IfABCis the given triangle, andCP_1,CP_2 are respectively the internal and external bisectors, thenABis divided harmonically byP1andP2.∴AP1:P1B=AP2:P2B.∴AP2:P2B=AP2-P1P2:P1P2-P2B,and this is the criterion for the harmonic progression still seen in many algebras. For, lettingAP2=a,P1P2=b,P2B=c, we havea/c= (a-b)/(b-c),which is also derived from taking the reciprocals ofa,b,c, and placing them in an arithmetical progression, thus:1/b- 1/a= 1/c- 1/b,whence (a-b)/ab= (b-c)/bc,or (a-b)/(b-c) =ab/bc=a/c.This is the reason why the lineABis said to be divided harmonically. The lineP1P2is also called theharmonic meanbetweenAP2andP2B, and the pointsA,P1,B,P2are said to form anharmonic range.It may be noted that ∠P2CP1, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratioCA:CBis given, andABis given, thenP1andP2are both fixed. HenceCmust lie on a semicircle withP1P2as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius.
IfABCis the given triangle, andCP_1,CP_2 are respectively the internal and external bisectors, thenABis divided harmonically byP1andP2.
∴AP1:P1B=AP2:P2B.∴AP2:P2B=AP2-P1P2:P1P2-P2B,
and this is the criterion for the harmonic progression still seen in many algebras. For, lettingAP2=a,P1P2=b,P2B=c, we have
a/c= (a-b)/(b-c),
which is also derived from taking the reciprocals ofa,b,c, and placing them in an arithmetical progression, thus:
1/b- 1/a= 1/c- 1/b,whence (a-b)/ab= (b-c)/bc,or (a-b)/(b-c) =ab/bc=a/c.
This is the reason why the lineABis said to be divided harmonically. The lineP1P2is also called theharmonic meanbetweenAP2andP2B, and the pointsA,P1,B,P2are said to form anharmonic range.
It may be noted that ∠P2CP1, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratioCA:CBis given, andABis given, thenP1andP2are both fixed. HenceCmust lie on a semicircle withP1P2as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius.
At this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, saying that such figures have "their sides proportional and their anglesequal." Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of "corresponding angles" we may say "homologous angles," but there seems to be no reason for using the less familiar word.
It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously similar of all figures,—two circles; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points,A1,A2,A3, ..., andB1,B2,B3, ..., as such thatA1A2,B1B2,C1C2, ... are concurrent inO, andA1O:A2O=B1O:B2O=C1O:C2O= ... Here the constant ratioA1O:A2Ois called theratio of similitude, andOis called thecenter of similitude. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilateralsrespectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead.
Among the interesting applications of similarity is the case of a shadow, as here shown, where the light is the center of similitude. It is also well known to most high school pupils that in a camera the lens reverses the image. The mathematical arrangement is here shown, the lens inclosing the center of similitude. The proposition may also be applied to the enlargement of maps and working drawings.
The propositions concerning similar figures have no particularly interesting history, nor do they present any difficulties that call for discussion. In schools where there is a little time for trigonometry, teachers sometimes find it helpful to begin such work at this time, since all of the trigonometric functions depend upon the properties of similar triangles, and a brief explanation of the simplest trigonometric functions may add a little interest to the work. In the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection.
It is a mistaken idea that geometry is a prerequisite to trigonometry. We can get along very well in teaching trigonometry if we have three propositions: (1) the one about the sum of the angles of a triangle; (2) the Pythagorean Theorem; (3) the one that asserts that two right triangles are similar if an acute angle of the one equals an acute angle of the other. For teachers who may care to make a little digression at this time, the following brief statement of a few of the facts of trigonometry may be of value:
In the right triangleOABwe shall letAB=y,OA=x,OB=r, thus adopting the letters of higher mathematics. Then, so long as ∠Oremains the same, such ratios asy/x,y/r, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we cally/rthesineofO, and we write sinO=y/r;x/rthecosineofO, and we write cosO=x/r;y/xthetangentofO, and we write tanO=y/x.Now becausesinO=y/r, thereforersinO=y;and because cosO=x/r, thereforercosO=x;and because tanO=y/x, thereforextanO=y.Hence, if we knew the values of sinO, cosO, and tanOfor the various angles, we could findx,y, orrif we knew any one of them.Now the values of the sine, cosine, and tangent (functionsof the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining themby actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:AngleSineCosineTangentAngleSineCosineTangent5°.087.996.08750°.766.6431.19210°.174.985.17655°.819.5741.42815°.259.966.26860°.866.5001.73220°.342.940.36465°.906.4232.14525°.423.906.46670°.940.3422.74830°.500.866.57775°.966.2593.73235°.574.819.70080°.985.1745.67140°.643.766.83985°.996.08711.43045°.707.7071.00090°1.000.000∞It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5° is 0.99619, and the sine of 85° is 0.99619. The entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently assigned the computing of a simpler table as a class exercise.Referring to the figure, if we know thatr= 30 and ∠O= 40°, then sincey=rsinO, we havey= 30 × 0.643 = 19.29. If we know thatx= 60 and ∠O= 35°, then sincey=xtanO, we havey= 60 × 0.7 = 42. We may also findr, for cosO=x/r, whencer=x/(cosO) = 60/0.819 = 73.26.
