CHAPTER XVI

If part of the curveAPBis known, takePas the mid-point. Then stretch the tape fromAtoBand drawPMperpendicular to it. Then swing the lengthAMaboutP, andPMaboutB, until they meet atL, and stretch the lengthABalongPLtoQ. This fixes the pointQ. In the same way fix the pointC. Points on the curve can thus be fixed as near together as we wish. The chordsAB,PQ,BC, and so on, are equal and are equally distant from the center.

Theorem.A line perpendicular to a radius at its extremity is tangent to the circle.

The enunciation of this proposition by Euclid is very interesting. It is as follows:

The straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further, the angle of the semicircle is greater and the remaining angle less than any acute rectilineal angle.

The first assertion is practically that of tangency,—"will fall outside the circle." The second one states, substantially, that there is only one such tangent, or, as we say in modern mathematics, the tangent is unique. The third statement relates to the angle formed by the diameterand the circumference,—a mixed angle, as Proclus called it, and a kind of angle no longer used in elementary geometry. The fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity. This gives rise to a difficulty that seems to have puzzled many of Euclid's commentators, and that will interest a pupil: As the circle diminishes this angle apparently increases, while as the circle increases the angle decreases, and yet the angle is always stated to be zero. Vieta (1540-1603), who did much to improve the science of algebra, attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time. He said that a circle was a polygon of an infinite number of sides (which it cannot be, by definition), and that, a tangent simply coincided with one of the sides, and therefore made no angle with it; and this view was also held by Galileo (1564-1642), the great physicist and mathematician who first stated the law of the pendulum.

Theorem.Parallel lines intercept equal arcs on a circle.

The converse of this proposition has an interesting application in outdoor work.

Suppose we wish to run a line throughPparallel to a given lineAB. With any convenient pointOas a center, andOPas a radius, describe a circle cuttingABinXandY. DrawPX. Then withYas a center andPXas a radius draw an arc cutting the circle inQ. Then run the line fromPtoQ.PQis parallel toABby the converse of the above theorem, which is easily shown to be true for this figure.

Theorem.If two circles are tangent to each other, the line of centers passes through the point of contact.

There are many illustrations of this theorem in practical work, as in the case of cogwheels. An interesting application to engineering is seen in the case of two parallel streets or lines of track which are to be connected by a "reversed curve."

If the lines areABandCD, and the connection is to be made, as shown, fromBtoC, we may proceed as follows: DrawBCand bisect it atM. ErectPO, the perpendicular bisector ofBM; andBO, perpendicular toAB. ThenOis one center of curvature. In the same way fixO'. Then to check the work apply this theorem,Mbeing in the line of centersOO'. The curves may now be drawn, and they will be tangent toAB, toCD, and to each other.

At this point in the American textbooks it is the custom to insert a brief treatment of measurement, explaining what is meant by ratio, commensurable and incommensurable quantities, constant and variable, and limit, and introducing one or more propositions relating to limits. The object of this departure from the ancient sequence, which postponed this subject to the book on ratio and proportion, is to treat the circle more completely in Book III. It must be confessed that the treatment is not as scientific as that of Euclid, as will be explained under Book III, but it is far better suited to the mind of a boy or girl.

It begins by defining measurement in a practical way, as the finding of the number of times a quantity of any kind contains a known quantity of the same kind. Ofcourse this gives a number, but this number may be a surd, like √2. In other words, the magnitude measured may be incommensurable with the unit of measure, a seeming paradox. With this difficulty, however, the pupil should not be called upon to contend at this stage in his progress. The whole subject of incommensurables might safely be postponed, although it may be treated in an elementary fashion at this time. The fact that the measure of the diagonal of a square, of which a side is unity, is √2, and that this measure is an incommensurable number, is not so paradoxical as it seems, the paradox being verbal rather than actual.

It is then customary to define ratio as the quotient of the numerical measures of two quantities in terms of a common unit. This brings all ratios to the basis of numerical fractions, and while it is not scientifically so satisfactory as the ancient concept which considered the terms as lines, surfaces, angles, or solids, it is more practical, and it suffices for the needs of elementary pupils.

"Commensurable," "incommensurable," "constant," and "variable" are then defined, and these definitions are followed by a brief discussion of limit. It simplifies the treatment of this subject to state at once that there are two classes of limits,—those which the variable actually reaches, and those which it can only approach indefinitely near. We find the one as frequently as we find the other, although it is the latter that is referred to in geometry. For example, the superior limit of a chord is a diameter, and this limit the chord may reach. The inferior limit is zero, but we do not consider the chord as reaching this limit. It is also well to call the attention of pupils to the fact that a quantity may decrease towards its limit as well as increase towards it.

