9. _And when the straight lines containing the angle cut off a circ.u.mference, the angle is said to stand upon that circ.u.mference._
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 circ.u.mference 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 "circ.u.mference" 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 s.p.a.ce 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" the polygon 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 cla.s.s, 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.
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It may also be mentioned to a cla.s.s 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_, _outside_ the figure. Furthermore, if we consider a quadrilateral as a figure formed by four intersecting lines, _AC_, _CF_, _BE_, and _EA_, it is apparent that this _general quadrilateral_ has six vertices, _A_, _B_, _C_, _D_, _E_, _F_, and three diagonals, _AD_, _BF_, and _CE_. Such broader ideas of geometry form the basis of what is called modern elementary geometry.
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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 with greatest 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 second a 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 subst.i.tute 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 the _Monist_, 1903, p. 303.
FOOTNOTES:
[53] Free use has been made of W. B. Frankland, "The First Book of Euclid"s "Elements,"" Cambridge, 1905; T. L. Heath, "The Thirteen Books of Euclid"s "Elements,"" Cambridge, 1908; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen Bucher," Leipzig, 1901.
[54] For a facsimile of a thirteenth-century MS. containing this definition, see the author"s "Rara Arithmetica," Plate IV, Boston, 1909.
[55] Our slang expression "The cart before the horse" is suggestive of this procedure.
[56] Loc. cit., Vol. II, p. 94.
CHAPTER XIII
HOW TO ATTACK THE EXERCISES
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 what he 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 a.n.a.lysis, 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 century B.C.
For a theorem, the method of a.n.a.lysis consists in reasoning as follows: "I can prove this proposition if I can 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 I _can_ prove 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. a.n.a.lysis 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 a.n.a.lysis and found that he _could_ prove it; he then applied synthesis and _did_ prove it.
For a problem, the method of a.n.a.lysis 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 triangle _ABC_, to draw _PQ_ parallel to the base _AB_, cutting the sides in _P_ and _Q_, so that _PQ_ shall equal _AP_ + _BQ_.
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=a.n.a.lysis.= a.s.sume the problem solved.
Then _AP_ must equal some part of _PQ_ as _PX_, and _BQ_ must equal _QX_.
But if _AP_ = _PX_, what must [L]_PXA_ equal?
[because] _PQ_ is || _AB_, what does [L]_PXA_ equal?
Then why must [L]_BAX_ = [L]_XAP_?
Similarly, what about [L]_QBX_ and [L]_XBA_?
=Construction.= Now reverse the process. What may we do to [Ls]
_A_ and _B_ in order to fix _X_? Then how shall _PQ_ be drawn?
Now give the proof.
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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 from _A_ and _B_ is _PP"_, and the locus of points equidistant from _B_ and _C_ is _QQ"_. These can easily be shown to intersect, as at _O_. Then _O_, being equidistant from _A_, _B_, and _C_, is also on the perpendicular bisector of _AC_. Therefore these bisectors are concurrent in _O_.