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

. (page 1 of 21)
Online LibraryLawrence S. (Lawrence Sluter) BensonGeometry : 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 1 of 21)
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Author of "The Truth of the Bible Upheld,"— London, 18G4; "Geometrical Disquisi-
tions," — London, 1864 ; " Scientific Disquisitions Concerning the
Circle and Ellipse," 1862.
Member of the New York Association for the Advancement of Science and Art ; Hon.
Mem. Phi Eappa Society, University of Georgia ; Brothers' Society,
Yale College, etc., etc.

[all rights reserved.]





Rev. Thomas A. Boone, Professor in Carolina Female College, An-
sonville, North Carolina, writes :

" Your new work on the Elements op Geometry (Book First) haa
been submitted to the President of Carolina Female College. He has
examined it critically, and indorses it as an evident advancement of the
science, in that it simplifies and meets the capacity of learners, retains
all the essentials of the science, and is equally as competent for mental
discipline as the old Eeductio Ad Abmrdum."


.-, 183 Wjjaiaar ^treeln Nw Y. . •


Entered, according to Act of Congress, in the year 1867, by


In the Cleric's Office of the District Court of the United States for the Southern

District of New York.




Sm — In permitting me to inscribe to you this Treatise of Elementary
Geometry, you do me great lionor. Your experience and success as a
Teaclier and an Author will readily enable you to give a full scrutiny to
the design and compass of this volume. Much originality can not be
expected in a subject which has been, for more than two thousand yeare,
enriched by a great number of eminent men ; but in these days of practi-
cability, a modification of this science may be attempted, as you have
yourself thought proper to do, with a view of utilizing the important
principles of Geometry, and presenting them in such a manner that
though " no royal road to Geometry" can be found, the path to a know-
ledge of it may be rendered so clear that the impediments wiU be in tihe
learner himself. And to remove much diflBculty in acquiring an easy
acquaintance with its numerous theorems and problems, I have thought
proper to exclude the inelegant Bedrictio ad absurdum from the methods
of geometrical reasoning which you have expressed — " a consummation
most devoutly to he wislied" and which accomplishment, resulting from my
labors, I now present for the benefit and use of those whose education is
in the future.

I hav3 the honor to be,

Very respectfully, yours,

Lawrence S. Benson.
Vvw York, AprU mh, 1867.


The College op the City of New York, »
Cor. Lexington Avenue and 23d Street. 1

New York, January Zd, 1867.

I have had several interviews with Mr. Lawrence S. Benson on
scientific subjects, and from his conversation, together with tlie Essays
which he has published, I esteem him an excellent scholar and fine
mathematician. He has a desire to establish the Elements of Euclid in
all canes, independently of the demonstration known as the Beductio ad
dbsurdum, " a consummation devoutly to be wished."

Whatever aid or advice you can render him in the furtherance of this
object will tend to the advancement of true science.

Yours truly,


Rooms op the New York Association for the Advancement )
of Science and Art, February )i8th, 18G7. I

Extract from the transactions of the Association for the Advancement
of Science and the Arts :

" At a meeting of the New York Association held February 25, 1867,
a paper on a new method of demonstrating the propositions of Geometry,
denominated the Direct Method, in place of the one now in use,;ijid called
the Indirect Method, was read by Lawrence S. Benson, Esq., which
method the writer proposes to introduce into Schools and Academies.

" After the reading of the paper, and the discussion of its merits, the
subject was referred to Professor Fox, Principal of the Department of
Free Schools of Cooper Union, and to Professor Cleveland Abbe, for ex-
amination and report. It was also moved and carried that the Report
when received be referred to the Section on Physical Science for final

" The Section, after reading the Report of Professor Fox, the letter of
Professor Abbe, and the opinion of Professor Docharty, who had been
invited to examine the work, feel justified in commending this work as
worthy of patronage. Professor Fox in his Report says : ' The design
of arranging the Definitions, Axioms, and Propositions of Geometry, so
as to use only the Direct Method of demonstration, is a good one, and
when arranged in the form of a neat elementary text-book, will doubtless
do much good, as the Direct Method is much more easily understood thaa
the Lidirect Method, by beginners.'