In the right triangleOABwe shall letAB=y,OA=x,OB=r, thus adopting the letters of higher mathematics. Then, so long as ∠Oremains the same, such ratios asy/x,y/r, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we call
y/rthesineofO, and we write sinO=y/r;x/rthecosineofO, and we write cosO=x/r;y/xthetangentofO, and we write tanO=y/x.
Now because
sinO=y/r, thereforersinO=y;and because cosO=x/r, thereforercosO=x;and because tanO=y/x, thereforextanO=y.
Hence, if we knew the values of sinO, cosO, and tanOfor the various angles, we could findx,y, orrif we knew any one of them.
Now the values of the sine, cosine, and tangent (functionsof the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining themby actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:
It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5° is 0.99619, and the sine of 85° is 0.99619. The entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently assigned the computing of a simpler table as a class exercise.
Referring to the figure, if we know thatr= 30 and ∠O= 40°, then sincey=rsinO, we havey= 30 × 0.643 = 19.29. If we know thatx= 60 and ∠O= 35°, then sincey=xtanO, we havey= 60 × 0.7 = 42. We may also findr, for cosO=x/r, whencer=x/(cosO) = 60/0.819 = 73.26.
Therefore, if we could easily measure ∠Oand could measure the distancex, we could find the height of abuildingy. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. We may then measure the distance to the building and apply the formulay=xtanO.
A Quadrant of the Sixteenth Century Finaeus's "De re et praxi geometrica," Paris, 1556A Quadrant of the Sixteenth CenturyFinaeus's "De re et praxi geometrica," Paris, 1556
It should always be understood that expensive apparatus is not necessary for such illustrative work. The telescope used on the transit is only three hundred yearsold, and the world got along very well with its trigonometry before that was invented. So a little ingenuity will enable any one to make from cheap protractors about as satisfactory instruments as the world used before 1600. In order that this may be the more fully appreciated, a few illustrations are here given, showing the old instruments and methods used in practical surveying before the eighteenth century.
A Quadrant of the Seventeenth CenturyA Quadrant of the Seventeenth Century
The illustration onpage 236shows a simple form of the quadrant, an instrument easily made by a pupil whomay be interested in outdoor work. It was the common surveying instrument of the early days. A more elaborate example is seen in the illustration, onpage 237, of a seventeenth-century brass specimen in the author's collection.[77]
A Quadrant of the Seventeenth Century Bartoli's "Del modo di misurare," Venice, 1689A Quadrant of the Seventeenth CenturyBartoli's "Del modo di misurare," Venice, 1689
Another type, easily made by pupils, is shown in the above illustration from Bartoli, 1689. Such instruments were usually made of wood, brass, or ivory.[78]
Instruments for the running of lines perpendicular to other lines were formerly common, and are easily made. They suffice, as the following illustration shows, for surveying an ordinary field.
Surveying Instrument of the Eighteenth Century N. Bion's "Traité de la construction ... des instrumens de mathématique," The Hague, 1723Surveying Instrument of the Eighteenth CenturyN. Bion's "Traité de la construction ... des instrumens de mathématique," The Hague, 1723
The Quadrant Used for Altitudes Finaeus's "De re et praxi geometrica," Paris, 1556The Quadrant Used for AltitudesFinaeus's "De re et praxi geometrica," Paris, 1556
The quadrant was practically used for all sorts of outdoor measuring. For example, the illustration from Finaeus, on this page, shows how it was used for altitudes, and the one reproduced onpage 240shows how it was used for measuring depths.
A similar instrument from the work of Bettinus is given onpage 241, the distance of a ship being found by constructing an isosceles triangle. A more elaborate form, with a pendulum attachment, is seen in the illustration from De Judaeis, which also appears onpage 241.
