Lawrence S. (Lawrence Sluter) Benson. # Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson online

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Online Library → Lawrence S. (Lawrence Sluter) Benson → Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson → online text (page 12 of 21)

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together, have to BAG the same ratio

as the sum of the squares of AB, AG

to the square of BC, that is, the ratio

of equality. From these equals take

the segments AFB, AGG, and there

remain the luncs DF, EG equal to the triangle ABC.

Cor. If the legs AB, AC be equal, the arcs AB'B, AGG are

equal, and each of them an arc of a quadrant ; also the radius

drawn from A is perpendicular to BC ; and since the halves of

equals are equal, each lune is equal to half of the triangle

ABC.

-If, therefore, ABC be a qnadi'ant, and

on its choi'd a semicircle be described, the

lune comprehended between the circum-

ferences is equal to the triangle ABC ;

and since (II. 13) a square can be found

equal to ABC, we can thus effect the

quadrature of a space, ADC, bounded

by arcs of circles.

Prop. XXV. â€” Prob. â€” To find the area of a circle.

Sc/io. 1. The approximate area of a circle can be found by

means of the twentieth proposition of this book, by what is

called the method of exha'ustions, giving an error in excess ;

viz., the approximate area thus obtained is square of radius

multiplied by .3.1415926, etc.

Geometry being an exact science, and its conclusions being

derived from accurate principles, the apj)roximate area for the

circle is not consistent with the strictness of geometrical rea-

soning, and the area of the circle must be established exactly

before it can be regarded a geometrical truth. The reason why

the method of e haustions gives the a]yjyroxim,ate result, is beÂ«

cause â€” by the twentieth proposition of this book â€” the circum-

scribed and inscribed regular and similar polygons about the

circle are supposed^ by continually doubling the number of their

140 THE ELEMENTS OF [BOOK V.

sides, to he made equivalent to the circle; but Carnot, in his

Reflexions sur la Metaphysique da Calcul Infinitesimal^

states, "That the ancient geometers did 'not consider it con-

sistent with the strictness of o-eometrical reasnTiing to re>2:ard

curve lines as polygons of a great number of sides." Now,

the area of any regular ])olygon is the rectangle of its a])othem

and semi-perimeter; but this area is derived from the sixth cor-

ollary of the twenty-third proposition of the first book â€” since

it has been shown in the first corollary of the twentieth propo-

sition of the same book, that any rectilineal figure can be

divided into as many triangles as the figure has sides ; there-

fore, in case of a regular polygon, when triangles are formed ia

it by straight lines drawn from the center to the extremities of

the several sides of the polygon, the area of the polygon be-

comes by the tenth axiom of the firj^t book equivalent to the

sum of these triangles ; hence (I. 23, cor. 6) each triangle is the

rectangle of the apothem of the polygon and a semi-side of the

polygon; therefore the area of the polygon is (I. ax. 10) the

rectangle of its apothem and its semi-perimeter. Since (I. 23,

C01-. 4) the aiea of a triangle is derived from the properties of

parallel straight lives, and any polygon has its sides straight

lines (I. def 12), the pre perties o^ parallel straight lines are

applicable to all polygons; but the circle being formed by a

curve li7ie, the properties of parallel straight lines are not ap-

plicable to it ; hence the reason is evident why the ancient

geometers objected to the curve line being regarded a polygon

of a great number of sides. Euclid, in his ^/e??<e/i^<f, endeav-

ored to sustain the proposition, that the circle is the I'cctangle of

its radius and semicircumfereiice, by what is called the indirect^

apogogic, or Meductio ad ahsurdwm, method. Now, every

true pro])osition can be directly demonstrated, and a fair test

of the truth or falsity of this proposition can be in the success

or failure of it being directly demonstrated. I have given the

d reel demonstrations for every other ])roposition in geometry;

but I can not do so in this case â€” therefore I believe the proj)0-

sition fallacious. Archimedes has shown that the relation of

diameter to the circumference of a circle expressed in numbers,

to be as 7 to 22 â€” which is practically correct. Among isoperi-

metrical figures, the circle contains the greatest area ; there-

BOOK v.] EUCLID AND LEGENDRE. 141

fore when 22 expresses the circumference of a circle, the perim-

eter of its equivalent square must he greater than 22; and if a

cube be inechauically constructed upon a base whose perimeter

is 24.2487 + , it will be equivalent to a cylinder of same height,

the diameter of whose base is 7.

Now, when 24.2487+ expresses the perimeter of a square,

each of its sides (I. 23, cor. 1) will be 6.0621 + ; and its area

â– will be 36.75, or three times square of the radius of the circle.

Hence we get by mechanical construction less than what is ob-

tained by the method of exhaustions. The geometrical con-

firmation of the mechanical construction is given in the second

corollary to the seventeenth proposition of the sixth book.

Scho. 2. Euclid has endeavored to demonstrate that the cir-

cle is the rectangle of circumference ajid semi-i'adius. Now,

the square equal to circle is somewhere between the inscribed

and circumscribed squares, and its area is equal to its perime-

ter multiplied by less than semi-radius; consequently the rect-

angle of circumference and semi-radius will produce more than

area of circle.

{^Or Thomson^ s Eficlid, Appendix, DooJc I., Prop. JTJTXZX)

" The area of a circle is equal to the rectangle under its ra-

dius, and a straight line equal to half its circumference. Let

AB be the radius of the circle BC ; the area of BC is equal to

the rectangle under AB and a straight line D equal to half the

circumference.