" L. D. Gale, M.D.,
Oen. Sec. of the New York Association for the
Advancement of Science and tlie Art»."


By way of preface, I will state what I have done in this edition, and
explain the reason why I have done so. I have used such propositions
only which are required to substantiate the principal theorems and
problems by which the principles of Geometry have practical applica-
tions in Trigonometry, Surveying, Mechanics, Engineering, Navigation,
and Astronomy. I have generalized the various propositions found in
the school editions of Geometry, and where particular cases arise under
such general propositions, I have given the demonstrations for them. I
have arranged the propositions to give the Direct Method of demonstra-
tion in place of the Beductio ad ahsurdum or Indirect Method.

My reasons for the foregoing changes are obvious to the experienced
mind ; considering the extent and variety of modem education, the time
devoted for the pupils to acquire rudimental knowledge becomes en-
croached upon in order to make them acquainted with its numerous
modifications ; and for the pupils to obtain such knowledge of the rudi-
ments as will enable them to see the practical applications throughout
all their extent and variety, which are very great in these days of ad-
vancement and civilization, the rudiments which were taught centuries
ago must be so abbreviated as to contain the essentials only. When materials
for instructing the mind were scant, there were no opportunities to make
close selection of them ; but now, when those materials are plentiful, a
judicious selection of tlie best becomes imperatively necessary. And
when geometrical principles have become extended by the Algebraic
Analysis, and have been made practicable by Trigonometry, Surveying,
Mechanics, Engineering, Navigation, and Astronomy, the mere mental
exercises, which were regarded so beneiicent by the ancients, are unsuited
for this practical age, which is continually bent on progress, while the
intellect is sufficiently exercised by utilizing modern acquisitions; for
this reason, I have reduced the number of propositions substantiating the
principles of Geometiy, and I have classified them in such a manner that
particular cases are enunciated by general propositions, a change which
is likely to impress on the pupils the accuracy of geometrical principles,
as they will be shown that geometrical principles are the same in all
cases and under every circumstance.

Many of the best geometers have objected to the Beductio ad dbsurdum
in Geometry, while all geometers prefer the Direct Method of demonstra-


tion. Any true proposition is susceptible of being directly demonstrated.
And without entering into the merits or demerits of the Reductio ad ab'
surdum, I have exchided it from geometrical reasoning, and have used
the Direct Method only, a change which agrees with tlie spirit of the age,
and fulfills the requirements of progress. I have omitted the various
diagrams usually put among the definitions of Geometry, because when
a magnitude is properly defined, the learner has a better conception of
it from the definition than any diagram can give him ; and the omission
of the diagrams will assist the mental exercise and cultivate the under-
standing of the learner, which is the great object of geometrical study;
and if the learner be made to draw the diagrams from the definitions, he
will be better instructed than if they be given by the author. The
time is not far distant when geometrical science may be attempted with-
out using diagrams in the demonstrations. The diagrams are auxiliaries
to the mind in the ascertainment of truth ; they are not necessary to the
existence of truth, and " Geometry considers all bodies in a state of ab-
straction, very different from that in which they actually exist, and the
truths it discovers and demonstrates are pure abstractions, hypothetical
truths." Hence, diagrams are like the pebbles used by Indians in count-
ing, or other means of computing before the principle of numeration was
discovered ; that when the intellect of man becomes more highly ex-
panded and cultivated, diagrams will be regarded necessary to the first
conceptions of geometrical knowledge, but altogether unsuited to a high
development of geometrical science.

I am greatly indebted to Hon. S. S. Randall, City Superintendent of
the Board of Education of New York, for many valuable suggestions in
the demonstrations and present arrangement of this work ; and also under
many obligations to Professor Docharty, of the College of the City of New
York ; and to L. D. Gale, M.D., General Secretary of the New York Asso-
ciation for the Advancement of Science and Art, Cooper Union.

