diff --git a/docs/Newton-Raphson.ipynb b/docs/Newton-Raphson.ipynb
index 9c8a880..706da9e 100644
--- a/docs/Newton-Raphson.ipynb
+++ b/docs/Newton-Raphson.ipynb
@@ -10,15 +10,6 @@
     "Cf. [\"Why Functional Programming Matters\" by John Hughes](https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf)"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "from notebook_preamble import J, V, define"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -47,15 +38,78 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "    a n over / + 2 /\n",
-    "    a n a    / + 2 /\n",
-    "    a n/a      + 2 /\n",
-    "    a+n/a        2 /\n",
-    "    (a+n/a)/2\n",
+    "Starting with $\\frac{(a_i+\\frac{n}{a_i})}{2}$ we can derive the Joy expression to compute it using abstract dummy variables to stand in for actual values.  First undivide by two:\n",
     "\n",
+    "$(a_i+\\frac{n}{a_i})$ `2 /`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Then unadd terms:\n",
+    "\n",
+    "$a_i$ $\\frac{n}{a_i}$ `+ 2 /`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Undivide again:\n",
+    "\n",
+    "$a_i$ $n$ $a_i$ `/ + 2 /`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Finally deduplicate the $a_i$ term:\n",
+    "\n",
+    "$a_i$ $n$ `over / + 2 /`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let's try out this function `over / + 2 /` on an example:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "  5 36 • over / + 2 /\n",
+      "5 36 5 • / + 2 /\n",
+      "   5 7 • + 2 /\n",
+      "    12 • 2 /\n",
+      "  12 2 • /\n",
+      "     6 • \n",
+      "\n",
+      "6"
+     ]
+    }
+   ],
+   "source": [
+    "clear\n",
+    "\n",
+    "5 36 [over / + 2 /] trace"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "The function we want has the argument `n` in it:\n",
     "\n",
-    "    F == n over / + 2 /"
+    "    n over / + 2 /"
    ]
   },
   {