" For if the rectangle AB. D be not equal to the circle BC, it

is equal to a circle either greater or less than BC. First, sup-

pose, if possible, the rectangle AB. D to be the area of a circle

EF, of which the radius AE is greater than AB." Here Euclid

is inconsistent with his own proposition : at the very stait he

bases his argument upon a contradiction. He premises that

the area of a circle is equal to the rectangle under its radius,

and a straight line equal to half its circumference ; then sujy-

pose, i. 6., asks to be granted for the sake of argument, that

that same rectangle is equal to a larger circle. Why does he

resort to this subterfuge? It will be said to show the Heductio

ad absurduni ; very well, let us follow his argument : "and let

GHK be a regular polygon described about the circle BC, such

142 THE ELKMENT8 OF [bOOK V.

that its sides flo not meet the circumference of EF. Then, by

dividing this polygon into triangles by radii drawn to G, H,

K, etc., it would be seen that its area is equal to the rectangle

under AB and half its perimeter. But the perimeter of the

polygon is greater than the ciixiumference of BC, and therefore

the area of the polygon is greater than the rectangle AB. D ;

that is, by hypothesis, than the area of the circle EF, which is

absurd." What is absurd ? That the circle EF is greater than

the polygon GHK, etc., or Euclid's argument? The absurdity

is in considering the area of a circle equal to a larger circle.

An argument based upon absurdity must necessarily lead to

absurdity, which in fact has been the case. When Euclid sup-

posed, i. e., asked to be granted for the cake of argument, AB.

D= circle EF, it does not prove the area of polygon greater

than the area of circle EF, because he at the start supposed

AB. D = circle EF, and consistently with his hypothesis and

his argument, it must be so to the end ; therefore, consistently

with his argument and his hypothesis, AB. D is greater than

the area of polygon GHK, etc. The first part of Euclid's prop-

osition is nothing more than a demonstration to prove the area

of a circle is greater than the area of a polygon drawn within

the circle. And the second part of Euclid's proposition is

nothing more than a demonstration to prove a circle less than

the circumscribing polygon. This proposition of Euclid is

very sophistical, and consequently its fallacy has been imde-

tected, owing no doubt to the repute of Euclid, and to the sup-

position that Euclid argued from axioms, and consistently with

the principles of geometry, which he did ; but in this instance

he deceived himself, and consequently all those who believe

him the oracle of geometry. When he attempted to prove

AB. D=area of circle BC, it was contradictory to his argument

to suppose AB.'D = area of circle EI*'; because when he based

his argument upon the premies that AB. D=:area of circle EF,

consistency demanded that he should stand by his premise, and

not forsake it as soon as it led to an absurdity, and judge a cir-

cle less than a polygon within a cii'cle. Tlie absurdity is in

his own argument, to base it upon a supposition which he knew

was inconsistent with his proposition, and the inconsistence to

drop his premiss when he perceived it led to an absurdity ; as

BOOK v.] EUCLID AND LEGENDRE. 143

AB. D is less than the circle EF, it is a very fallacious argu-

ment, when based on the supposition that they are equal, and

it leads to an absurdity ; and very inconsistent with geometri-

cal reasoning for Euclid to drop at the conclusion of his argu-

ment the very premiss upon which he based his argument.

Every method of demonstration, as well as that method termed

Heductio adalsurdttm, require that the premiss which is adopt-

ed at the start be retained to the conclusion. And when Euclid

adopted AB. D=circle EF at the commencement of his demon-

stration, consistence of reason and science demanded that he

fihould have kept it to the conclusion, and then there would

have been no absurdity, but a demonstration to prove that the

polygon GHK, etc., is less than circle EF. But Euclid had

in his mind AB. D = circle BC ; forgetting that he had adopt-

ed AB. D= circle EF, and had stiU to prove AB. D= circle

BC.

END OF BOOK FIFTH.

BOOK SIXTH.

ON THE PLANE AND S0LID3.

DEFINITIONS.

1. A STRAIGHT line is said to be perpendicular to a plane

when it makes ri'jrht ansjles with all straiiiht lines meeting' it iu

that plane.

2. The inclination of two planes which meet one another is

the angle contained by two straight lines drawn from any point

of their common section at right angles to it, one upon each

plane. The angle which one plane makes >.ith another is

Boraetimes called a dihedral angle.

3. If that angle be a right angle, the planes arc perpendicu-

lar to one another.

4. Parallel planes are such as do not meet one another,

though produced ever so far in every direction.

5. A solid angle is that which is made by more than two

plane angles meeting in one point, and not lying in the same

plane.

If the number of plane angles be three, the solid angle is tri-

hedral ; if four, tetrahedral ; if more than io\x\\ polyhedral.

6. Kpolyfiedron is a solid figure contained by plane figures.

If it be contained by four plane figures, it is called a tetrahe-

dron ; if by six, a hexahedron ; if by eight, an octahedron ; if

by twelve, a dodecahedron ; if by twenty, an icosahedron^ etc.

1. A regular body^ or regular polyhedron^ is a solid con-

tained by plane figures, which are all equal and similar.

8. Of solid figures contained by planes^ those are similar

which have all their solid angles equal, each to each, and which

are contained by the same number of similar plane figures, simi-

larly situated.

9. A pyra^nid is a solid figure contained by one plane figure

called its base, and by three or more triangles meeting in a

point without the plane, called the vertex of the pyramid.

BOOK VI.] EUCLID AND LEGENDEE. 145

10. A ]jrism is a solid fijjiire, the ends or hases of \vliich are

parallel, and are equal and similar plane figures, and its otlier

boundaries are parallelograms. One of tliese parallelograms

also is sometimes regarded as the base of the j)rism.

11. Pyramids and prisms are said to be triangular wlien

their bases are triangles ; quadrangular, when their bases are

quadrilaterals ; pentagonal, when ihey are pentagons, etc.

12. The altitude of a pyramid is the perpendicular drawn

from its vertex to its base; and the altitude of a prism is either

a perpendicular drawn from any point in one of its ends or

bases, to the other; or a perpendicular to one of its bounding

parallelograms from a point in the line opposite. The first of

these altitudes is sometimes called the length of the prism.

13. A prism, of which the ends or bases are perpendicular to

the other sides, is called a right pris7n y any other is an ohluiue

pristn.

14. A parallelopiped is a prism of which the bases are par-

allelograms.

15. A parallelopiped of which the bases and the other sides

are rectangles, is said to be rectangular.