Lawrence S. Benson,

61 MoETON Street, City or New Yobk,
April mh, 1867.




1. A definition is the precise term by which one thing is dis-
tinguished from all other things.

2. Mathematics is that science which treats of those abstract
quantities known as numbers, symbols, and magnitudes.

3. Geometry is that branch of Mathematics where the ex-
tensions of magnitudes are considered without regard to the
actual existence of those magnitudes.

4. A m^agnitude has one or more of three dimensions, viz.,
length, breadth, and thickness.

5. Geometers define a point, position without magnitude;
but to give a point position, would entitle it to the three dimen-
sions of magnitude, whereas a point in Geometry expresses no

6. A diagram, represents the abstractions of magnitudes,
whereby their dimensions are determined, and geometrical
reasoning conducted without regard to the actual properties of
those magnitudes.

'7. A line expresses length only, and is capable of two con-
ditions — it can be straight or curved ; when its length is always
in one direction, it is straight ; but when there is a continual
variation in the direction of its length, it is curved, or in brevity
called a curve.

Scholiiim. A straight line can not be defined as having all
its points in the same direction, because the points of a line are
its extremities, and the extremities of a curved line can be


placed on a straight line, and in this case the definition would
not distinguish a straight line from a curve. And if a line be
regarded composed of points, this would infer that a point has
dimension ; but the intersection of lines is a point, which, how-
ever, does not give position to the point, because a line is au
abstraction, and position implies actual existence.

8. A surface expresses an inclosure by not less than three
straight lines, or by one curved line, or by one straight line and
one curved line ; consequently a surface has breadth and length,
and the extremities of surfaces are lines, and the intersection of
one surface with another is a line.

Scho. A plane surface, or sometimes called a plane^ is one in
which any line can be drawn wholly in the surface; and a
c*rved surface is one in which a curve only can be drawn
wholly in the surface in the direction of the curvature.

9. A volume or solid expresses an inclosure made by sur-
faces, and has breadth, length, and thickness ; the extremities
of a volume are surfaces, and the intersection of one volume
with another is a surface.

10. An angle is formed by two straight lines meeting each
other ; the point of intersection of the lines is called the vertex
of the ano-le. When one straicjht line meets another straight
line, so as to make two adjacent angles, these angles are right
angles when they are equal ; and when one angle is greater than
the other angle, the greater angle is an obtuse angle, and the
less angle is an acute angle. The straight line which makes
the two adjacent angles equal is the perpendicular to the other
6trai<;ht line.

11. When two straight lines on the same plane never meet
each other on whichever side they be produced, they are called
parallel lines.

12. Rectilinear surfaces are contained by straight lines, and
are called polygons ; when a polygon has three sides, it is a
triangle ; when it has four sides, it is a quadrilateral ; when
it has five sides, it is z, pentagon ; when it has six sides, it is a
hexagon / when it has seven sides, it is a heptagon / when it
has eight sides, it is an octagon ; when it has nine sides, it is an
enneagon / when it has ten sides, it is a decagon ; and so on,
being distinguished by particular names derived from the Greek


language, denoting the number of angles formed by the sides.
The straight line drawn through two remote angles of a poly-
gon of four or more sides, is a diagonal.

13. When the triangle has its three sides equal, it is equi'
lateral; when two of its sides are equal, it is isosceles; and
when its sides are unequal, it is scalene. When its angles are
equal, it is eqidangular y when one of its angles is a right angle,
it is right-angled ; when one of its angles is an obtuse angle, it
is obtuse-angled ; and when all its angles are acute, it is acute-

14. When a quadrilateral has its opposite sides parallel, it is
2l parallelogram ; when it has two sides only parallel, or none
of its sides parallel, it is a trapezium.

15. When a parallelogram has a right angle, it is a rectangle ;
when it has two adjacent sides equal, but no right angle, it is a
rhombus. When the rectangle has its sides equal, it is a
square / and when its opposite sides only are equal, it is an ob-
long. When the parallelogram has its opposite sides only
equal, and no right angle, it is a rhomboid.

16. A plane surface contained by one line is a circle when
every part of the line is equally distant from a point in the sur-
face ; the point is the center of the circle, and the line is the

17. The straight line drawn from the center to the circum-
ference is the radius ; the straight line drawn from one part of
the circumference through the center to another part of the
circumference is the diameter, which divides the circle and cir-
cumference each into two equal parts. When the straight line
does not pass through the center, it is a chord.

18. That portion of the circle contained by the semicircnm-
ference and diameter is a semicircle; and that portion con-
tained by the chord and a part of the circumference is a seg-
ment ; a part of the circumference is an arc.

19. If the vertex of an angle be the center of a circle, that
part of the circumference intercepted by the sides of the angle
will give the value of the angle ; hence, the angle is measured
by an arc when its vertex is the center of the circle, liut
when the vertex is in the circumference, the angle is subtended
by the arc intercepted by its sides ; hence, equal angles will be


measured by equal arcs, and subtended by equal arcs ; therefore
equal arcs measure or subtend equal angles.

20. Two arcs are supplementary when both together are
equivalent to the semicircumferenoe. And two angles are sup-
plementary when both together are equivalent to two right
angles, and complementary when equivalent to one right

21. Things are equal when they have equal magnitudes and
when they coincide in all respects; and are equivalent when
they have equal magnitudes, but do not coincide in all respects.

22. The term, each to each, or sometimes respectively, is a
limiting expression, and is used to denote the equality of lines
or magnitudes taken in the same order; for without this quali-
fication, two lines or magnitudes said to be equal to two other
lines or magnitudes, would imply that their sums are equal,
when it would be desirous of meaning that they are equal in
the same order in which they are expressed — a difference very
important in the demonstration of a proposition.

23. A proposition is demonstrated by superpositio7i when
one figure is supposed applied to another, which is done in the
first case of the third proposition of this book.

24. One proposition is the converse of another when, in the
language of logic, the subject of the latter is the predicate of
the former, and the predicate of the latter is the subject of the


1. From the foregoing definitions, it is shown that the
straight line and curve have certain relations, uses, and prop-
erties which are important to be known. And in order that
these relations, uses, and properties may be satisfactorily inter-
preted, there are certain terms, expressive of certain facts or
states of knowledge, by means of which the mind intuitively
perceives a connection between the things known and those for
elucidation, such as axioms^ hypotheses^ and postulates/ as
demonstrations, theorems, problems, and lemmas,' as corollaries
and scholiums. With the assistance of these, the mind is
carried step by step in all its investigation of extension, and is
able to discover by such investigation the properties, uses, and
relations of geometrical magnitudes. They are the data by


which the hidden truths are revealed. Upon them a system of
logic or argumentation is conducted, and by the conformity of
the arguments and conclusions with the accepted truths, we
have the science of Geometry.

2. Proposition in Geometry is a general term, expressing the
subjects to be considered, and is either a problem or theorem.
When it is the first, there is something required to be per-
formed, such as drawing a line or constructing a figure ; and
whatever points, lines, angles, or other magnitudes are given to
efiect the purpose, they are the data of the problem ; and when
it is the latter, a truth is proposed for demonstration, and
whatever is assumed or admitted to be true, and from which
the proof is to be derived, is the hypothesis.

3. Demonstration consists in evident deductions from clear
premises, whereby the conclusion corroborates the premises and
shows the argumentativeness of the deductions. In the course
of demonstration, reference is often made to some previous
proposition or definition.

4. Sometimes inferences arise involving another principle,
but do not require any long process of reasoning to establish
their truth — these are corollaries. Any remark made from the
demonstration of a proposition is a scholium. A proposition
which is preparatory to one or more propositions, and is of no
other use, is a lemma.

