CHAPTER XIII

2.Tangent.A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Teachers who prefer to use "circumference" instead of "circle" for the line should notice how often such phrases as "cut the circle" and "intersecting circle" are used,—phrases that signify nothing unless "circle" is taken to mean the line. So Aristotle uses an expression meaning that the locus of a certain point is a circle, and he speaks of a circle as passing through "all the angles." Our word "touch" is from the Latintangere, from which comes "tangent," and also "tag," an old touching game.

3.Tangent Circles.Circles are said to touch one another which, meeting one another, do not cut one another.

The definition has not been looked upon as entirely satisfactory, even aside from its unfortunate phraseology. It is not certain, for instance, whether Euclid meant that the circles could not cut at some other point than that of tangency. Furthermore, no distinction is made between external and internal contact, although both forms are used in the propositions. Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's.

4.Distance.In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn to them from the center are equal.

It is now customary to define "distance" from a point to a line as the length of the perpendicular from the point to the line, and to do this in Book I. In higher mathematics it is found that distance is not a satisfactory term to use, but the objections to it have no particular significance in elementary geometry.

5.Greater Distance.And that straight line is said to be at a greater distance on which the greater perpendicular falls.

Such a definition is not thought essential at the present time.

6.Segment.A segment of a circle is the figure contained by a straight line and the circumference of a circle.

The word "segment" is from the Latin rootsect, meaning "cut." So we have "sector" (a cutter), "section" (a cut), "intersect," and so on. The word is not limited to a circle; we have long spoken of a spherical segment, and it is common to-day to speak of a line segment, to which some would apply a new name "sect." There is little confusion in the matter, however, for the context shows what kind of a segment is to be understood, so that the word "sect" is rather pedantic than important. It will be noticed that Euclid here uses "circumference" to mean "arc."

7.Angle of a Segment.An angle of a segment is that contained by a straight line and a circumference of a circle.

This term has entirely dropped out of geometry, and few teachers would know what it meant if they should hear it used. Proclus called such angles "mixed."

8.Angle in a Segment.An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it tothe extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.

Such an involved definition would not be usable to-day. Moreover, the words "circumference of the segment" would not be used.

9.And when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.

10.Sector.A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.

There is no reason for such an extended definition, our modern phraseology being both more exact (as seen in the above use of "circumference" for "arc") and more intelligible. The Greek word for "sector" is "knife" (tomeus), "sector" being the Latin translation. A sector is supposed to resemble a shoemaker's knife, and hence the significance of the term. Euclid followed this by a definition of similar sectors, a term now generally abandoned as unnecessary.

It will be noticed that Euclid did not use or define the word "polygon." He uses "rectilinear figure" instead. Polygon may be defined to be a bounding line, as a circle is now defined, or as the space inclosed by a broken line, or as a figure formed by a broken line, thus including both the limited plane and its boundary. It is not of any great consequence geometrically which of these ideas is adopted, so that the usual definition of a portion of a plane bounded by a broken line may be taken as sufficient for elementary purposes. It is proper to call attention, however, to the fact that we may have cross polygons of various types, and that the line that "bounds" thepolygon must be continuous, as the definition states. That is, in the second of these figures the shaded portion is not considered a polygon. Such special cases are not liable to arise, but if questions relating to them are suggested, the teacher should be prepared to answer them. If suggested to a class, a note of this kind should come out only incidentally as a bit of interest, and should not occupy much time nor be unduly emphasized.

It may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure, and that the modern idea, which is met in higher mathematics, leads to a modification of earlier concepts. For example, here is a quadrilateral with one of its diagonals,BD,outsidethe figure. Furthermore, if we consider a quadrilateral as a figure formed by four intersecting lines,AC,CF,BE, andEA, it is apparent that thisgeneral quadrilateralhas six vertices,A,B,C,D,E,F, and three diagonals,AD,BF, andCE. Such broader ideas of geometry form the basis of what is called modern elementary geometry.

The other definitions of plane geometry need not be discussed, since all that have any historical interest have been considered. On the whole it may be said that our definitions to-day are not in general so carefully considered as those of Euclid, who weighed each word withgreatest skill, but they are more teachable to beginners, and are, on the whole, more satisfactory from the educational standpoint. The greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small.

