The formation of the regular polygon in symmetry and sympathy, so to speak, with the circle, has ever been a favorite subject of geometric effort. For instance, an isosceles triangle, having its angles at the base each double the angle at the vertex, when inscribed in a circle, will give in the measurement of its base one side of an inscribed pentagon. Then again, an equilateral triangle inscribed in like manner, gives a rule for the construction of a hexagon as well as for the laying off chords measuring a third, a twelfth, a twentyfourth, and other multiples. The inscribing of a square divides it into two, four, eight, sixteen parts, or as far as we choose to bisect the sides. It will be apparent from what is thus shown, that by combining the subsections of triangle, square, pentagon, and hexagon, with those of other regular figures which may be inscribed, an almost infinite variety of polygons, and of course corresponding arcs and angles will be obtained. There are some equations, however, which do not flow readily from such combination. The ancient geometers met with no success in dividing the circle into 7, 9, 11, 13 parts, and until the beginning of the present century it was believed the circumference could not by pure geometric methods be divided into 17 equal parts. This last problem was accomplished by Gauss, an eminent German mathematician, and its solution, it is almost needless to say, attracted universal attention at the time. It even went further, and revived an interest in many other inquiries which had begun to be regarded as vain and fruitless. There are two problems of world-wide celebrity closely associated with researches of the character just indicated, which deserve more than passing attention. The first is called the trisection of an angle, or the division of any given segment of a circle into three equal parts by means of pure geometry. It is, perhaps, within bounds to say that this proposition, simple as it may at first seem, has taxed the ingenuity and wasted the mental labor of more men of genius than any other ever advanced among men. A favorite topic long before analysis possessed Athens or science sat enthroned at Alexandria, a thesis equally at Bagdad and Cordova, a foil to the keen intellects of Fermat, Hobbes, Pascal, it has withstood every attack of elementary geometry. Of course there are some angles which

do admit of this geometric trisection as the right angle of 90°, which, as radius measures 60°, may be divided into three equal parts by laying that off from each extremity of the arc. Again, an angle at the centre of a circle may be trisected when a line drawn from the extremity of one side to intersect the other side produced has cut off from it externally by the circumference a segment equal to radius. As the vertex of every angle may be made the center of a circle, this would give an easy mode of solution, but for the difficulty so to draw the line that the segment cut off shall be equal to radius. This, however, has suggested a mechanical contrivance for effecting trisection, whose method is sufficiently ingenious to be worthy of a presentation.

The accompanying diagram will serve to explain it. Let ABC be the angle it is required to trisect. With the center B and any radius, describe a circle cutting AB and BC. Bisect the angle ABC, and draw BL. Produce AB till it cuts the circle in G. Then, on the edge of a ruler mark off bd equal to radius, and lay the ruler at the point G, moving it until the point b cuts the

HBC.

B

A

circle, the point d meanwhile traveling in the line BL. Join B and H the point of intersection. Bisect ABH by BK. Then the angle ABC will be trisected, and ABK = KBH = Still again, trisection was early effected and proven by bringing into requisition a novel curve, called the conchoid, first made known by Nicomedes. To apply this curve, make the vertex the pole, let fall a perpendicular from the extremity of one side upon the other as the rule, from the foot of the perpendicular on that side produced lay off a modulus equal to twice the other side, and describe the curve. Then, through the extremity of the first side draw a parallel to the other side intersecting the curve. Join the point of intersection with the vertex and the line will cut off one

third of the angle. After which, bisect the remaining arc. Though centuries of failure have signalized the attempt at trisection by mere application of line and circle, though other methods furnish ample means of accomplishing the object, and though it may be doubted if any practical benefit would result to mathematics from such solution; yet the fascination which lingers around it will ever lure new generations of thinkers to wrestle with the problem. And who shall say that it will never be achieved? The third part of every angle is capable of numerical expression, and can be always designated graphically. The association too of line and circle is infinitely variable, as has been sufficiently seen, whilst no limits have yet been assigned to their relation in position, or the finding of new properties by intersection. Why then, may there not arise in some now unforeseen manner, by contact or transversal, or projection, an element of figure which shall furnish the key to this solution? Of one thing we may certainly be sure, which is, that no argument short of demonstrated impossibility will ever be accepted as conclusive against it by modern genius, which has unlocked so many secrets inscrutable to the past.

