“Five of you shall give chase a hundred, and a hundred shall chase ten thousand...” (Leviticus 26:8)
Second Reading: Evil Increases Linearly, but the Good Grows Exponentially
Our reading contains a verse that initially seems to be part of our blessings. When we need to pursue our enemies, 5 of us will overcome 100 of theirs and 100 of ours will vanquish 10,000 of theirs. This is great! It is not even a linear progression as Rashi points out, because the ratio of 100 to 5 is 1 to 20 and the ratio between 10,000 to 100 is 1 to 100. So what is the secret of our great “success?” Rashi’s answer is that when many follow the commandments it is not commensurate with the few that follow them. In other words, there is a quantity multiplier to our strength. The more Jews keep God’s commandments the more exponentially powerful we become.
This all sounds great, until we compare this verse with another in parashat Ha’azinu describing how in times of our weakness the enemy fights against us: “How is it that 1 [of theirs] chases 1000 of ours and 2 [of the enemy] make flee 10,000 of ours.”
Commentaries on the Torah ask: there two, when we are being pursued by our enemies (ostensibly because we have not kept our covenant with God), the enemy seems to become exponentially powerful. The verse starts with a ratio of 1000 to 1 and then continues to a ratio of 5000 to 1. The enemy too seems to enjoy the same 5 times multiplier in its strength. But, even before the multiplier, we are shocked to learn that they only need 1 of theirs to chase 1000 of ours, whereas when we are more powerful, we need 5 to chase only 100. This contradicts the well-established Torah principle, “the measure of good is always greater than the measure of punishment.”
Various answers have been offered by the commentaries, but they do not seem to satisfactorily solve the problem.
A Linguistic Solution
One possible solution looks at the end result. The verse describing our strength ends with “your enemies will fall before you by the sword.” But in the time of our weakness, even though we may be pursued, the Torah does not say that we will perish by the sword. In the end, even though they are more powerful, they do not have the power to obliterate us.
Another point to make is the change in verb in parashat Ha’azinu. Initially, the enemy chases us, but then it makes us flee. Being made to flee is weaker than being chased, as the latter as we see in our parashah can lead to death by the sword, but only fleeing implies escaping.
A Mathematical Analysis and Solution
Since the Torah is using ratios between numbers to describe the two realities of our strength and our weakness, we should analyze them mathematically. The main question we need to ask is why are the two sets of numbers chosen not the same? In strength the Torah chooses (5, 100) and (100, 10000). In our weakness, it uses (1, 1000) and (2, 10000). Let’s call these our data points and treat them as coordinates on an x-y coordinate system. The only number that is common to both sets of data points is 10000, and more about that later.
The second question is why is it that when describing the strength coordinates, the Torah does not start by telling us how many a single one of us will chase?
Linear versus Exponential
Contemplating these two verses a simple mathematical solution presents itself. The verse in our parashah describing our strength is based on exponential growth, while the verse in Deuteronomy describing our weakness is based on linear growth. In other words, to extrapolate the function describing the latter, we would plot the two coordinates (1, 1000) and (2, 10000) and draw a straight line through them. If we wanted to know how many 3 of the enemy could chase, we would continue the line to the right and determine that when x is 3, y would be 19000.
But with regard to our verse describing our strength, we are looking to plot an exponential function with (5, 100) and (100, 10000) as 2 of its points. However, 2 points are not enough to describe an exponential function. We need one more piece of information.
One possibility would be to just guess how many of the enemy 1 of ours could chase. We could then plot the exponential function and demonstrate the exponential growth of the good measure of strength. Even though the exponential function starts out slower than the linear function of Deuteronomy, the function describing our weakness, eventually, the exponential function will overtake the line and will uphold the principle that “the measure of good is always greater than the measure of punishment.” By recognizing that our verse is speaking about an exponential function while the verse in Deuteronomy is describing a linear function, we have already provided a solution to our problem.
However, we can do even better. We noticed that the only coordinate that is common to both functions is 10000. Perhaps, we say, the Torah is implying that when both functions reach a value of 10000, that is when the exponential function overtakes the linear one. In other words, when the x coordinate of our exponential function is 100 (and its y coordinate is 10000), that is when the derivative (the rate of change) of the exponential function overtakes the constant rate of change of the linear function.
With this new piece of information, it is possible to extract the exponential function that describes the growth of strength. We leave this to the capable reader, simply asking: how many of the enemy can a single 1 of ours chase according to the exponential function?
A Possible Solution
Using basic algebra and a little bit of differentiation, we arrived at the following exponential function that satisfies all our conditions: f(x) = 78.49e0.04847x
Solving for x=1, we find that 1 of ours will be able to chase 82.38 of theirs. The decimal fraction is part of the nature of the exponential function and in fact all the other values are not exact integers.
Here is a plot of this exponential function for good measure:
