Hello again, students. Today we will be looking at a more practical side to Arithmancy. The study of numbers has long been influential in the magic of potion-making. Discovering the reasons behind steps in brewing will often make you better potioneers.

In truth, potions are some of the most extraordinary magic that wizardkind has discovered. It is a field of combination, taking the disciplines of Astronomy and Arithmancy, and applying them to the magics of Creatures and Herbology. To this day, technomancers are hard at work discovering new ways to apply foreign magics to potions, using Arithmancy to justify their findings.

A simple example of how you might utilize Arithmancy in potion making is found in the action of stirring. You might remember from our first lesson that the Eighth Sympathetical represents "right," while the Ninth Sympathetical represents "left." These also stand for "clockwise" and "anti-clockwise," respectively. Groups of stirring will often be based on this, relying on basic math with the Eighth and Ninth Sympatheticals to infuse the solution with the brewer's magic. In fact, it is the Arithmantic properties of stirring a potion that make it magical at all, as opposed to a poisonous stew.

Let's take for example the basic Cure for Boils potion. I'll transcribe the recipe here for those who have forgotten it.

Finely crush 6 snake fangs into powder

Add 2 measures (half) of snake fang to cauldron

Add 1 finely sliced Punguous Onion

Add scoop of dried nettles

Place mixture on the fire

Add a dash of Flobberworm mucous and stir vigorously anti-clockwise 26 times

The potion should change color to red

Add 1 powdered ginger root and the other half of snake fangs

Stir vigorously clockwise 4 times

Add 3 pickled Shrake spines

Stir gently clockwise 4 times, so as to not agitate the spines

The potion should change color to lavender as the spines dissolve

Drop in 3 horned slugs

Wait 33-44 minutes

Take potion off the fire and add porcupine quills

Stir anti-clockwise 26 times

The potion should settle at a light blue color and have a thin viscosity

There are quite a few things in this simple potion that build off your basic knowledge of Sympatheticals. I'm not asking you to know the properties of the ingredients and how their innate magics interact, but a few obvious things should stick out at you.

All of the ingredients were an odd amount, at either 1 or 3. The snake fangs are initially at 6, but are divided in half so that they become two additions of an odd number. Ideally, counting the individual nettles before adding them and using an odd amount of them would strengthen the potion as well. Can anyone tell me why an odd amount of ingredients is significant? Generally, when dealing with the concepts of "odd" and "even," we represent them with the numbers one and two, respectively. So, knowing that using an odd number of ingredients is invoking the First Sympathetical, does any reasoning come to mind?

The answer is that the First Sympathetical represents rejuvenation, which is important in just about any healing potion. However, due to their complex nature, you will not always find healing potions to use an entirely odd numbered list of ingredients. In fact, part of the reason the Cure for Boils is marked a beginner's level potion is because it does this. The more a potion (or spell) follows common Arithmantic laws (that is, the less exceptions and Null spaces it has,) the easier it is to perform.

The other piece of information I'd like you to take from this potion is the stirring. As we talked about, both clockwise and anti-clockwise are representative of different effects. You may notice that in this potion, you stir anti-clockwise 26 times twice, and clockwise 4 times twice. This makes 54 total stirs anti-clockwise and 8 stirs clockwise. You could theoretically only stir the potion once at 54 rotations, and again with 8 rotations in the other direction. That creates the same basic Arithmantic effect. However, there is a compounding effect from splitting the amount of stirs in half and doing it twice. For every time a potion's stirring is cut in half, its potency is increased, as per the Second Sympathetical of Strength.

To get back on track, there are 54 anti-clockwise stirs. This is 9 times 6, a simple multiplication in this case. Many potions will use operations like the ninth root or something similar to create the correct sympathetic effect. This potion uses 9 times 6, using anti-clockwise stirs to invoke the Sixth Sympathetical of Memory. In this way, the potion "knows" what to restore a person's appearance to, after getting rid of the boils. Furthermore, with 8 clockwise stirs, the math on this one comes out as 1 times 8, using clockwise stirs to invoke the First Sympathetical of Rejuvenation. This compounds with the overall Arithmantic property of odd numbers to increase the healing capabilities.

There are dozens more Arithmantic properties to consider in this potion, including the lack of plant life, relying entirely on creature magic, the natures of the actual creatures themselves, the amount of color changes, the waiting duration and significance of that time. On top of that, there are considerations like what each creature's magic manifests as in those particular pieces of their body, how the Arithmantic strings combine together, the effects of astronomy on this potion, the brewer's intentions and awareness of the task, and so on. In fact, fully analyzing this potion for all of its intricacies (mind you, this is one of the simplest potions on the Hogwarts curriculum) would likely be a full OWL essay question.

Next time, we look at some more advanced combinations of Sympatheticals and what they mean in conjunction, depending on the operations involved.