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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16i

THE ELEMENTS OF

[book VI.

fixed axis BC, it will generate a cylincler (VI. def. 24) ; the
semicircle BNC will generate a sphere (VI. def. 17) ; and the
triangle BGC will generate a cone (VI. def. 21). The cone
Avill be one third the cylinder (VI. 17), and the sphere will bo
two thirds the same cylinder (VI. 17, cor 1).

The triangle BOP having one half altitnde and one half base
of the triangle BGC, will generate a cone
one eighvh of the cone generated by the
triangle BGC (VI. 16, cor. 3) ; hence, one
twelfth of the cylinder generated by the
square BENP;and the cone generated by
the triangle BNP is one half cone gener-
ated by the triangle BGC (VI. 16, cor. 1) ;
hence, four times cone generated by the
triangle BOP. And the hemisphere gen-
erated by the quadrant BNP is two thirds
cylinder generated by the square BENP
VI. 17, cor. 1), or eight times cone gen-
erated by the triangle BOP.

Let the triangle BSN be described on BN, equal to the tri-
angle BON (I. 23,and 15, cor. 4). Then the trapezium BSNP
will generate a solid equivalent to the sum of a cylinder one
half cylinder generated by the square BENP, and a cone one
sixth of the same cylinder, or eight times the cone generated
by the triangle BOP, making a solid equivalent to the hemis-
and same altitude BP. But the triangle BNP is common to
both the trapezium B>NP and the quadiant BNP, and gener-
ates in each case the solid equivalent to four times cone gener-
ated by the tiiangle BOP ; therefore the segment BN and the
triangle BSN generate an equivalence of solid, or four times
cone generated by the triangle BOP; consequently the seg-
ment BN and the triangle BSN are equivalent (I. ax. 1).

Again : the triangle BNP generates a cone one third the
cylinder generated by the square BENP (VI. 17), and the
quadrant BNP generates a heniisphere two thiids of the same
cylinder (VI. 17, cor. 1). The triangle BNP is one half the
square BlilNP.

Now, the trapezium BSNP, equivalent to three fourths of the

BOOK VI.] EUCLID AND LEGENDRE. 165

gquiire, on same raflius aiul altitude as the square, generates
a solid two thirds of the solid generated by the sqnuiv, and the
BENP generates an equivalent solid with the trapezium
BSNP. That an equivalence of surfaces ujion the same radius
â€˘will generate an equivalence of solids can be illustrated by-
taking a trapezium greater than the trapezium BSNP, having
tlie same radius. It can easily be shown that the greater trape-
zium generates a greater solid than the less trapezium, and in
a similar manner it can be shown that a less trapezium than the
trapezium BSNP generates a less solid ; hence, very evidently,
when a greater surface upon same radius generates a greater
solid, and a less surface generates a less solid, equivalent
surfaces must generate equivalent solids on the same radius;
and, conversely, when we have equivalent solids generated
npon the same radius, the generatirig surfaces are equivalent ;
therefore (I. ax. l) the quadrant BNP is three fourths of the
square BENP, or the semicircle BNC is three fourths of the
parallelogram BEGC, or any circle is three fourths of the cir-
cumscribing square, or Three Times Square of Badius.
Hence, we have a geometrical confirmation of the mechanical
construction in scholium to twenty-fifth proposition of book
fifth.

< TIIER'WISR :

Tlie triangle BGC generates a cone one third (VI. 17) of
the cylinder generated by tlie rectangle BEGC, and the semi-
cii'cle BXC genei'ates a sphere two thirds ( VI. 1 7, cor. 1 ) of the
same cylinder ; the s] here is the mean between the cone and
cylinder; therefore the semicircle is evidently the meanhe-
tween the triangle BGC and the rectangle BEGC, or three
fourths of the rectangle BEGC ; or any circle is three fourths
square of its diameter, or three times square of its radius.

