35^ 



SCIENCE. 



[Vol. XXII. No. 569 



would destroy the larger number of the root hairs, still 

 formed crystals as usual. Then root pressure must be 

 entirely wanting, as well as osmotic activity in plants at 

 this stage. Neither can the elevation of the water be due 

 to "negative pressure," since the portion of the stem 

 above the crystal-forming part may be split, or broken, or 

 cut entirely away, without affecting the formation of the 

 crystals. 



Capillary force is the only means by which the water 

 may be carried from the ground up through the plant to 

 where it forms crystals. The constant absorption and 

 evaporation by the dessicating tissues limit the region of 

 saturation and confine the formation of crystals to the 

 basal portion of the stems. The size and arrangement 

 of the medullary cells favor the lateral conduction of the 

 water by reason of their greater caj^illary power. The 

 portion of water at the jieripheral ends of the rays is 

 frozen and in expanding is forced outward. The portions 

 which replace it are in tui-n frozen, and the successive in- 

 crements thus formed give the length and account for the 

 perpendicular striations of the ice riband. This is sug- 

 gested by Professor Leconte, though he compares the 

 whole ray with the capillary pores of the soil in its ac- 

 tion. A temperature of several degrees below freezing 

 point is necessary to overcome the capillary force, and 

 freeze the water in the rays, which results in the splitting 

 of the stem. 



So far as can be learned from an examination of the 

 stems of the "frost jjlants," the only structural conditions 

 necessary are lai-ge and numerous vessels, thin-walled 

 medullary cells in a well marked ray, and a bark easily 

 split longitudinally. The category of plants furnishing 

 these conditions is by no means small. And it seems 

 highly probable that frost phenomena may be exhibited 

 by any of these j)lants which maj' pass through the death 

 stage at the season affording the necessary conditions of 

 temperature and moisture. 



I am indebted to Prof. Lester F. "Ward for some of the 

 references given above, as well as for other helpful sug- 

 gestions. 



QUANTITATIVE COMPARISONS: A COMMON 

 ERROR OF LANGUAGE. 



BY GEORGE H. JOHNSON, SC. D., ST. LOUIS, MO. 



In expressing the degrees in which any object — using 

 the word in its broadest or metaphysical sense — pos- 

 sesses a certain attribute or characteristic there must be 

 understood a unit of comparison or measurement. To 

 be comprehensible, this unit must be subject to the as- 

 sociative law of mathematics ; that is to say, if sub- 

 tracted from itself the remainder must be nothing, or 

 the zero of the scale of comparision, if added to itself the 

 sum must be twice itself, and if from the unit — supposed 

 positive — there be subtracted a quantity greater than 

 itself, the remainder must be negative. These facts, 

 which seem so axiomatic as to make their statement 

 superfluous, are frequently overlooked even by some 

 eminent speakers and writers. 



If we say that A is twice as long as B, we make B 

 ;he unit of comparison and affirm that the length of B is 

 oontained twice in that of A, or, no length being the 

 zero of linear measurement, the length of B is one unit 

 and that of A is two units. Similarly, if we say that A 

 is three-halves longer than B we have : 



Length A = length B + 3/2 length B = 5/2 length B ; 

 and if A is three-halves shorter than B we have : 



Length A = length B - 3/2 length B = - 1/2 length B. 



Now such a negative can occur only as indicative of 

 reversed direction or position relative to the zero, and 

 when no direction or position is assumed as positive the 



negative, as well as its imaginary roots, expresses the 

 impossible. For example, when we say it is twice as 

 far from A to B as from A to C, we have no reference 

 to the positions or directions of the lines A B and A C, 

 but only to their relative lengths, and a negative ex- 

 pression under these conditions is impossible in any 

 system of mathematics. 



A photographer advertised that by an improved pro- 

 cess he could take pictures thirty times quicker than 

 by the old process. Here, if T is the time required by 

 the old process and T' the time required by the new 

 process, we have : 



T' = T-30 T = -29 T; 

 the negative T being the algebraic expression for "less 

 than no time." Granting the claim of the advertise- 

 ment, itnecessarilly follows that the passage of time could 

 be stopped or reversed at our pleasure and the rapidity 

 of its backward flight would be determined only by the 

 number of photographs taken by the new process in a 

 unit of time. Amateur photographers will doubtless be 

 pleased to know that they have the fountain of eternal 

 youth so easily within their reach ! It is true, however, 

 that if an arbitrary assumption be made in regard to 

 the zero of the scale of "quickness" the claim of the ad- 

 vertisement may be verified. For example, if we agree 

 to take one second, s, as the zero of measurements, all 

 increments constituting slowness and all decrements 

 quickness, Q, then if T = 59/60 s we haveQ = 1/60 s 

 and O' = 30/60 s, whence 



T' = T — Q' =29/603 ; 

 so that the time by the new process would be nearly 

 half the time by the old process. But the "thirty times 

 quicker" was doubtless intended to mean one-thirtieth 

 of the time, and so was a notable example of an unsuc- 

 cessful and absurd attempt to make a quantitative state- 

 ment. 



A more remarkable example, because it occurred in a 

 carefully written essay by an eminent scientist describ- 

 ing a variable star, is as follows : 



"On April 27 it had become invisible in the great 

 telescope. It was then one hundred and sixty thousand 

 times fainter than it was at the time of discovery." 



Now it is evident what would be meant by saying 

 that it was one hundred and sixty thousand times 

 brighter at one time than another, because brightness 

 is an essentially positive quality whose quantity is de- 

 pendent upon if not proportional to the amount of lum- 

 inous energy eminating from the bod)^ ; but faintness is 

 a negative quality expressing only the absence of bright- 

 ness ; hence if there was no lack of brightness in the 

 star when discovered, faintness at an}^ other time could 

 not be expressed comparatively by using any positive 

 factor however large. 



Considering the quotation grammatical!)' the star is 

 said to be ' 'fainter" in the comparative degree ; hence it is 

 evident that it was first faint in the positive degree, and 

 since no unit of faintness is used in photometry we can 

 only assume that the brightness of the star in its posi- 

 tive condition of faintness as observed at discovery is 

 the unit of comparison ; hence when it was one hundred 

 and sixty thousand times fainter it must have been 

 (160,000-1) times less bright than an invisible body — 

 since the latter, without luminous energy, has no bright- 

 ness and presumably one unit of faintness. 



After the author of the statement quoted has shown 

 that 160,000 times fainter is equivalent to 1/160,000 as 

 bright, which is doubtless what he meant, I will show 

 that a liability of $1.00 is the same thing as assets of 

 $159,999.00; and such a blessed discovery for insolvent 

 debtors and their creditors would have so many degrees 

 of brightness as to quite outshine any variable star! 



