangles have the same numerical relation

as their bases. Why is this Proposition less general than Euc. vi. I ?

7. Are triangles which have one angle of one equal to one angle of

another, and the sides about two other angles proportionals, necessarily

similar ?

8. What are the conditions, considered by Euclid, under which two

triangles are similar to each other ?

9. Apply Euc. VI. 2, to trisect the diagonal of a parallelogram.

10. When are three lines said to be in harmonical proportion? If

both the interior and exterior angles at the vertex of a triangle (Euc. vi.

3, A.) be bisected by lines which meet the base, and the base produced,in

D, G ; the segments BGy GD, GC of the base shall be in Harmonical pro-

portion.

11. If the angles at the base of the triangle in the figure Euc. vi, J,

be equal to each other, how is the proposition modified ?

12. Under what circumstances will the bisecting line in the fig. Euc.

VI. A, meet the base on the side of the angle bisected ? Shew that there

is an indeterminate case.

300 EUCLIU'S ELEMENTS.

13. State some of the uses to which. Euc. vi. 4, may be applied.

14. Apply Euc. VI. 4, to prove that the rectangle contained by the

segments of any chord passing through a given point within a circle is

BOOK VI. PROP. XXXUI. 291

PROPOSITION XXXIII. THEOREM.

In equal circleSy angles, whether at the centers or circumferences, have

the same ratio which the circumferences on which they stand have to one

another : so also have the sectors.

Let ABC, DEFhe equal circles; and at their centers the angles

BGC, EIIF, and the angles BAC, EDF, at their circumferences.

As the circumference BC to the circumference FF, so shall the

angle BGC be to the angle EIIF, and the angle BAG to the

angle ^Di^;

and also the sector BGC to the sector EIIF,

Take any number of circumferences CK, KL, each equal to BC,

and any number whatever FM, MN, each equal to EF:

and join GK, GL, HM, UN.

Because the circumferences BC, CK, KL are all equal,

the angles BGC CGK, KGL are also all equal : (ill. 27.)

therefore what multiple soever the circumference BL is of the cir-

cumference BC, the same multiple is the angle BGL of the angle

BGC:

for the same reason, whatever multiple the circumference EN is of

the circumference EF^ the same multiple is the angle EHN of the

angle EHF:

and if the circumference BL be equal to the circumference EN,

the angle BGL is also equal to the angle EHN; (ill. 27.)

and if the circumference BL be greater than EN,

likewise the angle BGL is greater than EHN-, and if less, less:

therefore, since there are four magnitudes, the two circumferences

BC, EF, and the two angles BGC, EHF; and that of the circum-

ference BC, and of the angle BGC, have been taken any equimultiples

whatever, viz. the circumference BL, and the angle BGL : and of the

circumference EF, and of the angle EHF, any equimultiples what-

ever, viz. the circumference EN, and the angle EHN:

and since it has been proved, that if the circumference BL be greater

than EN)

the angle BGL is greater than EHN;

and if equal, equal ; and if less, less ;

therefore as the circumference BC to the circumference EF, so is the

angle BGC to the angle EHF: (v. def. 5.)

but as the angle BGC is to the angle EHFy so is the angle B AC to

the angle EDF: (v. 15.)

for each is double of each ; (ill. 20.)

therefore, as the circumference BCh to EF, so is the angle BGC to

the angle EHF, and the angle B AC to the angle EDF.

02

^92 Euclid's elements.

Also, as the circumference ^C to JEJF, so shall the sector BGChe

to the sector EHF.

