BOOK I AND ITS PROPOSITIONS
Having considered the nature of the geometry that we have inherited, and
some of the opportunities for improving upon the methods of presenting
it, the next question that arises is the all-important one of the
subject matter, What shall geometry be in detail? Shall it be the text
or the sequence of Euclid? Few teachers have any such idea at the
present time. Shall it be a mere dabbling with forms that are seen in
mechanics or architecture, with no serious logical sequence? This is an
equally dangerous extreme. Shall it be an entirely new style of geometry
based upon groups of motions? This may sometime be developed, but as yet
it exists in the future if it exists at all, since the recent efforts in
this respect are generally quite as ill suited to a young pupil as is
Euclid's "Elements" itself.
No one can deny the truth of M. Bourlet's recent assertion that
"Industry, daughter of the science of the nineteenth century, reigns
to-day the mistress of the world; she has transformed all ancient
methods, and she has absorbed in herself almost all human activity."[57]
Neither can one deny the justice of his comparison of Euclid with a
noble piece of Gothic architecture and of his assertion that as modern
life demands another type of building, so it demands another type of
geometry.
But what does this mean? That geometry is to exist merely as it touches
industry, or that bad architecture is to replace the good? By no means.
A building should to-day have steam heat and elevators and electric
lights, but it should be constructed of just as enduring materials as
the Parthenon, and it should have lines as pleasing as those of a Gothic
facade. Architecture should still be artistic and construction should
still be substantial, else a building can never endure. So geometry must
still exemplify good logic and must still bring to the pupil a feeling
of exaltation, or it will perish and become a mere relic in the museum
of human culture.
What, then, shall the propositions of geometry be, and in what manner
shall they answer to the challenge of the industrial epoch in which we
live? In reply, they must be better adapted to young minds and to all
young minds than Euclid ever intended his own propositions to be.
Furthermore, they must have a richness of application to pure geometry,
in the way of carefully chosen exercises, that Euclid never attempted.
And finally, they must have application to this same life of industry of
which we have spoken, whenever this can really be found, but there must
be no sham and pretense about it, else the very honesty that permeated
the ancient geometry will seem to the pupil to be wanting in the whole
subject.[58]
Until some geometry on a radically different basis shall appear, and of
this there is no very hopeful sign at present, the propositions will be
the essential ones of Euclid, excluding those that may be considered
merely intuitive, and excluding all that are too difficult for the pupil
who to-day takes up their study. The number will be limited in a
reasonable way, and every genuine type of application will be placed
before the teacher to be used as necessity requires. But a fair amount
of logic will be retained, and the effort to make of geometry an empty
bauble of a listless mind will be rejected by every worthy teacher. What
the propositions should be is a matter upon which opinions may justly
differ; but in this chapter there is set forth a reasonable list for
Book I, arranged in a workable sequence, and this list may fairly be
taken as typical of what the American school will probably use for many
years to come. With the list is given a set of typical applications, and
some of the general information that will add to the interest in the
work and that should form part of the equipment of the teacher.
An ancient treatise was usually written on a kind of paper called
papyrus, made from the pith of a large reed formerly common in Egypt,
but now growing luxuriantly only above Khartum in Upper Egypt, and near
Syracuse in Sicily; or else it was written on parchment, so called from
Pergamos in Asia Minor, where skins were first prepared in parchment
form; or occasionally they were written on ordinary leather. In any case
they were generally written on long strips of the material used, and
these were rolled up and tied. Hence we have such an expression as
"keeping the roll" in school, and such a word as "volume," which has in
it the same root as "involve" (to roll in), and "evolve" (to roll out).
Several of these rolls were often necessary for a single treatise, in
which case each was tied, and all were kept together in a receptacle
resembling a pail, or in a compartment on a shelf. The Greeks called
each of the separate parts of a treatise _biblion_ ([Greek: biblion]), a
word meaning "book." Hence we have the books of the Bible, the books of
Homer, and the books of Euclid. From the same root, indeed, comes Bible,
bibliophile (booklover), bibliography (list of books), and kindred
words. Thus the books of geometry are the large chapters of the subject,
"chapter" being from the Latin _caput_ (head), a section under a new
heading. There have been efforts to change "books" to "chapters," but
they have not succeeded, and there is no reason why they should succeed,
for the term is clear and has the sanction of long usage.
THEOREM. _If two lines intersect, the vertical angles are equal._
This was Euclid's Proposition 15, being put so late because he based the
proof upon his Proposition 13, now thought to be best taken without
proof, namely, "If a straight line set upon a straight line makes
angles, it will make either two right angles or angles equal to two
right angles." It is found to be better pedagogy to assume that this
follows from the definition of straight angle, with reference, if
necessary, to the meaning of the sum of two angles. This proposition on
vertical angles is probably the best one with which to begin geometry,
since it is not so evident as to seem to need no proof, although some
prefer to rank it as semiobvious, while the proof is so simple as easily
to be understood. Eudemus, a Greek who wrote not long before Euclid,
attributed the discovery of this proposition to Thales of Miletus (_ca._
640-548 B.C.), one of the Seven Wise Men of Greece, of whom Proclus
wrote: "Thales it was who visited Egypt and first transferred to
Hellenic soil this theory of geometry. He himself, indeed, discovered
much, but still more did he introduce to his successors the principles
of the science."
The proposition is the only basal one relating to the intersection of
two lines, and hence there are no others with which it is necessarily
grouped. This is the reason for placing it by itself, followed by the
congruence theorems.
There are many familiar illustrations of this theorem. Indeed, any two
crossed lines, as in a pair of shears or the legs of a camp stool, bring
it to mind. The word "straight" is here omitted before "lines" in
accordance with the modern convention that the word "line" unmodified
means a straight line. Of course in cases of special emphasis the
adjective should be used.
THEOREM. _Two triangles are congruent if two sides and the included
angle of the one are equal respectively to two sides and the included
angle of the other._
This is Euclid's Proposition 4, his first three propositions being
problems of construction. This would therefore have been his first
proposition if he had placed his problems later, as we do to-day. The
words "congruent" and "equal" are not used as in Euclid, for reasons
already set forth on page 151. There have been many attempts to
rearrange the propositions of Book I, putting in separate groups those
concerning angles, those concerning triangles, and those concerning
parallels, but they have all failed, and for the cogent reason that such
a scheme destroys the logical sequence. This proposition may properly
follow the one on vertical angles simply because the latter is easier
and does not involve superposition.
As far as possible, Euclid and all other good geometers avoid the proof
by superposition. As a practical test superposition is valuable, but as
a theoretical one it is open to numerous objections. As Peletier pointed
out in his (1557) edition of Euclid, if the superposition of lines and
figures could freely be assumed as a method of demonstration, geometry
would be full of such proofs. There would be no reason, for example, why
an angle should not be constructed equal to a given angle by superposing
the given angle on another part of the plane. Indeed, it is possible
that we might then assume to bisect an angle by imagining the plane
folded like a piece of paper. Heath (1908) has pointed out a subtle
defect in Euclid's proof, in that it is said that because two lines are
equal, they can be made to coincide. Euclid says, practically, that if
two lines can be made to coincide, they are equal, but he does not say
that if two straight lines are equal, they can be made to coincide. For
the purposes of elementary geometry the matter is hardly worth bringing
to the attention of a pupil, but it shows that even Euclid did not cover
every point.
Applications of this proposition are easily found, but they are all very
much alike. There are dozens of measurements that can be made by simply
constructing a triangle that shall be congruent to another triangle. It
seems hardly worth the while at this time to do more than mention one
typical case,[59] leaving it to teachers who may find it desirable to
suggest others to their pupils.
[Illustration]
Wishing to measure the distance across a river, some boys
sighted from _A_ to a point _P_. They then turned and measured
_AB_ at right angles to _AP_. They placed a stake at _O_,
halfway from _A_ to _B_, and drew a perpendicular to _AB_ at
_B_. They placed a stake at _C_, on this perpendicular, and in
line with _O_ and _P_. They then found the width by measuring
_BC_. Prove that they were right.
This involves the ranging of a line, and the running of a line at right
angles to a given line, both of which have been described in Chapter IX.
It is also fairly accurate to run a line at any angle to a given line by
sighting along two pins stuck in a protractor.
THEOREM. _Two triangles are congruent if two angles and the included
side of the one are equal respectively to two angles and the included
side of the other._
Euclid combines this with his Proposition 26:
If two triangles have the two angles equal to two angles
respectively, and one side equal to one side, namely, either
the side adjoining the equal angles, or that subtending one of
the equal angles, they will also have the remaining sides equal
to the remaining sides, and the remaining angle to the
remaining angle.
He proves this cumbersome statement without superposition, desiring to
avoid this method, as already stated, whenever possible. The proof by
superposition is old, however, for Al-Nair[=i]z[=i][60] gives it and
ascribes it to some earlier author whose name he did not know. Proclus
tells us that "Eudemus in his geometrical history refers this theorem to
Thales. For he says that in the method by which they say that Thales
proved the distance of ships in the sea, it was necessary to make use of
this theorem." How Thales did this is purely a matter of conjecture, but
he might have stood on the top of a tower rising from the level shore,
or of such headlands as abound near Miletus, and by some simple
instrument sighted to the ship. Then, turning, he might have sighted
along the shore to a point having the same angle of declination, and
then have measured the distance from the tower to this point. This
seems more reasonable than any of the various plans suggested, and it is
found in so many practical geometries of the first century of printing
that it seems to have long been a common expedient. The stone astrolabe
from Mesopotamia, now preserved in the British Museum, shows that such
instruments for the measuring of angles are very old, and for the
purposes of Thales even a pair of large compasses would have answered
very well. An illustration of the method is seen in Belli's work of
1569, as here shown. At the top of the picture a man is getting the
angle by means of the visor of his cap; at the bottom of the picture a
man is using a ruler screwed to a staff.[61] The story goes that one of
Napoleon's engineers won the imperial favor by quickly measuring the
width of a stream that blocked the progress of the army, using this very
method.
[Illustration: SIXTEENTH-CENTURY MENSURATION
Belli's "Del Misurar con la Vista," Venice, 1569]
This proposition is the reciprocal or dual of the preceding one. The
relation between the two may be seen from the following arrangement:
Two triangles are congruent if two _sides_ and the included
_angle_ of the one are equal respectively to two _sides_ and
the included _angle_ of the other.
Two triangles are congruent if two _angles_ and the included
_side_ of the one are equal respectively to two _angles_ and
the included _side_ of the other.