The Quadrant Used for Depths Finaeus's "Protomathesis," Paris, 1532The Quadrant Used for Depths Finaeus's "Protomathesis," Paris, 1532
A Quadrant of the Sixteenth Century De Judaeis's "De quadrante geometrico," Nürnberg, 1594A Quadrant of the Sixteenth CenturyDe Judaeis's "De quadrante geometrico," Nürnberg, 1594
The Quadrant Used for Distances Bettinus's "Apiaria universae philosophiae mathematicae," Bologna, 1645The Quadrant Used for DistancesBettinus's "Apiaria universae philosophiae mathematieae," Bologna, 1645
The quadrant finally developed into the octant, as shown in the following illustration from Hoffmann, and this in turn developed into the sextant, which is now used by all navigators.
The Octant Hoffmann's "De Octantis," Jena, 1612The OctantHoffmann's "De Octantis," Jena, 1612
In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given onpage 243shows how in early days a simple device was used for this purpose. Two similar triangles are formed in this way, and we have only to measure the height of the eye above the ground, and the distances of the mirror from the tower and the observer, to have three terms of a proportion.
All of these instruments are easily made. The mirror is always at hand, and a paper protractor on apiece of board, with a plumb line attached, serves as a quadrant. For a few cents, and by the expenditure of an hour or so, a school can have almost as good instruments as the ordinary surveyor had before the nineteenth century.
The Speculum Finaeus's "De re et praxi geometrica," Paris, 1556The SpeculumFinaeus's "De re et praxi geometrica," Paris, 1556
A well-known method of measuring the distance across a stream is illustrated in the figure below, where the distance fromAto some pointPis required.
Run a line fromAtoCby standing atCin line withAandP. Then run two perpendiculars fromAandCby any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some pointD, onCD, toP, putting a stake atB. Then run the perpendicularBE. SinceDE:EB=BA:AP, and since we can measureDE,EB, andBAwith the tape, we can compute the distanceAP.
Run a line fromAtoCby standing atCin line withAandP. Then run two perpendiculars fromAandCby any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some pointD, onCD, toP, putting a stake atB. Then run the perpendicularBE. SinceDE:EB=BA:AP, and since we can measureDE,EB, andBAwith the tape, we can compute the distanceAP.
There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. Other figures are suggested onpage 244, and the triangles need not be confined to those having a right angle.
A very simple illustration of the use of similar triangles is found in one of the stories told of Thales. It is related that he found the height of the pyramids by measuring their shadow at the instant when his own shadow just equaled his height. He thus had the case of two similar isosceles triangles. This is an interesting exercise which may be tried about the time that pupils are leaving school in the afternoon.
Another application of the same principle is seen in a method often taken for measuring the height of a tree.
The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in whichAB=BC. He holdsABlevel and walks toward the tree until he just sees the top alongAC. Then becauseAB=BC,andAB:BC=AD:DE,the height aboveDwill equal the distanceAD.Questions like the following may be given to the class:1. What is the height of the tree in the picture if the triangle is 5 ft. 4 in. from the ground, andADis 23 ft. 8 in.?2. Suppose a triangle is used which hasAB= twiceBC. What is the height ifAD= 75 ft.?
The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in whichAB=BC. He holdsABlevel and walks toward the tree until he just sees the top alongAC. Then because
AB=BC,andAB:BC=AD:DE,
the height aboveDwill equal the distanceAD.
Questions like the following may be given to the class:
1. What is the height of the tree in the picture if the triangle is 5 ft. 4 in. from the ground, andADis 23 ft. 8 in.?
2. Suppose a triangle is used which hasAB= twiceBC. What is the height ifAD= 75 ft.?
There are many variations of this principle. One consists in measuring the shadows of a tree and a staff at the same time. The height of the staff being known, the height of the tree is found by proportion. Another consists in sighting from the ground, across a mark on an upright staff, to the top of the tree. The height of the mark being known, and the distances from the eye to the staff and to the tree being measured, the height of the tree is found.
An instrument sold by dealers for the measuring of heights is known as the hypsometer. It is made of brass, and is of the form here shown. The base is graduated in equal divisions, say 50, and the upright bar is similarly divided. At the ends of the hinged radius are two sights. If the observer stands 50 feet from a tree and sights at the top, so that the hinged radius cuts the upright bar at 27, then he knows at once that the tree is 27 feet high. It is easy for a class to make a fairly good instrument of this kind out of stiff pasteboard.
