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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BOOK III.]

EUCLID AND LEGENDRE.

89

For MD, EII, which (const.) are parallel, are equal to one
another, because each of them is equal to KB; therefore (I. 15,
cor. B) Dll is parallel to ]\1E. Now (III. 11), the angle AME
in a semicircle is a right angle; and therefore (I. 16) the angles
ADII, DUE are right angles, and (III. 8, cor.) DH touches
both the ciicles, since it is perpendicular to the radii AD, EH.

ScJio. In the figure the circles lie on the same side of the
tangent, which is therefore exterior to them ; but the tangent
can be transverse, or lie between the circles. It is plain also,
tliat in these figures, by using the point L instead of M, an-
other exterior and another transverse tangent would be ob-
tained ; and this will always be so, when each of the circles
lies wholly without the other, and does not touch it. But if
the circles touch one another externally, the two transverse
tangents coalesce into a single line passing through their point
of contact ; if they cut one another, there will be two exterior
tangents, but no transverse one ; if one of the circles touch the
other internally, they carf have only one common tangent, and
this passes through their point of contact ; and, lastly, if one of
them lie wholly within the other without touching it, they can
have no common tangent.

If the circles be equal, the points of contact of the exterior
common tangents are the extremities of the diameters perpen-
dicular to the line joining the centers ; for (I. 15, cor. 3, and 1 6)
the lines connecting the extremities of these toward the same
parts are perpendicular to the diameters, and therefore (III. 8,
cor.) they touch the circles.

Prop. XXV.â€” Prob.â€” 7b insa-ibe a7iy regular polygon in a
given circle.

Let ABDC be the given circle ; it is
required to inscribe any polyg<m in it.

Since (I. 9, cor.) all the angles formed
by any number of straight lines inter-
secting each other in a common point
are equivalent to four right angles, and
(I, 20, cor. 1) any rectilinear figure can
"be divided into as many triangles as the
figure has sides, by straight lines joining the extremities of the

90 THE ELEMENTS OF [BOOK m.

sides with a poiut within the figure â€” a regular polygon being
equilateral â€” the straight lines connecting the extremities of
those sides with the center of the regular polygon will divide
the regular polygon into equal triangles ; for if ABC be a regu-
lar polygon of three sides, AB is equal BC, and also equal to
CA, and draw from E (III. 1, cor. 1), the center of the polygon,
EA, EB, and EC. Now the triangles AEB, BEC, and CEA,
having their bases equal (hyp.), their other sides common,
(const.) are equal; therefore (I. 4) the angles AEB, BEC, and
CEA are equal ; but those angles are equivalent to form right
angles (I. 20, cor. 1) ; hence, each is one third of four right
angles, or two thirds of two right angles ; consequently, a reg-
ular polygon can be inscribed in a circle by making an angle
on the diameter (I. 13) with the center of the circle the vertex
of the angle, equal to that part of four right angles that the
regular polygon has sides, viz. : if the regular polygon has four
sides, each angle at the center will be one fourth of four right
angles ; if five sides, one fifth of fourS-ight angles; and so on.
Or, by bisecting (I. 5) the angle or the arc (III. 11), a regular
polygon of double the number of sides can be inscribed in the
circle. Having by these means the angle at the center of a
regular polygon, the chord of the arc intercepted by the sides
of the angles will be the side of the regular polygon required,
from which (I. 12 and 13) the regular polygon can be inscribed
in the given circle, which was to be done.

Cor. 1. If the sides of the angles be produced beyond the
circumference (I. post. 2), and parallels to the exterior sides of
the polygon (I. 18) be drawn touching the circle (III. 9), then
a similar and regular polygon can be described about the given
circle.

Cor, 2. Find the center of a given regular polygon (III. 1,
cor. 1), and a circle can be inscribed in the polygon, or circum-
scribed about it, by taking the straight line drawn from the
center of the polygon to an extremity of the side of the poly-
gon i'or a radius lor the circumscribed circle, and a line drawn
from the center to the point of bisection of the side for a radius
of the inscribed circle.

Cor. 3. Since the triangles AEB, BEC, and AEC are equiv-
alent respectively to the rectangles (I. 23, cor. 6) under the ra-

BOOK in.] EUCLID AND LEGENDRE. 01

dius of an inscribed circle and the halves of AB, BC, and CA,
it follows (IL 1) that the area of ABC is equivalent to tlie rect-
angle under the radius and half the perimeter. Hence, if the
sides be given in numbers, the length of the radius may be
computed by calculating (II. 12, scho.) the area, and dividing
it by half the sum of the sides, or its double by the sum of the
sides.