In general, to every proposition involving _points_ and _lines_ there is
a reciprocal proposition involving _lines_ and _points_ respectively
that is often true,--indeed, that is always true in a certain line of
propositions. This relation is known as the Principle of Reciprocity or
of Duality. Instead of points and lines we have here angles (suggested
by the vertex points) and lines. It is interesting to a class to have
attention called to such relations, but it is not of sufficient
importance in elementary geometry to justify more than a reference here
and there. There are other dual features that are seen in geometry
besides those given above.
THEOREM. _In an isosceles triangle the angles opposite the equal sides
are equal._
This is Euclid's Proposition 5, the second of his theorems, but he adds,
"and if the equal straight lines be produced further, the angles under
the base will be equal to one another." Since, however, he does not use
this second part, its genuineness is doubted. He would not admit the
common proof of to-day of supposing the vertical angle bisected, because
the problem about bisecting an angle does not precede this proposition,
and therefore his proof is much more involved than ours. He makes
_CX_ = _CY_, and proves [triangles]_XBC_ and _YAC_ congruent,[62] and
also [triangles]_XBA_ and _YAB_ congruent. Then from [L]_YAC_ he takes
[L]_YAB_, leaving [L]_BAC_, and so on the other side, leaving [L]_CBA_,
these therefore being equal.
[Illustration]
This proposition has long been called the _pons asinorum_, or bridge of
asses, but no one knows where or when the name arose. It is usually
stated that it came from the fact that fools could not cross this
bridge, and it is a fact that in the Middle Ages this was often the
limit of the student's progress in geometry. It has however been
suggested that the name came from Euclid's figure, which resembles the
simplest type of a wooden truss bridge. The name is applied by the
French to the Pythagorean Theorem.
Proclus attributes the discovery of this proposition to Thales. He also
says that Pappus (third century A.D.), a Greek commentator on Euclid,
proved the proposition as follows:
Let _ABC_ be the triangle, with _AB_ = _AC_. Conceive of this
as two triangles; then _AB_ = _AC_, _AC_ = _AB_, and [L]_A_ is
common; hence the [triangles]_ABC_ and _ACB_ are congruent, and
[L]_B_ of the one equals [L]_C_ of the other.
This is a better plan than that followed by some textbook writers of
imagining [triangle]_ABC_ taken up and laid down on _itself_. Even to
lay it down on its "trace" is more objectionable than the plan of
Pappus.
THEOREM. _If two angles of a triangle are equal, the sides opposite the
equal angles are equal, and the triangle is isosceles._
The statement is, of course, tautological, the last five words being
unnecessary from the mathematical standpoint, but of value at this stage
of the student's progress as emphasizing the nature of the triangle.
Euclid stated the proposition thus, "If in a triangle two angles be
equal to one another, the sides which subtend the equal angles will also
be equal to one another." He did not define "subtend," supposing such
words to be already understood. This is the first case of a converse
proposition in geometry. Heath distinguishes the logical from the
geometric converse. The logical converse of Euclid I, 5, would be that
"_some_ triangles with two angles equal are isosceles," while the
geometric converse is the proposition as stated. Proclus called
attention to two forms of converse (and in the course of the work, but
not at this time, the teacher may have to do the same): (1) the complete
converse, in which that which is given in one becomes that which is to
be proved in the other, and vice versa, as in this and the preceding
proposition; (2) the partial converse, in which two (or even more)
things may be given, and a certain thing is to be proved, the converse
being that one (or more) of the preceding things is now given, together
with what was to be proved, and the other given thing is now to be
proved. Symbolically, if it is given that _a_ = _b_ and _c_ = _d_, to
prove that _x_ = _y_, the partial converse would have given _a_ = _b_
and _x_ = _y_, to prove that _c_ = _d_.
Several proofs for the proposition have been suggested, but a careful
examination of all of them shows that the one given below is, all things
considered, the best one for pupils beginning geometry and following
the sequence laid down in this chapter. It has the sanction of some of
the most eminent mathematicians, and while not as satisfactory in some
respects as the _reductio ad absurdum_, mentioned below, it is more
satisfactory in most particulars. The proof is as follows:
[Illustration:
=Given the triangle ABC, with the angle A equal to the angle B.=]
_To prove that_ _AC_ = _BC_.
=Proof.= Suppose the second triangle _A'B'C'_ to be an exact
reproduction of the given triangle _ABC_.
Turn the triangle _A'B'C'_ over and place it upon _ABC_ so that _B'_
shall fall on _A_ and _A'_ shall fall on _B_.
Then _B'A'_ will coincide with _AB_.
Since [L]_A'_ = [L]_B'_, Given
and [L]_A_ = [L]_A'_, Hyp.
[therefore][L]_A_ = [L]_B'_.
[therefore]_B'C'_ will lie along _AC_.
Similarly, _A'C'_ will lie along _BC_.
Therefore _C'_ will fall on both _AC_ and _BC_, and hence at their
intersection.
[therefore]_B'C'_ = _AC_.
But _B'C'_ was made equal to _BC_.
[therefore]_AC_ = _BC_. Q.E.D.
If the proposition should be postponed until after the one on the sum of
the angles of a triangle, the proof would be simpler, but it is
advantageous to couple it with its immediate predecessor. This simpler
proof consists in bisecting the vertical angle, and then proving the
two triangles congruent. Among the other proofs is that of the _reductio
ad absurdum_, which the student might now meet, but which may better be
postponed. The phrase _reductio ad absurdum_ seems likely to continue in
spite of the efforts to find another one that is simpler. Such a proof
is also called an indirect proof, but this term is not altogether
satisfactory. Probably both names should be used, the Latin to explain
the nature of the English. The Latin name is merely a translation of one
of several Greek names used by Aristotle, a second being in English
"proof by the impossible," and a third being "proof leading to the
impossible." If teachers desire to introduce this form of proof here, it
must be borne in mind that only one supposition can be made if such a
proof is to be valid, for if two are made, then an absurd conclusion
simply shows that either or both must be false, but we do not know which
is false, or if only one is false.
THEOREM. _Two triangles are congruent if the three sides of the one are
equal respectively to the three sides of the other._
It would be desirable to place this after the fourth proposition
mentioned in this list if it could be done, so as to get the triangles
in a group, but we need the fourth one for proving this, so that the
arrangement cannot be made, at least with this method of proof.
This proposition is a "partial converse" of the second
proposition in this list; for if the triangles are _ABC_ and
_A'B'C'_, with sides _a_, _b_, _c_ and _a'_, _b'_, _c'_, then
the second proposition asserts that if _b_ = _b'_, _c_ = _c'_,
and [L]_A_ = [L]_A'_, then _a_ = _a'_ and the triangles are
congruent, while this proposition asserts that if _a_ = _a'_,
_b_ = _b'_, and _c_ = _c'_, then [L]_A_ = [L]_A'_ and the
triangles are congruent.
The proposition was known at least as early as Aristotle's time. Euclid
proved it by inserting a preliminary proposition to the effect that it
is impossible to have on the same base _AB_ and the same side of it two
different triangles _ABC_ and _ABC'_, with _AC_ = _AC'_, and
_BC_ = _BC'_. The proof ordinarily given to-day, wherein the two
triangles are constructed on opposite sides of the base, is due to Philo
of Byzantium, who lived after Euclid's time but before the Christian
era, and it is also given by Proclus. There are really three cases, if
one wishes to be overparticular, corresponding to the three pairs of
equal sides. But if we are allowed to take the longest side for the
common base, only one case need be considered.
Of the applications of the proposition one of the most important relates
to making a figure rigid by means of diagonals. For example, how many
diagonals must be drawn in order to make a quadrilateral rigid? to make
a pentagon rigid? a hexagon? a polygon of _n_ sides. In particular, the
following questions may be asked of a class:
[Illustration]
1. Three iron rods are hinged at the extremities, as shown in
this figure. Is the figure rigid? Why?
2. Four iron rods are hinged, as shown in this figure. Is the
figure rigid? If not, where would you put in the fifth rod to
make it rigid? Prove that this would accomplish the result.
[Illustration]
Another interesting application relates to the most ancient form of
leveling instrument known to us. This kind of level is pictured on very
ancient monuments, and it is still used in many parts of the world.
Pupils in manual training may make such an instrument, and indeed one is
easily made out of cardboard. If the plumb line passes through the
mid-point of the base, the two triangles are congruent and the plumb
line is then perpendicular to the base. In other words, the base is
level. With such simple primitive instruments, easily made by pupils, a
good deal of practical mathematical work can be performed. The
interesting old illustration here given shows how this form of level was
used three hundred years ago.
[Illustration: EARLY METHODS OF LEVELING
Pomodoro's "La geometria prattica," Rome, 1624]
[Illustration]
Teachers who seek for geometric figures in practical mechanics will find
this proposition illustrated in the ordinary hoisting apparatus of the
kind here shown. From the study of such forms and of simple roof and
bridge trusses, a number of the usual properties of the isosceles
triangle may be derived.
THEOREM. _The sum of two lines drawn from a given point to the
extremities of a given line is greater than the sum of two other lines
similarly drawn, but included by them._
It should be noted that the words "the extremities of" are necessary,
for it is possible to draw from a certain point within a certain
triangle two lines to the base such that their sum is greater than the
sum of the other two sides.
[Illustration]
Thus, in the right triangle _ABC_ draw any line _CX_ from _C_
to the base. Make _XY_ = _AC_, and _CP_ = _PY_. Then it is
easily shown that _PB_ + _PX_ > _CB_ + _CA_.
[Illustration]
It is interesting to a class to have a teacher point out that,
in this figure, _AP_ + _PB_ < _AC_ + _CB_, and _AP'_ + _P'B_ <
_AP_ + _PB_, and that the nearer _P_ gets to _AB_, the shorter
_AP_ + _PB_ becomes, the limit being the line _AB_. From this
we may _infer_ (although we have not proved) that "a straight
line (_AB_) is the shortest path between two points."
THEOREM. _Only one perpendicular can be drawn to a given line from a
given external point._
THEOREM. _Two lines drawn from a point in a perpendicular to a given
line, cutting off on the given line equal segments from the foot of the
perpendicular, are equal and make equal angles with the perpendicular._
THEOREM. _Of two lines drawn from the same point in a perpendicular to a
given line, cutting off on the line unequal segments from the foot of
the perpendicular, the more remote is the greater._
THEOREM. _The perpendicular is the shortest line that can be drawn to a
straight line from a given external point._
These four propositions, while known to the ancients and incidentally
used, are not explicitly stated by Euclid. The reason seems to be that
he interspersed his problems with his theorems, and in his Propositions
11 and 12, which treat of drawing a perpendicular to a line, the
essential features of these theorems are proved. Further mention will be
made of them when we come to consider the problems in question. Many
textbook writers put the second and third of the four before the first,
forgetting that the first is assumed in the other two, and hence should
precede them.