Such further definitions as are needed in the theory of limits are now introduced. Among these is "area of a circle." It might occur to some pupil that since a circle is a line (as used in modern mathematics), it can have no area. This is, however, a mere quibble over words. It is not pretended that the line has area, but that "area of a circle" is merely a shortened form of the expression "area inclosed by a circle."

The Principle of Limits is now usually given as follows: "If, while approaching their respective limits, two variables are always equal, their limits are equal." This was expressed by D'Alembert in the eighteenth century as "Magnitudes which are the limits of equal magnitudes are equal," or this in substance. It would easily be possible to elaborate this theory, proving, for example, that ifxapproachesyas its limit, thenaxapproachesayas its limit, andx/aapproachesy/aas its limit, and so on. Very much of this theory, however, wearies a pupil so that the entire meaning of the subject is lost, and at best the treatment in elementary geometry is not rigorous. It is another case of having to sacrifice a strictly scientific treatment to the educational abilities of the pupil. Teachers wishing to find a scientific treatment of the subject should consult a good work on the calculus.

Theorem.In the same circle or in equal circles two central angles have the same ratio as their intercepted arcs.

This is usually proved first for the commensurable case and then for the incommensurable one. The latter is rarely understood by all of the class, and it may very properly be required only of those who show some aptitude in geometry. It is better to have the others understand fully the commensurable case and see the natureof its applications, possibly reading the incommensurable proof with the teacher, than to stumble about in the darkness of the incommensurable case and never reach the goal. In Euclid there was no distinction between the two because his definition of ratio covered both; but, as we shall see in Book III, this definition is too difficult for our pupils. Theon of Alexandria (fourth centuryA.D.), the father of the Hypatia who is the heroine of Kingsley's well-known novel, wrote a commentary on Euclid, and he adds that sectors also have the same ratio as the arcs, a fact very easily proved. In propositions of this type, referring to the same circle or to equal circles, it is not worth while to ask pupils to take up both cases, the proof for either being obviously a proof for the other.

Many writers state this proposition so that it reads that "central angles aremeasured bytheir intercepted arcs." This, of course, is not literally true, since we can measure anything only by some thing, of the same kind. Thus we measure a volume by finding how many times it contains another volume which we take as a unit, and we measure a length by taking some other length as a unit; but we cannot measure a given length in quarts nor a given weight in feet, and it is equally impossible to measure an arc by an angle, and vice versa. Nevertheless it is often found convenient todefinesome brief expression that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus wedefine"area of a circle," even when we use "circle" to mean a line; and so we may define the expression "central angles are measured by their intercepted arcs" to mean that central angles have the same numerical measure as these arcs. This is done by most writers, and is legitimate as explaining an abbreviated expression.

Theorem.An inscribed angle is measured by half the intercepted arc.

In Euclid this proposition is combined with the preceding one in his Book VI, Proposition 33. Such a procedure is not adapted to the needs of students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle standing on the same arc. Since Euclid never considered an angle greater than 180°, his inscribed angle was necessarily less than a right angle. The first one who is known to have given the general case, taking the central angle as being also greater than 180°, was Heron of Alexandria, probably of the first centuryA.D.[68]In this he was followed by various later commentators, including Tartaglia and Clavius in the sixteenth century.

One of the many interesting exercises that may be derived from this theorem is seen in the case of the "horizontal danger angle" observed by ships.

If some dangerous rocks lie off the shore, andLandL'are two lighthouses, the angleAis determined by observation, so thatAwill lie on a circle inclosing the dangerous area. AngleAis called the "horizontal danger angle." Ships passing in sight of the two lighthousesLandL'must keep out far enough so that the angleL'SLshall be less than angleA.

To this proposition there are several important corollaries, including the following:

1.An angle inscribed in a semicircle is a right angle.This corollary is mentioned by Aristotle and is attributed

to Thales, being one of the few propositions with which his name is connected. It enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board, a pencil being held at the vertex.

A more practical use for it is made by machinists to determine whether a casting is a true semicircle. Taking a carpenter's square as here shown, if the vertex touches the curve at every point as the square slides around, it is a true semicircle. By a similar method a circle may be described by sliding a draftsman's triangle so that two sides touch two tacks driven in a board.

Another interesting application of this corollary may be seen by taking an ordinary paper protractorACB, and fastening a plumb line atB. If the protractor is so held that the plumb line cuts the semicircle atC, thenACis level because it is perpendicular to the vertical lineBC. Thus, if a class wishes to determine the horizontal lineAC, while sighting up a hill in the directionAB, this is easily determined without a spirit level.