The other problem referred to is the quadrature of the circle, a sphynx riddle, which has come down to us from the Magi. This, it is well to premise, has a two-fold meaning which has often led to confusion. One is the geometrical quadrature; the other is the numerical quadrature. In the first the proposition is to construct a square that shall be exactly equal to a given circle. This has never been accomplished. It can only be said concerning it, as of the trisection of an angle, that there is no sufficient reason why it should not be possible as a geometrical construction. A square and a circle may be exactly equal, because any square may be constantly increased, so that from being less than the given circle it will become greater, and of course in such transition will pass a point of perfect equality. The rule only of such construction remains undiscovered. This can scarcely arise from the mere fact of the incommensurability between radius and circumference, for the same quality obtains between parts of some other geometrical figures very easy of construction, and in constructing which that incommensurability disappears. Indeed, between areas

bounded by arcs of circles, having different diameters, which may be taken to imply a different unit of space measure, and between like areas and triangular spaces, which, as we know, may be reduced to squares, such absolute equalities are known to exist. Thus, in the subjoined diagram representing a circle, with different semi-circles described upon subdivisions of the diameter, the areas 1, 2, 3, 4, may be proved equal each to each. The other illustration is one of the most striking landmarks of ancient geometry. It has been desiggnated as the lunes of Hippocrates. Thus: on AB de

scribe a semicircle AFDB, and in it erect a right triangle ADB. Then, on the sides AD and DB describe the semicircles AGD and DHB. The two figures AGDF and DHBE are lunes, and the sum of their areas is equal to the area of the triangle ADB. For the areas of any two semicircles are to each other as the squares of their diameters, and from the right triangle ADB we have AB2 = AD2 + DB2, hence, the sum of the semicircles on AD and DB is equal to the semi-circle on AB. Diminishing both members of this equation by the sum of the segments AFD and DEB, leaves the sum of the lunes equal to the triangle. This was followed by the achievement of Archimedes effecting the quadrature of the parabola, a grand step in advance, because it was the first curvilinear space legitimately squared. Even admitting then, as has been shown by Bernouilli, that there can exist no finite algebraic formula solving this problem, does it follow that there can be no geometric construction? Of the other form it may be said that it too has taxed the efforts of the greatest geometers of the world, from Archimedes to Leibnitz. The problem is to discover the numerical measure of the surface of a circle from the measured length of its diameter. The solution of this phase of the quadrature is really imprac

ticable, and can be proved to be so, owing to intrinsic difficulties in the relation of numbers. Nor is there any inconsistency between saying that the exact numerical measure of the surface of a circle does not exist though a geometrical square equal to it does exist: for the exact numerical value of ✔2 does not exist, yet 2 represents the diagonal of a square whose side is fig. 1, and which diagonal is clearly an existent geometrical line. This problem rests of course on the ratio of the diameter to the circumference. The ancients who first attempted the solution made an approximation which placed it between 310 and 310. The Hindoos and Persians seem to have reached a still closer ratio of 1 to 3.1416, even before its dissemination in Europe. Within recent times attention to the subject has been revived, and much patient labor bestowed in the hope that the decimal would run out, or take upon itself the form of a repetend. Ludolf Van Cluellen carried the fraction to the thirty-fifth place of figures, and had the number inscribed upon his tomb in St. Peter's churchyard at Leyden. This is the so-called "Ludophean number." How close even this approximation is to a true result may be conceived of when it is stated that if the distance from the earth to the nearest fixed star were taken as a radius with which to describe a circle, this decimal of thirty-five places would enable us to calculate the vast circumference so generated to within a space of less than the thickness of a hair. Consider again, however, that since that time an unremitting perseverance has extended the decimal fraction first to two hundred places, then to five hundred, and lastly, through the labors of Mr. William Shanks, of England, to six hundred and seven places. Thus, whilst no absolute solution has been or can be arrived at, for the two may be demonstrated as incommensurable, yet the line of separation has become certainly invisible, if not inconceivable.

The consideration so far given has been to figures supposed to lie in one plane, or space of two dimensions, but geometry applies itself also to magnitudes in space of three dimensions. This involves the intersection of planes, the construction of solid angles and the mensuration of volumes and capacities. It differs but little in its principles from those already made familiar in the propositions discussed.