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

. (page 26 of 38)
Font size Let Ai : 02 ' ' A2 : a^y
and let A^ be the greatest, and consequently a^ the least.
Then shall Ai-\- a^> a^ + ^3.
Since Ai : % :: ^3 : a^,

â€˘ "1^1 = :^
* ' tto a^ '

Multiply these equals by ~ ,

. ÂŁi _ ^

â€˘ â€˘ ^3 ~ ^4 '

subtract 1 from each of these equals,

.â€˘ :^ ~ 1 = ^ _ 1
A3 tti

or ^^ ~^3 = <^2 - Â«4
^3 O4

Multiplying these equals by â€” - â€” ,
. ^1 - ^3 _ :4

* * Og â€” Â©4 a4 *

but -i = -^ ,

% *4

-^1 ~ A^ ^ ^1

but ^1 > Oo, *.* Ai is the greatest of the four magnitudes,

.*. also Ai â€” A^ > a2 â€” a^,

add A^ + a^ to each of these equals,

.-. ^1 + 04 > Â©2 + ^3Â»

" The whole of the process in the Fifth Book is purely logical, that is,
the whole of the results are virtually contained in the definitions, in the
manner and sense in which metaphysicians (certain of them) imagine all
the results of mathematics to be contained in their definitions and hypo-
theses. No assumption is made to determine the truth of any conse-
quence of this definition, which takes for granted more about number or
magnitude than is necessary to imderstand the definition itself. The

QUESTIONS ON BOOK V. 257

latter being once understood, its results are deduced by inspection â€” of
itself only, without the necessity of looking at any thing else. Hence,
a great distinction between the fifth and the preceding books presents
itself. The first four are a series of propositions, resting on diff"erent fun-
damental assumptions ; that is, about diflerent kinds of magnitudes.
The fifth is a definition and its developement ; and if the analogy by which
names have been given in the preceding Books had been attended to, the
propositions of that Book would have been called corollaries of the defini-
tion." â€” Connexion of Number and Magnitude, by Professor De Morgan, p. 56.
The Fifth Book of the Elements as a portion of Euclid's System of
Geometry ought to be retained, as the doctrine contains some of the most
important characteristics of an efi'ective instrument of intellectual Educa-
tion. This opinion is favoured by Dr. Barrow in the following expressive
terms : " There is nothing in the whole body of the Elements of a more
Bubtile invention, nothing more solidly established, or more accurately
handled than the doctrine of proportionals."

QUESTIONS ON BOOK Y.

1. Explain and exemplify the meaning of the terms, multiple, sub-
multiple, equimultiple,

2. What operations in Geometry and Arithmetic are analogous ?

3. What are the different meanings of the term measure in Geometry ?
When are Geometrical magnitudes said to have a common measure?

4. When are magnitudes said to have, and not to have, a ratio to one
another? What restriction does this impose upon the magnitudes in
regard to their species f

5. When are magnitudes said to be commensurable or incommensur-
able to each other ? Do the definitions and theorems of Book v, include
incommensurable quantities ?

6. What is meant by the term geometrical ratio f How is it represented ?

7. Why does Euclid give no independent definition of ratio ?

8. What sort of quantities are excluded from Euclid's idea of ratio,
and how does his idea of ratio differ from the Algebraic definition ?

9. How is a ratio represented Algebraically? Is there any distinction
between the terms, a ratio of equality, and equality of ratio?

10. In what manner are ratios, in Geometry, distinguished from each
other as equal, greater, or less than one another? What objection is
there to the use of an independent definition (properly so called) of ratio
in a system of Geometry ?

1 1 . Point out the distinction between the geometrical and algebraical
methods of treating the subject of proportion.

12. What is the geometrical definition of proportion ? Whence arises
the necessity of such a definition as this ?

1 3. Shew the necessity of the qualification " any whatever'* in Euclid's
definition of proportion.