THEOREM. _Two right triangles are congruent if the hypotenuse and a side
of the one are equal respectively to the hypotenuse and a side of the
other._
THEOREM. _Two right triangles are congruent if the hypotenuse and an
adjacent angle of the one are equal respectively to the hypotenuse and
an adjacent angle of the other._
As stated in the notes on the third proposition in this sequence,
Euclid's cumbersome Proposition 26 covers several cases, and these two
among them. Of course this present proposition could more easily be
proved after the one concerning the sum of the angles of a triangle, but
the proof is so simple that it is better to leave the proposition here
in connection with others concerning triangles.
THEOREM. _Two lines in the same plane perpendicular to the same line
cannot meet, however far they are produced._
This proposition is not in Euclid, and it is introduced for educational
rather than for mathematical reasons. Euclid introduced the subject by
the proposition that, if alternate angles are equal, the lines are
parallel. It is, however, simpler to begin with this proposition, and
there is some advantage in stating it in such a way as to prove that
parallels exist before they are defined. The proposition is properly
followed by the definition of parallels and by the postulate that has
been discussed on page 127.
A good application of this proposition is the one concerning a method of
drawing parallel lines by the use of a carpenter's square. Here two
lines are drawn perpendicular to the edge of a board or a ruler, and
these are parallel.
THEOREM. _If a line is perpendicular to one of two parallel lines, it is
perpendicular to the other also._
This, like the preceding proposition, is a special case under a later
theorem. It simplifies the treatment of parallels, however, and the
beginner finds it easier to approach the difficulties gradually, through
these two cases of perpendiculars. It should be noticed that this is an
example of a partial converse, as explained on page 175. The preceding
proposition may be stated thus: If _a_ is [perp] to _x_ and _b_ is
[perp] to _x_, then _a_ is || to _b_. This proposition may be stated
thus: If _a_ is [perp] to _x_ and _a_ is || to _b_, then _b_ is [perp]
to _x_. This is, therefore, a partial converse.
These two propositions having been proved, the usual definitions of the
angles made by a transversal of two parallels may be given. It is
unfortunate that we have no name for each of the two groups of four
equal angles, and the name of "transverse angles" has been suggested.
This would simplify the statements of certain other propositions; thus:
"If two parallel lines are cut by a transversal, the transverse angles
are equal," and this includes two propositions as usually given. There
is not as yet, however, any general sanction for the term.
THEOREM. _If two parallel lines are cut by a transversal, the
alternate-interior angles are equal._
Euclid gave this as half of his Proposition 29. Indeed, he gives only
four theorems on parallels, as against five propositions and several
corollaries in most of our American textbooks. The reason for increasing
the number is that each proposition may be less involved. Thus, instead
of having one proposition for both exterior and interior angles, modern
authors usually have one for the exterior and one for the interior, so
as to make the difficult subject of parallels easier for beginners.
THEOREM. _When two straight lines in the same plane are cut by a
transversal, if the alternate-interior angles are equal, the two
straight lines are parallel._
This is the converse of the preceding theorem, and is half of Euclid I,
28, his theorem being divided for the reason above stated. There are
several typical pairs of equal or supplemental angles that would lead to
parallel lines, of which Euclid uses only part, leaving the other cases
to be inferred. This accounts for the number of corollaries in this
connection in later textbooks.
Surveyors make use of this proposition when they wish, without using a
transit instrument, to run one line parallel to another.
[Illustration]
For example, suppose two boys are laying out a tennis court and
they wish to run a line through _P_ parallel to _AB_. Take a
60-foot tape and swing it around _P_ until the other end rests
on _AB_, as at _M_. Put a stake at _O_, 30 feet from _P_ and
_M_. Then take any convenient point _N_ on _AB_, and measure
_ON_. Suppose it equals 20 feet. Then sight from _N_ through
_O_, and put a stake at _Q_ just 20 feet from _O_. Then _P_ and
_Q_ determine the parallel, according to the proposition just
mentioned.
THEOREM. _If two parallel lines are cut by a transversal, the
exterior-interior angles are equal._
This is also a part of Euclid I, 29. It is usually followed by several
corollaries, covering the minor and obvious cases omitted by the older
writers. While it would be possible to dispense with these corollaries,
they are helpful for definite reference in later propositions.
THEOREM. _The sum of the three angles of a triangle is equal to two
right angles._
Euclid stated this as follows: "In any triangle, if one of the sides be
produced, the exterior angle is equal to the two interior and opposite
angles, and the three interior angles of the triangle are equal to two
right angles." This states more than is necessary for the basal fact of
the proposition, which is the constancy of the sum of the angles.
The theorem is one of the three most important propositions in plane
geometry, the other two being the so-called Pythagorean Theorem, and a
proposition relating to the proportionality of the sides of two
triangles. These three form the foundation of trigonometry and of the
mensuration of plane figures.
The history of the proposition is extensive. Eutocius (_ca._ 510 A.D.),
in his commentary on Apollonius, says that Geminus (first century B.C.)
testified that "the ancients investigated the theorem of the two right
angles in each individual species of triangle, first in the equilateral,
again in the isosceles, and afterwards in the scalene triangle." This,
indeed, was the ancient plan, to proceed from the particular to the
general. It is the natural order, it is the world's order, and it is
well to follow it in all cases of difficulty in the classroom.
Proclus (410-485 A.D.) tells us that Eudemus, who lived just before
Euclid (or probably about 325 B.C.), affirmed that the theorem was due
to the Pythagoreans, although this does not necessarily mean to the
actual pupils of Pythagoras. The proof as he gives it consists in
showing that _a_ = _a_', _b_ = _b_', and _a_' + _c_ + _b_' = two right
angles. Since the proposition about the exterior angle of a triangle is
attributed to Philippus of Mende (_ca._ 380 B.C.), the figure given by
Eudemus is probably the one used by the Pythagoreans.
[Illustration]
There is also some reason for believing that Thales (_ca._ 600 B.C.)
knew the theorem, for Diogenes Laertius (_ca._ 200 A.D.) quotes
Pamphilius (first century A.D.) as saying that "he, having learned
geometry from the Egyptians, was the first to inscribe a right triangle
in a circle, and sacrificed an ox." The proof of this proposition
requires the knowledge that the sum of the angles, at least in a right
triangle, is two right angles. The proposition is frequently referred to
by Aristotle.
There have been numerous attempts to prove the proposition without the
use of parallel lines. Of these a German one, first given by Thibaut in
the early part of the eighteenth century, is among the most interesting.
This, in simplified form, is as follows:
[Illustration]
Suppose an indefinite line _XY_ to lie on _AB_. Let it swing
about _A_, counterclockwise, through [L]_A_, so as to lie on
_AC_, as _X'Y'_. Then let it swing about _C_, through [L]_C_,
so as to lie on _CB_, as _X''Y''_. Then let it swing about _B_,
through [L]_B_, so as to lie on _BA_, as _X'''Y'''_. It now
lies on _AB_, but it is turned over, _X'''_ being where _Y_
was, and _Y'''_ where _X_ was. In turning through [Ls]_A_, _B_,
and _C_ it has therefore turned through two right angles.
One trouble with the proof is that the rotation has not been about the
same point, so that it has never been looked upon as other than an
interesting illustration.
Proclus tried to prove the theorem by saying that, if we have two
perpendiculars to the same line, and suppose them to revolve about their
feet so as to make a triangle, then the amount taken from the right
angles is added to the vertical angle of the triangle, and therefore the
sum of the angles continues to be two right angles. But, of course, to
prove his statement requires a perpendicular to be drawn from the vertex
to the base, and the theorem of parallels to be applied.
Pupils will find it interesting to cut off the corners of a paper
triangle and fit the angles together so as to make a straight angle.
This theorem furnishes an opportunity for many interesting exercises,
and in particular for determining the third angle when two angles of a
triangle are given, or the second acute angle of a right triangle when
one acute angle is given.
Of the simple outdoor applications of the proposition, one of the best
is illustrated in this figure.
[Illustration]
To ascertain the height of a tree or of the school building,
fold a piece of paper so as to make an angle of 45 deg.. Then walk
back from the tree until the top is seen at an angle of 45 deg.
with the ground (being therefore careful to have the base of
the triangle level). Then the height _AC_ will equal the base
_AB_, since _ABC_ is isosceles. A paper protractor may be used
for the same purpose.
Distances can easily be measured by constructing a large equilateral
triangle of heavy pasteboard, and standing pins at the vertices for the
purpose of sighting.
[Illustration]
To measure _PC_, stand at some convenient point _A_ and sight
along _APC_ and also along _AB_. Then walk along _AB_ until a
point _B_ is reached from which _BC_ makes with _BA_ an angle
of the triangle (60 deg.). Then _AC_ = _AB_, and since _AP_ can be
measured, we can find _PC_.
Another simple method of measuring a distance _AC_ across a stream is
shown in this figure.
[Illustration]
Measure the angle _CAX_, either in degrees, with a protractor,
or by sighting along a piece of paper and marking down the
angle. Then go along _XA_ produced until a point _B_ is reached
from which _BC_ makes with _A_ an angle equal to half of angle
_CAX_. Then it is easily shown that _AB_ = _AC_.
A navigator uses the same principle when he "doubles the angle on the
bow" to find his distance from a lighthouse or other object.
[Illustration]
If he is sailing on the course _ABC_ and notes a lighthouse _L_
when he is at _A_, and takes the angle _A_, and if he notices
when the angle that the lighthouse makes with his course is
just twice the angle noted at _A_, then _BL_ = _AB_. He has
_AB_ from his log (an instrument that tells how far a ship goes
in a given time), so he knows _BL_. He has "doubled the angle
on the bow" to get this distance.
It would have been possible for Thales, if he knew this proposition, to
have measured the distance of the ship at sea by some such device as
this:
[Illustration]
Make a large isosceles triangle out of wood, and, standing at
_T_, sight to the ship and along the shore on a line _TA_,
using the vertical angle of the triangle. Then go along _TA_
until a point _P_ is reached, from which _T_ and _S_ can be
seen along the sides of a base angle of the triangle. Then
_TP_ = _TS_. By measuring _TB_, _BS_ can then be found.
