CHAPTER XVI

THE LEADING PROPOSITIONS OF BOOK III

In the American textbooks Book III is usually assigned to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion, and to compare the ancient and the modern theories. The subject is treated by Euclid in his Book V, and an anonymous commentator has told us that it "is the discovery of Eudoxus, the teacher of Plato." Now proportion had been known long before the time of Eudoxus (408-355 B.C.), but it was numerical proportion, and as such it had been studied by the Pythagoreans. They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry. It was because of this that Eudoxus worked out a theory of geometric proportion that was independent of number as an expression of ratio.

The following four definitions from Euclid are the basal ones of the ancient theory:

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Let magnitudes which have the same ratio be called proportional.[76]

Of these, the first is so loose in statement as often to have been thought to be an interpolation of some later writer. It was probably, however, put into the original for the sake of completeness, to have some kind of statement concerning ratio as a preliminary to the important definition of quantities in the same ratio. Like the definition of "straight line," it was not intended to be taken seriously as a mathematical statement.

The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included.

The third definition is the essential one of the ancient theory. It defines what is meant by saying that magnitudes are in the same ratio; in other words, it defines a proportion. Into the merits of the definition it is not proposed to enter, for the reason that it is no longer met in teaching in America, and is practically abandoned even where the rest of Euclid's work is in use. It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers.

As compared with the above treatment, the one now given in textbooks is unscientific. We define ratio as "the quotient of the numerical measures of two quantities of the same kind," and proportion as "an equality of ratios."

But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when we multiply a ratio by [sqrt]5, what is the meaning of this operation? If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we assign to "quotient"? If it is the number that shows how many times one number is contained in another, how many _times_ is [sqrt]3 contained in [sqrt]2? If to multiply is to take a number a certain number of times, how many times do we take it when we multiply by [sqrt]5? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to [sqrt]5 times? It will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geometric magnitudes, while the ancient treatment is independent of this theory.

Educationally, however, we are forced to proceed as we do. Just as Dedekind's theory of numbers is a simple one for college students, so is the ancient theory of proportion; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college classes. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus.

The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alternately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the proposition concerning "composition and division" is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry.

We shall now consider a few of the principal propositions of Book III.

THEOREM. _If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally._

In addition to the usual proof it is instructive to consider in class the cases in which the parallel is drawn through the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex.

THEOREM. _The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides._

The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed together at this time. The proposition relating to the exterior angle was recognized by Pappus of Alexandria.

If _ABC_ is the given triangle, and _CP__{1}, _CP__{2} are respectively the internal and external bisectors, then _AB_ is divided harmonically by _P__{1} and _P__{2}.

[therefore]_AP__{1} : _P__{1}_B_ = _AP__{2} : _P__{2}_B_.

[therefore]_AP__{2} : _P__{2}_B_ = _AP__{2} - _P__{1}_P__{2} : _P__{1}_P__{2} - _P__{2}_B_,

and this is the criterion for the harmonic progression still seen in many algebras. For, letting _AP__{2} = _a_, _P__{1}_P__{2} = _b_, _P__{2}_B_ = _c_, we have

_a_/_c_ = (_a_ - _b_)/(_b_ - _c_),

which is also derived from taking the reciprocals of _a_, _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 line _AB_ is said to be divided harmonically. The line _P__{1}_P__{2} is also called the _harmonic mean_ between _AP__{2} and _P__{2}_B_, and the points _A_, _P__{1}, _B_, _P__{2} are said to form an _harmonic range_.

It may be noted that [L]_P__{2}_CP__{1}, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio _CA_ : _CB_ is given, and _AB_ is given, then _P__{1} and _P__{2} are both fixed. Hence _C_ must lie on a semicircle with _P__{1}_P__{2} as 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 angles equal." 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, _A__{1}, _A__{2}, _A__{3}, ..., and _B__{1}, _B__{2}, _B__{3}, ..., as such that _A__{1}_A__{2}, _B__{1}_B__{2}, _C__{1}_C__{2}, ... are concurrent in _O_, and _A__{1}_O_ : _A__{2}_O_ = _B__{1}_O_ : _B__{2}_O_ = _C__{1}_O_ : _C__{2}_O_ = ... Here the constant ratio _A__{1}_O_ : _A__{2}_O_ is called the _ratio of similitude_, and _O_ is called the _center 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 quadrilaterals respectively 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 triangle _OAB_ we shall let _AB_ = _y_, _OA_ = _x_, _OB_ = _r_, thus adopting the letters of higher mathematics. Then, so long as [L]_O_ remains the same, such ratios as _y_/_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_/_r_ the _sine_ of _O_, and we write sin _O_ = _y_/_r_;