Join BC, CK, and in the circumferences, BC, CK, take any points

X, O, and join BX, XC, CO, OX,

Then, because in the triangles GBC, GCK,

the two sides BG, GC are equal to the two CG, GX each to each,

and that they contain equal angles ;

the base ^Cis equal to the base CX, (l. 4.)

and the triangle GBC to the triangle GCX:

and because the circumference BC is equal to the circumference CX,

the remaining part of the whole circumference of the circle ABC, is

equal to the remaining part of the whole circumference of the same

circle : (ax. 3.)

therefore the angle BXC is equal to the angle COX; (ill. 27.)

and the segment BXC is therefore similar to the segment COX;

(III. def. 11.)

and they are upon equal straight lines, BC, CX:

but similar segments of circles upon equal straight lines, are equal

to one another: (lil. 24.)

therefore the segment BXC is equal to the segment COX :

and the triangle BGCwas proved to be equal to the triangle CGX;

therefore the whole, the sector BGC, is equal to the whole, the

sector CGX:

for the same reason, the sector XGZ is equal to each of the sectors

BGC, CGX:

in the same manner, the sectors EHF, FH3Â£, MHN may be

proved equal to one another :

therefore, what multiple soever the circumference BL is of the circum-

ference BC, the same multiple is the sector BGL of the sector BGC;

and for the same reason, whatever multiple the circumference EN

is of EF, the same multiple is the sector EHN of the sector

EHF:

and if the circumference BL be equal to EX, the sector BGL is

equal to the sector EHN;

and it the circumference BL be greater than EN, the sector BGL

is greater than the sector EHN;

and if less, less ;

since, then, there are four magnitudes, the two circumferences BC,

EF, and the two sectors BGC, EHF, and that of the circumference

BC, and sector BGC, the circumference BL and sector BGL are any

equimultiples whatever; and of the circumference EF, and sector

EHF, the circumference EN, and sector EHN are any equimultiples

whatever ;

BOOK VI. PROP. B, C. 293

and since it has been proved, that if the circumference BL be greater

than EN, the sector BGL is greater than the sector EIIN ;

and if equal, equal ; and if less, less :

therefore, as the circumference ^Cis to the circumference JEF, so

is the sector BGC to the sector EHF. (v. def. 5.)

Wherefore, in equal circles, &c. Q.E.D.

PROPOSITION B. THEOREM.

If an angle of a triangle be bisected by a straight line which likeioise cuts

the base ; the rectangle contavied by the sides of the triangle is equal to the

rectangle contained by the segments of the base, together with the square on

the straight line which bisects the angle.

Let ABC he a triangle, and let the angle BA C be bisected by the

straight line AD.

The rectangle BA, AC shall be equal to the rectangle BD, DC^

together with the square on AD.

//

E

Describe the circle A CB about the triangle, (iv. 5.)

and produce AD to the circumference in E, and join EC.

Then because the angle BAD is equal to the angle CAE, (hyp.)

and the angle ABD to the angle AEC, (ill. 21.)

for they are in the same segment ;

the triangles ABD, AEC are equiangular to one another : (l. 32.)

therefore as BA to AD, so is EA to AC-, (vi. 4.)

and consequently the rectangle BA, ^ C is equal to the rectangle EA,

AD, (VI. 16.)

that is, to the rectangle ED, DA, together with the square on ^D ;

(II. 3.)

but the rectangle ED, DA is equal to the rectangle BD, DC\ (III. 35.)

therefore the rectangle BA, AC'ys, equal to the rectangle BD, DC,

together with the square on AD.

Wherefore, if an angle, &c. Q. E. D.

PROPOSITION C. THEOREM.

If from any angle of a triangle, a straight line be drawn 'perpendicular to

the base ; the rectangle contained by the sides of the triangle is equal to the

rectangle contained by the perpendicular and the diameter of the circle de-

scribed about the triangle.

Let ABC ha a triangle, and AD the perpendicular from the angle

A to the base B C.

The rectangle i?^, ^Cshall be equal to the rectangle containedby

AD and the diameter of the circle described about the triangle.

294:

EUCLID S ELEMENTS.

Describe the circle jiCJB about the triangle, (iv. 5.) and draw its

diameter AE, and join JEC.

Because the right angle J5DA is equal to the angle JECA in a

semicircle, (iii. 31.)

and the angle ABD equal to the angle -4^C in the same segment;

(III. 21.) the triangles ABD, AjECare equiangular:

therefore as BA to AI), so is UA to AC; (vi. 4.)

and consequently the rectangle BA, AC is equal to the rectangle ^A,

AD. (vi. 16.) If therefore from any angle, &c. Q.E.D.