THEOREM. _The sum of two sides of a triangle is greater than the third
side, and their difference is less than the third side_.
If the postulate is assumed that a straight line is the shortest path
between two points, then the first part of this theorem requires no
further proof, and the second part follows at once from the axiom of
inequalities. This seems the better plan for beginners, and the
proposition may be considered as semiobvious. Euclid proved the first
part, not having assumed the postulate. Proclus tells us that the
Epicureans (the followers of Epicurus, the Greek philosopher, 342-270
B.C.) used to ridicule this theorem, saying that even an ass knew it,
for if he wished to get food, he walked in a straight line and not along
two sides of a triangle. Proclus replied that it was one thing to know
the truth and another thing to prove it, meaning that the value of
geometry lay in the proof rather than in the mere facts, a thing that
all who seek to reform the teaching of geometry would do well to keep in
mind. The theorem might simply appear as a corollary under the postulate
if it were of any importance to reduce the number of propositions one
more.
If the proposition is postponed until after those concerning the
inequalities of angles and sides of a triangle, there are several good
proofs.
[Illustration]
For example, produce _AC_ to _X_,
making
_CX_ = _CB_.
Then [L]_X_ = [L]_XBC_.
[therefore] [L]_XBA_ > [L]_X_.
[therefore] _AX_ > _AB_.
[therefore] _AC_ + _CB_ > _AB_.
The above proof is due to Euclid. Heron of Alexandria (first century
A.D.) is said by Proclus to have given the following:
[Illustration]
Let _CX_ bisect [L]_C_.
Then [L]_BXC_ > [L]_ACX_.
[therefore] [L]_BXC_ > [L]_XCB_.
[therefore] _CB_ > _XB_.
Similarly, _AC_ > _AX_.
Adding, _AC_ + _CB_ > _AB_.
THEOREM. _If two sides of a triangle are unequal, the angles opposite
these sides are unequal, and the angle opposite the greater side is the
greater._
Euclid stated this more briefly by saying, "In any triangle the greater
side subtends the greater angle." This is not so satisfactory, for there
may be no greater side.
THEOREM. _If two angles of a triangle are unequal, the sides opposite
these angles are unequal, and the side opposite the greater angle is the
greater._
Euclid also stated this more briefly, but less satisfactorily, thus, "In
any triangle the greater angle is subtended by the greater side."
Students should have their attention called to the fact that these two
theorems are reciprocal or dual theorems, the words "sides" and
"angles" of the one corresponding to the words "angles" and "sides"
respectively of the other.
It may also be noticed that the proof of this proposition
involves what is known as the Law of Converse; for
(1) if _b_ = _c_, then [L]_B_ = [L]_C_;
(2) if _b_ > _c_, then [L]_B_ > [L]_C_;
(3) if _b_ < _c_, then [L]_B_ < [L]_C_;
therefore the converses must necessarily be true as a matter of
logic; for
if [L]_B_ = [L]_C_, then _b_ cannot be greater than _c_ without
violating (2), and _b_ cannot be less than _c_ without
violating (3), therefore _b_ = _c_;
and if [L]_B_ > [L]_C_, then _b_ cannot equal _c_ without
violating (1), and _b_ cannot be less than _c_ without
violating (3), therefore _b_ > _c_;
similarly, if [L]_B_ < [L]_C_, then _b_ < _c_.
This Law of Converse may readily be taught to pupils, and it has several
applications in geometry.
THEOREM. _If two triangles have two sides of the one equal respectively
to two sides of the other, but the included angle of the first triangle
greater than the included angle of the second, then the third side of
the first is greater than the third side of the second, and conversely._
[Illustration]
In this proposition there are three possible cases: the point _Y_ may
fall below _AB_, as here shown, or on _AB_, or above _AB_. As an
exercise for pupils all three may be considered if desired. Following
Euclid and most early writers, however, only one case really need be
proved, provided that is the most difficult one, and is typical. Proclus
gave the proofs of the other two cases, and it is interesting to pupils
to work them out for themselves. In such work it constantly appears that
every proposition suggests abundant opportunity for originality, and
that the complete form of proof in a textbook is not a bar to
independent thought.
The Law of Converse, mentioned on page 190, may be applied to the
converse case if desired.
THEOREM. _Two angles whose sides are parallel, each to each, are either
equal or supplementary._
This is not an ancient proposition, although the Greeks were well aware
of the principle. It may be stated so as to include the case of the
sides being perpendicular, each to each, but this is better left as an
exercise. It is possible, by some circumlocution, to so state the
theorem as to tell in what cases the angles are equal and in what cases
supplementary. It cannot be tersely stated, however, and it seems better
to leave this point as a subject for questioning by the teacher.
THEOREM. _The opposite sides of a parallelogram are equal._
THEOREM. _If the opposite sides of a quadrilateral are equal, the figure
is a parallelogram._
[Illustration]
This proposition is a very simple test for a parallelogram. It is the
principle involved in the case of the common folding parallel ruler, an
instrument that has long been recognized as one of the valuable tools
of practical geometry. It will be of some interest to teachers to see
one of the early forms of this parallel ruler, as shown in the
illustration.[63] If such an instrument is not available in the school,
one suitable for illustrative purposes can easily be made from
cardboard.
[Illustration: PARALLEL RULER OF THE SEVENTEENTH CENTURY
San Giovanni's "Seconda squara mobile," Vicenza, 1686]
A somewhat more complicated form of this instrument may also be made by
pupils in manual training, as is shown in this illustration from Bion's
great treatise. The principle involved may be taken up in class, even if
the instrument is not used. It is evident that, unless the workmanship
is unusually good, this form of parallel ruler is not as accurate as the
common one illustrated above. The principle is sometimes used in iron
gates.
[Illustration: PARALLEL RULER OF THE EIGHTEENTH CENTURY
N. Bion's "Traite de la construction ... des instrumens de
mathematique," The Hague, 1723]
THEOREM. _Two parallelograms are congruent if two sides and the included
angle of the one are equal respectively to two sides and the included
angle of the other._
This proposition is discussed in connection with the one that follows.
THEOREM. _If three or more parallels intercept equal segments on one
transversal, they intercept equal segments on every transversal._
These two propositions are not given in Euclid, although generally
required by American syllabi of the present time. The last one is
particularly useful in subsequent work. Neither one offers any
difficulty, and neither has any interesting history. There are, however,
numerous interesting applications to the last one. One that is used in
mechanical drawing is here illustrated.
[Illustration]
If it is desired to divide a line _AB_ into five equal parts,
we may take a piece of ruled tracing paper and lay it over the
given line so that line 0 passes through _A_, and line 5
through _B_. We may then prick through the paper and thus
determine the points on _AB_. Similarly, we may divide _AB_
into any other number of equal parts.
Among the applications of these propositions is an interesting one due
to the Arab Al-Nair[=i]z[=i] (_ca._ 900 A.D.). The problem is to divide
a line into any number of equal parts, and he begins with the case of
trisecting _AB_. It may be given as a case of practical drawing even
before the problems are reached, particularly if some preliminary work
with the compasses and straightedge has been given.
Make _BQ_ and _AQ'_ perpendicular to _AB_, and make _BP_ = _PQ_
= _AP'_ = _P'Q'_. Then [triangle]_XYZ_ is congruent to
[triangle]_YBP_, and also to [triangle]_XAP'_. Therefore
_AX_ = _XY_ = _YB_. In the same way we might continue to produce
_BQ_ until it is made up of _n_ - 1 lengths _BP_, and so for _AQ'_,
and by properly joining points we could divide _AB_ into _n_
equal parts. In particular, if we join _P_ and _P'_, we bisect
the line _AB_.
[Illustration]
THEOREM. _If two sides of a quadrilateral are equal and parallel, then
the other two sides are equal and parallel, and the figure is a
parallelogram._
This was Euclid's first proposition on parallelograms, and Proclus
speaks of it as the connecting link between the theory of parallels and
that of parallelograms. The ancients, writing for mature students, did
not add the words "and the figure is a parallelogram," because that
follows at once from the first part and from the definition of
"parallelogram," but it is helpful to younger students because it
emphasizes the fact that here is a test for this kind of figure.
THEOREM. _The diagonals of a parallelogram bisect each other._
This proposition was not given in Euclid, but it is usually required in
American syllabi. There is often given in connection with it the
exercise in which it is proved that the diagonals of a rectangle are
equal. When this is taken, it is well to state to the class that
carpenters and builders find this one of the best checks in laying out
floors and other rectangles. It is frequently applied also in laying out
tennis courts. If the class is doing any work in mensuration, such as
finding the area of the school grounds, it is a good plan to check a few
rectangles by this method.
An interesting outdoor application of the theory of parallelograms is
the following:
[Illustration]
Suppose you are on the side of this stream opposite to _XY_,
and wish to measure the length of _XY_. Run a line _AB_ along
the bank. Then take a carpenter's square, or even a large book,
and walk along _AB_ until you reach _P_, a point from which you
can just see _X_ and _B_ along two sides of the square. Do the
same for _Y_, thus fixing _P_ and _Q_. Using the tape, bisect
_PQ_ at _M_. Then walk along _YM_ produced until you reach a
point _Y'_ that is exactly in line with _M_ and _Y_, and also
with _P_ and _X_. Then walk along _XM_ produced until you reach
a point _X'_ that is exactly in line with _M_ and _X_, and also
with _Q_ and _Y_. Then measure _Y'X'_ and you have the length
of _XY_. For since _YX'_ is [perp] to _PQ_, and _XY'_ is also
[perp] to _PQ_, _YX'_ is || to _XY'_. And since _PM_ = _MQ_,
therefore _XM_ = _MX'_ and _Y'M_ = _MY_. Therefore _Y'X'YX_ is
a parallelogram.
The properties of the parallelogram are often applied to proving figures
of various kinds congruent, or to constructing them so that they will be
congruent.
[Illustration]
For example, if we draw _A'B'_ equal and parallel to _AB_,
_B'C'_ equal and parallel to _BC_, and so on, it is easily
proved that _ABCD_ and _A'B'C'D'_ are congruent. This may be
done by ordinary superposition, or by sliding _ABCD_ along the
dotted parallels.
There are many applications of this principle of parallel translation in
practical construction work. The principle is more far-reaching than
here intimated, however, and a few words as to its significance will now
be in place.
The efforts usually made to improve the spirit of Euclid are trivial.
