28 IRON ORES OF IRON SPRINGS DISTRICT, UTAH. 
Analyses of fresh and altered rocks show clearly the net results of 
the alteration, but they tell little of the extent to which the transfers 
of materials have been additions or subtractions, unless it is assumed 
that one substance has remained constant during the change. A 
variety of inferences are therefore possible, depending upon which 
substance is selected as a basis against which to measure the changes 
of the other constituents of the rock. Some substances are easily 
eliminated as possible constants; for instance, calcium carbonate, 
in the alteration of a limestone to a silicated rock; for if this material 
were assumed to remain constant in amount during the alteration, 
it would require so large an addition of other substances to develop 
the proportions in the altered rock that the volume would be increased 
beyond the possibilities of the situation, even allowing for increased 
density. But other constituents are not so satisfactorily eliminated 
by such reasoning, and there arises uncertainty and error in the 
selection from among the various possibilities of the particular sub- 
stance to serve as a basis of comparison of two analyses. If each 
substance in turn is considered as constant, the work of calculation 
becomes tedious and the probabilities of the situation are not easily 
discerned in the mass of figures and tables obtained. 
To meet these difficulties in inferring from analyses the real nature 
of the rock alterations, the writers present graphic comparisons of 
the analyses (Pis. IX-XII, A), from which it is possible, by assump- 
tions of constancy of any constituent during alteration, to read the 
percentage gains and losses of other constituents as a result of altera- 
tion, and, what is more important, to see at a glance what the 
probable constants are, without losing sight of the other possibilities. 
In the accompanying diagrams a circle with If -inch radius is 
divided by radii into 100 sectors, each sector therefore representing 
1 per cent of the total area of the circle. It is also divided into a 
series of annuli by concentric circles so spaced a that the area of each 
annulus is equal to 5 per cent of the total area of the circle. The 
subdivisions of the area, bounded by the sectorial radii and the con- 
centric circles, are therefore equivalent areas and each represents 
0.05 per cent of the total area of the circle. 
On the base diagram thus constructed, in PL IX, B, for example, 
the areas of sectors shown in different colors represent percentage 
weights of constituents of the altered limestone. Thus silica (analy- 
sis H) constitutes about 50 per cent of the altered rock. Of the fresh 
rock silica (average of analyses A and B) makes up about 7.5 per cent. 
The ratio of the two is 0.15, and hence the silica of the fresh rock is 15 
per cent of the silica of the altered rock. This is indicated on the 
diagram by the area of the silica sector cut off by the black circle 
a The area varies as the square of the radius. 
