MEASUREMENTS 51 



whether we know the units or not, but we cannot write an equation 

 M g^V cm^ because grams are not cubic centimeters. We can say, 

 however, M g=^kV cm'\ if we let fe be a constant. Any increase in V 

 brings a corresponding increase in M. Multiplication by the constant 

 must bring about an identity in dimensions, and therefore k must have 

 dimensions of its own. 



M g = k g/cm-'' X V cm-^ 



This particular constant is called density. If M were expressed in pounds 

 and V in gallons, the relationship between mass and volume would be 

 the same, but k would have a different numerical value and different 

 dimensions. 



Some biological measurements must be expressed in rather complex 

 dimensions. A measurement of rate of metabolism, for example, might 

 be in terms of microliters of oxygen used per gram of cells per hour or 

 fA. 02/g of cells X hr. It would be improper to vvnrite this as fA/g/hr be- 

 cause such expressions give even the mathematician fits. 



The main word of caution to be offered here is to remember the dimen- 

 sions. A measurement of millimeters of pressure change is not identical 

 to fA of O2 unless an appropriate correction is made. If you count scale 

 divisions on a dial, the divisions have meaning only if you know what 

 they stand for. Biological laboratories frequently use electrical recording 

 devices, but the record produced is in chart paper divisions until the 

 proper dimensional corrections are applied. Calculations can be prop)- 

 erly interpreted only if the dimensions are changed along with the 

 numbers. 



Indirect Measurement: Probably even more in biology than in physics, 

 certain measurements must be made indirectly. You cannot very well 

 measure the concentration of a substance inside a cell by ordinary chem- 

 istry without destroying the cell. Stephen Hales measured the blood 

 pressure of a horse directly, but his techniques cannot be used on 

 humans. 



Ingenuity on the part of the experimenter will often yield an indirect 

 measurement of a quantity. A plant cell contains an amount of water 

 which is not directly measurable, but generally the volume of the cell 

 can be estimated under a microscope. It might be guessed that the 

 amount of water is some function of the volume, W = f(V), where the 

 relationship might be proportional, logarithmic, or some more complex 

 function. If you have reason to suppose that water content is proportional 

 to volume, then W = kV, and it becomes possible to compare two cells. 