They ordinarily relate to some commonplace change of sequence, to some
slight change in language, or to some narrow line of applications. Such
attempts require no particular thought and yield no very noticeable
result. But there is a possibility, remote though it may be at present,
that a geometry will be developed that will be as serious as Euclid's
and as effective in the education of the thinking individual. If so, it
seems probable that it will not be based upon the congruence of
triangles, by which so many propositions of Euclid are proved, but upon
certain postulates of motion, of which one is involved in the above
illustration,--the postulate of parallel translation. If to this we join
the two postulates of rotation about an axis,[64] leading to axial
symmetry; and rotation about a point,[65] leading to symmetry with
respect to a center, we have a group of three motions upon which it is
possible to base an extensive and rigid geometry.[66] It will be through
some such effort as this, rather than through the weakening of the
Euclid-Legendre style of geometry, that any improvement is likely to
come. At present, in America, the important work for teachers is to
vitalize the geometry they have,--an effort in which there are great
possibilities,--seeing to it that geometry is not reduced to mere froth,
and recognizing the possibility of another geometry that may sometime
replace it,--a geometry as rigid, as thought-compelling, as logical,
and as truly educational.
THEOREM. _The sum of the interior angles of a polygon is equal to two
right angles, taken as many times less two as the figure has sides._
This interesting generalization of the proposition about the sum of the
angles of a triangle is given by Proclus. There are several proofs, but
all are based upon the possibility of dissecting the polygon into
triangles. The point from which lines are drawn to the vertices is
usually taken at a vertex, so that there are _n_ - 2 triangles. It may
however be taken within the figure, making _n_ triangles, from the sum
of the angles of which the four right angles about the point must be
subtracted. The point may even be taken on one side, or outside the
polygon, but the proof is not so simple. Teachers who desire to do so
may suggest to particularly good students the proving of the theorem for
a concave polygon, or even for a cross polygon, although the latter
requires negative angles.
Some schools have transit instruments for the use of their classes in
trigonometry. In such a case it is a good plan to measure the angles in
some piece of land so as to verify the proposition, as well as show the
care that must be taken in reading angles. In the absence of this
exercise it is well to take any irregular polygon and measure the angles
by the help of a protractor, and thus accomplish the same results.
THEOREM. _The sum of the exterior angles of a polygon, made by producing
each of its sides in succession, is equal to four right angles._
This is also a proposition not given by the ancient writers. We have,
however, no more valuable theorem for the purpose of showing the nature
and significance of the negative angle; and teachers may arouse a great
deal of interest in the negative quantity by showing to a class that
when an interior angle becomes 180 deg. the exterior angle becomes 0, and
when the polygon becomes concave the exterior angle becomes negative,
the theorem holding for all these cases. We have few better
illustrations of the significance of the negative quantity, and few
better opportunities to use the knowledge of this kind of quantity
already acquired in algebra.
[Illustration]
In the hilly and mountainous parts of America, where irregular-shaped
fields are more common than in the more level portions, a common test
for a survey is that of finding the exterior angles when the transit
instrument is set at the corners. In this field these angles are given,
and it will be seen that the sum is 360 deg.. In the absence of any outdoor
work a protractor may be used to measure the exterior angles of a
polygon drawn on paper. If there is an irregular piece of land near the
school, the exterior angles can be fairly well measured by an ordinary
paper protractor.
The idea of locus is usually introduced at the end of Book I. It is too
abstract to be introduced successfully any earlier, although authors
repeat the attempt from time to time, unmindful of the fact that all
experience is opposed to it. The loci propositions are not ancient. The
Greeks used the word "locus" (in Greek, _topos_), however. Proclus, for
example, says, "I call those locus theorems in which the same property
is found to exist on the whole of some locus." Teachers should be
careful to have the pupils recognize the necessity for proving two
things with respect to any locus: (1) that any point on the supposed
locus satisfies the condition; (2) that any point outside the supposed
locus does not satisfy the given condition. The first of these is called
the "sufficient condition," and the second the "necessary condition."
Thus in the case of the locus of points in a plane equidistant from two
given points, it is _sufficient_ that the point be on the perpendicular
bisector of the line joining the given points, and this is the first
part of the proof; it is also _necessary_ that it be on this line, i.e.
it cannot be outside this line, and this is the second part of the
proof. The proof of loci cases, therefore, involves a consideration of
"the necessary and sufficient condition" that is so often spoken of in
higher mathematics. This expression might well be incorporated into
elementary geometry, and when it becomes better understood by teachers,
it probably will be more often used.
In teaching loci it is helpful to call attention to loci in space
(meaning thereby the space of three dimensions), without stopping to
prove the proposition involved. Indeed, it is desirable all through
plane geometry to refer incidentally to solid geometry. In the
mensuration of plane figures, which may be boundaries of solid figures,
this is particularly true.
It is a great defect in most school courses in geometry that
they are entirely confined to two dimensions. Even if solid
geometry in the usual sense is not attempted, every occasion
should be taken to liberate boys' minds from what becomes the
tyranny of paper. Thus the questions: "What is the locus of a
point equidistant from two given points; at a constant distance
from a given straight line or from a given point?" should be
extended to space.[67]
The two loci problems usually given at this time, referring to a point
equidistant from the extremities of a given line, and to a point
equidistant from two intersecting lines, both permit of an interesting
extension to three dimensions without any formal proof. It is possible
to give other loci at this point, but it is preferable merely to
introduce the subject in Book I, reserving the further discussion until
after the circle has been studied.
It is well, in speaking of loci, to remember that it is entirely proper
to speak of the "locus of a point" or the "locus of points." Thus the
locus of a _point_ so moving in a plane as constantly to be at a given
distance from a fixed point in the plane is a circle. In analytic
geometry we usually speak of the locus of a _point_, thinking of the
point as being anywhere on the locus. Some teachers of elementary
geometry, however, prefer to speak of the locus of _points_, or the
locus of _all points_, thus tending to make the language of elementary
geometry differ from that of analytic geometry. Since it is a trivial
matter of phraseology, it is better to recognize both forms of
expression and to let pupils use the two interchangeably.
FOOTNOTES:
[57] Address at Brussels, August, 1910.
[58] For a recent discussion of this general subject, see Professor
Hobson on "The Tendencies of Modern Mathematics," in the _Educational
Review_, New York, 1910, Vol. XL, p. 524.
[59] A more extended list of applications is given later in this work.
[60] Ab[=u]'l-'Abb[=a]s al-Fadl ibn H[=a]tim al-Nair[=i]z[=i], so called
from his birthplace, Nair[=i]z, was a well-known Arab writer. He died
about 922 A.D. He wrote a commentary on Euclid.
[61] This illustration, taken from a book in the author's library,
appeared in a valuable monograph by W. E. Stark, "Measuring Instruments
of Long Ago," published in _School Science and Mathematics_, Vol. X, pp.
48, 126. With others of the same nature it is here reproduced by the
courtesy of Principal Stark and of the editors of the journal in which
it appeared.
[62] In speaking of two congruent triangles it is somewhat easier to
follow the congruence if the two are read in the same order, even though
the relatively unimportant counterclockwise reading is neglected. No one
should be a slave to such a formalism, but should follow the plan when
convenient.
[63] Stark, loc. cit.
[64] Of which so much was made by Professor Olaus Henrici in his
"Congruent Figures," London, 1879,--a book that every teacher of
geometry should own.
[65] Much is made of this in the excellent work by Henrici and
Treutlein, "Lehrbuch der Geometrie," Leipzig, 1881.
[66] Meray did much for this movement in France, and the recent works of
Bourlet and Borel have brought it to the front in that country.
[67] W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary
Schools," Board of Education circular (No. 711), p. 8, London, 1909.
CHAPTER XV
THE LEADING PROPOSITIONS OF BOOK II
Having taken up all of the propositions usually given in Book I, it
seems unnecessary to consider as specifically all those in subsequent
books. It is therefore proposed to select certain ones that have some
special interest, either from the standpoint of mathematics or from that
of history or application, and to discuss them as fully as the
circumstances seem to warrant.
THEOREMS. _In the same circle or in equal circles equal central angles
intercept equal arcs; and of two unequal central angles the greater
intercepts the greater arc_, and conversely for both of these cases.
Euclid made these the twenty-sixth and twenty-seventh propositions of
his Book III, but he limited them as follows: "In equal circles equal
angles stand on equal circumferences, whether they stand at the centers
or at the circumferences, and conversely." He therefore included two of
our present theorems in one, thus making the proposition doubly hard for
a beginner. After these two propositions the Law of Converse, already
mentioned on page 190, may properly be introduced.
THEOREMS. _In the same circle or in equal circles, if two arcs are
equal, they are subtended by equal chords; and if two arcs are unequal,
the greater is subtended by the greater chord_, and conversely.
Euclid dismisses all this with the simple theorem, "In equal circles
equal circumferences are subtended by equal straight lines." It will
therefore be noticed that he has no special word for "chord" and none
for "arc," and that the word "circumference," which some teachers are so
anxious to retain, is used to mean both the whole circle and any arc. It
cannot be doubted that later writers have greatly improved the language
of geometry by the use of these modern terms. The word "arc" is the
same, etymologically, as "arch," each being derived from the Latin
_arcus_ (a bow). "Chord" is from the Greek, meaning "the string of a
musical instrument." "Subtend" is from the Latin _sub_ (under), and
_tendere_ (to stretch).
It should be noticed that Euclid speaks of "equal circles," while we
speak of "the same circle or equal circles," confining our proofs to the
latter, on the supposition that this sufficiently covers the former.
THEOREM. _A line through the center of a circle perpendicular to a chord
bisects the chord and the arcs subtended by it._
This is an improvement on Euclid, III, 3: "If in a circle a straight
line through the center bisects a straight line not through the center,
it also cuts it at right angles; and if it cuts it at right angles, it
also bisects it." It is a very important proposition, theoretically and
practically, for it enables us to find the center of a circle if we know
any part of its arc. A civil engineer, for example, who wishes to find
the center of the circle of which some curve (like that on a running
track, on a railroad, or in a park) is an arc, takes two chords, say of
one hundred feet each, and erects perpendicular bisectors. It is well to
ask a class why, in practice, it is better to take these chords some
distance apart. Engineers often check their work by taking three chords,
the perpendicular bisectors of the three passing through a single
point. Illustrations of this kind of work are given later in this
chapter.
THEOREM. _In the same circle or in equal circles equal chords are
equidistant from the center, and chords equidistant from the center are
equal._
This proposition is practically used by engineers in locating points on
an arc of a circle that is too large to be described by a tape, or that
cannot easily be reached from the center on account of obstructions.