PROPOSITION D. THEOREM.

The rectangle contained by the diagonals of a quadrilateral figure inscribed

in a circle, is equal to both the rectangles contained by its opposite sides.

'Lei AB CD he any quadrilateral figure inscribed in a circle, and

'^om AC, BD.

The rectangle contained by A C, BD shall be equal to the two

rectangles contained by AB, CD, and by AD, BC.

Make the angle ABE equal to the angle DBC: (I. 23.)

add to each of these equals the common angle EBB,

then the angle ABD is equal to the angle EBC:

and the angle BDA is equal to the angle B CE, because they are

in the same segment : (IIL 21.)

therefore the triangle ABD is equiangular to the triangle BCE:

wherefore, ;as ^Cis to CE, so is BD to DA ; (vi. 4.)

and consequently the rectangle BC, AD is equal to the rectangle

BD, CE : (VL 16.)

again, because the angle ABE Is equal to the angle DBC, and the

angle BAE to the angle BDC, (in. 21.)

the triangle ABE is equiangular to the triangle BCD :

therefore as BA to AE, so is BD to DC;

wherefore the rectangle BA, DC is equal to the rectangle BD, AE:

but the rectangle BC, AD has been shewn to be equal

to the rectangle BD, CE ;

therefore the whole rectangle AC, BD is equal to the rectangle

AB, DC, together with the rectangle AD, BC. (il. 1.)

Therefore the rectangle, &c. Q. E. D.

This is a Lemma of CI. Ptolemseus, in page 9 of his MeydXtj Syvra^is.

NOTES TO BOOK YL

In this Book, tlie theory of proportion exhibited in the Fifth Book, is

applied to the comparison of the sides and areas of plane rectilineal figures,

both of those which are similar, and of those which are not similar.

Def. I. In defining similar triangles, one condition is sufficient, namely,

that similar triangles are those which have their three angles respectively

equal ; as in Prop. 4, Book vi, it is proved that the sides about the equal

angles of equiangular triangles are proportionals. But in defining similar

figures of more than three sides, both of the conditions stated in Def. i,

are requisite, as it is obvious, for instance, in the case of a square and a

rectangle, which have their angles respectively equal, but have not their

sides about their equal angles proportionals.

The following definition has been proposed : ** Similar rectilineal

figures of more than three sides, are those which may be divided into the

same number of similar triangles," This definition would, if adopted,

require the omission of a part of Prop. 20, Book vi.

Def. III. To this definition may be added the following :

A straight line is said to be divided harnioyiically, when it is divided

into three parts, such that the whole line is to one of the extreme segments,

as the other extreme segment is to the middle part. Three lines are in

harmonical proportion, when the first is to the third, as the diflerence be-

tween the first and second, is to the difference between the second and

third ; and the second is called a harmonic mean between the first and third.

The expression â€¢ harmonical proportion' is derived from the following

fact in the Science of Acoustics, that three musical strings of the same

material, thickness and tension, when divided in the manner stated in the

definition, or numerically as 6, 4, and 3, produce a certain musical note,

its fifth, and its octave.

Def. IV. The term altitude, as applied to the same triangles and paral-

lelograms, will be different according to the sides which may be assumed

as the base, unless they are equilateral.

Prop. I. In the same manner may be proved, that triangles and paral-

lelograms upon equal bases, are to one another as their altitudes.

Prop. A. When the triangle ABC is isosceles, the line which bisects

the exterior angle at the vertex is parallel to the base. In all other cases,

if the line which bisects the angle BAC cut the base BCin the point G,

then the straight line BD is harmonically divided in the points Gr, C.

For EG is to GC as BA is to AC ; (vi. 3.)

and BD is to DC as BA is to AC, (vi. a.)

therefore BD is to DC as BG is to GC,

but BG = BD - DG, and GC = GD - DC.