Since teachers are occasionally disturbed over the form in which definitions are stated, it is well to say a few words upon this subject. There are several standard types that may be used. (1) We may use the dictionary form, putting the word defined first, thus: "Right triangle. A triangle that has one of its angles a right angle." This is scientifically correct, but it is not a complete sentence, and hence it is not easily repeated when it has to be quoted as an authority. (2) We may put the word defined at the end, thus: "A triangle that has one of its angles a right angle is called a right triangle." This is more satisfactory. (3) We may combine (1) and (2), thus: "Right triangle. A triangle that has one of its angles a right angle is called a right triangle." This is still better, for it has the catchword at the beginning of the paragraph.

There is occasionally some mental agitation over the trivial things of a definition, such as the use of the words "is called." It would not be a very serious matter if they were omitted, but it is better to have them there. The reason is that they mark the statement at once as a definition. For example, suppose we say that "a triangle that has one of its angles a right angle is a right triangle." We have also the fact that "a triangle whose base is the diameter of a semicircle and whose vertex lies on the semicircle is a right triangle." The style of statement is the same, and we have nothing in the phraseology to show that the first is a definition and the seconda theorem. This may happen with most of the definitions, and hence the most careful writers have not consented to omit the distinctive words in question.

Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter.

1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them.

2. Admit no obscure or equivocal terms without defining them.

3. Use in the definitions only terms that are perfectly understood or are there explained.

4. Omit no necessary principles without general agreement, however clear and evident they may be.

5. Set forth in the axioms only those things that are in themselves perfectly evident.

6. Do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it.

7. Prove whatever is in the least obscure, using in the demonstration only axioms that are perfectly evident in themselves, or propositions already demonstrated or allowed.

8. In case of any uncertainty arising from a term employed, always substitute mentally the definition for the term itself.

Bibliography.Heath, Euclid, as cited; Frankland, The First Book of Euclid, as cited; Smith, Teaching of Elementary Mathematics, p. 257, New York, 1900; Young, Teaching of Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in theMonist, 1903, p. 303.

The old geometry, say of a century ago, usually consisted, as has been stated, of a series of theorems fully proved and of problems fully solved. During the nineteenth century exercises were gradually introduced, thus developing geometry from a science in which one learned by seeing things done, into one in which he gained power by actually doing things. Of the nature of these exercises ("originals," "riders"), and of their gradual change in the past few years, mention has been made in Chapter VII. It now remains to consider the methods of attacking these exercises.

It is evident that there is no single method, and this is a fortunate fact, since if it were not so, the attack would be too mechanical to be interesting. There is no one rule for solving every problem nor even for seeing how to begin. On the other hand, a pupil is saved some time by having his attention called to a few rather definite lines of attack, and he will undoubtedly fare the better by not wasting his energies over attempts that are in advance doomed to failure.

There are two general questions to be considered: first, as to the discovery of new truths, and second, as to the proof. With the first the pupil will have little to do, not having as yet arrived at this stage in his progress. A bright student may take a little interest in seeing whathe can find out that is new (at least to him), and if so, he may be told that many new propositions have been discovered by the accurate drawing of figures; that some have been found by actually weighing pieces of sheet metal of certain sizes; and that still others have made themselves known through paper folding. In all of these cases, however, the supposed proposition must be proved before it can be accepted.

As to the proof, the pupil usually wanders about more or less until he strikes the right line, and then he follows this to the conclusion. He should not be blamed for doing this, for he is pursuing the method that the world followed in the earliest times, and one that has always been common and always will be. This is the synthetic method, the building up of the proof from propositions previously proved. If the proposition is a theorem, it is usually not difficult to recall propositions that may lead to the demonstration, and to select the ones that are really needed. If it is a problem, it is usually easy to look ahead and see what is necessary for the solution and to select the preceding propositions accordingly.

But pupils should be told that if they do not rather easily find the necessary propositions for the construction or the proof, they should not delay in resorting to another and more systematic method. This is known as the method of analysis, and it is applicable both to theorems and to problems. It has several forms, but it is of little service to a pupil to have these differentiated, and it suffices that he be given the essential feature of all these forms, a feature that goes back to Plato and his school in the fifth centuryB.C.