[Illustration]
If part of the curve _APB_ is known, take _P_ as the mid-point.
Then stretch the tape from _A_ to _B_ and draw _PM_
perpendicular to it. Then swing the length _AM_ about _P_, and
_PM_ about _B_, until they meet at _L_, and stretch the length
_AB_ along _PL_ to _Q_. This fixes the point _Q_. In the same
way fix the point _C_. Points on the curve can thus be fixed as
near together as we wish. The chords _AB_, _PQ_, _BC_, and so
on, are equal and are equally distant from the center.
THEOREM. _A line perpendicular to a radius at its extremity is tangent
to the circle._
The enunciation of this proposition by Euclid is very interesting. It is
as follows:
The straight line drawn at right angles to the diameter of a
circle at its extremity will fall outside the circle, and into
the space between the straight line and the circumference
another straight line cannot be interposed; further, the angle
of the semicircle is greater and the remaining angle less than
any acute rectilineal angle.
The first assertion is practically that of tangency,--"will fall outside
the circle." The second one states, substantially, that there is only
one such tangent, or, as we say in modern mathematics, the tangent is
unique. The third statement relates to the angle formed by the diameter
and the circumference,--a mixed angle, as Proclus called it, and a kind
of angle no longer used in elementary geometry. The fourth statement
practically asserts that the angle between the tangent and circumference
is less than any assignable quantity. This gives rise to a difficulty
that seems to have puzzled many of Euclid's commentators, and that will
interest a pupil: As the circle diminishes this angle apparently
increases, while as the circle increases the angle decreases, and yet
the angle is always stated to be zero. Vieta (1540-1603), who did much
to improve the science of algebra, attempted to explain away the
difficulty by adopting a notion of circle that was prevalent in his
time. He said that a circle was a polygon of an infinite number of sides
(which it cannot be, by definition), and that, a tangent simply
coincided with one of the sides, and therefore made no angle with it;
and this view was also held by Galileo (1564-1642), the great physicist
and mathematician who first stated the law of the pendulum.
THEOREM. _Parallel lines intercept equal arcs on a circle._
The converse of this proposition has an interesting application in
outdoor work.
[Illustration]
Suppose we wish to run a line through _P_ parallel to a given
line _AB_. With any convenient point _O_ as a center, and _OP_
as a radius, describe a circle cutting _AB_ in _X_ and _Y_.
Draw _PX_. Then with _Y_ as a center and _PX_ as a radius draw
an arc cutting the circle in _Q_. Then run the line from _P_ to
_Q_. _PQ_ is parallel to _AB_ by the converse of the above
theorem, which is easily shown to be true for this figure.
THEOREM. _If two circles are tangent to each other, the line of centers
passes through the point of contact._
There are many illustrations of this theorem in practical work, as in
the case of cogwheels. An interesting application to engineering is seen
in the case of two parallel streets or lines of track which are to be
connected by a "reversed curve."
[Illustration]
If the lines are _AB_ and _CD_, and the connection is to be
made, as shown, from _B_ to _C_, we may proceed as follows:
Draw _BC_ and bisect it at _M_. Erect _PO_, the perpendicular
bisector of _BM_; and _BO_, perpendicular to _AB_. Then _O_ is
one center of curvature. In the same way fix _O'_. Then to
check the work apply this theorem, _M_ being in the line of
centers _OO'_. The curves may now be drawn, and they will be
tangent to _AB_, to _CD_, and to each other.
At this point in the American textbooks it is the custom to insert a
brief treatment of measurement, explaining what is meant by ratio,
commensurable and incommensurable quantities, constant and variable, and
limit, and introducing one or more propositions relating to limits. The
object of this departure from the ancient sequence, which postponed this
subject to the book on ratio and proportion, is to treat the circle more
completely in Book III. It must be confessed that the treatment is not
as scientific as that of Euclid, as will be explained under Book III,
but it is far better suited to the mind of a boy or girl.
It begins by defining measurement in a practical way, as the finding of
the number of times a quantity of any kind contains a known quantity of
the same kind. Of course this gives a number, but this number may be a
surd, like [sqrt]2. In other words, the magnitude measured may be
incommensurable with the unit of measure, a seeming paradox. With this
difficulty, however, the pupil should not be called upon to contend at
this stage in his progress. The whole subject of incommensurables might
safely be postponed, although it may be treated in an elementary fashion
at this time. The fact that the measure of the diagonal of a square, of
which a side is unity, is [sqrt]2, and that this measure is an
incommensurable number, is not so paradoxical as it seems, the paradox
being verbal rather than actual.
It is then customary to define ratio as the quotient of the numerical
measures of two quantities in terms of a common unit. This brings all
ratios to the basis of numerical fractions, and while it is not
scientifically so satisfactory as the ancient concept which considered
the terms as lines, surfaces, angles, or solids, it is more practical,
and it suffices for the needs of elementary pupils.
"Commensurable," "incommensurable," "constant," and "variable" are then
defined, and these definitions are followed by a brief discussion of
limit. It simplifies the treatment of this subject to state at once that
there are two classes of limits,--those which the variable actually
reaches, and those which it can only approach indefinitely near. We find
the one as frequently as we find the other, although it is the latter
that is referred to in geometry. For example, the superior limit of a
chord is a diameter, and this limit the chord may reach. The inferior
limit is zero, but we do not consider the chord as reaching this limit.
It is also well to call the attention of pupils to the fact that a
quantity may decrease towards its limit as well as increase towards it.
Such further definitions as are needed in the theory of limits are now
introduced. Among these is "area of a circle." It might occur to some
pupil that since a circle is a line (as used in modern mathematics), it
can have no area. This is, however, a mere quibble over words. It is not
pretended that the line has area, but that "area of a circle" is merely
a shortened form of the expression "area inclosed by a circle."
The Principle of Limits is now usually given as follows: "If, while
approaching their respective limits, two variables are always equal,
their limits are equal." This was expressed by D'Alembert in the
eighteenth century as "Magnitudes which are the limits of equal
magnitudes are equal," or this in substance. It would easily be possible
to elaborate this theory, proving, for example, that if _x_ approaches
_y_ as its limit, then _ax_ approaches _ay_ as its limit, and _x/a_
approaches _y/a_ as its limit, and so on. Very much of this theory,
however, wearies a pupil so that the entire meaning of the subject is
lost, and at best the treatment in elementary geometry is not rigorous.
It is another case of having to sacrifice a strictly scientific
treatment to the educational abilities of the pupil. Teachers wishing to
find a scientific treatment of the subject should consult a good work on
the calculus.
THEOREM. _In the same circle or in equal circles two central angles have
the same ratio as their intercepted arcs._
This is usually proved first for the commensurable case and then for the
incommensurable one. The latter is rarely understood by all of the
class, and it may very properly be required only of those who show some
aptitude in geometry. It is better to have the others understand fully
the commensurable case and see the nature of its applications, possibly
reading the incommensurable proof with the teacher, than to stumble
about in the darkness of the incommensurable case and never reach the
goal. In Euclid there was no distinction between the two because his
definition of ratio covered both; but, as we shall see in Book III, this
definition is too difficult for our pupils. Theon of Alexandria (fourth
century A.D.), the father of the Hypatia who is the heroine of
Kingsley's well-known novel, wrote a commentary on Euclid, and he adds
that sectors also have the same ratio as the arcs, a fact very easily
proved. In propositions of this type, referring to the same circle or to
equal circles, it is not worth while to ask pupils to take up both
cases, the proof for either being obviously a proof for the other.
Many writers state this proposition so that it reads that "central
angles are _measured by_ their intercepted arcs." This, of course, is
not literally true, since we can measure anything only by some thing, of
the same kind. Thus we measure a volume by finding how many times it
contains another volume which we take as a unit, and we measure a length
by taking some other length as a unit; but we cannot measure a given
length in quarts nor a given weight in feet, and it is equally
impossible to measure an arc by an angle, and vice versa. Nevertheless
it is often found convenient to _define_ some brief expression that has
no meaning if taken literally, in such way that it shall acquire a
meaning. Thus we _define_ "area of a circle," even when we use "circle"
to mean a line; and so we may define the expression "central angles are
measured by their intercepted arcs" to mean that central angles have the
same numerical measure as these arcs. This is done by most writers, and
is legitimate as explaining an abbreviated expression.
THEOREM. _An inscribed angle is measured by half the intercepted arc._
In Euclid this proposition is combined with the preceding one in his
Book VI, Proposition 33. Such a procedure is not adapted to the needs of
students to-day. Euclid gave in Book III, however, the proposition (No.
20) that a central angle is twice an inscribed angle standing on the
same arc. Since Euclid never considered an angle greater than 180 deg., his
inscribed angle was necessarily less than a right angle. The first one
who is known to have given the general case, taking the central angle as
being also greater than 180 deg., was Heron of Alexandria, probably of the
first century A.D.[68] In this he was followed by various later
commentators, including Tartaglia and Clavius in the sixteenth century.
One of the many interesting exercises that may be derived from this
theorem is seen in the case of the "horizontal danger angle" observed by
ships.
[Illustration]
If some dangerous rocks lie off the shore, and _L_ and _L'_ are
two lighthouses, the angle _A_ is determined by observation, so
that _A_ will lie on a circle inclosing the dangerous area.
Angle _A_ is called the "horizontal danger angle." Ships
passing in sight of the two lighthouses _L_ and _L'_ must keep
out far enough so that the angle _L'SL_ shall be less than
angle _A_.
To this proposition there are several important corollaries, including
the following:
1. _An angle inscribed in a semicircle is a right angle._ This corollary
is mentioned by Aristotle and is attributed to Thales, being one of the
few propositions with which his name is connected. It enables us to
describe a circle by letting the arms of a carpenter's square slide
along two nails driven in a board, a pencil being held at the vertex.
[Illustration]
A more practical use for it is made by machinists to determine whether a
casting is a true semicircle. Taking a carpenter's square as here shown,
if the vertex touches the curve at every point as the square slides
around, it is a true semicircle. By a similar method a circle may be
described by sliding a draftsman's triangle so that two sides touch two
tacks driven in a board.
[Illustration]
Another interesting application of this corollary may be seen
by taking an ordinary paper protractor _ACB_, and fastening a
plumb line at _B_. If the protractor is so held that the plumb
line cuts the semicircle at _C_, then _AC_ is level because it
is perpendicular to the vertical line _BC_. Thus, if a class
wishes to determine the horizontal line _AC_, while sighting up
a hill in the direction _AB_, this is easily determined without
a spirit level.
