THE HIVE-BEE. 451 



Any one with the proper instruments aould measure the angles 

 for himself, and the calculations of a mathematician like Koenig 

 would hardly be questioned. However, Maclaurin, the well- 

 known Scotch mathematician, was not satisfied. The two re- 

 sults very nearly tallied with each other, but not quite, and he 

 felt that in a mathematical question precision was a necessity. 

 So he tried the whole question himself, and found Maraldi's 

 measurement correct, namely, 109° 28', and 70° 32'. 



He then set to work at the problem which was worked out by 

 Koenig, and found that the true theoretical angles were 109° 28', 

 and 70° 32', precisely corresponding with the actual measurement 

 of the bee-cell. 



Another question now arose. How did this discrepancy oc- 

 cur ? How could so excellent a mathematician as Koenig make 

 so grave a mistake ? On investigation, it was found that no 

 blame attached to Koenig, but that the error lay in the book of 

 logarithms which he used. Thus a mistake in a mathematical 

 work was accidentally discovered by measuring the angles of a 

 bee-cell — a mistake sufficiently great to have caused the loss of a ship 

 whose captain happened to use a copy of the same logarithmic tables 

 for calculating his longitude. 



Now let us see how this beautiful lozenge is made. . There is 

 not the least difficulty in drawing it. Make any square, ABCD 

 (Fig. 3), and draw the diagonal AC. 



Produce BA toward F and AD, both ways to any distance. 



Make AE and AG equal to AC, and make AF equal to AB. 

 Join the points EFGrB, and you have the required figure. 



Now comes a beautiful point. If we take AB as 1, being one 

 side of the square on which the lozenge is founded, AE and AG 

 will be equal to V2, and EF, FG, GB, and BE, will be equal to 

 ■y/S, as can be seen at a glance by any one who has advanced as 

 far as the 47th proposition of the first book of Euclid. 



Perhaps some of my readers may say that all these figures may 

 be very true, but that they do not show how the cell is formed. 

 If the reader will refer to Fig. 4, he will see how the theory maj'^ 

 be reduced to practice. After he has drawn the lozenge-shaped 

 figure which has just been described, let him draw upon card- 

 board nine of them, as is shown in the illustration (Fig. 4). Then 

 let him cut out the figure, and draw his penknife half through 

 the cardboard at all the lines of junction. He will then find that 

 the cardboard will fold into an exact model of a bee-cell, the three 