14. Must magnitudes that are proportional be all of the same kind ?

15. To what objection has Euc. v. def. 5, been considered liable ?

16. Point out the connexion between the more obvious definition of
proportion and that given by Euclid, and illustrate clearly the nature of

17. Why may not Euclid's definition of proportion be superseded in

ÂŁ58 EUCLID'S ELEMENTS.

a system of Geometry by the following: "Four quantities are propor-
tionals, when the first is the same multiple of the second, or the same
part of it, that the third is of the fourth ?"

18. Point out the defect of the following definition: Â«â€˘ Four magni-
tudes are proportional when equimultiples may be taken of the first and
the third, and also of the second and fourth, such that the multiples of
the first and second are equal, and also those of the third and fourth."

19. Apply Euclid's definition of proportion, to shew that if four quan-
tities be proportional, and if the first and the third be divided into the
same arbitrary number of equal parts, then the second and fourth will either
be equimultiples of those parts, or will lie between the same two suc-
cessive multiples of them.

20. The Geometrical definition of proportion is a consequence of the
Algebraical definition ; and conversely.

21. What Geometrical test has Euclid given to ascertain that four
quantities are not proportionals ? What is the Algebraical test ?

22. Shew in the manner of Euclid, that the ratio of 15 to 17 is greater
than that of n to 13.

23. How far may the fifth definition of the fifth Book be regarded as
an axiom ? Is it convertible ?

24. Def. 9, Book v. ** Proportion consists of three terms at least."
How is this to be understood ?

25. Define duplicate ratio. How does it appear from Euclid that the
duplicate ratio of two magnitudes is the same as that of their squares ?

26. When is a ratio compounded of any number of ratios ? What is
the ratio which is compounded of the ratios of 2 to 5, 3 to 4, and 5 to 6 ?

27. By what process is a ratio found equal to the composition of two
or more given ratios? Give an example, where straight lines are the
magnitudes which express the given ratios.

28. What limitation is there to the alternation of a Geometrical pro-
portion ?

29. Explain the construction and sense of the phrases, ex (squali,
and ex cequali in proportione perturhata, used in proportions,

30. Exemplify the meaning of the word homologous as it is used in
the Fifth Book of the Elements.

31. Why, in Euclid v. 11, is it necessary to prove that ratios which
are the same with the same ratio, are the same Avith one another ?

32. Apply the Geometrical criterion to ascertain, whether the four
lines of 3, 5, 6, 10 units are proportionals.

33. Prove by taking equimultiples according to Euclid's definition,
that the magnitudes 4, 5, 7, 9, are not proportionals.

34. Give the Algebraical proofs of Props. 1 7 and 1 8, of the Fifth Book,

35. What is necessary to constitute an exact definition ? In the de-
monstration of Euc. V. 18, is it legitimate to assume the converse of the
fifth definition of that Book ? Does a mathematical definition admit of
proof on the principles of the science to which it relates r

36. Explain why the properties proved in Book v, by means oi straight
lines, are true of any concrete 7nagnitudes.

37. Enunciate Euc. v. 8, and illustrate it by numerical examples,

38. Prove Algebraically Euc. v. 25.

39. Shew that when four magnitudes are proportionals, they cannot,
when equally increased or equally diminished by any other magnitude,
continue to be proportionals.

40. What grounds are there for the opinionthat Euclid mtended to
exclude the idea of numerical measures of ratios in his Fifth Book,

41. What is the object of the Fifth Book of Euclid's Elements r

BOOK YI.

DEFINITIONS.
L

Similar rectilineal figures are those which have their several
angles equal, each to each, and the sides about the equal angles pro-
portionals.

II.

" Reciprocal figures, viz. triangles and parallelograms, are such as
have their sides about two of their angles proportionals in such a
manner, that a side of the first figure is to a side of the other, as the
remaining side of the other is to the remaining side of the first."

III.

A straight line is said to be cut in extreme and mean ratio, when
the whole is to the greater segment, as the greater segment is to the
less.

rv.

The altitude of any figure is the straight line drawn from its vertex
perpendicular to the base.

PROPOSITION I. THEOREM.

Triangles and parallelograms of the same altitude are one to the other as
their bases.