An interesting application of the theorem relating to similar triangles is this: Extend your arm and point to a distant object, closing your left eye and sighting across your finger tip with your right eye. Now keep the finger in the same position and sight with your left eye. The finger will then seem to be pointing to an object some distance to the right of the one at which you were pointing. If you can estimate the distance between these two objects, which can often be done with a fair degree of accuracy when there are houses intervening, then you will be able to tell approximately your distance from theobjects, for it will be ten times the estimated distance between them. The finding of the reason for this by measuring the distance between the pupils of the two eyes, and the distance from the eye to the finger tip, and then drawing the figure, is an interesting exercise.
Perhaps some pupil who has read Thoreau's descriptions of outdoor life may be interested in what he says of his crude mathematics. He writes, "I borrowed the plane and square, level and dividers, of a carpenter, and with a shingle contrived a rude sort of a quadrant, with pins for sights and pivots." With this he measured the heights of a cliff on the Massachusetts coast, and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and distances in which he is interested.
Theorem.If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:
1.The triangles thus formed are similar to the given triangle, and are similar to each other.
2.The perpendicular is the mean proportional between the segments of the hypotenuse.
3.Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side.
To this important proposition there is one corollary of particular interest, namely,The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter. By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please.
Thus we may makeAD= 2 in.,DB= 3 in., and erectDC⊥ toAB. Then the length ofDCwill be √6 in., and we may find √6 approximately by measuringDC.
Thus we may makeAD= 2 in.,DB= 3 in., and erectDC⊥ toAB. Then the length ofDCwill be √6 in., and we may find √6 approximately by measuringDC.
Furthermore, if we introduce negative magnitudes into geometry, and letDB= +3 andDA= -2, thenDCwill equal √(-6). In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities, as is done in the graphic treatment of imaginaries in algebra.
Furthermore, if we introduce negative magnitudes into geometry, and letDB= +3 andDA= -2, thenDCwill equal √(-6). In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities, as is done in the graphic treatment of imaginaries in algebra.
It is an interesting exercise to have a class find, to one decimal place, by measuring as above, the value of √2, √3, √5, and √9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of √(-6), √(-7), ..., may be found in the same way.
A practical illustration of the value of the above theorem is seen in a method for finding distances that is frequently described in early printed books. It seems to have come from the Roman surveyors.
If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant pointBand a pointAnear the stick, then the two triangles are similar. HenceAD:DC=DC:DB. Hence, ifADandDCare measured,DBcan be found. The experiment is an interesting and instructive one for a class, especially as the square can easily be made out of heavy pasteboard.
If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant pointBand a pointAnear the stick, then the two triangles are similar. HenceAD:DC=DC:DB. Hence, ifADandDCare measured,DBcan be found. The experiment is an interesting and instructive one for a class, especially as the square can easily be made out of heavy pasteboard.
Theorem.If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other.
Theorem.If from a point without a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment.
Corollary.If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn.
These two propositions and the corollary are all parts of one general proposition:If through a point a line is drawn cutting a circle, the product of the segments of the line is constant.
IfPis within the circle, thenxx'=yy'; ifPis on the circle, thenxandybecome 0, and 0 ·x'= 0 ·y'= 0; ifPis atP3, thenxandy, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; ifPis atP4, theny=y'and we havexx'=y2, orx:y=y:x', as stated in the proposition.
IfPis within the circle, thenxx'=yy'; ifPis on the circle, thenxandybecome 0, and 0 ·x'= 0 ·y'= 0; ifPis atP3, thenxandy, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; ifPis atP4, theny=y'and we havexx'=y2, orx:y=y:x', as stated in the proposition.
We thus have an excellent example of the Principle of Continuity, and classes are always interested to consider the result of lettingPassume various positions. Among the possible cases is the one of two tangents from an external point, and the one wherePis at the center of the circle.
Students should frequently be questioned as to the meaning of "product of lines." The Greeks always used "rectangle of lines," but it is entirely legitimate to speak of "product of lines," provided we define the expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter.
Theorem.The square on the bisector of an angle of a triangle is equal to the product of the sides of this anglediminished by the product of the segments made by the bisector upon the third side of the triangle.
This proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known.
For, in this figure, leta= 3,b= 5, andc= 6.Then ∵x:y=b:a, andy= 6 -x,we havex/(6 -x) = 5/3.∴ 3x= 30 - 5x.∴x= 3 3/4,y= 2 1/4.By the theorem,z2=ab-xy= 15 - (8 7/16) = 6 9/16.∴z= √(6 9/16) = 1/4 √105 = 2.5+.