For a theorem, the method of analysis consists in reasoning as follows: "I can prove this proposition if Ican prove this thing; I can prove this thing if I can prove that; I can prove that if I can prove a third thing," and so the reasoning runs until the pupil comes to the point where he is able to add, "but Icanprove that." This does not prove the proposition, but it enables him to reverse the process, beginning with the thing he can prove and going back, step by step, to the thing that he is to prove. Analysis is, therefore, his method of discovery of the way in which he may arrange his synthetic proof. Pupils often wonder how any one ever came to know how to arrange the proofs of geometry, and this answers the question. Some one guessed that a statement was true; he applied analysis and found that hecouldprove it; he then applied synthesis anddidprove it.

For a problem, the method of analysis is much the same as in the case of a theorem. Two things are involved, however, instead of one, for here we must make the construction and then prove that this construction is correct. The pupil, therefore, first supposes the problem solved, and sees what results follow. He then reverses the process and sees if he can attain these results and thus effect the required construction. If so, he states the process and gives the resulting proof. For example:

In a triangleABC, to drawPQparallel to the baseAB, cutting the sides inPandQ, so thatPQshall equalAP+BQ.Analysis.Assume the problem solved.ThenAPmust equal some part ofPQasPX, andBQmust equalQX.But ifAP=PX, what must ∠PXAequal?∵PQis ||AB, what does ∠PXAequal?Then why must ∠BAX= ∠XAP?Similarly, what about ∠QBXand ∠XBA?Construction.Now reverse the process. What may we do to ⦞A andBin order to fixX? Then how shallPQbe drawn? Now give the proof.

In a triangleABC, to drawPQparallel to the baseAB, cutting the sides inPandQ, so thatPQshall equalAP+BQ.

Analysis.Assume the problem solved.

ThenAPmust equal some part ofPQasPX, andBQmust equalQX.

But ifAP=PX, what must ∠PXAequal?

∵PQis ||AB, what does ∠PXAequal?

Then why must ∠BAX= ∠XAP?

Similarly, what about ∠QBXand ∠XBA?

Construction.Now reverse the process. What may we do to ⦞A andBin order to fixX? Then how shallPQbe drawn? Now give the proof.

The third general method of attack applies chiefly to problems where some point is to be determined. This is the method of the intersection of loci. Thus, to locate an electric light at a point eighteen feet from the point of intersection of two streets and equidistant from them, evidently one locus is a circle with a radius eighteen feet and the center at the vertex of the angle made by the streets, and the other locus is the bisector of the angle. The method is also occasionally applicable to theorems. For example, to prove that the perpendicular bisectors of the sides of a triangle are concurrent. Here the locus of points equidistant fromAandBisPP', and the locus of points equidistant fromBandCisQQ'. These can easily be shown to intersect, as atO. ThenO, being equidistant fromA,B, andC, is also on the perpendicular bisector ofAC. Therefore these bisectors are concurrent inO.

These are the chief methods of attack, and are all that should be given to an average class for practical use.

Besides the methods of attack, there are a few general directions that should be given to pupils.

1. In attacking either a theorem or a problem, take the most general figure possible. Thus, if a proposition relates to a quadrilateral, take one with unequal sides and unequal angles rather than a square or even a rectangle. The simpler figures often deceive a pupil into feeling that he has a proof, when in reality he has one only for a special case.

2. Set forth very exactly the thing that is given, using letters relating to the figure that has been drawn. Then set forth with the same exactness the thing that is to be proved. The neglect to do this is the cause of a large per cent of the failures. The knowing of exactly what we have to do and exactly what we have with which to do it is half the battle.

3. If the proposition seems hazy, the difficulty is probably with the wording. In this case try substituting the definition for the name of the thing defined. Thus instead of thinking too long about proving that the median to the base of an isosceles triangle is perpendicular to the base, draw the figure and think that there is given

AC=BC,AD=BD,

and that there is to be proved that

∠CDA= ∠BDC.

Here we have replaced "median," "isosceles," and "perpendicular" by statements that express the same idea in simpler language.