16. A cube is a rectangular parallelopiped, which has all its

six sides squares.

17. A sphere is a solid figure described by the revolution of

a semicircle about its diameter, which remains unmoved.

18. The axis of a sphere is the fixed straight line about

which the semicircle revolves.

19. The center of a sphere is the same as that of the generat-

ing semicircle.

20. A diameter of a sphere is any straight line which passes

through the center, and is terminated both ways by its surface.

21. A cone is a solid figure described by the revolution of a

riirht-aniiled triangle about one of the legs, which remains fixed.

If the fixed leg be equal to the other leg, the cone is called

a right-angled cone ; if it be less than the other leg, an obtuse-

angled, and if greater, an acute-angled cone.

22. The axis of a cone is the fixed straight line about which

the triangle revolves.

23. The base of a cone is the circle described by the leg

â– which revolves.

10

146

THE ELEMENTS OF

[book VI.

24. A cylinder is a solid figure described by the revolution

of a rectangle about one of its sides, which remains fixed.

25. The axis of a cylinder is the fixed straight line about

which tlie rectano-le revolves.

26. The bases or ends of a cylinder are the circles described

by the two revolving opposite sides of the rectangle.

27. Similar cones and cylinders are those which have their

axes and the diameters of their bases proportionals.

PROPOSITIONS.

Prop. I. â€” Theor. â€” One part of a straight line can not be in

a plane and another part above it.

J^et EFGH be a plane, then the straight line AB will be

wholly in the plane. By def 1,

Book VI., and def 7, Book L, AB,

being a straight line in the plane

EFGH, is wholly in that plane,

and can not have one part in the

plane and another part above it.

Cor. 1. Hence two straight lines

which cut one another are in the same plane ; so also are

three straight lines which meet one another, not in the same

point.

Cor. 2. Hence, if two planes cut one another, their common

section is a straight line.

Prop. H. â€” Theor, â€” If a straight li?7Â£ be perpendicular to

each of two straight lines at their point of intersection^ it is

also perpendicvlar to the plane in which they are.

Let the straight line EF be perpendicular to each of the

straight lines AB, CD at their intersection E ; EF is also per-

pendicular to the plane passing through AB, CD.

T.ake the straight lines EB, EC equal to one another, and

join BC ; in BC and EP" take any points G and F, and join

EG, FB, FG, FC. Then, in the triangles BEF, CEF, BE is

equal to CE; EF common; and the angles BEF, CEF are

equal, being (hyp.) right angles; therefore (I. 3) BF is equal

to CF. The triangle BFC is therefore isosceles ; and (IL 5,

BOOK VI.]

EUCLID AND LKGENDRE.

IIT

cor, 5) the square of BF is equivalent to the square of FG and the

rectangle BG.GC. Foi" the same reason, because (const.) the

triangle BEG is isosceles, the square of BE

is equivalent to the square GE and the rect-

angle BG.GC. To each of these add the

square of EF ; then tlie squaies of BE, EF

are equivalent to the squares of GE, EF,

and the rectangle BG.GC, But (T. 24, cor,

1) the squares of BE, EFare equivalent to

the square of V>F, because BEF is a right

angle; and it has been shown that the square of BF is equiva-

lent to the .square of FG and the rectangle BG.GC; therefore

the square of FG and the rectangle BG.GC are equivalent to

the squares of GE, EF, and the rectangle BG.GC. Take the

rectangle BG.GC from each, and there remains the square of

FG, equivalent to the squares of GE, EF ; wherefore (I. 24, cor.)

FEG is a right angle. In the same manner it would be proved

that EF is perpendicular to any other straight line drawn

through E in ihe plane passing through AB, CD. But (VI.

def 1) a straight line is perpendicular to a plane when it makes

right angles with all straight lines meeting it in that plane;

therefore EF is perpendicular to the plane of AB, CD. Where-

fore, if a straight line, etc.

Cor. Hence (VI. def. l) if three straight lines meet all in

one point, and a straight line be perpendicular to each of them

at that point, the three straight lines are in the same plane,

Pkop, III. â€” Theor, â€” Tf tioo straight lines he perpendicular

to the same plane, they are parallel to one another.

Let the straight lines AB, CD be at right angles to the same

plane BDE; AB is parallel to CD.

Let them meet the plane in the points B,

D ; join BD, and draw DE perpendicular to

BD in the plane BDE ; make DE equal to

AB, and join BE, AE, AD. Then, because

AB is perpendicular to the plane, each of the

angles ABD, ABE is (VL def 1) a right an-

gle. For the same reason, CDB, CDE are

right angles. And because AB is equal to

148

THE ELKMENTS OF

[lIOOK VI.

DE, BD common, and the angle ABD equal to BDE, AD is

equal (I. 3) to DE.

A-^ain : in the triangles ABE, ADE, AB is equal to DE, BE

to AD, and AE common ; therefore (I. 4) the angle ABE is

equal to EDA ; but ABE is a right angle ; therefore EDA is

also a right angle, and ED perpendicular to DA ; it is also

perpendicular to each of the two BD, DC; therefore (VI. 2,

cor.) these three straight lines DA, DB, DC are all in the same

plane. But (VI. 1, cor. 1) AB is in the plane in which are BD,

DA ; therefore AB, BD, DC are in one plane. Now (hyp.)

each of the angles ABD, BDC is a right angle ; theiefore (L

16, cor. 1) AB is parallel to CD. Wherefore, etc.

Cor. 1. Hence (I. def. 11) if two straight lin^s be parallel,

the straight line drawn from any point in the one to any point

in the other is in the same plane with the parallels.

Cor. 2. Hence, also, if one of two parallel straight lines be

perpendicular to a plane, the other is also perpendicular to it.

Also, two straight lines which are each of them T)arall(l to

the same straight line, and are not both in the same plane with

it, are parallel to one another.