Cor. 4. Since the angles formed about the center of a circle
are together equivalent to four right angles (I. 9, cor.), and
since the angles of a triangle are together equivalent to two
right angles (I. 20), it follows that when an equiangular trian-
gle is formed (I. 2) having a vertex at the center and the radius
for a side, each angle of the triangle is one tliird of two riglit
angles, or one sixth of four right angles ; hence, six equal
angles can be formed (I. 1.3) about the center of a circle, and
since the sides about tliese angles are intercepted by the cir-
cumference (const.), they are equal (I. def 16) ; and (I. 2, cor.)
an equiangular triangle being also equilateral, the side of the
triangle opposite the angle at the center of the circle is equal
to the radius of the circle (T. ax. 1) ; therefore the radius can
be made to subtend six equal arcs of the circumference.

Cor. 5. When in the case of a general proposition to descrV^e
a circle touching three given straight Imes ichich do vot pass
through the same point, and which are not all parallel to one
another. If two of the lines be parallel, there may evideiidy
be two equal circles, one on each side of the line falling on the
parallels, each of which will touch the three given lines, and
their centers will be the intersections of the lines bisectimr the
angles made by the parallels with the third line. But if the
lines form a triangle by their intersections, there will be four
circles touching them; one, inscribed, and the others each
touching one side externally and the other two produced. The
centers of the external circles will be the intersections of the
lines bisecting two exterior angles; and the line bisecting the
remote interior angle will pass through the same point. The
method of proof is tl.e same as that given in the proposition.

Cor. 6. If straight lines be drawn from the center of one of
the external circles to the vertices of the triangle, the three tri-
angles formed by the sides of the triangle and the straight lines

92 THE ELEMENTS OF [bOOK HI.

are respectively equivalent to the rectangles (I. 15, cor. 4)
under the radius of that circle and halves of tlie sides of the
triangle. And if the triangle formed by the b.ide of the original
triangle nearest the center of the circle, and the lines drawn to
the vertices at the extremities of that side, be taken from the
sum of the two other triangles, there remains the original tri-
angle equivalent to the rectangle under the radius, and the ex-
cess of half the sum of the two other sides of the orin-inal trian-
gle above half the side nearest the center of the circle, or, which
is the same thing, to the rectangle under the radius, and the
excess of half the perimeter of the original triangle above the
Bide nearest the center of the circle. The radius, therefore, of
any of the external circles may be computed by dividing the
area of the original triangle by the excess of half its perimeter
above the side which the circle touches externally.

Scho. The polygons considered in this proposition, and those
which may be derived from them by the process of bisecting
the angles or arcs subtending the sides of the polygons, are the
only ones till lately which geometers have been able to de-
scribe by elementary geometry, that is, by means of the
straight line and circle. M. Gauss, of Gottingen, in \n?, Disqui-
sitioiies Arithmeticoe, has shown that by elementary geometry,
every regular poly. on may be inscribed in a circle, the number
of whose sides is a power of 2 increased by unity, and is a
prime number, or a number which can not be produced by the
multiplication of two whole numbers, such as 17 â€” the fourth
power of 2 increased by unity â€” and polygons of 257 and C5537
sides. But the investigation is too complex and difficult for an
ordinary school text-book.

END OP b60K third.

BOOK FOURTH.*

THE GENERAL THEORY OF PROPORTION".

DEFINITIONS.

1. A LESS number or magnitude is said to measure a greater,
or to be a measure^ a part^ or a suhmuUxple of tlie greater, when
the less is contained a certain number of times, exactly, in the
greater; and

2. The greater is said to be a multiple of the less.

3. Magnitudes which can be compared in respect of quantity,
that is, which are either equal to one another, or unequal, are
Baid to be of the sam,e kind, or homogeneous.

Scho. 1. Thus, lines, whether straight or curved, are magni-
tudes of the same kind, or are homogeneous, since they may be
equal or unequal. In like manner, surfaces, solids, and angles
form three other classes of homogeneous magnitudes. On the
contrary, lines and surfaces, lines and angles, surfaces and
solids, etc., are heterogeneous. Thus, it is obviously improper
to say, that the side and area of a square are equal to one an-
other, or are unequal. So likewise we cannot say that the area
and one of the angles of a triangle are equal to each other, or
are unequal ; and they are therefore heterogeneous.

4. If there be two magnitudes of the same kind, the relation
which one of them bears to the other in respect of quantity, is
called its ratio to the other.