It follows from this corollary, as the pupil has already found, that the
mid-point of the hypotenuse of a right triangle is equidistant from the
three vertices. This is useful in outdoor measuring, forming the basis
of one of the best methods of letting fall a perpendicular from an
external point to a line.
[Illustration]
Suppose _XY_ to be the edge of a sidewalk, and _P_ a point in
the street from which we wish to lay a gas pipe perpendicular
to the walk. From _P_ swing a cord or tape, say 60 feet long,
until it meets _XY_ at _A_. Then take _M_, the mid-point of
_PA_, and swing _MP_ about _M_, to meet _XY_ at _B_. Then _B_
is the foot of the perpendicular, since [L]_PBA_ can be
inscribed in a semicircle.
2. _Angles inscribed in the same segment are equal._
[Illustration]
By driving two nails in a board, at _A_ and _B_, and taking an
angle _P_ made of rigid material (in particular, as already
stated, a carpenter's square), a pencil placed at _P_ will
generate an arc of a circle if the arms slide along _A_ and
_B_. This is an interesting exercise for pupils.
THEOREM. _An angle formed by two chords intersecting within the circle
is measured by half the sum of the intercepted arcs._
THEOREM. _An angle formed by a tangent and a chord drawn from the point
of tangency is measured by half the intercepted arc._
THEOREM. _An angle formed by two secants, a secant and a tangent, or two
tangents, drawn to a circle from an external point, is measured by half
the difference of the intercepted arcs._
These three theorems are all special cases of the general proposition
that the angle included between two lines that cut (or touch) a circle
is measured by half the sum of the intercepted arcs. If the point passes
from within the circle to the circle itself, one arc becomes zero and
the angle becomes an inscribed angle. If the point passes outside the
circle, the smaller arc becomes negative, having passed through zero.
The point may even "go to infinity," as is said in higher mathematics,
the lines then becoming parallel, and the angle becoming zero, being
measured by half the sum of one arc and a negative arc of the same
absolute value. This is one of the best illustrations of the Principle
of Continuity to be found in geometry.
PROBLEM. _To let fall a perpendicular upon a given line from a given
external point._
This is the first problem that a student meets in most American
geometries. The reason for treating the problems by themselves instead
of mingling them with the theorems has already been discussed.[69] The
student now has a sufficient body of theorems, by which he can prove
that his constructions are correct, and the advantage of treating these
constructions together is greater than that of following Euclid's plan
of introducing them whenever needed.
Proclus tells us that "this problem was first investigated by
Oenopides,[70] who thought it useful for astronomy." Proclus speaks of
such a line as a gnomon, a common name for the perpendicular on a
sundial, which casts the shadow by which the time of day is known. He
also speaks of two kinds of perpendiculars, the plane and solid, the
former being a line perpendicular to a line, and the latter a line
perpendicular to a plane.
It is interesting to notice that the solution tacitly assumes that a
certain arc is going to cut the given line in two points, and only two.
Strictly speaking, why may it not cut it in only one point, or even in
three points? We really assume that if a straight line is drawn through
a point within a circle, this line must get out of the circle on each
of two sides of the given point, and in getting out it must cut the
circle twice. Proclus noticed this assumption and endeavored to prove
it. It is better, however, not to raise the question with beginners,
since it seems to them like hair-splitting to no purpose.
The problem is of much value in surveying, and teachers would do well to
ask a class to let fall a perpendicular to the edge of a sidewalk from a
point 20 feet from the walk, using an ordinary 66-foot or 50-foot tape.
Practically, the best plan is to swing 30 feet of the tape about the
point and mark the two points of intersection with the edge of the walk.
Then measure the distance between the points and take half of this
distance, thus fixing the foot of the perpendicular.
PROBLEM. _At a given point in a line, to erect a perpendicular to that
line._
This might be postponed until after the problem to bisect an angle,
since it merely requires the bisection of a straight angle; but
considering the immaturity of the average pupil, it is better given
independently. The usual case considers the point not at the extremity
of the line, and the solution is essentially that of Euclid. In
practice, however, as for example in surveying, the point may be at the
extremity, and it may not be convenient to produce the line.
[Illustration]
Surveyors sometimes measure _PB_ = 3 ft., and then take 9 ft.
of tape, the ends being held at _B_ and _P_, and the tape being
stretched to _A_, so that _PA_ = 4 ft. and _AB_ = 5 ft. Then
_P_ is a right angle by the Pythagorean Theorem. This theorem
not having yet been proved, it cannot be used at this time.
A solution for the problem of erecting a perpendicular from the
extremity of a line that cannot be produced, depending, however, on the
problem of bisecting an angle, and therefore to be given after that
problem, is attributed by Al-Nair[=i]z[=i] (tenth century A.D.) to Heron
of Alexandria. It is also given by Proclus.
[Illustration]
Required to draw from _P_ a perpendicular to _AP_. Take _X_
anywhere on the line and erect _XY_ [perp] to _AP_ in the usual
manner. Bisect [L]_PXY_ by the line _XM_. On _XY_ take _XN_ = _XP_,
and draw _NM_ [perp] to _XY_. Then draw _PM_. The proof
is evident.
These may at the proper time be given as interesting variants of the
usual solution.
PROBLEM. _To bisect a given line._
Euclid said "finite straight line," but this wording is not commonly
followed, because it will be inferred that the line is finite if it is
to be bisected, and we use "line" alone to mean a straight line.
Euclid's plan was to construct an equilateral triangle (by his
Proposition 1 of Book I) on the line as a base, and then to bisect the
vertical angle. Proclus tells us that Apollonius of Perga, who wrote the
first great work on conic sections, used a plan which is substantially
that which is commonly found in textbooks to-day,--constructing two
isosceles triangles upon the line as a common base, and connecting their
vertices.
PROBLEM. _To bisect a given angle._
It should be noticed that in the usual solution two arcs intersect, and
the point thus determined is connected with the vertex. Now these two
arcs intersect twice, and since one of the points of intersection may be
the vertex itself, the other point of intersection must be taken. It is
not, however, worth while to make much of this matter with pupils.
Proclus calls attention to the possible suggestion that the point of
intersection may be imagined to lie outside the angle, and he proceeds
to show the absurdity; but here, again, the subject is not one of value
to beginners. He also contributes to the history of the trisection of an
angle. Any angle is easily trisected by means of certain higher curves,
such as the conchoid of Nicomedes (_ca._ 180 B.C.), the quadratrix of
Hippias of Elis (_ca._ 420 B.C.), or the spiral of Archimedes (_ca._ 250
B.C.). But since this problem, stated algebraically, requires the
solution of a cubic equation, and this involves, geometrically, finding
three points, we cannot solve the problem by means of straight lines and
circles alone. In other words, the trisection of _any_ angle, by the use
of the straightedge and compasses alone, is impossible. Special angles
may however be trisected. Thus, to trisect an angle of 90 deg. we need only
to construct an angle of 60 deg., and this can be done by constructing an
equilateral triangle. But while we cannot trisect the angle, we may
easily approximate trisection. For since, in the infinite geometric
series 1/2 + 1/8 + 1/32 + 1/128 + ..., _s_ = _a_ / (1 - _r_), we have
_s_ = 1/2 / 3/4 = 2/3. In other words, if we add 1/2 of the angle, 1/8
of the angle, 1/32 of the angle, and so on, we approach as a limit 2/3
of the angle; but all of these fractions can be obtained by repeated
bisections, and hence by bisections we may approximate the trisection.
The approximate bisection (or any other division) of an angle may of
course be effected by the help of the protractor and a straightedge. The
geometric method is, however, usually more accurate, and it is
advantageous to have the pupils try both plans, say for bisecting an
angle of about 49 1/2 deg..
[Illustration]
Applications of this problem are numerous. It may be desired, for
example, to set a lamp-post on a line bisecting the angle formed by two
streets that come together a little unsymmetrically, as here shown, in
which case the bisecting line can easily be run by the use of a
measuring tape, or even of a stout cord.
A more interesting illustration is, however, the following:
[Illustration]
Let the pupils set a stake, say about 5 feet high, at a point
_N_ on the school grounds about 9 A.M., and carefully measure
the length of the shadow, _NW_, placing a small wooden pin at
_W_. Then about 3 P.M. let them watch until the shadow _NE_ is
exactly the same length that it was when _W_ was fixed, and
then place a small wooden pin at _E_. If the work has been very
carefully done, and they take the tape and bisect the line
_WE_, thus fixing the line _NS_, they will have a north and
south line. If this is marked out for a short distance from
_N_, then when the shadow falls on _NS_, it will be noon by sun
time (not standard time) at the school.
PROBLEM. _From a given point in a given line, to draw a line making an
angle equal to a given angle._
Proclus says that Eudemus attributed to Oenopides the discovery of the
solution which Euclid gave, and which is substantially the one now
commonly seen in textbooks. The problem was probably solved in some
fashion before the time of Oenopides, however. The object of the problem
is primarily to enable us to draw a line parallel to a given line.
Practically, the drawing of one line parallel to another is usually
effected by means of a parallel ruler (see page 191), or by the use of
draftsmen's triangles, as here shown, or even more commonly by the use
of a T-square, such as is here seen. This illustration shows two
T-squares used for drawing lines parallel to the sides of a board upon
which the drawing paper is fastened.[71]
[Illustration]
[Illustration]
An ingenious instrument described by Baron Dupin is illustrated below.
[Illustration]
To the bar _A_ is fastened the sliding check _B_. A movable
check _D_ may be fastened by a screw _C_. A sharp point is
fixed in _B_, so that as _D_ slides along the edge of a board,
the point marks a line parallel to the edge. Moreover, _F_ and
_G_ are two brass arms of equal length joined by a pointed
screw _H_ that marks a line midway between _B_ and _D_.
Furthermore, it is evident that _H_ will draw a line bisecting
any irregular board if the checks _B_ and _D_ are kept in
contact with the irregular edges.
Book II offers two general lines of application that may be introduced
to advantage, preferably as additions to the textbook work. One of these
has reference to topographical drawing and related subjects, and the
other to geometric design. As long as these can be introduced to the
pupil with an air of reality, they serve a good purpose, but if made a
part of textbook work, they soon come to have less interest than the
exercises of a more abstract character. If a teacher can relate the
problems in topographical drawing to the pupil's home town, and can
occasionally set some outdoor work of the nature here suggested, the
results are usually salutary; but if he reiterates only a half-dozen
simple propositions time after time, with only slight changes in the
nature of the application, then the results will not lead to a
cultivation of power in geometry,--a point which the writers on applied
geometry usually fail to recognize.