5. And for the establishment of a proposition, there are four
things required, viz. : the general enunciation, the particular
enunciation, the construction, and the demonstration.

6. The hypotheses of demonstration are known as axiom, and
postulate; the former is assumed to prove the truth of a theorem,
and the latter is granted to pei'form the requisites of a

7. An axiom is so evidently clear, that no process of reason-
ing can make it more clear ; its truth is so easily recognized by
the human mind, that so soon as the terms by which it is ex-
pressed are understood, it is admitted; for instance, it is as-
sumed as


1. Things which are equal to the same, or to equals, are equal
to one another.


2. If equals or the same be added to equals, the wholes are

3. If equals or the same be taken from equals, the remainders
are equal.

4. If equals or the same be added to unequals, the wholes
are unequal.

5. If equals or the same be taken from unequals, the re-
mainders are unequal.

6. Things which are doubles of the same, or of equals, are
equal to one another.

I. Things which are halves of the same, or of equals, are
equal to one another.

8. Magnitudes which exactly coincide with one another are

9. The whole is greater than its part.

10. The whole is equal to all its parts taken together.

II. All right angles are equal to one another.

12. If a straight line meet two other straight lines which are
in the same plane, so as to make the two interior angles on the
same side of it, taken together, less than two right angles, these
straight lines shall at length meet upon that side, if they be
continually produced.

These are the self-evident truths used by Euclid for geomet-
rical demonstration ; but if the first eleven be considered for
awhile, it will be seen that they can be reduced to two general
axioms, viz., things which are equal to the same are equal, and
things which are not equal to the same are unequal ; because
when we add, subtract, multiply, or divide equals, the equality
in each case is not destroyed; hence in each case equal to
one another. And when we add unequals to or subtract un-
equals from equals, the sums or remainders are not equal to the
same, hence unequal to one another. And magnitudes which
exactly coincide with one another are equal to the same, hence
equal ; a whole and a part are not equal to the same, hence are
unequal ; Avhile a whole and all its parts are equal to the same,
hence are equal. From the definition of right angles, it is
seen that when a straight line meets another straight line, so as
to make the two adjacent angles formed by them equal to one


another, the two adjacent angles are right angles; then these
two right angles are equal ; and since all right angles agree with
the definition, they are equal to the same thing, hence equal to
one another. But the twelfth axiom is not self-evident, be-
cause the converse has been demonstrated, viz.: that two
straisrht lines which meet one another make with any third line
the interior angles less than two right angles. Geometers perceiv-
ing this blemish in the Elements of Euclid, have endeavored in
many ways to remove it, but without complete success. They
employed three methods for this purpose : 1, By adopting a
new definition of parallel lines. 2. By introducing a new
axiom. 3. By reasoning from the definition of parallel lines,
and the properties of lines already demonstrated.* The diffi-
culty with parallel lines is, that geometers have confounded a
definition with a proposition. Definition 11 is perfectly legiti-
mate, as it simply defines what kind of lines are parallel ; but
when it is inferred from it that these lines are equally distant
from each other, this is no axiomic inference, because the curve
and its asymptote are two lines which never meet, however far
they be produced on the same plane, but they are not equally
distant from each other ; hence the inference that parallel lines
are equally distant, embodies a question which requires a dem-
onstration to establish ; and to establish this question has given
perplexity to geometers, for though they have proven the lines
equally distant at particular points, they have not proven them
so at every point; and here consists the incompleteness of their
demonstrations, and here is required some general demonstra-
tion which will embrace every part of the lines, however so far
they be produced on the same plane.f

8. A postulate is a problem so easy to perform that it does
not require any explanation of the manner of doing it, so that
the geometer reasonably expects the method to be known ; for
instance, it is granted as —

* See notes to Playfair's Euclid, Legendre's Geometry, Leslie's Geom-
etry, the ea-cursus to the tirst book of Camerer's Euclid, Berlin, 1825 ;

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Online LibraryLawrence S. (Lawrence Sluter) BensonGeometry : 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 1 of 21)