Wherefore BD is to DC as BD - DG is to GD - DC.

Hence BD, DG, DC, are in harmonical proportion.

Prop. IV is the first case of similar triangles, and corresponds to the

third case of equal triangles, Prop. 26, Book i.

1

EUCLID S ELEMENTS.

Sometimes the sides opposite to the equal angles in two equiangular

triangles, are called the correspondmg sides, and these are said to be pro-

portional, which is simply taking the proportion in Euclid alternately.

The term homologous (o^aoXoyos), has reference to the places the sides

of the triangles have in the ratios, and in one sense, homologous sides may

be considered as corresponding sides. The homologous sides of any two

similar rectilineal figures will be found to be those which are adjacent to

two equal angles in each figure.

Prop. V, the converse of Prop, iv, is the second case of similar triangles,

and corresponds to Prop. 8, Book r, the second case of equal triangles.

Prop. VI is the third case of similar triangles, and corresponds to Prop.

4, Book I, the first case of equal triangles.

The property of similar triangles, and that contained in Prop. 47, Book

I, are the most important theorems in Geometry.

Prop. VII is the fourth case of similar triangles, and corresponds to the

fourth case of equal triangles demonstrated in the note to Prop. 26, Book i.

Prop. IX. The learner here must not forget the diiFerent meanings of

the word part, as employed in the Elements. The word here has the

same meaning as in Euc. v. def. 1.

It may be remarked, that this proposition is a more simple case of the

next, namely. Prop. x.

Prop. XI. This proposition is that particular case of Prop, xii, in which

the second and third terms of the proportion are equal. These two

problems exhibit the same results by a Geometrical construction, as are

obtained by numerical multiplication and division.

Prop. XIII. The difierence in the two propositions Euc. ii. 14, and

Euc. VI. 13, is this : in the Second Book, the problem is, to make a rect-

angular figure or square equal in area to an irregular rectilinear figure,

in which the idea of ratio is not introduced. In the Prop, in the Sixth

Book, the problem relates to ratios only, and it requires to divide a line

into two parts, so that the ratio of the whole line to the greater segment

may be the same as the ratio of the greater segment to the less.

The result in this proposition obtained by a Geometrical construction,

is analogous to that which is obtained by the multiplication of two

numbers, and the extraction of the square root of the fjroduct.

It may be observed, that half the sum of ^i^ and BC is called the

Arithmetic mean between these lines ; also that BD is called the Geo-

metric mean between the same lines.

To find two mean proportionals between two given lines is impossible

by the straight line and circle. Pappus has given several solutions of

this problem in Book iii, of his Mathematical Collections ; and Eutocius

has given, in his Commentary on the Sphere and Cylinder of Archimedes,

ten different methods of solving this problem.

Prop. XIV depends on the same principle as Prop, xv, and both may

easily be demonstrated from one diagram. Join DF, FE, EG in the fig.

to Prop. XIV, and the figure to Prop, xv is formed. We may add, that

there docs not appear any reason why the pro]3erties of the triangle and

parallelogram should be here separated, and not in the first proposition of

the Si:ith Book.

Prop. XV holds goou wnen one angle of one triangle is equal to the

defect from what the corresponding angle in the other wants of two right

angles.

This theorem will perhaps be more distinctly comprehended by the

learner, if he will bear in mind, that four magnitudes are reciprocally

NOTES TO BOOK VI. 297

proportional, when the ratio compounded of these ratios is a ratio of

equality.

Prop. XVII is only a particular case of Prop, xvi, and more properly,

might appear as a corollary: and both are cases of Prop. xiv.

Algebraically, Let AB, CD, E, F, contain a, b, c, d units respectively.

Then, since a, i, c, d are proportionals, .â€¢-=-.

b (i

Multiply these equals by bd, .'. ad = be,

or, the product of the extremes is equal to the product of the means.

And conversely. If the product of the extremes be equal to the pro-

duct of the means,

or ad = be,

then, dividing these equals by bd, /. t = -, ,

or the ratio of the first to the second number, is equal to the ratio of the

third to the fourth.