_x_/_r_ the _cosine_ of _O_, and we write cos _O_ = _x_/_r_;

_y_/_x_ the _tangent_ of _O_, and we write tan _O_ = _y_/_x_.

Now because

sin _O_ = _y_/_r_, therefore _r_ sin _O_ = _y_;

and because cos _O_ = _x_/_r_, therefore _r_ cos _O_ = _x_;

and because tan _O_ = _y_/_x_, therefore _x_ tan _O_ = _y_.

Hence, if we knew the values of sin _O_, cos _O_, and tan _O_ for the various angles, we could find _x_, _y_, or _r_ if we knew any one of them.

Now the values of the sine, cosine, and tangent (_functions_ of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them by 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:

============================================================ ANGLE | SINE |COSINE|TANGENT|| ANGLE | SINE |COSINE|TANGENT ------+------+------+-------++-------+------+------+-------- 5° | .087 | .996 | .087 || 50° | .766 | .643 | 1.192 ------+------+------+-------++-------+------+------+-------- 10° | .174 | .985 | .176 || 55° | .819 | .574 | 1.428 ------+------+------+-------++-------+------+------+-------- 15° | .259 | .966 | .268 || 60° | .866 | .500 | 1.732 ------+------+------+-------++-------+------+------+-------- 20° | .342 | .940 | .364 || 65° | .906 | .423 | 2.145 ------+------+------+-------++-------+------+------+-------- 25° | .423 | .906 | .466 || 70° | .940 | .342 | 2.748 ------+------+------+-------++-------+------+------+-------- 30° | .500 | .866 | .577 || 75° | .966 | .259 | 3.732 ------+------+------+-------++-------+------+------+-------- 35° | .574 | .819 | .700 || 80° | .985 | .174 | 5.671 ------+------+------+-------++-------+------+------+-------- 40° | .643 | .766 | .839 || 85° | .996 | .087 |11.430 ------+------+------+-------++-------+------+------+-------- 45° | .707 | .707 | 1.000 || 90° | 1.00 | .000 |[infinity] ============================================================

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 that _r_ = 30 and [L]_O_ = 40°, then since _y_ = _r_ sin _O_, we have _y_ = 30 × 0.643 = 19.29. If we know that _x_ = 60 and [L]_O_ = 35°, then since _y_ = _x_ tan _O_, we have _y_ = 60 × 0.7 = 42. We may also find _r_, for cos _O_ = _x_/_r_, whence _r_ = _x_/(cos _O_) = 60/0.819 = 73.26.

Therefore, if we could easily measure [L]_O_ and could measure the distance _x_, we could find the height of a building _y_. 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 formula _y_ = _x_ tan _O_.

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 years old, 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.

The illustration on page 236 shows a simple form of the quadrant, an instrument easily made by a pupil who may be interested in outdoor work. It was the common surveying instrument of the early days. A more elaborate example is seen in the illustration, on page 237, of a seventeenth-century brass specimen in the author's collection.[77]

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.

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 on page 240 shows how it was used for measuring depths.

A similar instrument from the work of Bettinus is given on page 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 on page 241.

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.

In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given on page 243 shows 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 a piece 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.

A well-known method of measuring the distance across a stream is illustrated in the figure below, where the distance from _A_ to some point _P_ is required.

Run a line from _A_ to _C_ by standing at _C_ in line with _A_ and _P_. Then run two perpendiculars from _A_ and _C_ by 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 point _D_, on _CD_, to _P_, putting a stake at _B_. Then run the perpendicular _BE_. Since _DE_ : _EB_ = _BA_ : _AP_, and since we can measure _DE_, _EB_, and _BA_ with the tape, we can compute the distance _AP_.