For, in this figure, leta= 3,b= 5, andc= 6.
Then ∵x:y=b:a, andy= 6 -x,we havex/(6 -x) = 5/3.∴ 3x= 30 - 5x.∴x= 3 3/4,y= 2 1/4.By the theorem,z2=ab-xy= 15 - (8 7/16) = 6 9/16.∴z= √(6 9/16) = 1/4 √105 = 2.5+.
Theorem.In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side.
This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circumscribed circle in terms of the three sides.
For if we designate the sides bya,b, andc, as usual, and letCD=dandPB=x, then(CP)2=a2-x2=b2- (c-x)2.∴a2-x2=b2-c2+ 2cx-x2.∴x= (a2-b2+c2) / 2c.∴ (CP)2=a2- ((a2-b2+c2) / 2c)2.ButCP·d=ab.∴d= 2abc/ √(4a2c2- (a2-b2+c2)2).
For if we designate the sides bya,b, andc, as usual, and letCD=dandPB=x, then
(CP)2=a2-x2=b2- (c-x)2.∴a2-x2=b2-c2+ 2cx-x2.∴x= (a2-b2+c2) / 2c.∴ (CP)2=a2- ((a2-b2+c2) / 2c)2.ButCP·d=ab.∴d= 2abc/ √(4a2c2- (a2-b2+c2)2).
This is not available at this time, however, because the Pythagorean Theorem has not been proved.
These two propositions are merely special cases of the following general theorem, which may be given as an interesting exercise:
If ABC is an inscribed triangle, and through C there are drawn two straight lines CD, meeting AB in D, and CP, meeting the circle in P, with angles ACD and PCB equal, then AC × BC will equal CD × CP.
Fig. 1 is the general case whereDfalls betweenAandB. IfCPis a diameter, it reduces to the second figure given onpage 249. IfCPbisects ∠ACB, we have Fig. 3, from which may be proved the proposition given at the foot ofpage 248. IfDlies onBAproduced, we have Fig. 2. IfDlies onABproduced, we have Fig. 4.This general proposition is proved by showing that ⧌ADCandPBCare similar, exactly as in the second proposition given onpage 249.
Fig. 1 is the general case whereDfalls betweenAandB. IfCPis a diameter, it reduces to the second figure given onpage 249. IfCPbisects ∠ACB, we have Fig. 3, from which may be proved the proposition given at the foot ofpage 248. IfDlies onBAproduced, we have Fig. 2. IfDlies onABproduced, we have Fig. 4.
This general proposition is proved by showing that ⧌ADCandPBCare similar, exactly as in the second proposition given onpage 249.
These theorems are usually followed by problems of construction, of which only one has great interest, namely,To divide a given line in extreme and mean ratio.
The purpose of this problem is to prepare for the construction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called "the golden section," and is probably "the section" mentioned by Proclus when he says that Eudoxus "greatly added to the number of the theorems whichPlato originated regarding the section." The expression "golden section" is not old, however, and its origin is uncertain.
If a lineABis divided in golden section atP, we haveAB×PB= (AP)2.Therefore, ifAB=a, andAP=x, we havea(a-x) =x2,orx2+ax-a2= 0;whencex= -a/2 ±a/2√5=a(1.118 - 0.5)= 0.618a,the other root representing the external point.That is,x= about 0.6a, anda-x= about 0.4a, andais therefore divided in about the ratio of 2 : 3.
If a lineABis divided in golden section atP, we have
AB×PB= (AP)2.
Therefore, ifAB=a, andAP=x, we have
a(a-x) =x2,orx2+ax-a2= 0;whencex= -a/2 ±a/2√5=a(1.118 - 0.5)= 0.618a,
the other root representing the external point.
That is,x= about 0.6a, anda-x= about 0.4a, andais therefore divided in about the ratio of 2 : 3.
There has been a great deal written upon the æsthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern.
Book IV treats of the area of polygons, and offers a large number of practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made in this chapter to suggest practical problems to the teachers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important.
Theorem.Two rectangles having equal altitudes are to each other as their bases.
Euclid's statement (Book VI, Proposition 1) was as follows:Triangles and parallelograms which are under the same height are to one another as their bases. Our plan of treating the two figures separately is manifestly better from the educational standpoint.
In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geometry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the importantthing is that the commensurable case should be proved and the incommensurable one recognized.
Euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was necessary. The French geometer, Legendre, gave a rigorous proof byreductio ad absurdum. In America the pupils are hardly ready for these proofs, and so our treatment by limits is less rigorous than these earlier ones.
Theorem.The area of a rectangle is equal to the product of its base by its altitude.
The easiest way to introduce this is to mark a rectangle, with commensurable sides, on squared paper, and count up the squares; or, what is more convenient, to draw the rectangle and mark the area off in squares.
It is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rectangle is no criterion as to the area. Thus, if a rectangle has an area of 1 square foot and is only 1/440 of an inch high, the perimeter is over 2 miles. The story of how Indians were induced to sell their land by measuring the perimeter is a very old one. Proclus speaks of travelers who described the size of cities by the perimeters, and of men who cheated others by pretending to give them as much land as they themselves had, when really they made only the perimeters equal. Thucydides estimated the size of Sicily by the time it took to sail round it. Pupils will be interested to know in this connection that of polygons having the same perimeter and the same number of sides, the one having equal sides and equal angles is the greatest, and that of plane figures having the same perimeter, the circle is the greatest. These facts were known to the Greek writers, Zenodorus (ca.150B.C.) and Proclus (410-485A.D.).
The surfaces of rectangular solids may now be found, there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so.
Theorem.The area of a parallelogram is equal to the product of its base by its altitude.
The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory.
Theorem.The area of a triangle is equal to half the product of its base by its altitude.
Of course, the Greeks would never have used the wording of either of these two propositions. Euclid, for example, gives this one as follows:If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.As to the parallelogram, he simply says it is equal to a parallelogram of equal base and "in the same parallels," which makes it equal to a rectangle of the same base and the same altitude.
The number of applications of these two theorems is so great that the teacher will not be at a loss to find genuine ones that appeal to the class. Teachers may now introduce pyramids, requiring the areas of the triangular faces to be found.
The Ahmes papyrus (ca.1700B.C.) gives the area of an isosceles triangle as ½bs, wheresis one of the equalsides, thus takingsfor the altitude. This shows the primitive state of geometry at that time.
Theorem.The area of a trapezoid is equal to half the sum of its bases multiplied by the altitude.
An interesting variation of the ordinary proof is made by placing a trapezoidT', congruent toT, in the position here shown. The parallelogram formed equalsa(b+b'), and thereforeT=a· (b+b')/2.The proposition should be discussed for the caseb=b', when it reduces to the one about the area of a parallelogram. Ifb'= 0, the trapezoid reduces to a triangle, andT=a·b/2.
An interesting variation of the ordinary proof is made by placing a trapezoidT', congruent toT, in the position here shown. The parallelogram formed equalsa(b+b'), and therefore
T=a· (b+b')/2.
The proposition should be discussed for the caseb=b', when it reduces to the one about the area of a parallelogram. Ifb'= 0, the trapezoid reduces to a triangle, andT=a·b/2.
This proposition is the basis of the theory of land surveying, a piece of land being, for purposes of measurement, divided into trapezoids and triangles, the latter being, as we have seen, a kind of special trapezoid.
The proposition is not in Euclid, but is given by Proclus in the fifth century.
The term "isosceles trapezoid" is used to mean a trapezoid with two opposite sides equal, but not parallel. The area of such a figure was incorrectly given by the Ahmes papyrus as ½(b+b')s, wheresis one of the equal sides. This amounts to takings=a.
The proposition is particularly important in the surveying of an irregular field such as is found in hilly districts. It is customary to consider the field as a polygon, and to draw a meridian line, letting fall perpendiculars upon it from the vertices, thus forming triangles and trapezoids that can easily be measured. An older plan, but one better suited to the use of pupils who may be working only with the tape, is given onpage 99.
Theorem.The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.
This proposition may be omitted as far as its use in plane geometry is concerned, for we can prove the next proposition here given without using it. In solid geometry it is used only in a proposition relating to the volumes of two triangular pyramids having a common trihedral angle, and this is usually omitted. But the theorem is so simple that it takes but little time, and it adds greatly to the student's appreciation of similar triangles. It not only simplifies the next one here given, but teachers can at once deduce the latter from it as a special case by asking to what it reduces if a second angle of one triangle is also equal to a second angle of the other triangle.
It is helpful to give numerical values to the sides of a few triangles having such equal angles, and to find the numerical ratio of the areas.
Theorem.The areas of two similar triangles are to each other as the squares on any two corresponding sides.