Similarly may be shewn, that if - = - ; then ad=b*.

d

And conversely, if adâ€”b^\ then t = -, â€¢

o a

Prop. XVIII. Similar figures are said to be similarly situated, when

their homologous sides are parallel, as when the figures are situated on

the same straight line, or on parallel lines ; but when similar figures are

situated on the sides of a triangle, the similar figures are said to be similarly

situated when the homologous sides of each figure have the same re-

lative position with respect to one another ; that is if the bases on which

the similar figures stand, were placed parallel to one another, the re-

maining sides of the figures, if similarly situated, would also be parallel

to one another.

Prop. XX. It may easily be shewn, that the perimeters of similar

polygons, are proportional to their homologous sides.

Prop. XXI. This proposition must be so understood as to include all

rectilineal figures whatsoever, which require for the conditions of simila-

rity another condition than is required for the similarity of triangles.

See note on Euc. vi. Def. i.

Prop. XXIII. The doctrine of compound ratio, including duplicate and

triplicate ratio, in the form in which it was propounded and practised by

the ancient Geometers, has been almost wholly superseded. However

satisfactory for the purposes of exact reasoning the method of expressing

the ratio of two surfaces, or of two solids by two straight lines, may be in

itself, it has not been found to be the form best suited for the direct ap-

plication of the results of Geometry. Almost all modern writers on Geo-

metry and its applications to every branch of the Mathematical Sciences,

have adopted the algebraical notation of a quotient -4 i5 : BC ; or of a

AB

fraction â€” - ; for expressing the ratio of two lines AB, BC : as well as that

x> G

of a product AB x BC, or AB .BC, for the expression of a rectangle.

The want of a concise and expressive method of notation to indicate the

proportion of Geometrical Magnitudes in a form suited for the direct ap-

plication of the results, has doubtless favoured the introduction of Alge-

braical symbols into the language of Geometry. It must be admitted,

Jiowever, that such notations in the language of pure Geometry are liable

o5

298

to very serious objections, chiefly on the ground that pure Geometry does

not admit the Arithmetical or Algebraical idea of a product or a quotient

into its reasonings. On the other hand, it may be urged, that it is not

the employment of symbols which renders a process of reasoning pecu-

liarly Geometrical or Algebraical, but the ideas which are expressed by

them. If symbols be employed in Geometrical reasonings, and be under-

stood to express the magnitudes themselves and the conception of their Geo-

metrical ratio, and not any measures, or numerical values of them, there

would not appear to be any very great objections to their use, provided

that the notations employed were such as are not likely to lead to mis-

conception. It is, however, desirable, for the sake of avoiding confusion

of ideas in reasoning on the properties of number and of magnitude, that

the language and notations employed both in Geometry and Algebra

should be rigidly defined and strictly adhered to, in all cases. At the

commencement of his Geometrical studies, the student is recommended

not to employ the symbols of Algebra in Geometrical demonstrations.

How far it may be necessary or advisable to employ them when he fully

understands the nature of the subject, is a question on which some diffe-

rence of opinion exists.

Prop. XXV. There does not appear any sufficient reason why this pro-

position is placed between Prop. xxiv. and Prop. xxvi.

Prop. XXVII. To understand this and the three following proposi-

tions more easily, it is to be observed :

1 . â€¢' That a parallelogram is said to be applied to a straight line, when

it is described upon it as one of its sides. Ex. gr. the parallelogram JC

is said to be applied to the straight line JB.

2. But a parallelogram ^E is said to be applied to a straight line

JB, deficient by a parallelogram, when JD the base of JE is less than

y4B, and therefore ^E is less than the parallelogram ^C described upon

AB in the same angle, and between the same parallels, by the parallelo-

gram DC; and DC is therefore called the defect of JE.

3. And a parallelogram //G is said to be applied to a straight line

JB, exceeding by a parallelogram, when .VF the base of ^G is greater

than AB, and therefore JG exceeds AC the parallelogram described

upon AB in the same angle, and between the same parallels, by the

parallelogram 5G."â€” Simson.

