By doubling the charge, however, we merely double the number of particles. Powdering finely is much more effective; for, since the weight of a particle varies as the cube of the diameter, halving the diameter of the particles increases their number eight-fold. If, now, we modify our illustration by assuming the particles to have only one-sixth the diameter of a full-stop (which would represent a powder of a fineness not unusual in ores prepared for assaying), we should multiply the number of particles by 200 (6 × 6 × 6 = 216). We should then reasonably expect a 14-lb. parcel of the powder to give as safe a sample as a ton of the sand would give; and portions of a size fit for crucible work, say 50 or 100 grams, would be as safe as 10 or 20-lb. samples of the coarser stuff. For example, 60 grams of such powder would contain, for a 5-dwt. ore, about 100 particles; and in the majority of cases the error due to sampling would be less than 10 or 12 grains to the ton, and would only occasionally exceed a pennyweight. With richer ores the actual deviation stated as so much to the ton of ore might be greater, but it would represent a smaller proportion, stated in percentage of the gold actually present, and would ultimately fall within the limits of unavoidable error. It will be seen that the size of the quartz particles has no direct bearing on the argument; and, in fact, the coarseness of the quartz only interferes by preventing the uniform mixing of the sand and by binding together several particles of gold; in this last case, particles so united must, of course, count as one larger particle. Now, there are some natural ores in which the gold particles are all very small; with these fine powdering and mixing yields a product from which a sample may be safely taken. Then, again, in "tailings," before or after treatment with cyanide, we have a similar material, inasmuch as the coarser gold has been removed by previous amalgamation. With these, it is not unusual to take the portion for assay without any further powdering, since they are poor in gold, and have already been stamped and passed through a sieve of say thirty holes to the inch (linear).