The first term, or magnitude, is called the antecedent of the
ratio, and the second the consequent.

5. If there be four magnitudes, and if any like multiples
whatever be taken of the first and third, and any whatever of

* The accurate but prolix method of Euclid is substituted by the fol-
lowing more concise method, by employing the notations and simple
principles of Algebra. See Sup. to Book V. Thombon's Euclid.

^^ THE ELEMENTS OF [bOOK IV.

the soconrl anfl fourth ; and if, according as the multiple of the
first is gi-eater than the multiple of the second, equal to it, or
less, the multiple of the third is also greater than the multiple
of the fourth, equal to it, or less ; then the first of the magni-
tudes is said to have to the second the same ratio that the third
has to the fourth.

6. Magnitudes which have the same ratio are called propor-
tionals ; and equality or identity of ratios constitutes /irojoor-
tion or analogy.

When magnitudes are proportionals, the relation is expressed
briefly by saying, that the first is to the second, aÂ« the third to
the fourth, the fifth to the sixth, and so on.

1. When of the multiples of four magnitudes, taken as in the
fifth definition, the multiple of the first is greater than that of
the second, but the multiple of the third is not greater than
that of the fourth ; then the first is said to have to the second
a greater ratio than the third has to the fourth ; and, on the
conti-ary, the third is said to have to the iomth a. less ratio than
the first has to the second.

8. When there are three or more magnitudes of the same
kind, such that the ratios of the first to the second, of the sec-
ond to the third, and so on, whatever may be their number, are
all equal; the magnitudes are said to be continual propor-
tionals.

9. The second of three continual proportionals is said to be a
mean proportional between the other two.

10. When there is any number of magnitudes of the same
kind, greater than two, the first is said to have to the last the
ratio compounded of the ratio which the first has to the second,
and of the latio which the second has to the third, and of the
ratio which the third has to the fourth, and so on to the last
magnitude.

For exam].]e, if A, B, C, D be four magnitudes of the same
kind, the first, .>, is said to have to the last, D, the ratio com-
pounded of tlie ratios of A to B, B to C, and C to D.

11. When iln-ee magnitudes are continual proportionals, the
ratio of tlie first to the third is said to be duplicate of the ratio
of the first Xo the second, or of the second to the third.

12. When iour magnitudes are continual proportionals, the

BOOK rv.] EUCLID AND LEGENDEE. 95

ratio of the first to the fourth is said to be triplicate of the ratio
of the first to the second, of the ratio of the second to the third,
or of tlie ratio of the third to the fourth.

Scho. 2. In continual proportionals, by their own nature, and
that of compound ratio, the ratio of the first to the third is
compounded of two equal ratios ; and the ratio of the first to
the fourth, of three equal ratios ; and hence we see the reason
and the propriety of calling the first duplicate ratio, and the
second triplicate. It is plain, that on similar principles, the
ratio of the first to the fifth would be said to be quadruplicate
of the ratio of the first to the second, of the second to the third,
etc., and thus we might form other similar terms at pleasure.

The terras subduplicate, subtriplicate^ and sesquiplicate^
which are sometimes employed by mathematical wiiters, are
easily understood after the explanations given above. In con-
tinual proportionals, the ratio of the first terra to the second is
said to be subcJuplicate of the ratio of the first to the third, and
subtriplicate of that of the first to the fourth. Again : if there
be four continual proportionals, the ratio of the first to the
fourth is said to be sesquiplicate of the ratio of the first to the
third ; or, which amounts to the same, the ratio which is com-
pounded of another ratio and its subduplicate, is sesquiplicate
of that ratio.

1 3. In proportionals, the antecedent terms are called homolo-
gous to one another, as also the consequents to one another.

Geometers make iise of the following technical words to
denote diiferent modes of deriving one proportion from an-
other, by changing either the order or the magnitudes of the
terms.

14. Alternately : this word is used when there are four pro-
portionals of the same kind ; and it is inferred that the first has
the same ratio to the third which the second has to the fourth ;
or that the first is to the third as the second to the fourth ; as
is shown in the fourth proposition of this book.

15. By inversion: when there are four proportionals, and it
is inferred that the second is to the first as the fourth to the
third. Prop. 3, Book IV.

16. By composition: when there are four proportionals, and
it is inferred, that the first, together with the second, is to the

96 THE ELEMENTS OF [BOOK IV.

second as the tliii-fl, together with the fourth is to the fourth.
Tenth Prop., Book IV.

