For convenience of comparison I let them begin with the same number and for simplicity I have taken 2 for this initial term; observe that in the (_GP_) each term is got from the preceding term by _multiplying_ by 2 and that in the (_AP_) each term is got from its predecessor by adding 2; in the first series the multiplier 2 is called the common _ratio_ and in the second series the repeatedly added 2 is called the common _difference_; it is again for the convenience of comparison that I have chosen the same number for both common ratio and common difference and for the sake of simplicity that I have taken for this number the easy number 2. Other choices would be logically just as good. Why have I introduced these two series? Because they serve to illustrate perfectly two widely different _laws of progress_—two laws representing vastly different _rates_ of growth, increase, or _advancement_. Do not fail to observe in this connection the following two facts. One of them is that the magnitude of the terms of any geometric progression whose ratio (no matter how small) is 2 or more will overtake and surpass the magnitude of the corresponding terms of any arithmetical progression, no matter how large the common difference of the latter may be. The other fact to be noted is that the greater the ratio of a geometric progression, the more rapidly do its successive terms increase; so that the terms of one geometric progression may increase a thousand or a million or a billion times faster than the corresponding terms of another geometric progression. As any geometric progression (of ratio equal to 2 or more), no matter how slow, outruns every arithmetic progression, no matter how fast, so one geometric progression may be far swifter than another one of the same type. To every one it will be obvious that the two progressions differ in pace; and that the difference between their corresponding terms becomes increasingly larger and larger the farther we go; for instance, the sum of the first six terms of the geometrical progression is 126, whereas the sum of the first six terms of the arithmetical progression is only 42, the difference between the two sums being 84; the sum of 8 terms is 510 for the (_GP_) and 72 for the (_AP_), the difference between these sums (of only 8 terms each) being 438, already much larger than before; if now we take the sums of the first 10 terms, they will be 2046 and 110 having a difference of 1936; etc., etc.