The Right Dose: The Math Between Medicine and Poison

Some numbers decide whether a medicine heals you or kills you. This is the story of one of them.

In the 1780s, an English doctor named William Withering learned that a common wildflower — the purple foxglove — could pull a dying patient back from heart failure. There was just one problem: the same flower, in a slightly larger amount, could stop a heart entirely. Cure and poison were the very same plant. The only difference between them was a number.

This episode is about the math Withering needed to tell those two numbers apart — ratios and proportions — and about why it's the same math you already use every time you scale a recipe, mix a drink or read a map. 

By the end, you'll come away with three tools you can actually use:

• Ratios — what they really are, and why a ratio is a relationship that survives a change in size
• Equivalent ratios and proportions — the machine for scaling anything up or down without breaking it
• The unit rate — "find the one first," that makes proportion problems easy to do in your head

You'll also pick up a quiet superpower for reading the world: listen for the word "per." Miles per hour, dollars per pound, milligrams per kilogram — every time you hear "per," someone has handed you a ratio, and this episode shows you what to do with it.

And if the phrase "constant of proportionality" ever turns up on a worksheet at your kitchen table, you'll be the one at the table who can say: oh, that's just the "per." We already know this one.

Story Problem teaches real math and real science through true stories — for parents helping kids with schoolwork, and for anyone who was ever made to feel they weren't a "math person." You were. You are.   

⏱️ Chapters
00:00 — Welcome
00:53 — A woman who cannot breathe
01:39  — The secret in the hedgerow
02:16 — The problem of "how much"
06:35 — What a ratio actually is
08:25 — Find the one first
11:20 — What Withering actually did
13:44 — The flower is still working
14:37 — The dose makes the poison

✏️ Try it yourself
Three quick problems on this episode's skill are waiting at storyproblem.org — they take about two minutes and they're the fastest way to make it stick.

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Hosted by Dr. Behi Rabbani.

Transcript

SPEAKER_01 0:07

Hello, thank you again for joining us. I'm Behe Rabani. I'm the host of Story Problem. This is a show where real math and real science can be found inside real stories. Today we're going to talk about a very important idea, proportional reasoning. And my hope is that by the end of this episode, we'll come away understanding three things ratios, equivalent ratios, and unit rate. Last time we found the math beating inside your heart. Today we're going to find the math that decides whether medicine heals you or can be lethal. To be able to better understand these concepts, I want to start with a story. A woman cannot breathe. She is drowning, and not in water. Her ankles have swollen until the skin becomes tense and shiny. Her belly is tight and full, and when she lies down, she can't even get her breath. So she sleeps sitting up if she sleeps at all. The doctors of her day had a name for it, drop C, and almost nothing to offer her. They can name this condition, but they really have no tools to help it or treat it. And what they cannot see is where the fluid is actually coming from. It's coming from her heart, a weak, failing heart, letting fluid back up into her body like water behind a failing dam. And the thing that will save her, if anything does, is a flower. A tall, elegant flower with speckled purple bells growing wild in the hedgerows outside her window. The same flower that, in a slightly larger amount, could stop her heart entirely. The very same plant is both cure and poison. And the only difference between them is a number. That number is what today is really about. So let's talk about the right dose. At this time in the late 1700s, intellectual life was alive with the idea that the universe followed rules that could be studied and discovered. The same mathematics used to understand the motion of the moon could also explain forces, machines, and new steam engines that were transforming everyday life. Aeronauts attempting to cross the English Channel in balloons were the mega rock stars of the day. And the story goes that Withering heard about an old woman out in the country of Shropshire, a folk healer. She had a family recipe passed down for treating exactly the kind of swelling that left doctors shrugging. People who'd been given up on were somehow getting better. Now, most physicians at the time would have ignored a country remedy. Withering did the opposite. He got hold of the recipe and found it was a jumble of some 20 different herbs. But Withering knew his plants. He looked at the list of 20 and reasoned most of these do nothing. They're padding, filler, just tradition passed along. Only one of them could be doing the real work, and only one of them was powerful enough to be dangerous. The foxglove digitalis. He'd found the active ingredient, which sounds like the end of the story, but it's actually where the where the really hard part begins. Because knowing that foxglove works is almost useless without knowing how much. And how much is not a botany question. It's a math question. It's a question this whole episode is about. Here's what made foxglove so terrifying to use. Give too little and nothing happens. The heart stays weak, the flood keeps rising, and the patient does not survive. Now, with the added unease that the cure was sitting right there, just out of reach. Give too much and the foxglove turns on you. It is, after all, a poison. The heart, instead of steadying, lurches into a dangerous rhythm. And today we know these rhythms as something called complete heart block and bidirectional ventricular tachycardia. Life-threatening heart rhythms, both. There are significant GI side effects as well. And strangest of all, the world takes on this yellow-green tint, as if someone had drawn a colored film across the patient's eyes. There's even a lingering theory that Vincent Van Gogh, well known for his blazing yellows, may have been under the plant's influence, though that one will leave as a maybe. So the safe amount and the deadly amount sit very close together, closer than for almost any medicine that you can name. The space between not enough and too much was very narrow, and a physician had to land within that window, exactly in a patient who was already dying. And it gets worse because the plant itself was different when you picked it. A leaf picked in spring isn't as strong as a leaf picked in autumn. A leaf grown in one soil isn't as strong as one grown in another. So a spoonful of foxglove was a meaningless instruction. One spoonful might carry twice the punch of the next. Withering was trying to find the right dose of medicine that was never quite the same strength twice. To have any hope, he needed to make the strength predictable. He needed to fix the relationship between the amount of leaf and the amount of medicine inside it. He needed a ratio. You use it in your kitchen. Say you're making lemonade for your kids. One cup lemon juice, one cup sugar, four cups water. Now their friends show up and you need a whole jug. To keep it tasting right, you have to scale up all the ingredients. More lemon juice, more sugar, and water all together, keeping them in step. It's two cups lemon juice, two cups sugar, eight cups water, etc. A bigger jug, but the same drink. That relationship, one to two or one to four, is a ratio. And here's the key: a ratio is the relationship that changes in step with a change in size. The numbers can grow one to four, can become three to twelve, can become ten to forty. But the relationship, the taste stays exactly the same. That's what makes a ratio different from the percentages we talked about last episode. A percentage was a snapshot, a part of the whole, just kind of frozen. A ratio is a relationship you can stretch. When two ratios describe the same relationship, like one to four and three to twelve, we call that a proportion. A proportion simply tells us that both amounts have been scaled by the same factor. So the relationship between them has stayed the same. It's how you double a recipe, it's how a map makes a country small enough to fit on a page. Every real world distance is reduced by the same scale, so the places stay correct, the correct distance apart from one another. And it's how a doctor scales a dose. Not memorize a formula. Here's this one. When you're staring at a ratio, find the one first, the amount per one. That is the unit rate. Here's what I mean. Suppose a hot chocolate recipe says six spoons of cocoa