Scho. The same has been proved (I. 1 7) respecting straight

lines in the same plane; therefore, universally, straight hues

â€¢which are parallel to the same straight line, are parallel to one

another.

Prop. TV. â€” Tfieor. â€” If two^ straight lines meeting one an-

other be parallel to two others that meet one another^ and are

not in the same plane with the first two ; tlie first two and the

other two contain equal angles.

Let the straight lines AB, BC, which meet one another, be

parallel to DE, EF, which also meet one

another, but are not in tlie same ])lane

with AB, BC ; the angle ABC is equal to

DEF.

Take BA, BC, ED, EF, all equal to one

another, and join AD, CF, BE, AC, DF.

Because BA is equal and parallel to ED,

thereft)re AD is (I. 15, cor. l) both equal

and parallel to BE. For the same reason, CF is equal and

^

N

E

\

D

f

BOOK VI.]

EUCLID AND LEGKNDKK.

149

parallel to BE. Therefore AD and CF Ix'ing each of them

parallel to BE, are (VI. 3, cor, 2) parallel to one another.

They are also (I. ax, 1) equal; and AC, DF join them toward

the same parts; and therefore (I. 15, cor. 1) AC is equal and

parallel to DF. And because AB, BC are equal to DE, EF,

and AC to DF, the angle ABC is equal (I. 4) to DEF. There-

fore, if two stiMight line?, etc.

Schu. Or supplemental ones, as will be plain after the de-

monstration here given, if AB be produced through 15. This

generalizes the third corollary to the sixteenth proposition of

the first book.

Prop. V. â€” Prob. â€” To clrmo a straight line perpendicular to

a plane, from a given point above it.

Let A be the given point above the plane BII ; it is required

to draw from A a perpendicular to BH.

In the plane draw any straight line BC, and (I. 8) from A

draw AD perpendicular to BC. Then, if AD be also perpen-

dicular to the plane BH, the thing required is done. But if it

be not, from D (I. 7) draw DE,

in the i)lane BII, at right angles

to BC ; from A draw AF perpen-

dicular to DE; and through F

draw (I. 18) Gil parallel to^ BC.

Then, because BC is at right an-

gles to ED and DA, BC is at

right angles (VI. 2) to the plane

passing through ED, DA ; and

GH being parallel to BC, is also (VT. 3, cor. 2) at right angles

to the plane through ED, DA ; and it is therefore perjjendicular

(VI. def 1) to every straight line meeting it in that ])lane; GM

is consequently perpendicular to AF. Therefore AF is per-

pendicular to each of the straight lines Gil, DE ; and conse-

quently (VI. 2) to the plane BHj wherefore AF is the perpen-

dicular required.

Prop. VI. â€” ^Prob. â€” To draw a straight line perpendicular to

a given plane from a point given in the plane.

150

THE ELEMENTS OF

[book VI.

D

B

Let A be the point given in the plane ; it is required to draw

a perpendicular from A to the plane.

From any point B, above the plane, draw (VI. 5) BC per-

pendicular to it ; if this pass through A, it is

the perpendicular required. If not, from A

draw (I. 18) AD parallel to BC. Then, be-

cause AD, CB are parallel, and one of them,

BC, is at right angles to tlie given plane, the

other, AD, is also (VI. 3, cor. 2) at right angles

to it.

Scho. From the same point in a given plane there can not

be two straight lines drawn perpendicular to the plane upon

the same side of it ; and there can be but one perpendicular to

a plane from a point above it.

Cor. Hence planes to which the same straight line is perpen-

dicular, are i^arallel to one another.

E

Prop. VII. â€” Theor. â€” Two planes are parallel^ if tico

straiglit lines which meet one another on one of them be parallel

to two which meet on the other.

Let the straight lines AB, BC meet on the plane AC, and

DE, EF on the plane DF ; if AB, BC be parallel to DE, EF,

the plane AC is parallel to DF.

From B draw (VI. 3, cor. 2) BG perpendicular to the plane

DF, and let it meet that plane in G ; and through G draw (L

18) Gil parallel to ED, and GK to

EF. Then, because BG is perpendicu-

lar to the plane DF, each of the angles

BGII, BGK is (VI. def 1) a right an-

gle ; and because (VI. 3, cor. 2) BA is

parallel to Gil, each of thom being par-

allel to DE, the angles GBA, BGII are

together equal (1. 1 6, cor. 1 ) to two right

angles. But BGII is a risjht aniile ;

therefore, also, GBA is a right angle, and GB perpendicular to

BA. For the same reason, (iB is pei-pendicular to BC. Since,

therefore, GB is perpendicular to BA, BC, it is perpendicular

(VL 2) to the plane AC; and (const.) it is perpendicular to the

plane DF. But (VI. 6, cor.) planes to which the same straight

BOOK VI.]

EUCLID AND LEGENDRE.

151

line is perpendicular are parallel to one another; therefore the

planes AC, DF are parallel. Wherefore, two planes, etc.

Cor. 1. Hence, if two parallel planes be cut by another plane,

their common sections with it are parallels.

Cor. 2. If a straight line be perpendicular to a plane, every

plane which passes through it is perpendicular to that plane.

C-yr. 3. Hence, if two planes cutting one another be each

perpendicular to a third plane, their common sectiiBU is perpen-

dicular to the same plane.

Paop. VIH. â€” Theor, â€” If two straight lines be cut by parallel

planes, they are cut in the same ratio.

Let the straight lines AB, CD be cut by the parallel planes

GH, KL, MN, in the points A, E, B ; C, F, D; as AE : EB : :

CF : FD.

Join AC, BD, AD, and let AD meet KL

in X ; join also EX, XF. Because the

two parallel planes KL, MN are cut by

the plane EBDX, the common sections

EX, BD are (VL 7, cor.) parallel. For

the same reason, because GH, KL are cut

by the plane AXFC, the common sections

AC, XF are parallel. Then (V. 2) because

EX is parallel to BD, a side of the triangle

ABD, AE : EB : : AX : XD ; and be- jf

cause XF is parallel to AC, a side of the triangle ADC, AX :

as the sum of the squares of AB, AG

to the square of BC, that is, the ratio

of equality. From these equals take

the segments AFB, AGG, and there

remain the luncs DF, EG equal to the triangle ABC.