OTHERWISE :

Circles are to one another as the squares described on their
diameters (V. 14) ; consequently squares are to one another as
the circles described on their sides; therefore there is an
equality of proportion (V. 24) ; hence, we derived the arith*
metical proportion :

166 THE ELEMENTS OF [boOK YI.

Rectangle BEGC, semicircle BNC, triangle BGC.

The sum of extremes is equivalent to twice the mean ; there-
fore we have â€”

Rectangle BEGC -f- triangle BGCo2 semicircle BNC; or,
semicircle BNCOi rectangle BEGC + i triangle BGC.

Also, the difference between first and second terms of an
arithmetical proportion is the same as the difference between
the second and third terms, as the diffei'ence between third and
fourth terms, and so on ; hence we have â€”

Rectangle BEGCâ€” semicircle BNCOsemicircle BNC -tri-
angle BGC ; therefore we get segment BXOtriangle BSN,
or one fourth square BENP; consequently, eircle =0 three
fourth square of diameter, or tliree times square of radius.

Cor. 3. Archimedes discovered the proi)ortion 1, 2, 3 be-
tween the cone, sjihere, and cylinder of similar dimensions; but
from the previous corollary we obtain the proportion 1, 2, 3,
4 for the cone, sphere, cylinder, and cube of similar diuiensions;
because the cube is eight times cube of radius of th^ sjyhere
(Vr. 10) ; the cylinder is six times cube of radius of the sphere
(VI. 17, cor, 2) ; the sphere infour times cube of radius (f the
sphere (VI. 17, cors. 1 and 2) ; and the cone is twice ctibe of
radius of the sphere (VI, 17, cors, 1 and 2). Hence the sphere
is the mean proportional of the cone, and the cube circum-
scribing the si)here, or one half the circumscribiug cube;
therefore the surface of the sphere is fur times the area of
one of its great circles, or two thirds the surface of the circum-
scribing cylinder. Hence, there is the identical proportion be-
tween the surfaces of the sphere and cylinder as there is
between \}\q solidities of the s[ihere and cylinder.

Scho. 1. Therefore the second corollary gives the solution
to the long mooted and much vexed question of the Quadra-
ture of the Circle, showing that the perplexity of it arose from
the uii geometrical sitjyposition (V. 25, scho.) that "the circle is
a regular polygon of an infinite inind)er of sides." Hence it is
evident that all conclusions derived fi'oni a fdlacious su])])0-
sition will give ])erplexity so long as the snpposilion is main-
tained, and must necessarily involve conti'adiction*^ to the rigor
of geometrical reasoning. And wlicii demonstrations are con-
ducted consistently with established definitions, axioms, and

BOOK VI.] EUCLID AND LKGKNDRE. 16T

propositions, all conclusions derived from tlieni are unimpeach-
able, and arc valuable to a system of scientific truths.

iicho. 2. Geometry, like all other sciences, is based upon
cvviiuuj'undamencal i)rinciples, and a close examination of tliia
science reveals the fact that, throughout its whole extent and
its vai'ious applications, tlie principle that the siraljht line is
the shortest line bcttoeen tv:o fjiven points is the fundamental
])rinciple of the science ; by this principle the dimensions of
magnitudes are determined, distances of objects made known,
and other useful and practical results ascertained. iSince this
jjrinciple is so important, it would be interesting to inquire the
reason why it has such manifest usefulness. The angle is a
magnitude contained l)y the intersection of two straifjht lii.es^
and the polygon is another magnitude bounded by three or
moYG straight lines ^ lience we see how intimate the connec-
tion between the straight line and the angle and polygon ;
therefore we find that the functions of the angle are straight
lines, such as sines, co-sines, tangents, etc. ; and the properties
of the polygon ai"e defined by straigld lines, such as its jjerime-
ter and apothem ; therefore in all rectilineal magnitudes we
discover a use for the straight line above all other lines, and
evidently the principle of the straight line has a i)eculiar force
to all rectilineal figures; consequently we adopt x}w straight
line as a means of measure for all rectilineal magnitudes. The
adoption of this m^eans of measure constitutes the straight line
a standard by which all measurements of rectilineal magni-
tudes are compared. Hence very naturally there is a consist-
ency between the measurements and other properties of recti-
lineal magnitudes.

