92 MEMOIRS OF THE NEW YORK BOTANICAL GARDEN 



outer series, still other arrangements are also found. Braun 

 notes more specifically the possible variations in the inner group, 

 which, however, need not concern us here. 



In my study of the colonies of Pediastrum all the measurements 

 of angles, arcs, etc., were made on photomicrographs such as are 

 reproduced in the figures. The photographs were from four to 

 ten cm. in diameter in the material here discussed. The magni- 

 fications are given in connection with the figures. In some cases 

 the original print as taken was enlarged by rephotographing it in 

 order to make it easier to measure the angles. This was done in 

 the case of both figures la and 2a. 



The difficulties in the way of measuring the angles in such 

 small figures are, of course, very great. A transparent xylonite 

 protractor with radii from the center marking the angles of 120° 

 was used and the angles were read by laying on a ruler. All 

 three angles about a point of intersection of the cell walls were 

 read with a single placing of the protractor so that the sum of 

 the angles about such a point was always made 360°, a result 

 which, of course, would not be achieved if the protractor was 

 placed anew for each of the angles to be measured. No claim 

 can be made for great accuracy in the measurements. The 

 breadth and frequently the vagueness in the outline of the cell 

 walls in the photographs precluded the possibility of getting a 

 high degree of consistency in the measurements even when re- 

 peated many times. There is no question, however, that much 

 more accurate results can be obtained by measuring photographs 

 than by attempting to measure the angles directly on the organism 

 under the microscope. And the data obtained are certainly con- 

 vincing on the general point that where, as is so regularly the 

 case in the i6-celled colonies, the cells in their contacts form groups 

 of three, the angles of intersection of their walls fluctuate about 

 120° with deviations which correlate directly with the number of 

 cells in the colony, their inherited form, etc. The measurement 

 of the angle which each cell of the colony subtends could be made 

 with greater accuracy. 



The colonies of Pediastrum tend to conform to the principle of 

 least surfaces both in the shape of their cells and in the form of 

 the group as a whole. Each cell tends to become as nearly 

 spherical as is consistent with its inherited form tendencies and 