makes three mugs. And you want to make a lot. Let's say five mugs. Don't panic. Don't reach for algebra or an equation. Just find the one first. Six spoons, three mugs. How much for one mug? Divide two spoons per mug. That's it. That's the hard part done. Now, one mug is two spoons. So five mugs is two times five, ten spoons. Once you know the per one, every other amount is a single step away. And here's a small piece of vocabulary offered only because you may run into it. That per one number, two spoons per mug, has a formal name in school textbooks. They call it the constant of proportionality. It's the fixed number that connects the mugs to cocoa. However many mugs you want, multiply by two to find the number of spoons. So if the phrase constant of proportionality shows up in a sixth or seventh grade worksheet, don't let the name make it sound harder than it is. It simply means the amount for one. In this case, two spoons for one mug. We already found it when we calculated the unit rate. Ratios come up in everyday conversation all the time. One easy way to recognize them is to listen for the word per miles per hour, dollars per pound, milligrams per kilogram. The little word per is Latin and it means roughly for each one. So whenever you hear per, someone has already done the hard part for you. They've found the one, they've handed you a unit rate, 60 miles per hour, just means in one hour, 60 miles. Scale it to any number of hours you like. So this week, listen for the word per. When you hear it, you're usually hearing one amount compared with another. And that means you're dealing with a ratio. He could not control the weather, the soil, or the natural strength of every foxglove plant, but he could make the preparation more consistent. He gathered the leaves at the same stage of growth, dried them in the same way, ground them into a powder, and weighed that powder instead of a vague spoonful. That brought him to the kind of relationship we've been talking about in math: a dependable amount of medicine for each dose. Instead of some foxglove, he could say, in effect, this much powder per dose. That per dose amount acted like a constant of proportionality. It was the fixed number connecting the amount of powder to the dose being given. Once that number was established, Withering could scale the treatment carefully. One measured dose, then another, while watching the patient's response. It was not perfectly predictable because Foxglove was still a plan and each patient responded differently, but Withering had replaced the spoonful with a measured unit. He had created a relationship he could observe, compare, and adjust. Doctors have a word for that careful creeping up. We call it titration, titration of medications and doses. But you can just call it what it is. It's proportional reasoning. Amazingly, Withering did this for 10 years. He kept records like a scientist, not a folk healer. 163 patients, their cases written down honestly, with failures alongside the cures. And in 1785, he published all of it in a book with a very 18th century title, An Account of the Foxglove and Some of its Medical Uses. Here's the part I love. Withering did not discover foxglove. Country healers had known it for generations. The old woman in Shropshire knew it. What Withering discovered was the dose, the proportion, the relationship between the plant and the patient that turned poison into one of the most important heart medicines in history. This was a landmark in modern therapeutics and clinical pharmacology. He took a natural remedy and studied it systematically. He standardized the preparation, recognized its toxic effects, and adjusted treatment according to the patient's response. And here is why this isn't just a history lesson. That same foxglove, refined, purified, measured with instruments Withering could only have dreamed of, is still in medicine cabinets today. Its descendant is a drug called digoxin, drawn from a close cousin of Withering's flower. We still use it for, in certain cases, for failing hearts and to control high heart rates in conditions such as atrial fibrillation. And we still handle it with enormous care, exactly for the reason Withering learned the hard way. The gap between the helpful dose and harmful one is famously, dangerously narrow. It's one of the few drugs that we measure the level in a patient's body to make sure we've given enough, but not too much. 240 years later, the story hasn't changed. Same flower, same math. There's a line, older even than Withering himself, from a physician named Paracelsus. He said, in effect, the dose makes the poison. Meaning there's no such thing as a substance that's simply safe or simply toxic. Everything is a poison at some amount. Nothing is a poison at a small enough one. Even water, even air, the same foxglove could help a patient in a small, carefully measured dose and harm that patient if too much was given. What mattered was not just the plant, but how much of it entered the patient's body and how that patient responded. And once you start seeing it, you can't stop. It's the coffee that's one scoop too strong, the paint that's mixed a shade off, that interest rate that decides what alone really costs, the recipe that feeds four or forty, depending only on how faithfully you hold the relationship. Proportion is the grammar underneath almost everything we measure. Here's a thing I most want you to take to walk away with. You've been reasoning proportionally your whole life. Every time you've doubled a recipe, split a bill in three ways. Maybe you compared two boxes at the store to see which gave you more for your money. You found the price for the same amount or the price per unit and chose the better deal. That was you using ratios and unit rates. Nobody had to hand you a formula to keep the lemonade tasting right. You already knew. So this week, listen for the word per. And when you see two amounts being compared, start by founding the amount for one. That's what Withering was trying to do with Foxglove, replace a vague spoonful with a measured dose. And it's the same step you use with recipes, prices, speeds, and other ratios. You may already use this kind of math every day. Now you know what it's called and how to do it on purpose. This has been Story Problem. And if this made an old idea feel new, follow the show wherever you're listening and come find the episode notes. And a few problems to try yourself at www.storyproblem.org. See you then.