Now, when w(^ examine the circle or portions of the circle,
as the segments, sectors, arcs, etc., we at once discover a nou'
coincidence between the curve which bounds them and the
straight line which bounds rectilineal magnitudes ; hence, very
evidently, the superficies of curvilmear spaces require a pecu-
liar coni-ection between tiiem and the bounding curve, as there
is a peculiar connection between the superficies of rectilineal
spaces and the bounding straight lines. In other words, since
"vve derive the measurements and other properties of rectilineal
magnitudes from the principle of the straight line^ so we nmst

V

168

THE ELEMKNTS OF

[book VI.

determine tlie measurements and other properties of curvilinear
magnitudes from the prini-iple of tlie curve, and thus we per-
ceive wh^, when we endeavor to obtain the area of circle by
the method of exhaustions, using the straight line as a mean\$
of measure, we can get the a2:>proxirnate area only, and ahy
it is necessai-y to obtain accurate results to use the [trniciple of
the curm. Geometry, in its pi'esent state, is the science of iho
straight line, and the introduction of the principle of t!ie curve
into geometrical consideration would usher in a distinct science^
but eminently useful in solving problems of cui'vilincar spaces
and boundaries which were before unsolved, inasmuch as the
approximate results only were given for them.

The method of exhaustions is applicable to rectilineal magni-
tudes, and its results are consistent with the ])rinciple of the
Btraight line, because the straight line is adopted as a means
and standard oi' measurement; but since the straight line and
curve do not coincide, the principle and propeities of the
straight line are not applicable to curvilinear spaces or bound
aries ; hence, what is true in one case, becomes absurd in the
other.

Prop. XVIIT. â€” Tiieou. â€” 77te sections of a solid I y parallel
planes are similar figures.

Let the prism JMN be cut by the two parallel planes AD,
FK ; their sections with it are similar figures.

For (VI. 7, cor.) the sections have parallel sides (T. 15, cor.
2). The figures AD, FK, thei-efore, have their
sides similar, each to each. Their several an-
gles are also (VI. 4) equal ; for they ai"e con-
tained by straight lines wliich are parallel ;
and therefore the figures are similar.

Cor. 1. A section of a prism by a plane par-
allel to the base is equal and bimilar to the
base.

Scho. 1. Since (VI. def. 24) a cylinder is de-
scribed by the revolution of a rectangle about
one of its sides, it is plain that any straight
line in the rectangle perpendicular to the fixed
line will describe a circle parallel to the base ; and hence

BOOK VI.] EUCLID AND LEGENDEE. 169

every section of a cylinder by a plane parallel to the base is a
circle equal to the base.

Cor. 2. The section of a pyramid by a plane parallel to its
base is a fiijure similar to the base.

Scho. 2. Since (VI. def. 21) a cone is described by the revo-
lution of a right-angled triangle about one of its legs, it is plain
that any straight line in the triangle perpendicular to the fixed
leg will describe a circle parallel to the base; and the radius
of that circle will be to the radius of the base as the altitude of
the cone cut off to that of the whole cone.

Cor. 3. A section of a sphere by a plane is a circle.

Since the radii of the sphere are all equal, each of them being
equal to the radius of the describing semicircle, it is plain that
if the section pass through the center, it is a circle of the same
radius as the sphere. But if the plane do not pass through the
center, draw (VI. 5) a perpendicular to it from the center, and
draw any number of radii of the sphere to the intersection of
its surface with the plane. These radii, which are equal, are
the hypothenuses of right-angled triangles, which have the per-
pendicular from the center as a common leg; and therefore (I.
24, cor. 2) their other legs are all equal ; wherefore the section
of the sphere by the plane is a circle, the center of which is the
point in which the perpendicular cuts the plane.