Let the triangles ABC, A CD, and the parallelograms JEC, CF,
have the same altitude,

viz. the perpendicular drawn from the point A to BD or BD pro-
duced.
As the base BCis to the base CJD, so shall the triangle ABC he to
the triangle A CD,

and the parallelogram UC to the parallelogram CF.

260 Euclid's elements.

HUB C I> ix

Produce BD both ways to the points H, X,
and take any number of straight lines BGj GH, each equal to the

base BC', (L 3.)

and DK, KL, any number of them, each equal to the base CD ;

and join AG, AH, AK, AL.

Then, because CB, BG, GH, are all equal,

the triangles AUG, AGB, ABC, are all equal : (l. 38.)

therefore, whatever multiple the base ^Cis of the base BC,

the same multiple is the ti'iangle AHC of the triangle ABC:

for the same reason whatever multiple the base ZC is of the base CD,

the same multiple is the triangle ALC o^ the triangle ADC:

and if the base HC be equal to the base CL,
the triangle AHC is also equal to the triangle ALC: (l. 38.)

and if the base HC be greater than the base CL,

likewise the triangle AHC is, greater than the triangle ALC)

and if less, less ;

therefore since there are four magnitudes,

viz. the two bases BC, CD, and the two triangles ABC, A CD;

and of the base BC, and the triangle ABC, the first and third, any

equimultiples whatever have been taken,

viz. the base HC and the triangle AHC;
and of the base CD and the triangle ACD, the second and fourth,
have been taken any equimultiples whatever,

viz. the base CL and the triangle ALC;
and since it has been shewn, that, if the base HC be greater than
the base CL,
the triangle AHC is greater than the triangle ALC;
and if equal, equal ; and if less, less ;
therefore, as the base jBCis to the base CD, so is the triangle ABC
to the triangle A CD. (v. def. 5.)
And because the parallelogram CJE is double of the triangle ABC,
(L 41.)

and the parallelogram CF double of the triangle A CD,
and that magnitudes have the same ratio which their equimultiples

have; (v. 15.)
as the triangle ABC is to the triangle A CD, so is the parallelogram

jE'C to the parallelogram CF;
and because it has been shewn, that, as the base ^Cis to the base

CD, so is the triangle ABC to the triangle A CD ;
and as the triangle ABC is to the triangle ACD, so is the paralle-
logram EC to the parallelogram CF;
therefore, as the base J5C is to the base CD, so is the parallelogram
^Cto the parallelogram CF. (V. 11.)

Wherefore, triangles, &c. q.e.d.

BOOK VI. PROP. II. S6l

Cor. From this it is plain, that triangles and parallelograms that
have equal altitudes, are to one another as their bases.

Let the figures be placed so as to have their bases in the same
straight line ; and having drawn perpendiculars from the vertices of
the triangles to the bases, the straight line which joins the vertices is
parallel to that in which their bases are, (l. 33.) because the perpen-
diculars are both equal and parallel to one another. (l. 28.) Then, if
the same construction be made as in the proposition, the demonstration
will be the same.

PROPOSITION II. THEOREM.

If a straight line he drawn parallel to one of the sides of a triangle,
it shall cut the other sides, or these produced, proportionally : and conversely,
if the sides, or the sides produced, be cut proportionally, the straight line
which joins tfie points of section shall be parallel to the remaining side of the
triangle.

Let DjE be drawn parallel to BC, one of the sides of the triangle ABC.
Then BD shall be to DA, as CE to EA,

Join BE, CD.
Then the triangle BDE is equal to the triangle CDE, (l. 37.)
because they are on the same base DE, and between the same
parallels DE, BC;

but ADE is another triangle ;
and equal magnitudes have the same ratio to the same magnitude;

(V. 7.)
therefore, as the triangle BDE is to the triangle ADE, so is the
triangle CDE to the triangle ADE:
but as the triangle BDE to the triangle ADE, so is BD to DA, (vi. 1.)
because, having the same altitude, viz. the perpendicular drawn
from the point E to AB, they are to one another as their bases ;
and for the same reason, as the triangle CDE to the triangle ADE,
so is CE to EA :

therefore, as BD to DA, so is CE to EA. (v. IL)
Next, let the sides AB, AC of the triangle ABC, or these sides
produced, be cut proportionally in the points D, E, that is, so that
BD may be to DA as CE to EA, and join DE.