Bibliography.Petersen, Methods and Theories for the Solution of Geometric Problems of Construction, Copenhagen, 1879, a curious piece of English and an extreme view of the subject, but well worth consulting; Alexandroff, Problèmes de géométrie élémentaire, Paris, 1899, with a German translation in 1903; Loomis, Original Investigation; or, How to attack an Exercise in Geometry, Boston, 1901; Sauvage, Les Lieux géométriques en géométrie élémentaire, Paris, 1893; Hadamard, Leçons de géométrie, p. 261, Paris, 1898; Duhamel, Des Méthodes dans les sciences de raisonnement, 3eéd., Paris, 1885; Henrici and Treutlein, Lehrbuch der Elementar-Geometrie, Leipzig, 3. Aufl., 1897; Henrici, Congruent Figures, London, 1879.

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 façade. 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 treatisebiblion(βιβλίον), 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 Latincaput(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.640-548B.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 onpage 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 oflines 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 fromAto a pointP. They then turned and measuredABat right angles toAP. They placed a stake atO, halfway fromAtoB, and drew a perpendicular toABatB. They placed a stake atC, on this perpendicular, and in line withOandP. They then found the width by measuringBC. 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 inChapter 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-Nairīzī[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 towerto 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 thatone 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.

Sixteenth-Century Mensuration Belli's "Del Misurar con la Vista," Venice, 1569Sixteenth-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 twosidesand the includedangleof the one are equal respectively to twosidesand the includedangleof the other.Two triangles are congruent if twoanglesand the includedsideof the one are equal respectively to twoanglesand the includedsideof the other.

Two triangles are congruent if twosidesand the includedangleof the one are equal respectively to twosidesand the includedangleof the other.

Two triangles are congruent if twoanglesand the includedsideof the one are equal respectively to twoanglesand the includedsideof the other.

In general, to every proposition involvingpointsandlinesthere is a reciprocal proposition involvinglinesandpointsrespectively 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 thereforehis proof is much more involved than ours. He makesCX=CY, and proves ⧌XBCandYACcongruent,[62]and also ⧌XBAandYABcongruent. Then from ∠YAChe takes ∠YAB, leaving ∠BAC, and so on the other side, leaving ∠CBA, these therefore being equal.

This proposition has long been called thepons 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 centuryA.D.), a Greek commentator on Euclid, proved the proposition as follows:

LetABCbe the triangle, withAB=AC. Conceive of this as two triangles; thenAB=AC,AC=AB, and ∠Ais common; hence the ⧌ABCandACBare congruent, and ∠Bof the one equals ∠Cof the other.

This is a better plan than that followed by some textbook writers of imagining ⧍ABCtaken up and laid down onitself. 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 "sometriangles 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 thata=bandc=d, to prove thatx=y, the partial converse would have givena=bandx=y, to prove thatc=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 thesequence 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 thereductio ad absurdum, mentioned below, it is more satisfactory in most particulars. The proof is as follows:

Given the triangle ABC, with the angle A equal to the angle B.Given the triangle ABC, with the angle A equal to the angle B.

To prove thatAC=BC.

Proof.Suppose the second triangleA'B'C'to be an exact reproduction of the given triangleABC.

Turn the triangleA'B'C'over and place it uponABCso thatB'shall fall onAandA'shall fall onB.

ThenB'A'will coincide withAB.Since ∠A'= ∠B', Givenand ∠A= ∠A', Hyp.∴∠A= ∠B'.∴B'C'will lie alongAC.Similarly,A'C'will lie alongBC.

ThereforeC'will fall on bothACandBC, and hence at their intersection.

∴B'C'=AC.ButB'C'was made equal toBC.∴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 consistsin bisecting the vertical angle, and then proving the two triangles congruent. Among the other proofs is that of thereductio ad absurdum, which the student might now meet, but which may better be postponed. The phrasereductio ad absurdumseems 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 areABCandA'B'C', with sidesa,b,canda',b',c', then the second proposition asserts that ifb=b',c=c', and ∠A= ∠A', thena=a'and the triangles are congruent, while this proposition asserts that ifa=a',b=b', andc=c', then ∠A= ∠A'and the triangles are congruent.

The proposition was known at least as early as Aristotle's time. Euclid proved it by inserting a preliminaryproposition to the effect that it is impossible to have on the same baseABand the same side of it two different trianglesABCandABC', withAC=AC', andBC=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 ofnsides. In particular, the following questions may be asked of a class:

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