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.

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.

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.

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:

_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:

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.

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.

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.

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_.

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.

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.

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:

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.

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°. Then walk back from the tree until the top is seen at an angle of 45° 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.

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°). 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.

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.

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:

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.

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:

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._

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._

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.

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.

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.

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_.

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:

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.

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° 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.

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°. 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] Méray 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.