17. By division : when there are four proportionals, avd it
is inferred that the exjcess of the first above the second is to the
second, as the excess of the third above the fourth is to the
foui-th. Tenth Prop., Book IV.

18. My coyiveraion : when there are four proportionals, and
it is inferred that the first is to its excess above tlie second, as
the third to its excess above the fourth. Eleventh Prop., Book
IV.

Sc'ho. The substance of the five preceding definitions may be
exhibited briefly in the following manner, the signs + and â€”
denoting addition and subtraction, as has been explained
already at the beginning of the second book:

Let A: B:: C:D;
Alternately, A : C : : B : D ;

By inversion, B : A : : D : C ^

By composition, A + B : B : : C + D : D ;

By division. Aâ€” B : B : : Câ€” D : D ;

By conversion, A : A â€” B : : C : C â€” D.

19. Ex mquo^ or ex equali (scil. distantid)^ from equality of
distance: when there is any number of magnitudes more than
two, and as many others, which, taken two in the one rank, and
two in the other, in direct order, have the same ratio; and it is
inferred that the first has to the last of the first rank the same
ratio which the first of the other rank has to the last. This is
demonstrated in the thirteenth proposition of this book.

20. Ex cequo^ inversely : when there are three or more mag-
nitudes, and as many others, which taken two and two in a
cross order, have the same ratio ; that is, when the first magni-
tude is to the second in the first rank, as tlie last but one is to
the last in the second rank; and the second to the third of the
first rank, as the last but two is to the last but one of the second
rank, and so on ; and it is inferred, as in the preceding defini-
tion, that the first is to the last of the first rank, as the first to
the last of the other rank. This is proved in the fourteenth
proposition of this book.

BOOK IV.] EUCLID AND LEGENDRE. 97

AXIOMS.

1. Like multiples of the same, or of equal magnitudes, are
equal to one another.

2. Those magnitudes of which the same, or equal magni-
tudes, are like multiples, are equal to one another.

3. A multiple of a greater magnitude is greater than the
same multiple of a less.

4. That magnitude of which a multiple is greater than the
same multiple of another, is greater than that other magnitude.

EXPLANATION OF SIGNS.

1. The product arising from multiplying one number by an-
other is expressed by writing the letters representing them, one
after the other, without any sign between them ; and some-
times by placing between them a point, or the sign x.

2. A product is called a power ^ when tlie factors are all the
same. Thus, AA, or as it is generally written, A^, is called the
second power, or the square of A ; AAA, or A', the third
power, or cube of A; AAA A, or K\ its fourth power, etc.

In relation to these powers, A is called their root. Thus, A
is the second or square root of A'^, the third or cube root of A%
the fourth root of A^, etc. In like manner, the second or square
root of A is a number Avhich, when multiplied by itself, pro-
duces A; the th rd or cube ro ^ of A is such a number, that if
it be multiplied by itself, and the product by the same root
again, the final product will be A. The square root of A is
denoted by -/A or A*, its cube root by -^A, or A*, its fourth
root by Ai, etc.

3. The quotient arising from dividing one number by an-
other is denoted by writing the dividend as the numerator of a
fraction, and the divisor as its denominator.

4. The signs =, =0, >, <, signify respectively equal to^
equivalent to, greater than, less than.

PROPOSITIONS.

Prop. I. â€” Treor, â€” If there be four numbers such that the
quotients obtained by dividing the first by the second, and the

1

98 THE ELEMENTS OF [bOOK IV.

third hy the fourth, are equal ; the first has to the second the
same ratio that the third has to the fourth.

Let A, B, C, D be four magnitudes, such that oâ€” fj > ^^^^

A: B:: C :D.

For, let m and n be any whole numbers, and multiply the

AC

fractions â€” and ^ by m, and divide the product by n y then

â€” =- = â€” ^-. Now, if mA be srreater than riB, mC will also be
nB riD

greater than nD ; for, if this were not so, â€” rr would not be

equal to -y:- In like manner it might be shown, that if mA

be equal to 7iB, mC will be equal to iiQ ; and that if raK be

less than nB, rriG will be less than wD. But wA, raQ are any

like multiples whatever of A, C ; and ?iB, 7^D any whatever of

B, D ; and therefore (IV. def 5) A : B : : C : D. Therefore, if

there be four numbers, etc.

AC 1

Scho. This proposition is the same, when ^ or Y\â€”P ^^ ^iP

being a whole number.