8cho. 3. All the sections through the center are equal to one
another, and are greater than the others. The former are
therefore called great circles, the latter small or less circles.

tScho. 4. A straight line drawn through the center of a circle
of the sphere perpendicular to its plane is a diameter of the
sphere. The extremities of this diameter are called the poles
of the circle. It is plain (I. 24, cor. 2) that chords drawn in
the sphere from either pole of a circle to the circumfeience are
all equal ; and therefore (III. 16, cor. 3) that arcs of great cir-
cles between the pole and circumference are likewise equal.

Scho. 5. The pyramid or cone cut off from another pyramid
or cone by a plane parallel to the base is similar (VI. defs. 8
and 27) to the whole pyi-amid or cone.

Prop. XIX. â€” Theor. â€” If the altitixde of a parallelopipedy
and the length and p rpendicular breadth of its base be all

170 THE ELKMKNTS OF [bOOK VI.

d'vided into parts equal to one another^ the continued p oduct
of the numler of parts in the three lines is the nvmhtr ofctiba
contained in the parcdldopiped, each cube havinrj the side of its
base equal to one (f the parts.

P'irst, suppose the ])nralk'loi)ipefl to 1)8 rcctanccular, Tlien
])laiies panvlU'l to the base i)assiiii; tliroiiu'h the points ot'seetion
of the nh.itude will evidently divide the so'id into as many-
equal solids as there ave parts in the altitude; and each of
tliese partial solids will l)e composed of as many cuius as the
base contains squares, each equal to a base of one of the cubes.
13ut (I. 23, cor. 4) the number of these squares is the product
cf the length and breadth of the base; and hence the entire
number of cubes will be equal to the product of the three
dimensions, the length, breadth, and altitvide.

If the base be not rectanj,ndar, its area (I. 23, cor. 5) will be
the product of its lenjxth and j.erpcndicular breadth; and it is
evident that the product of this by the altitude will be the
number of cubes as before.

Lastly, if the iusistiuii lines be not perpendicular to tlie base,
still the oblique parallelopi])ed is equal (VI. 15, and cor. 1) to
a rectano-ularone of the same altitude; and therefore the num-
ber of cubes will be found as before, by multiplying the area
of the base by the altitude.

Cor. 1. Hence it is evident that the volume, or numerical
solid C07itent, or, as it is also called, the solidify, of a parallelo-
piped is the product of its altittule and the area of its base
both expressed in numbers; and it is ])lain that the same
holds in i-egard to any prism whatever, and also in regard to
cylinders.

Cor. 2. The content of a pyramid or cone is found by multi-
plying the area of the base by the altitude, and tak'ug a third
of the product. For (VI. 17) a pyramid is a third part of a
prism, and (VI. 17) a cone a third jtart of a cylinder, of the
same base and altitude.

An easy method of comjmting the content of a truncated
pyramid or cone, that is, the frustum which remains when a
part is cut from the top by a plane parallel to the base, may be
thus investigated by the help of algebra. The solid cut off is
(VI. 18, scho. 3) similar to the whole; and tlierefore the areas

BOOK VI.] EUCLID AND LEOENDRE. 171

of their bases will be proportional to the squares of their cor-
respondino^ dimensions, and consequently to the squares of
tlieir altitudes. Hence puttins^ V to denote the volume or con-
tent of the frustum, 11 and B the altitudes and base of the
whole solid, and h and h those of the solid cut oif, if we put
^IP to denote B, since B : 6 : : H* : A", or B : 6 : : qW \ qh^^
we shall have (IV. 2, cor. 1) b = qh'^-^ and therefore (VI. 19, cor.
2) the contents of the whole cone and the part cut off are equal
respectively to ^^â– IP and ^qlf \ wherelbre \ = ^q(\ÂĄâ€”h')^ ov^
by resolvinnf the second member into factors, V=:i'/(H-+ HA
+ /r-) {\l-h)=^{qW+q\lh + qh') (H-A). Now ^H^ is equal
to B, qli' to 6, qWh to a mean proportional between them,
and II â€” h to the height of the fi'ustum. Hence, to findthe con-
tent of a truncated pyramid or cone, add to<jet/ter the areas of
its two bases aud a mean proportional between theni^ mxdtiply
the sum by the height of the frustum, anddivide the productby 3,
This admits of convenient modifications in particular cases.
Thus, if the bases be squares of which S and s are sides, and if
a be, the altitude of the frustum, we shall have

V=ia(S'+Ss-f-0=^Â»(3Ss+S-â€” 2Ss+5');
or, Vz^ia{3S5+(Sâ€” s)^}=a{S5+i(Sâ€” s)'}.