Then DE shall be parallel to BC.

because as BD to DA, so is CE to EA ;

and as BD to DA, so is the triangle BDE to the triangle ADE-, (vi. 1.)

and as CE to EA, so is the triangle CDE to the triangle ADE;

therefore the triangle BDE is to the triangle ADE, as the triangle

CDE to the triangle ADE; (v. 11.)

262 Euclid's elements.

that is, the triangles ÂŁDE, CDE have the same ratio to the triangle
therefore the triangle BDE is equal to the triangle CDE: (v. 9.)
and they are on the same base DE:
but equal triangles on the same base and on the same side of it, are
between the same parallels ; (i. 39.)

therefore BE is parallel to BC.
Wherefore, if a sti'aight line, &c. q.e.d.
Â»

PROPOSITION III. THEOREM.

If the angle of a triangle be divided into two equal angles, by a straight
line xohich also cuts the base ; the segments of the base shall have the same
ratio which the other sides of the triangle have to one another: and con-
versely, if the segments of the base have the same ratio which the other sides
of the triangle have to one another ; the straight line drawn from the vertex to
the point of section ^ divides the vertical angle into tioo equal angles.

Let ABC he a triangle, and let the angle BAChe divided into two
equal angles by the straight line AD.

Then BD shall be to DC, as BA to AC

Through the point Cdraw C^ parallel to DA, (l 31.)

and let BA produced meet CE in E.

Because the straight line ^C meets the parallels AD, EC,

the angle ACE is equal to the alternate angle CAD : (l. 29.)

but CAD, by the hypothesis, is equal to the angle BAD ;

wherefore BAD is equal to the angle A CE. (ax. 1.)

Again, because the straight line BAE meets the parallels AD, EC,

the outward angle BAD is equal to the inward and opposite angle

AEC: (I. 29.)
but the angle A CE has been proved equal to the angle BAD ;

therefore also A CE is equal to the angle AEC, (ax. 1.)
and consequents, the side AE is equal to the side AC: (l. 6.)
and because AD is drawn parallel to EC, one of the sides of the tri-
angle BCE,

therefore BD is to DC, as BA to AE: (vi. 2.)
but AE is equal to ^C;
therefore, as BD to DC, so is BA to AC. (v. 7.)
Next, let BD be to DC, as BA to A C, and join AD.
Then the angle BA C shall be divided into two equal angles by the

The same construction being made ;
because, as BD to DC, so is BA to A C;

BOOK VI. PROP. Ill, A. 263

and as J5D to DC, so is I^A to AJS, because AD is parallel to J3C',

(YI. 2.)

therefore JBA is to AC, as JB A to AD: (v. 11.)

consequently AC is equal to AD, (v. 9.)

and therefore the angle ADC is equal to the angle ACD: (l. 5.)

but the angle ADCis equal to the outward and opposite angle DAD;

and the angle ACD is equal to the alternate angle CAD : (l. 29.)

wherefore also the angle DAD is equal to the angle CAD ; (ax. 1.)

that is, the angle BACis cut into two equal angles by the straight

Therefore, if the angle, &c. q.e.d.

PROPOSITION A. THEOREM.

If the outward angle of a triangle made by producing one of its sides,
he divided into two equal angles, by a straight line, zchich also cuts the base
2}roduced ; the segments between the dividing line and the extremities of the
base, have the same ratio ichich the other sides of the triangle have to one
another : and conversely, if the segments of the base produced have the same
tatio which the other sides of the triangle have j the straight line drawn from
the vertex to the point of section divides the outward angle of the triangle
into two equal angles.

Let ABC he a triangle, and let one of its sides BA be produced to D;
and let the outward angle CAD be divided into tw-o equal angles by
the straiijht line AD which meets the base produced in D,
'Then BD shall be to DC, as BA to AC.

