208 



. KNOWLEDGE ♦ 



[March 13, 1885. 



himself by teariug open Lis mother's breast, and then 

 struck out upon the ocean, on whose waters he swam to 

 and fro for eight years, till at last he reached a nameless, 

 treeless, voiceless land. 



Tlins tlio IjirLli of Wilinamoinen ; 

 Tbus t'lie birth of tbo great siuger ; 

 Born he was of mi.i;hty parent ; 

 Burn of Ilmatar, hia mother. 



Then, after much pondering, he wondered who should sow 

 the life-giving seed, when Pellerwoinen appeared, scatter- 

 ing seeds broadcast, whence sprang all kinds of trees, the 

 berried junipers, the cone-laden lirs, the water-nouii-.hed 

 alder.s, " every tree after its kind," save the oak. Now in 

 a, far-off, mist-veiled meadow, Wainamoinen espied five 

 maidens mowing, and, as they raked the grass together, one 

 Tursa, a giant, rose from the ocean, and igaited the gras^ 

 by rubbing it, then wrapped an acorn in a love-leaf, and 

 buried it in the ashes. Quioklj'' grew and spread the 

 oak therefrom until its boughs arrested the clouds in 

 their cour.ses, and its leaves blotted out the light of 

 yun and moon. Wiiinamoinen, "old and steadfast," won- 

 dered whether hei'o or giant might be fovmd who could 

 fell the tree, and, in his nted, went to his mother (we find 

 the heroes of the Kalevala repairing to their mothers in all 

 times of trouble), and besought her to lend him the ooem's 

 strength to rid the earth of the evil-working tree. Then 

 out of the sea came a copper-clad dwaif, a span in length, 

 who held in bis hand a copper axe. Making known his 

 business to Wainamoinen, who laughed scornfully at the 

 i'Jea of so small a man doing so great a deed, the dwarf 

 suddenly became a giant, whos" bead reached to the clouds. 

 Planting his foot firm in oartli, he struck the oak with his 

 axe, when the mighty tree staggered and fell. Its trunk 

 lay eastward, its top jiointed westward, i's leaves bent 

 southward, and its branches northward. He who took a 

 twig had with it abiding happiness; he who gathered from 

 the top had magic power for ever; with l>im who plucked 

 a leaf dwelt everlasting charm ; while tlie splinters which 

 were drifted to the feet of the Northland maiden were 

 fashioned into arrows by the sorcerer's art. And the earth 

 was again touched by the sunshine, the clouds drifted 

 unhindered, the birds sang, the grass and flowers grew — 



All tliicGTS fair and lovely flourished, 

 All tbin<i:s save the one most ])recious 

 Fruit of fruits, the golden barley. 



Then Wainiimliinea, walking by the sea-shore, found seven 

 grains in the sand, and these he carried to be sown near 

 Kaleva's spring and Osmo's field. As he went thither a 

 titmouse sang from the tree-top : 



Onnio's barley will not Honrish, 

 Nor Kaleva's oats be ripened ; 

 For as yet untitled the soil is, 

 And uncleared is yet the forest. 



Then Wainiimoinen, after the manner of culture-heroes of 

 other races, cut down the trees, sparing only one bii'ch- 

 tree, that therein the birds might make melody and the 

 cuckoo build its nest. For this a grateful ea^le brought 

 him fire wherewith to burn the felled trees, and as the hero 

 cast the barley grains into the cleared ground he invoked 

 the sleepins; caith to awake and nourish the seed, and 

 prayed to Ukko, Heaven- Father, to water it with the rain- 

 clouds : 



Bid them all their sweetness sprinkle, 



That the ear may lift its treasure, 



And the corn .with plenty rustle. 



Ukko granted his prayer, and in three days the barley was 

 ready for the reaper. Then came the cuckoo, bird of 

 spring, and, learning why the birch-tree was spared, did 



Wainamoinen's bidding, and sang with a voice that 

 gladdened the hearts of men. 



Among the many barbaric myths accounting for the fitful, 

 and startling, and turbulent in Nature, the wind and 

 thunder are alike conceived of as caused by a mon.ster bird, 

 whose course is as the lightning-flash, and who is the 

 messenger between heaven and earth, with the gift of fire 

 to men. 



In the eagle, as fire-bringer, a note in primitive thought 

 is struck, although, as will be seen hereafter, fire itself is 

 personified, eluding pursuit and working havoc. Another 

 note is also struck, but softer in contrast to the often harsh 

 melody of the Kalevala, in this communion of man with 

 the bird of the air and the beast of the field, as sharers of 

 a common life, given by the great Earth-mother. The 

 presence of this in myth and folk-tale may suggest to us 

 that if we have gained much through larger knowledge of 

 her works and ways, the familiarity has tended to deaden 

 sympathy with all life lower than our own, and that to 

 quicken this anew within us is to touch the science of to-day 

 with the poetry of the past. 



HOW TO TRISECT AN ANGLE. 



By Alfred Jukes Allen. 



ONE of the problems made famous in the history of 

 Cfcometry by its ))ersisteut refusal to admit of any 

 solution, even at the hands of the most skilful mathema- 

 ticians, is that which requires the trisection or division 

 into three equal parts, by Euclidian methods, of any given 

 angle. The Postulates of Euclid, it will be remembered, 

 allow only the use of an instrument for drawing straight 

 lines (such as a ruler), and of a pair of compasses for 

 describing circles. It is now admitted — indeed, it can be 

 proved— that with these instruments alone, the solution of 

 the above problem is impossible, and that no combination 

 of circles and straight lines can effect the desirt d end. If, 

 however, we assume our ability to describe other curves 

 besides the circle, we find no ditficulty at all. For instance, 

 the combination in a suitable manner of a circle with a 

 certain curve called an hyperbola " of eccentricity equal to 

 2," gives us a very ready means of trisecting an angle, 

 provided we have our hyperbola already accurately 

 described. Let us now see if the problem can be solved by 

 the use of special instruments. 



The simplest possible mathematical instrument is a rod 

 turning round a fixed pivot, with a pencil or other tracer 

 fixed at some point of its length — a primitive form, in fact, 

 of a pair of compasses. By means of such an instrument 

 very correct circles can be described. A coirect straiylil 

 line, however, if supposed to be drawn by means of a ruler, 

 requires the previous production of a straight edge on the 

 same — a matter of no s-mall difficulty. Such was, doubtless, 

 the manner in which Euclid supposed straight lines to be 

 drawn, but it has been found of late years that the same 

 result may be obtained by means of rods suitably pivoted 

 together, and not involving such accurate workmanship. 

 And further, by other combinations of simple rotation 

 about a pivot, the trisection of angles, which defied the 

 efforts of geometers provided only with rule and conifiasses, 

 has been successfully accomplished. Ir. is instruments of 

 the latter class that we are now to consider. 



The "fan" invented for the purpose by Professor 

 Sylvester is shown in Fig. 1. It consists of six rods 

 pivoted together at O, the four alternate ones of which 

 are for convenience prolonged. To each rod, at a given 

 distance from O, is pivoted a link, all (except the two end 



