The Teaching of Geometry Part 16

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:

[Ill.u.s.tration]

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_=""><>

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._

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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 circ.u.mlocution, 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._

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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 ill.u.s.tration.[63] If such an instrument is not available in the school, one suitable for ill.u.s.trative purposes can easily be made from cardboard.

[Ill.u.s.tration: PARALLEL RULER OF THE SEVENTEENTH CENTURY

San Giovanni's "Seconda squara mobile," Vicenza, 1686]

A somewhat more complicated form of this instrument may also be made by pupils in manual training, as is shown in this ill.u.s.tration from Bion's great treatise. The principle involved may be taken up in cla.s.s, 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 ill.u.s.trated above. The principle is sometimes used in iron gates.

[Ill.u.s.tration: PARALLEL RULER OF THE EIGHTEENTH CENTURY

N. Bion's "Traite de la construction ... des instrumens de mathematique," The Hague, 1723]

THEOREM. _Two parallelograms are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other._

This proposition is discussed in connection with the one that follows.

THEOREM. _If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal._

These two propositions are not given in Euclid, although generally required by American syllabi of the present time. The last one is particularly useful in subsequent work. Neither one offers any difficulty, and neither has any interesting history. There are, however, numerous interesting applications to the last one. One that is used in mechanical drawing is here ill.u.s.trated.

[Ill.u.s.tration]

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 pa.s.ses through _A_, and line 5 through _B_. We may then p.r.i.c.k 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 compa.s.ses 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_.

[Ill.u.s.tration]