Cor. If the legs AB, AC be equal, the arcs AB'B, AGG are

equal, and each of them an arc of a quadrant ; also the radius

drawn from A is perpendicular to BC ; and since the halves of

equals are equal, each lune is equal to half of the triangle

ABC.

-If, therefore, ABC be a qnadi'ant, and

on its choi'd a semicircle be described, the

lune comprehended between the circum-

ferences is equal to the triangle ABC ;

and since (II. 13) a square can be found

equal to ABC, we can thus effect the

quadrature of a space, ADC, bounded

by arcs of circles.

Prop. XXV. â€” Prob. â€” To find the area of a circle.

Sc/io. 1. The approximate area of a circle can be found by

means of the twentieth proposition of this book, by what is

called the method of exha'ustions, giving an error in excess ;

viz., the approximate area thus obtained is square of radius

multiplied by .3.1415926, etc.

Geometry being an exact science, and its conclusions being

derived from accurate principles, the apj)roximate area for the

circle is not consistent with the strictness of geometrical rea-

soning, and the area of the circle must be established exactly

before it can be regarded a geometrical truth. The reason why

the method of e haustions gives the a]yjyroxim,ate result, is beÂ«

cause â€” by the twentieth proposition of this book â€” the circum-

scribed and inscribed regular and similar polygons about the

circle are supposed^ by continually doubling the number of their

140 THE ELEMENTS OF [BOOK V.

sides, to he made equivalent to the circle; but Carnot, in his

Reflexions sur la Metaphysique da Calcul Infinitesimal^

states, "That the ancient geometers did 'not consider it con-

sistent with the strictness of o-eometrical reasnTiing to re>2:ard

curve lines as polygons of a great number of sides." Now,

the area of any regular ])olygon is the rectangle of its a])othem

and semi-perimeter; but this area is derived from the sixth cor-

ollary of the twenty-third proposition of the first book â€” since

it has been shown in the first corollary of the twentieth propo-

sition of the same book, that any rectilineal figure can be

divided into as many triangles as the figure has sides ; there-

fore, in case of a regular polygon, when triangles are formed ia

it by straight lines drawn from the center to the extremities of

the several sides of the polygon, the area of the polygon be-

comes by the tenth axiom of the firj^t book equivalent to the

sum of these triangles ; hence (I. 23, cor. 6) each triangle is the

rectangle of the apothem of the polygon and a semi-side of the

polygon; therefore the area of the polygon is (I. ax. 10) the

rectangle of its apothem and its semi-perimeter. Since (I. 23,

C01-. 4) the aiea of a triangle is derived from the properties of

parallel straight lives, and any polygon has its sides straight

lines (I. def 12), the pre perties o^ parallel straight lines are

applicable to all polygons; but the circle being formed by a

curve li7ie, the properties of parallel straight lines are not ap-

plicable to it ; hence the reason is evident why the ancient

geometers objected to the curve line being regarded a polygon

of a great number of sides. Euclid, in his ^/e??<e/i^<f, endeav-

ored to sustain the proposition, that the circle is the I'cctangle of

its radius and semicircumfereiice, by what is called the indirect^

apogogic, or Meductio ad ahsurdwm, method. Now, every

true pro])osition can be directly demonstrated, and a fair test

of the truth or falsity of this proposition can be in the success

or failure of it being directly demonstrated. I have given the

d reel demonstrations for every other ])roposition in geometry;

but I can not do so in this case â€” therefore I believe the proj)0-

sition fallacious. Archimedes has shown that the relation of

diameter to the circumference of a circle expressed in numbers,

to be as 7 to 22 â€” which is practically correct. Among isoperi-

metrical figures, the circle contains the greatest area ; there-

BOOK v.] EUCLID AND LEGENDRE. 141

fore when 22 expresses the circumference of a circle, the perim-

eter of its equivalent square must he greater than 22; and if a

cube be inechauically constructed upon a base whose perimeter

is 24.2487 + , it will be equivalent to a cylinder of same height,

the diameter of whose base is 7.

Now, when 24.2487+ expresses the perimeter of a square,

each of its sides (I. 23, cor. 1) will be 6.0621 + ; and its area

â– will be 36.75, or three times square of the radius of the circle.

Hence we get by mechanical construction less than what is ob-

tained by the method of exhaustions. The geometrical con-

firmation of the mechanical construction is given in the second

corollary to the seventeenth proposition of the sixth book.

Scho. 2. Euclid has endeavored to demonstrate that the cir-

cle is the rectangle of circumference ajid semi-i'adius. Now,

the square equal to circle is somewhere between the inscribed

and circumscribed squares, and its area is equal to its perime-

ter multiplied by less than semi-radius; consequently the rect-

angle of circumference and semi-radius will produce more than

area of circle.

{^Or Thomson^ s Eficlid, Appendix, DooJc I., Prop. JTJTXZX)

" The area of a circle is equal to the rectangle under its ra-

dius, and a straight line equal to half its circumference. Let

AB be the radius of the circle BC ; the area of BC is equal to

the rectangle under AB and a straight line D equal to half the

circumference.