It follows from this corollary, as the pupil has already found, that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices. This is useful in outdoor measuring, forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line.

SupposeXYto be the edge of a sidewalk, andPa point in the street from which we wish to lay a gas pipe perpendicular to the walk. FromPswing a cord or tape, say 60 feet long, until it meetsXYatA. Then takeM, the mid-point ofPA, and swingMPaboutM, to meetXYatB. ThenBis the foot of the perpendicular, since ∠PBAcan be inscribed in a semicircle.

2.Angles inscribed in the same segment are equal.

By driving two nails in a board, atAandB, and taking an anglePmade of rigid material (in particular, as already stated, a carpenter's square), a pencil placed atPwill generate an arc of a circle if the arms slide alongAandB. This is an interesting exercise for pupils.

Theorem.An angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs.

Theorem.An angle formed by a tangent and a chord drawn from the point of tangency is measured by half the intercepted arc.

Theorem.An angle formed by two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured by half the difference of the intercepted arcs.

These three theorems are all special cases of the general proposition that the angle included between two lines that cut (or touch) a circle is measured by half the sum of the intercepted arcs. If the point passes from within the circle to the circle itself, one arc becomes zero and the angle becomes an inscribed angle. If the point passes outside the circle, the smaller arc becomes negative, having passed through zero. The point may even "go toinfinity," as is said in higher mathematics, the lines then becoming parallel, and the angle becoming zero, being measured by half the sum of one arc and a negative arc of the same absolute value. This is one of the best illustrations of the Principle of Continuity to be found in geometry.

Problem.To let fall a perpendicular upon a given line from a given external point.

This is the first problem that a student meets in most American geometries. The reason for treating the problems by themselves instead of mingling them with the theorems has already been discussed.[69]The student now has a sufficient body of theorems, by which he can prove that his constructions are correct, and the advantage of treating these constructions together is greater than that of following Euclid's plan of introducing them whenever needed.

Proclus tells us that "this problem was first investigated by Œnopides,[70]who thought it useful for astronomy." Proclus speaks of such a line as a gnomon, a common name for the perpendicular on a sundial, which casts the shadow by which the time of day is known. He also speaks of two kinds of perpendiculars, the plane and solid, the former being a line perpendicular to a line, and the latter a line perpendicular to a plane.

It is interesting to notice that the solution tacitly assumes that a certain arc is going to cut the given line in two points, and only two. Strictly speaking, why may it not cut it in only one point, or even in three points? We really assume that if a straight line is drawn througha point within a circle, this line must get out of the circle on each of two sides of the given point, and in getting out it must cut the circle twice. Proclus noticed this assumption and endeavored to prove it. It is better, however, not to raise the question with beginners, since it seems to them like hair-splitting to no purpose.

The problem is of much value in surveying, and teachers would do well to ask a class to let fall a perpendicular to the edge of a sidewalk from a point 20 feet from the walk, using an ordinary 66-foot or 50-foot tape. Practically, the best plan is to swing 30 feet of the tape about the point and mark the two points of intersection with the edge of the walk. Then measure the distance between the points and take half of this distance, thus fixing the foot of the perpendicular.

Problem.At a given point in a line, to erect a perpendicular to that line.

This might be postponed until after the problem to bisect an angle, since it merely requires the bisection of a straight angle; but considering the immaturity of the average pupil, it is better given independently. The usual case considers the point not at the extremity of the line, and the solution is essentially that of Euclid. In practice, however, as for example in surveying, the point may be at the extremity, and it may not be convenient to produce the line.

Surveyors sometimes measurePB= 3 ft., and then take 9 ft. of tape, the ends being held atBandP, and the tape being stretched toA, so thatPA= 4 ft. andAB= 5 ft. ThenPis a right angle by the Pythagorean Theorem. This theorem not having yet been proved, it cannot be used at this time.

A solution for the problem of erecting a perpendicular from the extremity of a line that cannot be produced, depending, however, on the problem of bisecting an angle, and therefore to be given after that problem, is attributed by Al-Nairīzī (tenth centuryA.D.) to Heron of Alexandria. It is also given by Proclus.

Required to draw fromPa perpendicular toAP. TakeXanywhere on the line and erectXY⊥ toAPin the usual manner. Bisect ∠PXYby the lineXM. OnXYtakeXN=XP, and drawNM⊥ toXY. Then drawPM. The proof is evident.

These may at the proper time be given as interesting variants of the usual solution.

Problem.To bisect a given line.