[Illustration]
One of the simple applications of this book relates to the rounding of
corners in laying out streets in some of our modern towns where there is
a desire to depart from the conventional square corner. It is also used
in laying out park walks and drives.
[Illustration]
The figure in the middle of the page represents two streets,
_AP_ and _BQ_, that would, if prolonged, intersect at _C_. It
is required to construct an arc so that they shall begin to
curve at _P_ and _Q_, where _CP_ = _CQ_, and hence the "center
of curvature" _O_ must be found.
The problem is a common one in railroad work, only here _AP_ is
usually oblique to _BQ_ if they are produced to meet at _C_, as
in the second figure on page 218. It is required to construct
an arc so that the tracks shall begin to curve at _P_ and _Q_,
where _CP_ = _CQ_.
[Illustration]
The problem becomes a little more complicated, and correspondingly more
interesting, when we have to find the center of curvature for a street
railway track that must turn a corner in such a way as to allow, say,
exactly 5 feet from the point _P_, on account of a sidewalk.
[Illustration]
The problem becomes still more difficult if we have two roads of
different widths that we wish to join on a curve. Here the two centers
of curvature are not the same, and the one road narrows to the other on
the curve. The solutions will be understood from a study of the figures.
The number of problems of this kind that can easily be made is
limitless, and it is well to avoid the danger of hobby riding on this
or any similar topic. Therefore a single one will suffice to close this
group.
[Illustration]
If a road _AB_ on an arc described about _O_, is to be joined
to road _CD_, described about _O'_, the arc _BC_ should
evidently be internally tangent to _AB_ and externally tangent
to _CD_. Hence the center is on _BOX_ and _O'CY_, and is
therefore at _P_. The problem becomes more real if we give some
width to the roads in making the drawing, and imagine them in a
park that is being laid out with drives.
It will be noticed that the above problems require the erecting of
perpendiculars, the bisecting of angles, and the application of the
propositions on tangents.
A somewhat different line of problems is that relating to the passing of
a circle through three given points. It is very easy to manufacture
problems of this kind that have a semblance of reality.
[Illustration]
For example, let it be required to plan a driveway from the
gate _G_ to the porch _P_ so as to avoid a mass of rocks _R_,
an arc of a circle to be taken. Of course, if we allow pupils
to use the Pythagorean Theorem at this time (and for metrical
purposes this is entirely proper, because they have long been
familiar with it), then we may ask not only for the drawing,
but we may, for example, give the length from _G_ to the point
on _R_ (which we may also call _R_), and the angle _RGO_ as
60 deg., to find the radius.
A second general line of exercises adapted to Book II is a continuation
of the geometric drawing recommended as a preliminary to the work in
demonstrative geometry. The copying or the making of designs requiring
the describing of circles, their inscription in or circumscription about
triangles, and their construction in various positions of tangency, has
some value as applying the various problems studied in this book. For a
number of years past, several enthusiastic teachers have made much of
the designs found in Gothic windows, having their pupils make the
outline drawings by the help of compasses and straightedge. While such
work has its value, it is liable soon to degenerate into purposeless
formalism, and hence to lose interest by taking the vigorous mind of
youth from the strong study of geometry to the weak manipulation of
instruments. Nevertheless its value should be appreciated and conserved,
and a few illustrations of these forms are given in order that the
teacher may have examples from which to select. The best way of using
this material is to offer it as supplementary work, using much or
little, as may seem best, thus giving to it a freshness and interest
that some have trouble in imparting to the regular book work.
The best plan is to sketch rapidly the outline of a window on the
blackboard, asking the pupils to make a rough drawing, and to bring in a
mathematical drawing on the following day.
[Illustration]
It might be said, for example, that in planning a Gothic window
this drawing is needed. The arc _BC_ is drawn with _A_ as a
center and _AB_ as a radius. The small arches are described
with _A_, _D_, and _B_ as centers and _AD_ as a radius. The
center _P_ is found by taking _A_ and _B_ as centers and _AE_ as
a radius. How may the points _D_, _E_, and _F_ be found? Draw
the figure. From the study of the rectilinear figures suggested
by such a simple pattern the properties of the equilateral
triangle may be inferred.
The Gothic window also offers some interesting possibilities in
connection with the study of the square. For example, the illustration
given on page 223 shows a number of traceries involving the construction
of a square, the bisecting of angles, and the describing of circles.[72]
[Illustration]
The properties of the square, a figure now easily constructed by the
pupils, are not numerous. What few there are may be brought out through
the study of art forms, if desired. In case these forms are shown to a
class, it is important that they should be selected from good designs.
We have enough poor art in the world, so that geometry should not
contribute any more. This illustration is a type of the best medieval
Gothic parquetry.[73]
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
Even simple designs of a semipuzzling nature have their advantage in
this connection. In the following example the inner square contains all
of the triangles, the letters showing where they may be fitted.[74]
Still more elaborate designs, based chiefly upon the square and circle,
are shown in the window traceries on page 225, and others will be given
in connection with the study of the regular polygons.
[Illustration]
Designs like the figure below are typical of the simple forms, based on
the square and circle, that pupils may profitably incorporate in any
work in art design that they may be doing at the time they are studying
the circle and the problems relating to perpendiculars and squares.
[Illustration]
Among the applications of the problem to draw a tangent to a given
circle is the case of the common tangents to two given circles. Some
authors give this as a basal problem, although it is more commonly given
as an exercise or a corollary. One of the most obvious applications of
the idea is that relating to the transmission of circular motion by
means of a band over two wheels,[75] _A_ and _B_, as shown on page 226.
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
The band may either not be crossed (the case of the two exterior
tangents), or be crossed (the interior tangents), the latter allowing
the wheels to turn in opposite directions. In case the band is liable to
change its length, on account of stretching or variation in heat or
moisture, a third wheel, _D_, is used. We then have the case of tangents
to three pairs of circles. Illustrations of this nature make the
exercise on the drawing of common tangents to two circles assume an
appearance of genuine reality that is of advantage to the work.
[Illustration]
FOOTNOTES:
[68] This is the latest opinion. He is usually assigned to the first
century B.C.
[69] See page 54.
[70] A Greek philosopher and mathematician of the fifth century B.C.
[71] This illustration and the following two are from C. Dupin,
"Mathematics Practically Applied," translated from the French by G.
Birkbeck, Halifax, 1854. This is probably the most scholarly attempt
ever made at constructing a "practical geometry."
[72] This illustration and others of the same type used in this work are
from the excellent drawings by R. W. Billings, in "The Infinity of
Geometric Design Exemplified," London, 1849.
[73] From H. Kolb, "Der Ornamentenschatz ... aus allen Kunst-Epochen,"
Stuttgart, 1883. The original is in the Church of Saint Anastasia in
Verona.
[74] From J. Bennett, "The Arcanum ... A Concise Theory of Practicable
Geometry," London, 1838, one of the many books that have assumed to
revolutionize geometry by making it practical.
[75] The figures are from Dupin, loc. cit.
CHAPTER XVI
THE LEADING PROPOSITIONS OF BOOK III
In the American textbooks Book III is usually assigned to proportion. It
is therefore necessary at the beginning of this discussion to consider
what is meant by ratio and proportion, and to compare the ancient and
the modern theories. The subject is treated by Euclid in his Book V, and
an anonymous commentator has told us that it "is the discovery of
Eudoxus, the teacher of Plato." Now proportion had been known long
before the time of Eudoxus (408-355 B.C.), but it was numerical
proportion, and as such it had been studied by the Pythagoreans. They
were also the first to study seriously the incommensurable number, and
with this study the treatment of proportion from the standpoint of
rational numbers lost its scientific position with respect to geometry.
It was because of this that Eudoxus worked out a theory of geometric
proportion that was independent of number as an expression of ratio.
The following four definitions from Euclid are the basal ones of the
ancient theory:
A ratio is a sort of relation in respect of size between two
magnitudes of the same kind.
Magnitudes are said to have a ratio to one another which are
capable, when multiplied, of exceeding one another.
Magnitudes are said to be in the same ratio, the first to the
second and the third to the fourth, when, if any equimultiples
whatever be taken of the first and third, and any
equimultiples whatever of the second and fourth, the former
equimultiples alike exceed, are alike equal to, or alike fall
short of, the latter equimultiples respectively taken in
corresponding order.
Let magnitudes which have the same ratio be called
proportional.[76]
Of these, the first is so loose in statement as often to have been
thought to be an interpolation of some later writer. It was probably,
however, put into the original for the sake of completeness, to have
some kind of statement concerning ratio as a preliminary to the
important definition of quantities in the same ratio. Like the
definition of "straight line," it was not intended to be taken seriously
as a mathematical statement.
The second definition is intended to exclude zero and infinite
magnitudes, and to show that incommensurable magnitudes are included.
The third definition is the essential one of the ancient theory. It
defines what is meant by saying that magnitudes are in the same ratio;
in other words, it defines a proportion. Into the merits of the
definition it is not proposed to enter, for the reason that it is no
longer met in teaching in America, and is practically abandoned even
where the rest of Euclid's work is in use. It should be said, however,
that it is scientifically correct, that it covers the case of
incommensurable magnitudes as well as that of commensurable ones, and
that it is the Greek forerunner of the modern theories of irrational
numbers.
As compared with the above treatment, the one now given in textbooks is
unscientific. We define ratio as "the quotient of the numerical measures
of two quantities of the same kind," and proportion as "an equality of
ratios."
But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when
we multiply a ratio by [sqrt]5, what is the meaning of this operation?
If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we
assign to "quotient"? If it is the number that shows how many times one
number is contained in another, how many _times_ is [sqrt]3 contained in
[sqrt]2? If to multiply is to take a number a certain number of times,
how many times do we take it when we multiply by [sqrt]5? We certainly
take it more than 2 times and less than 3 times, but what meaning can we
assign to [sqrt]5 times? It will thus be seen that our treatment of
proportion assumes that we already know the theory of irrationals and
can apply it to geometric magnitudes, while the ancient treatment is
independent of this theory.
Educationally, however, we are forced to proceed as we do. Just as
Dedekind's theory of numbers is a simple one for college students, so is
the ancient theory of proportion; but as the former is not suited to
pupils in the high school, so the latter must be relegated to the
college classes. And in this we merely harmonize educational progress
with world progress, for the numerical theory of proportion long
preceded the theory of Eudoxus.