" For if the rectangle AB. D be not equal to the circle BC, it

is equal to a circle either greater or less than BC. First, sup-

pose, if possible, the rectangle AB. D to be the area of a circle

EF, of which the radius AE is greater than AB." Here Euclid

is inconsistent with his own proposition : at the very stait he

bases his argument upon a contradiction. He premises that

the area of a circle is equal to the rectangle under its radius,

and a straight line equal to half its circumference ; then sujy-

pose, i. 6., asks to be granted for the sake of argument, that

that same rectangle is equal to a larger circle. Why does he

resort to this subterfuge? It will be said to show the Heductio

ad absurduni ; very well, let us follow his argument : "and let

GHK be a regular polygon described about the circle BC, such

142 THE ELKMENT8 OF [bOOK V.

that its sides flo not meet the circumference of EF. Then, by

dividing this polygon into triangles by radii drawn to G, H,

K, etc., it would be seen that its area is equal to the rectangle

under AB and half its perimeter. But the perimeter of the

polygon is greater than the ciixiumference of BC, and therefore

the area of the polygon is greater than the rectangle AB. D ;

that is, by hypothesis, than the area of the circle EF, which is

absurd." What is absurd ? That the circle EF is greater than

the polygon GHK, etc., or Euclid's argument? The absurdity

is in considering the area of a circle equal to a larger circle.

An argument based upon absurdity must necessarily lead to

absurdity, which in fact has been the case. When Euclid sup-

posed, i. e., asked to be granted for the cake of argument, AB.

D= circle EF, it does not prove the area of polygon greater

than the area of circle EF, because he at the start supposed

AB. D = circle EF, and consistently with his hypothesis and

his argument, it must be so to the end ; therefore, consistently

with his argument and his hypothesis, AB. D is greater than

the area of polygon GHK, etc. The first part of Euclid's prop-

osition is nothing more than a demonstration to prove the area

of a circle is greater than the area of a polygon drawn within

the circle. And the second part of Euclid's proposition is

nothing more than a demonstration to prove a circle less than

the circumscribing polygon. This proposition of Euclid is

very sophistical, and consequently its fallacy has been imde-

tected, owing no doubt to the repute of Euclid, and to the sup-

position that Euclid argued from axioms, and consistently with

the principles of geometry, which he did ; but in this instance

he deceived himself, and consequently all those who believe

him the oracle of geometry. When he attempted to prove

AB. D=area of circle BC, it was contradictory to his argument

to suppose AB.'D = area of circle EI*'; because when he based

his argument upon the premies that AB. D=:area of circle EF,

consistency demanded that he should stand by his premise, and

not forsake it as soon as it led to an absurdity, and judge a cir-

cle less than a polygon within a cii'cle. Tlie absurdity is in

his own argument, to base it upon a supposition which he knew

was inconsistent with his proposition, and the inconsistence to

drop his premiss when he perceived it led to an absurdity ; as

BOOK v.] EUCLID AND LEGENDRE. 143

AB. D is less than the circle EF, it is a very fallacious argu-

ment, when based on the supposition that they are equal, and

it leads to an absurdity ; and very inconsistent with geometri-

cal reasoning for Euclid to drop at the conclusion of his argu-

ment the very premiss upon which he based his argument.

Every method of demonstration, as well as that method termed

Heductio adalsurdttm, require that the premiss which is adopt-

ed at the start be retained to the conclusion. And when Euclid

adopted AB. D=circle EF at the commencement of his demon-

stration, consistence of reason and science demanded that he

fihould have kept it to the conclusion, and then there would

have been no absurdity, but a demonstration to prove that the

polygon GHK, etc., is less than circle EF. But Euclid had

in his mind AB. D = circle BC ; forgetting that he had adopt-

ed AB. D= circle EF, and had stiU to prove AB. D= circle

BC.

END OF BOOK FIFTH.

BOOK SIXTH.

ON THE PLANE AND S0LID3.

DEFINITIONS.

1. A STRAIGHT line is said to be perpendicular to a plane

when it makes ri'jrht ansjles with all straiiiht lines meeting' it iu

that plane.

2. The inclination of two planes which meet one another is

the angle contained by two straight lines drawn from any point

of their common section at right angles to it, one upon each

plane. The angle which one plane makes >.ith another is

Boraetimes called a dihedral angle.

3. If that angle be a right angle, the planes arc perpendicu-

lar to one another.

4. Parallel planes are such as do not meet one another,

though produced ever so far in every direction.

5. A solid angle is that which is made by more than two

plane angles meeting in one point, and not lying in the same

plane.

If the number of plane angles be three, the solid angle is tri-

hedral ; if four, tetrahedral ; if more than io\x\\ polyhedral.

6. Kpolyfiedron is a solid figure contained by plane figures.

If it be contained by four plane figures, it is called a tetrahe-

dron ; if by six, a hexahedron ; if by eight, an octahedron ; if

by twelve, a dodecahedron ; if by twenty, an icosahedron^ etc.

1. A regular body^ or regular polyhedron^ is a solid con-

tained by plane figures, which are all equal and similar.

8. Of solid figures contained by planes^ those are similar

which have all their solid angles equal, each to each, and which

are contained by the same number of similar plane figures, simi-

larly situated.

9. A pyra^nid is a solid figure contained by one plane figure

called its base, and by three or more triangles meeting in a

point without the plane, called the vertex of the pyramid.

BOOK VI.] EUCLID AND LEGENDEE. 145

10. A ]jrism is a solid fijjiire, the ends or hases of \vliich are

parallel, and are equal and similar plane figures, and its otlier

boundaries are parallelograms. One of tliese parallelograms

also is sometimes regarded as the base of the j)rism.

11. Pyramids and prisms are said to be triangular wlien

their bases are triangles ; quadrangular, when their bases are

quadrilaterals ; pentagonal, when ihey are pentagons, etc.

12. The altitude of a pyramid is the perpendicular drawn

from its vertex to its base; and the altitude of a prism is either

a perpendicular drawn from any point in one of its ends or

bases, to the other; or a perpendicular to one of its bounding

parallelograms from a point in the line opposite. The first of

these altitudes is sometimes called the length of the prism.

13. A prism, of which the ends or bases are perpendicular to

the other sides, is called a right pris7n y any other is an ohluiue

pristn.

14. A parallelopiped is a prism of which the bases are par-

allelograms.

15. A parallelopiped of which the bases and the other sides

are rectangles, is said to be rectangular.

16. A cube is a rectangular parallelopiped, which has all its

six sides squares.