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 on page 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 which _AB_ = _BC_. He holds _AB_ level and walks toward the tree until he just sees the top along _AC_. Then because

_AB_ = _BC_, and _AB_ : _BC_ = _AD_ : _DE_,

the height above _D_ will equal the distance _AD_.

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, and _AD_ is 23 ft. 8 in.?

2. Suppose a triangle is used which has _AB_ = twice _BC_. What is the height if _AD_ = 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 the objects, 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 make _AD_ = 2 in., _DB_ = 3 in., and erect _DC_ [perp] to _AB_. Then the length of _DC_ will be [sqrt]6 in., and we may find [sqrt]6 approximately by measuring _DC_.

Furthermore, if we introduce negative magnitudes into geometry, and let _DB_ = +3 and _DA_ = -2, then _DC_ will equal [sqrt](-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 [sqrt]2, [sqrt]3, [sqrt]5, and [sqrt]9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of [sqrt](-6), [sqrt](-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 point _B_ and a point _A_ near the stick, then the two triangles are similar. Hence _AD_ : _DC_ = _DC_ : _DB_. Hence, if _AD_ and _DC_ are measured, _DB_ can 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_.

If _P_ is within the circle, then _xx'_ = _yy'_; if _P_ is on the circle, then _x_ and _y_ become 0, and 0 · _x'_ = 0 · _y'_ = 0; if _P_ is at _P__{3}, then _x_ and _y_, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if _P_ is at _P__{4}, then _y_ = _y'_ and we have _xx'_ = _y_^2, or _x_ : _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 letting _P_ assume various positions. Among the possible cases is the one of two tangents from an external point, and the one where _P_ is 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 angle diminished 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, let _a_ = 3, _b_ = 5, and _c_ = 6.

Then [because] _x_ : _y_ = _b_ : _a_, and _y_ = 6 - _x_,

we have _x_/(6 - _x_) = 5/3.

[therefore] 3_x_ = 30 - 5_x_.

[therefore] _x_ = 3 3/4, _y_ = 2 1/4.

By the theorem, _z_^2 = _ab_ - _xy_ = 15 - (8 7/16) = 6 9/16. [therefore] _z_ = [sqrt](6 9/16) = 1/4 [sqrt]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 by _a_, _b_, and _c_, as usual, and let _CD_ = _d_ and _PB_ = _x_, then

(_CP_)^2 = _a_^2 - _x_^2 = _b_^2 - (_c_ - _x_)^2. [therefore] _a_^2 - _x_^2 = _b_^2 - _c_^2 + 2_cx_ - _x_^2. [therefore] _x_ = (_a_^2 - _b_^2 + _c_^2) / 2_c_. [therefore] (_CP_)^2 = _a_^2 - ((_a_^2 - _b_^2 + _c_^2) / 2_c_)^2.

But _CP_ · _d_ = _ab_. [therefore] _d_ = 2_abc_ / [sqrt](4_a_^2_c_^2 - (_a_^2 - _b_^2 + _c_^2)^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 where _D_ falls between _A_ and _B_. If _CP_ is a diameter, it reduces to the second figure given on page 249. If _CP_ bisects [L]_ACB_, we have Fig. 3, from which may be proved the proposition given at the foot of page 248. If _D_ lies on _BA_ produced, we have Fig. 2. If _D_ lies on _AB_ produced, we have Fig. 4.

This general proposition is proved by showing that [triangles]_ADC_ and _PBC_ are similar, exactly as in the second proposition given on page 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 which Plato originated regarding the section." The expression "golden section" is not old, however, and its origin is uncertain.

If a line _AB_ is divided in golden section at _P_, we have

_AB_ × _PB_ = (_AP_)^2.

Therefore, if _AB_ = _a_, and _AP_ = _x_, we have

_a_(_a_ - _x_) = _x_^2, or _x_^2 + _ax_ - _a_^2 = 0; whence _x_ = - _a_/2 ± _a_/2[sqrt]5 = _a_(1.118 - 0.5) = 0.618_a_,

the other root representing the external point.

That is, _x_ = about 0.6_a_, and _a_ - _x_ = about 0.4_a_, and _a_ is 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.

FOOTNOTES:

[76] For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited.

[77] These two and several which follow are from Stark, loc. cit.

[78] The author has a beautiful ivory specimen of the Sixteenth century.