Hence, to find the content (f the frustum of a square pyramid^
to the rectangle under the sides of its bases add a third of the
square of their difference, and mxdtiply the sum. by the height.
It would be shown in like manner (V. 25, scho. and VI. IV,
cor. 2), that if II and r be the radii of the bases of the frustum
of a cone, and a its altitude,

N '^Za{Vxr\-\^â€”rY\.

Solidity of cylinder, 3 x 11" x a.

Solidity of cone, R^Xflf.

Solidity of sphere, is 4R'.

Solidity of spherical sector, is 211' X or.

Solidity of spherical segment, when it has two bases, is

f R'4-r)xa+ia';

and when it has but one base,

^R-'xÂ« + ^a'.

Cor. 3. The content of a polyhedron may be found by divid-

172

THE ELEMENTS OF

[book VI.

ing it into pyramids, and adding togetlicr tlieir contents. The
division into pyramids may be matle either by phuies passini^
throng!) tlie vertex of one of the soliil angles, or by planes
passing throngh a point within tlie body.

Piiop. XX. â€” Theor. â€” The surfaces of two similar polyhe-
drons may be divLdedinto the same number cf similar triangles
similarly situated.

This fullows immediately from the definition (VT. def. 8) of
similar bodies bounded by planes, if the sides or faces of the
polyhedron be triangles; and any face in the one, and the cor-
responding face in the other, which are not triangles, are yet
similar, and may be divided (I. 20) into the same number of
similai- tiiangles similarlj' situated.

Cor. Hence it would be shown, as in tlie fourteenth proposi-
tion of the fifth bo()k, that the surfaces of the polyhedions are
proportional to any two of their ^similar triangles ; and there-
fore they are to one another in the duplicate ratio of the
homologous sides of those triangles, that is, of the edges or in-
tersections of the similar planes. Hence also the surfaces are
proportional (V. 14, coi'. 2) to the squares of the edges.

Prop. XXI. â€” Theor. â€” Triangular pyramids are similar, if
two faces in one of them be similar to twj faces in the other^
each to each, and their inclinations equ<d.

Let ABC, abc be the bases, and D, d the vertices of two tn-
angular pyramids, in which ABC, DBC are respectively similar
to abc, dbc, and the inclination of ABC, DBC equal to that of
abc, dbc ^ the pyi'amids are similar.

To demonstrate this, it is sufficient to show that the triangles
j^ AI>D, ACD are similar to ahd^

acd, i\)Y then the solid angles
(Vi. 9, cor. 3) will be equal, each
to each, and (VI. def 8) the pyr-
amids similar. Since the plane
angles at B and b are equal, the
inclinations of ABC, DBC, and
of vhc, dbc, are (VI. 9, cor, 2) equal ; therefore ABl), abd are
equal. Then (hyj..) DB : BC : ; c/i : ic, and BC ; BA : : dc :

BOOK VI.] EUCLID AND LEGENDRE. 173

ba/ whence, ex ceqnn, DB : BA : : clb : ha ; and therefore (V.
6) the trianirles ABD, al>d are equiangular, and consequently
Bimilar ; and it would be proved in the same maimer that
ACD, acd are similar. Therelure (VI. def. 8) the pyramids
are similar.

Cor. Hence triangular pyramids are similar, if three faces of
one of them be respectively similar to three faces of the other.

In the triangular pyramids ABCD, ahcd (see the preceding
figure), let the faces ABC, ABD, DBC be similar to ahc^ abd,
doc, each to each ; the pyramids are similar.