Both among Euclid's Theorems and Problems, cases occur in which

the hypotheses of the one, and the data or qusesita of the other, are

restricted within certain limits as to magnitude and position. The

determination of these limits constitutes the doctrine of Maxima and

Minima, Thus : â€” The theorem Euc. vi. 27 is a case of the maximum

value which a figure fulfilling the other conditions can have ; and the

succeeding proposition is a problem involving this fact among the

conditions as a part of the data, in truth, perfectly analogous to Euc. i.

20, 22 ; wherein the limit of possible diminution of the sum of the two

sides of a triangle described upon a given base, is the magnitude of

the base itself: the limit of the side of a square which shall be equal to

the rectangle of the two parts into which a given line may be divided,

is half the line, as it appears from Euc. ii. 5 : â€” the greatest line that can

be drawn from a given point within a circle, to the circumference,

Euc. III. 7, is the line which passes through the center of the circle ;

and the least line which can be so drawn from the same point, is the part

produced, of the greatest line between the given point and the circum-

ference. Euc. III. 8, also affords another instance of a maximum and a

minimum when the given point is outside the given circle.

QUESTIONS ON BOOK VI. 299

Prop. XXXI. This proposition is the general case of Prop. 47, Book i,

for any similar rectilineal figure described on the sides of a right-angled

triangle. The demonstration, however, here given is wholly independent

ofEuc. I. 47.

Prop. XXXIII. In the demonstration of this important proposition,

angles greater than two right angles are employed, in accordance with

the criterion of proportionality laid down in Euc. v. def. 5.

This proposition forms the basis of the assumption of arcs of circles

for the measures of angles at their centers. One magnitude may be as-

sumed as the measure of another magnitude of a different kind, when the

two are so connected, that any variation in them takes place simultane-

ously, and in the same direct proportion. This being the case with

angles at the center of a circle, and the arcs subtended by them ; the

arcs of circles can be assumed as the measures of the angles they subtend

at the center of the circle.

Prop. B. The converse of this proposition does not hold good when

the triangle is isosceles.

QUESTIONS ON BOOK YI.

1 . Distinguish between similar figures and equal figures.

2. "What is the distinction between homologoy^s sides ^ and equal sides

in Geometrical figures ?

3. What is the number of conditions requisite to determine similarity

of figures ? Is the number of conditions in Euclid's definition of similar

figures greater than what is necessary ? Propose a definition of similar

figures which includes no superfluous condition.

4. Explain how Euclid makes use of the definition of proportion in

Euc. VI. 1.

5. Prove that triangles on the same base are to one another as their

altitudes.

6. If two triangles of the same altitude have their bases unequal,

and if one of them be divided into w equal parts, and if the other contain

n of those parts ; prove that the triangles have the same numerical relation

as their bases. Why is this Proposition less general than Euc. vi. I ?

7. Are triangles which have one angle of one equal to one angle of

another, and the sides about two other angles proportionals, necessarily

similar ?

8. What are the conditions, considered by Euclid, under which two

triangles are similar to each other ?

9. Apply Euc. VI. 2, to trisect the diagonal of a parallelogram.

10. When are three lines said to be in harmonical proportion? If

both the interior and exterior angles at the vertex of a triangle (Euc. vi.

3, A.) be bisected by lines which meet the base, and the base produced,in

D, G ; the segments BGy GD, GC of the base shall be in Harmonical pro-

portion.

11. If the angles at the base of the triangle in the figure Euc. vi, J,

be equal to each other, how is the proposition modified ?

12. Under what circumstances will the bisecting line in the fig. Euc.

VI. A, meet the base on the side of the angle bisected ? Shew that there

is an indeterminate case.

300 EUCLIU'S ELEMENTS.

13. State some of the uses to which. Euc. vi. 4, may be applied.

14. Apply Euc. VI. 4, to prove that the rectangle contained by the

segments of any chord passing through a given point within a circle is

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38