Through Cdraw OF parallel to AD: (l. 31.)

and because the straight line A C meets the parallels AD, FC,

the angle ^Ci^ is equal to the alternate angle CAD: (l. 29.)

therefore also DAD is equal to the angle ACF. (ax. 1.)

the outward angle DAD is equal to the inward and opposite angle

CFA: (1.29.)

but the angle ^Ci^has been proved equal to the angle DAD;

therefore also the angle ^Ci'^is equal to the angle CFA ; (ax. 1.)

and consequently the side AF \s equal to the side AC: (l. 6.)

and because AD is parallel to jPC, a side of the triangle BCF,

therefore BD is to DC, as BA to AF: (vi. 2.)

I but ^i^is equal to AC;

therefore, as BD is to DC, so is BA to A C. (v. 7.)
Next, let BD be to DC, as BA to AC, and join AD.

264 Euclid's elements.

and that JSJD is also to DC, as JBA to AF; (vi. 2.)

therefore BA is to AC, as ^BA to AF: (v. 11.)

wherefore AC is equal to AF, (v. 9.)

and the anj^le AFC equal to the angle A CF: (l. 5.)

but the angle ^i'X^ is equal to the outward angle FAD, (i. 29.)

and the angle A CF to the alternate angle CAD ;

Wherefore, if the outward, &c. Q.E.D.

PROPOSITION IV. THEOREM.

The sides ahout the equal angles of equiangular triangles are proportionals ;
and those which are opposite to the equal angles are homologous sides, that is,
are the antecedents or coiisequents of the ratios.

Let ABC, DCFhe equiangular triangles, having the angle ABC
equal to the angle DCF, and the angle ACB to the angle DEC; and
consequently the angle BA C equal to the angle CDF. (l. 32.)

The sides about the equal angles of the triangles ABC, Z)CJ? shall

be proportionals ;
and those shall be the homologous sides which are opposite to the
equal angles.

c E

Let the triangle DCFhe placed, so that its side CF may be con-
tiguous to BC, and in the same straight line with it. (i 22.)

Then, because the angle BCA is equal to the angle CFD, (hyp.)

add to each the angle ABC-,
therefore the two angles ABC, BCA are equal to the two angles

ABC, CFD: (ax. 2.)

but the angles ABC, BCA are together less than two right angles ;

(I. 17.)

therefore the angles ABC, CFD are also less than two right angles :

wherefore BA, FD if produced will meet : (I. ax. 12.)

let them be produced and meet in the point F:

then because the angle ABC is equal to the angle DCF, (hyp.)

^i^is parallel to CD ; (l. 28.)

and because the angle A CB is equal to the angle DFC,

AC is parallel to FF: (i. 28.)

thereibre FA CD is a parallelogram ;

and consequently ^i^is equal to CD, and AC to FD : (l. 34.)

and because A Cis parallel to FF, one of the sides of the triangle FBF,

BA is to AF, &s BC to CF: (VL 2.)

but ^i^ is equal to CD ;

therefore, as BA to CD, so is ^Cto C^: (v. 7.)

and alternately, as AB to BC, so is DC to CF; (v. 16.)

I

BOOK VI. PROP. IV, V. 265

again, because CD is parallel to BF,

as BC to CE, so is FD to DE: (vi. 2.)

but FD is equal to AC;

therefore, as i? (7 to CE, so is ^Cto DE-, (v. 7.)

and alternately, as BC to CA, so CE to ^D : (v. 16.)

therefore, because it has been proved that AB k to BC,Qia DC to CE,

and as J5Cto CA, so CE to ^D,

ex asquali, BA is to ^ C, as CD to D^. (v. 22.)

Therefore the sides, &c. Q. e. d.

PROPOSITION Y. THEOREM.

If the sides of two triangles, about each of their angles, he proportio7ials,
the triangles shall be equiangular ; and the equal angles shall be those which
are opposite to the homologous sides.

Let the triangles ABC, DEFh^xve their sides proportionals,

so that AB is to BC, as DE to EF;

and BC to CA, as EF to FD-,

and consequently, ex sequali, BA to A C, as ED to DF.