Euclid said "finite straight line," but this wording is not commonly followed, because it will be inferred that the line is finite if it is to be bisected, and we use "line" alone to mean a straight line. Euclid's plan was to construct an equilateral triangle (by his Proposition 1 of Book I) on the line as a base, and then to bisect the vertical angle. Proclus tells us that Apollonius of Perga, who wrote the first great work on conic sections, used a plan which is substantially that which is commonly found in textbooks to-day,—constructing two isosceles triangles upon the line as a common base, and connecting their vertices.

Problem.To bisect a given angle.

It should be noticed that in the usual solution two arcs intersect, and the point thus determined is connected with the vertex. Now these two arcs intersect twice, and since one of the points of intersection may be the vertexitself, the other point of intersection must be taken. It is not, however, worth while to make much of this matter with pupils. Proclus calls attention to the possible suggestion that the point of intersection may be imagined to lie outside the angle, and he proceeds to show the absurdity; but here, again, the subject is not one of value to beginners. He also contributes to the history of the trisection of an angle. Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca.180B.C.), the quadratrix of Hippias of Elis (ca.420B.C.), or the spiral of Archimedes (ca.250B.C.). But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection ofanyangle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. Thus, to trisect an angle of 90° we need only to construct an angle of 60°, and this can be done by constructing an equilateral triangle. But while we cannot trisect the angle, we may easily approximate trisection. For since, in the infinite geometric series 1/2 + 1/8 + 1/32 + 1/128 + ...,s=a÷ (1 -r), we haves= 1/2 ÷ 3/4 = 2/3. In other words, if we add 1/2 of the angle, 1/8 of the angle, 1/32 of the angle, and so on, we approach as a limit 2/3 of the angle; but all of these fractions can be obtained by repeated bisections, and hence by bisections we may approximate the trisection.

The approximate bisection (or any other division) of an angle may of course be effected by the help of the protractor and a straightedge. The geometric method is, however, usually more accurate, and it is advantageousto have the pupils try both plans, say for bisecting an angle of about 49-1/2°.

Applications of this problem are numerous. It may be desired, for example, to set a lamp-post on a line bisecting the angle formed by two streets that come together a little unsymmetrically, as here shown, in which case the bisecting line can easily be run by the use of a measuring tape, or even of a stout cord.

A more interesting illustration is, however, the following:

Let the pupils set a stake, say about 5 feet high, at a pointNon the school grounds about 9A.M., and carefully measure the length of the shadow,NW, placing a small wooden pin atW. Then about 3P.M.let them watch until the shadowNEis exactly the same length that it was whenWwas fixed, and then place a small wooden pin atE. If the work has been very carefully done, and they take the tape and bisect the lineWE, thus fixing the lineNS, they will have a north and south line. If this is marked out for a short distance fromN, then when the shadow falls onNS, it will be noon by sun time (not standard time) at the school.

Problem.From a given point in a given line, to draw a line making an angle equal to a given angle.

Proclus says that Eudemus attributed to Œnopides the discovery of the solution which Euclid gave, and which is substantially the one now commonly seen in textbooks. The problem was probably solved in some fashion before the time of Œnopides, however. The object of the problem is primarily to enable us to draw a line parallel to a given line.

Practically, the drawing of one line parallel to another is usually effected by means of a parallel ruler (seepage 191), or by the use of draftsmen's triangles, as here shown, or even more commonly by the use of a T-square, such as is here seen. This illustration shows two T-squares used for drawing lines parallel to the sides of a board upon which the drawing paper is fastened.[71]

An ingenious instrument described by Baron Dupin is illustrated below.

To the barAis fastened the sliding checkB. A movable checkDmay be fastened by a screwC. A sharp point is fixed inB, so that asDslides along the edge of a board, the point marks a line parallel to the edge. Moreover,FandGare two brass arms of equal length joined by a pointed screwHthat marks a line midway betweenBandD. Furthermore, it is evident thatHwill draw a line bisecting any irregular board if the checksBandDare kept in contact with the irregular edges.

Book II offers two general lines of application that may be introduced to advantage, preferably as additions to the textbook work. One of these has reference to topographical drawing and related subjects, and the other to geometric design. As long as these can be introduced

to the pupil with an air of reality, they serve a good purpose, but if made a part of textbook work, they soon come to have less interest than the exercises of a more abstract character. If a teacher can relate the problems in topographical drawing to the pupil's home town, and can occasionally set some outdoor work of the nature here suggested, the results are usually salutary; but if he reiterates only a half-dozen simple propositions time after time, with only slight changes in the nature of the application, then the results will not lead to a cultivation of power in geometry,—a point which the writers on applied geometry usually fail to recognize.

One of the simple applications of this book relates to the rounding of corners in laying out streets in some of our modern towns where there is a desire to depart from the conventional square corner. It is also used in laying out park walks and drives.