For (V. def 1) AD : DB â– .-. ad : dh, and DB : DC : : cf5 :
dc ; whence, ex aequo, AD : DC : : ad '. dc. Also DC : CB
: : dc : cb, and CB : CA : : cb : ca ; whence, ex ceqi/o, DC :
CA : : dc : ca ; and therefore (V. 5) the triangles ADC, adc
are equiangular, and (VI. 9, cor. 3, and def. 8) the pyramids
are similar.

Prop. XXII. â€” Tiieor. â€” Similar polyhedrons may be divid-
ed nfo the same riumfer of triangular pyratnids, similar, each
to each, and similarly situated.

Let ABCDEFG and ahcdefg be similar polyhedrons, having
the solid angles equal which are marked with the corresponding
large and small letters ; they may be divided into the same
number of similar triangular jjyramids similarly situated.

The surfaces of the polygons may be divided (VI. 20) into
the same number of sim lar triangles, similarly situated; then
planes passing through any two corresponding solid angles, A,
a, and through the sides of all these triangles, except those

forming the solid angles. A, a, will divide the polyhedrons into
triangular pyramids, similar to one another, and similarly sit-
uated.

174

TnE ELEMENTS OF

[book VT.

The pyramids thus formocl have each one solid anole at the
common vertex A or a ; and these solid angles may be of three
classes: Is^ those which have two of their faces coincidins
with faces of one of the polyhedi'ons; 2f/, those which have
only one face coinciding ; and 3o?, those which lie wholly with-
in the solid angle A or a. Now those of the first kind in one
of the polyhedrons are similar to the corresponding ones in the
other, by the corollary to the twenty-first proposition of this
Look ; and those of the second kind by the twenty-first
From the polyhedrons take two of these similar ])yramids, and
the remaining bodies will be similar, as the boundaries common
to them and the pyramids are (VI. 21, and cor.) similar trian-
gles ; and their other boun<laries are similar, being faces of the
proposed polyhedrons. Also the solid angles of the remaining
bodies are equal, as some of them are angles of the primitive
polyhedrons, and the rest are either trihedral angles which are
contained by equal plane angles, or may be divided into such.
From these remaining bodies other similar triangular pyramids
may be taken in a similar manner, and the process may be con-
tinued till only two similar triangular pyramids remain ; and
thus the polyhedrons are resolved into the same number of
eimilar triangular pyramids.

Prop. XXIII. â€” Pitoc. â€” To find the diameter of a given
sphere.

Let A he any point in the surface of the civen sphere, and
take any three jjoints B, C, I) at equal distances from A. De-
ecribe the triangle hcd having he equal to the distance or chord
BC, cd equal to CD, and hd to BD. Find e the center of the

circle described about hcd^ and join he; draw a^perpendicular
to he^ and make ha eqtial to BA ; draw ^perpendicular to 6a,
and af is equal to the diameter of the sphere.

BOOK VI."] EUCLID AND LEGENDRE. 175

Conceive a circle to be described through BCD, and E to be
its center; that circle will evidently be the section of the
sphere by a plane throui;h B, C, D ; and it will be equal to
the circle described about bed. Conceive the diameter AEF
to be drawn, and liA, BE, BF to be joined. Then, in the
right-angled triangles ABE, abe^ the sides AB, BE are respect-
ively equal to ab^ be, and therefore (I. 24, cor. 2) the angles A,
a are equal.

Again : in the right-angled triangles ABF, abf, the angles
A, a are equal, and also the sides AB, ab/ hence (I. 14) the
Bides AF, a/' are equal; that is, a/ is equal to the diameter of
the sphere.

Pkop. XXIV. â€” TiiEOR. â€” T/ie angle of a spherical triangle is
the angle formed by the tangents of the arcs forming the
fpherical angle, and is measured by the arc of a great circle
described from the vertex as apcle, and intercepted by the sides,
produced if necessary.

Let BAC be a spherical angle formed by the arcs AB and
AC, then it is the same as the angle EAD fonncd by the tan-
gents EA ajid DA, and is measured by the arc
of a great circle intercepted by the arcs AB
and AC, produced if necessary. The tangents