17. A sphere is a solid figure described by the revolution of

a semicircle about its diameter, which remains unmoved.

18. The axis of a sphere is the fixed straight line about

which the semicircle revolves.

19. The center of a sphere is the same as that of the generat-

ing semicircle.

20. A diameter of a sphere is any straight line which passes

through the center, and is terminated both ways by its surface.

21. A cone is a solid figure described by the revolution of a

riirht-aniiled triangle about one of the legs, which remains fixed.

If the fixed leg be equal to the other leg, the cone is called

a right-angled cone ; if it be less than the other leg, an obtuse-

angled, and if greater, an acute-angled cone.

22. The axis of a cone is the fixed straight line about which

the triangle revolves.

23. The base of a cone is the circle described by the leg

â– which revolves.

10

146

THE ELEMENTS OF

[book VI.

24. A cylinder is a solid figure described by the revolution

of a rectangle about one of its sides, which remains fixed.

25. The axis of a cylinder is the fixed straight line about

which tlie rectano-le revolves.

26. The bases or ends of a cylinder are the circles described

by the two revolving opposite sides of the rectangle.

27. Similar cones and cylinders are those which have their

axes and the diameters of their bases proportionals.

PROPOSITIONS.

Prop. I. â€” Theor. â€” One part of a straight line can not be in

a plane and another part above it.

J^et EFGH be a plane, then the straight line AB will be

wholly in the plane. By def 1,

Book VI., and def 7, Book L, AB,

being a straight line in the plane

EFGH, is wholly in that plane,

and can not have one part in the

plane and another part above it.

Cor. 1. Hence two straight lines

which cut one another are in the same plane ; so also are

three straight lines which meet one another, not in the same

point.

Cor. 2. Hence, if two planes cut one another, their common

section is a straight line.

Prop. H. â€” Theor, â€” If a straight li?7Â£ be perpendicular to

each of two straight lines at their point of intersection^ it is

also perpendicvlar to the plane in which they are.

Let the straight line EF be perpendicular to each of the

straight lines AB, CD at their intersection E ; EF is also per-

pendicular to the plane passing through AB, CD.

T.ake the straight lines EB, EC equal to one another, and

join BC ; in BC and EP" take any points G and F, and join

EG, FB, FG, FC. Then, in the triangles BEF, CEF, BE is

equal to CE; EF common; and the angles BEF, CEF are

equal, being (hyp.) right angles; therefore (I. 3) BF is equal

to CF. The triangle BFC is therefore isosceles ; and (IL 5,

BOOK VI.]

EUCLID AND LKGENDRE.

IIT

cor, 5) the square of BF is equivalent to the square of FG and the

rectangle BG.GC. Foi" the same reason, because (const.) the

triangle BEG is isosceles, the square of BE

is equivalent to the square GE and the rect-

angle BG.GC. To each of these add the

square of EF ; then tlie squaies of BE, EF

are equivalent to the squares of GE, EF,

and the rectangle BG.GC, But (T. 24, cor,

1) the squares of BE, EFare equivalent to

the square of V>F, because BEF is a right

angle; and it has been shown that the square of BF is equiva-

lent to the .square of FG and the rectangle BG.GC; therefore

the square of FG and the rectangle BG.GC are equivalent to

the squares of GE, EF, and the rectangle BG.GC. Take the

rectangle BG.GC from each, and there remains the square of

FG, equivalent to the squares of GE, EF ; wherefore (I. 24, cor.)

FEG is a right angle. In the same manner it would be proved

that EF is perpendicular to any other straight line drawn

through E in ihe plane passing through AB, CD. But (VI.

def 1) a straight line is perpendicular to a plane when it makes

right angles with all straight lines meeting it in that plane;

therefore EF is perpendicular to the plane of AB, CD. Where-

fore, if a straight line, etc.

Cor. Hence (VI. def. l) if three straight lines meet all in

one point, and a straight line be perpendicular to each of them

at that point, the three straight lines are in the same plane,

Pkop, III. â€” Theor, â€” Tf tioo straight lines he perpendicular

to the same plane, they are parallel to one another.

Let the straight lines AB, CD be at right angles to the same

plane BDE; AB is parallel to CD.

Let them meet the plane in the points B,

D ; join BD, and draw DE perpendicular to

BD in the plane BDE ; make DE equal to

AB, and join BE, AE, AD. Then, because

AB is perpendicular to the plane, each of the

angles ABD, ABE is (VL def 1) a right an-

gle. For the same reason, CDB, CDE are

right angles. And because AB is equal to

148

THE ELKMENTS OF

[lIOOK VI.

DE, BD common, and the angle ABD equal to BDE, AD is

equal (I. 3) to DE.

A-^ain : in the triangles ABE, ADE, AB is equal to DE, BE

to AD, and AE common ; therefore (I. 4) the angle ABE is

equal to EDA ; but ABE is a right angle ; therefore EDA is

also a right angle, and ED perpendicular to DA ; it is also

perpendicular to each of the two BD, DC; therefore (VI. 2,

cor.) these three straight lines DA, DB, DC are all in the same

plane. But (VI. 1, cor. 1) AB is in the plane in which are BD,

DA ; therefore AB, BD, DC are in one plane. Now (hyp.)

each of the angles ABD, BDC is a right angle ; theiefore (L

16, cor. 1) AB is parallel to CD. Wherefore, etc.

Cor. 1. Hence (I. def. 11) if two straight lin^s be parallel,

the straight line drawn from any point in the one to any point

in the other is in the same plane with the parallels.

Cor. 2. Hence, also, if one of two parallel straight lines be

perpendicular to a plane, the other is also perpendicular to it.

Also, two straight lines which are each of them T)arall(l to

the same straight line, and are not both in the same plane with

it, are parallel to one another.

Scho. The same has been proved (I. 1 7) respecting straight

lines in the same plane; therefore, universally, straight hues

â€¢which are parallel to the same straight line, are parallel to one

another.