The figure in the middle of the page represents two streets,APandBQ, that would, if prolonged, intersect atC. It is required to construct an arc so that they shall begin to curve atPandQ, whereCP=CQ, and hence the "center of curvature"Omust be found.The problem is a common one in railroad work, only hereAPis usually oblique toBQif they are produced to meet atC, as in the second figure onpage 218. It is required to construct an arc so that the tracks shall begin to curve atPandQ, whereCP=CQ.

The problem is a common one in railroad work, only hereAPis usually oblique toBQif they are produced to meet atC, as in the second figure onpage 218. It is required to construct an arc so that the tracks shall begin to curve atPandQ, whereCP=CQ.

The problem becomes a little more complicated, and correspondingly more interesting, when we have to find the center of curvature for a street railway track that must turn a corner in such a way as to allow, say, exactly 5 feet from the pointP, on account of a sidewalk.

The problem becomes still more difficult if we have two roads of different widths that we wish to join on a curve. Here the two centers of curvature are not the same, and the one road narrows to the other on the curve. The solutions will be understood from a study of the figures.

The number of problems of this kind that can easily be made is limitless, and it is well to avoid the dangerof hobby riding on this or any similar topic. Therefore a single one will suffice to close this group.

If a roadABon an arc described aboutO, is to be joined to roadCD, described aboutO', the arcBCshould evidently be internally tangent toABand externally tangent toCD. Hence the center is onBOXandO'CY, and is therefore atP. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives.

It will be noticed that the above problems require the erecting of perpendiculars, the bisecting of angles, and the application of the propositions on tangents.

A somewhat different line of problems is that relating to the passing of a circle through three given points. It is very easy to manufacture problems of this kind that have a semblance of reality.

For example, let it be required to plan a driveway from the gateGto the porchPso as to avoid a mass of rocksR, an arcof a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length fromGto the point onR(which we may also callR), and the angleRGOas 60°, to find the radius.

A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the describing of circles, their inscription in or circumscription about triangles, and their construction in various positions of tangency, has some value as applying the various problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of compasses and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formalism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few illustrations of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work.

The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day.

It might be said, for example, that in planning a Gothic window this drawing is needed. The arcBCis drawn withAas a center andABas a radius. The small arches are described withA,D, andBas centers andADas a radius. The center _P_ is found by takingAandBas centers andAEas a radius. How may the pointsD,E, andFbe found? Draw the figure. From the study of the rectilinear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred.

The Gothic window also offers some interesting possibilities in connection with the study of the square. For example, the illustration given onpage 223shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles.[72]

The properties of the square, a figure now easily constructed by the pupils, are not numerous. What few there are may be brought out through the study of art forms, if desired. In case these forms are shown to a class, it is important that they should be selected from good designs. We have enough poor art in the world, so that geometry should not contribute any more. This illustration is a type of the best medieval Gothic parquetry.[73]

Gothic Designs employing Circles and Bisected AnglesGothic Designs employing Circles and Bisected Angles

Even simple designs of a semipuzzling nature have their advantage in this connection. In the following example the inner square contains all of the triangles, the letters showing where they may be fitted.[74]

Still more elaborate designs, based chiefly upon the square and circle, are shown in the window traceries onpage 225, and others will be given in connection with the study of the regular polygons.

Designs like the figure below are typical of the simple forms, based on the square and circle, that pupils may profitably incorporate in any work in art design that they may be doing at the time they are studying the circle and the problems relating to perpendiculars and squares.

Among the applications of the problem to draw a tangent to a given circle is the case of the common tangents to two given circles. Some authors give this as a basal problem, although it is more commonly given as an exercise or a corollary. One of the most obvious applications of the idea is that relating to the transmission of circular motion by means of a band over two wheels,[75]AandB, as shown onpage 226.

Gothic Designs employing Circles and Bisected AnglesGothic Designs employing Circles and Bisected Angles

The band may either not be crossed (the case of the two exterior tangents), or be crossed (the interior tangents), the latter allowing the wheels to turn in opposite directions. In case the band is liable to change its length, on account of stretching or variation in heat or moisture, a third wheel,D, is used. We then have the case of tangents to three pairs of circles. Illustrations of this nature make the exercise on the drawing of common tangents to two circles assume an appearance of genuine reality that is of advantage to the work.

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 (408-355B.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 equimultipleswhatever 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]

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 equimultipleswhatever 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 √2 by √3? And when we multiply a ratio by √5, what is the meaning of this operation? If we say that √2 : √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 manytimesis √3 contained in √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 √5? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to √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 theproposition 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.

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