THEOREM. _In the same circle or in equal circles equal chords are equidistant from the center, and chords equidistant from the center are equal._
This proposition is practically used by engineers in locating points on an arc of a circle that is too large to be described by a tape, or that cannot easily be reached from the center on account of obstructions.
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If part of the curve _APB_ is known, take _P_ as the mid-point.
Then stretch the tape from _A_ to _B_ and draw _PM_ perpendicular to it. Then swing the length _AM_ about _P_, and _PM_ about _B_, until they meet at _L_, and stretch the length _AB_ along _PL_ to _Q_. This fixes the point _Q_. In the same way fix the point _C_. Points on the curve can thus be fixed as near together as we wish. The chords _AB_, _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 s.p.a.ce between the straight line and the circ.u.mference 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 a.s.sertion 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 diameter and the circ.u.mference,--a mixed angle, as Proclus called it, and a kind of angle no longer used in elementary geometry. The fourth statement practically a.s.serts that the angle between the tangent and circ.u.mference is less than any a.s.signable quant.i.ty. 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.
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Suppose we wish to run a line through _P_ parallel to a given line _AB_. With any convenient point _O_ as a center, and _OP_ as a radius, describe a circle cutting _AB_ in _X_ and _Y_.
Draw _PX_. Then with _Y_ as a center and _PX_ as a radius draw an arc cutting the circle in _Q_. Then run the line from _P_ to _Q_. _PQ_ is parallel to _AB_ by 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 pa.s.ses through the point of contact._
There are many ill.u.s.trations 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."
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If the lines are _AB_ and _CD_, and the connection is to be made, as shown, from _B_ to _C_, we may proceed as follows: Draw _BC_ and bisect it at _M_. Erect _PO_, the perpendicular bisector of _BM_; and _BO_, perpendicular to _AB_. Then _O_ is one center of curvature. In the same way fix _O"_. Then to check the work apply this theorem, _M_ being in the line of centers _OO"_. The curves may now be drawn, and they will be tangent to _AB_, to _CD_, 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 quant.i.ties, 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 quant.i.ty of any kind contains a known quant.i.ty of the same kind. Of course this gives a number, but this number may be a surd, like [sqrt]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 [sqrt]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 quant.i.ties 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 cla.s.ses 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 quant.i.ty 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 if _x_ approaches _y_ as its limit, then _ax_ approaches _ay_ as its limit, and _x/a_ approaches _y/a_ as 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 cla.s.s, and it may very properly be required only of those who show some apt.i.tude in geometry. It is better to have the others understand fully the commensurable case and see the nature of 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 century A.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 are _measured by_ their 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 to _define_ some brief expression that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus we _define_ "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 century A.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.
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If some dangerous rocks lie off the sh.o.r.e, and _L_ and _L"_ are two lighthouses, the angle _A_ is determined by observation, so that _A_ will lie on a circle inclosing the dangerous area.
Angle _A_ is called the "horizontal danger angle." Ships pa.s.sing in sight of the two lighthouses _L_ and _L"_ must keep out far enough so that the angle _L"SL_ shall be less than angle _A_.
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.
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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.
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Another interesting application of this corollary may be seen by taking an ordinary paper protractor _ACB_, and fastening a plumb line at _B_. If the protractor is so held that the plumb line cuts the semicircle at _C_, then _AC_ is level because it is perpendicular to the vertical line _BC_. Thus, if a cla.s.s wishes to determine the horizontal line _AC_, while sighting up a hill in the direction _AB_, 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.
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Suppose _XY_ to be the edge of a sidewalk, and _P_ a point in the street from which we wish to lay a gas pipe perpendicular to the walk. From _P_ swing a cord or tape, say 60 feet long, until it meets _XY_ at _A_. Then take _M_, the mid-point of _PA_, and swing _MP_ about _M_, to meet _XY_ at _B_. Then _B_ is the foot of the perpendicular, since [L]_PBA_ can be inscribed in a semicircle.
2. _Angles inscribed in the same segment are equal._
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By driving two nails in a board, at _A_ and _B_, and taking an angle _P_ made of rigid material (in particular, as already stated, a carpenter"s square), a pencil placed at _P_ will generate an arc of a circle if the arms slide along _A_ and _B_. 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 pa.s.ses from within the circle to the circle itself, one arc becomes zero and the angle becomes an inscribed angle. If the point pa.s.ses outside the circle, the smaller arc becomes negative, having pa.s.sed through zero.
The point may even "go to infinity," 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 ill.u.s.trations of the Principle of Continuity to be found in geometry.