A C
Cor, If AD=BC, by dividing by B and D, we get ^=t^>

and therefore, by this proposition, A : B : : C : D. Hence, if
the product of two numbers be equal to that of two others, the
one pair may be taken as the extremes and the other as the
means of an analogy.

Prop. II. â€” Theor. â€” If any four numbers he proportional^

and if the first be divided by the second, and the third by the

fourth, the quotients are equal.

A C
Let A : B : : C : D ; then ^= jt

If A and B be whole numbers, let the first and third terms
be multiplied by B, and the second and fourth by A, and the
products are AB, AB, BC, AD. Now, since the first and sec-
ond of these are the same, the third and fourth are (IV. def. 6)
equal; that is, AD=BC; and by dividing these by B and D,

A C

we find (IV. ax, 2) =t = ^.

BOOK IV.] EUCLID AND LEGENDEE. 99

If A and B be fractions, let A=â€” , and B=-, so that niAâ€”

' in n

E, and nB=F; the numerators and denominators E, F, m, n

E F

beinii whole numbers. Then (hyp.) â€” : - : : C : D. Multiply

the first and third of these by mF, and the second and fourth
l)y wE, and the products are EF, EF, mYQ, and nED. Now,
the first and second of these being the same, the third and
fourth (IV. def 5) are equal ; that is, nED=mFC, or mnP^ â€”
wiftBC, since E=imA, and F=wB. Hence, by dividing these
by m and w, we get (IV. ax. 2) ADi=BC ; and the rest of the
proof is the same as in the first case. Therefore, if any four
numbers, etc.

Scho. 1. If either A or B be a whole number, the proof is in-
cluded in the second part of the demonstration given above.
Thus, if A be a whole number, we have simply E = A and m=:
1, and everything will proceed as above. The proof would also
be readily obtained by substituting for B as before, but retain-
inc: A unchanged.

If A and B be incommensurable, such as the numbers ex-
pressing the lengths of the diagonal and side of a square, the
lengths of the diameter and circumference of a circle, etc., their
ratios may be approximated as nearly as we please. Thus the
diagonal of a square is to its side, as |f : 1, nearly ; as \%\ : 1,
more nearly; as \%\\ : 1, still more nearly, etc. Hence, in
such cases we can have no hesitation in admitting the truth of
the proposition, as we see that it holds with respect to numbers
the ratio of which differs from that of the proposed numbers by
a quantity which may be rendered as small as we please â€”

smaller, in fact, than anything that can be assigned.

A 1

Scho. 2. This proposition is the same, when T>=i? or -, p

being a whole number.

Scho. 3. From this proposition and the foregoing, it appears,
that if two fractions be equal, the numerator of the one is to its
denominator as the numerator of the other to its denominator ;
and that if the first and second of four proportional numbers be
made the numerator and denominator of one fraction, and the
third and fourth those of another, the two fractions are equal.

100 THE ELEMENTS OF [bOOK TV.

This is the same in substance as that the two expressions, A :

A C
B : : C : D, and |T=t-o are equivalent, and may be used for

one another.

Cor. 1. It appears in the demonstration of this proposition,
that AD = BC; that is, if four numbers be pT'oportionals, the
product of the extremes is equal to the product of the means.
Hence, if the product of the means be di\ided by one of the ex-
tremes, the quotient is the other; and thus we have a proof of
the ordinary ai'ithmetical rule for finding a fourth proportional
to three uiven numV)ers.

Cor. 2. It is evident, that if A be greater than B, C must be
greater than D ; if equal, equal ; and if less, less ; as otherwise

=5" and =^ could not be equal.

A C
Cor. 3. If A : B : : C : D, and consequently â€”=y^, by multi-

plying these fractions byâ€”, we get â€” 7^= â€” :Fr, or mA : wB : :

mC : nT>.

A

Cor. 4. If A be greater than B, the fraction â€” is evidently

T> C C

greater than -^, and the fraction -^ less than ry ; that is, of two

unequal numbers, the greater has a greater ratio to a third than
the less has ; and a thii'd number has a greater ratio to the
less than it has to the greater.

A B

Cor. 5. Conversely, if -^ be greater than -r^, A is greater than

â€” be less than =,
A B'

B ; and, if â€” be less than z^, A is also greater than B.

Prop. III. â€” Theor. â€” If four numbers he proportionals, they
are proportionals also when taken inversely.

If A : B : : C : D ; then, inversely, B : A : : D : C.

For (IV. 2, cor. 1) BC = AD ; and hence by dividing by A

and C, we obtain -r=p5 or (I^- 2, scho. 2) B : A : : D : C.