Prop. TV. â€” Tfieor. â€” If two^ straight lines meeting one an-

other be parallel to two others that meet one another^ and are

not in the same plane with the first two ; tlie first two and the

other two contain equal angles.

Let the straight lines AB, BC, which meet one another, be

parallel to DE, EF, which also meet one

another, but are not in tlie same ])lane

with AB, BC ; the angle ABC is equal to

DEF.

Take BA, BC, ED, EF, all equal to one

another, and join AD, CF, BE, AC, DF.

Because BA is equal and parallel to ED,

thereft)re AD is (I. 15, cor. l) both equal

and parallel to BE. For the same reason, CF is equal and

^

N

E

\

D

f

BOOK VI.]

EUCLID AND LEGKNDKK.

149

parallel to BE. Therefore AD and CF Ix'ing each of them

parallel to BE, are (VI. 3, cor, 2) parallel to one another.

They are also (I. ax, 1) equal; and AC, DF join them toward

the same parts; and therefore (I. 15, cor. 1) AC is equal and

parallel to DF. And because AB, BC are equal to DE, EF,

and AC to DF, the angle ABC is equal (I. 4) to DEF. There-

fore, if two stiMight line?, etc.

Schu. Or supplemental ones, as will be plain after the de-

monstration here given, if AB be produced through 15. This

generalizes the third corollary to the sixteenth proposition of

the first book.

Prop. V. â€” Prob. â€” To clrmo a straight line perpendicular to

a plane, from a given point above it.

Let A be the given point above the plane BII ; it is required

to draw from A a perpendicular to BH.

In the plane draw any straight line BC, and (I. 8) from A

draw AD perpendicular to BC. Then, if AD be also perpen-

dicular to the plane BH, the thing required is done. But if it

be not, from D (I. 7) draw DE,

in the i)lane BII, at right angles

to BC ; from A draw AF perpen-

dicular to DE; and through F

draw (I. 18) Gil parallel to^ BC.

Then, because BC is at right an-

gles to ED and DA, BC is at

right angles (VI. 2) to the plane

passing through ED, DA ; and

GH being parallel to BC, is also (VT. 3, cor. 2) at right angles

to the plane through ED, DA ; and it is therefore perjjendicular

(VI. def 1) to every straight line meeting it in that ])lane; GM

is consequently perpendicular to AF. Therefore AF is per-

pendicular to each of the straight lines Gil, DE ; and conse-

quently (VI. 2) to the plane BHj wherefore AF is the perpen-

dicular required.

Prop. VI. â€” ^Prob. â€” To draw a straight line perpendicular to

a given plane from a point given in the plane.

150

THE ELEMENTS OF

[book VI.

D

B

Let A be the point given in the plane ; it is required to draw

a perpendicular from A to the plane.

From any point B, above the plane, draw (VI. 5) BC per-

pendicular to it ; if this pass through A, it is

the perpendicular required. If not, from A

draw (I. 18) AD parallel to BC. Then, be-

cause AD, CB are parallel, and one of them,

BC, is at right angles to tlie given plane, the

other, AD, is also (VI. 3, cor. 2) at right angles

to it.

Scho. From the same point in a given plane there can not

be two straight lines drawn perpendicular to the plane upon

the same side of it ; and there can be but one perpendicular to

a plane from a point above it.

Cor. Hence planes to which the same straight line is perpen-

dicular, are i^arallel to one another.

E

Prop. VII. â€” Theor. â€” Two planes are parallel^ if tico

straiglit lines which meet one another on one of them be parallel

to two which meet on the other.

Let the straight lines AB, BC meet on the plane AC, and

DE, EF on the plane DF ; if AB, BC be parallel to DE, EF,

the plane AC is parallel to DF.

From B draw (VI. 3, cor. 2) BG perpendicular to the plane

DF, and let it meet that plane in G ; and through G draw (L

18) Gil parallel to ED, and GK to

EF. Then, because BG is perpendicu-

lar to the plane DF, each of the angles

BGII, BGK is (VI. def 1) a right an-

gle ; and because (VI. 3, cor. 2) BA is

parallel to Gil, each of thom being par-

allel to DE, the angles GBA, BGII are

together equal (1. 1 6, cor. 1 ) to two right

angles. But BGII is a risjht aniile ;

therefore, also, GBA is a right angle, and GB perpendicular to

BA. For the same reason, (iB is pei-pendicular to BC. Since,

therefore, GB is perpendicular to BA, BC, it is perpendicular

(VL 2) to the plane AC; and (const.) it is perpendicular to the

plane DF. But (VI. 6, cor.) planes to which the same straight

BOOK VI.]

EUCLID AND LEGENDRE.

151

line is perpendicular are parallel to one another; therefore the

planes AC, DF are parallel. Wherefore, two planes, etc.

Cor. 1. Hence, if two parallel planes be cut by another plane,

their common sections with it are parallels.

Cor. 2. If a straight line be perpendicular to a plane, every

plane which passes through it is perpendicular to that plane.

C-yr. 3. Hence, if two planes cutting one another be each

perpendicular to a third plane, their common sectiiBU is perpen-

dicular to the same plane.

Paop. VIH. â€” Theor, â€” If two straight lines be cut by parallel

planes, they are cut in the same ratio.

Let the straight lines AB, CD be cut by the parallel planes

GH, KL, MN, in the points A, E, B ; C, F, D; as AE : EB : :

CF : FD.

Join AC, BD, AD, and let AD meet KL

in X ; join also EX, XF. Because the

two parallel planes KL, MN are cut by

the plane EBDX, the common sections

EX, BD are (VL 7, cor.) parallel. For

the same reason, because GH, KL are cut

by the plane AXFC, the common sections

AC, XF are parallel. Then (V. 2) because

EX is parallel to BD, a side of the triangle

ABD, AE : EB : : AX : XD ; and be- jf

cause XF is parallel to AC, a side of the triangle ADC, AX :

Online Library → Lawrence S. (Lawrence Sluter) Benson → Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson → online text (page 12 of 21)