Then the triangle ^ 5 C shall be equiangular to the triangle DEF,

and the angles which are opposite to the homologous sides shall be

equal, viz. the angle ABC equal to the angle DEF, and BCA to

EFD, and also ^^Cto EDF,

B C G

At the points E, F, in the straight line EF, make the angle FEG
equal to the angle ^^(7, and the angle EFG equal to BCA: (l. 23.)
wherefore the remaining angle EGF, is equal to the remaining
angle BA C, (i. 32.)
and the triangle GEFh therefore equiangular to the triangle ABC:
consequently they have their sides opposite to the equal angles pro-
portional : (vi. 4.)

wherefore, SiS A B to BC, so is GE to EF-,

but as AB to BC, so is DE to EF; (hyp.)

therefore as DE to EF, so GE to EF; (v. 11.)

that is, DE and GE have the same ratio to EF,

and consequently are equal, (v. 9.)

For the same reason, DFis equal to FG :

and because, in the triangles DEF, GEF, DE is equal to EG, and

EF is common,
the two sides DE, EF are equal to the two GE, EF, each to each ;
and the base DF is equal to the base GF;
therefore the angle DEF is equal to the angle GEF, (l. 8.)
and the other angles to the other angles which are subtended by the

equal sides ; (i. 4.)
therefore the angle DFE is equal to the angle GFE, and EDF to
EGF,

N

^66 Euclid's elements.

and because the angle DEF is equal to the angle GUF,

and G^^i^ equal to the angle ABC; (constr.)

therefore the angle ABCis equal to the angle DFF: (ax. 1.)

for the same reason, the angle A CB is equal to the angle I)FF,

and the angle at A equal to the angle at D :

therefore the triangle ABC is equiangular to the triangle DFF.

Wherefore, if the sides, &c. q.e.d.

PROPOSITION VI. THEOREM.

Tf two triangles have one angle of the one equal to one angle of the other ^
and the sides about the equal angles proportionals^ the triangles shall be
equiangular, and shall have those angles equal which are opposite to the
homologous sides.

Let the triangles ABC, JDFFhaYe the angle BA Cm the one equal
to the angle EDFin the other, and the sides about those angles pro-
portionals ; that is, BA to A C, as ED to DF,

Then the triangles -4 ^C, D^i^ shall be equiangular, and shall have
the angle ^^C equal to the angle DEF, and -4C'i>* to DFE.
A i>

\

G

At the points D, F, in the straight line DF, make the angle FDG
equal to either of the angles BA C, EDF-, {i. 23.)

and the angle DFG equal to the angle A CB :
wherefore the remaining angle at B is equal to the remaining angle
at G : (L 32.)
and consequently the triangle Z)6^i^is equiangular to the triangle ^J5C;
therefore as BA to AC, so is GD to DF: (vi. 4.)
but, by the hypothesis, as BA to A C, so is ED to DF;
therefore as ED to DF, so is GD to DF; (v. 11.)
wherefore ED is equal to DG; (v. 9.)
and JDi^ is common to the two triangles EDF, GDF:
therefore the two sides ED, DF are equal to the two sides GD, DF,
each to each ;

and the angle EDF is equal to the angle GDF; (constr.)
wherefore the base EF is equal to the base FG, (I. 4.)
and the triangle EDF to the triangle GDF,
and the remaining angles to the remaining angles, each to each,
which are subtended by the equal sides :

therefore the angle DFG is equal to the angle DFEy

and the angle at G to the angle at E;

but the angle DFG is equal to the angle A CB ; (constr.)

therefore the angle ACB is equal to the angle BEE; (ax. 1.)

and the angle BACis equal to the angle EDF: (hyp.)
wherefore also the remaining angle at B is equal to the remaining

angle at E; (i. 32.)
therefore the triangle ABC is equiangular to the triangle DEF.
Wherefore, if two triangles, &c. q.e.d.

BOOK VI. PROP. VII. 267

PROPOSITION VII. THEOREM.

If two triangles have one angle of the one equal to one angle of the other ^
and the sides about two other angles proportionals ; then, if each of the
remaining angles be either less, or not less, than a right angle, or if one of